APPLYING MONTE CARLO SIMULATION IN REAL ESTATE … ISEL… · APPLYING MONTE CARLO SIMULATION IN...

School of Architecture Urban Planning Construction Engineering

Master of Science in Management of Built Environment

APPLYING MONTE CARLO SIMULATION IN REAL

ESTATE CAPITAL BUDGETING FOR INVESTMENT

EVALUATION

Supervisor: Liala Baiardi

Author: Marco Isella

Matriculation number: 897863

Academic year 2018/2019

iii

Contents

Figures Index ..................................................................................................................... vi

Tables Index .................................................................................................................... viii

Abstract ............................................................................................................................. x

Introduction .................................................................................................................... xiii

PART 1 – ITALIAN REAL ESTATE MARKET OUTLOOK .............................................. 2

PART 2.1 - BUILDING BLOCKS IN CAPITAL BUDGETING .......................................... 8

2.1.1 - The Choice To Invest ............................................................................................... 8

2.1.1.1 - Time Value of Money .............................................................................................................................. 8

2.1.1.2 - Present Value and Future Value ............................................................................................................. 9

2.1.1.3 - Opportunity Cost of Capital .................................................................................................................. 10

2.1.1.4 - Discounted Cash Flow – DCF ................................................................................................................. 10

2.1.2 - Cash flows Estimation ........................................................................................... 11

2.1.2.1 - Relevant Cash Flows .............................................................................................................................. 11

2.1.2.2 - Project Cash Flows ................................................................................................................................. 12

2.1.3 - Investment Criteria ............................................................................................... 14

2.1.3.1 - Net Present Value - NPV ........................................................................................................................ 14

2.1.3.2 - Internal Rate of Returns – IRR............................................................................................................... 15

2.1.3.3 - Profitability Index – PI............................................................................................................................ 16

2.1.3.4 – Payback Time ......................................................................................................................................... 17

2.1.3.5 - Modified Internal Rate of Return – MIRR ............................................................................................ 17

PART 2.2 – INVESTMENT RISK ............................................................................. 19

2.2.1 - Some Definitions About Risk and Risk Management ............................................. 19

2.2.2 - Returns of Real Estate Assets ................................................................................ 21

2.2.2.1 - Nominal and Real Returns ..................................................................................................................... 25

2.2.2.2 - Is It Better to Use Nominal or Real Interest Rates? ............................................................................ 26

2.2.3 - Risk and Returns Relationship ............................................................................... 27

2.2.3.1 - Security Market Line .............................................................................................................................. 28

2.2.3.2 - Leverage and Risk .................................................................................................................................. 31

2.2.3.3 - Risk-free and Risk Premium .................................................................................................................. 32

2.2.4 - Cap Rate Determinants ......................................................................................... 34

iv

2.2.5 - Measuring Risk ...................................................................................................... 35

2.2.5.1 - The Range ............................................................................................................................................... 35

2.2.5.2 - Standard Deviation ................................................................................................................................ 36

2.2.5.3 - Semi-standard Deviation ....................................................................................................................... 37

2.2.5.4 - Value at Risk – VaR................................................................................................................................. 37

2.2.5.5 - Annualizing Volatility and Returns ........................................................................................................ 38

2.2.6 - Risk-adjusted Performances Measures ................................................................. 39

2.2.6.1 - Coefficient of Variation ......................................................................................................................... 39

2.2.6.2 - Information Ratio ................................................................................................................................... 40

2.2.6.3 - Sharpe Ratio ........................................................................................................................................... 40

2.2.6.4 - Treynor Ratio .......................................................................................................................................... 40

2.2.6.5 - Sortino Ratio ........................................................................................................................................... 40

2.2.7 - Covenants on Debt ................................................................................................ 41

2.2.7.1 - Interest Coverage Ratio – ICR ............................................................................................................... 41

2.2.7.2 - Loan to value – LTV ................................................................................................................................ 42

2.2.7.3 - Debt Service Coverage Ratio – DSCR .................................................................................................... 42

2.2.8 - Cost of Capital ....................................................................................................... 42

2.2.8.1 - Cost of Equity ......................................................................................................................................... 42

2.2.8.2 - Cost of Debt and Mortgages ................................................................................................................. 45

2.2.8.3 - Weighted Average Cost of Capital ........................................................................................................ 51

2.2.9 - Sources of Risk Identification in the Real Estate Investment ................................. 52

2.2.9.1 - Market Risk ............................................................................................................................................. 52

2.2.9.2 - Operative Risk ........................................................................................................................................ 53

2.2.9.3 - Interest Rates Risk ................................................................................................................................. 53

2.2.9.4 - Legal Compliance Risk ........................................................................................................................... 53

2.2.9.5 - Asset Concentration Risk ....................................................................................................................... 54

2.2.9.6 - Strategic Risk .......................................................................................................................................... 54

2.2.9.7 - Other Risk Classification ........................................................................................................................ 54

PART 2.3 ANALYSIS TOOLS AND TECNIQUES ....................................................... 55

2.3.1 - Project Analysis ..................................................................................................... 55

2.3.1.1 - Scenario Analysis.................................................................................................................................... 55

2.3.1.2 - Sensitivity Analysis ................................................................................................................................. 56

2.3.1.3 - Break-Even Analysis ............................................................................................................................... 57

2.3.1.4 - Monte Carlo Simulation ........................................................................................................................ 57

2.3.2 - Risk assessment .................................................................................................... 59

PART 3 – CASE STUDY ......................................................................................... 63

3.1 - Case Study Asset ...................................................................................................... 63

3.2. - Cash Flows Estimation in Practice ........................................................................... 64

3.2.1 - Potential Gross Income ............................................................................................................................ 64

v

3.2.2 - Estimated Rental Value – ERV .................................................................................................................. 64

3.2.3 - Vacancy Length ......................................................................................................................................... 71

3.2.4 - Vacancy Allowance ................................................................................................................................... 71

3.2.5 - Operative Expenses .................................................................................................................................. 73

3.2.6 - Capital Expenditure................................................................................................................................... 77

3.2.7 - Reversion Value ......................................................................................................................................... 78

3.2.8 - Discount Rate ............................................................................................................................................ 79

3.3 - Project Analysis ........................................................................................................ 86

3.3.1 – Most Likely DCF ........................................................................................................................................ 86

3.3.2 - Applied Scenario Analysis ......................................................................................................................... 87

3.3.3 - Applied Sensitivity Analysis ...................................................................................................................... 90

3.3.4 - Applied Break-even Analysis .................................................................................................................... 93

3.4 - Run the Monte Carlo Simulation .............................................................................. 94

Conclusions ...................................................................................................................... 98

Bibliography ................................................................................................................... 102

Attachments .................................................................................................................. 107

vi

Figures Index

Figure 1. Debt and GDP growth rate and rates differential – Italy 2000-2017 .................................................... 2

Figure 2. Value added and measures of productivity, Italian whole economy - 1996-2017, percentage

variations ................................................................................................................................................................... 3

Figure 3. Normalized transaction (NNT) percentage change of the Italian real estate market 2004-2017 ...... 3

Figure 4. Investment volume in the Italian real estate market by origins of the capital ..................................... 4

Figure 5. Real estate investments in Italy by location and sector ......................................................................... 4

Figure 6. Stock of grade A offices - % of total stock ............................................................................................... 5

Figure 7. SGR and number of operative funds in Italy 2007-2017 ........................................................................ 6

Figure 8. Selected indicators of Italian REIF’s performances during the period 2005-2010 ............................... 6

Figure 9. The circular flows diagram ....................................................................................................................... 9

Figure 10. Typical cash flows pattern of an investment....................................................................................... 15

Figure 11. Risk management process .................................................................................................................... 21

Figure 12. Typical peaked profile of a maintenance plan costs over the asset’s life cycle ............................... 23

Figure 13. NCREIF index components of returns .................................................................................................. 24

Figure 14. Deflated Euribor 3 months interest rates – three different deflation assumption ........................... 26

Figure 15. Probability distribution of returns of three assets .............................................................................. 28

Figure 16. Systematic and unsystematic risks ...................................................................................................... 29

Figure 17. Security market line .............................................................................................................................. 30

Figure 18. Risk and return relationship in USA major asset class (1970-2003) .................................................. 33

Figure 19. S&P 500 daily returns in 2018 .............................................................................................................. 36

Figure 20. Returns distribution and 95% confidence interval .............................................................................. 37

Figure 21. β estimation for Adobe and Amazon stocks calculated on monthly returns from January 2010 to

July 2019 .................................................................................................................................................................. 44

Figure 22. Average Overall Effective Rate – AOER – for mortgage loans with fixed and variables rate

compared with Eurirs 10 years and Euribor 1 month .......................................................................................... 46

Figure 23. Interest-only loan – total payment and interest expense (figures in €) ............................................ 48

Figure 24. Constant-Amortization Mortgage – total payment and interest expense (figures in €) ................. 49

Figure 25. Constant-Payment Mortgage – total payment and interest expense (figures in €) ........................ 49

Figure 26. Adjustable Rate Mortgage – total payment and interest expense (figures in €) ............................. 50

Figure 27. Balloon mortgage – total payment and interest expense (figures in €) ........................................... 51

Figure 28. Example of sensitivity analysis ............................................................................................................. 56

Figure 29. Inputs variables aggregation with Monte Carlo simulation - 5,000 iterations................................. 58

Figure 30. Derivation of probability distribution from empirical observations .................................................. 60

Figure 31. Asset geolocalization ............................................................................................................................ 63

Figure 32. Vodafone Village main facade ............................................................................................................. 63

Figure 33. Inflation trend (5 years moving average) in Italy 2000-2018 ............................................................ 65

Figure 34. Performance decay over time .............................................................................................................. 67

Figure 35. Effect of maintenance on performances degradation and useful life length ................................... 68

Figure 36. Indexes of the nominal ERV components - trend and effects sum ................................................... 69

Figure 37. ERV - trend and simulated .................................................................................................................... 70

Figure 38. 5 ERV simulations with trend ............................................................................................................... 70

Figure 39. ERV probability distribution in year 1 and 10 obtained with a Monte Carlo simulation – 5,000

iteration ................................................................................................................................................................... 70

vii

Figure 40. Poisson distributions with λ equals to 1, 2, 3 ...................................................................................... 71

Figure 41. Maintenance expenses profile over building life cycle ....................................................................... 76

Figure 42. Capex profile over building life cycle .................................................................................................... 76

Figure 43. Regression analysis for β estimation using monthly returns of COIMA and FTSE MIB (left) and FTSE

EPRA NAREIT Index (right) – January 2017-July 2019 .......................................................................................... 82

Figure 44. NPV and IRR for the three scenarios .................................................................................................... 89

Figure 45. NPV tornado chart with +30% and -30 % variation ............................................................................ 92

Figure 46. IRR tornado chart with +30% and -30 % variation ............................................................................. 92

Figure 47. Sensitivity of risky variables .................................................................................................................. 93

Figure 48. Monte Carlo simulation outcomes distributions – NPV (left) and IRR (right) – 20,000 thousand of

iterations ................................................................................................................................................................. 96

Figure 49. Monte Carlo simulation NPV frequency distribution of "exercised" (left) and "not exercised break-

option" (right) .......................................................................................................................................................... 96

Figure 50. NPV bimodal distribution given by the two scenarios sum ................................................................ 97

viii

Tables Index

Table 1. Italy's national account selected figures ................................................................................................... 2

Table 2. Survey of Italy's REIFs’ balance sheet main figures in 2003-2016 .......................................................... 5

Table 3. IRR e NPV of investments A and B with a discount rate of 10% ............................................................ 16

Table 4. IRR e NPV of investments A and B with a discount rate of 25% ............................................................ 16

Table 5. PI and NPV of investments A, B and C ..................................................................................................... 17

Table 6. IRR reinvestments assumptions break-down.......................................................................................... 18

Table 7. Capex sub-categories ............................................................................................................................... 24

Table 8. Portfolio expected returns and beta for different levels of asset allocation ........................................ 30

Table 9. The effect of leverage on returns ............................................................................................................ 32

Table 10. Historical records of USA major asset classes ...................................................................................... 33

Table 11. Elements the determine the current market price of a property ........................................................ 35

Table 12. Milan central apartment sub-market annualized returns and volatility (2015-2019) ...................... 38

Table 13. Comparison of two project with different scale ................................................................................... 41

Table 14. Interest-only loan (figures in €) ............................................................................................................. 48

Table 15. Constant-Amortization Mortgage (figures in €)................................................................................... 48

Table 16. Constant-Payment Mortgage (figures in €) .......................................................................................... 49

Table 17. Adjustable Rate Mortgage (figures in €) .............................................................................................. 50

Table 18. Balloon mortgage (figures in €) ............................................................................................................. 51

Table 19. Risks classification .................................................................................................................................. 54

Table 20. Project A scenario analysis .................................................................................................................... 55

Table 21.Break-even point calculations on NPV ................................................................................................... 57

Table 22. Most likely values for the income statement of firm A for the next year ........................................... 58

Table 23. Assumptions of distributions and parameters of firm A income statement variables ...................... 58

Table 24. Inflation Forecasts .................................................................................................................................. 64

Table 25. Historical Eurostat HICP inflation in Italy 2008-2018 .......................................................................... 65

Table 26. Historical Istat CPI inflation in Italy 2000-2018.................................................................................... 65

Table 27. Milan office rent values and growth rates at Q2 2019........................................................................ 66

Table 28. Assumed real growth rate of the market rent in the south-west Milan periphery sub-market ....... 66

Table 29. ERV simulation applying effects sum and formula (47) ....................................................................... 69

Table 30. Vodafone Village operative costs in 2017 and 2018 ........................................................................... 73

Table 31. Example of building components technical card.................................................................................. 75

Table 32. Study case maintenance expenses - years 1-25 ................................................................................... 76

Table 33. Annual Capex ratio for NCREIF office buildings, 1978-2014 ............................................................... 77

Table 34. Study case capex - years 1-25................................................................................................................ 77

Table 35. U.S. cap rate survey - H1 2019. Suburban offices markets in Tier I cities .......................................... 79

Table 36. Going-out cap rate spread estimation .................................................................................................. 79

Table 37. risk-free rate estimation using 10 years and 12 months Italian government bonds rates ............... 80

Table 38. Risk premium estimation with the Empirical Historical Method......................................................... 81

Table 39. Average levered and unlevered β of a selected basket of firms in the real estate sector in western

Europe ...................................................................................................................................................................... 82

Table 40. Average excess of return of FTSE MIB vs 12 months BOT - 2002-2018 ............................................. 83

Table 41. Different methods WACC estimation and average .............................................................................. 85

Table 42. Asset general information ..................................................................................................................... 86

ix

Table 43. DCFA assumption and formulas ............................................................................................................ 86

Table 44. Most Likely DCF excerpt ......................................................................................................................... 87

Table 45. Different scenarios ERV trend calculations ........................................................................................... 88

Table 46. Annualized standard deviation calculation for the sub-market "D25" OMI zone in Milan ............... 89

Table 47. Sensitivity analysis summary ................................................................................................................. 91

Table 48. Break-even analysis ................................................................................................................................ 94

Table 49. Assumptions on variables' distribution and parameters ..................................................................... 95

Table 50. Comparison between Most likely and Monte Carlo NPVs ................................................................... 99

x

Abstract

L’oggetto di questa tesi è la valutazione di un asset immobiliare considerato come opportunità

d’investimento. Le ipotesi che stanno alla base del seguente elaborato sono due; la prima è che il metodo

tradizionale di valutazione degli asset immobiliari dell’income approach applicato per mezzo di una

discounted cash flows analisi (in seguito DCFA o DCF) non è uno strumento capace di fornire indicazioni

affidabili ad un investitore immobiliare, in quanto non permette di quantificare la rischiosità dell’asset che

sta valutando. Questa mancanza porta a valutazioni del prezzo incorrette, dal momento che uno dei principi

cardine dell’economia finanziaria è che asset più rischiosi hanno prezzi bassi e rendimenti relativamente

elevati rispetto ad asset meno rischiosi, che hanno prezzi alti e rendimenti relativamente bassi. La seconda

ipotesi è che rappresentare le funzioni obiettivo della DCFA – sia che essi siano il prezzo, il valore attuale

netto, il tasso interno di ritorno, o altre – come una variabile aleatoria (o stocastica) permette di quantificare

la rischiosità di un asset e quindi fornire indicazioni più precise sia al fine della stima del prezzo (argomento

che non verrà trattato) che per quanto riguarda la valutazione della convenienza dell’investimento in un

asset immobiliare. Per riuscire in questo intento è stato utilizzato il tradizionale metodo del DCF in

combinazione con la simulazione Monte Carlo, grazie alla quale è stato possibile aggregare i rischi derivanti

dalle singole variabili contenute nel DCF ed ottenere una descrizione quantitativa del rischio globale

dell’asset. Le variabili del DCF devono necessariamente essere rese stocastiche per poter applicare il

metodo Monte Carlo, dunque devono essere scelti i parametri e il tipo di distribuzione che meglio le

rappresenta. La peculiarità della parte applicativa della tesi è rappresentata dalla presenza di una break-

option nel contratto di locazione, la quale viene incorporata nel DCF seguendo il modello sviluppato da C.

Amédée‐Manesme et all (2013), per poterne verificare gli effetti sulla redditività e sulla rischiosità dell’asset.

I risultati raggiunti applicando il metodo brevemente appena descritto sono che la presenza della break-

option crea due scenari completamente disgiunti, i quali portano a far assumere una forma bimodale alla

distribuzione di probabilità delle variabili aleatorie dell’NPV e dell’IRR, dovuta proprio alla compresenza nel

suo dominio di valori derivanti da uno scenario in cui la break-option viene esercitata e uno in cui non lo è.

Grazie all’uso del Monte Carlo è stato possibile stimare la probabilità di avvenimento dei due scenari e

quantificarne sia i valori attesi di performances che i livelli di rischio. Le ipotesi sono state verificate tramite

l’applicazione degli strumenti e della metodologia appena descritta per mezzo di un caso studio in cui viene

analizzato un immobile di proprietà di COIMA RES conosciuto come Vodafone Village; un edificio per uffici

con una superficie commerciale di circa 46 mila metri quadri localizzato nella periferia sud-ovest di Milano.

xi

Abstract – English version

The object of this thesis is the evaluation of a real estate asset considered as an investment opportunity.

The hypotheses underlying the following paper are two; the first is that the traditional method of valuing

real estate assets of the income approach applied by means of a discounted cash flows analysis (hereinafter

DCFA or DCF) is not a tool capable of providing reliable indications to a real estate investor since it does not

allow to quantify the riskiness of the asset being evaluated. This lack leads to incorrect price evaluations,

since one of the key principles of the financial economy is that riskier assets have low prices and relatively

high returns compared to less risky assets, which have high prices and relatively low returns. The second

hypothesis is that representing the objective functions of the DCFA - whether they are the price, the net

present value, the internal rate of return, or others - as a random variable (or stochastic) allows to quantify

the riskiness of an asset and therefore provide more precise indications both for the purpose of estimating

the price (a topic that will not be dealt with) and as regards the evaluation of the convenience of investing

in a real estate asset. To achieve this, the traditional DCF method was used in combination with the Monte

Carlo simulation, thanks to which it was possible to aggregate the risks deriving from the individual variables

contained in the DCF and obtain a quantitative description of the global risk of the asset. The DCF variables

must necessarily be made stochastic in order to apply the Monte Carlo method, therefore the parameters

and the type of distribution that best represents them must be chosen. The peculiarity of the applicative

part of the thesis is represented by the presence of a break-option in the lease contract, which is

incorporated into the DCF following the model developed by C. Amédée-Manesme et all (2013) and so be

able to verify the effects on profitability and on the riskiness of the asset. The results achieved by applying

the method just briefly described above are that the presence of the break-option creates two completely

disjointed scenarios, which lead the NPV and IRR random variables to be represented by a bimodal

probability distribution. This id due to the coexistence in the same domain of values deriving from a scenario

in which the break-option is exercised and one in which it is not. Using the Monte Carlo method it was

possible to firstly separate two scenario, then estimate their probability of occurrence and to quantify both

the expected values of performances and the levels of risk. The hypotheses were verified through the

application of the tools and methodology just described by means of a case study in which is analysed a

property owned by COIMA RES known as Vodafone Village; that is an office building with a commercial area

of approximately 46 thousand square meters located in the south-west suburbs of Milan.

xiii

Introduction

The following thesis has as its main object the development of a "stochastic" discounted cash flows analysis

(hereinafter DCFA or DCF) as it presents in input, and returns in output, random variables. This approach

was designed thinked starting from one of the fundamental concepts of the financial economy, namely that

with the same expected return, riskier assets - or more volatile, characterized by a greater standard

deviation of the probability distribution of returns - have less demand on the market, which lowers the price

and increases potential returns; vice versa for less risky assets. Considering this assumption true, it is clear

that the real estate valuation that does not contain any type of indication on the riskiness of the asset, is

partial and incomplete and can lead to the overestimation of the asset value, since the riskiness is not

known. The objective of the following paper is therefore to develop a tool called stochastic discounted cash

flows, or stochastic DCF, through the application of the Monte Carlo method that allows to represent the

objective functions of the DCF as random variables and therefore to be able to quantify the risk of an asset.

To do this it is necessary to choose the type and parameters of the most suitable probability distributions

to represent the single variables that make up the DCF. In the applicative part, a particular emphasis was

given to the analysis of the effects that the presence of a break-option has on the profitability and riskiness

of the asset using the model developed by C. Amédée-Manesme et all (2013). The structure of the thesis is

divided into three major parts. The first is a brief outlook on the Italian real estate sector and its recent

developments and evolutions; the second is the theoretical part and a last is dedicated to a case study. As

far as the theoretical part is concerned, the starting point is the treat of the “building blocks” of the capital

budgeting, because if the main instrument of evaluation is the DCF, this means that it was first of all

necessary to identify the relevant cash flows for the analysis and then find a proper discount rate for

actualized the future values. To this follow a brief explanation of the classic performance evaluation tools

such as NPV and IRR (but not only) in order to be able to evaluate the result given by the DCF, focusing in

particular on the assumption about the reinvestment of the net cash flow on which the NPV and the IRR are

built. In the second part of the theoretical discussion, rist of all I tried to give some definition of what risk is,

then I deepened the relationship between risk and return and the reason why they are inextricably linked

and the conseguences that this link generates on the relationship between price and expected return. In

the last part of the theoretical discussion I described the classic investment analysis tools such as the

scenario and the sensitivity analysis, focusing in particular on the Monte Carlo simulation. From a

methodological point of view, for reasons of greater clarity I presented easy examples for each tool I

explained. The third part of the thesis is the practical one, in which I developed a case study analysing the

Vodafone Village building owned by COIMA RES looking at it as an investment possibility and trying to

understand if it is worth the trouble to invest in it. The first part of the case study is focused on the

explanation of how cash flows are been estimated, on the models applied and the assumption made. The

analysis tool are used to extrapolate information from the “deterministic” DCF and then I used the Monte

Carlo to prepare the stochastic DCF and analyse the effect that the presence of a break-option in the leasing

contract generates of the DCF’s objective functions.

PART 1

ITALY OUTLOOK

2

PART 1 – ITALIAN REAL ESTATE MARKET OUTLOOK

Looking at the main figures of Italian public finance, debt and GDP, it can be seen that after the 2008 “sub-

prime” crisis, public debt grew at rates higher than GDP. After the difficult 2008-2013 period, marked by

the global crisis of 2008-2009 and the subsequent “sovereign debt crisis” of 2011, Italy is back to a path of

growth, albeit weak, of productivity - Figure 2 - and of GDP - Figure 1 and Table 1.

Table 1. Italy's national account selected figures

2000 2001 2002 2003 2004 2005 2006 2007 2008

Government debt A 1.303 1.360 1.372 1.397 1.450 1.519 1.588 1.606 1.671

Debt growth rate [%] B=At/At-1-1 1,37 4,42 0,84 1,87 3,73 4,76 4,57 1,14 4,06

GDP C 1.299 1.346 1.391 1.448 1.490 1.548 1.610 1.632 1.573

GDP growth rate [%] D= Ct/Ct-1-1 4,81 3,61 3,34 4,15 2,86 3,94 3,94 1,40 -3,63

Debt/GDP [%] E=A/C 100,29 101,08 98,64 96,49 97,31 98,08 98,67 98,41 106,27

rate differential [%] F=B-D -3,44 0,81 -2,49 -2,27 0,88 0,82 0,63 -0,26 7,69

2009 2010 2011 2012 2013 2014 2015 2016 2017

Government debt A 1.770 1.852 1.908 1.990 2.070 2.137 2.173 2.220 2.269

Debt growth rate [%] B=At/At-1-1 5,91 4,61 3,03 4,31 4,03 3,24 1,69 2,16 2,19

GDP C 1.605 1.637 1.613 1.605 1.622 1.652 1.690 1.727 1.757

GDP growth rate [%] D= Ct/Ct-1-1 2,01 2,05 -1,48 -0,54 1,07 1,87 2,28 2,22 1,71

Debt/GDP [%] E=A/C 110,33 113,09 118,27 124,03 127,65 129,37 128,62 128,54 129,14

rate differential [%] F=B-D 3,90 2,56 4,51 4,84 2,95 1,37 -0,60 -0,06 0,48

Notes: Debt stands for Government consolidated gross debt, figures are in billions of current euro. GDP stands for Gross domestic

product at market prices, figures are in billions of current euro.

Source: author’s elaboration based on Eurostat database

Figure 1. Debt and GDP growth rate and rates differential – Italy 2000-2017

Note: Debt-GDP growth rate differential is calculated subtracting the GDP growth rate from the Debt growth rate. A positive value

indicate that the debt has grown faster than the GDP.

Source: author’s elaboration based on Eurostat database

-6,00%

-4,00%

-2,00%

0,00%

2,00%

4,00%

6,00%

8,00%

10,00%

Debt growth rate GDP growth rate Debt-GDP growth rate differential

3

Figure 2. Value added and measures of productivity, Italian whole economy - 1996-2017, percentage variations

Note: The leasing activities of real estate, families and cohabitation, international organizations and bodies and all the economic

activities that are part of the institutional sector of Public Administrations are excluded from the field of observation.

Source: [1] Istat, «Misure di produttività. Anni 1995-2017,» 2018

As can be seen from Figure 3, the real estate sector also comes back to increasing number of transactions

only after 2013-2015, after the "pause" of the 2008-2013 period due to the sub-prime mortgage crisis and

then to that of sovereign debts. In addition to transactions, investments were also start again in 2013 -

Figure 4 - driven mainly by foreign investors, to which domestic investors have also joined since 2015-2016.

As can be seen from Figure 5, Milan is by far the most attractive real estate market for investors, followed

by Rome, which in any case collects investment volumes equal to about a third of those in the Lombard

capital. The sectors considered the best by investors are the office and retail, which share the same two-

thirds of the total pie.

Figure 3. Normalized transaction (NNT) percentage change of the Italian real estate market 2004-2017

Source: [2] Pwc, «Real Estate Market Overview. Italy 2018,» 2018

4

Figure 4. Investment volume in the Italian real estate market by origins of the capital

Source: [3] CBRE, «Real Estate Market Outlook. Italia,» 2018

Figure 5. Real estate investments in Italy by location and sector


Despite the relative dynamism of the Milanese real estate sector, a report edited by PwC says that in Milan:

"Office demand is strong, [...] If you have a good product in Milan, there is a queue of tenants because there

is a construction backlog of 10 years."1 All this is confirmed by the annual report of 2018 of COIMA RES,

which shows the graph in Figure 6, which highlights the lack of quality assets in Milan compared to other

European metropolises, moreover in a leading sector such as the offices. This means that despite the strong

growth observed from 2013 onwards, other investments are also needed in the coming years to double the

share of high-quality office stocks, in order to align Milan with other high-ranking cities in Europe.

1 [48] PwC, «Emerging Trends in Real Estate.® Creating an impact. Europe 2019,» 2019. Pag. 57

5

Figure 6. Stock of grade A offices - % of total stock

Source: [4] COIMA RES, «Relazione finanziaria annuale,» 2018

Table 2. Survey of Italy's REIFs’ balance sheet main figures in 2003-2016

Real estate funds: market structure

Years # of funds Total assets Debts NAV Leverage

Mln of € Real estate Total

2003 19 5.141 3.718 573 4.414 1,16 2004 31 12.309 10.520 3.979 8.084 1,52 2005 61 18.326 15.215 6.019 11.859 1,55 2006 119 27.248 22.110 9.890 16.384 1,66 2007 174 36.058 30.434 13.453 21.531 1,67 2008 229 42.390 36.791 16.630 24.446 1,73 2009 267 47.517 40.936 19.517 26.306 1,81 2010 281 47.771 41.678 19.347 26.846 1,78

… … … … … … … 2013 361 55.212 47.963 19.821 32.846 1,67 2014 395 58.367 50.239 18.511 37.529 1,55 2015 417 60.338 51.502 17.342 40.526 1,49 2016 439 64.526 54.890 18.232 43.777 1,47

Source: author’s elaboration based on [5] Banca d'Italia, «Focus sull’industria dei fondi immobiliari retail,» 2017 and [6] M. L. Bianchi

e A. Chiabrera, «Italian real estate investment funds: market structure and risk measurement,» Questioni di Economia e Finanza,

April 2012

Table 2 shows the main balance sheet statistics and the number of Real Estate Investment Funds - REIFs -

operating in Italy from 2003 until 2016. The sector has grown steadily throughout the period considered,

both as regards the total number of operational funds, and for the net asset value and also for the financial

leverage used, at least until 2009. From Figure 7 it is possible to find confirmation of what reported in Table

2, and to observe a progressive reduction of the overall number of SGR that happens contextually to the

strong growth in the number of funds. “Real estate funds in Italy represent about 3% of the Italian asset

management market. […] During 2017, the weight of the first 5 SGRs, which amounted to 32.8 bn, has slightly

decreased from 47.9% to 46.3%. The main component of Italian real estate funds reserved funds accounting

for 96% over the total amount. […] The increase of the total number of reserved funds also during 2017 is

not related to the growth of the number of asset management companies but from the greater efficiency

achieved by the companies in relation to a larger amount of asset under management.”2 The paper "Italian

2 [2] Pwc, «Real Estate Market Overview. Italy 2018,» 2018. Pag. 38

6

real estate investment funds: market structure and risk measurement" contains some statistics, reported in

Figure 8, on the performances of Italian REIFs in the period 2005-2010 and it is noted that despite the

increase in leverage (and funds fees) ROE was overall decreasing, due to the adverse market conditions

highlighted by the overview presented so far. “We also find that the ROE relative to retail products has been

in general less volatile, although it has suffered from the negative cycle in recent years. However, in the long-

term it remains, in most cases, positive. As expected, the recent financial crisis has affected both the financial

drivers and the market value of REIF assets.”3

Figure 7. SGR and number of operative funds in Italy 2007-2017


Figure 8. Selected indicators of Italian REIF’s performances during the period 2005-2010

Note: The median values are in black, the mean values in blue, and the strips between the red lines contains 70 per cent of the

analyzed funds. Non-development funds active for at least two years are considered. One-year moving average of the 6 month

Euribor is reported in the chart of the cost of debt.

Source: Italian real estate investment funds: market structure and risk measurement.

3 [6] M. L. Bianchi e A. Chiabrera, «Italian real estate investment funds: market structure and risk measurement,» Questioni di Economia e Finanza, April 2012. Pag. 15

7

PART 2

INVESTMENT EVALUATION

8

PART 2.1 - BUILDING BLOCKS IN CAPITAL BUDGETING

2.1.1 - The Choice To Invest

An individual investor (or a company or any economic agent) according to classical microeconomic theory

makes decisions aimed at maximizing his profit, by maximizing the difference between revenues and costs.

The ability to generate income passes through the purchase of labor, capital and other inputs in the factor

markets to combine them together and produce goods and services, which are "brought" to the market of

goods and services to be sold and, through exchange, collect revenues. These steps are depicted in Figure

9. To be able to achieve his goal, the investor will have to make a multitude of choices about the use of the

resources at his disposal - which as they are scarce are susceptible to alternative uses - to make sure that

the costs that emerge from their use are more than offset by revenues, so as to obtain a profit.4 Investment

is a type of economic choice that does not concern the immediate maximization of the well-being of the

individual, but on the contrary entails the need to sustain only costs in the present, with the expectation of

obtaining greater benefits in the future. In this thesis I will deal precisely with the evaluation of an

investment by a private economic agent which has as its object a real estate property, which intends to

know if the game is worth the trouble and therefore needs analysis tools that allow him to find an answer.

The main elements that constitute an investment can be summarized in:

• The direct cost of the investment

• The indirect cost of the investment, that is the opportunity cost of capital

• The time value of money

• A series of future expected risky cash flows

2.1.1.1 - Time Value of Money

Time value of money means that time is money, or that time has a value that can be represented in

monetary terms; this means that a euro today is worth more than one euro tomorrow, as is commonly said.

This is mainly due to the fact that money has an opportunity cost defined by the interest rate. First of all,

we need to define the concept of opportunity cost, which is nothing but "implicit cost of capital, [...] it

reflects the income that could have been realized if the capital had been used in its next best alternative

way."5 As far as regards the interest rate, it can be defined as the cost paid by the debtor expressed as a

percentage of the sum borrowed. If an individual lends money for a certain period of time, we assume 1000

€ for a year, at the expiry of the loan the sum returned will be 1000 € plus interest, or the cost that the

debtor was willing to bear in order to have the money immediately . This explains why a euro today is worth

more than one euro tomorrow, or why having the euro available tomorrow we give up the possibility of

being able to use it in the best alternative investment available and the relative profit. Nevertheless, there

are two other factors that determine the time value of money6: i) inflation: a euro today is worth more than

4 [23] P. Krugman e R. Wells, Microeconomics, 2th edition, Worth Publishers, 2009. Pag. 6. “Why do individuals have to make choices?

The ultimate reason is that resources are scarce.” 5 Ibidem 6 [24] S. Vishwanath , Corporate Finance. Theory and Practice, 2th edition, New Dehli: Sage Publications, 2007

9

one euro tomorrow because inflation destroys the purchasing power of real goods and services; ii) cash

flow risk: the euro we have today is safe, while the euro we should have tomorrow is risky. Due to these

reasons described above it is not possible to directly compare cash flows received and paid in different time

periods, but as briefly mentioned above, the interest rate can be used to convert future cash flows to their

present value.

Figure 9. The circular flows diagram

Source: [7] N. G. Mankiw, Principles of Microeconomics, 5th edition, Cengage Learning, 2008.

2.1.1.2 - Present Value and Future Value

If it is true that € 1 today does not count as € 1 tomorrow, it is however true that € 1 today is worth (1 + x)

€ tomorrow. To solve the equation we need to understand what is the opportunity cost of the capital we

pay by giving up € 1 today. “The idea of present discounted value arose because we wanted to be able to

convert money at one point in time. "The interest rate" is the return on an investment that allows us to

transfer funds in this way ";7 in other words, by applying the interest rate to the present value as in equation

(1), it is possible to obtain the future value and therefore find that sum of future money equivalent to the

sum of current money in the case of an investment lasting only one period.

𝐹𝑉 = 𝑃𝑉(1 + 𝑟) (1)

7 [25] H. R. Varian, Intermediate Microeconomics A Modern Approach, 9th Edition, W. W. Norton & Company, Inc., 2014. Pag 201

10

If we consider investing € 100 for a year at an interest rate of 5%, the future value will be 100(1 + r) = € 105;

if this sum is always reinvested for one year at the 5% interest rate, the future value will be 105 (1 + r) = €

110.25. Thanks to this process of continuous reinvestment of the interests earned, the value of the

investment grows at a compound rate; the interest rate that describes this dynamic is called compounded

interest. After t periods the future value of the investment is given by the formula (2):

𝐹𝑉 = 𝑃𝑉(1 + 𝑟)𝑛 (2)

The procedure that allows to find the present value of a future cash flow is called discounting, while

capitalization is used to find the future value of a present cash flow.

2.1.1.3 - Opportunity Cost of Capital

How can these concepts be applied to the evaluation of an investment? We assume that an individual owns

a real estate property inherited from the current value of € 300,000 and that with an expenditure of €

70,000 for restructuring he can sell the building for € 400,000; the jobs would last one year and the interest

rate in that period is 10%. For simplicity, it is assumed that all cash flows are risk-free. Should the investment

be made? At first glance it would seem so, as against an expenditure of € 70,000 it could return a revenue

of € 400,000, but our investor did not consider the opportunity cost of capital. First, the value of the

property must be considered as a cost of investment, although it is an implicit cost, it’s a fundamental input

and it has certainly not rained from the sky without any cost being incurred, it is not a free lunch. Secondly,

we must consider the alternative of selling the property in the current state and investing in securities that

promise a 10% interest rate and which could therefore give a return of € 30,000 within a year; this is also

an opportunity cost, as it is the profit that is renounced by choosing restructuring rather than investing in

the financial market. The final value of the property is discounted using the opportunity cost of the capital,

or the interest rate of 10%, in order to check which present value would be needed to obtain € 400,000

after a year with a 10% return. The present value is € 363.636 against a total expenditure of € 370,000, so

the investment should not be considered, in fact with an investment of € 370,000 in securities with a 10%

interest rate, after one year the assets of the investor would amount to € 407,000.

