Mathematical Methods in Economics and · PDF fileMathematical Methods in Economics and Finance...

Mathematical Methodsin Economics and Finance

Editor and Managing Director:Elio Canestrelli

Guest Editors:Marco CorazzaClaudio Pizzi

Editorial Board:Elio CanestrelliMarco CorazzaPaola Ferretti

Editorial Assistants:Diana BarroMartina Nardon

Advisory Board:Anna Rita Bacinello, TriesteAntonella Basso, VeneziaPaolo Bortot, VeneziaGiovanni Castellani, VeneziaFrancesco Mason, VeneziaGraziella Pacelli, AnconaPaolo Pianca, VeneziaBruno Viscolani, Padova

Contents

Global asset return in pension funds: a dynamical risk analysis . . . . 1Sergio Bianchi and Alessandro Trudda

A tale of two systems: Winners and Losers when moving from

defined benefit to defined contribution pensions . . . . . . . . . . . . . . . . . . . . 17Evert Carlsson, Karl Erlandzon and Jonas Gustavsson

A stochastic model for the analysis of demographic risk in

pay-as-you-go pension funds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Alessandro Fiori Maccioni

A policyholder’s utility indifference valuation model for the

guaranteed annuity option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Matheus R. Grasselli and Sebastiano Silla

Mergers, acquisitions, and innovation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81Hidenobu Hirata

Risk indicators in equity markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101Leonard MacLean, Giorgio Consigli, Yonggan Zhao and

William Ziemba

High watermarks of market risks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119Bertrand Maillet, Jean-Philippe Medecin and Thierry Michel

Preface

This is the second of two special issues of Mathematical Methods in Economics

and Finance devoted to the International Conference MAF 2008 – Mathematical

and Statistical Methods for Actuarial Sciences and Finance held in Venice (Italy)

from March 26 to 28, 2008.

The conference has been the first international edition of a biennial national

series begun at 2004, which was born by a brilliant belief of the colleagues –

and friends – of the Department of Economics and Statistical Sciences of the

University of Salerno: the idea following which the cooperation between mathe-

maticians and statisticians in working in actuarial sciences, in insurance and in

finance can improve research on these topics. The proof of the goodness of this

belief has consisted in the wide participation to these events.

This issue collects a series of original papers freely submitted to the journal

by Contributors of the conference and, following the usual praxis, each peer

reviewed by at least two anonymous referees.

Guest Editors (and co-Chairs of the conference):

Marco Corazza and Claudio Pizzi

Global asset return in pension funds:a dynamical risk analysis

Sergio Bianchi1 and Alessandro Trudda2

1 D.I.Me.T. and L.I.S.A.,University of Cassino,

Via S. Angelo, 03043 Cassino, [email protected]

2 D.E.I.R., University of Sassari,Via Torre Tonda n. 34, 07100 Sassari, Italy

[email protected]

Abstract. The aim of the paper is to develop a technique for rebal-ancing pension fund portfolios in function of their pointwise level of risk.The performance of pension funds is often measured by their global assetreturns because of the latter’s influence on periodic contributions and/orfuture benefits. However, in periods of market crisis attention is focusedon the risk level given their social security (and not speculative) function.We describe the process of the global asset return by a multifractionalBrownian motion using the function H(t) to detect high or low volatil-ity phases. A procedure is carried out to balance the asset compositionwhen the established local degree of risk is exceeded. The application iscarried out on portfolios obtained in accordance with Italian regulationsregarding investment limits.

Keywords. Pension funds, risk control, multifractional Brownian mo-tion.

M.S.C. classification. 28A80, 91B28, 91B30.

J.E.L. classification. C22; G11; G23.

1 Introduction

Pension funds mainly have a social security function of reimbursing workers’savings in the form of a life annuity. This involves an accurate and prudent assetallocation scheme administration.

It has been debated that post retirement benefit plans should have limita-tions on their asset allocation, based on the risk profile of the different financialinstruments available on the financial market.

The performance of pension funds is usually measured in terms of returnsrather than risk. Risks are taken into account especially during market crises,when losses in the portfolio of financial instruments of the fund could lead todepreciations in the accrued contributions.

2 Sergio Bianchi and Alessandro Trudda

Some studies (see [24]) show that post 2001 bankruptcies of US pension fundshad their roots in the actuarial evaluation techniques rather than in asset losses,if long-term stock return is considered. According to [5] and [17], post retirementbenefit plans, pertaining to the ‘first pillar’ of a pension system, should not investin high-risk financial instruments because this would lead to problems related tomoral hazard and to the evaluation of ‘superfluous risk’.

Trudda [27] proposes an application to the pension funds of Italian profes-sional Orders, in which marginal increments in global asset return appear tostrongly reduce the default probability. He also shows that there is an incentiveto take superfluous risks in the case of a slackening of regulations.

More recently, Otranto and Trudda [18] have supported the idea that thereis a need for a classification of the various degrees of risk for pension funds.They propose a cluster analysis based on the GARCH volatility of the rates ofreturn. In [19] another methodology is carried out distinguishing between twokinds of risk for pensions funds: constant risk and time-varying risk. Althoughthe method provides a satisfactory ex post risk analysis, the large lag necessaryto get reliable estimates weakens its employment in practical applications whena timely response is required.

Bikker, Broaders and Drew [8] study the impact of stock market performanceon the investment policy of Dutch pension funds and show that their investmentpolicies are partially driven by the cyclical performance of the stock market.In addition they point out that pension funds respond asymmetrically to stockmarket shocks: rebalancing is much stronger after negative equity returns.

Stewart [26] analyzes the increasing tendency of pension funds to invest inhedge funds. He observes that in many cases the real risk is not correctly per-ceived. This is due to an inefficient regulating system and, in several countries,the absence of risk monitoring instrument.

In many cases the rules on pension funds investments are derived from thesame laws that regulate investment companies, considering their speculativefunction. These regulations often indicate a qualitative restriction without lim-iting the quantitative measurement of the risk.

In Italy the regulating system for pension funds establishes non restrictiverules in the investment portfolio composition. Pension funds can invest in liquidassets, stocks, share of common investment funds. There are some restrictionsabout investments in equity and bonds traded in the over the counter marketsand/or in non OECD countries.

In this paper we concentrate on the investment risk: a dynamical analysisof pension fund’s portfolios is performed by estimating the pointwise regularityof the return series, assuming that these can be modeled by a multifractionalGaussian process, using the function H(t) to detect high or low volatility phases.In this framework, the estimator we use quantifies the pointwise degree of theobservations’ departure from independence. In a more general way we estimatethe local smoothness of a signal representing the portfolio quote. The intuition isthat this can well synthesize the local degree of risk of a given asset or portfolio.Provided that the window of estimation is sufficiently small, it should be possible

Global asset return in pension funds. . . 3

to build a warning system for monitoring the risk in pension funds. In this way,we can monitor the risk evolution after a short time using few daily data, thanksto the good rate of convergence of the estimator. In the paper we develop thesystem and describe the appropriate techniques for the automatic compositionof the fund’s portfolio in case of infringement of the given risk thresholds.

In the application three investment portfolios are simulated respecting Italianfinancial laws to show how the levels of risk obtained can be very different.

The paper is organized as follows: in section 2 we recall the main propertiesof the model we assume to generate the price dynamics. In section 3 the esti-mator of the pointwise regularity of the process is discussed. Section 4 concernsthe analytical relationship between the portfolio’s H(t) and the H(t)’s of itsindividual assets. In section 5 we develop an analytical approach through theanalysis of the variable H(t). The purpose is the control of risk by a continuousmonitoring and rebalancing policy using optimal portfolio definition. An appli-cation of three simulated portfolios with different risk degrees is carried out insection 6; we use the introduced approach to evaluate the the levels of risk overthe time and to develop a rebalancing technique, through the analysis of thevariable H(t). Finally some conclusion are discussed in section 7.

2 The model

In the following we will assume the log price dynamics to be described by a ver-satile process: the multifractional Brownian motion (mBm). A convenient wayto introduce the mBm is recalling its very well-known special case: the fractionalBrownian motion (fBm). Defined in a celebrated paper by Mandelbrot and VanNess [16], the fBm is characterized by a slowly decaying autocorrelation functiondepending on the parameter H ∈ (0, 1], named Hurst exponent. Following thedefinition that can be found in [9], the process has moving average representation

BH(t) = CπK(2H)1/2∫

R

ft(s)dB(s) (1)

with

ft(s) =1

Γ(

H + 12

)

|t− s|H− 12 1]−∞,t](s)− |s|H− 1

2 1]−∞,0](s)

where B(·) stands for the ordinary Brownian motion, C is a positive constant

and K is the function defined on ]0, 2[ as K(α) = Γ (α + 1)sin απ

2

π . The processis self-similar3 of parameter H and has stationary increments. Its covariancefunction reads as3 We recall that the process X(t), t ∈ T is said self-similar with parameter H if for

any α > 0 X(αt) d= αHX(t), where the equality holds for the finite-dimensional

distributions of the process (see e.g. [25]).


E (BH(t)BH(s)) =c2

2

(

|t|2H + |s|2H − |t− s|2H)

(2)

The fBm can be generalized by allowing H to vary over time. This extension− known as multifractional Brownian motion (mBm) (see [20], [21], [1]) − hasthe following representation

MH(t)(t) = CπK(2H(t))1/2∫

R

ft(s)dB(s) (3)

with

ft(s) =1

Γ(

H(t) + 12

)

|t− s|H(t)− 12 1]−∞,t](s)− |s|H(t)− 1

2 1]−∞,0](s)

where H : [0,∞) → (0, 1] is required to be a Holder function of order 0 < η ≤ 1to ensure the continuity of the motion.

Notice that since H(t) is the punctual Holder exponent of the mBm at pointt, the process is locally asymptotically self-similar with index H(t) (see, e.g.,[6]) in the sense that, denoted by Z(t, au) := MH(t+au)(t + au) −MH(t)(t) theincrement process of the mBm at time t and lag au, it holds

lima→0+

a−H(t)Z(t, au)d= BH(t)(u), u ∈ R. (4)

The above distributional equality indicates that at any point t there existsan fBm with parameter H(t) tangent to the mBm. Moreover, since BH(t)(u) ∼

N (0, C2u2H(t)), the infinitesimal increment of the mBm at time t, normalized byaH(t), normally distributes with mean 0 and variance C2u2H(t) (u ∈ R , a → 0+).

The increments of the mBm are no longer stationary nor self-similar; despitethis, the process is extremely versatile since the time dependency of H is usefulto model phenomena whose punctual regularity is time changing.

From a financial viewpoint one can think of H(t) in a suggestive way asa ”memory” function, i.e. as the degree of confidence the investors nourish inthe past. High values of H(t) correspond to trends (or low volatility phases),i.e. to periods in which the past information weighs in the investors’ tradingdecisions; low values of H(t) are associated to high volatility periods, in whichprices display an antipersistent or mean reverting behaviour because of the quickbuy-and-sell activity that is typically induced by uncertainty. Standard financialtheory is recovered whenH = 1

2 , case in which the mBm reduces to the Brownianmotion. The level of risk coupled with a financial time series is therefore framedinto a dynamical perspective in which it can change from point to point, evenin a strong way. What makes the difference here is not much and not only thetype of investment (bond, stock, derivatives) but the time.


3 Pointwise estimation of the Holderian regularity of themBm

Given a sample path of the mBm, one of the main problems is estimating thefunction H(t) from actual data. To deal with this problem one could think atadapting the traditional estimators ofH available in literature in order to shadowthe dynamics of H(t). The weakness of this approach resides in the fact that verylarge samples are needed to get reliable estimates and in over a long time-spanH is likely to change even widely. So, more efficient estimators are needed inthe case of the mBm. An answer to this problem is provided by Bianchi [7],who develops the work of Peltier ad Levy Vehel ([20]) and defines a family of”moving-window” estimators of H(t) based on the k-th absolute moment of aGaussian random variable of mean zero and given variance VH (the variance ofthe unit lag increment of a mBm). Given a series of length N and a window oflength δ, the estimator has the form

Hkδ,N (t) =

log(

2k/2Γ(

k+12

)

Vk/2H

)

− log(√

πδ

∑t−1j=t−δ |Xj+1,N −Xj,N |k

)

k log (N − 1)(5)

for j = t− δ, ..., t− 1; t = δ + 1, ..., N ; k ≥ 1.

Thanks to its good rate of convergence O(

δ−12 (logN)

−1)

, (5) allows reliable

estimates even for very short δ′s. The family of estimators (5) was proved to becorrect and normally distributed as

Hkδ,N (t) ∼ N

(

H(t),π

δk2 ln2 (N − 1) 2k(

Γ(

k+12

))2σ2

)

(6)

σ2 being the variance of a Gaussian random variable defined as a proper rescaledsum. Toilsome computations show that whenH = 1

2 the variance of the estimatorreduces to

V ar(Hkδ,N (t)) =

√π

δk2 ln2 (N − 1)[

Γ(

k+12

)]2 ·(

Γ

(

2k + 1

2

)

− 1√π

[

Γ

(

k + 1

2

)]2)

(7)

and the optimal value of k is deduced by minimizing the last relation. So onefinds that the minimum of (7) takes place when k = 2, value which will be usedin the empirical application discussed below. An idea of the way the estimator(5) work is provided by Figure 1. Panel (a) shows a sample path generated by amBm with sinusoidal functional parameter (four periods were considered, withH(t) ranging in the interval [0.2, 0.8]); panel (b) shows the variations of the signal(notice the bursts of variance corresponding to low values of H(t)); finally, inpanel (c) the continuous line is the functional parameter and the zigzagged lineis the functional parameter estimated by filtering the original signal through (5),setting δ = 30.