2.1.1.4 - Discounted Cash Flow – DCF

Discounted cash flow, hereinafter DCF, is a method of valuing investments based on the discounting of a

series of expected cash flows generated by an asset or investment when the duration is longer than a single

period. The fundamental elements are i) the amount of net cash flows, ii) the distribution of flows over time

and iii) the discount rate. With regard to the cash flow dimension, it is essentially defined by the calculation

of the net cash flows for each period, algebraically adding the cash in-flow and the cash out-flows;

distribution is reported based on the income and expenses of each period, while the discount rate is

calculated as the opportunity cost of capital. The formula (3) is shown below.

11

𝑉 = ∑𝑁𝐶𝐹𝑡

(1+𝑟)𝑡𝑇𝑡=0 +

𝑅𝐸𝑉𝑡

(1+𝑟)𝑇 (3)

V: present value of expected cash flows

NCFt: are the net cash flows expected each year

r: opportunity cost of capital

REVt: reversion value, or collection received at the end of the investment from the disposal of the asset

T: final year of the investment

2.1.2 - Cash flows Estimation

“The effect of taking a project is to change the firm’s overall cash flows today and in the future. To evaluate

a proposed investment, we must consider these changes in the firm’s cash flows and then decide whether

they add value to the firm. The first (and most important) step, therefore, is to decide which cash flows are

relevant.”8

2.1.2.1 - Relevant Cash Flows

Also called incremental cash flows, the relevant cash flows "consist of any change in the firm's future cash

flows that are a direct consequence of taking the project." 9 Especially if a company is large it could be very

difficult to assess all the cash flows if we were to have to evaluate a project, but fortunately it is necessary

to evaluate only the incremental cash flows, or the difference between the cash flows obtained by the

company with the investment compared to the cash flows obtained without the investment.

Sunk Costs

The sunk costs are costs for which you have already paid or you to which you have already contracted the

obligation to pay, so there is no possibility of acting to avoid making that expense. For the definition that

has been given of relevant cash flows, the sunk costs do not fall into this category, therefore they should

not be considered when analyzing a project.

Opportunity Cost

The opportunity cost, as defined above, is the best possible benefit that you have to give up when you

decide to use a resource. Economists usually say "there is no such thing as a free lunch" to express the

concept of opportunity cost. From the point of view of identifying relevant cash flows this means that any

input that is used for a project is susceptible to alternative uses and therefore its use must be taken into

account considering the right cost.

Side Effects

8 [11] S. A. Ross, R. W. Westerfield e B. D. Jordan, Fundamentals of Corporate Finance, 10th edition, The McGraw-Hill Companies, Inc., 2013. Pag. 306 9 Ibidem

12

It is possible that the introduction of a new product drains demand from the other products of a company,

causing a negative impact in terms of cash flows on the entire portfolio. This phenomenon is called erosion

or cannibalism and must be recognized as relevant cash outflow caused by the project, as its direct

consequence.

Net Working Capital

Usually a project requires to expand current assets such as inventories, account receivables and account

payables. An expansion of net working capital must be seen as a loan, since on this immobilized capital no

interest is received and if the project is not implemented it would be earned by the company. Once the

project is completed, the inventory is generally sold, the account receivables collected and the debts paid,

going to eliminate the working capital increase sustained by the company to finance the project.

Financiang Costs

“In analyzing a proposed investment, we will not include interest paid or any other financing costs such as

dividends or principal repaid because we are interested in the cash flow generated by the assets of the project

[…] our goal in project evaluation is to compare the cash flow from a project to the cost of acquiring that

project in order to estimate NPV. The particular mixture of debt and equity a firm actually chooses to use in

financing a project is a managerial variable and primarily determines how project cash flow is divided

between owners and creditors. This is not to say that financing arrangements are unimportant. They are just

something to be analyzed separately.”10 In other words, the important thing is the evaluation of the cash

flows generated by the assets and the cost incurred to purchase them; how these cash flows will have to be

distributed between equity and debt is another topic that needs to be treated separately.

2.1.2.2 - Project Cash Flows

The free cash flows generated by the assets are given by the following formula:

Project free CF = operating CF − capital spending − changes in net working capital (4)

Operating Cash Flows

The operating cash flow formula is as follows:

Operating CF = EBIT + Depreciation − Taxes (5)

EBIT = Revenues − COGS − Opex − Depreciation (5.1)

10 Ibidem

13

Since only the cash expenses are considered, the depreciation has to be added back because of iys nature

of accrued expense. It should be noted that real estate properties purchased as an investment are not

subject to depreciation.

Current and Non-current Capital Expenditures

Capital expenditure includes both investment costs in net working capital and those for the purchase of

non-current assets. As far as net working capital is concerned, only the variation is considered, that is the

difference between the quantity at the beginning of the year and that at the end; this serves to compensate

for the fact that revenues are considered completely cash. A simple example allows to explain the

mechanism in the best way.

Income Statement, figures are in €

Revenues 500

Costs (310)

Net income 190

Source: [11] S. A. Ross, R. W. Westerfield e B. D. Jordan, Fundamentals of Corporate Finance, 10th edition, The McGraw-Hill

Companies, Inc., 2013

Figures are in € Beginning of Year End of Year Change

Account Receivable 880 910 +30

Account Payableble 550 605 +55

Net working capital 330 305 -25



Assuming for simplicity that the capital expenditure is zero, applying formula (5) we obtain: operating CF =

190 - 0 - (-25) = 215 €. By adding the variation of the net working capital it is possible to compensate the

fact that the values of the revenues and costs expressed in the income statement are non-cash. An increase

of € 30 in accounts receivables means that of the € 500 of revenues, 470 have been cashed and 30 are yet

to be collected, so the total cash inflows are 500 - 30 = € 470. An increase of € 55 in payables accounts

means that of € 310 of costs, 255 have been paid and 55 remain to be paid, so the total cash outflows are

310 - 55 = € 255. The net cash flows at the end of the year will therefore be 470 - 255 = € 215, or the same

value obtained by applying the formula (5).

Asset Disposition

At the end of the investment period, the assets acquired at the beginning can be sold. In the specific case

of real estate investments, the profit component deriving from the terminal value of the asset at the end of

the holding period occupies a substantial part of the total returns and is therefore an item that must be

considered carefully. The cash flows obtained from the sale of the asset must be considered net of any costs

incurred during the sale and taxes. The estimate of the relevant cash flows of the project will be made in

the case study.

14

2.1.3 - Investment Criteria

“An investment is worth undertaking if it creates value for its owners. In the most general sense, we create value by identifying an investment worth more in the marketplace than it costs us to acquire. […] This is what capital budgeting is all about—namely, trying to determine whether a proposed investment or project will be worth more, once it is in place, than it costs.”11 When we talk about investment that is worth more than the costs, we must consider all the costs, therefore also the opportunity cost, remembering that a euro today is worth more than one euro tomorrow and that a secure euro is worth more than a risky euro. Investments usually promise a risky euro tomorrow in exchange for a certain amount of expenditure today. To compare these two values the discount rate is used, whose estimation will be discussed in chapter 4.

2.1.3.1 - Net Present Value - NPV

The net present value (henceforth NPV) allows us to verify whether the present value of a series of cash

flows generated by an investment is greater, less than or equal to the sum used for the investment and

therefore to understand if value has been created. The NPV formula (6) is like the DCF formula (3) with the

addition of the negative cash flow due to the initial investment expense. This means that for each period

we should estimate the net cash flows generated by business activities and discount them by using an

adequate discount rate to define their present value. Figure 10 shows the typical pattern of an investment,

that consist in a large initial expense, a series of relatively small positive cash flows with respect to the

investment and a relatively large cash flow defined as terminal value, calculated as disinvestment of the

assets purchased at start of the useful life of the investment or as present value of the cash flows generated

beyond the time horizon considered in the capital budgeting.

𝑁𝑃𝑉 = − 𝐼 + ∑𝑁𝐶𝐹𝑡

(1+𝑘)𝑡𝑇𝑡=0 (6)

NCFt: expected net cash flow for each period t

T: end of the investment period

I: initial investment

k: opportunity cost of capital

The NPV decision rule says that if it is greater than zero, the cash flows generated by the initial investment

have a size and distribution over time such as to offset the opportunity cost of the investor, or the required

return rate and therefore the investment should be accepted. Conversely, if NPV is lower than zero it is not

worth the trouble and the investor should refuse the investment. It should be pointed out that an NPV less

than or equal to zero does not mean that the profit obtained from the investment is zero, because the cash

flows are discounted so as to be "weighed" for their riskiness. A negative NPV means that the return is not

large enough to offset the risk, as it was quantified in the discount rate, so an NPV equal to 0 does not imply

the rejection of the investment, as it means that the present value of the flows of cash is large enough to

offset the risk incurred, based on the discount rate used. The weaknesses of the NPV are that it does not

consider the amount of capital employed of the initial investment relative to the investor's possibilities, it

11 [11] S. A. Ross, R. W. Westerfield e B. D. Jordan, Fundamentals of Corporate Finance, 10th edition, The McGraw-Hill Companies, Inc., 2013. Pag. 267

15

does not give any information about the period in which the investment reaches break-even and requires

the estimate of the opportunity cost of capital, which is not always easy.

Figure 10. Typical cash flows pattern of an investment

Source: http://jukebox.esc13.net/untdeveloper/RM/RM_L9_P4/RM_L9_P4_New2.html

2.1.3.2 - Internal Rate of Returns – IRR

L’IRR is the discount rate that equates the discounted cash in-flows and discounted cash out-flows; in

other words, it is the discount rate that makes the NPV equal to zero if used as a discount rate. 12 The

formula is:

𝑁𝑃𝑉 = ∑𝑁𝐶𝐹𝑡

(1+𝐼𝑅𝑅)𝑡𝑇𝑡=0 = 0 (7)

In practice, the IRR is the rate of return that, on average, we expect the investment generates in each period.

The decision rule is to accept only investments with IRR greater than the opportunity cost of the capital or

of the return rate requested by the investor. The IRR decision rule does not always agree with the NPV

decision rule: we assume that you are in the situation of having to decide between the two mutually

exclusive investments A and B; the IRR can be used to compare the two investments. Looking at Table 3,

where the NPV was calculated with a discount rate of 10 percent, the choice would fall on investment B if

we were to look at the NPV, while A it would be preferable if we look at the IRR. If we use a discount rate of

25 percent, as described in Table 4, the NPV of alternative B would become negative, since its IRR is less

than 25 and the investment A remains the only rational choice. The bottom line is that IRR and NPV can

make conflicting judgments. This depends on the reinvestment hypothesis: the NPV assumes that the cash

flows are reinvested at the cost of capital and therefore at the discount rate, while the IRR is reinvested at

the internal rate of return.13 The fact that the NPV or the IRR prevails in the valuation of an investment

depends on the use made of the cash flows received and the return that can be obtained. If, for example,

one can expect to obtain a return equal to the opportunity cost of capital from the re-investment of returns,

12 [11] S. A. Ross, R. W. Westerfield e B. D. Jordan, Fundamentals of Corporate Finance, 10th edition, The McGraw-Hill Companies, Inc., 2013. Pag. 280 13 [12] F. Fabozzi e P. Peterson, Capital Budgeting: Theory and Practice, John Wiley & Sons, Inc., 2002. Pag. 89

http://jukebox.esc13.net/untdeveloper/RM/RM_L9_P4/RM_L9_P4_New2.html

16

in investments with IRR higher than the cost of capital, the IRR generates an overestimation of the returns

on the investment. If the returns are reinvested at the opportunity cost of the capital, the NPV must be

used. In a real estate investment, year-end profits cannot be reinvested in the same asset that generated

them, due to the specificity and uniqueness of the asset itself, so the NPV should be considered more

informative in the selection of alternative projects.

Table 3. IRR e NPV of investments A and B with a discount rate of 10%

Investment IRR NPV

A 28,65% 516.315€

B 22,79% 552.620€ Sources: [12] F. J. Fabozzi, P. P. Peterson, Capital Budgeting: Theory and Practice, John Wiley & Sons, Inc., 2002

Table 4. IRR e NPV of investments A and B with a discount rate of 25%

Investment IRR NPV

A 28,65% 75.712 €

B 22,79% -67.520 € Sources: [12] F. J. Fabozzi, P. P. Peterson, Capital Budgeting: Theory and Practice, John Wiley & Sons, Inc., 2002

Other information provided by the IRR is the timing of the cash flows: investments with cash flows closer

over time generate higher IRR, as a higher discount rate will be required to equal outgoing cash flows to

incoming cash flows. With regard to the riskiness of the cash flows, we can assume that we have two

alternative investments that produce the same NCF but have two different risks and therefore two different

discount rates. The IRR of the two investments will be the same as the cash flows are the same both for

timing and size, but the NPV of the less risky investment will be greater. In summary, the IRR cannot see the

riskiness of the investments, while the NPV can; the IRR is able to give information about the timing of the

cash flows, as well as the NPV. In general, the IRR should be used as the first level of alternative investment

screening, to eliminate those with capital costs greater than the rates of return, after which the NPV should

be used to select investments that maximize investor wealth.

2.1.3.3 - Profitability Index – PI

The profitability index is calculated with the following formula:

𝑃𝐼 = 𝑃𝑉 𝑜𝑓 𝑓𝑢𝑡𝑢𝑟𝑒 𝑐𝑎𝑠ℎ 𝑖𝑛𝑓𝑙𝑜𝑤𝑠

𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 (8)

When NPV is zero, PI is 1. PI tells the investor how many euros his wealth increases for each euro invested.

For indices greater than one the investment creates value, for indices between zero and one the investment

destroys value. The decision rule of the PI should therefore be to choose the investment with the greatest

ratio, but this rule does not always lead to take the alternative that creates greater value, especially when

comparing investments with different scales. This fact is represented in Table 5.

17

Table 5. PI and NPV of investments A, B and C

Investment PV outflows PV inflows PI NPV

A 10.000 € 16.000 € 1.6 6.000 €

B 10.000 € 15.000 € 1.5 5.000 €

C 20.000 € 28.000 € 1.4 8.000 € Sources: [12] F. J. Fabozzi, P. P. Peterson, Capital Budgeting: Theory and Practice, John Wiley & Sons, Inc., 2002

If one chooses on the basis of the greater PI, the investment A should be undertaken, while if one chooses

by looking at the NPV, the choice would fall on the investment C. Looking at investments A and B, which

have the same scale, one sees that the PI gives the same information as the NPV. When investments have

different scales, the PI should not be used, while on an equal scale it says the same things as the NPV, so it

is redundant.

2.1.3.4 – Payback Time

“A project’s payback period is found by counting the number of years it takes before the cumulative cash

flow equals the initial investment.”14 The decision rule used is that a shorter payback is better than a longer

one. The problem is that this indicator does not tell us anything about the cash flows after break-even, so

you cannot know any measure of investment profitability, despite this the payback is useful in industries

characterized by extremely high depreciation rates of the assets, such as the electronic equipment industry,

which therefore require rapid payback periods since within one or two years it will be necessary to innovate

again with new investments.15 In sectors where the investment is long-lasting, such as real estate,

repayment is not very useful.

2.1.3.5 - Modified Internal Rate of Return – MIRR

To explain the MIRR it is necessary to return to the hypothesis of reinvestment underlying the IRR: to say

that an investment has an IRR of 28.62% means that all net cash flows obtained at the end of each period

are reinvested with a compounded rate of return of 28, 62% for the whole life cycle of the investment, in

table 6 this situation was hypothesized.

r = √𝐹𝑉

𝑃𝑉

𝑡− 1 (9)

In the hypothesis that the NCFs would not be reinvested every year with a compound rate until the end of

the life cycle of the investment, the rate of return would be:

FT = 40 € * 5 = 200 €

14 [17] R. A. Brealey, S. C. Myers e F. Allen, Principles of Corporate Finance, 10th edition, The McGraw-Hill Companies, Inc., 2011. Pag. 105 15 [12] F. Fabozzi e P. Peterson, Capital Budgeting: Theory and Practice, John Wiley & Sons, Inc., 2002

18

PV = 100 €

r = √𝐹𝑇

𝑃𝑉

𝑡− 1 = √

200

100

5− 1 = 14.87 %

The return rate of this investment in which the annual NCFs are not reinvested is defined as modified

internal rate of return or MIRR. The MIRR can be modified by acting on the return rate assumed for the

reinvestment of annual NCFs; in fact the compound return of 14.87 per cent was obtained assuming a

reinvestment of 0 per cent, but it could be another value. In this way it is possible to make the hypothesis

of reinvestment more likely, for example, assuming that the reinvestment rate is that of an annual

government bond. The decision rule of the MIRR is to accept projects with MIRR greater than the

opportunity cost of capital.

Table 6. IRR reinvestments assumptions break-down

Years 0 1 2 3 4 5

NCF - 100,00 € 40,00 € 40,00 € 40,00 € 40,00 € 40,00 €

NPV 51,60 €

IRR 28,6%

Each year NCF is reinvested at 28,6% of rate of return

year 1 40,00 € 51,45 € 66,18 € 85,12 € 109,49 €

year 2 40,00 € 51,45 € 66,18 € 85,12 €

year 3 40,00 € 51,45 € 66,18 €

year 4 40,00 € 51,45 €

year 5 40,00 €

FV (CF at 5th years sum) 352,23 €

PV 100,00 €

Compound rate of return – r (using formula 9) 28,6% Source: author’s elaboration

19

PART 2.2 – INVESTMENT RISK

Until now it has been implicitly assumed that the expected cash flows are "safe", that is there are no

possibilities that the amount and timing of their collection or payment would be different than expected.

However, it is clear that being these cash flow events that belong to the future, they cannot be considered

as secure at 100% and, on the contrary, they are much more likely to differ from what was expected. Since

the present value of a series of cash flows depends on their size, the timing with which they occur and the

discount rate with which they are discounted, a variation in one of this parameters cause a change in the

performances of the investment, so it is necessary to understand how the risk impacts on the main elements

of the DCF and so on the profitability of an investment.

2.2.1 - Some Definitions About Risk and Risk Management

Risk can be defined as:

• Risk – when an outcome may or may not occur, but its probability of occurring is known. Uncertainty

– when an outcome may or may not occur and its probability of occurring is not known.16

• Risk: effect of uncertainty on objectives. Risk is often expressed in terms of a combination of the

consequences of an event (including changes in circumstances) and the associated likelihood of

occurrence.17

• A risk can be defined as an uncertain event or circumstance that, if it occurs, will affect the outcome

of a programme/project.18

• Risk is generally referred to as: the uncertainty expressed through the significance and likelihood of

events and their outcomes that could have a material effect on the goals of a real estate development

organization over a stated time horizon.19

• Risk is a possible future event combining the probability or frequency of occurrence of a defined

threat or opportunity and the magnitude of the consequences of that occurrence.20

Risk management can be defined as:

• Enterprise risk management is a process, effected by an entity’s board of directors, management

and other personnel, applied in strategy setting and across the enterprise, designed to identify

potential events that may affect the entity, and manage risk to be within its risk appetite, to provide

reasonable assurance regarding the achievement of entity objectives.21

16 Sloman (1995) cited in [31] P. Loizou e N. French, «Risk and uncertainty in development: A critical evaluation of using the Monte Carlo simulation method as a decision tool in real estate development projects,» Journal of Property Investment & Finance, vol. 30, n. 2, pp. 198-210, 2012 17 [34] International Organization for Standardization (ISO), «Risk management — Principles and guidelines. ISO 31000:2009(E),» ISO, 2009 18 [16] Royal Institution of Chartered Surveyors (RICS), «Management of risk, 1st edition,» Royal Institution of Chartered Surveyors (RICS), London, 2015 19 [13] W. GleiBner e T. Wiegelmann, «Quantitative methods for risk management in the real estate development industry. Risk measures, risk aggregation and performance measures,» Journal of Property Investment & Finance, vol. 30, n. 6, pp. 612-630, 2012 20 [32] International Organization for Standardization (ISO), «ISO/IEC Guide 73:2002 Risk management -- Vocabulary -- Guidelines for use in standards,» International Organization for Standardization (ISO), 2002 21 [33] Committee of Sponsoring Organizations of the Treadway Commission (COSO), «Enterprise Risk Management — Integrated Framework. Executive Summary,» Committee of Sponsoring Organizations of the Treadway Commission, 2004

20

• Risk management: coordinated activities to direct and control an organization with regard to risk.

Risk management framework: set of components that provide the foundations and organizational

arrangements for designing, implementing, monitoring, reviewing and continually improving risk

management throughout the organization.22

Risk is defined as a possible event whose probability is known, which causes a deviation of the observed

results from those expected and can produce both a negative and a positive impact on the output-related

objectives. The uncertainty differs from the risk in that the probability of an event is not known, while for

the risk it is. Risk has a quantitative nature, while uncertainty is something more subjective. Risk

management is the entire process of identifying, quantifying, mitigating and monitoring risks.23

According to GleiBner and Wiegelmann (2012) there are some common elements, highlighted in Figure 11,

in the literature analyzed by them regarding risk management: “Four core elements in common […]. The

goal of the risk identification process is to identify possible risks, which may affect, either negatively or

positively, the objectives of the business and the activity under analysis. Risk assessment is defined as the

overall process of risk analysis and risk evaluation and helps in determining which risks have a greater

consequence and impact than others as well as the probability of the event occurring. This is followed by the

risk control phase, which evaluates whether the level of risk found during the assessment process requires

management attention. Risk monitoring is the periodic tracking of risks and reviews the effectiveness of the

treatment plan.”24 In summary, the activities that are part of the risk management process are:

• Risk identification: identification of sources of risk that may have a negative (and positive) impact

on the objective of the business.

• Risk assessment: quantification of impact and probability of risks identified

• Risk control: decide if avoid, minimize, transfer or accept risk

• Risk monitoring: periodic tracking and check of effectiveness of actions undertaken from feedback.

As already specified in the introduction, the perimeter of this thesis will only include the part of

identification and assessment of the risk and I will not deal with control and monitoring at all. Looking at

Figure 11 three distinct phases can be identified: the definition of objectives, the identification of risks and

the management of risks. The objectives of a business activity can be the most diverse and are generally

expressed with indicators, both quantitative and qualitative. As stated at the beginning of this chapter, the

ultimate goal, the end game of an investor as well as of a company, is to maximize profit; this means that in

the preliminary evaluation phase of a project, objectives will be set about the potential gains that this can

give. In the risk identification phase it will be necessary to understand which are the sources from which

those events could be generated that could lead to a deviation from what was expected. The point is that

there will always be forecast errors, but only those that exceed the maximum deviation value set a priori,

whose extent defines the risk tolerance, or risk appetite, of the investor must be considered risky. The

magnitude of the deviation that you are willing to accept is the price that you have to pay in order to get

the profit you are aiming for “is what you must give up in order to get an item you want - the opportunity

cost of that item.”25 The definition of opportunity cost tells us that not only obtaining a profit has a cost,

22 [34] International Organization for Standardization (ISO), «Risk management — Principles and guidelines. ISO 31000:2009(E),» ISO, 2009 23 [16] Royal Institution of Chartered Surveyors (RICS), «Management of risk, 1st edition,» Royal Institution of Chartered Surveyors (RICS), London, 2015 24 [4] W. GleiBner e T. Wiegelmann, «Quantitative methods for risk management in the real estate development industry. Risk measures, risk aggregation and performance measures,» Journal of Property Investment & Finance, vol. 30, n. 6, pp. 612-630, 2012 25 [23] P. Krugman e R. Wells, Microeconomics, 2th edition, Worth Publishers, 2009, pag. 7

21

that is the possibility of variance between observations and expectations, but also that the greater the profit

that one aims to obtain, the greater the variance that one must be willing to bear. This is why we refer to

the relationship between risk and return as a trade-off.

Figure 11. Risk management process

Sources: [13] W. GleiBner e T. Wiegelmann, «Quantitative methods for risk management in the real estate development industry.

Risk measures, risk aggregation and performance measures» Journal of Property Investment & Finance, vol. 30, n. 6, pp. 612-630,

2012

2.2.2 - Returns of Real Estate Assets

To understand what returns the best thing to do is solve equation (1) for r:

𝑟 = 𝐹𝑉

𝑃𝑉− 1 (10)

Substantially, the returns are the rate of profit on an investment. In real estate, the two sources of income

from an investment come from the annual income and from the reversion value, or from the receipt

obtained from the disinvestment of the property at the end of the holding period. Adding these elements

into the "conceptual framework" of equation (10), we obtain what is reported in equation (11). The first

element on the right hand of the equation is profit from the rent, which is generally called income yield,

while the second element indicates the return obtained from the price change of the asset, usually defined

capital gain.

22

𝑟 = 𝑁𝑂𝐼𝑡

𝑃𝑡 +

𝑃𝑡+1 − 𝑃𝑡

𝑃𝑡 (11)

𝑟 = 𝑖𝑛𝑐𝑜𝑚𝑒 𝑦𝑖𝑒𝑙𝑑 + 𝑐𝑎𝑝𝑖𝑡𝑎𝑙 𝑔𝑎𝑖𝑛 (11.1)

The income yield can be calculated using both the annual income and the net operating income (NOI), or

the annual income net of operating costs incurred during the period over which the return was calculated;

in the first case we will talk about gross yield, in the second case of net yield. We can consider the NOI as

the EBITDA of the property. The reports of the largest and most well-known real estate brokers, such as

Kushman & Wakefield or CBRE, report net yields, as investors are interested in knowing the income they

will have available after have paid for the recurring costs.26 All this implies that the operating costs of each

property, the share of these costs repaid by the tenants to the owner and the interest expense incurred

should be known; obviously this is not possible, and approximations and hypotheses are made to provide

information that remains however roughly correct. In the accounting of a real estate between the NOI and

the free cash flow actually available to the owners at the end of each period there are, in addition to the

interest, capital expenditure, usually called capex. The estimate of this cost item, although extremely

important, is difficult to generalize and predict, as it depends on a multitude of factors, including the age of

the building and the owner's choices in relation to market demand. In any case, the costs incurred can vary

considerably from one period to another, in fact the typical curve of a maintenance plan is characterized by

a "peak" profile over time, as illustrated in Figure 12, due to the need to group together their various

activities for reasons of site efficiency.27

Within the discourse on the returns of a real estate asset, this premise serves to explain that in order to

correctly defining the net returns of a real estate investment it is necessary to take into account, in addition

to the operating expenses, also the capex and their effect on the value of the property; the net yield

calculated using the NOI in the numerator is useful to provide a rough indication of the level and dynamics

of cap rates in the market, but does not return an indicator of effective return for landlords, which must

pay capex, interest and taxes and only after making a net profit. To better understand how capex impacts

on returns, it is useful to consider a hypothetical value added transaction on a low quality property. The

price initially paid is low and the capex incurred for the restructuring are huge, but allow the property to be

sold at a much higher price than the purchase price, showing a significant capital gain. Considering only the

NOI, purchase price and resale price, as the formula (11) would suggest, the return would be high, but is

this procedure correct? The answer is no, because you didn’t consider capex. The NCREIF, National Council

of Real Estate Investment Fiduciaries, has developed a method of calculating total returns r (14) separating

between income returns y (12) and capital returns g (13).28

26 The following is the footnote used by Kushman & Wakefield in its quarter snapshot [26]: (*) Yields are calculated on a net basis as reported below: Net Yield = NOI (1) / PP (2) 1. Net operating income – after deducting all non-recoverable expenditure 2. Purchasing price – excluding transafer costs, tax and legal fees. With respect to the yield data provided, in light of the changing nature of the market and the costs implicit in any transaction, such as financing, these are very much a guide only to indicate the approximate trend and direction of prime initial yield levels and should not be used as a comparable for any particular property or transaction without regard to the specifics of the property. 27 [8] S. Han, C. Hyun e J. Kim, «Minimizing Fluctuation of the Maintenance, Repair, and Rehabilitation Cost Profile of a Building» Journal of Performance of Constructed Facilities, vol. 30, n. 3, 2015 28 [27] National Council of Real Estate Investment Fiduciaries, «NCREIF,» [Online]. Available: https://www.ncreif.org/public_files/Users_Guide_to_NPI.pdf

23

𝑦 = 𝑁𝑂𝐼

𝑉𝑡− 1

2(𝑃𝑆−𝐶𝐼)−

1

3𝑁𝑂𝐼

(12)

𝑔 = 𝑉𝑡+1− 𝑉𝑡+(𝑃𝑆−𝐶𝐼)

𝑉𝑡− 1


1

3𝑁𝑂𝐼

(13)

𝑟 = 𝑉𝑡+1− 𝑉𝑡+(𝑃𝑆−𝐶𝐼)+𝑁𝑂𝐼

𝑉𝑡− 1


1

3𝑁𝑂𝐼

(14)

NOI: “net operating income is gross rental income plus any other income less operating expenses - utilities,

maintenance, taxes, property management, insurance”.29

Vt: value of the property at the beginning of the period

Vt+1: value of the property at the end of the period

PS: partial sales, ovvero “sale of a portion of the property”30

CI: capital improvements, that is “expenditures for ongoing costs of business not accounted for in the Net

Operating Income (NOI) including leasing commissions, tenant improvements, and other expenditures for

replacement of long-lived equipment or physical structures. […] Major capital expenditures for expansions

or renovations […] occasional, highdollar-value capital expenditures that alter the physical, functional, or

economic condition of a property.”31 For greater clarity, the different types of capex were organized in Table

7 to clearly show the two sub-categories distinct from the NCREIF.32

Figure 12. Typical peaked profile of a maintenance plan costs over the asset’s life cycle

Source: [8] S. Han, C. Hyun e J. Kim, «Minimizing Fluctuation of the Maintenance, Repair, and Rehabilitation Cost Profile of a

Building,» Journal of Performance of Constructed Facilities, vol. 30, n. 3, 2015

29 Ibidem 30 Ibidem 31 Ibidem 32 [28] M. S. Young, J. D. Fisher e J. D’Alessandro, New NCREIF Value Index and Operations Measures, NCREIF, 2016

24

Table 7. Capex sub-categories

CAPEX

A - Recurring capex B - Extraordinary capex

Leasing commissions Building expansion

Tenant improvement Large renovations

Equipment replacement High-dollar valued activties

Replacement of physical structure Alteration of physical, functional,

economic state of the building

Typical recurring capex Sources: author’s elaboration based on NCREIF classifications

Figure 13. NCREIF index components of returns

Source: [9] NCREIF, «Fourth Quarter 2018 NCREIF Indices Review,» 2018. [Online]. Available:

https://www.ncreif.org/globalassets/public-site/webinar--education-page-images/webinars/webinar-slides-4q-2018-v-4.pdf

The expenses that fall under category A are considered "capital improvement" and used directly in formula

(14), while if they belong to sub-category B they are subject to the "filtering rule", which is nothing but the

exclusion of a cost item from the total sum of capital improvement, if this has an absolute value, expressed

in relation to the initial market value of the property, greater than 10%.33 If a building is in a state of

significant transformation due to the performance of relatively expensive activities, it is excluded from the

index prepared by the NCREIF, to re-enter once having stabilized its value.

The moral of this story is that the comparison between the returns of alternative investments, together

with the comparison of risks, is one of the fundamental activities from the analysis of an investment and as

I have tried to argue, in the real estate sector, it is not a process as immediate as it could be for stock and

bonds, but requires attention in the hypotheses underlying the calculation and the method by which the

various cost items are considered.

33 [29] NCREIF, NCREIF Research Corner - New NCREIF Indices – New Insights, 2015

https://www.ncreif.org/globalassets/public-site/webinar--education-page-images/webinars/webinar-slides-4q-2018-v-4.pdf

25

2.2.2.1 - Nominal and Real Returns

Once clarified what the returns of an investment are, it is necessary to consider the distinction between

nominal and real returns. The difference between the two rates is simply that nominal rates are not adjusted

for inflation, while real rates are. Inflation is basically the increase in the prices of consumer goods and

services; Istat defines it as “a process of continuous, generalised increase in the prices of goods and services

for household consumption. Rising inflation refers to a situation in which the rate of price increases is

accelerating, while falling inflation describes the case in which prices are rising but at a decreasing rate.”34

In other words, inflation rate is the rate of change for the period of the Consumer Price Index (CPI), or

“prices of a basket of goods and services, which is representative of households consumer spending in a

specific year.”35 The price increase leads to a decrease in the value of money, or in the quantity of goods

and services that can be purchased with a certain amount of money. To better understand this part, it is

useful to give a simple example. Let's assume that prices rise by 5% a year, so the inflation rate is 5% and an

investment is estimated that promises to return € 115.50 in a year against an initial cost of € 100; the

nominal rate of return on the investment is therefore 15.5%. What is the effect of inflation on this

investment? To understand the difference between nominal and real values, it is useful to give examples

with quantities of real goods: we assume that a pizza costs € 5 at the beginning of the year and therefore

with € 100 we can buy 20 pizzas. Due to inflation after a year a pizza will cost 5% more, or € 5.25 and then

with the € 115.50 of the future value it will be possible to buy € 115.50 / 5.25 = 22 pizzas. Measured in

pizzas, our investment will give a return rate of (22/20) - 1 = 10%. The nominal return rate, measured in

monetary value, is 15.5%, while the real return rate, measured in real goods, is 10%.36 To sum up, we can

say that “The nominal rate on an investment is the percentage change in the number of dollars you have.

The real rate on an investment is the percentage change in how much you can buy with your dollars—in

other words, the percentage change in your buying power.”37 From this simple example we can guess why

investors are interested in real returns instead of nominal ones. The one just described is the so-called Fisher

Effect and is mathematically formalized in the formula (15).

(1 + 𝑖𝑡) = (1 + 𝑟𝑡)(1 + 𝜋𝑒) (15)

it: current nominal rate

rt: current real rate

πe: expected inflation rate

The current nominal rate is therefore given by the combined effect of the real rate required by investors

today and expected inflation during the investment period. Since the money is immobilized today and the

investment bears fruit in the future, given the same real returns required, the higher the expected inflation,

the higher the current nominal yields required. Returning to the example above, a nominal return rate of

15.5% with inflation of 5%, gives you a real return of (1 + r) = (1 + i) / (1+ π), then r = 1.155 /1.05 - 1 = 10%.

34 [30] ISTAT, «Consumer prices: data and information,» [Online]. Available: https://www.istat.it/it/archivio/17484 35 [30] Ibidem 36 [11] S. A. Ross, R. W. Westerfield e B. D. Jordan, Fundamentals of Corporate Finance, 10th edition, The McGraw-Hill Companies, Inc., 2013 37 [11] Ibidem

26

The formula (15) is useful because we are required to define the discount rate suitable for discounting

expected cash flows from an investment to assess the current value, taking into consideration both the

actual return required and the expected inflation and choose given the specific needs of each investment

valuation, always remembering to be consistent in the treatment of inflation and then using nominal rates

to discount nominal exchanges and real rates for real exchanges.

2.2.2.2 - Is It Better to Use Nominal or Real Interest Rates?

As already explained in the previous paragraph, the use of nominal or real discount rates is irrelevant

provided that they remain consistent using nominal rates to discount nominal flows and real rates for real

cash flows.

Figure 14. Deflated Euribor 3 months interest rates – three different deflation assumption

1) Three-month interbank rates minus annual consumer price inflation

2) For each period, the latest available forecast at that time is used. Consensus Economics reports, on a monthly basis, inflation

forecasts for the euro area as annual averages. Every January (year t), another year (t+1) is added to the forecast. The measure of

expected inflation based on these forecasts is calculated in the following way: in the first six months of each year, the average of

the forecast for the same year and the coming year is used. In the second half of each year, only the forecast for the coming year

is used.

Sources: [10] European Central Bank, «Recent developments in real interest rates in the euro area,» April 2001

“In constructing real interest rates, one difficult measurement issue is how inflation expectations, which are

not observable, are computed. The simplest approach, which may work well over shorter horizons, is to

assume that expectations simply reflect past developments, so that the best forecast of future inflation is

its most recent level. […] another approach often pursued is to subtract the level of current actual inflation

(rather than estimates of expected inflation) from the nominal interest rate, on the assumption that ex post

27

expectations are on average in line with actual outturns.”38 As explained with the formula (15), nominal

rates depend on both the actual rates required and the expected inflation, the quantity of which is not

known and may vary depending on the basket of goods considered and therefore on the type of indicator

used; those published by the European Central Bank are the HICP - Harmonized Index of Consumer Prices,

the CPI - Consumer Price Index and the PPI - Producer Price Index. The Euribor and the Eurirs are nominal

rates, so if you want to use real discount rates in your DCF you should adjust them for inflation, but, as I

tried to argue, this operation requires making choices and assumptions that can lead to significantly

different results, as can be seen in Figure 14. In short, using nominal cash flows and therefore nominal

discount rates is the solution that requires fewer arbitrary choices on the part of the investor, especially

with regard to estimating the discount rate, so in the case study I will use cash flows and nominal discount

rates.

2.2.3 - Risk and Returns Relationship

As has just been described, an investment is risky when the expected returns are not completely certain ex

ante and there is a degree of uncertainty about their future values, while an investment is risk-free when it

provides the absolute certainty of giving an equal return as expected. In the Figure 15 three different assets

have been represented, A, B, C, putting on the horizontal axis the possible values of the returns that these

assets can give and on the vertical axis the probability that each value can be reached. Asset A is risk-free

because it gives 100 percent probability that the return is 10 percent and no deviation from this value is

possible; this absolute certainty is reflected in the absence of standard deviation of the probability

distribution. By investing in B it is possible to obtain returns that deviate from the expected value and for

the definition of risk that has been given, B is a risky asset, even if it is not possible to have losses, or returns

below 0 per cent. C is also a risky asset and we see that its probability distribution of returns has greater

dispersion than B, so we can say that C is riskier than B. As was observed at the end of the previous

paragraph, in the case of risky investments the great is the variance that one is willing to accept, the greater

the chances of earning, but looking at Figure 15 we can see how the deviation can be both positive and

negative: “the more risky (higher standard deviation) the asset, the greater also is the ‘‘upside’’ possibility.