Fig. 1. Estimation of the Holderian function of a simulated mBm

4 The portfolio’s H(t)

In this section we derive the portfolio’s H(t) and write it as a function of theH(t)’s of individual assets.As usual, let

Πj =N∑

s=1

αsXs,j (8)

denote a portfolio ofN assets, each characterized by its own functional parameter

sHkδ,q,n(t) (s = 1, ..., n) and with unit variance at time n.

We set k = 2 because it is easy to show that this value minimizes the estima-tor’s variance when H = 1/2. In this way the estimator of the portfolio’s H(t)can be written as a function of sH

kδ,q,n(t) and it reads as:


ΠH2δ,q,n(t) = −

ln

(

N∑

s=1α2s

(

n−1q

)−2sH2δ,q,n(t)

+

2 ln(

n−1q

)

+2N−1∑

p=1

N∑

r=p+1αpαr

(

n−1q

)−(pH2δ,q,n(t)+rH

2δ,q,n(t))

ρp,r,δ

)

2 ln(

n−1q

)

(9)

with ρp,r,δ :=

t−1∑

j=t−δ

|dXp,j,q||dXr,j,q|√

t−1∑

j=t−δ

|dXp,j,q|2t−1∑

j=t−δ

|dXr,j,q|2

In fact, let dΠj,q = Πj+q −Πj =N∑

s=1αsdXs,j,q denote the portfolio’s incre-

ments, where dXs,j,q := Xs,j+q −Xs,j . One has

ΠH2δ,q,n(t) = −

ln

t−1∑

j=t−δ

∣

∣

∣

∣

N∑

s=1

αsdXs,j,q

∣

∣

∣

∣

2

K2(δ−q+1)

2 ln(

n−1q

) =

= −ln

N∑

s=1

α2s

t−1∑

j=t−δ

dX2s,j,q+2

t−1∑

j=t−δ

N−1∑

p=1

N∑

r=p+1

αpαr|dXp,j,q||dXr,j,q|

K2(δ−q+1)

2 ln(

n−1q

)

From (5) it readily follows that

t−1∑

j=t−δ

dX2j

K2(δ − q + 1)=

(

n− 1

q

)−2H2δ,q,n(t)

and therefore

ΠH2δ,q,n(t) = −

ln

N∑

s=1

α2s

(

n−1q

)

−2sH2δ,q,n(t)

+2

t−1∑

j=t−δ

N−1∑

p=1

N∑

r=p+1

αpαr|dXp,j,q||dXr,j,q|K2(δ−q+1)

2 ln(

n−1q

)


ΠH2δ,q,n(t) = −

ln

(

N∑

s=1α2s

(

n−1q

)−2sH2δ,q,n(t)

+

2 ln(

n−1q

)

+2

(

t−1∑

j=t−δ

α1α2|dX1,j,q||dX2,j,q|+...+t−1∑

j=t−δ

αN−1αN |dXN−1,j,q||dXN,j,q|)

K2(δ−q+1)

2 ln(

n−1q

)

(10)

A more insightful way of writing relation (10) again exploits (5), from whichit is easy

−pH2δ,q,n(t) =

ln

t−1∑

j=t−δ

|dXp,j,q|2

K2(δ−q+1)

2 ln(

n−1q

) and −r H2δ,q,n(t) =

ln

t−1∑

j=t−δ

|dXr,j,q|2

K2(δ−q+1)

2 ln(

n−1q

)

Summing up side by side we get

−pH2δ,q,n(t)−r H

2δ,q,n(t) =

ln

t−1∑

j=t−δ

|dXp,j,q|2t−1∑

j=t−δ

|dXr,j,q|2

K4(δ−q+1)2

2 ln(

n−1q

)

and therefore

(

n− 1

q


2δ,q,n(t))

=

√

∑t−1j=t−δ |dXp,j,q|

2∑t−1j=t−δ |dXr,j,q|

2

K2(δ − q + 1).

from which it follows:

(

n− 1

q


2δ,q,n(t))

ρp,r,δ =

t−1∑

j=t−δ

|dXp,j,q| |dXr,j,q|

K2(δ − q + 1)

and by substituting in (10) one gets (9), where the factor ρ clearly representsthe correlation of the absolute increments of the process.

5 The dynamic optimization problem

In order to cope with the optimization problem we use the relationship betweenthe portfolio’s H(t) and the functions H(t) of each asset included in the portfolio


itself as developed in the previous Section. The procedure intervenes when theH(t) value decreases under a fixed threshold. This dynamic approach, combinedwith a control on the level of return, is based on the assumption that it existsan inverse relation between the value of the portfolio’s H(t) and its exposure torisk. In other words, since high values of H(t) are indicative of trends, once theprocedure excluded that the trend is negative, one can use this information torebalance the portfolio in order to control its level of risk. Let us denote by L,B and S the liquidity, the bond and the stock components of the portfolio, byΠX the set of indexes pertaining to the investment of type X (X = L, B orS), by N the number of assets in the portfolio, by αX the (fixed) ratio of assetsof type X, by rΠ(t) the log price of asset k at time t and ∆ the length of thewindow. Equipped with this notation, the risk-minimizer portfolio manager hasto solve the following constrained problem

maxαk

ΠHk1,q,n(t)

s.t.∑

k∈ΠLαk = αL

∑

k∈ΠSαk = αS

∑

k∈ΠBαk = αB

αk ≥ 0

αL + αS + αB = 1

∑th=t−∆

rΠ(h)∆+1 ≥ φ; φ ≥ 0

(11)

which means to determine the vector (α1, ..., αN ) defining the H-optimal port-folio.

At each time t the algorithm checks whether the portfolio’sH(t) is lower thanthe fixed threshold. If not the portfolio is mantained, otherwise it is rebalancedusing the new weights determined by solving the optimization problem. For theconstrained optimization we used an extension of primal interior point methods,which applies sequential quadratic programming techniques to a sequence ofbarrier problems. Trust regions are used to ensure the robustness of the iterationand to allow using the second order derivatives. The software was developed inMatLab environment at the L.I.S.A. 4, using the function fmincon (see [23])already implemented in the optimization toolbox.

4 The computer lab for advanced scientific computing operating within D.I.Me.T. Theauthors aim a special thank to dr. Augusto Pianese and to dr. Alexandre Pantanellafor the algorithm implementation.


6 Application

In order to estimate the risk dynamic, we started using three portfolios complyingwith the Italian laws on pension funds investments. The analyzed portfolios werecharacterized by strong differences in terms of returns variability and thereforein the risk profile. In the maximum and medium risk portfolios were includedinvestments in stock components traded in over the counter markets and in nonOECD countries. We used daily data from 30/09/2003 to 19/04/2007.

Table 1 shows the fixed ratios of the portfolios’ components (αL, αB andαS).

Table 1. Composition of the three portfolios

Portfolio Liquidity Bonds Stocks

Minimum risk 10% 90% 0%Medium risk 10% 70% 20%Maximum risk 10% 40% 50%

The portfolio composition respected the investment limits imposed by Italianregulation (over the counter and non OECD assets). The minimum risk portfoliowas characterized by a larger investment of 90% in bonds (Bond PE 22.5%, ArcaMM 22.5%, Arca RR 22.5%, Arca TE 22.5%), 10% liquidity (Libor 5%, ArcaBT 5%), with a very low standard deviation value showing the low risk profileoffered.The medium risk portfolio was composed of 70% bonds (Bond PE 17.5%, ArcaMM 17.5%, Arca RR 17.5%, Arca TE 17.5%), 10% liquid assets (Libor 5%,Arca BT 5%) and 20% stocks (Mibtel 2.8%, Ibm 2.8%, Nasdaq 2.8%, DowChem2.8%, Ibovespa 2%, Shangai 2%, Google 2.8%, Kospi 2%. It included stockscomponent traded in OECD unregulated markets (2% Kospi) and in non OECDregulated markets (2% Ibovespa, 2% Shangai) This portfolio was characterizedby an intermediate risk.The third portfolio (maximum risk) was composed of 10% liquid assets (Libor5%, Arca BT 5%), 40% bonds (Bond BDPE 10%, Arca MM 10%, Arca RR10%, Arca TE 10% ), 50% stocks (Mibtel 8.8%, IBM 8.8 %, Nasdaq 8.8%, DownChem 8.8%, Ibovespa 2%, Shangai 2%, Google 8.8%, Kospi 2%). It includedstocks component traded in OECD unregulated markets (2% Kospi) and in nonOECD regulated markets (2% Ibovespa) In spite of its strong bonds component,this portfolio presents high returns variability expressed by an high standarddeviation value.

Figure 2 displays the global asset return and the estimated H(t) values ofthe three initial portfolios. As expected, the return increases with the risk of theportfolios (panel (a)) whereas high values of H(t) are associated with a low apriori risk (panel (b)).


In order to define a rebalancing strategy, we developed a procedure work-ing as follows: given a threshold H∗, at each time we test whether the currentestimation of H(t) is below the fixed threshold, which means that - under theassumptions of our model - the portfolio is going subject to an excess risk. In thiscase, we rebalance the portfolio solving the optimization problem (11). Other-wise we maintain the current portfolio. Notice that the last constraint of problem(11) is meant to guarantee a minimum positive return φ for the portfolio; thecondition is necessary because the sole H(t) does not give information about thedirection of the local trend, which can be negative as well as positive. The strat-

Fig. 2. Global asset return and the estimated H(t) values of the three initial portfolios

egy described above was applied to the three portfolios with different thresholds0.75 ≤ H∗ ≤ 0.90 and φ given by the daily rate of return equal to the five-yearsaverage Gross Domestic Product (GDP), using ∆ = 0. Obviously, the number ofrebalancings strongly depends on the thresholds H∗ and φ (they increase withthe former and decrease with the latter). An example of the results producedby the strategy is shown in Figures 3-6, obtained setting H∗ = 0.85. Figure 3displays the values of H(t) for the rebalanced portfolio (continuous line) andfor the initial portfolio (dotted line). The vertical bars below indicate the timesin which the rebalancing has occurred. Observe that a convenient choice of theassets heavily modifies the risk profile, even of .2817 (at day 279). It is obviousthat the reduction of risk reflects in a lower return, as shown in Figure 4 whichdisplays the global asset return of the initial and the rebalanced portfolios (themaximum difference is under 0.1 on a time horizon of three years). Figure 5displays the risk-return profile of the portfolios; since differently from the tra-


ditional Markowitz’s model here we use H(t) as a proxy of the risk level, therisk-dominant rebalanced portfolios are located in the upper right area of thegraph. It is apparent the effect of the rebalancing strategy, which forces upwardH(t) (solid squares) with respect to the values of the initial portfolio (empty cir-cles). Finally, Figure 6 shows the ’s of the rebalanced portfolios. In this regard,observe that the optimization problem (11) does not contain constraints aboutover the counter and non OECD assets. This means that the new portfolios gen-erally do not comply the limits imposed by Italian regulation concerning the apriori risky markets; nonetheless, they are less risky than the initial portfolio’.

Fig. 3. H(t) dynamics of the maximum risk portfolio with threshold H = 0.85

The analyzed portfolios deserve a couple of further comments. First, in allcases the values of H(t) are significantly far from the central value assumed bystandard financial theory. This is consistent with a number of works, but here -differently from what occurs in the case of single stocks or indexes - the valuesare also significantly above . Second, large variations characterize the estimates;for the minimum, the medium and the maximum risk portfolios the ranges arerespectively 0.162, 0.164, 0.136. Again, this is inconsistent with the models as-suming a constant value of H and strongly suggests a dynamical approach toportfolio management. Looking at things with more detail one realizes that theestimates of H(t) seem to cluster towards low values. This is reflected by thenegative skewness of the distributions: -0.61, -0.68 and -0.59 respectively for theminimum, the medium and the maximum risk portfolios.


Fig. 4. Global Return of the initial and rebalanced portfolios Maximum risk portfoliowith threshold H = 0.85

Fig. 5. Return-risk profile. Maximum risk portfolio threshold H = 0.85


Fig. 6. Weights of the portfolio. The assets are the following: 1−Libor; 2−Arca BT;3−Bond PE; 4−Arca MM; 5−Arca RR; 6−Arca TE; 7−MibTel; 8−IBM; 9−Nasdaq10−DowChem; 11−Ibovespa; 12−Shangai; 13−Google; 14−Kospi.

7 Concluding remarks and further developments

The September 2001 market crisis caused the failure of some pension funds inthe USA and Europe. A debate about the financial investment limits and the riskstructure of pension funds was opened. Several analysis highlights the tendencyof the Funds to increase the portfolios risk in order to obtain higher values of theexpected global asset return. Some economic theories study phenomena like as amoral hazard problem because of accounting rules which encourage Pension Cor-poration to assume excessive risk. Many authors emphasize that pension fundshave to maintain a prudent profile because the social function (in particular forthe first pillar) prevails over the speculative function. In general, financial lawsuse mutual fund regulations to determine the limits of investments in risky fi-nancial instruments. Moreover, regulations are often qualitative and do not usequantitative methods.In order to investigate the regulation potency, in our applications we use invest-ment portfolios compliant with the Italian laws on pension funds. The resultshighlights how Italian pension fund regulation permits investments whit verydifferent risk degrees.A dynamic approach is introduced in order to constantly balance the invest-ment portfolio to control the risk evolution. The risk dynamic is analyzed usinga multifractional Brownian motion to describe the log price of the global assetportfolios. We use the function H(t) to evaluate the volatility level in the instantt: when the estimation of H(t) is below the fixed threshold H∗, an optimization


problem is applied to rebalance the portfolio over the time in order to controlthe volatility of global asset return. It’s important to note that the procedurerespond a volatility changes in a quick time using only the lag data because ofthe convergence proprieties of H(t). The applications show that using this pro-cedure to control the excess of risk, a cost in term of lower global asset return ispayed. An interesting development of this work will be to investigate the rela-tionship between the level of maximum volatility required H∗ and the reductionof returns using our strategy.

References

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and Applications, SIAM Review, 10 (1968) 422-43717. McClurken: Testing a Moral Hazard Hypothesis in Pension Investment. Washing-

ton University, Washington (2006)


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A tale of two systems: Winners and Losers whenmoving from defined benefit to defined

contribution pensions⋆

Evert Carlsson, Karl Erlandzon, and Jonas Gustavsson

Centre for Finance & Department of Economics, School of Business, Economics &Law, Goteborg University

PO Box 640, S-405 30 Goteborg, Swedenevert.carlsson, [email protected]

[email protected]

Abstract. There is a trend among employers to prefer Defined Contri-bution instead of Defined Benfit pension plans, since the former transferall risks associated with investment return, longevity, etc from the em-ployer to the employee. However, Defined Contribution plans also allowthe individual to enter into positions contingent on the individual sit-uation. This paper investigates the individual welfare consequences ofdifferent plans. We used the recent transition from defined benefit to de-fined contribution for white-collar workers in Sweden as the benchmarkfor our analysis. The framework for our analysis is a life cycle model ofa borrowing-constrained individual’s consumption- and portfolio choicesin the presence of uncertain labour income. The main result is that indi-viduals with the characteristic of a low expected pre-retirement incomerelative to average income and high variance in earnings are winners(men with university degree in the private sector), and that those withthe opposite characteristic (women with university degree in the publicsector) are losers.

Keywords. Life-cycle, portfolio choice, defined contribution, definedbenefit, income process.

M.S.C. classification. 62M45, 65C05, 68T05.

J.E.L. classification. D31, D91, G11, G23, H24, J31.

⋆ We would like to thank Lennart Flood for helping us with numerous discussionsabout income definitions and LINDA problems and Rolf Poulsen as well as partici-pants at the seminar of the Centre for Finance for comments and discussions. Theusual disclaimer applies. The authors gratefully acknowledge the financial supportof the Centre For Finance. Erlandzon and Gustavsson also gratefully acknowledgesupport from Stiftelsen Bankforskningsinstitutet and Stiftelsen Centrum for Finans,respectively.

18 Evert Carlsson et al.

1 Introduction

Defined Contribution—DC pension plans1 are now often the preferred pensionsystem among employers. This is not very surprising since the shift from De-fined Benefit—DB2, transfers all the risks associated with investment return andlongevity from the employer to the employee. However, there are also several ad-vantages for the individual with a DC plan: it allows the individual to enter intospecific positions, which reduces the consequences of forcing all individuals intoone-size-fits-all, in terms of risk and return characteristics; it facilitates porta-bility when the agent transfers from one employer to another; and not the leastit assuages the risks of lower wages in the final years of employment. Which ofthese systems that are beneficial to the individual is very state dependent andmerits this research.

In this paper we have analysed the welfare consequences for the individualwhen transferring from a DB to a DC system. As a benchmark for this anal-ysis, we have chosen the recently negotiated transfer from a DB to a DC planbetween private white-collar workers union and their employers in Sweden, cf.[21]. This analysis is even more pertinent, since this transfer will most likely bethe blueprint for a similar future settlement for the public employees.

The main result is that individuals with the characteristic of a low expectedpre-retirement income relative to average income and high variance in earningsare winners (men with university degree in the private sector), and that thosewith the opposite characteristic (women with university degree in the publicsector) are losers.

Our analysis draws heavily on the literature highlighting: life-cycle savingand consumption, [20] and [15]; and portfolio-choice, [22] and [19]. [11], [6] and[16] created life-cycle models with uncertain wages and borrowing constraints;which showed that market-incompleteness is important when explaining individ-ual choice and welfare effects. [10] and others extended the model with portfolio-choice between a risk-free and a risky asset. [2] added a mandatory pensionscheme to the model.

The introduction of non-tradeable human capital into the intertemporal lifecycle model with portfolio choice and consumption, creates an asset that willinfluence—how much the individual saves and the optimal portfolio choice insavings. These choices depend on the expected individual dividend profile fromhuman capital and associated uncertainties, but also on the characteristics ofother assets; primarily private savings, pension savings and housing.

Labour generates two types of dividends: wages and pension contributions. Inthis paper we estimated the income process that should be used as the underlyingfor calculating the derivatives—net wages and pension contributions; keeping the“dividends” from human capital separate from other types of asset-income.

[7], [16], [10], and other similar earlier studies generally treated returns fromhuman capital as equal to earned income net of return from private savings.

1 A DC plan accumulates a proportion of every salary as a contribution.2 A DB plan pays a proportion of final salary as a pension.

A tale of two systems. . . 19

Such a wide definition lead to some double-counting, for those who retire earlyor receive pension benefits dependent on contributions during their working life.In our definition of returns from human capital, we only included income thatstem from individual productivity and insurances against, e.g. disability, parentalleave, unemployment etc.—not from early withdrawals from retirement savings.

We are interested in the expected income profile as the underlying for pensioncontributions and taxes, which influence the individuals future choices in termsof saving and portfolio allocation. It is therefore natural to model individualsrather than households3, since pension contributions and taxes are primarilydependent on the individual instead of family incomes.

The remainder of the paper is organised as follows: Section 2 describes themodel, while Section 3 describes the optimisation problem, and Section 4 thecalibration of the model. Section 5 discusses the results, and the final Section 6summarizes and draws some conclusions.

2 The Model

2.1 Individual preferences

We assume that an individual maximise the expected utility over their adultlife-cycle, which starts at the age of τ0, and dies no later than at the age of T .We assume that an individual has constant relative-risk-aversion preferences fora single non-durable consumption good—Cτ .

Individual preferences at time—m are defined as

C1−γm

1− γ+ Em

T∑

τ=m+1

δτ−m

τ−2∏

j=m

pj

pτ−1C1−γ

τ

1− γ+ b(1− pτ−1)

D1−γτ

1− γ

, (1)

γ is the coefficient of relative risk aversion, pτ is the one-year age-contingentsurvival-probability, δ is the discount factor, b is the bequest parameter and Dτ

is the bequest amount.

2.2 The labour-income process

Following [7], we assume that the individual income process during workinglife—Lit, is exogenously given by

log(Lit) = liτ = f(τ,Ziτ ) + viτ + εiτ , τ ≤ K, (2)

3 Furthermore, female labour-participation and divorce rates are high, which—together with an age-difference between man and wife—could have obscured theexpected-wage profile if estimated on family data. When estimating on family data,the educational status, age and retirement date is typically defined by the head ofhousehold only.


where—f(τ, Ziτ ) is a deterministic function of individual i′s age—τ , and a vectorof the individual characteristics4—Z, where—K is the retirement age, and—viτis given by

viτ = viτ−1 + uiτ , (3)

where the permanent shock—uiτ ∼ N(0, σ2u) is independent from the idiosyn-

cratic temporary shock—εiτ ∼ N(0, σ2εk). The permanent shock—uiτ , consists

of a group aggregate component—ξkτ ∼ N(0, σ2ξk) as well as an idiosyncratic

component—ωiτ ∼ N(0, σ2ωk),

uiτ = ξkτ + ωiτ . (4)

2.3 Assets

There are two assets, one risky and one risk-free asset with after-tax real log-returns equal of reτ and rf respectively. Excess return is defined as

reτ − rf = µe + ητ , (5)

where the noise—η is correlated with the group-aggregate innovation in perma-nent labour-income—ξk, which allows for a group specific sensitivity to the riskyasset,

[

ξ

η

]

∼ N

([

− 12σ

2ξ

− 12σ

2η

]

,

[

Σ σξη

σ′

ξη σ2η

])

. (6)

2.4 Past and present mandatory savings and retirement benefits

In the old system5, individuals have a defined-benefit and a defined contributionplan. The defined benefit plan has a payout of 10%, 65% and 32.5% of incomesat retirement6 in the intervals [0, 320), [320, 850), and [850, 1270) respectively7.

Payout from this plan is constant in real terms, and guaranteed for the re-mainder of life, PODBiτ ,

PODBiτ = 0.1min[

LPi64; 320

]

+0.65min

[

max(

LPi64 − 320; 0

)

; 850− 320]

+0.325min

[

max(

LPi64 − 850; 0

)

; 1270− 850]

.

(7)

The defined contribution plan has contributions at 4.5% of annual labour incomeup to 320.

4i.e. age, martial status, family size, and number and age of children.

5 Individuals born before 1979.6 In reality it depends on the wage during the five years prior to retirement. However,modelling this rule correctly would have necessitated additional state variables. Wetherefore approximate this by only including the permanent income changes untilretirement.

7 In the following, we express all amounts in thousands of SEK. The present exchangerate is circa 6 SEK / USD.


The new system is only based defined contributions (cf. [21]), with contribu-tions set to: 7% for annual incomes up to 320 and 30% for incomes above thislimit. Contributions to the defined contribution plans—DC are therefore,

DCiτ =

0.045min [Liτ ; 320] ,0.07min [Liτ ; 320] + 0.3max (Liτ − 320; 0) ,

if in old system,if in new system.

(8)

Individuals can choose the fraction, λ of the defined contribution wealth,DCW to allocate to the risky asset,

DCWiτ =

erf [

1 + λiτ−1(eµe+ητ − 1)

]

DCWiτ−1 +DCiτ , τ < 65,

erf [

1 + λiτ−1(eµe+ητ − 1)

]

DCWiτ−1 − PODCiτ , τ ≥ 65,(9)

where, PODC, is the mortality-adjusted annuity from the defined contributionplan.

Irrespective of system, all individuals also receive social security pensionbenefits—SS, which depend on the individual’s labour-income trajectory duringworking life. In [4], we modelled this system as state dependent and from thesimulated trajectories we have estimated a piece-wise linear retention-rate,

SSiτ = 0.4min[

LPi64; 320

]

+ 0.1min[

max(

LPi64 − 320; 0

)

; 850− 320]

, (10)

dependent on the permanent part of labour income—LPi64,

LPi64 = efk(τ,Zi64)+vi64 . (11)

All payouts from these pension plans are assumed to be forfeited in the eventof death.

2.5 Labour income and taxes

Wage and retirement income—L can now be defined as

Liτ =

eliτ , τ < 65,PODCiτ + PODBiτ + SSiτ ,

PODCiτ + SSiτ ,

τ ≥ 65 if in old system,τ ≥ 65 if in new system,

(12)

According to Swedish tax rules8, labour income and pension benefits aretaxed at a common rate, separate from capital income. To calculate net income—Lniτ , we first deduct a general allowance of 10; then a municipal tax of 30%; then

national tax of 20% on all income above 300; and finally, an additional nationaltax of 5% on income above 450. Net income is bounded below by the socialwelfare minimum-benefit and government-guaranteed minimum pension at 60.Therefore

8 We use the tax rules for incomes earned in 2003.


Lniτ = max[Liτ − 0.3max (Liτ − 10; 0)−

0.2max (Liτ − 300; 0)− 0.05max(Liτ − 450; 0); 60].(13)

All the threshold-values that create kinks in tax-rates and benefits9 are indexedto the expected growth in national labour income—µl, except the social welfareminimum benefit which is kept constant in real terms.

2.6 Private savings and consumption

An individual starts her optimisation life with initial wealth set to . In thefollowing pre-retirement years they receive wages, and in subsequent years re-tirement benefits. The individual has two control variables: the proportion ofcash on hand to consume—θτ , and what proportion of savings—ατ , to allocateto the risky asset. The cash on hand—disposable wealth, is therefore,

Xiτ =

erf [

1 + αiτ−1(eµe+ητ − 1)

]

[1− θiτ−1]Xiτ−1 + Lniτ , τ > τ0,

i + Lniτ , τ = τ0,

(14)

of which consumption is,Ciτ = θiτXiτ . (15)

There are also constraints on both borrowing and short-sales,

0 ≤ θiτ ≤ 1,0 ≤ αiτ ≤ 1.