That is, given two assets with the same expected return, the more risky asset will typically have a greater

chance of returning a larger profit than the less risky asset could.”39 Now, assuming you have to choose

between these three assets to make an investment, which would be the best asset? Since all three assets

have the same expected return, the best is certainly A, followed by B, making C inevitably the least attractive

choice. The preference of the asset A compared to the other two, if generalized to all investors who buy

and sell on the market, will ensure that the price of A increases and for the dynamics described by equation

(5) this causes a decrease in the final yield . At the same time the price of the other assets will decrease

given the lack of demand, increasing the potential returns. “In equilibrium (that is, when supply and demand

balances for both assets), the riskier asset must offer a higher mean return (ex ante) than the less risky asset.

This is perhaps the most fundamental point in the financial economic theory of capital markets: that

expected returns are (and should be) greater for more risky assets.”40

38 [40] European Central Bank, «ECB Monthly Bulletin,» March 1999

39 [14] D. M. Geltner, N. G. Miller, J. Clayton e P. Eichholtz, Commercial Real Estate. Analysis & Investments, 2nd edition, Oncourse Learning, 2006 40 Ibidem, pag. 186

28

Figure 15. Probability distribution of returns of three assets

Sources: [14] D. M. Geltner, N. G. Miller, J. Clayton e P. Eichholtz, Commercial Real Estate. Analysis & Investments, 2nd edition,

Oncourse Learning, 2006

The concept is worth repeating: relatively risky assets have relatively high returns and vice versa, which

means that investors demand higher returns as the risk grows. This relationship can be formalized in

equation (16).

𝐸(𝑟) = 𝑟𝑓 + 𝑟𝑝 (16)

E(r): expected return from the investment

rf: risk-free asset return

rp: risk premium, that is the extra return required by the investor to accept the greatest risk

2.2.3.1 - Security Market Line

The construction of the security market line, henceforth SML, allows us to better understand the

relationship between risk and return. Not all types of risks are remunerated by the market through higher

returns. One of the possible breakdowns of the overall risk is that which divides it into systematic risk, also

called market risk, and non-systematic risk or specific risk. “The systematic risk principle states that the

reward for bearing risk depends only on the systematic risk of an investment. The underlying rationale for

this principle is straightforward: Because unsystematic risk can be eliminated at virtually no cost (by

diversifying), there is no reward for bearing it. Put another way, the market does not reward risks that are

29

borne unnecessarily. The systematic risk principle has a remarkable and very important implication: The

expected return on an asset depends only on that asset’s systematic risk.”41

Figure 16. Systematic and unsystematic risks

Source: https://www.investopedia.com/managing-wealth/modern-portfolio-theory-why-its-still-hip/

The measure of market risk is β, which indicates how much systematic risk an asset has with respect to an

average risky asset, or rather, compared to the market portfolio, which by definition has β = 1. An asset with

β greater than 1 will therefore be more risky, and therefore potentially profitable compared to the market,

while an asset with β between 0 and 1 will be less risky than the market. Since β is a measure of the riskiness

of an asset, if an asset is risk-free its β will be 0. Also in this case, using a numerical example can help to

make comprehension easier. Let's assume that we have an asset A with a return of 20 percent and a β of

1.6, and that government bonds with a one-year maturity are currently traded at a rate of 8 percent. By

combining asset A and bonds with different weights it is possible to create a portfolio with variable amounts

of capital allocated to the two assets; if 100% of the capital is allocated to the risk-free asset, the return on

the portfolio will be equal to the risk-free return, if 100% of the capital is allocated to the asset A the return

on the portfolio will be equal to the return on A ; if the capital is partly allocated to A and partly to the risk-

free asset, the return on the portfolio will be equal to the weighted average of returns. The same argument

holds for β, since the β of the portfolio is given by the weighted sum of the β of the assets that make it up.

𝐸(𝑅𝑃) = 0.5 ∗ 𝐸(𝑅𝐴) + 0.5 ∗ 𝑅𝑓 =

𝐸(𝑅𝑃) = 0.5 ∗ 20% + 0.5 ∗ 8% = 14%

β𝑃 = 0.5 ∗ β𝐴 + 0.5 ∗ β𝑅𝐹 =

𝐸(𝑅𝑃) = 0.5 ∗ 1.6 + 0.5 ∗ 0 = 0.8

Repeating the calculations for different values of the weight of A in the portfolio, we obtain the values

contained in Table 8, which are then represented in Figure 17, that is the security market line of the asset

A. The slope of the SML is given by the formula (17), in which the risk premium of asset A is divided by β;

this is the reward-to-risk ratio, since it tells us how much is the risk premium for each unit of systematic risk

of the asset.

41 [11] S. A. Ross, R. W. Westerfield e B. D. Jordan, Fundamentals of Corporate Finance, 10th edition, The McGraw-Hill Companies, Inc., 2013- pag. 427

https://www.investopedia.com/managing-wealth/modern-portfolio-theory-why-its-still-hip/

30

Table 8. Portfolio expected returns and beta for different levels of asset allocation

Percentage of

Portfolio in Asset A Portfolio

Expected Return Portfolio Beta

0% 8% 0 25% 11% 0.4

50% 14% 0.8

75% 17% 1.2

100% 20% 1.6

125% 23% 2

150% 26% 2.4 Source: [11] S. A. Ross, R. W. Westerfield e B. D. Jordan, Fundamentals of Corporate Finance, 10th edition, The McGraw-Hill


Figure 17. Security market line



𝑆𝑙𝑜𝑝𝑒 𝑆𝑀𝐿 = E(R𝐴)− R𝑅𝐹

β𝐴 (17)

Should there be an asset B that has expected return levels greater than A for each value of β, the demand

for asset B would increase and that for A would decrease, causing the price of B to rise and thus reducing

its yield. In an active and well-functioning market all the assets traded have the same SML, which is therefore

the same as the entire market. Now suppose we consider a portfolio that contains all the assets of the

market, which we call “market portfolio”, of which we know the expected return E(R𝑀). Since this portfolio

contains all the assets of the market, it will have an average systematic risk, so β will be 1. By applying the

formula (17) I get:

𝑆𝑙𝑜𝑝𝑒 𝑆𝑀𝐿 = E(R𝑀) − R𝑅𝐹

β𝑀


1


Since we know that the SML of the market portfolio is equal to the SML of any market asset, we can write

the following formula:

0%

5%

10%

15%

20%

25%

30%

0 0,5 1 1,5 2 2,5 3

Exp

ecte

d R

etu

rns

Beta

31

E(R𝑀) − R𝑅𝐹 = E(R𝑖)− R𝑅𝐹

β𝑖 (18)

E(R𝑖): expected return of any asset

β𝑖: beta of any asset

The formula (18) can be rewritten (18.1) obtaining the formula of the Capital Asset Pricing Model - CAPM:42

E(R𝑖) = R𝑅𝐹 + β𝑖[E(R𝑀) − R𝑅𝐹] (18.1)

The bottom line is that SML is used to describe the relationship between expected returns and systematic

risk. The application of the SML to estimate the cost of equity is developed later in the Cost of Equity

paragraph and in practice in the case study.

2.2.3.2 - Leverage and Risk

Financial leverage indicates the share of debt compared to the total value of assets used by an investor or

a company. According to Geltner et all “the use of debt to finance an equity investment creates what is called

‘‘leverage’’ in the equity investment, because it allows equity investors to magnify the amount of underlying

physical capital they control.”43 The higher the leverage, the higher the value of the assets controlled by the

company will be high compared to the value of the equity: “financial leverage can dramatically alter the

payoffs to shareholders in the firm. […] acts to magnify gains and losses to shareholders.”44 It is precisely the

possibility of controlling large asset values with respect to the equity that amplifies the volatility of returns

on equity, hence the risk. Assume that an investor buys a real estate asset for € 10,000,000 using €

4,000,000 of his own capital and € 6,000,000 of debt raised through a mortgage. The investor's leverage is

2.5, as he owns an asset that costs five times of his assets. That is, leverage ratio is LR = (E + D) / E, where E

stands for equity and D for debt. To understand the amplifying effect of leverage on return on equity we

make an example with two scenarios, an optimist and a pessimist. The results are shown in Table 9. The

annual cash flows are the same, as the ownership is the same, but in the scenario with leverage, interest

must be paid due to the presence of the debt. It is important to emphasize that the net income between

the two scenarios is different, but the range between the positive and the negative scenario does not

change; this means that leverage has no amplifying effect on the net income, in fact the difference lies in

the return rates on the invested equity. In the lower part of the table we can see how the range of returns

with leverage is much greater than in the unlevered scenario, but not only; in fact the range of levered

returns is 2.5 times greater than the range of unlevered returns. The point is that leverage amplifies the

returns proportionally to the leverage ratio. Although the volatility increases proportionally, the returns do

not do the same, in fact considering the returns in the two optimistic scenarios, it can be calculated that

40/19 = 2.1. This means that the risk has increased more than the expected returns have increased. A further

observation that must be made is that the increase in the leverage increases the share of capital gain in the

42 for a more advanced explanation of the CAPM refer to [45] Z. Bodie, A. Kane e A. J. Marcus, Investments, 10th Edition, McGraw-Hill Education, 2014. Chapter 9 and [46] E. J. Elton, M. J. Gruber, S. J. Brown e W. N. Goetzmann, Modern Portfolio Theory and Investment Analysis, JohnWiley & Sons, Inc, 2014. Chapter 13 43 [14] D. M. Geltner, N. G. Miller, J. Clayton e P. Eichholtz, Commercial Real Estate. Analysis & Investments, 2nd edition, Oncourse Learning, 2006, pag. 298 44 [11] S. A. Ross, R. W. Westerfield e B. D. Jordan, Fundamentals of Corporate Finance, 10th edition, The McGraw-Hill Companies, Inc., 2013

32

contribution to the total return: in the unlevered scenario the share is 10/19 = 52.6%, while in the levered

scenario it is 25/40 = 62.5%.

Table 9. The effect of leverage on returns

Mln € No debt Leverage ratio 2,5

Pessimistic Optimistic Range Pessimistic Optimistic Range

Initial Value

(equity) 10 10 0 4 4 0

Cash flows 0.7 0.9 0.2 0.7 0.9 0.2

Interest 0 0 0 0.3 0.3 0

Net income 0.7 0.9 0.2 0.4 0.6 0.2

Ending value

(equity) 9 11 2 3 5 2

Income returns 7% 9% 2% 10% 15% 5%

Appreciation

returns -10% 10% 20% -25% 25% 50%

Total returns -3% 19% 22% -15% 40% 55%

Sources: author’s elaboration based on D. M. Geltner, N. G. Miller, J. Clayton, P. Eichholtz, Commercial Real Estate Analysis and

Investments, Second Edition, 2006

Summing up, the effects of leverage are:

• Decrease in net income

• The net income volatility does not change

• ROE’s volatility increases proportionally to the leverage ratio

• ROE increases proportionately less than volatility increases

• The share of capital gains on total returns is growing

2.2.3.3 - Risk-free and Risk Premium

The conditions that allow an asset to be defined as risk-free are that the actual return must be equal to the

expected return, it must be free of default and reinvestment risk.45 The first condition is typical of bonds, as

the debtor is bound to pay periodic sums pre-established by contract, but the default risk depends on the

financial strength of the debtor and not even the largest corporations can be defined absolutely risk-free by

default. For this reason, to satisfy the second condition, reference is made to the government bonds of the

most financially solid countries. As for the third, governments can issue bonds of two kinds, short and long

term; the former do not provide for coupon payments, unlike the latter, which therefore expose the investor

to uncertainty about the future interest rate at which he will invest the receipts received from interim

payments. “Because the government can always raise taxes to pay its bills, the debt represented by T-bills is

virtually free of any default risk over its short life. Thus, we will call the rate of return on such debt the risk-

free return, and we will use it as a kind of benchmark.”46

45 [24] S. Vishwanath , Corporate Finance. Theory and Practice, 2th edition, New Dehli: Sage Publications, 2007, pag. 74 46 [11] S. A. Ross, R. W. Westerfield e B. D. Jordan, Fundamentals of Corporate Finance, 10th edition, The McGraw-Hill Companies, Inc., 2013

33

Table 10. Historical records of USA major asset classes

Historical year-to-year nominal returns 1926-2010

Asset Class Average Total

Return Risk premium Standard Deviation

T-bill 3.76 0.00 3.14

Bonds 5.75 1.99 9.73

Real Estate* 9.91 3.61 9.02

Stock 12.47 8.72 20.30

Inflation 3.07 - 4.18 Note: (*) Returns are from 1970 to 2003

Source: author’s elaboration based on data of S. A. Ross, R. W. Westerfield, B. D. Jordan, Fundamentals of Corporate Finance,

10th edition, The McGraw-Hill Companies, Inc., 2013 and D. M. Geltner, N. G. Miller, J. Clayton e P. Eichholtz, Commercial Real

Estate. Analysis & Investments, 2nd edition, Oncourse Learning, 2006

Since government bonds with maturities of less than one year have at the same time the qualities of being

free of any risk on expected return, default risk and reinvestment risk, they are used as proxies for risk-free

returns. There is another, much more practical, reason for which short-term government bonds are used

as a reference for risk-free returns, that is there are no asset classes that have lower yields. Looking at Table

10, it can be seen that the T-bill have both smaller historical returns than the other asset classes and a lower

standard deviation. The complete table of this historical returns can be consulted in Attachment 1. The

historical results confirm the theory, or that the greater the risk of an asset, measured by the standard

deviation of returns, the greater are both the average returns and the risk-premium, namely the difference

between average returns of an asset class and average returns of the T-bill. Investors demand an extra

return, called excess return, to take a greater risk. From Figure 18 it can be seen that the real estate asset

class of the US market, gave lower returns than stocks, but with a much lower volatility and outperformed

long-term government bonds both for higher yield and for lower volatility.

Figure 18. Risk and return relationship in USA major asset class (1970-2003)

Source: author’s elaboration based on data from D. M. Geltner, N. G. Miller, J. Clayton e P.

Eichholtz, Commercial Real Estate. Analysis & Investments, 2nd edition, Oncourse Learning, 2006

T-Bill

BondsReal Estate (NCREIF)

Stocks (S&P500)

0

2

4

6

8

10

12

14

0 5 10 15 20

Exp

ecte

d r

etu

rn (

%)

Standard deviation (%)

34

2.2.4 - Cap Rate Determinants

Cap rate stands for rate capitalization and is the expected return from the first year of income generated

by a commercial property acquired as an investment; basically it is the income return of an investment

property.

𝐶𝑎𝑝 𝑟𝑎𝑡𝑒𝑡 =𝑁𝑂𝐼𝑡+1

𝐴𝑠𝑠𝑒𝑡 𝑚𝑎𝑟𝑘𝑒𝑡 𝑝𝑟𝑖𝑐𝑒𝑡 (19)

It is used as a discount rate for future income expected from a real estate asset in order to determine its

current market value. “The cap rate provides a summary measure of price paid per dollar of expected first

year property income.”47 In other words, the investor buys the future income that expects from the

property. The higher the willingness to pay for every euro of income, the lower the cap rate will be and vice

versa. The dynamics is absolutely the same as that described in the paragraph Risk and Return Relationship,

in fact the lower the risk perceived by the investor, the greater the demand and therefore the price of the

asset, so as to decrease the return on the investment and vice versa. In valuation practice, by applying the

so-called income approach, formula (3) is used to establish the current price of a real estate asset through

the sum of discounted expected cash flows. Assuming that cash flows grow at a constant rate over an infinite

period of time, it is possible to transform formula (3) into an annuity with constant growth. Solving the

equation (3) as a function of the cap rate the formula (20) is obtained. Since required returns contain a risk-

free component and a risk premium, formula (20) can be written as is in formula (20.1).

𝑁𝑂𝐼

𝑝= 𝑟 – 𝑔 (20)

𝑐𝑎𝑝 𝑟𝑎𝑡𝑒 = 𝑟𝑓 + 𝑟𝑝 – 𝑔 (20.1)

NOI: net operating income

p: market price

r: opportunity cost of capital

g: annual growth rate of the NOI

This report tells us that the drivers of the cap rate are three:

• Interest rates on government bonds

• Risk perceived by investors

• Rent expected growth rates

Low interest rates on government bonds make capital cheaper, making investors willing to accept lower

investment returns. Bullish expectations for the rent also push towards a greater willingness to pay, favoring

47 [36] J. clayton e L. Dorsey glass, «Cap Rate & Real Estate Cycles: A historical perspective with a look to the future,» Cornerstone Real Estate Advisers LLC, 2009

35

the reduction of the cap rate. Opposite movements increase the cap rate. This dynamic is summarized in

Table 11.

Table 11. Elements the determine the current market price of a property

Expected Rent Expected asset

price Interest rates Perceived risk Current price

Source: author’s elaboration

“It is important to note that risk-free comes from the capital markets, risk premium comes from both the

capital markets and the real estate space markets, while g comes from the real estate space or user market.

[…] It is also crucial to recognize the three cap rate ingredients are not independent of one another; cap rate

fluctuations result from a complex interaction of the three variables.”48 Equation (20.1) can be written in the

form of equation (20.2), which tells us that the risk premium, obtained from the differential between cap

rate and risk-free interest rate, is pushed down by expectations of increasing income and from the increase

in the risk appetite of economic agents. In moments of particular "euphoria" the high demand drives up the

price, increasing the weight of the capital gain with respect to the income yield in the overall return.

𝑐𝑎𝑝 𝑟𝑎𝑡𝑒 − 𝑟𝑓 = 𝑟𝑝 − 𝑔 (20.2)

2.2.5 - Measuring Risk

At the beginning of the chapter it was concluded that the risk, unlike uncertainty, has a quantitative nature.

Statistical measurement tools are needed to quantify the risk.

2.2.5.1 - The Range

“The range is a statistical measure representing how far apart are the two extreme outcomes of the

probability distribution. The range is calculated as the difference between the best and the worst possible

outcomes.”49

𝑅𝑎𝑛𝑔𝑒 = 𝑏𝑒𝑠𝑡 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒 − 𝑤𝑜𝑟𝑠𝑡 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒 (21)

48 [36] J. clayton e L. Dorsey Glass, «Cap Rate & Real Estate Cycles: A historical perspective with a look to the future,» Cornerstone Real Estate Advisers LLC, 2009, pag. 5-6 49 [12] F. Fabozzi e P. Peterson, Capital Budgeting: Theory and Practice, John Wiley & Sons, Inc., 2002. Pag. 134

36

2.2.5.2 - Standard Deviation

Standard deviation is an index of statistical dispersion; is represented with the letter σ. The standard

deviation formula is as follows:

𝜎 = √∑ (𝑥𝑖 − �̅�)2𝑁𝑖=1

𝑁 (22)

N: population of statistical units

�̅�: arithmetic mean of X

The standard deviation can be seen as an average of the deviations of the observed values compared to the

expected ones, that is the errors, to second power. He tells us then how, on average, the individual

observations are dispersed with respect to the expected value. In the economic and financial sphere, the

standard deviation is calculated using price periodic returns (days, weeks, etc.) and is called volatility. Figure

19 shows the daily returns of the S & P500 during 2018. We see how volatility increases and decreases with

a cyclical trend. When returns are more dispersed around the moving average, volatility is higher, when

returns are closer to the moving average, volatility is low.

Figure 19. S&P 500 daily returns in 2018

Sources: author’s elaboration on Investing.com database

-6,0%

-4,0%

-2,0%

0,0%

2,0%

4,0%

6,0%

Daily returns 10 days moving average

High volatility

low volatility

37

2.2.5.3 - Semi-standard Deviation

The standard deviation considers both positive and negative changes for the investor, called downside risk.

Since a distribution could be asymmetric, or because we want to know only the negative side, we can choose

to use the semi-standard deviation.

∀𝑥 < 𝑥𝑇 𝑠𝑒𝑚𝑖 𝑆𝐷 = √∑ (𝑥𝑖 − 𝑥𝑡)2𝑁𝑖=1

𝑁 (23)

𝑥𝑇 is an arbitrary target value. In short, the deviations are only calculated for values below an arbitrarily

established target level, which can be zero, the risk-free return rate or the minimum acceptable rate of

return. In doing so we go on to see how the observations below the target are dispersed on average

compared to the target itself.

2.2.5.4 - Value at Risk – VaR

“Value-at-Risk is a measure of the maximum potential change in value of a portfolio of financial instruments

with a given probability over a pre-set horizon. VaR answers the question: how much can I lose with x%

probability over a given time horizon.”50 To calculate the VaR it is first necessary to have a historical series

to calculate the periodic returns. Let us assume that they are daily returns, so we calculate the standard

deviation of the series of historical returns. Assuming that returns are normally distributed as in Figure 20,

if you want to find the one-day VaR with a 95% confidence interval, the calculation is VaR = 1.65 * σ * size,

where size is the total value of the 'investment.

Figure 20. Returns distribution and 95% confidence interval

Source: [15] J. Longerstaey e M. Spencer, «RiskMetricsTM — Technical Document, 4th edition,» J. P. Morgan, 1996

To apply VaR to real estate investment there is a need to increase the reference time horizon beyond the

single day or in any case the single period. An example can make things clear. Let's assume that a real estate

investment of 1 million is made and it is estimated that after 10 years the probability distribution of the

50 [15] J. Longerstaey e M. Spencer, «RiskMetricsTM — Technical Document, 4th edition,» J. P. Morgan, 1996

38

property value is normal with an average 1.5 million and a standard deviation of 0.5 million. We want to

know the maximum loss we can incur with 95% confidence. The final value of the investment is given by µ

- 1.65σ, or 1 - 1.65 * 0.5 = 0.175 million. The VaR is therefore 0.175 - 1 = -0.825 million. This means that

with 95% probability the potential loss in 10 years will not exceed 0.825 million, or that there is a 5% chance

of losing more than 0.825 million in 10 years.

2.2.5.5 - Annualizing Volatility and Returns

Annualize returns and volatility serves to be able to compare the performances of investments with

different durations bringing them back to unit values. If an investment has had a 10% return in 10 years it

is quite another thing than one that has given 10% in just one year. Table 12 shows the quarterly prices of

the apartments in the centre of Milan from the beginning of 2015 to the second quarter of 2019. AROR

means annualized rate of return and was calculated both with simple (24) and compound (25)

capitalization.51

Table 12. Milan central apartment sub-market annualized returns and volatility (2015-2019)

Quarters Market price

[€/mq] Returns

QI 2015 7.676 € QII 2015 7.757 € 1,06% QIII 2015 7.645 € -1,44%

QIV 2015 7.632 € -0,17%

QI 2016 7.642 € 0,13%

QII 2016 7.789 € 1,92%

QIII 2016 7.988 € 2,55%

QIV 2016 7.952 € -0,45%

QI 2017 7.872 € -1,01%

QII 2017 8.170 € 3,79%

QIII 2017 8.017 € -1,87%

QIV 2017 7.483 € -6,66%

QI 2018 7.972 € 6,53%

QII 2018 8.169 € 2,47%

QIII 2018 8.247 € 0,95%

QIV 2018 8.575 € 3,98%

QI 2019 8.905 € 3,85%

QII 2019 9.034 € 1,45%

n° of quarters 18

Average quarter return 1,00%

Quarterly volatility 2,98%

AROR simple 3,9%

AROR compound 3,69% ources: author’s elaboration based on data from https://www.immobiliare.it/mercato-immobiliare/lombardia/milano/centro/

51 [37] P. J. Kaufman, Trading System and Methods, 5th edition, John Wiley & Sons, Inc., 2013

39

ARORsimple = (𝐸𝑓

𝐸0− 1)

𝑃

𝑁 (24)

Ef: final equity

E0: initial equity

N: number of quarters of duration of the investment. In this case there are 18

P: number of periods in a year. In this case there are four quarters in a year

ARORcompound = (𝐸𝑓

𝐸0)

𝑃

𝑁− 1 (25)

The formula (26) shows how to calculate the annualized volatility starting from the periodic returns.52

Vannualized = σri√P (26)

2.2.6 - Risk-adjusted Performances Measures

Starting from the assumption that every investor wants to maximize profit and minimize risk, the

performance of an investment must be evaluated considering these two factors simultaneously. For any

given risk level chosen by the investor, the investment that offers the greatest return will always be

preferred. The formulas that will follow in this paragraph have the common element of having a risk

measure in the denominator and a measure of yield in the numerator.

2.2.6.1 - Coefficient of Variation

The coefficient of variation, also called relative standard deviation, is an indicator that makes it possible to

"standardize" the standard deviation and thus make comparable projects with different scales. Table 13 at

page 41 shows the cash flows of two different projects, A and B, each with its probability. Looking at the

value of the standard deviation of the two projects we would be led to say that A is more risky than B, but

this is due to the fact that A simply has a larger scale, in fact its expected value is more than ten times larger

than B Applying the formula of the coefficient of variation we see that instead it is B to be more risky than

A, since the amount of risk is relatively larger than the expected value.53

𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑣𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛 = 𝜎(𝑋)

𝐸(𝑋) (27)

52 Ibidem 53 [12] F. Fabozzi e P. Peterson, Capital Budgeting: Theory and Practice, John Wiley & Sons, Inc., 2002

40

2.2.6.2 - Information Ratio

Information ratio is calculated dividing annual returns by annual volatility, that is the volatility calculated

using annual returns. IR is a way with which an investor can evaluate how much units or returns are given

for each unit of risk.54

𝐼𝑅 = 𝑎𝑛𝑛𝑢𝑎𝑙𝑖𝑧𝑒𝑑 𝑟𝑒𝑡𝑢𝑟𝑛𝑠

𝑎𝑛𝑛𝑢𝑎𝑙𝑖𝑧𝑒𝑑 𝑣𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦 (28)

2.2.6.3 - Sharpe Ratio

Presented for the first time by William Sharpe, this formula allows us to go a little bit further than the basic

information ratio, by subtracting the risk-free rate from the annual returns and so isolate the excess of

return from the performances.55

𝑆𝑅 = 𝑎𝑛𝑛𝑢𝑎𝑙𝑖𝑧𝑒𝑑 𝑟𝑒𝑡𝑢𝑟𝑛𝑠 − 𝑟𝑖𝑠𝑘 𝑓𝑟𝑒𝑒 𝑟𝑎𝑡𝑒

𝑎𝑛𝑛𝑢𝑎𝑙𝑖𝑧𝑒𝑑 𝑣𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦 (29)

2.2.6.4 - Treynor Ratio

This ratio is also based on the isolation of the excess return, but the risk measure used is the portfolio beta,

or its relative volatility compared to the market. This way of measuring the risk / return ratio is more useful

if you want to add an asset in your portfolio, as it would allow you to see the results of better diversification

through a reduction in beta and therefore an increase in the ratio.56

𝑇𝑅 = 𝑎𝑛𝑛𝑢𝑎𝑙𝑖𝑧𝑒𝑑 𝑟𝑒𝑡𝑢𝑟𝑛𝑠 − 𝑟𝑖𝑠𝑘 𝑓𝑟𝑒𝑒 𝑟𝑎𝑡𝑒

𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑏𝑒𝑡𝑎 (30)

2.2.6.5 - Sortino Ratio

In this ratio the downside volatility is used as risk measure. In this way the focus is on the “bad side of the

volatility” and could be very useful when the distribution of returns is not symmetric.57

𝑆𝑅 = 𝑎𝑛𝑛𝑢𝑎𝑙𝑖𝑧𝑒𝑑 𝑟𝑒𝑡𝑢𝑟𝑛𝑠 − 𝑟𝑖𝑠𝑘 𝑓𝑟𝑒𝑒 𝑟𝑎𝑡𝑒

𝑑𝑜𝑤𝑛𝑠𝑖𝑑𝑒 𝑣𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦 (31)

54 [37] P. J. Kaufman, Trading System and Methods, 5th edition, John Wiley & Sons, Inc., 2013 55 Ibidem 56 Ibidem 57 Ibidem

41

Table 13. Comparison of two project with different scale

Project A Project B

Cash flow Probability Cash flow Probability

100.000,00 € 5% 9.000,00 € 5%

90.000,00 € 10% 8.000,00 € 10%

80.000,00 € 20% 7.000,00 € 20%

70.000,00 € 30% 6.000,00 € 30%

60.000,00 € 20% 5.000,00 € 20%

50.000,00 € 10% 4.000,00 € 10%

40.000,00 € 5% 3.000,00 € 5%

A B

Expected value 70.000,00 € 6.000,00 €

Standard deviation 21.602,47 € 2.160,25 €

Coefficient of variation 0,31 0,36 Sources: author’s elaboration

2.2.7 - Covenants on Debt

When an economic operator contracts a debt with a credit institution, say a bank, the contract often

includes "covenants" that the creditor uses to protect themselves. “Il covenant, letteralmente impegno, è

una clausola contrattuale che viene concordata in fase di definizione contrattuale e che riconosce alla banca

il diritto di rinegoziare o risolvere il contratto al verificarsi di eventi espressamente previsti nella stessa

clausola.”58 The following are the most common alliances in which any investor who has contracted a debt

to finance his project may have to comply. The subject is dealt with in this chapter because going beyond

the maximum threshold established by the pacts due to a sharp drop in asset prices or due to an increase

in interest rates, makes the investor find himself in a situation of emergency in which it is forced to disinvest,

contract new debts or find any other source of liquidity necessary to reduce the existing debt with the

creditor. To the difficult patrimonial situation or of the income are added the financial damages caused by

the actions necessary to stick with the agreed parameters. This eventuality certainly causes a negative

deviation in the expected result and therefore in line with the definition of risk given at the beginning of the

chapter.

2.2.7.1 - Interest Coverage Ratio – ICR

The interest coverage ratio is a covenant given by the ratio between EBIT and the total annual amount of

interest on the debt and measures the ability of the debtor to pay the interest with the income generated

by the commercial activity. In the event that this ratio is paid below a threshold set in the contract, the

debtor is obliged to repay part of the debt in order to reduce the interest expense.59

58 [38] G. Morri e M. Antonio, Finanziamento immobiliare. Finanziamenti strutturati, leasing, mezzanine e NPL, Egea, 2010, pag. 17 59 [39] A. Ciaramella e O. Tronconi, Real Estate Asset Management, Il Sole 24 Ore, 2012, pag. 170

42

2.2.7.2 - Loan to value – LTV

The LTV index is obtained by dividing the total amount of the outstanding debt by the market value of the

asset and makes it possible to assess the "health" of the loan guarantee. If the LTV rises above 100% it

means that the debt is greater than the market value of the asset and that the creditor's guarantee is small

compared to the loan granted and in the event of the debtor's default the creditor would not be able to

regain possession of the whole amount lent and would incur a loss.60 Real estate companies usually try to

keep this indicator below 40%.

2.2.7.3 - Debt Service Coverage Ratio – DSCR

The DSCR index is the ratio between the EBIT and the total annual debt service expenditure, both for the

interest rate and for the capital amortization. Indicates the investor's ability to repay the debt with the

income generated by the investment.61

2.2.8 - Cost of Capital

The cost of capital is basically the rate of return required by an investor to "put money" into a given project,

therefore it depends on the level of risk. Cost of capital, required rate of return and appropriate discount

rate are more or less the same things. “firm’s overall cost of capital will reflect the required return on the

firm’s assets as a whole. Given that a firm uses both debt and equity capital, this overall cost of capital will

be a mixture of the returns needed to compensate its creditors and those needed to compensate its

stockholders. In other words, a firm’s cost of capital will reflect both its cost of debt capital and its cost of

equity capital.”62 In fact, capital is generally made up of both equity and debt, and both shareholders and

debtors want to be adequately compensated for their risk taking.

2.2.8.1 - Cost of Equity

The cost of equity is given by the required return for the pure time value of money and for the risk taken,

so it is necessary to calculate risk-free and risk-premium and then add them together.

Risk free:

For the risk-free, the nominal rate of return of short-term government bonds can be used as a proxy, say 12

months, so as to consider both the return necessary to compensate for inflation - as already discussed in

the paragraph Nominal and real returns - and receive a real return. With reference to formula (15),

therefore, it is possible to say that the risk-free is obtainable by looking at the current nominal rate of 12-

month government bonds, as they already discount the real return requested by investors given current

market conditions and expected inflation. As far as the issuing nation of these securities is concerned, please

60 Ibidem, pag. 170 61 Ibidem, pag. 170 62 [11] S. A. Ross, R. W. Westerfield e B. D. Jordan, Fundamentals of Corporate Finance, 10th edition, The McGraw-Hill Companies, Inc., 2013, pag. 451

43

refer to the case study. According to Geltner et all [14] an alternative method to define the risk-free can be

the following: “In this context, the relevant risk-free component for use in a real estate investment’s OCC is

not the current T-bill yield but rather the average T-bill rate expected over the long-term investment horizon.

[…] it is important to recognize that long-term T-bonds are not riskless, and their yields reflect some risk

premium in the investment marketplace. To adjust for this, a simple approach is to subtract from the current

long-term T-bond yield the typical or average ‘‘yield curve effect,’’ that is, the difference between the yield

on long-term and short-term government debt that typically prevails in the capital market.”63 In short, it is

necessary to calculate the average historical spread between 10-year T-Bonds and T-Bill 3 or 12 months and

subtract this value from the current 10-year bond yield, so as to make it effectively risk free. The application

of this method will be developed in the case study.

Cost of equity:

“Cost of equity is the opportunity cost of capital for the investors who hold the firm’s shares.”64 Is more difficult to estimate because you cannot observe this rate on financial market as you can do with government bonds. There is no single method for calculating the cost of equity. Below I report some of the most widespread. Depending on the calculation method used, the values obtained may vary considerably; therefore all those values that are not likely because they are too low or too high are discarded and an average of the other values obtained is used. Constant Growth Perpetuity Model: a first possible approach is the "Constant growth perpetuity model" also known as the "Dividend growth model" in the equity world. In practice, the formula (3) used for the DCF calculation is applied and it is assumed that the annual cash flows will grow at a constant rate from today up to infinity. The formula for defining the present value is (32), which resolved according to the rate of return becomes the formula (32.1)

𝑃𝑉 = 𝐶𝐹𝑡

𝑟𝑒−𝑔 (32)

𝑟𝑒 = 𝐶𝐹𝑡

𝑃𝑉+ 𝑔 (32.1)

P0: current price CFt: annual expected cash flow g: expected net cash flows growth rate

re: rate of return on equity

The current price and the cash flows can be observed in the space market or calculated if the overall annual

costs are known. The growth rate g must be estimated. There are essentially two ways: the first is to consult

the publications of different agencies that carry out analyzes and forecasts on future price and rent trends

and take an average; the second is to look at the historical growth rates of the previous 5 or 10 years and

average them. The advantage of this method lies above all in its simplicity; the disadvantages that a

reasonable and stable growth rate needs as it is assumed to project it to infinity, the discount rate obtained

63 [14] D. M. Geltner, N. G. Miller, J. Clayton e P. Eichholtz, Commercial Real Estate. Analysis & Investments, 2nd edition, Oncourse Learning, 2006, pag. 251 64 [17] R. A. Brealey, S. C. Myers e F. Allen, Principles of Corporate Finance, 10th edition, The McGraw-Hill Companies, Inc., 2011, pag.

216

44

is very sensitive to the estimated rate of growth of the cash flows and that the risk is not explicitly taken

into consideration.

Historical Empirical Evidence: This method is essentially based on the calculation of a historical risk-premium

by averaging the year-to-year differentials between risk-free rates and total market returns.65 In other

words, the annual returns of an arbitrarily chosen time horizon must be calculated; the value used as proxy

of the risk-free to obtain the risk-premium of each year is subtracted from this value and finally the average

of all the risk-premium values found is defined so as to define the risk-premium expected for that market .

Both methods will be developed in practice in the case study.

Security Market Line: The idea for estimating the cost of equity using SML starts from the concept of

opportunity cost of the investment. When evaluating an investment it is necessary to consider the possible

alternatives, therefore the "real" investment or an investment with a similar level of risk in the financial

market. Using as a discount rate for a project, the return obtainable on a certain security traded on the

financial market with a similar level of risk, it is possible to calculate an NPV that indicates whether it is

better to invest in the project, if it is greater than zero, or if it is better to invest directly in the financial

security, in case the NPV is less than zero. The brief theoretical explanation of the SML in the section Risk

and Returns Relationship, shows how the expected return of an asset depends on three factors: risk-free,

market risk premium, systematic risk of asset - βE. These three elements must be estimated and inserted in

the formula (33) of the CAPM.

𝑟𝐸 = 𝑟𝑓 + 𝛽𝐸(𝑟𝑀 − 𝑟𝑓) (33)

For the estimation of the risk free I use the method just described above and for the risk premium use the

historical method. The estimate of β for securities normally traded on the stock market is made by plotting

on the x-axis the periodic returns of the market used as a benchmark and on y-axis the returns of the asset

of which we want to know β and then we calculate the regression of this scatter plot. Figure 21 shows an

example of what has just been described.