(16)

3 Optimisation

To simplify the calculation10, we introduce a decision rule that defines the assetallocation in the defined contribution account. This rule originates from [19] andstates that; in complete markets—the allocation to risky assets—λ, is dependenton the relative size of investable assets to total wealth. In our model, total wealthis the sum of: present value of human capital, cash on hand and expected after-tax11 DC wealth—DCW at. The present value of human capital is the sum of:income plus defined benefits and defined contributions, net of taxes and adjustedfor survival probabilities. Prior to retirement, the human capital is discountedwith the complete market rate—s,

9 This is similar to the US since the ”bend points” when calculating the primaryinsurance amounts (PIA) are adjusted by average earnings growth.

10 The portfolio choice in the DC-account and for private savings is highly interdepen-dent, making a simultaneous choice very complicated numerically.

11 The after-tax rate is set to the municipal-tax only, since this is typically the onlytax that an agent pays when in retirement.


s = rf −σ2

ξ

2 + βk(µe +

σ2η

2 ),

where βk = Cov(η,ξk)V ar(η) ,

(17)

and with the risk-free rate after retirement. Our decision rule is adjusted for theimplicit equity exposure through the present value of human capital—βk∆,

λ = min

µe [DCW at + (1− θ)X + PV (HC)]

γσ2η [DCW at + (1− θ)X]

−βk∆PV (HC)

DCW at + (1− θ)X; 1

,

(18)where ∆ is the change in present value of human capital from a group specificpermanent income shock—ξk.

The individual’s problem therefore has four state variables (τ , v, X andDCW ) and two choice variables (θ and α) as well as four stochastic variables(ǫ, ω, ξ and η). The value function of their intertemporal consumption andinvestment problem can then be written as

Vτ (Γτ ) = maxθτ , ατ

C1−γτ

1−γ+ δEτ

[

pτVτ+1 (Γτ+1) + (1− pτ ) bD

1−γτ+1

1−γ

]

Γτ = Xτ , vτ , DCWτ .(19)

The solution to this maximisation problem together with our decision-rulefrom 18 gives us the state dependent policy rules,

θτ = θkτ (Γτ ),ατ = αkτ (Γτ ),λτ = λkτ (Γτ ).

(20)

We solved the problem numerically by backward recursion from the finalyear—T , using by-now standard methods, cf. [18] and [10].

4 Calibration of parameters

4.1 Estimation of labour income process

Follwing [7], we modelled the log of real income as deterministic part with bothpermanent and temporary shocks. Their description of the income-process hasbeen used in several life cycle models, cf. [2], [10], [4], [9], [5] and [23]. Thedeterministic part of Equation 2 was estimated (cf. Appendix 6 for details)using a longitudinal panel of data—LINDA, (cf. [12] for details), that coversthe Swedish population in the age interval [28, 64] for fourteen years during1992− 2005, resulting in more than 1.4 million observations.

The data set augmented with wealth information, has recently received atten-tion in cf. [1], [3] and [14]. The data-set was divided into twelve non-intersectinggroups, depending on sex, education and sector (private and public). Using the


methodology of [7], we estimated the variances of the permanent σ2u and transi-

tory σ2ε components of shocks to income as specified in Equation 2 (cf. Appendix

6).

4.2 Individual parameters

We used a standard set of assumptions with respect to the individual parametersfor the reference case. First, we set the coefficient of relative risk aversion—γ to 5and the discount factor—δ to 0.98. The gender specific survival probabilities—p

were taken from the Swedish life-insurers when underwriting new policies,i.e.they are forward looking. The bequest parameter—b was set to 1. Adult lifeis divided into two intervals: working life [28, 64] and retirement [65, 100]. Theimportance of the risk aversion parameter—γ will be elaborated on when wereport on the sensitivity analysis in Section 5.4.

4.3 Assets and correlations

In the optimisation, we set the risk-free after-tax rate—rf to 1.5%, which isconsistent with the present gross return of less than 2% for long-dated index-linked bonds. The mean after-tax equity premium—µe was set to 3%, whichis lower than the historical average, but corresponds well with forward-lookingestimates (cf . [8], [13]). Because of uncertainty about the equity-premium, weanalysed its sensitivity in Section 5.4. Volatility ση was set to 17% for the riskyasset.

Next, we followed the procedure of [10] to estimate the correlation—kη be-

tween group specific permanent labour income shocks—ξkτ and lagged equityreturns—ητ−1. Table 5, shows the estimated correlation, using the returns onthe Swedish equity-index—OMX and on the 12-month Swedish Treasury Billsas proxies for risky returns and the risk-free rate respectively.

We also set the growth in average labour income—µl to 1.8%, which is the es-timate used by the National Social Insurance Board. Finally, the initial wealth— is set to 40, corresponding to the mean wealth for individuals at the age of28.

5 Results

5.1 Labour income process

For reference, we plotted the average of the simulated income profiles for some12

of the groups, cf. Figure 5.1.

12 In order to increase readablity, we omitted the groups with similar profiles to thegroup with the lowest income.


Three findings are notable: First, individuals with a university degree experi-enced a significantly faster income growth in mid-life than did the other groups,a result which matches stylised facts from the US, cf. [17], [16] and [10]. Secondly,at each level of education, men had higher income than did women, at all stagesof the life-cycle. Thirdly, that remunerations in the private sector was typicallyhigher than in the public sector.

Our results also show a strikingly lower permanent variance if the agent isemployed by a public vs. a private entity, whereas the temporary variance wassimilar, except for those with university degrees. After controlling for private vs.public sector, most of the gender differences in variance, that we found in ourprevious study ([4]) disappeared.

Figure 5.1 shows the large effect that a higher variance in the permanentcomponent for Men in Private sector will have on labour income variation atretirement, when compared to Females in Public sector; albeit both are groupswith a University degree.

We also note that the permanent shocks to income has the highest correlationwith the equity market for the privately employed with an university degree, andthat gender is less of an importance.

5.2 Winners & Losers

We simulated the individual behaviour from age—28 until 100 with 10000 tra-jectories. Contingent on their random experience, individuals chose responsesdetermined by the policy rules in Equation 20.

Since the change of pension system was negotiated by consenting adults—wewould expect that on average the two systems would generate similar benefits.However, under the new system, individuals have a much larger responsibilityfor the appropriate management of the DC-account, since the outcome restssolely with the employee. In Figure 5.2 we show (for the highest income group—Men with University degree in the private sector), the variation in size of theDC-accounts.

In order to discover to what extent this new pension system generated win-ners and losers, we evaluated the value function (Equation 19), for the differentgroups in the first period; using both the old and the new pension system. Foreach group, we then added an initial amount to the DC-account that was asso-ciated with the lowest value of the value function, until the value of the valuefunctions were equal for both pension systems. The results for a subset of thegroups are presented in Table 1.

Intuitively, we would expect the group with the highest expected final pre-retirement income relative to average income, to lose from the transition and vice

versa. Another factor, is that high uncertainty in final pre-retirement incomewill decrease the expected utility of a defined benefit pension. Men with anuniversity degree in the private sector has an early earnings career and a morepronounced decline in income prior to retirement. They are therefore the winnersfrom a transition. The gain for this group is increased, as they also have a highervariance in income, which makes their expected final income less certain.


Table 1. Initial amount in Old or New DC-account necessary to equalise the value tothe individual of the pension systems

Amount in KSEK Pension System

Old New

Private MenHigh school 19University 101

WomenHigh school 32University 12

Public MenUniversity 40

WomenUniversity 80

The defined-contribution system recently negotiated in the private sector is alikely blueprint for a potential change of system for those in the public sector aswell. Our analysis shows (cf. Table 1) that publicly employed would on averagelose and that this loss is most pronounced for women. Women typically havetheir earnings-career later in life (cf. Figure 5.1), and therefore have less benefitsfrom early contributions; and secondly that, the lower variance in earnings amongpublicly employed and will make the Defined Benefit pension closer to a risk-freeasset.

5.3 Effects on portfolio choice

The positive labour income profile and short-sales constraints will typically makeyounger individuals ”more” constrained, i.e. with an equity allocation quite dif-ferent from the complete market solution. Cash on hand is very small in com-parison to the human capital and since their DC-account cannot be used forprecaution, we get a maximum allocation to equities in the DC-account. Thedifference in equity allocation (α − λ) decrease—as cash on hand accumulatesand as the income profile flattens. On average, we therefore expect the allocationto equities in the defined contribution account vs. cash on hand to differ a lot inearly life and that the differences will decrease at a later stage.

Figure 5.3, shows the average equity proportion of the DC-account. Irrespec-tive of system, since mandatory savings in pension systems cannot be used forprecaution or bequest; there is no reason for young individuals to have anythingbut equities in the DC-account.

With increasing age, the combined effect of: the DC-account being a muchlarger proportion of total wealth in the new system and the old Defined Benefitpensions being less risky; will lead to a more conservative behaviour for an agentin the new system. After retirement, when the Defined Benefit benefits becomerisk-free and hence ∆ = 0 in Equation 18, we can identify a large increase in theequity exposure for an individual in the old system.


In Figure 5.3, we show the same profile, but now for cash on hand. There is alarge difference between the risky weight in cash on hand vs. DC-account in earlylife, for precautionary and bequest reasons. After retirement, with decreasingpresent value of human capital, there is a gradual decline in equity-exposuretowards the complete-market solution.

It is important to note that; the profiles reflect the simulated averages for oneindividual. Figure 5.3 shows some percentiles of equity exposure for an agent inthe old pension system. The large variation is solely due to the accumulated effectof individual experiences. If we in addition, also could account for differencesamong individuals in: e.g. risk aversion, discounting or expected equity premia;then the variation would most likely be even larger.

5.4 Sensitivity analysis

In order to analyse to what extent our results are parameter-dependent, weperformed a sensitivity analysis using the group whose benefits are most affectedby the change in pension systems—men with a university degree in the privatesector. Table 2 shows the initial amounts that the DC-account must be increasedwith, in order to equalise the value of the two pension systems, with respect tochanging risk-aversion and a higher equity-premium.

Table 2. Initial amount in the Old DC-account necessary to equalise the value of thepension systems to the individual, with respect to equity and risk premia for Men withUniversity degree in the Private sector

µe γ KSEK

Reference 3 5 101Low risk aversion 3 2 179High risk aversion 3 8 70High equity premium 4 5 106

In all cases, it was beneficial for this group to move into the new system.The result show that changes in the equity premium—µe does not have a largeimpact, whereas the benefit to the less risk-averse was increased substantially.

In Equation 18, we created a decision-rule for the equity share in the DC-account. If this rule is too crude, we would expect individuals to compensate forany such errors in the allocation of their private savings. We therefore ”tested”this rule by calculating the difference between the equity share in DC-accountand in cash on hand—(λiτ − αiτ ).

A priori, we would expect this difference to be small and show little variancefor unconstrained individuals when the precautionary motive is weak, e.g., afterretirement. Early in life, however, when individuals are borrowing-constrained,we know that differences between trajectories can be large. Figure 5.4 plots thisdifference and the variation after retirement is not very large, which indicatesthat our rule seems to work.


6 Conclusions

This paper aims to contribute to the understanding of the welfare effects of mov-ing from primarily a defined-benefit to a defined-contribution pension system,and the changes in optimal individual behaviour required by such a change. Thesetting is a borrowing-constrained individual’s consumption- and portfolio-choicein the presence of uncertain labour-income, with group-dependent labour-incomeprocesses and realistically-calibrated tax- and pension-systems. We found thatthose employed in the private sector had higher income-variance than those inthe public sector, while gender differences (after controlling for private vs. publicemployment) were small.

We have used the recently negotiated change from defined benefit to definedcontribution pension systems as a benchmark for our analysis. The finding wasthat agents with low expected final income relative to average income and thosewith high income variance are set to gain from this transfer. Winners are menwith an university degree within in the private sector, and losers would be womenin the public sector with an university degree. The value of the different systemsto the individual are dependent on the risk preferences, but will not change thepreference of system.

Introducing a defined contribution system will necessitate that the individualhas to manage the assets differently in private and pension savings; and thatthe differences in portfolio choice between agents due to individual situationsare relatively large, even if we do not account for differences in terms of risk-aversion, etc, between individuals. One-size-fits-all kind of life-cycle funds, wherethe equity allocation depend on age alone, will therefore not fulfil the individualdemand for advice about asset allocation.

Appendix: Data and Estimations

A.1 Estimation of the labour-income process

The data set was divided into twelve non-intersecting groups, depending onsex (Male, Female), education (Compulsory school-, High-school- or University-degree) and employer (Public, Private). The matrix of individual characteristics—Z, includes variables for the number of children in different age-intervals as wellsas a dummies for maritial status, age. Income was adjusted to real values bydeflating with the official consumer price-index. Measured income is an aggre-gate including gross wages, also all social security benefits (primarily income-compensation for unemployment, disability and childcare) and pension benefits.

To avoid double-counting, we deleted all observations where income includedvoluntary pension benefits, i.e., individuals above the age of 55 receiving pensionpay-outs at their own request. Pension benefits paid prior to age 55 can be con-sidered as insurance payouts and were therefore included. Progressive taxationwill induce most agents to make these early withdrawals only if the individualhas simultaneously reduced the ordinary wage income. Finally, we exclude an


observation if income is less than 100.000 SEK. Individuals with income lowerthan this level are assumed to be voluntarily unemployed.

The following random-effects linear model was used to estimate the deter-ministic function for each group,

lit = β0 + Zitβ+ϑi + eit,

eit = ρeit−1 + κit,

ϑi ∼ N(0, σ2ϑ),

κit ∼ N(0, σ2κ),

(21)

where—Zit are the nonstochastic regressors and β is the vector of regressioncoefficients. Estimation results are presented in Table 3.

Table 3. Labour Income Process: Coefficients from Regression

AR(1) Random effects Regression

Log real income #Children at age Marr- AR Std. in Std. in R2

2004 KSEK 1-2 3-5 6-17 ied=0 ρ fixed overall withinSingle=1 σϑ σe

Private MenCompulsory -.00599 .00076 -.00284 -.02846 .5305 .2378 .2378 .185High school -.00762 -.00242 .000027 -.02957 .5356 .2688 .1644 .235University -.00786 .00077 .00035 -.03185 .5469 .3784 .2096 .317

WomenCompulsory -.08666 -.04532 -.02430 .03215 .5433 .2223 .1338 .271High school -.12572 -.06728 -.03316 .03444 .5116 .2306 .1625 .300University -.17466 -.10077 -.05480 .01108 .4645 .3194 .2085 .328

Public MenCompulsory -.02575 -.00889 -.00838 -.04473 .4857 .2462 .1214 .213High school -.01951 -.01389 -.00617 -.03341 .5076 .2642 .1296 .267University -.00655 .00014 .00201 -.01905 .5394 .3190 .1449 .381

WomenCompulsory -.06417 -.03328 -.01257 .02817 .5480 .2018 .1152 .270High school -.10207 -.04860 -.02063 .03488 .5274 .1699 .1224 .376University -.13186 -.06781 -.02867 .03442 .5038 .2368 .1455 .443

We then calculate the deterministic component of labourincome—exp

fk(τ,Zkτ)

, adjusted for age dummies with the averages of thecharacteristics. This was then used to estimate a third-degree polynomial withrespect to age, cf. Equation 22 Table 4, and Figure 5.1,

exp

fk(τ,Zkτ)

=

3∑

m=0

akm(AGEτ − 18)m. (22)


Table 4. Coefficients in the age polynomial of the forward-looking income profile

Income profile, 2004 KSEK, (AGE-18)Constant Age Age2 Age3

a0 a1 a2 a3

Private MenCompulsory 187.4410 3.7553 -0.0149 -0.0009High school 192.5214 5.4665 -0.0449 -0.0008University 50.6314 26.8242 -0.5615 0.0028

WomenCompulsory 170.8340 0.0714 0.0986 -0.0020High school 200.2780 -2.5127 0.2277 -0.0037University 170.6430 5.3380 0.0553 -0.0027

Public MenCompulsory 150.9325 3.0467 0.0011 -0.0008High school 176.5778 2.5334 0.0325 -0.0013University 91.3953 14.9683 -0.2048 0.0000

WomenCompulsory 154.1382 0.4386 0.0563 -0.0012High school 181.6734 -2.1792 0.1781 -0.0027University 217.9771 -5.5608 0.4423 -0.0065

A.2 Variance Decomposition

We followed [7] in decomposing permanent and temporary variances. By combin-ing the error terms from Equation 2—vit + εit with the estimated residual—eitfrom Equation 21, we get:

∆eit(d) = eit+d−eit = (vit+d+εit+d)−(vit+εit) = (uit+d+...+uit)+(εit+d−εit)(23)

and consequentially the variance is,

V ar(∆eit(d)) = d · σ2u + 2 · σ2

ε . (24)

Following [7], we allowed for serial correlation in the transitory shock of theorder MA(2), and therefore excluded observations with a time distance less than3. OLS on Equation 24 was then used to estimate σ2

u and σ2ε .

A.3 Income correlation with the equities

We followed [10] in estimating the correlation between labour-income shocks andequity–returns. Using Equation 2, the first difference in l∗ikt = likt − fk(τ,Zikτ

)can be written as

l∗ikt = ξkt + ωikt+ ǫikt. (25)


Taking the average over individuals in each group gives us the group-aggregatecomponent,

l∗kt = ξkt. (26)

Finally, we estimated the correlations—ξkη, by applying OLS to,

l∗kt = βk(ret−1 − r

ft−1) + φt. (27)

Table 5 presents the result from this regression using the real return of theSwedish equity index OMX as a proxy for equity-returns—re and the real returnon 12-month Swedish Treasury Bill as the risk-free rate—rf .

Table 5. Variance decomposition and equity correlations

Number of Estimated Estimated Std. of the Correla-obs variance variance permanent tions with

of the of the aggregate Swedishpermanent transitory aggregate equitycomponent component component returns

σ2

ukσ2

εkσξk ξkη

Full sample 1 423 930 0.0211 0.40

Private Men 585 446Compulsory 140 413 0.0042 0.0152 0.0222 0.39High school 310 835 0.0054 0.0183 0.0229 0.40University 134 198 0.0098 0.0284 0.0270 0.61

Women 290 776Compulsory 67 364 0.0048 0.0104 0.0187 0.45High school 152 254 0.0054 0.0182 0.0173 0.45University 71 158 0.0079 0.0306 0.0258 0.51

Public Men 152 243Compulsory 17 039 0.0021 0.0083 0.0249 0.25High school 47 543 0.0029 0.0096 0.0219 0.23University 87 661 0.0044 0.0115 0.0216 0.24

Women 395 465Compulsory 41 921 0.0034 0.0082 0.0236 0.30High school 176 607 0.0030 0.0103 0.0207 0.25University 176 937 0.0038 0.0138 0.0233 0.22

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7. Carroll, C.D., Samwick, A.: The Nature of Precautionary Wealth. Journal of Mon-etary Economics 40 (1997) 41-71

8. Claus, J., Thomas, J.: Equity premia as low as three percent? Evidence fromanalysts’ earnings forecasts for domestic and international stock markets. Journalof Finance 61 (2001) 1629-1666

9. Cocco J.F.: Portfolio Choice in the Presence of Housing. Review of Financial Stud-ies 18 (2005) 535-567

10. Cocco, J.F, Gomes, F.J., Maenhout, P.J.: Consumption and Portfolio Choice overthe Life-Cycle. Review of Financial Studies 18 (2005) 491-533

11. Deaton, A.: Saving and liquidity constraints. Econometrica 59 (1991) 1221-124812. Edin, P., Fredriksson, P.: Longitudinal INdividual DAta for Sweden. Working paper

2000:19, University of Uppsala (2000)13. Fama, E.F., French, K.R.: The Equity Premium. Journal of Finance 62 (2002)

637-65914. Flood, L.R.: Can we afford the future? An evaluation of the new Swedish pen-

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16. Gourinchas, P-O., Parker, J.A.: Consumption over the life-cycle. Econometrica 70

(2002) 47-8917. Hubbard, R.G, Skinner, J., Zeldes, S.P.: Precautionary Saving and Social Insur-

ance. Journal of Political Economy 103(2) (1995) 360-39918. Judd, K.L.: Numerical Methods in Economics. MIT Press, Cambridge, MA (1998)19. Merton, R.C.: Optimum Consumption and Portfolio Choice in a Continuous-Time

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An Interpretation of Cross-section Data. In Kurihara, K. (ed.): Post KeynesianEconomics. Rutgers University Press, New Brunswick, NJ (1954)

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23. Zhou, J.: The Asset Location Puzzle: Taxes Matter. Working paper, Univerisity ofWestern Ontario (2006)


Fig. 1. Income profiles for different groups. Simulated real gross wages—L withoutproductivity change


Fig. 2. Income variation. 25th and 75th percentiles for simulated real gross wages—L

without productivity for Men and Females with University degrees employed in thePrivate and Public sector


Fig. 3. DC-account variation. 25th and 75th percentiles of the DC-account in the Oldand the New pension system for Males with University degree in the Private sector


Fig. 4. Simulated Average Equity Exposure in DC-Account. Simulated average equityshare—λ in the DC-account for Males with University degree in the Private sector, forthe old and new pension system


Fig. 5. Simulated Average Equity Exposure in Cash-on-Hand. Simulated average eq-uity share—α in the cash on hand-X for Males with University degree in the Privatesector, for the old and new pension system


Fig. 6. Variation in equity exposure in DC-Account. Simulated 5th, 25th, 75th and95th percentiles percentiles for the equity share—λ in the DC−account for Males withUniversity degree in the Private sector in the Old pension system


Fig. 7. Variation in the difference in equity exposure between DC-Account and cashon hand. Simulated 25thand 75th percentiles for the difference in equity share betweenDC−account—DCW and cash-on-hand—X and for Males with University degree inthe Private sector in the Old pension system—(λ− α)

A stochastic model for theanalysis of demographic risk inpay-as-you-go pension funds⋆

Alessandro Fiori Maccioni

Dipartimento di Economia, Impresa e Regolamentazione,Universita di Sassari,

Via Torre Tonda n. 34 - 07100 Sassari, [email protected]

Abstract. This research presents an analysis of the demographic riskrelated to future membership patterns in pension funds with restrictedentrance, financed under a pay-as-you-go scheme. The paper, therefore,proposes a stochastic model for investigating the behaviour of the demo-graphic variable ‘new entrants’ and the influence it exerts on the financialdynamics of such funds. Further information on pension funds of Ital-ian professional categories and an application to the Cassa Nazionale di

Previdenza e Assistenza dei Dottori Commercialisti are then provided.

Keywords. Pension funds, demographic risk, new entrants, Markovchain, professional categories.

M.S.C. classification. 11K45, 60J10, 60J20, 62N02.

J.E.L. classification. C15, C32, C53, G23, J11.

1 Introduction

The financial sustainability of a pension scheme depends not only on the time-length of the benefit to be paid, subject to longevity risk, but also on the correctquantification of the contributions to be received. The uncertainty related tofuture contributions primarily affects the retirement plans based on a pay-as-you-go financing system (PAYG), where pensions are directly funded by currentemployees’ salary deductions. Thus, an “intergenerational pact” compels younggenerations (composed by current and future contributors) to sustain the older

⋆ I wish to thank Prof. Ron W. Anderson and Prof. Alessandro Trudda for their pre-cious comments on an earlier version of this paper. I am also greatly indebted to theCNPADC for providing the data and would finally like to thank Dr Francesco Arzilli,Dr Claudio Detotto, Dr Simonetta Falchi, Dr Roberta Melis, Ms Lisa O’Donoughueand the two anonimous referees for their useful suggestions. This work would havenever been possible without the ‘Master and Back’ program and the support toresearch provided by the Regione Autonoma della Sardegna.

42 Alessandro Fiori Maccioni

age groups. For the financial self-sufficiency of such pension schemes, it is essen-tial that in the long term there should be equilibrium between the number ofpensioners and the number of workers. Due to the inversion of the production cy-cle, delivering financial equilibrium in the short-medium term via an increase inthe number of new members can be harmful in the long run, when contributors inthe preceding period will become pensioners. If the ratio contributors/pensionerswas to decrease in the future, due to an increase of an older population, therewould be ceteris paribus an increase of financial burden for the pension system;thus creating a disequilibrium in the pension scheme and the risk of financialdifficulties.

Public PAYG pension schemes are generally opened to different professionalgroups, in order to prevent financial unbalances due to a decline in a specific pro-fession (and the consequent decline in its contributors). Therefore, variations inthe number of contributors are mostly influenced by changes in the age structureof the population. On the other hand, private PAYG pension schemes can chooseto admit only a homogeneous class of people (e.g. employees of a specific firm,workers with given professional qualification, etc.). For such “closed” pensionfunds, the demographic risk related to the variable ‘new entrants’ is relevant be-cause changes in the job market may influence the number of contributors. Thisis the case of the self-administered PAYG pension funds of Italian professionalcategories.

Until the early nineties, social security and pension disbursement in Italy werepublicly funded and administered. Nearly all citizens, regardless of their income,were entitled to a pension that allowed them to approximately maintain the samesocio-economic status they enjoyed while employed. The right to a pension wasgenerously guaranteed as prescribed by the article 38 of the Italian Constitution.In 1994 the Legislative Decree No. 509 was passed, calling for the privatisationof certain sectors of social security and pension administration. Any professionalgroup organized as an Order/Board (such as lawyers, accountants, engineers,doctors, pharmacists, etc.) was to create and administer its own retirement fund.Members who work autonomously would deposit portions of their incomes duringtheir working years, and receive pensions upon retirement; meanwhile, memberswho work as employees would still be entitled to public pension cover. Thesenew financial institutions, called Casse di Previdenza e Assistenza dei Liberi

Professionisti, have been no longer dependent on governmental assistance. Thechange meant that, should a given fund reach a negative balance, there wouldbe no more financial backup from the public finances; thus retirees would haveno pensions available to them.