The major criticality of this method is that it is necessary to make arbitrary choices about the frequency of

the period on which the returns are calculated (weekly, monthly, quarterly), the overall time horizon of

65 [14] D. M. Geltner, N. G. Miller, J. Clayton e P. Eichholtz, Commercial Real Estate. Analysis & Investments, 2nd edition, Oncourse Learning, 2006, pag. 252

β = 1,36R² = 0,2782-30,00%

-20,00%

-10,00%

0,00%

10,00%

20,00%

30,00%

-15,0% -10,0% -5,0% 0,0% 5,0% 10,0% 15,0%Am

azo

n

NASDAQ

β = 1,22R² = 0,3879-20,0%

-15,0%

-10,0%

-5,0%

0,0%

5,0%

10,0%

15,0%

20,0%

25,0%

-15,0% -10,0% -5,0% 0,0% 5,0% 10,0% 15,0%

Ad

ob

e

NASDAQ

Figure 21. β estimation for Adobe and Amazon stocks calculated on monthly returns from January 2010 to July 2019

45

which we consider the returns (1, 3, 5, 7, 10 years) and the benchmark used to represent the market. Results

can vary considerably by varying one of these parameters. Assuming that the risk-free rate used is 2 percent

and that the market's risk premium is 7 percent, the return rate required to invest in Amazon will be 2% +

1.36 * 7% = 11.52% and for Adobe 2% + 1.22 * 7% = 10.54%. The β defined using the method represented

in Figure 21 refers to the company's equity and therefore is a β levered, or β equity. If the β levered is

estimated through financial market data and one wants to know the β unlevered, the formula (34) can be

used:

𝛽𝐴 = 𝛽𝐸 [𝐸

(1−𝜏)𝐷+𝐸] = 𝛽𝐸 [

1

1+(1−𝜏)𝐷

𝐸

] (34)66

𝛽𝐸: beta equity

𝛽𝐴: beta asset

𝜏: marginal tax rate

The advantages of this method are that it explicitly considers the risk and that it can also be applied to

companies that do not have constant dividend growth, while the disadvantages are that they serve beta

and risk premiums that are unstable, as they vary according to the periods considered for the calculation

and change in the future with the succession of new economic conditions.

2.2.8.2 - Cost of Debt and Mortgages

The cost of debt is easier to obtain than the cost of equity, as it does not have to be estimated; “is simply the interest rate the firm must pay on new borrowing, and we can observe interest rates in the financial markets.”67 Loans for real estate investments are often mortgages, whose interest rates are defined by two variables, namely the interbank interest rate and the spread added by the credit institution at which the investor has chosen to contract it, as shown in formula (35).

𝑟𝐷 = 𝑖𝑛𝑡𝑒𝑟𝑏𝑎𝑛𝑘 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 + 𝑑𝑒𝑓𝑎𝑢𝑙𝑡 𝑟𝑖𝑠𝑘 𝑝𝑟𝑒𝑚𝑖𝑢𝑚 (35)

Mortgages can be at a fixed or variable rate: for the fixed rate, the Eurirs (Euro Interest Rate Swap) is used as a reference plus a spread that serves the bank to cover its expenses and obtain a profit by offsetting the risk incurred, while for a variable rate mortgage, the Euribor is paid at one or three months plus a spread.68 Fixed rate mortgages represent a greater risk for the creditor, who could sustain losses if the interbank rate rose above the interest rate to which the credit was granted; this increased risk translates into a higher interest rate applied to the loan granted. For real estate mortgage rates I refer to the AOER (Average overall effective rate) indicator on mortgage loans published by the Bank of Italy: “The AOER, including fees, remuneration of any sort and expenses (excluding taxes and duties) refers to the annual interest rate charged by bank and financial intermediaries on like operations."69 Looking at figure 22, we can observe on the left

66 [12] F. Fabozzi e P. Peterson, Capital Budgeting: Theory and Practice, John Wiley & Sons, Inc., 2002. Pag. 146 67 [11] S. A. Ross, R. W. Westerfield e B. D. Jordan, Fundamentals of Corporate Finance, 10th edition, The McGraw-Hill Companies, Inc., 2013 pag. 502 68 «Gli indici Euribor e i mutui ipotecari,» [Online]. Available: https://it.euribor-rates.eu/euribor-ipotecari.asp 69 «Average overall effective rate (AOER),» [Online]. Available: https://www.bancaditalia.it/compiti/vigilanza/compiti-vigilanza/tegm/index.html?com.dotmarketing.htmlpage.language=1

46

the AOER trends on fixed-rate mortgages and the 10-year Eurirs, while on the right there is the AOER on variable-rate mortgages and the 1-month Euribor. The table with complete data is given in Attachment 2. Figure 22. Average Overall Effective Rate – AOER – for mortgage loans with fixed and variables rate compared

with Eurirs 10 years and Euribor 1 month

Source: author’s elaborations based on Banca d’Italia and European Central Bank Statistical Data Warehouse

The cost of debt is the return demanded by creditors to grant credit to the company, so in the face of a

negative cash flow, the creditor receives in the future a series of positive cash flows that include a share of

interest and a share of amortization, the pattern of which depends on the type of loan that has been decided

to contract in agreement between the two parties. According with Geltner et all [14], there are four main

rules for mortgage design:

• The interest due for each period is calculated by multiplying the applicable interest rate to the

outstanding principal balance.

𝐼𝑁𝑇𝑡 = 𝑂𝐿𝐵𝑡−1 ∗ 𝑟𝑡 (36)

• The principal is amortized with a periodic payment included in the total payment together with

the interest.

𝐴𝑀𝑂𝑅𝑇𝑡 = 𝑇𝑃𝑡 − 𝐼𝑁𝑇𝑡 (3)

• The OLB after each payment decreases by the amount of amortized debt.

𝑂𝐿𝐵𝑡 = 𝑂𝐿𝐵𝑡−1 − 𝐴𝑀𝑂𝑅𝑇𝑡 (38)

• The initial principal outstanding is equal to the initial sum borrowed from the debtor.

𝑂𝐿𝐵0 = 𝐿 (39)

These rules constitute a method by which it is possible to design different forms of mortgage, in order to

find a compromise between the debtor's needs and the creditor's profit and therefore to find the pattern

of payments that best fits the debtor's profit pattern over the course of the useful life of the investment. If,

for example, an investor expects to have low inflows in the first part of the useful life of the investment,

having to face large expenses for debt service, it could lead him to shortgage of liquidity, with the need, in

-1,00

0,00

1,00

2,00

3,00

4,00

5,00

6,00

7,00

III 2

004

II 20

05

I 200

6

IV 2

006

III 2

007

II 20

08

I 200

9

IV 2

009

III 2

010

II 20

11

I 201

2

IV 2

012

III 2

013

II 20

14

I 201

5

IV 2

015

III 2

016

II 20

17

I 201

8

IV 2

018

AOER mortage loan at variable rate and euribor QIII 2004 - QI2019

TEGM variable rate euribor 1 month

0,00

1,00

2,00

3,00

4,00

5,00

6,00

7,00

III 2

004

II 20

05

I 200

6

IV 2

006

III 2

007

II 20

08

I 200

9

IV 2

009

III 2

010

II 20

11

I 201

2

IV 2

012

III 2

013

II 20

14

I 201

5

IV 2

015

III 2

016

II 20

17

I 201

8

IV 2

018

AOER mortage loan at fixed rate and eurirs QIII 2004 -QI 2019

TEGM fixed rate eurirs 10 years

47

extreme cases, to have to apply for short-term loans to meet long-term debt service. The creditor himself

has all the interest that the debtor does not go into financial difficulties, since his profit is part of the latter's

profit. Below I report the main types of mortgage using as parameters values an annual interest rate of 3

per cent, then 0.25 per cent monthly, on a € 10,000,000 mortgage repayable within 10 years, then 120

months.

Interest-only loan: also called bullet; each month only interest is paid and there is no amortization of the

principal, which is fully repaid at the end. The interests can be either fixed, and therefore the expenditure

is equal every month, or they can be variable and therefore vary according to the variations of the 1-month

Euribor. This typology is used by those who intend to sell the property at the end of the loan, even if in case

of difficulty in finding a counterparty in the disinvestment or in the fall of the prices compared to the

moment of the subscription of the debt there can be problems. Table 14 shows the dynamics month after

month of the OLB, the total payments divided by interest and amortization of the principal.

Constant-Amortization Mortgage: the sum paid for the principal's amortization remains constant month

after month, while the interest expense varies depending on whether the rate is fixed or variable. To

calculate the monthly sum due for amortization, the total initial debt is divided by the number of periods in

which the debt must be paid and this amount is added to the interest expense monthly due. The quantities

of the example are shown in Table 15, while in Figure 24 we see the dynamics of interest and outstanding

debt.

Constant-Payment Mortgage: the total expense, amortization plus interest, is always the same every

month. This typology is advantageous both for the creditor and for the debtor from the point of view of

budgeting. The total TP expense is calculated using the formula (40). The amount of depreciation is defined

by subtracting the interest expense from the total expense. From Figure 25 we see that the expenditure for

interest decreases exponentially despite the total payment remaining constant, so as to progressively

reduce the advantages due to the fiscal deductibility of the interests with an overall expense that does not

decrease.

𝑃𝑇𝑡 = 𝑂𝐵𝐿0𝑟

1− 1

(1+𝑟)𝑁

(40)

Adjustable Rate Mortgage: this type of loan provides for the periodic adjustment of the interest rate, usually

every year, to adjust to the current interbank interest rate. The risk is transferred from the creditor to the

debtor, who will pay less in interest but will have a greater risk due to the unpredictability of future costs

he will have to face. The total monthly payment is recalculated at the beginning of each year using a

government bond with one-year maturity, plus a constant spread to define the interest rate; although the

rate profile is unpredictable, the loan must be fully amortized over a set period of time. This objective is

achieved using the formula (40), it is equal to the number of months remaining to the maturity of the loan

and the last value of the outstanding debt series is used. I hypothesize that the expectation on bond yields

with one-year maturity is growth in the next few years, notably 75 basis points for the first two years, 50

points in the following two years, 25 points for three years and 10 points in the last two. Looking at the

values shown in Table 17 it can be seen that the total payment remains constant within the year and varies

in the first period of each new year. The profile of the monthly payment of interest expenditure is shown in

Figure 26.

48

Table 14. Interest-only loan (figures in €)

Month OLB INT AMORT PT OLB end

period

1 10.000.000 100.000 - 100.000 10.000.000

2 10.000.000 100.000 - 100.000 10.000.000

3 10.000.000 100.000 - 100.000 10.000.000

4 10.000.000 100.000 - 100.000 10.000.000

… … … … … …

117 10.000.000 100.000 - 100.000 10.000.000

118 10.000.000 100.000 - 100.000 10.000.000

119 10.000.000 100.000 - 100.000 10.000.000

120 10.000.000 100.000 10.000.000 10.100.000 - Source: author’s elaboration based on Geltner et all [5]

Source: author’s elaboration based on Geltner et all [5]

Table 15. Constant-Amortization Mortgage (figures in €)


period 1 10.000.000 100.000 83.333 183.333 9.916.667

2 9.916.667 99.167 83.333 182.500 9.833.333

3 9.833.333 98.333 83.333 181.667 9.750.000

4 9.750.000 97.500 83.333 180.833 9.666.667

… … … … … …

117 333.333 3.333 83.333 86.667 250.000

118 250.000 2.500 83.333 85.833 166.667

119 166.667 1.667 83.333 85.000 83.333

120 83.333 833 83.333 84.167 0 Source: author’s elaboration based on Geltner et all [5]

10.100.000 €

90.000 €

95.000 €

100.000 €

105.000 €

110.000 €

115.000 €

120.000 €

1 61

11

62

12

63

13

64

14

65

15

66

16

67

17

68

18

69

19

61

01

10

61

11

11

6

PT INT

Figure 23. Interest-only loan – total payment and interest expense (figures in €)

49

Figure 24. Constant-Amortization Mortgage – total payment and interest expense (figures in €)


Table 16. Constant-Payment Mortgage (figures in €)


period

1 10.000.000 100.000 43.471 143.471 9.956.529

2 9.956.529 99.565 43.906 143.471 9.912.623

3 9.912.623 99.126 44.345 143.471 9.868.279

4 9.868.279 98.683 44.788 143.471 9.823.491

… … … … … …

117 559.819 5.598 137.873 143.471 421.946

118 421.946 4.219 139.251 143.471 282.694

119 282.694 2.827 140.644 143.471 142.050

120 142.050 1.421 142.050 143.471 0 Source: author’s elaboration based on Geltner et all [5]

Figure 25. Constant-Payment Mortgage – total payment and interest expense (figures in €)


- €

20.000 €

40.000 €

60.000 €

80.000 €

100.000 €

120.000 €

140.000 €

160.000 €

180.000 €

200.000 €

1 7

13

19

25

31

37

43

49

55

61

67

73

79

85

91

97

10

3

10

9

11

5

PT INT

-10000

10000

30000

50000

70000

90000

110000

130000

150000

1 7

13

19

25

31

37

43

49

55

61

67

73

79

85

91

97

10

3

10

9

11

5

PT INT

50

Table 17. Adjustable Rate Mortgage (figures in €)


period

1 10.000.000 116.667 38.600 155.266 9.961.400

2 9.961.400 116.216 39.050 155.266 9.922.350

3 9.922.350 115.761 39.506 155.266 9.882.844

… … … … … …

12 9.549.747 111.414 43.853 155.266 9.505.894

13 9.505.894 116.843 42.625 159.468 9.463.269

14 9.463.269 116.319 43.149 159.468 9.420.121

… … … … … …

24 9.007.116 110.712 48.756 159.468 8.958.361

25 8.958.361 115.712 47.652 163.364 8.910.709

26 8.910.709 115.097 48.267 163.364 8.862.441

… … … … … …

118 496.569 7.221 163.139 170.360 333.430

119 333.430 4.849 165.512 170.360 167.918

120 167.918 2.442 167.918 170.360 - Source: author’s elaboration based on Geltner et all [5]

Balloon mortgage: in this type of loan, maturity is set on a shorter time horizon than amortization, so

monthly payments are not large enough to fully repay the debt at maturity, an exceptional debt debited

which is repaid with a large final payment. Table 18 shows the total monthly payments, assuming a constant

amortization with a horizon of twenty years and maturity set at ten years. The last period, the outstanding

debt is about half of the initial debt; this is the balloon that is amortized with the last payment. The dynamics

of payments, which can be seen in Figure 27, is probably one of the most suitable forms of real estate

investment, as it offers a lower monthly total expenditure compared to CAM without a balloon, despite the

higher interest expense, and is well suited to the model of cash receipts provided by the debtor.

Figure 26. Adjustable Rate Mortgage – total payment and interest expense (figures in €)


€-

€20.000,00

€40.000,00

€60.000,00

€80.000,00

€100.000,00

€120.000,00

1 7

13

19

25

31

37

43

49

55

61

67

73

79

85

91

97

10

3

10

9

11

5

PT INT

51

Table 18. Balloon mortgage (figures in €)

Month OLB INT AMORT TP OLB end

period

1 10.000.000 100.000 41.667 141.667 9.958.333

2 9.958.333 99.583 41.667 141.250 9.916.667

3 9.916.667 99.167 41.667 140.833 9.875.000

4 9.875.000 98.750 41.667 140.417 9.833.333

… … … … … …

117 5.166.667 51.667 41.667 93.333 5.125.000

118 5.125.000 51.250 41.667 92.917 5.083.333

119 5.083.333 50.833 41.667 92.500 5.041.667

120 5.041.667 50.417 5.041.667 5.092.083 - Source: author’s elaboration based on Geltner et all [5]

Figure 27. Balloon mortgage – total payment and interest expense (figures in €)


2.2.8.3 - Weighted Average Cost of Capital

The WACC formula is as follows:

𝑊𝐴𝐶𝐶 = 𝑟𝑒𝐸

𝑉+ 𝑟𝑑

𝐷

𝑉 (41)

𝑟𝑒: return on equity

𝑟𝑑: return on debt

𝐸: market value of firm’s equity

𝐷: market value of firm’s debt

𝑇: tax rate

𝑉: market value of firm’s assets

5.092.083,33 €

€(10.000,00)

€10.000,00

€30.000,00

€50.000,00

€70.000,00

€90.000,00

€110.000,00

€130.000,00

€150.000,00

1 7

13

19

25

31

37

43

49

55

61

67

73

79

85

91

97

10

3

10

9

11

5

PT INT

52

Considering the taxes, the formula (42) of the WACC after tax must be used:


𝑉+ 𝑟𝑑(1 − 𝑇)

𝐷

𝑉 (42)

The problem with this formula is that if the cost of debt is lower than the cost of equity, you could think of

using large portions of debt to finance the project, as this would minimize the WACC. As seen in the Leverage

and risk section, an increase in the debt used corresponds to an increase in risk and therefore should

correspond to an increase in the discount rate. Since the WACC is a required return rate on both equity and

debt, it can be correctly defined as a return on assets. Replacing WACC with rA (return on asset) in formula

(41) and solving for rE, we obtain the following formula (41.1)

𝑟𝑒 = 𝑟𝑎𝑉

𝐸− 𝑟𝑑

𝐷

𝐸 (41.1)

2.2.9 - Sources of Risk Identification in the Real Estate Investment

In this section I present an overview of the particular risks of the real estate sector.

2.2.9.1 - Market Risk

Risk due to the fluctuation of prices and market rents of assets held in the portfolio due to changes in

macroeconomic variables, therefore external to the company, or to specific conditions of the buildings such

as maintenance status, location and local market and therefore internal. Write-downs of prices and rents

lead to an increase in the risk of vacancy and capital losses at the time of disposal of the asset.70

Vacancy Risk

The vacancy risk is the possibility that a tenant terminates the lease and leaves the real estate unit vacant,

causing a portion of the income from the investor's income to be lost. Lease agreements generally contain

break-options, which give the tenant the right, but not the duty, to terminate the contract unilaterally. The

reasons why a tenant decides to leave the space he occupies are multiple, including: relationship between

passing rent and estimated rental value, location, state of maintenance of the building, incidence of

management costs, current market offer, energy class, transfer costs.71 All these reasons could be

summarized as the relative price of one space over another: if at the same cost a property unit has a set of

characteristics such that it is perceived as "better" by the tenant, this at the time of break-option, will

terminate the contract to settle in relatively less expensive space. It is classified as market risk in that it is

linked to the fluctuation in the prices of rents in the market: if at the time a break option market rent is

lower than the rent roll defined by the contract , the tenant will be much more likely to exercise the option.

70 [4] COIMA RES, «Relazione finanziaria annuale,» 2018 [42] Assogestioni, «Mappatura dei rischi immobiliare nei fondi real estate» 2006 71 [39] O. Tronconi e A. Ciaramella, Real Estate Asset Management, Il Sole 24 Ore S.p.A., 2012

53

Disposal Risk

Risk due to the fluctuation in market prices of the asset held in the portfolio of which the reversion value is

unknown at the end of the planned holding period. The return on investment component due to capital

gains is always important in a real estate investment, but it is especially important in value added and

speculative investments, in which the holding period is short and the main aim of the investor is the capital

gain generated by the sale of the asset.

2.2.9.2 - Operative Risk

Operational risk can be defined as the variability of costs incurred for operating expenses generated by

property management. The risk varies according to the state of maintenance of the building and plant

components, the adequacy of the maintenance plan, the functional adequacy of the building with respect

to market demand. In essence, the operational risk emerges from the technical, economic and managerial

organization of management activities. In the Returns paragraph I have already defined how the distinction

between maintenance activities and capital expenditure is made, which serve the purpose of slowing down

the physical degradation of the building and therefore keeping the asset competitive with respect to the

market in order to guarantee its profitability. Not performing maintenance to save money is equivalent to

reducing the property's ability to generate income, increase the probability of vacancy and lower the profit

from capital gains at the end of the investment.

Capex

The capex item includes the activities described in the Returns paragraph. From the point of view of risk

management, what makes the difference is the frequency of the activities and their relevance as costs that

must be sustained. As for the activities defined as "Recurring capex", these are relatively inexpensive and

do not require long periods of time to be realized; they differ from ordinary maintenance activities since

they do not take place on an ongoing basis every year and are non-recurring. The activities defined as

"Extraordinary capex" differ from Recurring capex because they are even less frequent and more expensive.

Utilities

Utilities costs include electricity, water and gas. These is variable operating expenses since they usually grow

as the occupancy rate of the building increases.

2.2.9.3 - Interest Rates Risk

Risk due to the possible increase in interest rates, resulting in a reduction in the current value of assets, an

increase in the weight of debts and a reduction if cash flows due to an increase in interest expense.72

2.2.9.4 - Legal Compliance Risk

“Change in the regulatory framework that forces the company to redefine or reorganize its activities. The

legislation can concern the technical plant of buildings, as well as outsourcing contracts, the relationship

with tenants, taxation and more.”73

72 [4] COIMA RES, «Relazione finanziaria annuale,» 2018 [44] P. Loizou e N. French, «Risk and uncertainty in development: A critical evaluation of using the Monte Carlo simulation method as a decision tool in real estate development projects,» Journal of Property Investment & Finance, vol. 30, n. 2, pp. 198-210, 2012 73 [42] Assogestioni, «Mappatura dei rischi immobiliare nei fondi real estate,» 2006

54

2.2.9.5 - Asset Concentration Risk

Risk due to asset allocation strategy, with which a portfolio can be more or less diversified. Concentration

can be geographical, typological and sectorial. Diversifying means investing in assets whose return rates are

not correlated. This operationally translates into the purchase of assets located in different sub-markets,

different types and geographical areas, and rent them to tenants who operate in economic sectors that are

as independent as possible from one another.

2.2.9.6 - Strategic Risk

“Pure risk and business risk; this consists of the current or prospective risk of a fall in profits or capital,

resulting from changes in the operating environment or from incorrect corporate decisions, inadequate

implementation of decisions, poor reaction to changes in the competitive environment, customer behaviour

or technological developments.”74

2.2.9.7 - Other Risk Classification

Table 19. Risks classification

Type Source of Risk Possibilities

Rent level risk External Influenceable

Vacancy risk External Influenceable

Disposal risk External Influenceable

Taxation risk External Uncontrollable

Interest rate risk External Uncontrollable

Asset concentration Internal Controllable

Strategic risk Internal Controllable

Operative risk Internal Controllable

Capex risk Internal Controllable Source: Author’s elaboration based on [16] Royal Institution of Chartered Surveyors (RICS), «Management of risk, 1st edition,» Royal

Institution of Chartered Surveyors (RICS), London, 2015

Other types of risk classifications are also possible, for example based on the source of the risk and the

mitigation possibilities that the investor can adopt, as shown in Table 19.75 External risks are possible events

that potentially have an external origin with respect to the company, which therefore cannot directly control

them or prevent them at the root, but at most it can try to mitigate the impact on its activities if the risky

event occurs. The risks relating to the fluctuation of market prices, vacancy and disinvestment can be

influenced as they are correlated with the "quality" of the building and the economic-financial solidity of

the chosen tenants, so even if little, the investor has room counter reactions even on some external risks.

On the other hand, the change in taxation and interest rates are completely uncontrollable. All cost items

and strategic choices are instead directly controllable by the investor.

74 [4] COIMA RES, «Relazione finanziaria annuale,» 2018 75 [16] Royal Institution of Chartered Surveyors (RICS), «Management of risk, 1st edition,» Royal Institution of Chartered Surveyors (RICS), London, 2015

55

PART 2.3 ANALYSIS TOOLS AND TECNIQUES

2.3.1 - Project Analysis

Once all the cash flows have been estimated and entered into the DCF, the discount rate has been decided

and the NPV has been calculated, if the latter is positive, is it possible to accept the investment? Is the

information obtained sufficient to make a decision? Probably not. Further information that has to be

acquired are: what is the probability that the project will go as planned? If the investment goes bad, how

big would be the loss? If the underlying assumptions to forecasted cash flows are wrong, what happens?

First of all, we need to clarify the fact that the expected cash flows reported in the DCF are the expected

value of the variable for each period and the fact that the expected value differs from the observed one is

already taken into account in the discount rate, that has precisely this function. When a forecast is made,

what Ross et all [11] defines as forecasting risk, that is “the possibility that errors in projected cash flows will

lead to incorrect decisions. Also known as estimation risk. If the projections are seriously in error, then we

have a classic GIGO (garbage in, garbage out) system.”76 In short, we must consider the possibility of making

wrong predictions and try to understand what the effects of these errors could be. This process is called

"what-if analysis", and includes the scenario, sensitivity and simulation analysis.

2.3.1.1 - Scenario Analysis

An analysis scenario consists of calculating the project's NPV and IRR after assigning to each DCF variable

values that represent a certain type of scenario. Usually the considered scenarios are optimistic, pessimistic

and most likely, but by attributing specific values to the variables it is possible to represent innumerable

types of scenarios. In summary, the scenario analysis consists of varying all the variables, but of a single

value. For the sake of clarity I quote the factual example from R. A. Brealey et all [17] in which a scenario

analysis is made in the event of high oil prices for an electric scooter production project.

Table 20. Project A scenario analysis

Cash flows, years 1-10, milions of €

Base case High oil prices and

recession Expansion

Revenues 37.5 44.9 45

Variable cost (30) (35.9) (32)

Fixed cost (3) (3.5) (3)

Depreciation (1.5) (1.5) (1.5)

Pretax profit 3 4 8.5

Tax (50%) (1.5) (2) (4.25)

Net profit 1.5 2 4.25

Depreciation 1.5 1.5 1.5

Net cash flows 3 3.5 5.75

PV of cash flows @ 10% 18.4 21.5 35.3

NPV 3.4 6.5 20.3 Sources: [17] R. A. Brealey, S. C. Myers, F. Allen, Principles of Corporate Finance, tenth edition, The McGraw-Hill Companies, Inc.,

2011, pag. 246

76 [11] S. A. Ross, R. W. Westerfield e B. D. Jordan, Fundamentals of Corporate Finance, 10th edition, The McGraw-Hill Companies, Inc., 2013. Pag. 344

56

The initial investment is 15 million, the discount rate applied is 10 percent and it is assumed for simplicity

that the cash flows for each year are the same. Evaluating the scenario of rising oil prices, it can be seen

how the effects of increased sales can interact due to greater demand for alternative means of transport to

diesel or gasoline vehicles and increased production costs due to the increase in the price of a fundamental

input like oil. The advantages of using the scenario analysis are that it provides an evaluation of the results

of the projects under different business environment conditions and that by measuring the range between

the pessimistic and the optimistic scenario, a measure of project risk can be obtained. The disadvantages of

this method are that the sensitivities of the result are not identified with respect to the single variables, a

break-even point is not identified and that no information is given about the probability of the different

scenarios and therefore of the relative outcomes.

2.3.1.2 - Sensitivity Analysis

“Sensitivity analysis is the investigation of what happens to NPV when only one variable is changed.”77 The

difference with respect to the scenario analysis is that the sensitivity holds all the variables except one, to

which different values are assigned, while the scenario varies all the variables but of only one value.

Figure 28. Example of sensitivity analysis

Sources: author’s elaborations based on data in R. A. Brealey, S. C. Myers, F. Allen, Principles of Corporate Finance, tenth edition,

The McGraw-Hill Companies, Inc., 2011, pag. 246

The objective of the sensitivity is to identify the "critical" variables, that is those variables for which the

objective function is more sensitive, in the sense that a unitary variation of the input corresponds to a "large"

variation of the output. The more the result is sensitive to a variable, the more the forecasting risk

associated with it is high. Assuming we want to evaluate the fixed cost and taxes variables of the example

shown in Table 20, these are varied with respect to the base case from -10 to +10 per cent with steps of 2

per cent, leaving for simplicity the same values in each of the ten years of the DCF. The results shown in

Figure 28 show that the fixed cost variable is more critical, since its unit variation has a greater effect on the

77 [11] ibidem. Pag. 349

-40,0%

-30,0%

-20,0%

-10,0%

0,0%

10,0%

20,0%

30,0%

40,0%

-12,5% -7,5% -2,5% 2,5% 7,5% 12,5%

NP

V %

var

iati

on

s

Variable % variations

Fixed cost Taxes

57

NPV; graphically the greater sensitivity is given by a greater first derivative of the curve. This information

can be useful, for example, to decide whether it is better to invest to transfer production in jurisdictions

with lower corporate tax rates or to use those resources to reduce fixed costs. In any case, the investor is

informed that the fixed rate variable has a greater forecasting risk.

2.3.1.3 - Break-Even Analysis

The break-even analysis consists in searching for the break-even point, that is “the level of activity (or of any

variable, ndr) at which profitability is zero.”78 As an example I used the values of the base case of Table 21,

calculating the financial break-even for each variable, or those values that null the NPV. The values reported

in Table 21 show that both the revenues and the variable costs of the base case are slightly higher than the

break-even point; if the revenues for the next 10 years will be 36.38 million euros, the present value of the

cash flows will be equal to the value of the initial investment (15 million) and therefore the NPV will be 0.

This type of information is useful to understand how much the gone can go worse than expected before

you get to suffer a loss.

Table 21.Break-even point calculations on NPV

Item Base Case (mln €) NPV break-even

Revenues 37,5 36,38

Variable cost -30 -31,11

Fixed cost -3 -4,11 Source: author’s elaboration

Note: calculations are based on data from Table 20. The break-even is financial, so the objective function that is to be nulled is the

NPV

2.3.1.4 - Monte Carlo Simulation

The simulation analysis carried out with the Montecarlo method allows the scenario analysis to be combined

with the sensitivity, since the values of all the variables are varied for all the values of their probability

distribution. It is like a scenario analysis in which all the scenarios are considered or as a sensitivity in which

all the variables are made to vary simultaneously. The Montecarlo simulation consists of the generation of

n different results of the phenomenon studied so as to obtain the outcome as a probability distribution.

Initially the domain of the input variables is defined, that is the probability distribution and the parameters

of the random variables used are defined. For each of the n iterations, a value is randomly extracted from

each random variable that is part of the process to be simulated, so as to be able to perform a deterministic

calculation. The n final results are aggregated together into a single final result represented by a probability

distribution. In my case, a probability distribution is assigned to the single variables contained in the DCF.

The objective functions are calculated for n times, so the outputs, typically NPV and IRR, are represented as

probability distributions. The probability of extracting a certain input value from a variable is given by its

probability density function: if, for example, it is assumed that a variable has a uniform distribution, at each

iteration all values between the maximum and the minimum of the distribution will have the same

probability of being extracted; if, instead, we hypothesize that the probability distribution that best

describes a variable is gaussian, the values around the average will have a higher probability of being

78 [24] S. Vishwanath , Corporate Finance. Theory and Practice, 2th edition, New Dehli: Sage Publications, 2007. Pag. 214

58

extracted with respect to the values in the tails, which will therefore be extracted a smaller number of times.

For the sake of clarity, I will make a simple example. We hypothesize to analyze the income statement of

company A, whose forecasts about the most probable revenues, variable costs, fixed costs and net income,

of the next year are shown in Table 22, while in Table 23 are reported the types and parameters of the

random variables chosen to represent the variables of the income statement.

Table 22. Most likely values for the income statement of firm A for the next year

Most Likely

Revenues 10,00 €

Variable costs -5,00 €

Fixed costs -3,00 €

Net income 2,00 € Source: author’s elaboration

Table 23. Assumptions of distributions and parameters of firm A income statement variables

Input

variabile Distribution

Parameters

Mean St dev Min Max

Revenues Normal 10,00 € 2,00 €

Variable cost Normal 5,00 € 0,50 €

Fixed cost Uniform 2,00 € 4,00 € Source: author’s elaboration

Sources: author’s elaboration. Simple example of Monte Carlo simulation with 5,000 iterations

Mean 1,98

Standard error 0,03

Median 1,96

Mode 1,03

Standard deviation 2,12

Sample variance 4,51

Kurtosis 0,04

skewness 0,10

Intervall 15,71

Minimum value -4,88

Maxium value 10,82

Sum 9893,32

Count 5000

Bigger (1) 10,82

Smaller (1) -4,88

Confidence level (95,0%) 0,06

Net income

Figure 29. Inputs variables aggregation with Monte Carlo simulation - 5,000 iterations

0

50

100

150

200

250

-4,8

8

-3,0

9

-1,2

9

0,50

2,29

4,09

5,88

7,68

9,47

Freq

uen

cy

Net income

0

100

200

300

1,7

3,8

6,0

8,1

10,3

12,4

14,6

16,7

Freq

uen

cy

Revenues

0

50

100

150

200

250

3,30

3,74

4,17

4,60

5,04

5,47

5,91

6,34

Freq

uen

cy

Variable cost

0

20

40

60

80

100

2,00

2,26

2,51

2,77

3,03

3,29

3,54

3,80

Freq

uen

cy

Fixed cost

59

This method allows to express the objective functions as probability distributions and therefore to quantify

the risk. It is assumed that the expected value of the variable costs is half of the revenues and that it is

described by a normal distribution, as are the revenues. The fixed costs have a normal distribution with an

expected value of € 3. Figure 29 shows a summary of the Monte Carlo simulation for the simple

hypothesized example. The overall risk is contained in the probability distribution of net income for the

following year. Using the standard deviation, the risk is 2.12 €, using the range is 10.82 - (-4.88) = 15.7 €,

while using the value at risk the following year with 95 per cent confidence is µ - (µ - 1.65σ) = 1.98 - (1.98 -

1.65 * 2.12) = 1.98 + 1.52 = 3.5 €; this means that at 95 percent probability the net income will not differ

from the expected value more than € 3.5. Although the Monte Carlo simulation allows us to verify the

outcome of a large number of combinations of variables, its biggest limitation is that of not being able to

take into consideration the correlations and the cause-effect relationships that bind the variables.

2.3.2 - Risk assessment

In this paragraph I will go on to explain the method by which I intend to approach the quantification of risk.

The formula that is generally adopted is (43). Its meaning is that a rare event can be very risky when it has

an extremely severe impact when it occurs, as well as a very probable event can be considered risky even if

its impact per individual event is small. Extremely frequent and harmful events are obviously very risky, as

well as infrequent and little impacting events are negligible.

𝑅𝑖𝑠𝑘 = 𝑙𝑖𝑘𝑒𝑙𝑖ℎ𝑜𝑜𝑑 x 𝑖𝑚𝑝𝑎𝑐𝑡 (43)

To be able to quantify the risk it is necessary to have available the probability distribution of the variable.

According to GleiBner and Wiegelmann (2012) “the non-quantification of risks is not possible. […] In the

absence of other information, all potential situations are regarded as equally probable […]. This means that

all risks are quantifiable even if no information exists. […] where the probability distribution of probabilities

is known, they may be translated for an objective first-level probability that is implicitly present; and where,

however, no probability of any level is known for the validity of a probability distribution, a uniform

distribution that is known with certainty will have to be assumed.”79 In any case, uniform distribution is

assumed when there is no information. The use of random variables instead of variables defined by a single

value gives the advantage of being able to obtain the expected value and a risk measure from the same

common basis, that is the probability distribution. According to the method used by D. J. Gimpelevich [18]

the first step is therefore the collection of data regarding the variables that make up the business plan, so

as to be able to observe the distribution and then choose one of the various known probability distributions

that best represents the observable distribution in the data.80 I will give an example to describe the method

I intend to use in the case study. I hypothesize that the empirical observations regarding the variable X

indicate that in the past it assumed the values A, B, C, D, E with probability described by the distribution in

the left part of the Figure 30. At this point it is hypothesized that the observed distribution (left side of the

image) can be validly represented with a known probability distribution (right part of the image), in this case

79 [13] W. GleiBner e T. Wiegelmann, «Quantitative methods for risk management in the real estate development industry. Risk

measures, risk aggregation and performance measures,» Journal of Property Investment & Finance, vol. 30, n. 6, pp. 612-630, 2012 80 [18] D. J. Gimpelevich, «Simulation-based excess return model for real estate development,» Journal of Property Investment & Finance, vol. 29, n. 2, pp. 115-144, 2011

60

a triangular distribution, whose formulas are known, so one could be able to calculate the mean (44),

variance (45) and frequency (46) for each x.

(𝑋) = 𝑎 +𝑏 + 𝑐

3 (44)

σ2 = (𝑎2+𝑏2+𝑐2)−(𝑎𝑏+𝑎𝑐+𝑏𝑐)

18 (45)

𝑓(𝑥) =

{

2

𝑏−𝑎

𝑥−𝑎

𝑐−𝑎 𝑠𝑒 𝑎 ≤ 𝑥 < 𝑐

2

𝑏−𝑎 𝑠𝑒 𝑥 = 𝑐

2

𝑏−𝑎

𝑏−𝑥

𝑏−𝑐 𝑠𝑒 𝑐 < 𝑥 ≤ 𝑐

(46)

Sources: author’s elaboration

The risk will be quantified using one of the risk measures described in the Measuring Risk paragraph, that

is standard deviation, value at risk, semivariance, but also others. The point is that it is not so important to

measure the risk of a single variable, but rather the overall risk arising from the aggregation of the

individual risks (single random variables), represented by the probability distribution of the objective

function used to verify the profitability of the investment, such as the NPV or the IRR. According with

GleiBner and Wiegelmann (2012) “The quantification of risks starts with the quantitative description of the

risks by an appropriate probability distribution. As businesses, projects or entire companies are generally

subject to a number of risks; these must be aggregated to determine the overall risk. This requires the use

of Monte Carlo simulation, in which a large representative sample of risk-bearing possible future scenarios

is calculated. Risk-related information therefore adds value to the “traditional” organization or investment

planning. The total risk the frequency or probability distributions are derived from the so-called “risk

measures” such as standard deviation or value-at-risk.”81 In other words, the use of Monte Carlo

simulation makes it possible to aggregate individual risks and obtain an objective function expressed as a

probability distribution, from which it is possible to extrapolate both the expected value and the risk

measures, such as standard deviation or value at risk. The risk quantification takes place precisely through

81 [13] Ibidem

0%

5%

10%

15%

20%

25%

30%

35%

40%

45%

A B C D E

0%

5%

10%

15%

20%

25%

30%

35%

40%

45%

a c b

Figure 30. Derivation of probability distribution from empirical observations

61

the analysis of the probability distribution of the objective function of the DCF obtained with the Monte

Carlo simulation.