In this perspective, the present study addresses the evaluation of demographicrisk related to the variable ‘new entrants’ in PAYG pension systems. It startswith a brief review of recent literature on pension fund risk management (Sect. 2)and continues with the mathematical formalization of the problem (Sect. 3) andan application to the Cassa Nazionale di Previdenza e Assistenza dei Dottori

Commercialisti (Sects. 4 and 5). Finally, conclusions are drawn (Sect. 6).

A stochastic model for the analysis of demographic risk . . . 43

2 Risk management in pension funds: state of the art

Over the last few years a vast literature regarding management and regulationof risk in pension systems has developed. The main topics on quantitative re-search have been the development of stochastic models for longevity risk andglobal asset return. Recent financial scandals have also improved research ongovernmental regulations for life insurance institutions.

An introduction to longevity risk with a comprehensive literary review canbe found in [38] and [39]. Rigorous analyses of mortality projections have beenconducted by Lee and Carter in [23], Benjamin and Pollard in [5], Benjaminand Soliman in [6], Haberman and Renshaw in [19], Lee in [24], Olivieri in [29],Thatcher et al. in [45] and Olivieri and Pitacco in [34]. Joint analyses of bothfinancial and longevity risks have been proposed by Olivieri and Pitacco in [31]and by Coppola et al. in [13]. The securitisation of mortality risk has beenanalysed by Lin and Cox in [25] and by Cairns et al. in [10].

Several stochastic models for global asset return in pension funds have beenproposed; see for example Parker in [37], Cairns and Parker in [11], Blake et al. in[8] and [9]. Mandl and Mazurova in [26] use spectral decomposition of stationaryrandom sequences for assessing defined benefit pension schemes under randomlyfluctuating rates of return and numbers of entrants. Haberman in [17] identifiesa ‘contribution rate risk’ and considers as stochastic components both rate ofreturn and contribution rate. Gerrard et al. in [16] analyse the financial risk facedby members of defined contribution schemes both during the service period andafter retirement.

Stochastic analyses of new entrants in private pension schemes have beenproposed by Janssen and Manca in [20] and by Colombo and Haberman in [12].Sinn in [42], [43] and Abio et al. in [1] consider the age structure of future nationalpopulation as a prime risk factor in PAYG public pension systems. Angrisani etal. in [3] propose a demographic model for studying the impact on PAYG pensionsystems of future developments of the population. Bianchi et al. in [7] conducejoint demographic and behavioural analyses via dynamic microsimulation to testthe economic effects of pension reforms.

A vast literature on risk management policies has been developed followingdefaults on life insurance sector; see for example Plantin and Rochet in [40]. Adebated point is whether competition among pension funds and moral hazardcan expose funds to excessive risks that are not compatible with their social not-speculative function; see for example McClurken in [28]. Bader in [4] suggeststhat pension funds should avoid investing in specific sectors in the stock market.Ryan and Fabozzi in [41] study the defaults of US pension funds due to actu-arial losses and not to wrong portfolio investments. Trudda in [47] shows thatmarginal increments in global asset return appear to strongly reduce the defaultprobability of the pension fund of an Italian professional Order, thus generatingan incentive to take superfluous risks in case of lacking of regulations. Otrantoand Trudda in [36] urge the need for a risk rating system for pension funds andpropose a cluster analysis based on GARCH volatility of their rates of return.


3 A model for the evaluation of new entrants to apension fund with restricted entrance

The Population-Education-Profession (P-E-P) model, that we propose here, isa discrete-time stochastic model for the estimation of the new entrants in apension fund with restricted entrance, such as that of a professional category.The model is based on the study of variables related to the demographic evolutionof the population, the development of university instruction and the attractionof the profession. It can be used, with appropriate simplifications, to forecast theentrants in any kind of pension scheme. To the best of our knowledge, it is thefirst stochastic model specifically designed for the estimation of new members ofa professional category (and, subsequently, of its pension fund). An early versionof the model and a deterministic application are proposed in [15] and [47].

The intergenerational patterns of employment in a given professional groupdepend on different specific variables, both demographic (trend in population,trend in study choices, etc.) and economic (appeal of the profession, appeal ofthe firm, expected income, etc.). Thus, for a correct estimation of the futurecontributors to such a “closed” pension fund we should address the followingquestions:

– What will the demographic evolution of the reference population be?– What are the trends in the choices of study regarding the specific profession?– What is the attraction of the profession (or of the firm, in case of corporate

pension fund)?– How is the admission to the pension fund regulated (e.g. elective/compulsory

entrance)?

The stages that a potential contributor has to leave behind before enteringin the pension fund of a professional category have been represented in theMarkov chain in Fig. 1. Accordingly, we propose a model based on subsequentestimations of the population in different stages, as described in Table 1. TheMarkov chain is composed by the following states:

1. Belonging to the cohort of reference (e.g. Italian population aged 18-25);2. Having a high school diploma;3. Being enrolled in the required course of study (e.g. Bachelor degree in Law);4. Being graduated in the specific degree;5. Starting the training period required for taking the admission exam;6. Becoming a member of the professional category (e.g. becoming a Lawyer);7. Joining the pension fund of the professional category.

The pij(t) in the Markov chain represents the probability of transition from statei to state j at time t; h and k are the expected lengths for successfully completing,respectively, the course of study and the professional training period. At timet, each potential future contributor can only be in one state. An individual canmove to a greater state exclusively after fixed time periods (0, h or k time units)depending on the state itself.


Fig. 1. Markov chain for the estimation of future entrants to the pension fund of aprofessional category

Let us introduce the stochastic process POP (t) representing the populationin state 1 at time t, which is, in other words, the starting population of theMarkov chain, defined as:

POP (t) =

M∑

x=m

POP x(t) + σpop ǫ (1)

where POP x(t) is the expected value of the stochastic process POPx(t) repre-senting the national population of age x at time t; σpop is the standard deviationof∑

POP x(t); m and M are integer numbers indicating, respectively, the min-imum and maximum age of the cohorts considered in the potential population;ǫ is a standard normal variable. Equation 1 holds for

∑

POP x(t) ≥ −σpop ǫbecause population must be equal to a non-negative number.

Let Pij(t) be a stochastic process defined as:

Pij(t) =

0 if ǫ < −pij(t)

σij

pij(t) + σij ǫ if ǫ ≥ −pij(t)

σij

(2)

where pij(t) is the expected value of the probability of transition pij(t), σij is itsstandard deviation, and ǫ is a standard normal variable. The Pij(t) can be seenas an approximation of the transition probability pij(t), but not as a probabilityin itself because it can assume values higher than one.

Let NE(t) be a Markov process representing the number of new entrants tothe pension fund at time t, defined as:

NE(t) = POP (t− h− k)P12(t− h− k)P23(t− h− k)·


Table 1. Stages for estimating new entrants to the pension fund of a professionalcategory

Population of potential university students Regarding trends in population↓

Number of enrolments at university↓

Enrolments in the specific field of study Regarding trends in education↓

Graduations in the specific field of study↓

Graduates who start the training period↓

Trainees who complete the training period Regarding the appeal of the profession↓

New members of the professional category= Trainees who pass the admission exam

↓

New entrants to the pension fund Regarding entrance regulations

·P34(t− k)P45(t− k)P56(t)P67(t). (3)

To make the Markov chain more compliant with generally available statistics,we simplify it by merging some steps. Thus, we focus on the stochastic processesP13(t), P34(t), P46(t) and P67(t), and we rewrite (3) as:

NE(t) = POP (t− h− k)P13(t− h− k)P34(t− k)P46(t)P67(t). (4)

Equation (4) represents the fundamental formula of the P-E-P Model. Forsufficiently small values of the ratios pij(t)/σij and assuming the independenceof the random processes, it also results:

NE(t) ≈ [

M∑

x=m

POP x(t− h− k) + σpop ǫ ][p13(t− h− k) + σ13 ǫ ]·

· [p34(t− k) + σ34 ǫ ][p46(t) + σ46 ǫ ][p67(t) + σ67 ǫ ] (5)

E[NE(t)] ≈M∑

x=m

POP x(t− h− k) p13(t− h− k) p34(t− k) p46(t) p67(t) (6)

and:

V ar[NE(t)] ≈ [

M∑

x=m

POP x(t− h− k)]2 + σ2pop[p13(t− h− k)2 + σ2

13]·

· [p34(t− k)2 + σ234][p46(t)

2 + σ246][p67(t)

2 + σ267]. (7)


In pension funds of professional categories, the number of future new entrantscan be influenced by different variables, related to reference population, educa-tion choices, school completion, professional appeal and entrance regulations.The Population-Education-Profession model analyses such variables with fivemain stochastic processes. Indeed, POP (t) represents the national populationwho fulfil the university age requirements, P13(t) the rate of diffusion of univer-sity studies among the population, P34(t) the success rate in university study,P46(t) the admission rate in the professional category, and P67(t) the inscriptionrate to the category’s pension fund.

All of the parameters of the model can be estimated according to data thatis easily available to professional categories, coming from the Central Instituteof Statistic, from the Ministry of Education and from Professional Associations.Assuming that the period between t− 1 and t corresponds to the calendar yeary, we can approximate P13(t) as the ratio between university enrolments (in thespecific field of study) and the starting population at calendar year y. Accord-ingly, we can estimate P34(t) as the ratio between graduations at year y andenrolments in previous year y − h; P46(t) as the ratio between new members ofthe professional category at year y and graduates at previous year y − k; P67(t)as the ratio between new entrants to the pension fund and new members of thecategory at year y. Such ratios can sometimes have values higher than 1 (e.g.because of university reforms, of changes in the fund’s entrance regulations, etc.)and this is reflected in the definition of Pij(t).

The P-E-P model makes a breakthrough in the evaluation of new entrants topension funds of professional categories. Classic actuarial techniques are mostlybased on the time series analysis of past membership trends; this approach canunderestimate the probability of sudden changes in the demographic dynamics,because it only considers the past outcome (new entrants time series) but not howthis outcome had been generated. On the contrary, the P-E-P model can evaluatethe risk of abrupt changes in the different variables that influence the numberof future members (e.g. due to university reforms, decline in the professionalappeal, etc.). Accordingly, it can assign different levels of demographic risk infunds that presented similar new entrants time series.

4 A numerical application of the P-E-P model

We used the P-E-P model for estimating the demographic evolution of the CassaNazionale di Previdenza e Assistenza dei Dottori Commercialisti (CNPADC).The CNPADC is the private self-managed pension fund of Italian CharteredAccountants (Ordine dei Dottori Commercialisti) set up by Law n. 335/1995.It is a cash-balance pension fund financed with a pay-as-you-go system, so themoney collected during the year from contributors is immediately used for payingthe same year’s pensions, and only partially saved.

The demographic structure of the CNPADC is that of a “young” retirementfund, meaning that it has still not reached the long-term natural balance betweenthe number of contributors and pensioners. This is a consequence of the huge


increase in new memberships that has occurred since the mid-Nineties. In the pe-riod 1995-2005, there has been an increase of more than 100% in the total numberof members, that have risen from 21,762 to 44,706; the ratio workers/pensionershas increased from 7 to 9.6; instead the ratio between contribution revenues andbenefit expenses has been nearly constant, passing from 2.74 to 2.72.

The values of POP x(t) for the years 1998-2006 (thus, influencing the numberof new contributors in 2007-2015) have been taken from the official estimates ofthe Italian population published by the Italian Institute of Statistics (ISTAT).The values of POP x(t) and σpop for the period 2007-2050 have been estimatedon the basis of the National Demographic Forecasting of Italian population pub-lished by ISTAT.

The required degrees for being admitted to the Order of Chartered Accoun-tants have been considered the Laurea (triennale, specialistica and vecchio ordi-

namento) in Economics, Management and Business Administration. We focus onthe period that has followed the main reform of the Italian university system indecades, which has introduced the three-years Bachelor degree. The value of h,the average time for completing university has been considered equal to 5 years.The value of k, the average time for completing the professional training andpassing the exam, has been considered equal to 4 years. We assume m = 18 andM = 25, basing such assumption on the fact that more than 95% of universitypopulation in Italy is composed by students aged 18-25, and this ratio has beenreasonably constant over time in the last two decades.

The moments of P13(t) and P34(t) have been estimated according to historicaldata from ISTAT and the Italian Ministry of University (MIUR). The momentsof P46(t) and P67(t) have been estimated according to historical data fromMIUR,CNPADC and Fondazione Aristeia. For the evaluation of P67(t) we have alsoconsidered historical data on cancellations from the fund, so that the estimatesof new entrants can be considered net of cancellations.

Estimates of new entrants to the CNPADC fund have been made with theP-E-P model and tested with the Monte Carlo method. With 10,000 simulations,we have drawn the probability distribution of the future entrants, divided by sex,for the period 2006-2059. Tables 2 and 3 present, respectively, the parameters ofthe model and the expected results. Figures 2, 3 and 4 present the percentilesof the frequency distributions for new entrants obtained in the simulations (re-spectively: total, females and males).

Table 2. CNPADC pension fund: parameters of the P-E-P model

p1 3

(t) σ1 3 p3 4

(t) σ3 4 p4 6

(t) σ4 6 p6 7

(t) σ6 7

Females 0.0085 0.0007 0.5110 0.1996 0.0811 0.0291 0.6261 0.1088

Males 0.0090 0.0005 0.5110 0.1996 0.0893 0.0320 0.6388 0.1108

Notably, the highest values of new members are in the period 2006-2012.This is an effect of the reform of the university system, started in the academic


Table 3. New members of the CNPADC fund, expected values, years 2006-2059

Year Mal. Fem. Tot. Year Mal. Fem. Tot. Year Mal. Fem. Tot. Year Mal. Fem. Tot.2006 1,119 915 2,034 2020 595 516 1,111 2034 592 511 1,103 2048 561 447 1,0082007 1,330 1,088 2,418 2021 592 513 1,105 2035 593 512 1,106 2049 556 443 9992008 1,509 1,235 2,744 2022 591 511 1,102 2036 593 512 1,105 2050 552 440 9922009 1,565 1,280 2,845 2023 589 510 1,099 2037 592 511 1,103 2051 549 437 9862010 1,280 976 2,256 2024 585 506 1,091 2038 590 508 1,098 2052 546 435 9812011 1,166 904 2,071 2025 582 503 1,085 2039 586 504 1,090 2053 544 434 9782012 896 724 1,621 2026 578 499 1,077 2040 578 497 1,076 2054 543 432 9752013 713 573 1,286 2027 576 497 1,072 2041 570 490 1,060 2055 542 432 9742014 650 561 1,210 2028 574 495 1,068 2042 562 484 1,046 2056 542 432 9742015 621 541 1,162 2029 574 495 1,068 2043 554 476 1,030 2057 542 432 9742016 613 533 1,146 2030 576 497 1,072 2044 546 470 1,016 2058 542 432 9742017 606 526 1,132 2031 579 500 1,078 2045 539 464 1,003 2059 543 433 9762018 600 521 1,121 2032 584 504 1,088 2046 574 458 1,0322019 597 518 1,114 2033 588 508 1,106 2047 568 452 1,020

Fig. 2. Total new entrants to the CNPADC pension fund, percentiles of the frequencydistributions, years 2006-2059

Fig. 3. Female new entrants to the CNPADC pension fund, percentiles of the frequencydistributions, years 2006-2059


Fig. 4. Male new entrants to the CNPADC pension fund, percentiles of the frequencydistributions, years 2006-2059

year 2000/01; students who did not previously complete their Laurea (whichrequired from 4 to 6 years, depending on the field of study) have been allowed tobe re-enrolled to an equivalent post-reform Bachelor degree (Laurea triennale)without having to re-sit for the exams that they had already passed. Thus, a largenumber of students, who had previously left university without completing their4-6 years program, have quickly obtained a Bachelor degree. This phenomenonis reflected in the estimates of P13(t) and P34(t), which indicate that it will ceaseits effect from 2008 (thus, affecting the number of new contributors until 2012).

5 An analysis of the financial dynamics of the CNPADCfund

5.1 The model

The aim of this section is to estimate the effects of the variable ‘new entrants’ onthe financial dynamics of the CNPADC retirement fund in the period 2006-2046.

The fund value Vt, corresponding to the value of the net assets belonging tothe fund at time t, has been modelled with the following recursive equation:

Vt = Vt−1 · (1 + rt) + Ct +Rt −Bt − Et (8)

where rt is the nominal annual rate of return, Ct, Bt and Et represent respec-tively the amounts of contribution income, pension disbursement and admin-istrative expenses generated in the period [t − 1, t]. All of the cash flows areassumed to take place at the end of each period as this is more consistent withthe fund’s regulations.


5.2 Contribution income

The annual contribution income at time t has been estimated with the followingequation:

Ct =G∑

g=1

S∑

s=1

π∑

x=α+1

A∑

a=1

cg s x a(t) ·Ns x a(t) ,

∀(x, a) ∈ N×N − x > xb s t ∧ a ≥ ab s t (9)

where cg s x a(t) is the average contribution of type g paid at time t by an indi-vidual of sex s, age x and seniority a, and Ns x a(t) is the number of members ofthe fund alive at time t of sex s, age x and seniority a. The term G representsthe number of types of contributions, S the number of sex categories, α and πthe minimum and maximum potential age of contributors, and A the maximumpotential seniority. The terms xb s t and ab s t represent the retirement require-ments of age and seniority, in force at time t, for members of sex s to be entitledto a benefit of type b; thus, the Ns x a(t) considered in equation (9) are cohortsof active members.

It also results:cg s x a(t) = γg x a t ·Rg s x a(t) (10)

where γg x a t and Rg s x a(t) represent respectively the contribution rate and theexpected income amount for the determination of the contribution of type g dueat time t by individuals of sex s, age x and seniority a.

In the application we consider two main types of contributions of the fund:the soggettivo and the integrativo, determined annually as shares of, respectively,professional income and sales subject to Value Added Tax.

5.3 Pension disbursement

The annual pension disbursement at time t has been estimated with the followingequation:

Bt =D∑

d=1

S∑

s=1

ω∑

x=β+1

A∑

a=1

bd s x a(t) ·Ns x a(t) ,

∀(x, a) ∈ N×N : x > xb s t ∧ a ≥ ab s t (11)

where bd s x a(t) is the average contribution of type d received at time t by apensioner of sex s, age x and seniority a, and Ns x a(t) is the number of membersof the fund alive at time t of sex s, age x and seniority a. The term D representsthe number of types of benefits, β and ω the minimum and maximum potentialage of pensioners. The terms xb s t and ab s t represent the retirement requirementsof age and seniority in force at time t; thus, the Ns x a(t) considered in equation(11) are cohorts of retired members.

In the application we consider the two main types of benefits of the fund:the pensione di vecchiaia and the pensione unica contributiva, computed witha pro-rata mechanism in accordance with CNPADC regulations as amended bythe statutory reform of 2004.


5.4 Mortality rate

Let qs x(t) be the probability at time t that an individual of sex s and age x,still alive at time t, will die before time t+ 1, defined as:

qs x(t) =

0 if ǫ < − qs x(t)σs x

qs x(t) + σs x ǫ if − qs x(t)σs x

≤ ǫ ≤ 1−qs x(t)σs x

1 if ǫ > 1−qs x(t)σs x

(12)

where qs x(t) and σs x represent respectively the expected value and the standarddeviation of qs x(t), and ǫ is a standard normal variable.

We model qs x(t) as:

qs x(t) = (1 + µs x)t−t0 · qs x(t0) (13)

where qs x(t0) and µs x represent respectively an initial known value and theexpected annual rate of change of qs x(t).

The aim of the mortality model is to evaluate the impact of the accidentalcomponent of longevity risk on the financial dynamics of the CNPADC fund.Such component is due to random deviations from the expected mortality values.It is a simple stochastic model for which parameters can easily be estimatedfrom official data published by national institutes of statistics. The discrete timeapproach has been preferred since the time unit in the application is the year.

5.5 Rate of return

Let rt be a stochastic process representing the annual nominal rate of returndefined as:

rt = rt +Xt (14)

where rt is the expected nominal rate of return in the period [t− 1, t] and Xt isan AR(1) process defined as:

Xt = ϕXt−1 + σ ǫ (15)

with −1 < ϕ < 1, where ϕ and σ are the parameters of the process, and ǫ isa standard normal variable. The proposed model represents a discrete form ofthe Vasicek model, with properties of normality, stationarity, mean reversion,and finite variance. The prudential asset allocation of the CNPADC fund iscompatible with such properties.

The approach based on the Vasicek model seems to be particularily suitablefor describing the rate of return in first pillar pension funds such as the CNPADC.It takes into account the possibility of obtaining negative values, which is adesirable feature when modelling the rate of return. Indeed, pension funds ofItalian professional Orders can occasionally suffer financial losses although theirsocial not-speculative function. This has happened for example in 2008.


5.6 Technical assumptions

The model has been employed with the following assumptions.

Demographic hypotheses:

– Effective population of pensioners and contributors on the 1st of January2006, according to data from the CNPADC, divided by sex, age and seniority.

– Future entrants determined according to the P-E-P model as stated in theprevious section, age of entry 29.

– Initial mortality rates qs x(t0) equal to rates of the 2006 Italian mortalitytable published by ISTAT; values of µs x and σs x estimated according todata from ISTAT on Italian mortality in the period 1981-2006.

Financial hypotheses:

– Fund’s net assets equal to 2,067,793,989 euros on the 1st of January 2006,according to the 2005 financial report.

– Administrative costs of year 2006 equal to 28,447,830 euros, appreciatedin the following years at 5% nominal annual rate according to technicalassumptions in [2].

– Parameters for the estimation of the rate of return are X0 = 0, ϕ = −0.612and σ = 0.03667, according to results obtained by Melis in [27] for Italianfixed income investment funds.

– Inflation rate equal to Italian Government’s expectations exposed in thebudget DPEF 2007-2011, thus equal to 2% in 2006, to 1.7% in 2007, to2.1% in 2008, to 1.9% in 2009, and to 1.6% in the following years.

Contributions:

– Two types of contributions (soggettivo and integrativo) determined in accor-dance with CNPADC regulations. New entrants exercise the statutory rightof exemption from contributions for the first 3 years.

– Annual professional incomes and VAT sales equal, for each cohort of samesex and age, to the effective average values registered in 2005, appreciatedat inflation rate.

– Subjective contribution rate equal to 10.7%1 of annual professional income;sums paid under the defined-contribution scheme accrued at 3.4% nominalannual rate.

– Integrative contribution rate equal to 4% in 2006-2010 and successively to2% of annual professional VAT sales, according to the statutory reform of2004.

Pensions:

– Benefits paid to pensioners who retired before the 1st of January 2006 equalto the effective average values registered in 2006.

1 Subjective contribution rate varies electively between 10% and 17% of annual pro-fessional income. In 2005 the average rate was 10.71%.


– Two types of benefits (pensione di vecchiaia and unica contributiva) forpensioners who retire after the 1st of January 2006, determined with a pro-rata mechanism in accordance with CNPADC regulations as amended bythe statutory reform of 2004.

– All benefits appreciated annually at the inflation rate. Each cohort of contrib-utors retires immediatly after fulfilling the requirements. We do not considerbenefit reversion to survivors.

5.7 Results

The probabilistic structure of the fund value has been estimated with stochasticsimulation based on Monte Carlo techniques. This approach generates a rangeof outcomes which represents a probability distribution, conditional on the as-sumptions made. The number of outcomes has been 10,000 for each test made.

In the first test, the three main risk factors — mortality, new entrants andrate of return — are considered as stochastic variables. Results are presented inFig. 5 and indicate that, with 99.9% confidence level, the CNPADC fund willmaintain a positive value in the forecasting period. The probability distributionof the fund value is nearly standard, slightly leptokurtic and right-skewed. Thevalues of skewness and kurtosis tend to increase with the passing of time, asdemonstrated in Tab. 4.

The probability distribution of the total balance presents a large variation; itsmedian value reaches a peak of 900 millions in 2026 and then decreases, reachingits minimum, 15 millions, in 2042, as demonstrated in Fig. 6. The percentiledistribution of the pension balance indicates that this value will turn negative inthe period between 2033 and 2037, with 99.9% confidence level, because of theageing of the population; its probability distribution presents a low dispersionaround the median values, as demonstrated in Fig. 7. Returns on investments areexpected to partially cover the increase in pension disbursement, thus preventingfrom abrupt slumps in the total balance.

Fig. 5. Value of the CNPADC pension fund in million euros, percentiles of the fre-quency distributions, years 2006-2046


Fig. 6. Total balance of the CNPADC pension fund in million euros, percentiles of thefrequency distributions, years 2006-2046

Fig. 7. Pension balance of the CNPADC pension fund in million euros, percentiles ofthe frequency distributions, years 2006-2046

We have conducted three other tests in which each risk factor — mortality,rate of return and new entrants — is considered as stochastic variable while theothers are assumed deterministic. Finally, one last simulation has estimated theexpected financial dynamics of the fund considering all variables as deterministic.Results have been used to conduct a sensibility analysis of the fund, with thefollowing conclusions.

The effects on the financial dynamics of the fund of random deviations fromgiven mortality trends (that is, the accidental component of longevity risk) arenegligible, and account for less than 1% of dispersion around the median value.Indeed, this is a pooling risk and its effects tend to disappear in large pensionfunds such as the CNPADC.

Most of the variance in the fund value distribution seems to be described bythe stochastic behaviour of the rate of return. This can be inferred by examining


the similarities between the percentile distribution obtained in the first test, withall the three risk factors considered as stochastic, and the percentile distributionobtained under the hypotheses of stochastic rate of return and deterministicmortality and new entrants. The values are presented in Figs. 5 and 8.

Fig. 8. Value of the CNPADC pension fund in million euros, obtained consideringstochastically only the variable ‘rate of return’, percentiles of the frequency distribu-tions, years 2006-2046

Finally, the CNPADC fund seems to have a relatively low exposure to therisk related to future new entrants. This is suggested by the large impact of therate of return on the percentile distribution of the fund value. Nonetheless, thestochastic behaviour of new entrants describes almost all of the variation in thepension fund balance. This can be deduced by examining the similarities betweenthe variance in the pension balance distribution, obtained in the first test, and thefund value distribution obtained under the hypotheses of stochastic new entrantsand deterministic mortality and rate of return. The values are presented in Figs.9 and 7.