62

PART 3

CASE STUDY

63

PART 3 – CASE STUDY

3.1 - Case Study Asset

The asset that I will analyze for this case study is the Vodafone Village owned by COIMA RES. The building is

located in Milan, via Lorenteggio 240, as shown in Figure 31.

Source: Google Maps

Figure 32. Vodafone Village main facade


Figure 31. Asset geolocalization

64

3.2. - Cash Flows Estimation in Practice

3.2.1 - Potential Gross Income

Revenues from commercial property derive from rents collected; referring to M. Geltner et all [14] I will call

this item Potential gross income - PGI. If there is a rental contract between owner and tenant, the amount

of annuity that must be paid periodically (every month or quarter) is defined in the terms of the contract,

which almost always includes indexing with inflation. If there is no rental agreement, “revenue will be a

function of future leases that will likely be signed.”82 For the estimation of this variable I will apply two rules:

• Rule 1. If there is a lease contract:

𝑃𝐺𝐼𝑡 = 𝑃𝐺𝐼𝑡−1(1 + 𝜋) (47)

• Rule 2. If there isn’t a lease contract:

𝑃𝐺𝐼𝑡 = 𝐸𝑅𝑉𝑡−1(1 + 𝜋) (48)

3.2.2 - Estimated Rental Value – ERV

The estimate of the ERV must be made taking into consideration three elements: inflation, "real" market

trend and physical degradation of the asset.

Inflation

As can be seen from Table 26 and Figure 33, the inflation trend in Italy from 2000 to today is clearly bearish.

Although the economic policy objective of the European Central Bank regarding inflation is to keep it stable

at 2%, it would seem that the economic phenomenon in question is not so easily governable. As regards the

estimate of cash flows, I will use the inflation value of 1.2 per cent, as shown by the averages for the last

ten years both for Istat and Eurostat data.

Table 24. Inflation Forecasts

Inflation Forecasts in Italy

OECD IMF ECB Average

1% 1.2% 1.0% 1.06% Source: author’s elaboration based on [19] OECD, «Inflation forecast,» [Online]. Available: https://data.oecd.org/price/inflation-forecast.htm; [20] International Monetari Fund, «Italy at a glance,» [Online]. Available: https://www.imf.org/en/Countries/ITA; [21] Eurtopean Central Bank, «Eurosystem staff macroeconomic projections for the euro area countries,» 2019

82 [14] D. M. Geltner, N. G. Miller, J. Clayton e P. Eichholtz, Commercial Real Estate. Analysis & Investments, 2nd edition, Oncourse Learning, 2006

https://data.oecd.org/price/inflation-forecast.htm

https://data.oecd.org/price/inflation-forecast.htm

https://www.imf.org/en/Countries/ITA

65

Table 25. Historical Eurostat HICP inflation in Italy 2008-2018

Harmonised Indices of Consumer Prices in Italy 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018

HICP 90,4 91,1 92,6 95,3 98,4 99,7 99,9 100 99,9 101,3 102,5

% variation 0,77 1,65 2,92 3,25 1,32 0,2 0,10 -0,10 1,40 1,18

Average - 10 years 1,27


Source: author’s elaboration based on [19] Eurostat, «Eurostat. Your key to European statistics,» [Online]. Available:

https://ec.europa.eu/eurostat/data/database

Table 26. Historical Istat CPI inflation in Italy 2000-2018

Consumer Price Index for the whole collectivity

%

variation

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

2,55 2,75 2,50 2,69 2,21 1,92 2,12 1,85 3,33 0,81

2010 2011 2012 2013 2014 2015 2016 2017 2018

1,53 2,80 3,02 1,23 0,19 0,09 -0,10 1,20 1,19

Average from 2000 1,78


Average - 5 years 0,51 Source: author's elaboration based on [19] Istat, «Prezzi,» [Online]. Available: https://www.istat.it/it/prezzi?dati.

Figure 33. Inflation trend (5 years moving average) in Italy 2000-2018

Source: author’s elaboration based on [19] Istat, «Prezzi,» [Online]. Available: https://www.istat.it/it/prezzi?dati

Market Trend

As far as regards the market trend, I will use the estimates and considerations provided by some of the main

players in the Italian real estate market. In Table 27 the relevant element is the growth rate in the Milan

suburbs, that is where the Vodafone Village is located. In the last 5 years the compounded annual nominal

growth rate has been 4.2 percent, which means, according to the data reported in Table 25 and 26, about

3.7 percent net of inflation. Reported below there are two statements taken from annual reports of Jones

Lang Lasalle and COIMA:

-0,50%

0,00%

0,50%

1,00%

1,50%

2,00%

2,50%

3,00%

3,50%

20

00

20

01

20

02

20

03

20

04

20

05

20

06

20

07

20

08

20

09

20

10

20

11

20

12

20

13

20

14

20

15

20

16

20

17

20

18

CPI 5 Per. media mobile (CPI)

https://www.istat.it/it/prezzi?dati

https://www.istat.it/it/prezzi?dati

66

“The future supply for the next 2 years is lower than the average take-up of the last 5 years, both in Milan

than in Rome. The majority of the pipeline is speculative, but approximately 20-30% of this is very close to

being signed and will change into pre-let status over the short term.”83

“[…] a causa della scarsità di immobili di Grado A, che rappresentano solo il 10% del totale stock ad uso

ufficio, ma rappresentano oltre il 70% della domanda da parte dei conduttori. Questo squilibrio suggerisce

che la domanda potrebbe superare di 2-3 volte l’offerta di nei prossimi anni il che sosterrà la crescita dei

canoni a breve e medio termine.”84 The indication is that of an excess demand on the supply as regards the

years to come for the Milanese office sector, which suggests a possible increase in rents.

Table 27. Milan office rent values and growth rates at Q2 2019

Prime Office Rents in Milan

€/mq yr Growth 1 yr 5 yr CAGR

CBD 590 3,50% 4,40%

Centre 460 7,00% n/a

Semi Centre 350 9,40% 5,30%

Periphery 270 12,50% 4,20%

Hinterland 240 9,10% n/a Source: Kushman and Wakefield Q2 2019

The annual financial report of COIMA for the year 2018 states that the property "Vodafone Village" has

appreciated by 2.5 per cent since the acquisition in 2016, approximately 0.83 per cent per year on average,

and 0.4 per cent in 2018.85 The hypotheses about the discrepancy with respect to the Kushman & Wakefield

data are that i) the micro-area in which the building is located is growing less than in other areas of the

Milan suburbs, or ii) that a building of this size can hardly be appreciated 12 percent per year or in any case

have sudden price changes. Being COIMA’s data specific of the building, I consider these growth rates

already net of the depreciation rate due to the physical degradation of the asset, as I will explain in the

Degradation and Obsolescence paragraph; this means that the real growth rate should stand at 0.83

(nominal) + 1.67 = 2.5 per cent per year, where 1.67 is the annual depreciation rate due to the aging of the

asset, assuming a useful life of 70 years. Considering the acceleration of the growth rates of the rent

recorded in 2019 and the imbalance in the market in favor of the demand, I hypothesize a first phase of fast

growth followed by a slowdown and then a slight decrease in the market rents, as reported in Table 28.

Table 28. Assumed real growth rate of the market rent in the south-west Milan periphery sub-market

year 1 2 3 4 5 6 7 8 9 10 Trend (var %) 3.0 2.5 2.0 1.0 0.5 -0.1 -0.2 -0.3 -0.4 -0.5


Degradation and Obsolescence

Degradation of a building is defined as “progressive alteration of the physical integrity of the materials and

component with a related performance decay”.86 Over time, the physical-chemical actions caused by

83 [20] Jones Lang Lasalle, «Office Italy. Annual report 2018,» 2018. Pag. 23 84 [4] COIMA RES, «Relazione finanziaria annuale,» 2018. Pag. 11 85 Ibidem 86 [53] CIB, «W86 Building pathology

67

"degrading agents" such as humidity, solar radiation and wind, gradually lead to the deterioration of the

functional characteristics of the construction elements, on which the ability of the building component to

give performances depends and therefore satisfy user requests. For what has been said so far and for the

definition of building quality, that is “the set of properties and characteristics of the building (or of its parts)

which satisfy clear or implicit requirements through performances”,87 we understand how degradation

reduces building quality. The ability of a property to generate income is closely related to its quality,

therefore the degradation by reducing the quality decreases the property's ability to generate income due

to a lower willingness to pay for the user to acquire the right to use of a given space of lower quality. “The

simplistic assumption that rents will grow at the inflation rate will often be unrealistic, however. It tends to

ignore the effect of functional and economic depreciation of the building, […] The subject building must

compete against these newer structures in the same market, and this may force it to lower its rents relative

to the ‘‘top of the market’’ rent level.”88

Figure 34. Performance decay over time

Source: [20] B. Daniotti, Building durability, Cortina Libreria Milano, 2012

As shown in Figure 34, the performance decay continues until a lower limit is reached, a level at which the

performances of the building component are no longer high enough to satisfy a certain user requirement,

making so end the building's service life, which is defined as “period of time after installation during which

a building or its parts meets or exceeds the performances requirements upper or equal to the accepted

limits.”89 In short, degradation decreases the performance of the building, causing a reduction in quality and

therefore in profitability, until the end of its service life beyond which the building is "out of the market".

Maintenance - one whose possible definition is “stage collecting the repair or replacement operations of the

building parts to secure the right functioning during time, on the basis of management planning”90 - allows

to slow down the physical degradation process and therefore lengthen the service life of a building, as

shown in figure 35. The maintenance and replacement activities of components allow to extend the useful

life of the building both from a technical and economic point of view, thanks to "leaps" in the level of quality

due to realization of maintenance and capital expenditure activity, that increase the quality and therefore

the ERV of the asset. The theme of the relationship between maintenance and building quality is certainly

complex and goes beyond the scope of this thesis, therefore I will not consider this report. This means that

87 [20] B. Daniotti, Building durability, Cortina Libreria Milano, 2012 88 [14] D. M. Geltner, N. G. Miller, J. Clayton e P. Eichholtz, Commercial Real Estate. Analysis & Investments, 2nd edition, Oncourse Learning, 2006. Pag. 238 89 [20] B. Daniotti, Building durability, Cortina Libreria Milano, 2012 90 Ibidem

68

in modelling the ERV dynamic I will not take into consideration the effect of the expenses for ordinary

maintenance and capital expenditure carried out.

Figure 35. Effect of maintenance on performances degradation and useful life length

Source: [21] I. Flores-Colen e J. de Brito, «Discussion of proactive maintenance strategies in fa çades’ coatings of social housing,»

Journal of Building Appraisal, vol. 5, n. 3, pp. 223-240, 2010

As the building ages it becomes relatively less attractive due to the entry into the market of new buildings

that are more up-to-date and technologically more efficient, which, for the same rent, would therefore be

relatively less expensive. There are several ways to estimate a possible rate of decrease in price caused by

aging. You can analyze the history of the leases of a building and see how the rent has evolved over time,

comparing it with inflation and the market trend; another way is to compare buildings of the same type of

different ages located in the same market at a given time and detect the gap between rents. If for example,

there are two building that have an age gap of 12 year and they are rented at 12 €/sqm per month the

younger and at 10 €/sqm per month the older, the depreciation rate can be calculated applying the

compounded rate formula, so (10/12)1/12-1 = -0.015 = -1.5 %.91 As for my case study, I will assume that the

building suffers a linear degradation over a 70-year useful life, so that the ERV decreases at an annual rate

of 1/70 = 1.43 percent, without considering the effect of maintenance costs.

Effects Sum

By combining the effects of market trends, inflation and degradation it is possible to obtain an estimate of

the asset's ERV as shown in Figure 36. These three distinct effects are added together defining a specific

trend. Following C.O. Amédée ‐ Manesme et al (2012) it is possible to hypothesize that the price will oscillate

stocastically around this trend moving according to a geometric Brownian motion:92

𝑑𝑃𝑡

𝑃𝑡 = 𝜇𝑝𝑑𝑡 + 𝜎𝑝𝑑𝑊𝑡

𝑃 (49)

91 [14] D. M. Geltner, N. G. Miller, J. Clayton e P. Eichholtz, Commercial Real Estate. Analysis & Investments, 2nd edition, Oncourse Learning, 2006. Pag. 239 92 [54] C. Amédée‐Manesme, F. Barthélémy, M. Baroni e E. Dupuy, «Combining Monte Carlo simulations and options to manage the risk of real estate portfolios,» Journal of Property Investment & Finance, vol. 31, n. 4, pp. 360-389, 2013

69


Equation (49) says that prices can be modelled with a diffusion process, where the μP and σP parameters

correspond to trends and historical market volatility. Table 29 shows the hypothesized and calculated values

of the different effects, while Figure 37 graphically represent the result of the application of the model. The

trend is calculated as the sum of the three effects that act together to define the market price; the

randomized rate is calculated by applying the formula (49); the ERV is simulated by applying the rate of

change to the initial rent price estimated at the current time. This model is supposed to acts synergistically

with the Monte Carlo simulation, in order to be able to calculate different historical series, one for each

iteration of the simulation, and so be able to consider the effects of different possible paths that the price

can shows in the future. Always following C.O. Amédée-Manesme et al (2012), this model is particularly

useful if applied in combination with the vacancy allowance simulation as described in the next paragraph.

Table 29. ERV simulation applying effects sum and formula (49)

ERV st. dev. [%] A=2,47

ERV t0 [€] B=13,80

Years 0 1 2 3 4 5 6 7 8 9 10

Trend [%] C 3,00 2,50 2,00 1,00 0,50 -0,10 -0,20 -0,30 -0,40 -0,50

Trend [%] C 3,00 2,50 2,00 1,00 0,50 -0,10 -0,20 -0,30 -0,40 -0,50

Inflation [%] D 1,20 1,20 1,20 1,20 1,20 1,20 1,20 1,20 1,20 1,20

Degradation [%] E -1,43 -1,43 -1,43 -1,43 -1,43 -1,43 -1,43 -1,43 -1,43 -1,43

Effect sum [%] Ft=C+D+E 2,77 2,27 1,77 0,77 0,27 -0,33 -0,43 -0,53 -0,63 -0,73

Randomized rate [%] Gt=Ft+Aε~N(0,1) 5,64 0,73 3,12 -0,76 -1,39 -2,90 -1,64 -4,77 -2,88 -0,74

Simulated ERV [€] It=It-1(1+Gt) 13,80 14,58 14,69 15,14 15,03 14,82 14,39 14,15 13,48 13,09 12,99


0,80

0,85

0,90

0,95

1,00

1,05

0 1 2 3 4 5 6 7 8 9 10 11 12

Degradation

0,981,001,021,041,061,081,101,121,14

0 1 2 3 4 5 6 7 8 9 10 11 12

Inflation

Figure 36. Indexes of the nominal ERV components - trend and effects sum

0,98

1,00

1,02

1,04

1,06

1,08

1,10

0 1 2 3 4 5 6 7 8 9 10 11 12

Rent trend

0,94

0,96

0,98

1,00

1,02

1,04

1,06

1,08

1,10

1 2 3 4 5 6 7 8 9 10 11

Effects sum

70

Figure 37. ERV - trend and simulated


Figure 38. 5 ERV simulations with trend


Figure 39. ERV probability distribution in year 1 and 10 obtained with a Monte Carlo simulation – 5,000 iteration


12,50 €

13,50 €

14,50 €

15,50 €

16,50 €

1 2 3 4 5 6 7 8 9 10 11

Simulated ERV ERV trend

12,00 €

13,00 €

14,00 €

15,00 €

16,00 €

17,00 €

1 2 3 4 5 6 7 8 9 1 0 1 1

Serie1 Serie2 Serie3 Serie4 Serie5 Trend

0

50

100

150

200

250

300

10 12 14 16 18 20

Fre

qu

en

cy

ERV

year 1 year 10

71

Figure 38 shows 5 different price simulations carried out by applying the formula (49). The time series differ

from each other since the historical volatility, expressed by a standard deviation, is multiplied by a random

number extracted from a Gaussian probability distribution with mean 0 and standard deviation 1. Year after

year the uncertainty about the value of the price increases, as shown in Figure 39, in which the distribution

of ERV in year 1 is compared to the one in year 10, after having performed a simulation of 5,000 iterations;

the average distribution for year 10 is greater than in year 1 due to the bullish rent trend.

3.2.3 - Vacancy Length

The starting point for the modeling of the variable vavancy length was the model proposed by C.O. Amédée‐

Manesme et all (2012), in which the Poisson distribution is used - formula (50) - to generate different

vacancy length values at each iteration of the Monte Carlo simulation.

X~P(𝑋 = 𝑘) =λ𝑘𝑒−λ

𝑘! (50)

K: vacancy length

λ: average vacancy length

In Figure 40, we see the probability distributions of the random variable "vacancy length", with λ of value 1,

2, 3 from left to right.

3.2.4 - Vacancy Allowance

When considering potential revenues, it is good to assume that the spaces will not all be rented for the

entire duration of the investment, and therefore take into account the effect of the vacancy on the

profitability of the investment.

According to Geltner et all (2006) the most simple method is considering a “rent cycle” as sum of the average


length of signed contracts and average vacancy observed in a specific market and therefore calculate the

vacancy rate by dividing the average vacancy length by the length of the rental cycle. For example, if a

Figure 40. Poisson distributions with λ equals to 1, 2, 3

72

contract lasts 5 years on average, then 60 months, and between one contract and another pass on average

5 months, the rent cycle will be 65 months long and the vacancy rate will be 5/65 = 7.69%. At this point, the

vacancy allowance will be calculated by multiplying the vacancy rate by the potential gross income. My case

study considers the presence of a break-option in the lease. Using the approach theorized by C.O. Amédée

‐ Manesme et all (2012), it is possible to quantify the vacancy allowance jointly using a random variable to

represent the risk of vacancy and the Monte Carlo simulation. The factors that are taken into consideration

are: i) the evolution of the rent roll over time based on contract agreements, ii) the evolution of the ERV in

the reference sub-market, iii) the vacancy dynamics. I assumed that both tenant and landlords want to

maximize their well-being, so the former will want to minimize spending, while the latter will want to

maximize revenues. When the tenant has the right (but not the duty) to exercise the option, he will continue

to occupy the real estate unit if the rent roll is less than the ERV plus the transaction, agency and transfer

costs; vice versa will exercise the option. For each t, there are 3 possible values of the rent: a) in the first

year of the contract, the rent is approximately the market price, b) for the following years the annuity is

indexed according to what is defined in the contract, c) the unit real estate is vacant and the rent is 0.

∀𝑡, ∀𝑖, ∀𝑗 = {

𝐸𝑅𝑉𝑡,𝑖𝑗 (𝑎)

𝑓 (𝑅𝑒𝑛𝑡𝑡−1,𝑖 , ∆𝐼𝑡,𝑗𝑅𝑒𝑛𝑡𝑡,𝑖) (𝑏)

0 (𝑐)

(51)

𝑅𝑒𝑛𝑡𝑡−1,𝑖: rent roll for the space i at the time t.

𝐸𝑅𝑉𝑡,𝑖𝑗

: estimated market rent for the space i in the sub-market j at the time t

𝐼𝑡,𝑗𝑅𝑒𝑛𝑡𝑡,𝑖: index of 𝑅𝑒𝑛𝑡𝑡,𝑖 based on the contractual agreements.

Each space can be rented or vacant at each t. The idea of the model is to find a dynamic rule that defines:

i) if a real estate unit rented in t remains leased in t + 1 when there is the possibility of exercising a break-

option and ii) for how long a unit real estate remains vacant when the contract expires or when the break-

option is exercised. 𝑆𝑖 is the time period in which the break-option can be exercised, α is the cost of the

transfer.

Vacancy allowance first rule:

∀𝑖 = 1, . . . , 𝑛, ∀𝑖 = 1, . . . , 𝑚, ∀𝑡 ∈ 𝑆𝑖 , 𝑖𝑓 𝑅𝑒𝑛𝑡𝑡,𝑖

𝐸𝑅𝑉𝑡,𝑖𝑗+𝛼> 1, 𝑡ℎ𝑒𝑛: 𝑅𝑒𝑛𝑡𝑡+1,𝑖 = 0 (52)

If t ∉ 𝑆𝑖 the space remains automatically rented. 𝛼 can be zero.

Vacancy allowance second rule: the length of the vacancy is modeled with the Poisson distribution - formula

(50). As for the use I intend to make of this model, I will apply rule 1 with α = 0, so if in the year in which the

tenant can choose to exercise the break-option or not, ERV and rent roll will be compared. If rent roll and

ERV are such that the break-option is not exercised, the vacancy allowance will be 0 in all the years following

the option, since the contract continues and the space continues to remain occupied with 100 percent

probability.

73

∀𝑡 ∈ 𝑆𝑖, 𝑖𝑓 0 <𝑅𝑒𝑛𝑡 𝑟𝑜𝑙𝑙𝑡,𝑖

𝐸𝑅𝑉𝑡,𝑖𝑗 < 1, 𝑡ℎ𝑒𝑛: 𝑃𝐺𝐼𝑡+1,𝑖 = 𝑅𝑒𝑛𝑡 𝑟𝑜𝑙𝑙𝑡,𝑖(1 + 𝜋) 𝑎𝑛𝑑 𝑉𝐴 = 0 (53)

If the break-option is exercised the vacancy allowance of each period depends on the PGI and vacancy

length.

Year t1:

𝑖𝑓 0 < 𝑉𝐿 < 5, 𝑡ℎ𝑒𝑛 𝑉𝐴𝑡1 = 𝑉𝐿

4𝑃𝐺𝐼𝑡1

𝑖𝑓 𝑉𝐿 > 4, 𝑡ℎ𝑒𝑛 𝑉𝐴𝑡1 = 𝑃𝐺𝐼𝑡1

Year t2:

𝑖𝑓 0 < 𝑉𝐿 < 5, 𝑡ℎ𝑒𝑛 𝑉𝐴𝑡2 = 0

𝑖𝑓 4 < 𝑉𝐿 < 9, 𝑡ℎ𝑒𝑛 𝑉𝐴𝑡2 = (𝑉𝐿

4− 1) 𝑃𝐺𝐼𝑡2

𝑖𝑓 𝑉𝐿 > 8, 𝑡ℎ𝑒𝑛 𝑉𝐴𝑡2 = 𝑃𝐺𝐼𝑡2

Year t3:

𝑖𝑓 0 < 𝑉𝐿 < 5, 𝑡ℎ𝑒𝑛 𝑉𝐴𝑡3 = 0

𝑖𝑓 4 < 𝑉𝐿 < 9, 𝑡ℎ𝑒𝑛 𝑉𝐴𝑡3 = 0

𝑖𝑓 𝑉𝐿 > 8, 𝑡ℎ𝑒𝑛 𝑉𝐴𝑡3 = (𝑉𝐿

4− 2) 𝑃𝐺𝐼𝑡3

As the rules for the calculation of the PGI have been defined, the vacancy length has an impact on the choice

of what of the two rules apply: if VL is less than one year, it means that in the first year for a period of time

there will not be a contract and therefore PGI is calculated using rule 2 (48), while in the second year the

contract will be in place and rule 1 (47) will be applied. If VL is between 1 and 2 years, in the first year after

the break-option there will be no contract and rule 2 will be used, as in year 2, while in the third year there

will be a contract and rule 1 will be used and so on.

3.2.5 - Operative Expenses

This item includes property management, ordinary maintenance, utilities, insurance and property taxes.

Table 30. Vodafone Village operative costs in 2017 and 2018

Vodafone Village

Figures in thousand 2017 2018 Average

Annual Rent 13,877 (1) 13,964 (1)

Property Management -208 -1.50% -301 -2.17% -1.83%

Insurance -59 -0.43% -69 -0.50% -0.46%

IMU e TASI and other taxes -880 -6.34% -881 -6.35% -6.35%

Utilities -962 -6.93% -1,312 -9.45% -8.19%

Maintenance -199 -1.43% -892 -6.43% -3.93%

Total -2,308

-3,455

Recovery from Tenants 1,275 55.24% 2,448 70.85% 63.05% Note: (1) Costs are expressed as percentage of annual rent


74

I will consider insurance and taxes as fixed costs, property management and utilities as variable costs; the

maintenance subject matter will be treated separately. Table 30 shows the operating costs of the Vodafone

Village in the years 2017 and 2018.

Fixed Costs

For the estimation of these cash flows I used the specific data relating to the building object of the case

study reported by COIMA in its annual financial reports, visible in Table 30. In the DCF I will use the

percentages shown in the table below, which are multiplied by the PGI of the reference year to obtain the

cash flow values.

Operative Fixed Costs

Insurance -0.46%

IMU e TASI and other taxes -6.35%

Variable Costs

I consider utilities and property management expenses as variable costs as they depend on the vacancy of

the asset. In order to take in account this dynamic I will multiply the percentage shown in the table below

for the effective gross income so as to vary the expenses as the level of vacancy varies.

Operative Variable Costs

Utilities -8.19%

Property Management -1.83%

Maintenance

As regards the definition of maintenance costs, the distinction between maintenance and capital

expenditure is fundamental, since the former fall within the operating expenses and therefore contribute

to the calculation of net operating income, while the latter are counted after the NOI. NCREIF defines

operating expenses as “incurred during operation and maintenance of a property” and includes “repairs

and maintenance, insurance, management fees, utilities, supplies, property taxes” and capex as an expense

for “improve the real estate, extend its useful life, or improve future cash flows”.93 The main problem that

the term "maintenance" includes a very broad spectrum of activities, some of which can be confused with

capex. More current and specific literature [22] distinguishes capex in different sub-categories - as already

highlighted in the Returns paragraph - between “typical recurring expenses for ordinary repairs” and

“occasional high dollar value expenses that alter the physical, functional, or economic condition of a

property ”; other sources [58] define operating expenses as: "Operational activities may involve some

routine maintenance and minor repair work that are incidental to operations but do not include any

significant amount of maintenance or repair work."94 The main distinction that seems to emerge between

maintenance and capex it's the following:

• Maintenance activities: routine, planned, recurring, small-scale activities, tests and inspections;95

• Capex: improvements (activities aimed at improving asset performance), large substitutions (over

50% of the component), expansions, replacement of obsolete parts near the end of useful life.

93 [55] NCREIF, «Glossary of Terms,» [Online]. Available: https://epitest.ncreif.org/documents/event_docs/StPeteBeach2014/GlossaryofTermsv2TH.pdf. See definitions of Net Operating Income (“NOI”) and Capital Improvements 94 [58] Core Working Group Members, APPA, Federal facility Council, Holder, IFMA, NASFA, «Asset Lifecycle Model for Total Cost of Ownership Management. Framework, Glossary & Definitions». 95 Ibidem. Also exist “Emergency Maintenance” e “Unscheduled/Unplanned Maintenance”. This type of expense will not be taken into account.

https://epitest.ncreif.org/documents/event_docs/StPeteBeach2014/GlossaryofTermsv2TH.pdf

75

As regards the estimate of cash flows due to maintenance, I decided to use a case study developed in 2018

with some classmate (M. Carrara, D. Cifarelli, A. Dealexadris, G. Gaglione, H. Jingxuan, A. Yazdan, Z. Wubo)

which had as its objective the preparation of a maintenance plan for an office building in Milan of about

2,500 square meters in Via Pisani.

Table 31. Example of building components technical card

Floor

code operation description U.M

. €/U.M

frequenc

y (years)

note

s strategy

2C.24.770.0010

.a

cleaning - all

floors cleaning of floors and

horizontal surfaces in general mq 1.00

1 per

week

(1)

(2)

Predetermine

d

maintenance

1C.24.050.0040 cleaning more

deep

Cleaning of surfaces with

steam jet with detergents and

degreasers

sqm 7.67 1 per

month

(1)

(2)

Predetermine

d

maintenance

1C.24.050.0010

.c partial

refurbishment

- marble

Sandblasting of natural and

artificial stone surfaces using

mechanical methods with an

abrasive action, including

work surfaces and wall

assemblies

sqm 12.56

15

(1)

(2)

(3)

condition

based

maintenance

1C.24.050.0040

Cleaning of surfaces with

steam jet with detergents and

degreasers

sqm 7.67

- total - 20.23

1C.01.100.0010

.b

total

refurbishment

- PVC

Demolition of internal PVC

floors, including temporary

protective works, handling

with any means of the rubble

in the yard

sqm 5.71

15

(1)

(2)

(3)

condition

based

maintenance 1C.18.550.0030

Linoleum tile floor 2.5 cm

thick solid, variegated or

marbled, on a protected

surface with appropriate

treatment, laid with adhesive,

including the normal shaving

of a suitable screed, masonry

assistance with protection and

final cleaning

sqm 38.64

- total - 44.35

(1) R. Di Giulio, Manuale di manutenzione edilizia, p. 69

(2) J. R. Albano, La manutenzione degli edifici, p. 254

(3) The ground floor is in marble. The other floors are in linoleum

Source: auhtors’ elaboration (M. Carrara, D. Cifarelli, A. Dealexadris, G. Gaglione, H. Jingxuan, A. Yazdan, Z. Wubo)

This choice seems to be suitable as maintenance costs incorporate both materials and labor costs and

depend on the dynamism of a specific real estate market in a certain historical period, since it is logical to

assume that in a market where prices rise, the offer of new buildings is growing and therefore the existing

buildings must be well maintained and constantly updated to be able to remain competitive with respect to

the new assets that take over the market. In short, the total amount of maintenance costs depends on

economic factors specific to the region and the contingent situation of the real estate market.

From a methodological point of view, the estimate of the costs of maintenance activities was carried out by

preparing a "technical card", an example of which is shown in Table 31, for each building component using

the 2018 edition of the “Listino prezzi per l’esecuzione di opere pubbliche e manutenzioni” of the

municipality of Milan. The activities were therefore divided into "Capex" and "Maintenance" using the above

definitions, so as to be able to prepare the maintenance plan table shown in Attachment 3.1 and 3.2. To

make the expenditure "general", the costs were expressed in relation to the building's annual ERV,

estimated in 2018 at € 1,200,000, or € 480 / sqm per year. Expenses are shown in Table 32 and shown in

Figure 41. There are two observations to be made: i) maintenance costs are characterized by a "peak"

profile, which makes the average annual spending value not very useful, and ii) the highest expenditure

76

peak is registered at the year 10. The second observation is explained by Figure 42: at the year 15 and 20

the components begin to be replaced partially or completely and therefore these new components require

lower maintenance expenses in the following years.

Table 32. Study case maintenance expenses - years 1-25

ERV 2018 1.200.000 €

Years Maint exp Maint/ERV Years Maint exp Maint/ERV

1 3.356,60 € 0,28% 14 6.952,50 € 0,58%

2 5.020,50 € 0,42% 15 107.405,80 € 8,95%

3 21.085,72 € 1,76% 16 15.642,85 € 1,30%

4 13.710,85 € 1,14% 17 5.288,60 € 0,44%

5 101.565,63 € 8,46% 18 25.679,96 € 2,14%

6 13.230,90 € 1,10% 19 18.802,19 € 1,57%

7 3.356,60 € 0,28% 20 90.958,62 € 7,58%

8 38.888,95 € 3,24% 21 5.288,60 € 0,44%

9 10.568,66 € 0,88% 22 15.642,85 € 1,30%

10 116.244,75 € 9,69% 23 37.678,76 € 3,14%

11 31.116,81 € 2,59% 24 21.464,43 € 1,79%

12 23.853,25 € 1,99% 25 63.130,43 € 5,26%

13 15.805,66 € 1,32%

Note: Expenses refers to a 2,500 smq office building in Pisani street in Milano in 2018


Figure 41. Maintenance expenses profile over building life cycle


Figure 42. Capex profile over building life cycle


2,71%

0%

2%

4%

6%

8%

10%

12%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Maint/ERV Average

3,71%

0%

5%

10%

15%

20%

25%

30%

35%

40%

45%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Capex/ERV Average

77

As regards the estimate of the cash flows, I will use the percentage values shown in the "Maint/ERV" column

of Table 32, considering 2012 as year 1, because it is the year of construction of the Vodafone Village; since

the time horizon considered in the DCF will go from 2019 to 2030, I will use the values corresponding to the

years ranging from 8 to 19.

3.2.6 - Capital Expenditure

For the estimation of capex the company NCREIF has published the study "New NCREIF Value Index and

Operations Measures", edited by M.S. Young, J.D. Fisher and J. D’Alessandro (2016), in which a formula is

proposed to express the capex values relative to the market value of the property, the so-called capex ratio,

expressed by the formula (54).96 According to the surveys carried out by Young et all, for properties

belonging to the NCREIF index basket from 1978 to 2014, the average capex ratio value for office buildings

was 2.36 percent of the property's market value.

𝐶𝑋𝑅 = 𝐶𝐸𝑡

𝑀𝑉𝑡−1 (54)

𝐶𝐸𝑡: current year capital expenditure

𝑀𝑉𝑡−1: preceding year asset market value

Table 33. Annual Capex ratio for NCREIF office buildings, 1978-2014

Annual Capex Ratio

Mean 2.36

Standard deviation 0.75 Source: New NCREIF Value Index and Operations Measures

Table 34. Study case capex - years 1-25

ERV 2018 1.200.000 €

Year Capex Capex/ERV Year Capex Capex/ERV

1 € 0.00 0.00% 14 € 0.00 0.00%

2 € 0.00 0.00% 15 € 244,684.42 20.39%

3 € 0.00 0.00% 16 € 0.00 0.00%

4 € 0.00 0.00% 17 € 0.00 0.00%

5 € 81,983.01 6.83% 18 € 0.00 0.00%

6 € 0.00 0.00% 19 € 0.00 0.00%

7 € 0.00 0.00% 20 € 504,474.17 42.04%

8 € 0.00 0.00% 21 € 0.00 0.00%

9 € 0.00 0.00% 22 € 0.00 0.00%

10 € 81,983.01 6.83% 23 € 0.00 0.00%

11 € 0.00 0.00% 24 € 0.00 0.00%

12 € 0.00 0.00% 25 € 199,884.71 16.66%

13 € 0.00 0.00%

Note: Expenses refers to a 2,500 smq office building in Pisani street in Milano in 2018


96 [22] M. S. Young, J. D. Fisher e J. D’Alessandro, New NCREIF Value Index and Operations Measures, NCREIF, 2016

78

Using this capex ratio value for the estimation of the annual capex to be used in the DCF, as regards the first

year of the investment, the expenditure would be equal to 2.36% * 209.300 = 4.939 thousand €. Since the

annual rent roll of the period is € 14.132 thousand, the annual capex would be equal to 4.939 / 14.132 =

34.94% of the annual income. This value of expenditure seems excessive, especially if present at each year

of the investment. To estimate the cash flows due to the capex, I will use the values in the "Capex / ERV"

column shown in Table 34 which range from the year 8 to 19.

3.2.7 - Reversion Value

The reversion value is the amount that is obtained by selling the asset at the end of the investment's holding

period, net of any transaction costs and tax paid on any capital gains. To estimate this value, I use the

method of direct capitalization of the NOI at the end of the time horizon considered in the DCF by applying

the formula (19.1). It is important to note that the cap rate is calculated with the NOI of the year just ended

and the price of the previous year in which the asset was purchased; this means that to estimate the

reversion value it is necessary to project the NOI and the cap rate one year beyond the expected holding

period of the investment.

𝐴𝑠𝑠𝑒𝑡 𝑚𝑎𝑟𝑘𝑒𝑡 𝑝𝑟𝑖𝑐𝑒𝑇 =𝑁𝑂𝐼𝑇+1

𝑐𝑎𝑝 𝑟𝑎𝑡𝑒𝑇+1 (19.1)

“by using projected direct capitalization based on NOI, the reversion forecast is based on a projection of the

fundamental ability of the property to earn operating cash flow in the rental market, rather than a simple

extrapolation of the current investor’s purchase price of the property. […] It is normally most realistic to

project a going-out cap rate at least equal to, or slightly higher than, the going-in cap rate (that is, the cap

rate at the time of purchase), based on typical market cap rates. This is because, as buildings age, they

usually become more risky or less able to grow the rents they can charge (or more prone to needing capital

improvement expenditures). So cap rates of older buildings tend to be higher than those of otherwise similar

newer buildings. […] As buildings age, they tend to gradually evolve from ‘‘class A’’ quality to ‘‘class B’’

quality, with the resulting required expected investment return risk premium”97 In any case, if generally the

going-out cap rate should be greater than the going-in, one must to carefully consider the market conditions

at the time of purchase of the asset. During periods of rapid price expansion, cap rates tend to be low, as

noted in the Cap Rate Determinants paragraph, so using this value could lead to an underestimation of the

going-out cap rate and therefore overestimate the reversion value. If, on the other hand, the asset was

purchased in a phase of particularly high rents and now we are near the end of the contract in a market

characterized by lower rents, the going-in cap rate will be particularly large, leading to underestimating the

possible reversion value.98 A first possible approach for estimating the going-out cap rate consists in using

the depreciation rate to estimate the property's loss in value compared to today. Assuming a useful life of

70 years, the annual depreciation rate would be 1/70 = 1.43%. If the purchase price of the asset was 200

million with an income of 14 million, the going-in gross cap rate is 14/200 = 7%. After 10 years the

depreciation would bring the asset value to 200 * (1-1.43% * 10) = 200 * 0.857 = 171.4 million, bringing the

cap rate to 14 / 171.4 = 8.16%. As far as regards my case study, I will use the method applied by [23] M.

Hoesli and G. Morri and (2010) in which is estimated a spread between going-in and going-out cap rate. To

be able to quantify this spread, we need surveys that present cross-section analyzes about the different cap

97 [14] D. M. Geltner, N. G. Miller, J. Clayton e P. Eichholtz, Commercial Real Estate. Analysis & Investments, 2nd edition, Oncourse Learning, 2006. Pag. 244 98 Ibidem

79

rates of buildings of different ages that can be observed in a certain market at a given time. Since these

studies are not available for the Italian market, I decided to use a CBRE study for the first half of 2019 for

the US market. The data are shown in Table 35. The selected values are those of the office sector in suburban

contexts of the tier 1 cities. The hypothesis is that after 11 years the Vodafone Village moves from the high

end of the AA class to the low end of the A class or to the high end of the B class.

Table 35. U.S. cap rate survey - H1 2019. Suburban offices markets in Tier I cities

Class AA Class A Class B Class AA min-

Class A min

spread

Class AA min-

Class B min

spread Min Max Min Max Min Max

Boston 6,75 7,25 6,75 7,75 8,50 9,50 0,00 1,75

Chicago 6,75 7,25 7,50 8,75 8,75 10,50 0,75 2,00

Oakland 5,50 6,50 6,00 6,50 6,00 7,00 0,50 0,50

San Francisco 5,25 6,25 6,00 7,00 6,50 7,75 0,75 1,25

San Jose 5,25 6,50 6,00 6,75 6,50 7,75 0,75 1,25

New Jersey 6,00 6,50 7,00 7,50 8,25 8,75 1,00 2,25

Los Angeles 4,75 6,25 5,25 6,75 6,75 8,25 0,50 2,00

Orange Country 5,00 5,50 5,50 6,50 6,75 7,75 0,50 1,75

Seattle 5,25 5,75 6,00 6,50 7,00 7,50 0,75 1,75

Washington D.C. 5,25 6,00 6,50 7,25 7,50 8,50 1,25 2,25

Average 0,68 1,68

Source: author’s elaboration based on [24] CBRE, «U.S. Cap Rate Survey. Advanced Review H1 2019,» July 2019

For the estimation of the cash flows due to this variable, the NOI will be divided by the cap rate obtained

adding the spread calculated to the going-in cap rate. I chose to use the 1.68 percent spread for the

pessimistic scenario, the 0.68 percent for the optimistic one and the value of (0.68 + 1.68) / 2 = 1.18 for the

case base. As far as the Monte Carlo simulation is concerned, I will use the same as the average value used

in the case base and, assuming a gaussian distribution, I will use the value of (1.68-0.68) / 6 = 0.16% as the

standard deviation. As reported by COIMA RES in the 2018 annual report, the market value of the Vodafone

Village asset is 209.300 k with an annual income of 13.964 and net operating expenses of 1.009 k. This

means that the net going-in cap rate is equal to (13.964-1.009) / 209.300 = 6.19%, therefore the net going-

out cap rate used in the base case will be 6.19 + 1.18 = 7.37%

Table 36. Going-out cap rate spread estimation

Net

going-in

cap rate

Base case

spread

Best case

spread

Worst case

spread Distribution σ

Net going-

out cap

rate

6.19% 1.18% 0.68% 1.68% Gaussian 0.16% 7.37% Source: authors’ elaboration

3.2.8 - Discount Rate

As a discount rate I will use the WACC, therefore the risk-free rate, the cost of equity and the cost of debt

has to be estimated.

80

Risk-free

As already described in the Cost of Equity section in Part II, a risk-free proxy can be obtained simply as the

last available rate of short-term government bonds. For the case study I will use Italian BOTs with a maturity

of 12 months, whose average value in the fourth quarter of 2018 was 0.65 percent. The second method

proposed consists in calculating the average historical spread between bonds with 10-year maturity and

those with 12 months maturity, so as to detect the risk-premium on ten-year bonds, and therefore subtract

this spread from the current value of the ten-year bond to obtain the risk-free rate. This procedure is

reported in the excerpt from Table 37; the whole table can be consulted in Attachment 4. To estimate the

discount rate I will use the risk-free rate of 1.23.

Table 37. risk-free rate estimation using 10 years and 12 months Italian government bonds rates

Risk-free Rate Estimation BTP

10 yrs

BOT 12

month Spread

BTP

10 yrs

BOT 12

month Spread

BTP

10 yrs

BOT 12

month Spread

I 00 5,69 4,04 1,65 III 06 4,07 3,49 0,57 IV 12 4,67 1,72 2,95 II 00 5,56 4,63 0,93 IV 06 4,07 3,68 0,39 I 13 4,60 1,08 3,52

… … … … … … … … … … … … I 06 3,84 2,81 1,03 III 12 5,70 2,39 3,31 IV 18 3,14 0,65 2,49 II 06 4,30 3,17 1,13

Average 1,907

Risk-free 1,233

Note: 10 years bond rate are calculated averaging three end-month rate for each quarter. 12 months bond rate are calculated

averaging three mid-month (15th day) rate for each quarter.

Source: author’s elaboration based on Ministero dell’Economia e delle Finanze e Investing.com database

Cost of Equity

Per la stima del cost of equity ho introdotto tre diversi metodi nella parte II: il Constant Growth Perpetuity

Model, l’Historical Empirical Evidence e il metodo Security Market Line.

Method 1 - Constant Growth Perpetuity Model: this approach is based on the hypothesis that the annual

cash flows will grow at a constant rate from today up to infinity, causing the DCF formula to collapse in the

formula (32.1) given below.

𝑟𝑒 = 𝑁𝐶𝐹𝑡

𝑃𝑉+ 𝑔 (32.1)

This means that it is necessary to know the net cash flow of the first year as well as the market price; since

both values are known the only parameter that must be estimated is the growth rate of the NCF - g. Since

the final result of the cost of equity is extremely sensitive to the value of g, this estimate must be prudent

and made considering that it must be a long-term rate that includes both phases of growth and phases of

recession. The value of g used is the average of the market trend rates reported in Table 29, row “Effect

sum [%]”.

Net cash flows growth rate estimation re

2.77 2.27 1.77 0.77 0.27 -0.33 -0.43 -0.53 -0.63 -0.73 0.52

81

With regard to the present value, I refer to the market price of the asset estimated at 31 December 2018,

or € 209.3 million.99 NCF are calculated using the following formula:

𝑁𝐶𝐹 = 𝑅𝑒𝑛𝑡 𝑟𝑜𝑙𝑙 − 𝑛𝑒𝑡 𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑛𝑔 𝑒𝑥𝑝𝑒𝑛𝑠𝑒𝑠 − 𝑐𝑎𝑝𝑒𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑒𝑥𝑝𝑒𝑛𝑠𝑒𝑠

The rent roll for 2018, taken from COIMA's annual report for 2018, is € 13,964,000;100 net operating

expenses amounted to 1,009,000; the capex is assumed to be 2.36% of the present value, or 2.36% *

209,300,000 = € 4,939,480; to calculate the interest expenses use the interest rate of 2.05% as calculated

later in the Cost of Debt paragraph, assuming a loan to value of 35% and therefore a total debt of

73,255,000. The interest expense is 2.05% * 73.255.000 = € 1.501.727.

𝑁𝐶𝐹 = 13,964 - 1,009 - 4,939.48 - 1.501,73 = 6,513.79 thousand of €

𝑟𝑒 = 𝑁𝐶𝐹𝑡𝑃𝑉

+ 𝑔

𝑟𝑒 = 6,513.79

209,300+ 0.52% = 3.11% + 0.52% = 3.63%

Method 2 - Historical Empirical Evidence: the method is based on the application of the formula (16), for

which I will use the value of 1.23 percent for the risk-free estimate just above and the risk premium rate

whose estimation is performed as described in the excerpt contained in Table 38. The full table is reported

in attachment 5. The return rate given by the sum of risk free and risk premium is therefore 1.23 + 3.44 =

4.67%. Since the values of total returns have been calculated starting from market values, the return

obtained is a return on assets and not a return on equity and therefore it can be used as discount rate as is.

Table 38. Risk premium estimation with the Empirical Historical Method

Risk premium estimation using the Empirical Historical Method

BTP

10 yrs

BOT 12

month

Risk-

free

Market

Price

Cap

Rate

Capital

Returns

Total

Returns

Risk

Premium

A B C=A-B D E F=Dt+1/Dt-1 G=D+F H=G-C

SI - 2009 4,52% 1,38% 3,13% 2.300 € 6,40%

SII - 2009 4,06% 0,89% 3,17% 2.225 € 6,40% -3,26% 3,14% -0,03%

… … … … … … … … …

SI - 2017 2,22% -0,27% 2,48% 2.175 € 5,25% 0,00% 5,25% 2,77%

… … … … … … … … …

Average 2.47% 6.25% -0.34% 5.91% 3,44%

Note: 10 years bond rate are calculated averaging six end-month rate for each quarter. 12 months bond rate are calculated

averaging six mid-month (15th day) rate for each quarter. Cap rate value are taken from COIMA 2018 and refers to office sector

for good secondary districts. Market price are taken from Agenzia delle Entrate database and refers to the “D25” zone. Source: author’s elaboration

Risk-free Risk premium rE

1.23% 3.44% 4.67%

Method 3 - Security Market Line: this approach is based on the application of the formula (33), and since I

already know the value of the risk-free rate, the expected return rate of the market and the β must be

estimated. Let’s start with β. Usually to estimate the firm’s beta, is performed a regression of the scatter

plot obtained putting on the x axis the returns of a benchmark index used as proxy of the “market portfolio”

and on the y axis the firm’s returns. This operation is shown in Figure 43; the whole calculation are reported

99 [4, 61] COIMA RES, «Relazione finanziaria annuale,» 2018 100 Ibidem

82

in the Attachment 6. The return rates used are monthly, taken over a time period that goes from January

2017 to July 2019; the benchmarks used are the FTSE MIB index and the FTESE EPRA/NAREIT Developed

Europe index, published by the: “The FTSE EPRA Nareit Developed Europe Index is a subset of the FTSE EPRA

is designed to track the performance of listed real estate companies and REITS. By making the index

constituents free-float adjusted, liquidity, size and revenue screened, the series is suitable for use as basis for

investment products, such as derivatives and exchange traded funds (ETFs).”101 In practice, the index shows

the weighted performance of a basket of selected European companies in the real estate sector. The beta

values that emerge from this analysis are:

FTSE MIB EPRA Index

0.33 0.28

Source: author’s elaboration based on Investing.com database

Another way to estimate β is to refer to comparable companies. In Table 39 I reported the β values

contained in a database available on the internet (see Source of Table 39) which contains the βs estimated

for different sectors in Western Europe, both levered and unlevered.

Table 39. Average levered and unlevered β of a selected basket of firms in the real estate sector in western Europe

Industry Name Number of

firms Average

Unlevered Beta Average

Levered Beta

R.E.I.T. 238 0.41 0.68 Real Estate (Development) 18 0.87 1.19 Real Estate (General/Diversified) 11 1.33 1.36 Real Estate (Operations & Services) 59 0.95 1.35

Source: http://people.stern.nyu.edu/adamodar/New_Home_Page/dataarchived.html#region

101 [56] European Public Real Estate Association, «FTSE EPRA Nareit Developed Europe Index,» 2019

y = 0,332x + 0,0066R² = 0,1615

-10,00%

-5,00%

0,00%

5,00%

10,00%

15,00%

-15,00% -10,00% -5,00% 0,00% 5,00% 10,00%

CO

IMA

FTSE MIB

y = 0,2883x + 0,0074R² = 0,0622

-10,00%

-5,00%

0,00%

5,00%

10,00%

15,00%

-10,00% -5,00% 0,00% 5,00% 10,00% 15,00%

CO

IMA

FTESE EPRA/NAREIT Developed Europe

Figure 43. Regression analysis for β estimation using monthly returns of COIMA and FTSE MIB (left) and FTSE EPRA NAREIT Index

(right) – January 2017-July 2019

http://people.stern.nyu.edu/adamodar/New_Home_Page/dataarchived.html#region

83

Since companies have different capital structures, in this case it is better to refer to unlevered beta, as it is

not affected by the leverage, and then convert this value into a levered beta using the capital structure of

the company or the project you want to analyze. The formula for this operation is (34.1).

𝛽𝐸 = 𝛽𝐴 [1 +(1−𝜏)𝐷

𝐸] (34.1)

The sector most similar to that of COIMA RES is the first line "R.E.I.T.", therefore the value considered is

0.41. The value of βE will be calculated in the paragraph dedicated to the calculation of the WACC, since D

and E are not known for now.

The last possible approach is to refer to various finance-related internet sites, to see which beta values they

report for the company COIMA RES. The values of Yahoo Finance and Investing.com are shown in the table

below. The values are in line with those defined with the regression analysis reported in Figure 43. Taking

the average of these 4 values we obtain (0.33 + 0.28 + 0.21 + 0.24) / 4 = 0.26.

COIMA RES βE Yahoo Finance Investing FTSEMIB

regression EPRA regression Average

0.21 0.24 0.33 0.28 0.26

Now we need to estimate the excess return on the market. The first question is: should we use the excess

of returns from the real estate or securities market? As far as concerns this case study, I will use securities

market values, since beta measures the riskiness of a security compared to the stock market, allowing to

increase or reduce the return required according to the greater or lesser risk of a security compared to the

market average. Calculating the average return of the FTSE MIB compared to 12-month BOTs from 2002 to

2018, the calculated value of risk-free rate is -0.18 percent. Since this value makes no sense it will not be

considered.

Table 40. Average excess of return of FTSE MIB vs 12 months BOT - 2002-2018

FTSE MIB BOT 12 months Risk premium

2003 14,4% 2,21% 12,16% 2004 14,9% 2,17% 12,77%

2005 15,5% 2,23% 13,31%

2006 16,0% 3,29% 12,76%

2007 -7,0% 4,09% -11,04%

2008 -49,5% 3,77% -53,29%

2009 19,5% 1,14% 18,33%

2010 -13,2% 1,32% -14,55%

2011 -25,2% 3,22% -28,42%

2012 7,8% 2,32% 5,52%

2013 16,6% 0,97% 15,58%

2014 0,2% 0,48% -0,24%

2015 12,7% 0,07% 12,59%

2016 -10,2% -14,20% 4,00%

2017 13,6% -0,31% 13,93%

2018 -16,1% 0,16% -16,31%

Average 0,63% 0,81% -0,18% Note: returns are calculated using the end-month value of December

Source: author’s elaboration based on Investing.com database

84

Brealey et all (2012) in Principles of Corporate Finance 102 citing E. Dimson, PR Marsh, and M. Staunton,

Triumph of the Optimists: 101 Years of Investment Returns (Princeton, NJ: Princeton University Press, 2002)

report the risk premium of Italian stock exchange considering a period from 1900 to 2008. The value

obtained is 10.2 percent. Considering the very long-time horizon considered by Dimson et all, this value is

certainly more robust, and logically more solid, compared to the one I calculated, so I will use the value of

10.2 percent as the risk premium. Pending verification of COIMA's debt and equity market values, it is

possible to use the βE value of 0.26 to calculate a possible cost of equity using this approach. Assuming the

project is funded half with debt and half with equity, βE calculated should be as follow:

𝛽𝐸 = 𝛽𝐴 [1 +(1 − 𝜏)𝐷

𝐸]

𝛽𝐸 = 0.41 ∗ 2 = 0.82

𝑟𝐸 = 1.23 + 0.82(10.2 − 1.23) = 1.23 + 7.35 = 8.58%

Cost of Debt

The cost of debt is the yield to maturity of corporate debt. Since COIMA RES has never issued bonds, it is

not possible to observe the interest rate at which its debt is traded on the market, therefore the best thing

that can be done in this case is to look at the interest rate paid on all the debt of COIMA RES, considering

the book value. The total debt of COIMA RES at December 31, 2018 is € 306.12 million with an interest

payment of 6,295,000 €; the cost of debt is therefore 6,295,000 / 306,122,000 = 2.05%.

WACC

The formula for calculating the WACC is (42), given below


𝑉+ 𝑟𝑑(1 − 𝑇)

𝐷

𝑉

Since the tax rate of COIMA RES is zero thanks to its SIIQ status, the formula becomes:


𝑉+ 𝑟𝑑

𝐷

𝑉

All the values necessary for the calculation are known except E. Unlike debt, for the equity it is possible to

know the market value since COIMA RES is listed on the stock exchange. At 31 December 2018 the company

had issued 36,007,000 ordinary shares whose market price was € 7.10 for a total market capitalization of

36.007.000 * 7.10 = € 255.649,700.

Conceptually the WACC of the company is fine to evaluate a project if this is as risky as the company. “The

WACC for a firm reflects the risk and the target capital structure of the firm’s existing assets as a whole. As

a result, strictly speaking, the firm’s WACC is the appropriate discount rate only if the proposed investment

is a replica of the firm’s existing operating activities. In broader terms, whether or not we can use the firm’s

WACC to value a project depends on whether the project is in the same risk class as the firm.”103

The LTV is 35%, therefore D = 35, E = 65, V = 100. I will calculate the value of WACC for each method used

to estimate the cost of equity. The parameter values are summarized in the table below:

102 [17] R. A. Brealey, S. C. Myers e F. Allen, Principles of Corporate Finance, 10th edition, The McGraw-Hill Companies, Inc., 2011. Pag 161 103 [11] S. A. Ross, R. W. Westerfield e B. D. Jordan, Fundamentals of Corporate Finance, 10th edition, The McGraw-Hill Companies, Inc., 201. Pag. 462-463

85

ra/re D/V E/V rd

Method 1, 2, 3 0.35 0.65 2.05%

Method 1:

rE = 3.63%

𝑊𝐴𝐶𝐶 = 3.63% ∗ 0.65 + 2.05% ∗ 0.35 = 3.07%

Method 2: returns observed on the market referring to the property itself are clearly returns on the asset.

Since the WACC measures the required returns on the entire asset, this value can already be used as a

discount rate.

𝑟𝐴 = 𝑊𝐴𝐶𝐶 = 4.67%

Method 3:

Using the firm’s β of 0.26 the result is:

𝑟𝐸 = 𝑟𝑓 + 𝛽𝐸(𝑟𝑀 − 𝑟𝑓)

𝑟𝐸 = 1.23 + 0.26(10.2 − 1.23) = 1.23 + 2.33 = 3.56%

𝑊𝐴𝐶𝐶 = 3.56% ∗ 0.65 + 2.05% ∗ 0.35 = 3.03%

Using the firm’s β of 0.41 the result is:

𝛽𝐸 = 𝛽𝐴 [1 +(1 − 𝜏)𝐷

𝐸]

𝛽𝐸 = 0.41 [1 +35

65] = 0.41 ∗ 1.54 = 0.63

𝑟𝐸 = 1.23 + 0.63(10.2 − 1.23) = 1.23 + 5.65 = 6.88%

𝑊𝐴𝐶𝐶 = 6.88% ∗ 0.65 + 2.05% ∗ 0.35 = 5.19%

Come tasso di sconto utilizzerò la media di questi quattro valori, come riportato in Tabella 40.

Table 41. Different methods WACC estimation and average

WACC estimation

Method 1 Method 2 Method 3a Method 3a Average

3.07% 4.67% 3.03% 5.19% 3.99% Source: author’s elaboration

86

3.3 - Project Analysis

Once the necessary assumptions have been made to estimate the cash flows, it is possible to proceed with

the preparation of the DCF.

3.3.1 – Most Likely DCF

Table 42. Asset general information

General Figures

Location Lorenteggio street, 240, Milan Year of construction 2012 Year of acquisition 2016

Surfaces

Commercial surface 46,323 mq Occupancy 100%

Economics

2018 market value 209.3 mln 2018 rent roll 13.964 mln

Contract

The contract is a 15+6, from 2012 to 2027, year in which there is a break-option. The tenant cannot terminate the contract until 2027, otherwise it must pay the full amount of the fees not paid from the withdrawal upon

the expiry of the contract. Source: author’s elaboration

Table 43. DCFA assumption and formulas

DCFA assumptions

Initial investment A=209,300 k

ERV 2018 B=13,964 k growth rate in row Effect sum, Table 29

Potential gross income C=13,964 k indexed at the annual inflation of 1.2%

Vacancy length D=5 quarters

Vacancy allowance E

Effective gross income F=C-E

Insurance G=0.46%C

IMU and TASI H=5.00%C

Property management I=1.83%F

Utilities L=8.19%F

Maintenance M=X%C X = rate in Table 32 from year 8 to 19

Total operating expenses N=G+H+I+L+M

Recovery from tenants O=N*65%

Net operating expenses P=N+O

Net operating income Q=F-P

Capex R=Y%C Y = rate in Table 34 from year 8 to 19

Exit cap rate S=entry net yield+sprd sprd = Table 36, column Base case

spread Reversion value T=Q/S

Brokerage commissions U=3%T

Property before tax cash flows V=Q-R+T-U

Net cash flows Z=V COIMA ‘s tax rate = 0 Source: author’s elaboration

87

The assumptions made are that the property is purchased on 1 January 2019 at € 209.3 million with a rent

roll of 13,964 thousand €, for a duration of 8 years until 1 January 2027 in which there is a break-option and

therefore the possibility of the tenant to leave the contract or renew it for another 6 years. The assumption

made in the Vacancy Allowance paragraph is that if the ERV at the time of the break-option is less than the

rent roll, the tenant exits the contract. If the base case this situation occurs, and the asset is unoccupied for

a period equal to the expected duration of the vacancy, which for the assumptions described in the Vacancy

Length paragraph is 4.67 quarters, rounded to 5. The net cash flows are discounted using a 3.99% rate, as

calculated in the WACC paragraph.

The outcome of the DCF analysis is reported in the last two lines: the NPV has a positive value of 18,330,000

€, and the IRR is 5.05%. According to the rules of these two performance indicators, investment create value

for the company and should be accepted. The complete Most Likely DCF is reported in the Attachment 7.

Table 44. Most Likely DCF excerpt

“MOST LIKELY” DCF

0 1 … 9 10 11 12

Thousand of € 01/01/2018 01/01/2019 … 01/01/2027 01/01/2028 01/01/2029 01/01/2030

Initial investment -€ 209.300 …

ERV growth rate - Trend effect 3,00% … -0,40% -0,50% -0,60% 0,00%

ERV 13.964 € 14.351 € … 14.806 € 14.698 € 14.576 € 14.543 €

Potential gross income 13.964 € 14.132 € … 15.079 € 14.984 € 15.164 € 15.346 €

Vacancy allowance … 15.079 € 3.746 €

Effective gross income 14.132 € … 0 € 11.238 € 15.164 € 15.346 €

Operating expenses …

Insurance -65 € … -69 € -69 € -70 € -71 €

IMU and TASI -897 € … -957 € -951 € -963 € -974 €

Property management -259 € … 0 € -206 € -277 € -281 €

Utilities -1.157 € … 0 € -920 € -1.242 € -1.257 €

Maintenance/ERV 3,24% … 1,30% 0,44% 2,14% 1,57%

Maintenance -458 € … -196 € -66 € -325 € -241 €

Total operating expenses -2.836 € … -1.223 € -2.212 € -2.877 € -2.824 €

Recovery from tenants 1.844 € … 795 € 1.438 € 1.870 € 1.835 €

Net operating expenses -993 € … -428 € -774 € -1.007 € -988 €

Net operating income 13.139 € … -428 € 10.464 € 14.157 € 14.357 €

Capex/ERV 0,00% … 0,00% 0,00% 0,00% 0,00%

Capex 0 € … 0 € 0 € 0 € 0 €

Exit cap rate … 7,37%

Reversion value … 194.808 €

Brokerage commissions … 3,00%

Property before tax cash flows -209.300 € 13.139 € … -428 € 10.464 € 203.121 €

Taxes 0 € … 0 € 0 € 0 €

Net cash flows -209.300 € 13.139 € … -428 € 10.464 € 203.121 €

Discounted cash flows @ WACC -209.300 € 12.635 € … -301 € 7.076 € 132.083 €

NPV 18.330 € …

IRR 5,05% …


3.3.2 - Applied Scenario Analysis

For the scenario analysis I will vary the values of the cash flows assuming a best case and a worst case. To

be effective, this method of analysis needs the values of the "likely" variables, in the sense that the ranges

of variation between the worst and the best case must refer to historical values in order not to risk

hypothesising impossible scenarios, both in positive and in negative. If, for example, market rents in the last

cycle have contracted by 2 per cent on average per year, for 4 years, and then returned to growth of 3 per

88

cent a year, assume that prices fell by 10 per cent per year in the worst case can lead to considering unlikely,

if not impossible, eventualities. In short, it is necessary to have reliable historical data to carry out an

effective analysis scenario. Below are the assumptions about the cash flow values in the different scenarios.

ERV Trend

The hypothesis made to find a pessimistic and an optimistic scenario, is that each year the basic trend (that

is ERV trend used in the Most Likely DCF – Table 44 and Attachment 6) of ERV growth rates takes a lower or

higher value than a standard deviation. In other words, if the base case trend is the average, for the

pessimistic scenario series, values equal to the average minus one standard deviation are considered; for

the optimistic scenario one standard deviation is added to the "average trend". This hypothesis is formalized

in equations 55 and 56.

𝑡% 𝑝𝑒𝑠𝑠𝑖𝑚𝑖𝑠𝑡𝑖𝑐 = 𝑡% − 𝜎 (55)

𝑡% 𝑜𝑝𝑡𝑖𝑚𝑖𝑠𝑡𝑖𝑐 = 𝑡% + 𝜎 (56)

t%: ERV growth rate trend in the most likely scenario. The series is reported in the row “Most Likely” – Table

45, section “Trend”.

σ: annualized volatility of the south-west Milan periphery sub-market, classified as “D25” in the Agenzia

delle Entrate database. The value is obtained through the calculation reported in Table 45.

At the trend value I added the expected inflation and the effect of degradation, as I do for the calculations

of the most likely case. That said, the assumptions about growth rates in the different scenarios are shown

in Table 45.

Table 45. Different scenarios ERV trend calculations

0 1 2 3 4 5 6 7 8 9 10 11

Inflation – A 1,20 1,20 1,20 1,20 1,20 1,20 1,20 1,20 1,20 1,20 1,20

Degradation - B -1,43 -1,43 -1,43 -1,43 -1,43 -1,43 -1,43 -1,43 -1,43 -1,43 -1,43

Trend - C

Most Likely 3,00 2,50 2,00 1,00 0,50 -0,10 -0,20 -0,30 -0,40 -0,50 -0,60

Best case 7,92 7,42 6,92 5,92 5,42 4,82 4,72 4,62 4,52 4,42 4,32

Worst case -1,92 -2,42 -2,92 -3,92 -4,42 -5,02 -5,12 -5,22 -5,32 -5,42 -5,52

Effect sum – D= A+B+C

Most Likely 2,77 2,27 1,77 0,77 0,27 -0,33 -0,43 -0,53 -0,63 -0,73 -0,83

Best case 7,69 7,19 6,69 5,69 5,19 4,59 4,49 4,39 4,29 4,19 4,09

Worst case -2,15 -2,65 -3,15 -4,15 -4,65 -5,25 -5,35 -5,45 -5,55 -5,65 -5,75

Index - Et=Et-1(1+D)

Most Likely 1 1,02 1,05 1,07 1,07 1,08 1,07 1,07 1,06 1,06 1,05 1,04

Best case 1 1,07 1,15 1,23 1,3 1,36 1,43 1,49 1,5 1,62 1,69 1,76

Worst case 1 0,97 0,95 0,9 0,88 0,84 0,79 0,75 0,71 0,67 0,63 0,60


89

Operating Expenses and Capex

For operating expenses and capex, I assume that the costs may undergo variations with respect to the

estimated values; in the best case the variations are 15 percent lower and in the worst case 15 percent

higher. Insurance and taxes never change.

Table 46. Annualized standard deviation calculation for the sub-market "D25" OMI zone in Milan

D25 OMI zone, Milan, average rent H1 2006 - H1 2017

Period Average rent Semestral

returns Period

Average rent

Semestral returns

HI - 2006 8 HII - 2012 7,7 0,00% HII - 2006 7,5 -6,25% HI - 2013 7,7 0,00% HI - 2007 8,5 13,33% HII - 2013 7,7 0,00% HII - 2007 8,5 0,00% HI - 2014 7,5 -2,60% HI - 2008 8,5 0,00% HII - 2014 7,5 0,00% HII - 2008 8,5 0,00% HI - 2015 7,5 0,00% HI - 2009 8,5 0,00% HII - 2015 7,5 0,00% HII - 2009 8,25 -2,94% HI - 2016 7,5 0,00% HI - 2010 8 -3,03% HII - 2016 7,5 0,00% HII - 2010 8 0,00% HI - 2017 7,5 0,00% HI - 2011 7,7 -3,75% Average -0,24% HII - 2011 7,7 0,00% St dev 3,48% HI - 2012 7,7 0,00% Annualized st dev 4,92%

Note: Average rent is obtained averaging the minimum and maximum values of monthly rent.

Source: author’s elaboration based on Agenzia delle Entrate database

Reversion value

As already described in Table 36 the value of the going-out cap rate in the best case is 6.87 percent, while

in the worst case of 7.87 percent.

Scenario analysis results

Figure 44. NPV and IRR for the three scenarios


€ 18.330

€ 51.860

-€ 57.612-80.000 €

-40.000 €

0 €

40.000 €

80.000 €

Base Best Worst

NPV

5,05%

6,76%

-0,41%-2,00%

0,00%

2,00%

4,00%

6,00%

8,00%

Base Best Worst

IRR

90

Figure 44 shows the result of the scenario analysis. First of all, it should be noted that the pessimistic

scenario has a greater deviation than the base case of the optimistic scenario; this is due to the fact that

the base scenario incorporates bullish expectations of market rent. However, the scenario analysis turns

out to be really effective if the ranges used for the calculations in the two scenarios are plausible values

extrapolated from empirical observations, in order to don’t risk of considering too pessimistic scenarios or,

even worse, too optimistic ones. What is the probability that the variables act as described in the worst or

in the best scenario? The lack of happening probability of a scenario is the biggest limitation of this method.

A useful information that can be obtained from this analysis is the range of possible scenarios, making the

difference between the best and the worst scenario. The complete DCFs of the Best and Worst scenarios

are consultable in Attachment 8.1 and 8.2.

3.3.3 - Applied Sensitivity Analysis

Considering the DCF variables, those that can be varied in order to identify the sensitivity of NPV and IRR to

their variation, are the following:

Sensitivity analysis variables

Initial investment

ERV growth rate

Vacancy allowance (vacancy length)

Total operating expenses

Recovery from tenants

Capex

Exit cap rate

Initial investment

These variables will vary from +30 to -30 percent with 10 percent steps. For the ERV trend I varied a

coefficient with which I multiplied the percentages of annual growth rates; the number of quarters of

vacancy length was changed using the "half-quarter" as the minimum unit. The length of the vacancy has

important consequences on the value of the disinvestment, as it affects the value of the rent roll of the new

lease derived from the ERV prices in the year preceding the end of the period of vacancy. For total operating

expenses, as well as for capital expenditure, I varied the total amount of expenditure year by year; in the

recovery from tenants I changed the percentage of total operating expenses to be repaid; for the exit cap

rate I acted on the capitalization rate; for the initial investment I made the total expenditure vary. The results

of these calculations are shown in Table 47. Figures 45 and 46 show the ranking of the risky variables

produced considering the range of +30 and -30 per cent for the NPV and the IRR respectively; variables with

larger ranges indicate a greater sensitivity of the NPV and the IRR to the unitary variation of these variables,

which therefore contain a greater forecasting risk. In order to obtain the tornado charts, I classified the NPV

changes due to the variation of +30 and -30 per cent with respect to the basic NPV as positive or negative,

depending on whether they caused an increase or a reduction in the NPV. In Figures 45 and 46 (values used

for the preparation of the two tornado charts are shown in Attachment 9) the variables are ordered from

the most to the least risky, using as indicator the overall range given by the sum of the absolute value of the

two variations. Some observation can be made:

• the two most critical variables are the purchase price and the selling price of the asset, which

provides extremely useful information: the capital gain component remains fundamental also for

the asset class real estate, despite being considered as a type of income-based investment.

• As for the capex, it is worth remembering that, as shown in Table 34, the high expenses are placed

between the 15th and the 25th year after the construction of the building; this means that due to

91

the fact that the building was recently built the risk due to this variable is limited. When an investor

is comparing many different assets, the age is factor that cannot be neglected.

• NPV is quite sensitive with respect to recovery from tenants and this should kept in mind when the

landlord negotiates the terms of the contract with the tenant: gives incentives to the tenant through

lower recovery on the operative expenses has a significant weight on the final performances.

• NPV is not sensitive to ERV trend is not because the market is “suspended” through the contract,

that allows to eliminate the uncertainty related to the price fluctuations.

• The more the vacancy is close to the starting period of the investment the more it has a relevant

impact on the NPV. In my case study the break-option is 9th year of the investment over an holding

horizon of 11 years. Sensitivity analysis cannot find this relationship, but a landlord must knows that

the more he place break-options far from the present, the more him NPV will be higher.

Table 47. Sensitivity analysis summary

Trend ERV Vacancy lenght Total operating expenses Recovery from tenants NPV IRR NPV IRR NPV IRR NPV IRR

30% € 21.611 5,22% 30% € 14.664 4,84% 30% € 13.221 4,76% 30% € 28.062 5,57%

20% € 20.512 5,16% 20% € 15.886 4,91% 20% € 14.924 4,85% 20% € 24.656 5,40%

10% € 19.418 5,10% 10% € 17.108 4,98% 10% € 16.627 4,95% 10% € 21.736 5,22%

1 € 18.330 5,05% 5 € 18.330 5,05% -€ 2.836 € 18.330 5,05% 0,65 € 18.330 5,05%

-10% € 17.248 4,99% -10% € 19.552 5,11% -10% € 20.033 5,14% -10% 15.411 € 4,87%

-20% € 16.172 4,93% -20% € 20.775 5,18% -20% € 21.736 5,24% -20% € 12.005 4,69%

-30% € 15.101 4,87% -30% € 22.053 5,39% -30% € 23.440 5,33% -30% € 9.085 4,50%

Capex Exit cap rate Initial investment NPV IRR NPV IRR NPV IRR

30% € 17.379 4,99% 30% -€ 10.016 3,35% 30% -€ 44.460 1,82% 20% € 17.696 5,01% 20% -€ 2.103 3,86% 20% -€ 23.530 2,78% 10% € 18.013 5,03% 10% € 7.118 4,41% 10% -€ 2.600 3,85%

- € 18.330 5,05% 7,37% € 18.330 5,05% - € 18.330 5,05% -10% € 18.647 5,06% -10% € 32.045 5,77% -10% € 39.260 6,41% -20% 18.964 € 5,08% -20% € 48.945 6,58% -20% € 60.190 8,00% -30% € 19.281 5,10% -30% € 70.958 7,56% -30% € 81.120 9,88%


Figure 47 shows how the variables move at each step; Initial investment and exit cap rates have been placed

in a separate chart to be able to appreciate the variations of the other variables, as they are on a much

smaller scale. This representation is useful because allow to check for non-linear trend, as the exit cap rate

shows. The non-linearity of the variation of a risky variable is an information that you want to know, because

if it accelerates can cause loss greater than expected if a linear dynamic was assumed. In short, assuming a

linear relationship between the variable and the NPV variation can lead to severe underestimation of risk.

The ERV trend variable could assume a non-linear trend if the ERV were greater than the rent roll in the

year of the break-option, because if the option is exercised by the tenant is created a great discontinuity in

the series of potential gross income. This point could be underestimated because the low sensitivity of the

NPV to the vacancy length, but the big loss comes from the interrelation between the ERV and the reversion

value. Since the sensitivity analysis by definition cannot see this interrelation, this topic will be discussed in

more detail in the Monte Carlo Simulation paragraph.

92

Figure 45. NPV tornado chart with +30% and -30 % variation

Note: the base value of NPV are been subtracted from +30% and -30% values of NPV to find “Positive” and “Negative” variations. A

variation is been considered “positive” if the NPV increase, vice versa was considered “negative”


Figure 46. IRR tornado chart with +30% and -30 % variation

Note: the base value of NPV are been subtracted from +30% and -30% values of NPV to find “Positive” and “Negative” variations. A

variation is been considered “positive” if the NPV increase, vice versa was considered “negative”


€ 951

€ 3.281

€ 3.723

€ 5.109

€ 9.732

€ 52.628

€ 62.790

-€ 951

-€ 3.229

-€ 3.666

-€ 5.109

-€ 9.245

-€ 28.346

-€ 62.790

-€ 80.000 -€ 60.000 -€ 40.000 -€ 20.000 € 0 € 20.000 € 40.000 € 60.000 € 80.000

Capex

Trend ERV

Vacancy lenght



Exit cap rate

Initial investment

NPV

Positive

Negative

0,05%

0,18%

0,34%

0,29%

0,53%

2,51%

4,84%

-0,05%

-0,18%

-0,20%

-0,29%

-0,54%

-1,69%

-3,22%

-4,00% -3,00% -2,00% -1,00% 0,00% 1,00% 2,00% 3,00% 4,00% 5,00% 6,00%

Capex

Trend ERV

Vacancy lenght



Exit cap rate

Initial investment

IRR

Positive

Negative

93

Figure 47. Sensitivity of risky variables

Note: ΔNPV on the y-axis is calculated subtracting the base case NPV value from the value obtained by varying the parameter.


3.3.4 - Applied Break-even Analysis

The break-even analysis makes it possible to identify how much a variable must worsen or improve to ensure

that the performance indicator used reaches the break-even point. In accounting the break-even occurs

when the net profit is zero, in this case I consider the break-even as the point at which the NPV is nulled or

when the IRR becomes equal to the risk-free, indicating the cancellation of the risk premium obtained by

the investor. The results are shown in Table 48. Some observation can be made:

-€ 80.000

-€ 60.000

-€ 40.000

-€ 20.000

€ 0

€ 20.000

€ 40.000

€ 60.000

€ 80.000

- 3 0 % - 2 0 , 0 0 % - 1 0 , 0 0 % 0 , 0 0 % 1 0 , 0 0 % 2 0 , 0 0 % 3 0 , 0 0 %

ΔN

PV

VARIABLE % VARIATION

Initial investment Exit cap rate

-€ 15.000

-€ 10.000

-€ 5.000

€ 0

€ 5.000

€ 10.000

€ 15.000

- 3 0 % - 2 0 , 0 0 % - 1 0 , 0 0 % 0 , 0 0 % 1 0 , 0 0 % 2 0 , 0 0 % 3 0 , 0 0 %ΔN

PV

VARIABLE % VARIATION

Recovery from tenants Total operating expenses Vacancy lenght

Trend ERV Capex

94

• ERV trend in the first years of the holding period has to be bearish to set to zero the NPV. Knowing

that an investor can decide of don’t invest if he or she has a slightly negative expectation about the

space market for the for the years immediately following the purchase.

• Operating expenses has to be twice as high as the DCF to cancel the NPV. This give a good margin

of safety on this item forecast.

• If recovery from tenants are higher than 0.27 per cent of total operating expenses the NPV should

be positive. This information can be used for set the level of incentives give to the tenants on this

particular item.

• Capex has to be more than six time as forecast in the base case to cancel the NPV. This is due to the

fact that the asset is new and it doesn’t require high capex.

• As far as regard the initial investment, you know that you can pay up to -€ 227.630 before making 0

the NPV.

Looking at this data you must know that variables are considered one at a time, and this kind of analysis

completely ignore the sum of the effects of unfavorable movements of revenues and cost. Combinations of

effect can be considered using the next proposed technique, that is Monte Carlo simulation.

Table 48. Break-even analysis

Most Likely NPV break-even IRR = risk-free

Trend ERV (*) 1 -0,77 -5,31 Vacancy length 5 quarters > 3 years > 3 years Total operating expenses (*) 1 2,07 4,64 Recovery from tenants (*) 0,65 0,27 -0,63 Capex (*) 1 6,78 21,47 Exit cap rate 7,37% 8,66% 13,77% Initial investment -€ 209.300 -€ 227.630 -€ 284.699

Note: (*) Values of this variable in the base case are multiplied by the factor in the column “Most Likely” of this table. Of course,

when the factor is 1 the variable doesn’t change and is the same as is in the base case.


3.4 - Run the Monte Carlo Simulation

To run a Monte Carlo simulation it is necessary first of all to describe the DCF variables as probability

distributions, therefore it is necessary to define the type of distribution and the parameters that are to be

adopted.

ERV trend: for this variable I refer to the paragraph Effects Sum in which I described how I intend to use the

Brownian diffusion as in the model proposed by C.O. Amédée ‐ Manesme et al (2012). The average of the

price is defined by the trend, while at each iteration a value is extracted from a Gaussian distribution with

mean 0 and standard deviation 1, which is then multiplied by the standard deviation of 4.92 per cent, whose

calculation is reported in Table 46.

Vacancy length: the method by which the duration of the vacancy is modelled has been described in the

Vacancy Length paragraph. The triangular distribution used has a mode of 3, with a range from the minimum

of 1 to a maximum of 12.

Vacancy allowance: the algorithm is described in the Vacancy Allowance paragraph. The vacancy allowance

assumes values greater than zero only if the break-option is exercised. This means that if in the year in which

the tenant can exercise the break-option, the ERV is greater than the rent roll, the option is not exercised,

therefore the vacancy has a length equal to 0 and there is no vacancy allowance. If, on the contrary, in the

95

year of the break-option the ERV is less than the rent roll, the option is exercised and the space remains

vacant for a duration defined by the value extracted from the probability distribution of vacancy length. In

the periods in which there is vacancy, the vacancy allowance is calculated based on the potential gross

income of the period, or the ERV of the previous year; the assumption made is that since there is no longer

a contract, the PGI would be equal to the rent roll of a new contract, which is set on the basis of the nearest

consolidated market prices, that is the ERV of the previous year increased by inflation.

Total operating expenses: this variable is modelled by assuming a gaussian distribution with the values used

in the DCF most likely base as average and with a standard deviation of 10 percent.

Recovery from tenants: the hypothetical distribution is triangular with a maximum value of 70 per cent of

total operating expenses, a mode of 65 and a minimum of 50 per cent. These choices are based on the

hypothesis that there is a maximum limit beyond which the tenant does not intend to go, while on the

contrary it is possible that the owner grants more moderate repayments as an incentive.

Capex: for this variable the idea is that, for the data available to me, the expected values are very uncertain;

the probability that they are greater or less, even considerably, compared to those used in the most likely

case is high. All values that vary between +30 and -30 percent from del base case are equally possible, so I

decided to use a uniform distribution, with a minimum value given by the DCF most likely case capex

reduced by 30 percent and a maximum value gives by the base DCF capex increased by 30 percent.

Exit cap rate: the idea is that there is uncertainty about the relative building quality of the asset compared

to the other buildings that will be present on the market in 10 years and it is assumed a range of increase

of the going-in cap rate that allows to consider a more or less strong degradation. The topic was discussed

in more detail in the Reversion Value paragraph, with specific reference to Tables 35 and 36. The probability

distribution is Gaussian, the average value of the net going-out cap rate is 7.37 percent and the standard

deviation is 0.16 percent.

Table 49 summarizes the hypotheses described above.

Table 49. Assumptions on variables' distribution and parameters

Distribution Parameters

Trend ERV Gaussian Mean: ERV

trend St dev.: 4.92%

Vacancy length Triangular Min: 1 Mode: 3 Max: 12

Vacancy allowance Algorithm described in the paragraph ”Vacancy Allowance”

Total operating expenses Gaussian Mean: Base

case DCF values St dev: 10%

Recovery from tenants Triangular Min: 0.50% Mode: 0.65% Max: 0.70%

Capex Uniform Min: Base case

DCF values – 30% Max: Base case DCF

values + 30%

Exit cap rate Gaussian Mean = 7.37% St dev: 0.16%


Figure 48 shows the results of the simulation. How it is possible to observe the distribution of objective

functions is bimodal. The data of NPV and IRR bimodal distributions obtained from the Monte Carlo

simulation are reported in Attachment 10.1 and 10.2. To understand the reason for this "mix" I tried to run

the simulation by removing one variable at a time, being able to observe that the key factor is the break-

option: when you arrive at the year in which it is possible to exercise the break-option they create two

distinctly separate scenarios, one in which the option is exercised and one in which it is not; the first case is

unfavourable due to the vacancy period, the lower rent value set in the contract once a new tenant has

been found and for the negative capital gains obtained with reversion value. In the second case the contract

continues with the previous rent roll until the disposal of the asset. In Figure 49 are represented separately

96

the frequency distributions of the two scenarios. The case in which the option is not exercised seems always

profitable even if this safety has an opportunity cost, in fact you have to give up the potential upside of the

revenues that comes from a bullish market: when the break-option year arrives, if the ERV is bigger than

the contract, the tenants stays in the space to maximize its wealth and the landlord loses the opportunity

to sign a contract in a favourable market condition. Printing the two distributions in the same graph gives

the Figure 50, whose data obtained from the Monte Carlo simulation are reported in Attachment 11.1 and

11.2.


Figure 49. Monte Carlo simulation NPV frequency distribution of "exercised" (left) and "not exercised break-option"

(right)

0

100

200

300

400

500

600

700

800

900

1000

-0,6

3%

-0,2

7%

0,1

0%

0,4

6%

0,8

3%

1,2

0%

1,5

6%

1,9

3%

2,2

9%

2,6

6%

3,0

3%

3,3

9%

3,7

6%

4,1

2%

4,4

9%

4,8

6%

5,2

2%

5,5

9%

5,9

5%

6,3

2%

6,6

9%

Freq

uen

cy

IRR

0

100

200

300

400

500

600

700

-€ 5

4.33

2

-€ 4

8.96

2

-€ 4

3.59

2

-€ 3

8.22

3

-€ 3

2.85

3

-€ 2

7.48

3

-€ 2

2.11

3

-€ 1

6.74

3

-€ 1

1.37

3

-€ 6

.003

-€ 6

34

€ 4.

736

€ 10

.106

€ 15

.476

€ 20

.846

€ 26

.216

€ 31

.585

€ 36

.955

€ 42

.325

€ 47

.695

€ 53

.065

Freq

uen

cy

NPV

Figure 48. Monte Carlo simulation outcomes distributions – NPV (left) and IRR (right) – 20,000 thousand of iterations

0

100

200

300

400

-€ 6

4.95

4

-€ 5

5.56

8

-€ 4

6.18

3

-€ 3

6.79

7

-€ 2

7.41

2

-€ 1

8.02

6

-€ 8

.640

€ 74

5

€ 10

.131

€ 19

.516

€ 28

.902

€ 38

.288

€ 47

.673

€ 57

.059

Freq

uen

cy

NPV

Exercised break-option

0

50

100

150

200

250

300

€ 23

.195

€ 25

.305

€ 27

.414

€ 29

.524

€ 31

.633

€ 33

.742

€ 35

.852

€ 37

.961

€ 40

.070

€ 42

.180

€ 44

.289

€ 46

.398

€ 48

.508

Freq

uen

cy

NPV

Not exercised break-option

97

Figure 50. NPV bimodal distribution given by the two scenarios sum


Once separated into two scenarios, you can use the classic statistical tools for analysing the distributions,

assuming they are gaussian. From the simulation results it is possible to calculate the probability of the two

scenarios, dividing the number of iterations that goes under each scenario by the total number of iterations.

The "with break-option" scenario occurs in the 62 per cent of cases, rounding off; the "without break-

option" scenario in the remaining 38 per cent.

“Exercised” scenario

“Not exercised” scenario

Total

12,321 7,679 20,000

61.61% 38.40% 100%

0

100

200

300

400

500

600

700

-$ 4

7,2

50

-$ 4

3,5

00

-$ 3

9,7

50

-$ 3

6,0

00

-$ 3

2,2

50

-$ 2

8,5

00

-$ 2

4,7

50

-$ 2

1,0

00

-$ 1

7,2

50

-$ 1

3,5

00

-$ 9

,750

-$ 6

,000

-$ 2

,250

$ 1,

500

$ 5,

250

$ 9,

000

$ 12

,750

$ 16

,500

$ 20

,250

$ 24

,000

$ 27

,750

$ 31

,500

$ 35

,250

$ 39

,000

$ 42

,750

$ 46

,500

Alt

ro

Freq

uen

cy

NPV

not exercised break-option exercised break-option

98

Conclusions

The hypotheses from which this paper is born are:

1. The "deterministic" DCF on its own is not sufficient either as an instrument for evaluating an asset or

as an instrument for evaluating an investment. It is necessary to have a probability distribution of the

price and of the performances of the investment to be able to make rational choices that take into

consideration the relationship between returns and risk.104

2. By representing performances as probability distribution it is possible to quantify the risk of an asset.

Table 50 shows the results of the NPVs obtained from the Monte Carlo simulation and those of the

deterministic DCF, in which are also reported the values of the scenario analysis. Some observation can be

made:

• The deterministic DCF provides less data from which to extract useful information for the decision-

making process. With the sensitivity analysis, the risk of the individual variables can be obtained but

not the overall risk.

• In the deterministic DCF it is not known how much the investment is likely to give the expected

result. As described in the section Risk and Returns Relationship, given the same expected returns,

a riskier asset should cost less than an asset whose returns are more stable and predictable with

low error margins. If used as an investment valuation tool, the deterministic DCF leads to not

considering the risk and therefore overpricing an asset. This is the main reason why hypothesis 1 is

true: if the risk is not considered in the evaluation of an investment, it is not respected “the most

fundamental point in the financial economic theory of capital markets: that expected returns are

(and should be) greater for more risky assets.”105

• Using the Monte Carlo simulation it is possible to find probability of the occurrence of the two

scenarios. This allows to correctly quantify the expected value of the investment, that is the

weighted sum of the expected values of the two scenarios.

• With the Monte Carlo simulation it is possible to answer the question what is the probability that

the project is successful (NPV bigger than 0)? To do this it is sufficient to count the number of

iterations produced by the Monte Carlo simulation that return an NPV value greater than 0,

considered as the target of the investment. If the “with break-option” scenario occurs - 61.61% of

cases according to expectations on ERV and on the assumptions on which the model is based - there

is a 53.83% probability that the project will give an NPV greater than 0 , while in the "without break-

option" scenario, which occurs in 38.40% of cases, the probability of success is 100%. By applying

the formula (55) the overall probability of success of the project can be obtained, which is equal to

71.56%.

• The risk-return ratio, calculated with the coefficient of variation – formula (27) – of the break-option

scenario is 110 times greater than the scenario without break-option.

• Having the probability distribution, is possible to answer to the question how much money can be

loss if the investment would go wrong? To answer can be considered a certain confidence interval,

in this case 95%. The calculation can be performed either using the data obtained from the

simulation by sorting them in ascending order and taking the value in position (5% * # iterations)-

104 [54] The major traditional valuation methods, widely accepted by practitioners and academics, are: the cost approach, the income approach (discounted cash-flow) and the market approach. However, these traditional valuation methods suffer from many limitations. In particular, they all suffer from the same inherent disadvantage: they do not take proper account of risk, and they are too sensitive to specific parameters, such as the infinite growth rate of the cash flow.” C. Amédée‐Manesme, F. Barthélémy, M. Baroni e E. Dupuy, «Combining Monte Carlo simulations and options to manage the risk of real estate portfolios,» Journal of Property Investment & Finance, vol. 31, n. 4, pp. 360-389, 2013 105 [14] D. M. Geltner, N. G. Miller, J. Clayton e P. Eichholtz, Commercial Real Estate. Analysis & Investments, 2nd edition, Oncourse Learning, 2006. Pag. 186

99

th, or it can be assumed that the distribution is normal and use the formula µ - 1.65σ. in Table 50 I

used this last method. If the break-option occurs, the maximum loss that you must be willing to

bear is - € 23.632 k, while in the opposite case there is 95% probability that the NPV is greater than

€ 30,683 thousand, so there is no it is risk of loss.

• Using the data obtained from the probability distributions of the outcomes constructed by means

of the Monte Carlo simulation, it is possible to quantify the risk of an investment by using risk

indicators such as the standard deviation, the coefficient of variation, the value-at-risk or the range

. This allows us to prove that hypothesis 2 is correct.

𝑝% 𝑠𝑢𝑐𝑐𝑒𝑠𝑠: 𝑝% 𝑠𝑢𝑐𝑐𝑒𝑠𝑠 𝑤𝑖𝑡ℎ ∗ 𝑝% 𝑤𝑖𝑡ℎ + 𝑝% 𝑠𝑢𝑐𝑐𝑒𝑠𝑠 𝑤𝑖𝑡ℎ𝑜𝑢𝑡 ∗ 𝑝% 𝑤𝑖𝑡ℎ𝑜𝑢𝑡 (55)

Table 50. Comparison between Most likely and Monte Carlo NPVs

“Monte Carlo” DCF “Deterministic” DCF

With break-option

Without break-option

Total DCF Most

Likely Pessimistic

scenario Optimistic scenario

# iterations 12,321 7,679 20,000 - - -

Scenario’s probability

61.61% 38.40% 100% ? ? ?

Prob. of success 53.83% 100.00% 71,56% ? - -

Expected value € 1,371 € 36,758 € 14.958 € 18,330 -€ 57,612 € 51,860

Risk measures

Standard deviation € 15,153 € 3,682 - ? ? ?

Coeff. of variation 11.06 0.10 - ? ? ?

5th percentile (VaR) -€ 23,632 € 30,683 - ? ? ?

95th percentile € 26,373 € 42,833 - ? ? ?

Range € 50,004 € 12,149 - ? - -


Monte Carlo makes it possible to assess the result of the investment if the forecasted cash flows included

in the deterministic DCF are wrong, taking into consideration a distribution of probability of values and not

just a single one, but there always be what W. GleiBner and T. Wiegelmann (2012) call "meta risk", or the

risk of hypothesizing a probability distribution for incorrect variables: “In real business situations, because

information is incomplete and historical data is limited, it is often not easy to decide by what probability

distributions a risk may be quantitatively described in an adequate way.”106 In other words, there is the risk

of mistaking the "logical construction of the model", to assign wrong probability distributions to the

variables and to incorrectly estimate the expected cash flows; this is what is referred to when we talk about

GIGO - garbage-in garbage-out - effect. This risk can only be mitigated by increasing the skills of the decision

makers and the quality of the data used for the production of information, both as regards the ability to

estimate market prices and the ability to predict operational costs due to property management. The DCF

remains a useful analysis tool, but as a tool, it is the user's ability to make the difference.

106 [13] W. GleiBner e T. Wiegelmann, «Quantitative methods for risk management in the real estate development industry. Risk measures, risk aggregation and performance measures,» Journal of Property Investment & Finance, vol. 30, n. 6, pp. 612-630, 2012

100

The effect of a break-option on a rental contract is to:

• eliminate market risk for the landlord over the time period prior to the option. The elimination of

volatility has an opportunity cost, that is the up-side of the probability distribution; up-side profit is

traded for down-side risk. Cash flows up to the break-option period can be discounted at a risk-free

rate since they have to remunerate only the time value of money and inflation

• The structure of future cash flows is spread in the period in which the option can be exercised,

creating two separate paths: one in which the option is not exercised and one in which it is

exercised, each with its own cash flow pattern. The deepening of the options topic is not part of

this thesis, so I will not treat the topic.

Points of strength and weakness:

Strenght:

• The Monte Carlo simulation allows to quantify the riskiness of an asset through the aggregation of

the risks of the single variables that make up the DCF; express the price of the asset or the NPV as

a probability distribution; apply classical statistical tools to quantify the risk.

• It’s possible to find the probability of occurrence of incompatible events - such as the exercise or

non-exercise of the break-option - which can be used as input in other types of analysis, such as

real options.

• The structure of the contract is taken into consideration as an element that determines the value

of the asset.

Weakness:

• In the Monte Carlo simulation cause-effect relations between the variables are not taken into

account, so some iterations can contain values that could represent impossible scenarios.

• The model is based on the hypothesis that the tenant decides to exercise the break-option only in

the presence of a market rent lower than the contract rent roll, but there are numerous other

factors that can lead to a transfer that are not taken into account.

Further improvements:

• incorporate real options into the model and quantify the maximum amount of incentives that has

to be given in order to maximize the landlord’s wealth.

• Calculate the value of α in the formula (52), for example considering the elasticity of the demand.107

• Apply the model to an asset portfolio so as to verify the effect on an aggregate level.

107 [57] D. Barker, «Length of residence discounts, turnover, and demand elasticity. Should long-term tenants pay less than new tenants?,» Journal of Housing Economics, vol. 12, n. 1, pp. 1-11, 2003

102

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Attachments

108

Attachment 1 – Historical Year-to-Year Total Returns on United Stated Financial Markets 1926-2010

Historical Year-to-Year Total Returns on United Stated Financial Markets 1926-2010

Year Large

company Stocks

Long-Term Governement

Bonds

U.S. Treasury

Bills

Consumer Price Index

Year Large

company Stocks

Long-Term Governement

Bonds

U.S. Treasury

Bills

Consumer Price Index

1926 13,75 5,69 3,3 -1,12 1969 -8,43 -5,63 6,9 6,2 1927 35,7 6,58 3,15 -2,26 1970 3,94 18,92 6,5 5,57 1928 45,08 1,15 4,05 -1,16 1971 14,3 11,24 4,36 3,27 1929 28,8 4,39 4,47 0,58 1972 18,99 2,39 4,23 3,41 1930 -25,13 4,47 2,27 -6,4 1973 -14,69 3,3 7,29 8,71 1931 -43,6 -2,15 1,15 -9,32 1974 -26,47 4 7,99 12,34 1932 -8,75 8,51 0,88 -10,27 1975 37,23 5,52 5,87 6,94 1933 52,95 1,92 0,52 0,76 1976 23,93 15,56 5,07 4,86 1934 -2,31 7,59 0,27 1,52 1977 -7,16 0,38 5,45 6,7 1935 46,79 4,2 0,17 2,99 1978 6,57 -1,26 7,64 9,02 1936 32,49 5,13 0,17 1,45 1979 18,61 1,26 10,56 13,29 1937 -35,45 1,44 0,27 2,86 1980 32,5 -2,48 12,1 12,52 1938 31,63 4,21 0,06 -2,78 1981 -4,92 4,04 14,6 8,92 1939 21,43 3,84 0,04 0 1982 21,55 44,28 10,94 3,83 1940 -10,36 5,7 0,04 0,71 1983 22,56 1,29 8,99 3,79 1941 -12,02 0,47 0,14 9,93 1984 6,27 15,29 9,9 3,95 1942 20,75 1,8 0,34 9,03 1985 31,73 32,27 7,71 3,8 1943 25,38 2,01 0,38 2,96 1986 18,67 22,39 6,09 1,1 1944 19,49 2,27 0,38 2,3 1987 5,25 -3,03 5,88 4,43 1945 36,21 5,29 0,38 2,25 1988 16,61 6,84 6,94 4,42 1946 -8,42 0,54 0,38 18,13 1989 31,69 18,54 8,44 4,65 1947 5,05 -1,02 0,62 8,84 1990 -3,1 7,74 7,69 6,11 1948 4,99 2,66 1,06 2,99 1991 30,46 19,36 5,43 3,06 1949 17,81 4,58 1,12 -2,07 1992 7,62 7,34 3,48 2,9 1950 30,05 -0,98 1,22 5,93 1993 10,08 13,06 3,03 2,75 1951 23,79 -0,2 1,56 6 1994 1,32 -7,32 4,39 2,67 1952 18,39 2,43 1,75 0,75 1995 37,58 25,94 5,61 2,54 1953 -1,07 2,28 1,87 0,75 1996 22,96 0,13 5,14 3,32 1954 52,23 3,08 0,93 -0,74 1997 33,36 12,02 5,19 1,7 1955 31,62 -0,73 1,8 0,37 1998 28,58 14,45 4,86 1,61 1956 6,91 -1,72 2,66 2,99 1999 21,04 -7,51 4,8 2,68 1957 -10,5 6,82 3,28 2,9 2000 -9,1 17,22 5,98 3,39 1958 43,57 -1,72 1,71 1,76 2001 -11,89 5,51 3,33 1,55 1959 12,01 -2,02 3,48 1,73 2002 -22,1 15,15 1,61 2,38 1960 0,47 11,21 2,81 1,36 2003 28,89 2,01 0,94 1,88 1961 26,84 2,2 2,4 0,67 2004 10,88 8,12 1,14 3,26 1962 -8,75 5,72 2,82 1,33 2005 4,91 6,89 2,79 3,42 1963 22,7 1,79 3,23 1,64 2006 15,79 0,28 4,97 2,54 1964 16,43 3,71 3,62 0,97 2007 5,49 10,85 4,52 4,08 1965 12,38 0,93 4,06 1,92 2008 -37 41,78 1,24 0,09 1966 -10,06 5,12 4,94 3,46 2009 26,46 -25,61 0,15 2,72 1967 23,98 -2,86 4,39 3,04 2010 15,06 7,73 0,14 1,5 1968 11,03 2,25 5,49 4,72

Source: [8] S. A. Ross, R. W. Westerfield e B. D. Jordan, Fundamentals of Corporate Finance, 10th edition, The McGraw-Hill Companies, Inc., 2013

109

Attachment 2 – AOER (Average overall effective rate) on mortage loan at variable and fiexd rate; Interbank

rates (Euribor and Eurirs); spread between the two previous series

AOER

mortgage

loan at

variable rate

AOER

mortgage

loan at fixed

rate

Euribor 1

month

Eurirs 10

years

Spread

TEGM

variable

rate-euribor

Spread

TEGM fixed

rate-eurirs

III 2004 3,87 5,47 2,0768 4,32 1,79 1,15

IV 2004 3,84 5,43 2,1211 3,97 1,72 1,46

I 2005 3,86 5,56 2,106 3,74 1,75 1,82

II 2005 3,87 5,36 2,1042 3,54 1,77 1,82

III 2005 3,86 5,16 2,1114 3,39 1,75 1,77

IV 2005 3,82 5 2,2515 3,55 1,57 1,45

I 2006 3,85 4,97 2,4935 3,72 1,36 1,25

II 2006 4,16 5,14 2,7369 4,27 1,42 0,87

III 2006 4,42 5,3 3,0634 4,17 1,36 1,13

IV 2006 4,77 5,71 3,4705 4,03 1,3 1,68

I 2007 5,1 5,99 3,7036 4,24 1,4 1,75

II 2007 5,31 5,72 3,9589 4,54 1,35 1,18

III 2007 5,58 5,91 4,2823 4,64 1,3 1,27

IV 2007 5,71 6,06 4,3876 4,53 1,32 1,53

I 2008 5,75 6,08 4,228 4,38 1,52 1,7

II 2008 6 6,04 4,4096 4,78 1,59 1,26

III 2008 5,96 5,99 4,5397 4,9 1,42 1,09

IV 2008 6,3 6,3 3,8891 4,66 2,41 1,64

I 2009 5,45 5,39 1,6797 4,54 3,77 0,85

II 2009 4,58 4,42 0,9366 4,46 3,64 -0,04

III 2009 3,39 4,46 0,5243 4,19 2,87 0,27

IV 2009 3,25 5,19 0,4476 4,06 2,8 1,13

I 2010 2,92 5,36 0,4215 4,02 2,5 1,34

II 2010 2,63 5,17 0,4247 4,03 2,21 1,14

III 2010 2,56 4,99 0,6138 3,9 1,95 1,09

IV 2010 2,6 4,51 0,8096 4,2 1,79 0,31

I 2011 2,68 4,19 0,8632 4,78 1,82 -0,59

II 2011 2,79 4,68 1,2165 4,8 1,57 -0,12

III 2011 2,79 4,68 1,3808 5,49 1,41 -0,81

IV 2011 3,19 5,15 1,2443 6,61 1,95 -1,46

I 2012 3,3 5,12 0,6432 5,71 2,66 -0,59

II 2012 3,43 4,68 0,3942 5,79 3,04 -1,11

III 2012 3,66 4,75 0,1566 5,69 3,5 -0,94

IV 2012 4,34 5,51 0,1101 4,78 4,23 0,73

I 2013 3,92 5,34 0,1169 4,45 3,8 0,89

II 2013 4,06 5,43 0,1169 4,21 3,94 1,22

III 2013 4,01 5,42 0,127 4,46 3,88 0,96

IV 2013 3,68 5,09 0,1582 4,15 3,52 0,94

I 2014 3,88 5,11 0,2266 3,64 3,65 1,47

II 2014 3,81 5,11 0,2216 3,09 3,59 2,02

III 2014 3,73 5,17 0,0663 2,61 3,66 2,56

IV 2014 3,82 5,11 0,0136 2,23 3,81 2,88

I 2015 3,66 4,85 -0,0016 1,52 3,66 3,33

II 2015 3,47 4,5 -0,0472 1,79 3,52 2,71

110

III 2015 3,31 4,31 -0,0881 1,93 3,4 2,38

IV 2015 3,13 3,96 -0,1485 1,61 3,28 2,35

I 2016 2,97 3,6 -0,2588 1,49 3,23 2,11

II 2016 2,83 3,6 -0,3484 1,47 3,18 2,13

III 2016 2,72 3,39 -0,37 1,22 3,09 2,17

IV 2016 2,6 3,18 -0,3715 1,76 2,97 1,42

I 2017 2,5 3,04 -0,3718 2,25 2,87 0,79

II 2017 2,52 2,77 -0,3728 2,17 2,89 0,6

III 2017 2,47 2,65 -0,3723 2,15 2,84 0,5

IV 2017 2,43 2,79 -0,3709 1,89 2,8 0,9

I 2018 2,45 2,91 -0,3697 2,01 2,82 0,9

II 2018 2,43 2,94 -0,3709 2,23 2,8 0,71

III 2018 2,41 2,77 -0,3701 2,92 2,78 -0,15

IV 2018 2,34 2,67 -0,3685 3,28 2,71 -0,61

I 2019 2,28 2,55 -0,3672 2,76 2,65 -0,21

II 2019 2,27 2,54 -0,3737 2,52 2,64 0,02

Source: author’s elaborations based on Banca d’Italia and European Central Bank Statistical Data Warehouse

111

Attachment 3.1 – Study case maintenance plan used for the estimation of the annual maintenance

expense in the Table 32. Study case maintenance expenses.


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Attachment 3.2 – Study case maintenance plan used for the estimation of the annual capital expenditure

in the Table 34. Study case capex


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113

Attachment 4 – risk-free rate estimation using the average values of the yearly spread between BTP 10

years and BOT 12 months Italian government bonds.

Risk-free Rate Estimation BTP

10 yrs

BOT 12

month Spread

BTP

10 yrs

BOT 12

month Spread

BTP

10 yrs

BOT 12

month

Sprea

d

I 00 5,69 4,04 1,65 III 06 4,07 3,49 0,57 IV 12 4,67 1,72 2,95 II 00 5,56 4,63 0,93 IV 06 4,07 3,68 0,39 I 13 4,60 1,08 3,52 III 00 5,60 5,08 0,52 I 07 4,26 3,92 0,34 II 13 4,19 0,86 3,33 IV 00 5,41 4,96 0,45 II 07 4,61 4,21 0,41 III 13 4,46 1,16 3,31 I 01 5,15 4,47 0,68 III 07 4,60 4,22 0,37 IV 13 4,09 0,80 3,30 II 01 5,48 4,31 1,17 IV 07 4,48 4,01 0,48 I 14 3,52 0,67 2,85 III 01 5,18 4,06 1,11 I 08 4,41 3,72 0,69 II 14 2,93 0,58 2,35 IV 01 4,95 3,17 1,78 II 08 4,89 4,19 0,70 III 14 2,50 0,31 2,19 I 02 5,30 3,52 1,78 III 08 4,89 4,38 0,51 IV 14 2,10 0,35 1,75 II 02 5,33 3,74 1,59 IV 08 4,66 2,78 1,89 I 15 1,42 0,18 1,25 III 02 4,76 3,37 1,39 I 09 4,58 1,51 3,07 II 15 1,89 0,03 1,86 IV 02 4,62 2,84 1,78 II 09 4,45 1,26 3,19 III 15 1,82 0,05 1,76 I 03 4,20 2,35 1,85 III 09 4,08 0,85 3,23 IV 15 1,51 0,00 1,51 II 03 4,00 2,11 1,89 IV 09 4,04 0,93 3,11 I 16 1,36 -0,06 1,41 III 03 4,27 2,10 2,17 I 10 4,00 0,89 3,10 II 16 1,38 -0,11 1,50 IV 03 4,48 2,29 2,19 II 10 4,08 1,25 2,83 III 16 1,17 -0,18 1,35 I 04 4,26 2,05 2,21 III 10 3,86 1,40 2,46 IV 16 1,83 -0,22 2,05 II 04 4,50 2,19 2,31 IV 10 4,47 1,74 2,73 I 17 2,22 -0,24 2,46 III 04 4,29 2,24 2,05 I 11 4,80 2,01 2,79 II 17 2,21 -0,30 2,51 IV 04 3,94 2,21 1,73 II 11 4,80 2,08 2,72 III 17 2,10 -0,34 2,44 I 05 3,76 2,20 1,56 III 11 5,53 3,59 1,94 IV 17 1,85 -0,38 2,23 II 05 3,51 2,09 1,42 IV 11 6,74 5,20 1,54 I 18 1,96 -0,41 2,37 III 05 3,38 2,11 1,27 I 12 5,43 2,12 3,30 II 18 2,43 -0,07 2,50 IV 05 3,59 2,50 1,09 II 12 5,78 3,05 2,73 III 18 3,04 0,48 2,56 I 06 3,84 2,81 1,03 III 12 5,70 2,39 3,31 IV 18 3,14 0,65 2,49 II 06 4,30 3,17 1,13

Average 1,907

Risk-free 1,233

Source: author’s elaboration based on Ministero dell’Economia e delle Finanze e Investing.com database

114

Attachment 5 – Risk premium estimation as average of the year differential between the annual total

returns and risk-free rate. Risk-free rate is estimate taking the difference between the BTP 10 years and BOT

12 months maturity Italian government bonds.

Risk premium estimation using the empirical historical method

BTP

10 yrs

BOT 12

month

Risk-

free

Market

Price

Cap

Rate

Capital

Returns

Total

Returns

Risk

Premium

A B C=A-B D E F=Dt+1/Dt-1 G=D+F H=G-C

SI - 2009 4,52% 1,38% 3,13% 2.300 € 6,40%

SII - 2009 4,06% 0,89% 3,17% 2.225 € 6,40% -3,26% 3,14% -0,03%

SI - 2010 4,04% 1,07% 2,96% 2.225 € 6,30% 0,00% 6,30% 3,34%

SII - 2010 4,16% 1,57% 2,59% 2.300 € 6,30% 3,37% 9,67% 7,08%

SI - 2011 4,80% 2,04% 2,75% 2.300 € 6,40% 0,00% 6,40% 3,65%

SII - 2011 6,14% 4,40% 1,74% 2.300 € 6,40% 0,00% 6,40% 4,66%

SI - 2012 5,60% 2,59% 3,01% 2.300 € 7,10% 0,00% 7,10% 4,09%

SII - 2012 5,18% 2,05% 3,13% 2.275 € 7,10% -1,09% 6,01% 2,88%

SI - 2013 4,40% 0,97% 3,43% 2.275 € 7,10% 0,00% 7,10% 3,67%

SII - 2013 4,28% 0,98% 3,30% 2.250 € 7,10% -1,10% 6,00% 2,70%

SI - 2014 3,22% 0,62% 2,60% 2.250 € 6,25% 0,00% 6,25% 3,65%

SII - 2014 2,30% 0,33% 1,97% 2.250 € 6,25% 0,00% 6,25% 4,28%

SI - 2015 1,66% 0,11% 1,55% 2.250 € 5,50% 0,00% 5,50% 3,95%

SII - 2015 1,66% 0,03% 1,64% 2.175 € 5,50% -3,33% 2,17% 0,53%

SI - 2016 1,37% -0,09% 1,45% 2.175 € 5,50% 0,00% 5,50% 4,05%

SII - 2016 1,50% -0,20% 1,70% 2.175 € 5,50% 0,00% 5,50% 3,80%

SI - 2017 2,22% -0,27% 2,48% 2.175 € 5,25% 0,00% 5,25% 2,77%

SII - 2017 1,98% -0,36% 2,34% - 5,25% - - -

SI - 2018 2,20% -0,24% 2,44% - 5,00% - - -

SII - 2018 3,09% 0,57% 2,52% - 5,00% - - -

Average 2.47% 6.25% -0.34% 5.91% 3,44%

Note: 10 years bond rate are calculated averaging six end-month rate for each quarter. 12 months bond rate are calculated

averaging six mid-month (15th day) rate for each quarter. Cap rate value are taken from COIMA 2018 and refers to office

sector for good secondary districts. Market price are taken from Agenzia delle Entrate database and refers to the “D25”

zone.


115

Attachment 6 – Monthly returns calculation for FTSE MIN Index, FTSE EPRA/NAREIT Developed Europe Index

and COIMA RES. Values are been used to estimate COIMA RES’s β using both FTSE MIB and FTSE

EPRA/NAREIT Developed Europe Index

FTSE MIB FTSE EPRA/NAREIT Developed

Europe Index COIMA RES

Data Close Var. % Close Var. % Close Var. %

gen-17 18.590,73 -3,35% 2.121,77 -1,00% 6,83 5,40%

feb-17 18.913,28 1,73% 2.179,84 2,74% 7,05 3,22%

mar-17 20.492,94 8,35% 2.185,90 0,28% 7,33 3,97%

apr-17 20.609,16 0,57% 2.294,57 4,97% 7,235 -1,30%

mag-17 20.731,68 0,59% 2.404,57 4,79% 7,64 5,60%

giu-17 20.584,23 -0,71% 2.385,43 -0,80% 7,695 0,72%

lug-17 21.486,91 4,39% 2.448,96 2,66% 8 3,96%

ago-17 21.670,02 0,85% 2.476,34 1,12% 8,1 1,25%

set-17 22.696,32 4,74% 2.486,83 0,42% 7,85 -3,09%

ott-17 22.793,69 0,43% 2.471,24 -0,63% 8,65 10,19%

nov-17 22.368,29 -1,87% 2.543,78 2,94% 9 4,05%

dic-17 21.853,34 -2,30% 2.666,97 4,84% 8,985 -0,17%

gen-18 23.507,06 7,57% 2.735,89 2,58% 8,82 -1,84%

feb-18 22.607,61 -3,83% 2.528,82 -7,57% 8,68 -1,59%

mar-18 22.411,15 -0,87% 2.626,98 3,88% 8,46 -2,53%

apr-18 23.979,37 7,00% 2.677,58 1,93% 8,54 0,95%

mag-18 21.784,18 -9,15% 2.579,21 -3,67% 7,94 -7,03%

giu-18 21.626,27 -0,72% 2.575,34 -0,15% 8 0,76%

lug-18 22.215,69 2,73% 2.610,00 1,35% 7,96 -0,50%

ago-18 20.269,47 -8,76% 2.604,64 -0,21% 7,64 -4,02%

set-18 20.711,70 2,18% 2.519,89 -3,25% 7,98 4,45%

ott-18 19.050,22 -8,02% 2.378,38 -5,62% 7,68 -3,76%

nov-18 19.188,97 0,73% 2.352,04 -1,11% 7,02 -8,59%

dic-18 18.324,03 -4,51% 2.255,66 -4,10% 7,1 1,14%

gen-19 19.730,78 7,68% 2.496,44 10,67% 7,48 5,35%

feb-19 20.659,46 4,71% 2.447,00 -1,98% 7,9 5,61%

mar-19 21.286,13 3,03% 2.496,41 2,02% 8,02 1,52%

apr-19 21.881,33 2,80% 2.457,09 -1,58% 7,74 -3,49%

mag-19 19.802,11 -9,50% 2.419,26 -1,54% 7,58 -2,07%

giu-19 21.234,79 7,23% 2.391,03 -1,17% 7,78 2,64%

lug-19 21.735,70 2,36% 2.339,93 -2,14% 8,18 5,14%

Source: author’s elaboration based on Investing.com database.

116

Attachment 7 – Vodafone Village “Most Likely” DCFA.

Source: auhtor’s elaboration

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Init

ial

inve

stm

en

t‐€

209

.300

ERV

gro

wth

ra

te -

Tre

nd

eff

ect

3,00

%2,

50%

2,00

%1,

00%

0,50

%-0

,10%

-0,2

0%-0

,30%

-0,4

0%-0

,50%

-0,6

0%0,

00%

ERV

13.9

64 €

14.3

51 €

14.6

77 €

14.9

37 €

15.0

52 €

15.0

93 €

15.0

43 €

14.9

79 €

14.9

00 €

14.8

06 €

14.6

98 €

14.5

76 €

14.5

43 €

Pote

nti

al

gro

ss i

nco

me

13.9

64 €

14.1

32 €

14.3

01 €

14.4

73 €

14.6

46 €

14.8

22 €

15.0

00 €

15.1

80 €

15.3

62 €

15.0

79 €

14.9

84 €

15.1

64 €

15.3

46 €

Va

can

cy a

llo

wa

nce

15.0

79 €

3.74

6 €

Effe

ctiv

e gr

oss

inco

me

14.1

32 €

14.3

01 €

14.4

73 €

14.6

46 €

14.8

22 €

15.0

00 €

15.1

80 €

15.3

62 €

0 €

11.2

38 €

15.1

64 €

15.3

46 €

Op

era

tin

g e

xpe

nse

s

Insu

ran

ce‐6

5 €

‐66

€‐6

7 €

‐67

€‐6

8 €

‐69

€‐7

0 €

‐71

€‐6

9 €

‐69

€‐7

0 €

‐71

€

IMU

an

d T

AS

I‐8

97 €

‐908

€‐9

19 €

‐930

€‐9

41 €

‐953

€‐9

64 €

‐976

€‐9

57 €

‐951

€‐9

63 €

‐974

€

Pro

pe

rty

ma

na

gem

en

t‐2

59 €

‐262

€‐2

65 €

‐268

€‐2

71 €

‐275

€‐2

78 €

‐281

€0

€‐2

06 €

‐277

€‐2

81 €

Uti

liti

es

‐1.1

57 €

‐1.1

71 €

‐1.1

85 €

‐1.2

00 €

‐1.2

14 €

‐1.2

29 €

‐1.2

43 €

‐1.2

58 €

0 €

‐920

€‐1

.242

€‐1

.257

€

Ma

inte

na

nce

/ER

V3,

24%

0,88

%9,

69%

2,59

%1,

99%

1,32

%0,

58%

8,95

%1,

30%

0,44

%2,

14%

1,57

%

Ma

inte

na

nce

‐458

€‐1

26 €

‐1.4

02 €

‐379

€‐2

95 €

‐198

€‐8

8 €

‐1.3

75 €

‐196

€‐6

6 €

‐325

€‐2

41 €

Tota

l o

pe

rati

ng

exp

en

ses

‐2.8

36 €

‐2.5

33 €

‐3.8

38 €

‐2.8

44 €

‐2.7

90 €

‐2.7

23 €

‐2.6

43 €

‐3.9

60 €

‐1.2

23 €

‐2.2

12 €

‐2.8

77 €

‐2.8

24 €

Re

cove

ry f

rom

te

na

nts

1.84

4 €

1.64

6 €

2.49

5 €

1.84

9 €

1.81

3 €

1.77

0 €

1.71

8 €

2.57

4 €

795

€1.

438

€1.

870

€1.

835

€

Ne

t o

pe

rati

ng

exp

en

ses

‐993

€‐8

86 €

‐1.3

43 €

‐996

€‐9

76 €

‐953

€‐9

25 €

‐1.3

86 €

‐428

€‐7

74 €

‐1.0

07 €

‐988

€

Net

ope

rati

ng in

com

e13

.139

€13

.415

€13

.129

€13

.651

€13

.846

€14

.047

€14

.255

€13

.976

€-4

28 €

10.4

64 €

14.1

57 €

14.3

57 €

Cap

ex/

ERV

0,00

%0,

00%

6,83

%0,

00%

0,00

%0,

00%

0,00

%20

,39%

0,00

%0,

00%

0,00

%0,

00%

Cap

ex

0 €

0 €

‐988

€0

€0

€0

€0

€‐3

.132

€0

€0

€0

€0

€

Exit

ca

p r

ate

7,37

%

Re

vers

ion

va

lue

194.

808

€

Bro

kera

ge c

om

mis

sio

ns

3,00

%

Pro

pe

rty

be

fore

ta

x ca

sh f

low

s‐2

09.3

00 €

13.1

39 €

13.4

15 €

12.1

41 €

13.6

51 €

13.8

46 €

14.0

47 €

14.2

55 €

10.8

44 €

‐428

€10

.464

€20

3.12

1 €

Taxe

s0

€0

€0

€0

€0

€0

€0

€0

€0

€0

€0

€

Net

cas

h fl

ows

-209

.300

€13

.139

€13

.415

€12

.141

€13

.651

€13

.846

€14

.047

€14

.255

€10

.844

€-4

28 €

10.4

64 €

203.

121

€

Dis

cou

nte

d c

ash

flo

ws

@ W

AC

C‐2

09.3

00 €

12.6

35 €

12.4

05 €

10.7

96 €

11.6

73 €

11.3

86 €

11.1

08 €

10.8

40 €

7.93

0 €

‐301

€7.

076

€13

2.08

3 €

NPV

18.3

30 €

IRR

5,05

%

MO

ST L

IKEL

Y

117

Attachment 8.1 – Vodafone Village “Best Case” DCFA


Tho

usa

nd

of

€0

1/0

1/2

01

80

1/0

1/2

01

90

1/0

1/2

02

00

1/0

1/2

02

10

1/0

1/2

02

20

1/0

1/2

02

30

1/0

1/2

02

40

1/0

1/2

02

50

1/0

1/2

02

60

1/0

1/2

02

70

1/0

1/2

02

80

1/0

1/2

02

90

1/0

1/2

03

0

01

23

45

67

89

1011

12

Init

ial

inve

stm

en

t‐€

209

.300

ERV

gro

wth

ra

te -

Tre

nd

eff

ect

5,00

%4,

00%

3,00

%2,

00%

1,00

%-0

,05%

-0,1

0%-0

,15%

-0,2

0%-0

,25%

-0,3

0%0,

00%

ERV

13.9

64 €

14.6

30 €

15.1

82 €

15.6

03 €

15.8

79 €

16.0

02 €

15.9

57 €

15.9

05 €

15.8

44 €

15.7

77 €

15.7

01 €

15.6

18 €

15.5

82 €

Pote

nti

al

gro

ss i

nco

me

13.9

64 €

14.1

32 €

14.3

01 €

14.4

73 €

14.6

46 €

14.8

22 €

15.0

00 €

15.1

80 €

15.3

62 €

15.5

47 €

15.7

33 €

15.9

22 €

16.1

13 €

Va

can

cy a

llo

wa

nce

Effe

ctiv

e gr

oss

inco

me

14.1

32 €

14.3

01 €

14.4

73 €

14.6

46 €

14.8

22 €

15.0

00 €

15.1

80 €

15.3

62 €

15.5

47 €

15.7

33 €

15.9

22 €

16.1

13 €

Op

era

tin

g e

xpe

nse

s

Insu

ran

ce‐6

5 €

‐66

€‐6

7 €

‐67

€‐6

8 €

‐69

€‐7

0 €

‐71

€‐7

2 €

‐72

€‐7

3 €

‐74

€

IMU

an

d T

ASI

‐897

€‐9

08 €

‐919

€‐9

30 €

‐941

€‐9

53 €

‐964

€‐9

76 €

‐987

€‐9

99 €

‐1.0

11 €

‐1.0

23 €

Pro

pe

rty

ma

na

gem

en

t‐1

81 €

‐183

€‐1

85 €

‐188

€‐1

90 €

‐192

€‐1

94 €

‐197

€‐1

99 €

‐202

€‐2

04 €

‐206

€

Uti

liti

es

‐810

€‐8

20 €

‐830

€‐8

40 €

‐850

€‐8

60 €

‐870

€‐8

81 €

‐891

€‐9

02 €

‐913

€‐9

24 €

Ma

inte

na

nce

/ER

V3,

24%

0,88

%9,

69%

2,59

%1,

99%

1,32

%0,

58%

8,95

%1,

30%

0,44

%2,

14%

1,57

%

Ma

inte

na

nce

‐321

€‐8

8 €

‐982

€‐2

66 €

‐206

€‐1

39 €

‐62

€‐9

62 €

‐141

€‐4

8 €

‐239

€‐1

77 €

Tota

l o

pe

rati

ng

exp

en

ses

‐2.2

74 €

‐2.0

65 €

‐2.9

82 €

‐2.2

90 €

‐2.2

55 €

‐2.2

12 €

‐2.1

60 €

‐3.0

86 €

‐2.2

91 €

‐2.2

23 €

‐2.4

40 €

‐2.4

05 €

Re

cove

ry f

rom

te

na

nts

1.47

8 €

1.34

2 €

1.93

9 €

1.48

9 €

1.46

6 €

1.43

8 €

1.40

4 €

2.00

6 €

1.48

9 €

1.44

5 €

1.58

6 €

1.56

3 €

Ne

t o

pe

rati

ng

exp

en

ses

‐796

€‐7

23 €

‐1.0

44 €

‐802

€‐7

89 €

‐774

€‐7

56 €

‐1.0

80 €

‐802

€‐7

78 €

‐854

€‐8

42 €

Net

ope

rati

ng in

com

e13

.336

€13

.578

€13

.429

€13

.845

€14

.033

€14

.226

€14

.424

€14

.282

€14

.745

€14

.955

€15

.068

€15

.271

€

Cap

ex/

ERV

0,00

%0,

00%

6,83

%0,

00%

0,00

%0,

00%

0,00

%20

,39%

0,00

%0,

00%

0,00

%0,

00%

Cap

ex

0 €

0 €

‐840

€0

€0

€0

€0

€‐2

.663

€0

€0

€0

€0

€

Exit

ca

p r

ate

6,87

%

Re

vers

ion

va

lue

222.

291

€

Bro

kera

ge c

om

mis

sio

ns

3,00

%

Pro

pe

rty

be

fore

ta

x ca

sh f

low

s‐2

09.3

00 €

13.3

36 €

13.5

78 €

12.5

89 €

13.8

45 €

14.0

33 €

14.2

26 €

14.4

24 €

11.6

20 €

14.7

45 €

14.9

55 €

230.

690

€

Taxe

s0

€0

€0

€0

€0

€0

€0

€0

€0

€0

€0

€

Net

cas

h fl

ows

-209

.300

€13

.336

€13

.578

€12

.589

€13

.845

€14

.033

€14

.226

€14

.424

€11

.620

€14

.745

€14

.955

€23

0.69

0 €

Dis

cou

nte

d c

ash

flo

ws

@ W

AC

C‐2

09.3

00 €

12.8

24 €

12.5

56 €

11.1

95 €

11.8

39 €

11.5

39 €

11.2

49 €

10.9

68 €

8.49

7 €

10.3

69 €

10.1

13 €

150.

011

€

NPV

51.8

60 €

IRR

6,76

%

BES

T C

ASE

118

Attachment 8.2 – Vodafone Village “Worst Case” DCFA


Tho

usa

nd

of

€0

1/0

1/2

01

80

1/0

1/2

01

90

1/0

1/2

02

00

1/0

1/2

02

10

1/0

1/2

02

20

1/0

1/2

02

30

1/0

1/2

02

40

1/0

1/2

02

50

1/0

1/2

02

60

1/0

1/2

02

70

1/0

1/2

02

80

1/0

1/2

02

90

1/0

1/2

03

0

01

23

45

67

89

1011

12

Init

ial

inve

stm

en

t‐€

209

.300

ERV

gro

wth

ra

te -

Tre

nd

eff

ect

-2,1

5%-2

,65%

-3,1

5%-4

,15%

-4,6

5%-5

,25%

-5,3

5%-5

,45%

-5,5

5%-5

,65%

-5,7

5%

ERV

13.9

64 €

13.6

32 €

13.2

40 €

12.7

93 €

12.2

33 €

11.6

36 €

10.9

99 €

10.3

85 €

9.79

6 €

9.23

0 €

8.68

7 €

8.16

8 €

8.15

0 €

Pote

nti

al

gro

ss i

nco

me

13.9

64 €

14.1

32 €

14.3

01 €

14.4

73 €

14.6

46 €

14.8

22 €

15.0

00 €

15.1

80 €

15.3

62 €

9.91

3 €

9.34

1 €

8.79

2 €

8.89

7 €

Va

can

cy a

llo

wa

nce

9.91

3 €

9.34

1 €

8.79

2 €

Effe

ctiv

e gr

oss

inco

me

14.1

32 €

14.3

01 €

14.4

73 €

14.6

46 €

14.8

22 €

15.0

00 €

15.1

80 €

15.3

62 €

0 €

0 €

0 €

8.89

7 €

Op

era

tin

g e

xpe

nse

s

Insu

ran

ce‐6

5 €

‐66

€‐6

7 €

‐67

€‐6

8 €

‐69

€‐7

0 €

‐71

€‐4

6 €

‐43

€‐4

0 €

‐41

€

IMU

an

d T

ASI

‐897

€‐9

08 €

‐919

€‐9

30 €

‐941

€‐9

53 €

‐964

€‐9

76 €

‐629

€‐5

93 €

‐558

€‐5

65 €

Pro

pe

rty

ma

na

gem

en

t‐3

36 €

‐340

€‐3

44 €

‐348

€‐3

53 €

‐357

€‐3

61 €

‐365

€0

€0

€0

€‐2

12 €

Uti

liti

es

‐1.5

05 €

‐1.5

23 €

‐1.5

41 €

‐1.5

59 €

‐1.5

78 €

‐1.5

97 €

‐1.6

16 €

‐1.6

36 €

0 €

0 €

0 €

‐947

€

Ma

inte

na

nce

/ER

V3,

24%

0,88

%9,

69%

2,59

%1,

99%

1,32

%0,

58%

8,95

%1,

30%

0,44

%2,

14%

1,57

%

Ma

inte

na

nce

‐595

€‐1

64 €

‐1.8

23 €

‐493

€‐3

83 €

‐257

€‐1

14 €

‐1.7

87 €

‐168

€‐5

3 €

‐245

€‐1

82 €

Tota

l o

pe

rati

ng

exp

en

ses

‐3.3

98 €

‐3.0

00 €

‐4.6

94 €

‐3.3

98 €

‐3.3

24 €

‐3.2

33 €

‐3.1

26 €

‐4.8

35 €

‐843

€‐6

90 €

‐843

€‐1

.946

€

Re

cove

ry f

rom

te

na

nts

2.20

9 €

1.95

0 €

3.05

1 €

2.20

9 €

2.16

0 €

2.10

1 €

2.03

2 €

3.14

3 €

548

€44

8 €

548

€1.

265

€

Ne

t o

pe

rati

ng

exp

en

ses

‐1.1

89 €

‐1.0

50 €

‐1.6

43 €

‐1.1

89 €

‐1.1

63 €

‐1.1

31 €

‐1.0

94 €

‐1.6

92 €

‐295

€‐2

41 €

‐295

€‐6

81 €

Net

ope

rati

ng in

com

e12

.942

€13

.251

€12

.830

€13

.457

€13

.659

€13

.869

€14

.086

€13

.670

€-2

95 €

-241

€-2

95 €

8.21

6 €

Cap

ex/

ERV

0,00

%0,

00%

6,83

%0,

00%

0,00

%0,

00%

0,00

%20

,39%

0,00

%0,

00%

0,00

%0,

00%

Cap

ex

0 €

0 €

‐1.2

85 €

0 €

0 €

0 €

0 €

‐4.0

72 €

0 €

0 €

0 €

0 €

Exit

ca

p r

ate

7,87

%

Re

vers

ion

va

lue

104.

396

€

Bro

kera

ge c

om

mis

sio

ns

3,00

%

Pro

pe

rty

be

fore

ta

x ca

sh f

low

s‐2

09.3

00 €

12.9

42 €

13.2

51 €

11.5

45 €

13.4

57 €

13.6

59 €

13.8

69 €

14.0

86 €

9.59

8 €

‐295

€‐2

41 €

100.

969

€

Taxe

s0

€0

€0

€0

€0

€0

€0

€0

€0

€0

€0

€

Net

cas

h fl

ows

-209

.300

€12

.942

€13

.251

€11

.545

€13

.457

€13

.659

€13

.869

€14

.086

€9.

598

€-2

95 €

-241

€10

0.96

9 €

Dis

cou

nte

d c

ash

flo

ws

@ W

AC

C‐2

09.3

00 €

12.4

46 €

12.2

54 €

10.2

66 €

11.5

08 €

11.2

32 €

10.9

67 €

10.7

11 €

7.01

9 €

‐207

€‐1

63 €

65.6

57 €

NPV

-57.

612

€

IRR

-0,4

1%

WO

RST

CA

SE

119

Attachment 9 – Sensibility analysis maximum (Positive 30%) and minimum (Negative 30%) values of NPV

and IRR for each DCF’s risky variable. Variables are ranked in ascending order of risk using as indicator the

range between “Δ Positive” and “Δ Negative”.

Positive

30% Negative

30% Δ Positive Δ Negative Range

B C D=B-A E=C-A F=D-E

NPV Most Likely € 18.330

Capex € 19.281 € 17.379 € 951 -€ 951 € 1.902

Trend ERV € 21.611 € 15.101 € 3.281 -€ 3.229 € 6.510

Vacancy lenght € 22.053 € 14.664 € 3.723 -€ 3.666 € 7.390

Total operating expenses € 23.440 € 13.221 € 5.109 -€ 5.109 € 10.219

Recovery from tenants € 28.062 € 9.085 € 9.732 -€ 9.245 € 18.977

Exit cap rate € 70.958 -€ 10.016 € 52.628 -€ 28.346 € 80.974

Initial investment € 81.120 -€ 44.460 € 62.790 -€ 62.790 € 125.580

IRR base case 5,05%

Capex 5,10% 4,99% 0,05% -0,05% 0,11%

Trend ERV 5,22% 4,87% 0,18% -0,18% 0,36%

Vacancy lenght 5,39% 4,84% 0,34% -0,20% 0,55%

Total operating expenses 5,33% 4,76% 0,29% -0,29% 0,57%

Recovery from tenants 5,57% 4,50% 0,53% -0,54% 1,07%

Exit cap rate 7,56% 3,35% 2,51% -1,69% 4,20%

Initial investment 9,88% 1,82% 4,84% -3,22% 8,06%


120

Attachment 10.1 – Monte Carlo simulation output – 20,000 iterations. NPV Bimodal Frequency

Distribution – Data, Frequency and Cumulative Distribution. The two modes are highlighted

NPV Bimodal Frequency and Cumulative Distribution

NPV Freque

ncy %

cumulative NPV

Frequency

% cumulative

NPV Frequen

cy %

cumulative -€ 54.332 1 0,01% -€ 17.510 125 6,60% € 19.312 122 52,97% -€ 53.565 0 0,01% -€ 16.743 105 7,13% € 20.079 127 53,60% -€ 52.798 0 0,01% -€ 15.976 129 7,77% € 20.846 136 54,28% -€ 52.031 0 0,01% -€ 15.209 115 8,35% € 21.613 112 54,84% -€ 51.264 0 0,01% -€ 14.442 133 9,01% € 22.380 98 55,33% -€ 50.496 1 0,01% -€ 13.675 154 9,78% € 23.147 127 55,97% -€ 49.729 2 0,02% -€ 12.908 144 10,50% € 23.914 90 56,42% -€ 48.962 5 0,05% -€ 12.140 169 11,35% € 24.681 73 56,78% -€ 48.195 0 0,05% -€ 11.373 139 12,04% € 25.449 105 57,31% -€ 47.428 1 0,05% -€ 10.606 140 12,74% € 26.216 80 57,71% -€ 46.661 2 0,06% -€ 9.839 171 13,60% € 26.983 90 58,16% -€ 45.894 1 0,07% -€ 9.072 182 14,51% € 27.750 84 58,58% -€ 45.127 5 0,09% -€ 8.305 197 15,49% € 28.517 116 59,16% -€ 44.359 2 0,10% -€ 7.538 179 16,39% € 29.284 130 59,81% -€ 43.592 2 0,11% -€ 6.771 212 17,45% € 30.051 159 60,60% -€ 42.825 5 0,14% -€ 6.003 227 18,58% € 30.818 185 61,53% -€ 42.058 1 0,14% -€ 5.236 205 19,61% € 31.585 239 62,72% -€ 41.291 4 0,16% -€ 4.469 218 20,70% € 32.353 332 64,38% -€ 40.524 8 0,20% -€ 3.702 210 21,75% € 33.120 375 66,26% -€ 39.757 4 0,22% -€ 2.935 220 22,85% € 33.887 464 68,58% -€ 38.990 8 0,26% -€ 2.168 205 23,87% € 34.654 538 71,27% -€ 38.223 7 0,30% -€ 1.401 223 24,99% € 35.421 581 74,17% -€ 37.455 4 0,32% -€ 634 229 26,13% € 36.188 656 77,45% -€ 36.688 12 0,38% € 134 213 27,20% € 36.955 618 80,54% -€ 35.921 13 0,44% € 901 220 28,30% € 37.722 658 83,83% -€ 35.154 20 0,54% € 1.668 257 29,58% € 38.490 629 86,98% -€ 34.387 18 0,63% € 2.435 228 30,72% € 39.257 578 89,87% -€ 33.620 13 0,70% € 3.202 211 31,78% € 40.024 534 92,54% -€ 32.853 19 0,79% € 3.969 241 32,98% € 40.791 403 94,55% -€ 32.086 22 0,90% € 4.736 262 34,29% € 41.558 348 96,29% -€ 31.318 15 0,98% € 5.503 244 35,51% € 42.325 225 97,42% -€ 30.551 32 1,14% € 6.271 242 36,72% € 43.092 185 98,34% -€ 29.784 34 1,31% € 7.038 228 37,86% € 43.859 106 98,87% -€ 29.017 33 1,47% € 7.805 230 39,01% € 44.627 80 99,27% -€ 28.250 30 1,62% € 8.572 248 40,25% € 45.394 54 99,54% -€ 27.483 39 1,82% € 9.339 227 41,39% € 46.161 29 99,69% -€ 26.716 51 2,07% € 10.106 224 42,51% € 46.928 23 99,80% -€ 25.949 40 2,27% € 10.873 218 43,60% € 47.695 16 99,88% -€ 25.181 43 2,49% € 11.640 193 44,56% € 48.462 11 99,94% -€ 24.414 54 2,76% € 12.407 228 45,70% € 49.229 2 99,95% -€ 23.647 65 3,08% € 13.175 193 46,67% € 49.996 7 99,98% -€ 22.880 77 3,47% € 13.942 182 47,58% € 50.764 2 99,99% -€ 22.113 71 3,82% € 14.709 152 48,34% € 51.531 0 99,99% -€ 21.346 76 4,20% € 15.476 150 49,09% € 52.298 0 99,99% -€ 20.579 73 4,57% € 16.243 182 50,00% € 53.065 1 100,00% -€ 19.812 86 5,00% € 17.010 180 50,90% Other 1 100,00% -€ 19.044 84 5,42% € 17.777 163 51,71%

-€ 18.277 112 5,98% € 18.544 129 52,36%


121

Attachment 10.2 – Monte Carlo simulation output – 20,000 iterations. IRR Bimodal Frequency Distribution

– Data, Frequency and Cumulative Distribution. The two modes are highlighted

IRR Bimodal Frequency and Cumulative Distribution

IRR Frequen

cy %

cumulative IRR Frequency

% cumulative

IRR Frequen

cy %

cumulative

-0,63% 1 0,01% 1,88% 14 0,91% 4,39% 273 37,52% -0,58% 0 0,01% 1,93% 32 1,07% 4,44% 263 38,84% -0,53% 0 0,01% 1,98% 25 1,19% 4,49% 277 40,22% -0,48% 0 0,01% 2,03% 35 1,37% 4,54% 274 41,59% -0,43% 0 0,01% 2,09% 29 1,51% 4,60% 283 43,01% -0,37% 0 0,01% 2,14% 49 1,76% 4,65% 261 44,31% -0,32% 0 0,01% 2,19% 53 2,02% 4,70% 255 45,59% -0,27% 0 0,01% 2,24% 45 2,25% 4,75% 222 46,70% -0,22% 0 0,01% 2,29% 35 2,42% 4,80% 247 47,93% -0,16% 0 0,01% 2,35% 50 2,67% 4,86% 210 48,98% -0,11% 0 0,01% 2,40% 61 2,98% 4,91% 209 50,03% -0,06% 0 0,01% 2,45% 58 3,27% 4,96% 206 51,06% -0,01% 1 0,01% 2,50% 66 3,60% 5,01% 211 52,11% 0,05% 0 0,01% 2,56% 69 3,94% 5,07% 160 52,91% 0,10% 0 0,01% 2,61% 70 4,29% 5,12% 153 53,68% 0,15% 0 0,01% 2,66% 76 4,67% 5,17% 150 54,43% 0,20% 0 0,01% 2,71% 86 5,10% 5,22% 158 55,22% 0,25% 1 0,02% 2,76% 100 5,60% 5,27% 122 55,83% 0,31% 0 0,02% 2,82% 115 6,18% 5,33% 120 56,43% 0,36% 0 0,02% 2,87% 109 6,72% 5,38% 120 57,03% 0,41% 2 0,03% 2,92% 115 7,30% 5,43% 100 57,53% 0,46% 1 0,03% 2,97% 125 7,92% 5,48% 112 58,09% 0,52% 0 0,03% 3,03% 119 8,52% 5,54% 109 58,63% 0,57% 3 0,05% 3,08% 126 9,15% 5,59% 133 59,30% 0,62% 1 0,05% 3,13% 148 9,89% 5,64% 140 60,00% 0,67% 0 0,05% 3,18% 138 10,58% 5,69% 211 61,05% 0,73% 0 0,05% 3,24% 170 11,43% 5,75% 303 62,57% 0,78% 0 0,05% 3,29% 167 12,26% 5,80% 420 64,67% 0,83% 2 0,06% 3,34% 193 13,23% 5,85% 540 67,37% 0,88% 2 0,07% 3,39% 177 14,11% 5,90% 668 70,71% 0,93% 2 0,08% 3,44% 192 15,07% 5,95% 787 74,64% 0,99% 3 0,10% 3,50% 211 16,13% 6,01% 843 78,86% 1,04% 3 0,11% 3,55% 216 17,21% 6,06% 922 83,47% 1,09% 2 0,12% 3,60% 196 18,19% 6,11% 838 87,66% 1,14% 3 0,14% 3,65% 182 19,10% 6,16% 736 91,34% 1,20% 9 0,18% 3,71% 233 20,26% 6,22% 579 94,23% 1,25% 8 0,22% 3,76% 210 21,31% 6,27% 464 96,55% 1,30% 7 0,26% 3,81% 241 22,52% 6,32% 300 98,05% 1,35% 10 0,31% 3,86% 283 23,93% 6,37% 177 98,94% 1,41% 11 0,36% 3,92% 255 25,21% 6,43% 126 99,57% 1,46% 8 0,40% 3,97% 280 26,61% 6,48% 46 99,80% 1,51% 5 0,43% 4,02% 245 27,83% 6,53% 25 99,92% 1,56% 7 0,46% 4,07% 277 29,22% 6,58% 7 99,96% 1,61% 7 0,50% 4,12% 296 30,70% 6,63% 5 99,98% 1,67% 17 0,58% 4,18% 291 32,15% 6,69% 1 99,99% 1,72% 17 0,67% 4,23% 254 33,42% Other 3 100,00% 1,77% 13 0,73% 4,28% 245 34,65%

1,82% 21 0,84% 4,33% 302 36,16%


122

Attachment 11.1 – Monte Carlo simulation output – 20,000 iterations. NPV Frequency and Cumulative

Distribution. Isolation of the NPV output in the scenario in which the break-option is excercised.

“Break-option Exercised” NPV Frequency and Cumulative Distribution

Break-option

Exercised

Frequency

% cumulative

Break-option

Exercised Frequency

% cumulativ

e

Break-option

Exercised Frequency

% cumulativ

e

-€ 64.954 1 0,01% -€ 20.372 145 7,91% € 24.209 132 93,73%

-€ 63.781 0 0,01% -€ 19.199 159 9,20% € 25.382 122 94,72%

-€ 62.607 0 0,01% -€ 18.026 152 10,43% € 26.556 95 95,49%

-€ 61.434 0 0,01% -€ 16.853 172 11,83% € 27.729 91 96,23%

-€ 60.261 0 0,01% -€ 15.680 212 13,55% € 28.902 96 97,01%

-€ 59.088 0 0,01% -€ 14.506 202 15,19% € 30.075 69 97,57%

-€ 57.915 0 0,01% -€ 13.333 229 17,04% € 31.248 48 97,95%

-€ 56.741 0 0,01% -€ 12.160 234 18,94% € 32.422 41 98,29%

-€ 55.568 0 0,01% -€ 10.987 283 21,24% € 33.595 41 98,62%

-€ 54.395 0 0,01% -€ 9.814 283 23,54% € 34.768 38 98,93%

-€ 53.222 0 0,01% -€ 8.640 303 26,00% € 35.941 24 99,12%

-€ 52.049 0 0,01% -€ 7.467 324 28,63% € 37.114 20 99,29%

-€ 50.875 0 0,01% -€ 6.294 301 31,07% € 38.288 20 99,45%

-€ 49.702 0 0,01% -€ 5.121 310 33,58% € 39.461 14 99,56%

-€ 48.529 0 0,01% -€ 3.948 363 36,53% € 40.634 16 99,69%

-€ 47.356 1 0,02% -€ 2.774 335 39,25% € 41.807 10 99,77%

-€ 46.183 0 0,02% -€ 1.601 373 42,28% € 42.980 8 99,84%

-€ 45.009 3 0,04% -€ 428 362 45,22% € 44.154 6 99,89%

-€ 43.836 3 0,06% € 745 336 47,94% € 45.327 6 99,94%

-€ 42.663 6 0,11% € 1.918 361 50,87% € 46.500 0 99,94%

-€ 41.490 4 0,15% € 3.092 345 53,67% € 47.673 2 99,95%

-€ 40.317 4 0,18% € 4.265 379 56,75% € 48.846 1 99,96%

-€ 39.144 9 0,25% € 5.438 363 59,69% € 50.020 0 99,96%

-€ 37.970 12 0,35% € 6.611 354 62,57% € 51.193 1 99,97%

-€ 36.797 18 0,50% € 7.784 337 65,30% € 52.366 2 99,98%

-€ 35.624 25 0,70% € 8.958 333 68,01% € 53.539 0 99,98%

-€ 34.451 23 0,88% € 10.131 341 70,77% € 54.712 0 99,98%

-€ 33.278 33 1,15% € 11.304 333 73,48% € 55.886 0 99,98%

-€ 32.104 35 1,44% € 12.477 276 75,72% € 57.059 1 99,99%

-€ 30.931 39 1,75% € 13.650 310 78,23% € 58.232 0 99,99%

-€ 29.758 34 2,03% € 14.824 278 80,49% € 59.405 0 99,99%

-€ 28.585 45 2,39% € 15.997 265 82,64% € 60.578 0 99,99%

-€ 27.412 83 3,07% € 17.170 237 84,56% € 61.752 0 99,99%

-€ 26.238 67 3,61% € 18.343 251 86,60% € 62.925 0 99,99%

-€ 25.065 81 4,27% € 19.516 215 88,35% € 64.098 0 99,99%

-€ 23.892 82 4,93% € 20.690 204 90,00% Other 1 100,00%

-€ 22.719 105 5,79% € 21.863 187 91,52%

-€ 21.546 116 6,73% € 23.036 140 92,65%


123

Attachment 11.2 – Monte Carlo simulation output – 20,000 iterations. NPV Frequency and Cumulative

Distribution. Isolation of the NPV output in the scenario in which the break-option is not excercised.

“Break-option not exercised” NPV Frequency and Cumulative Distribution

Break-option not exercised

Frequency %

cumulative


Frequency %

cumulative


Frequency %

cumulative

€ 23.195 1 0,01% € 32.236 135 11,47% € 41.276 133 89,07%

€ 23.497 0 0,01% € 32.537 145 13,36% € 41.577 110 90,51%

€ 23.798 0 0,01% € 32.838 124 14,98% € 41.878 101 91,82%

€ 24.100 2 0,04% € 33.140 128 16,64% € 42.180 100 93,12%

€ 24.401 0 0,04% € 33.441 140 18,47% € 42.481 76 94,11%

€ 24.702 1 0,05% € 33.742 177 20,77% € 42.782 65 94,96%

€ 25.004 1 0,07% € 34.044 186 23,19% € 43.084 69 95,86%

€ 25.305 2 0,09% € 34.345 188 25,64% € 43.385 58 96,61%

€ 25.606 3 0,13% € 34.646 213 28,42% € 43.686 45 97,20%

€ 25.908 3 0,17% € 34.948 213 31,19% € 43.988 42 97,75%

€ 26.209 6 0,25% € 35.249 230 34,18% € 44.289 28 98,11%

€ 26.510 7 0,34% € 35.550 216 37,00% € 44.590 32 98,53%

€ 26.812 4 0,39% € 35.852 240 40,12% € 44.892 27 98,88%

€ 27.113 7 0,48% € 36.153 257 43,47% € 45.193 14 99,06%

€ 27.414 3 0,52% € 36.454 263 46,89% € 45.494 13 99,23%

€ 27.716 14 0,70% € 36.756 223 49,80% € 45.796 13 99,40%

€ 28.017 12 0,86% € 37.057 234 52,85% € 46.097 8 99,51%

€ 28.318 15 1,05% € 37.358 217 55,67% € 46.398 9 99,62%

€ 28.620 22 1,34% € 37.660 259 59,04% € 46.700 8 99,73%

€ 28.921 26 1,68% € 37.961 249 62,29% € 47.001 3 99,77%

€ 29.222 30 2,07% € 38.262 250 65,54% € 47.303 7 99,86%

€ 29.524 27 2,42% € 38.564 215 68,34% € 47.604 1 99,87%

€ 29.825 34 2,86% € 38.865 233 71,38% € 47.905 3 99,91%

€ 30.126 48 3,49% € 39.166 192 73,88% € 48.207 2 99,93%

€ 30.428 56 4,22% € 39.468 208 76,59% € 48.508 2 99,96%

€ 30.729 69 5,12% € 39.769 172 78,83% € 48.809 1 99,97%

€ 31.030 71 6,04% € 40.070 191 81,31% € 49.111 0 99,97%

€ 31.332 78 7,06% € 40.372 158 83,37% Other 2 100,00%

€ 31.633 98 8,33% € 40.673 153 85,36%

€ 31.934 106 9,71% € 40.974 152 87,34%


APPLYING MONTE CARLO SIMULATION IN REAL ESTATE … ISEL… · APPLYING MONTE CARLO SIMULATION IN...

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