6 Conclusions

In the present paper we have addressed the issue of the demographic risk relatedto future membership patterns in retirement funds with restricted entrance,financed under a pay-as-you-go scheme.

We have proposed a discrete-time Markov model for the estimation of newentrants in pension funds of professional categories, that highlights the inter-actions between demographic, economic and regulatory variables. The modelconsiders the effects of trends in population, trends in education choices, appealof the profession, and entrance regulations.

Numerical applications of the model have analysed the demographic andfinancial dynamics of the pension fund of Italian Chartered Accountants. De-mographic results have revealed the effects of a main reform in the university


Fig. 9. Value of the CNPADC pension fund in million euros, obtained consideringstochastically only the variable ‘new entrants’, percentiles of the frequency distribu-tions, years 2006-2046

system on the number of new entrants to the pension fund. Financial resultssuggest that the fund has a relatively low exposure to the risk related to futurenew entrants, and that its main risk factor is the rate of return. Instead, the riskof random deviations from expected mortality trends generates negligible effectsbecause of the large population of the fund.

Table 4. Value of the CNPADC fund: moments of the frequency distributions obtainedin the Monte Carlo simulation

Date: 1st Jan. 2010 1st Jan. 2015 1st Jan. 2020 1st Jan. 2025Avg. Value: 3,525,456,641 5,768,926,483 8,725,501,938 12,728,412,509St. Deviation: 143,505,878 310,161,641 542,057,540 873,711,092Skewness: 0.096 0.130 0.196 0.226Kurtosis: 0.054 0.052 0.119 0.127

Date: 1st Jan. 2030 1st Jan. 2035 1st Jan. 2040 1st Jan. 2045Avg. Value: 17,045,796,388 20,532,458,231 22,259,140,016 22,292,121,288St. Deviation: 1,337,223,372 1,930,828,426 2,597,537,506 3,271,388,475Skewness: 0.237 0.305 0.358 0.365Kurtosis: 0.116 0.198 0.266 0.234

Including other risk factors constitutes a main area of interest for furtherextensions. Specifically, the risk of regular deviations from expected mortalitytrends (that is, the systematic component of longevity risk) could be included.


Table 5. Expected cash flows of the CNPADC pension fund under deterministic as-sumptions, values in thousand euros, years 2006-2046

Value at Subjective Integrative Pension Pension Investm. Admin. Total Value atYear 1st January Contrib. Contrib. Disburs. Balance Returns Costs Balance 31st Dec.

A B C D E=B+C-D F G H=E+F-G I=A+H

2006 2,067,794 235,721 155,133 126,378 264,476 70,305 28,448 306,333 2,374,1272007 2,374,127 251,469 165,874 127,616 289,727 80,720 29,870 340,577 2,714,7042008 2,714,704 273,516 180,726 129,717 324,525 92,300 31,364 385,461 3,100,1652009 3,100,165 296,535 195,999 132,946 359,588 105,406 32,932 432,062 3,532,2272010 3,532,227 320,021 105,717 135,779 289,959 120,096 34,578 375,476 3,907,7032011 3,907,703 344,246 113,646 143,858 314,034 132,862 36,307 410,588 4,318,2912012 4,318,291 369,640 121,979 151,338 340,281 146,822 38,123 448,980 4,767,2722013 4,767,272 395,262 130,491 158,308 367,446 162,087 40,029 489,504 5,256,7762014 5,256,776 420,800 139,148 162,362 397,586 178,730 42,030 534,286 5,791,0612015 5,791,061 442,274 146,518 183,278 405,514 196,896 44,132 558,278 6,349,3402016 6,349,340 449,171 149,275 216,779 381,667 215,878 46,338 551,206 6,900,5462017 6,900,546 465,270 155,035 223,675 396,630 234,619 48,655 582,593 7,483,1392018 7,483,139 486,811 162,532 224,363 424,980 254,427 51,088 628,319 8,111,4572019 8,111,457 499,487 167,154 220,442 446,199 275,790 53,643 668,346 8,779,8032020 8,779,803 519,236 174,026 212,218 481,044 298,513 56,325 723,233 9,503,0362021 9,503,036 535,622 179,714 204,108 511,228 323,103 59,141 775,190 10,278,2262022 10,278,226 550,441 184,774 199,133 536,082 349,460 62,098 823,443 11,101,6702023 11,101,670 562,260 188,751 197,256 553,755 377,457 65,203 866,009 11,967,6782024 11,967,678 550,194 184,773 202,953 532,014 406,901 68,463 870,452 12,838,1302025 12,838,130 555,862 186,599 212,552 529,909 436,496 71,886 894,519 13,732,6492026 13,732,649 556,398 186,627 231,957 511,068 466,910 75,480 902,497 14,635,1462027 14,635,146 553,372 185,563 260,129 478,807 497,595 79,255 897,147 15,532,2932028 15,532,293 540,360 180,946 295,673 425,633 528,098 83,217 870,514 16,402,8072029 16,402,807 528,597 177,013 338,932 366,678 557,695 87,378 836,995 17,239,8022030 17,239,802 514,022 172,243 386,708 299,557 586,153 91,747 793,963 18,033,7662031 18,033,766 500,528 167,916 429,731 238,712 613,148 96,334 755,526 18,789,2922032 18,789,292 488,378 164,259 477,128 175,508 638,836 101,151 713,193 19,502,4852033 19,502,485 507,559 170,759 533,357 144,961 663,084 106,209 701,837 20,204,3222034 20,204,322 482,169 162,691 587,370 57,490 686,947 111,519 632,918 20,837,2402035 20,837,240 485,158 163,733 642,333 6,557 708,466 117,095 597,928 21,435,1682036 21,435,168 443,015 147,875 720,059 -129,169 728,796 122,950 476,677 21,911,8452037 21,911,845 418,296 139,557 799,093 -241,240 745,003 129,097 374,666 22,286,5112038 22,286,511 391,102 130,357 885,043 -363,585 757,741 135,552 258,605 22,545,1152039 22,545,115 364,139 121,175 952,225 -466,911 766,534 142,330 157,293 22,702,4092040 22,702,409 348,623 115,898 999,893 -535,372 771,882 149,446 87,064 22,789,4722041 22,789,472 337,174 111,961 1,030,520 -581,385 774,842 156,919 36,539 22,826,0112042 22,826,011 333,977 110,795 1,041,047 -596,274 776,084 164,764 15,046 22,841,0572043 22,841,057 336,503 111,606 1,029,905 -581,796 776,596 173,003 21,797 22,862,8542044 22,862,854 339,549 112,603 1,000,520 -548,368 777,337 181,653 47,316 22,910,1702045 22,910,170 343,496 113,914 977,384 -519,974 778,946 190,735 68,236 22,978,4072046 22,978,407 347,699 115,319 960,813 -497,795 781,266 200,272 83,199 23,061,606

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A policyholder’s utility indifference valuation

model for the guaranteed annuity option

Matheus R. Grasselli1 and Sebastiano Silla2

1 Dept. of Math and Stats, McMaster University1280, Main Street West, Hamilton, Ontario, Canada L8S 4K1

e-mail: [email protected] Dept. of Social Sciences, Polytechnic University of Marche

P.le Martelli 8, 60121 Ancona, Italy.e-mail: [email protected]

Abstract. Insurance companies often include very long-term guaran-tees in participating life insurance products, which can turn out to bevery valuable. Under a guaranteed annuity options (g.a.o.), the insurerguarantees to convert a policyholder’s accumulated funds to a life annuityat a fixed rated when the policy matures. Both financial and actuarialapproaches have been used to valuate of such options. In the presentwork, we present an indifference valuation model for the guaranteed an-nuity option. We are interested in the additional lump sum that thepolicyholder is willing to pay in order to have the option to convert theaccumulated funds into a lifelong annuity at a guaranteed rate.

Keywords. Indifference valuation, guaranteed annuity option, gao, in-complete markets, insurance, life annuity, annuitization, optimal asset al-location, retirement, longevity risky, optimal consumption/ investment,expected utility, stochastic control, Hamilton-Jacobi-Bellman equation.

M.S.C. classification. 91G10, 91G80.

J.E.L. classification. D91, G11, J26.

1 Introduction and literature review

Insurance companies often include very long-term guarantees in participatinglife insurance products, which can turn out to be very valuable. Guaranteedannuity options (g.a.o.) are options available to holders of certain policies thatare common in U.S. tax-sheltered plans and U.K. retirement savings. Underthese options, the insurer guarantees to convert a policyholder’s accumulatedfunds to a life annuity at a fixed rated when the policy matures. Comprehensiveintroductions to the design of such options are offered by O’Brien [43], Boyle& Hardy [8] [7], Hardy [19] and Milevsky [30]. For concreteness, we will focuson the analysis of a particular type of policy, but the framework we use can bereadily extended to more general products in this class.

62 Matheus R. Grasselli and Sebastiano Silla

1.1 The design of the policy

We analyze a standard contract designed as follows: at time t0 = 0 the pol-icyholder agrees to pay a continuous premium at a rate P for an insurancepolicy maturing at T . The premium is deemed to be invested in a money mar-ket account with continuously compounded interest rate r, and the policyholderreceives the corresponding accumulated funds A at time T . We are interested inthe additional lump sum L0 that the policyholder is willing to pay at time t0in order to have the option to convert the accumulated funds A into a lifelongannuity at a guaranteed rate h.

Between time t0 and time T , the liabilities associated with such guaranteedannuity options are related to changes in economic conditions and mortalitypatterns. A rational policyholder will only exercise the option at time T if itis preferable to the annuity rates prevailing in the market at that time. Asremarked by Milevsky and Promislow [33], the company has essentially grantedthe policyholder an option on two underlying stochastic variables: future interestrates and future mortality rates.

1.2 Literature review

The nature of guaranteed annuity options was firstly presented in Bolton et al.[6] and O’Brien [43]. The liabilities under guaranteed annuity options representan important factor that can influence the solvency of insurance companies. Ina stochastic framework, a first pioneering approach was proposed by Milevskiand Posner [32] and Milevsky and Promislow [33]. The literature concerning thevaluation of guaranteed annuity options in life insurance contracts has grownand developed in several directions. Both financial and actuarial approacheshandle implicit (“embedded”) options: while the formers are concerned withrisk-neutral valuation and fair pricing, the others focus on shortfall risk underan objective real-world probability measure. The interaction between these twoways was analyzed by Gatzert & King [16]. The seminal approach of Milevsky& Promislow [33] considered the risk arising both from interest rates and hazardrates. In this context, the force of mortality is viewed as a forward rate randomvariable, whose expectation is the force of mortality in the classical sense. On thesame line, the framework proposed by Dahl [14] described the mortality intensityby an affine diffusion process. Ballotta & Haberman [2] [3] analyzed the behaviorof pension contracts with guaranteed annuity options when the mortality risk isincluded via a stochastic component governed by an Ornstein-Uhlenbeck process.Then, Biffis & Millossovich [4] proposes a general framework that examines someof the previous contributions. For an overview on stochastic mortality, longevityrisk and guaranteed benefits, see also Cairns et al. [9] [10], Pitacco [51] [50]and Schrager [53]. Finally, a different approach, based on the annuity price, wasoffered by Wilkie [58] and Pelsser [48] [49]. In particular, Pelsser introduceda martingale approach in order to construct a replicating portfolio of vanillaswaptions. We also mention the related contributions of Bacinello [1], Olivieri &Pitacco [45], [46], [47] and Pitacco [26], Olivieri [44].

Policyholder’s utility indifference valuation for . . . 63

1.3 Objective of the paper

The present paper considers, for the first time to our knowledge, an indifferencemodel to value guaranteed annuity options. The indifference model proposed herecan capture at once the incompleteness characterizing the insurance market andthe theory of the optimal asset allocation in life annuities toward the end of thelife cycle.

The priciple of equivalent utility is built around the investor’s attitude to-ward the risk. Approaches based on this paradigm are now common in financialliterature concerning incomplete markets. In a dynamic setting the indifferencepricing methodology was initially proposed by Hodges and Neuberger [20], whointroduced the concept of reservation price. For an overview, we address thereader to the following contributions and to the related bibliography: Carmona[11], Musiela and Zariphopoulou [42], Zariphopoulou [63]. Recently Young andZariphopoulou [62] and Young [61], applied the principle of equivalent utility todynamic insurance risk.

Our argument is inspired by the theory on the optimal asset allocation inlife annuities toward the end of the life cycle. For instance, we refer to Milewsky[40], [41], Milewsky & Young [36], [37], [38], [39], Milevsky et al. [31] and Blakeet al. [5]. The model is developed in two stages. First we compare two strategiesat time T , when the policyholder is asked to decide whether or not she wantsto exercise the guaranteed annuity option. Next, we go back to t0 and comparethe expected utility arising from a policy with the guaranteed annuity optionagainst a policy where no implicit options are included.

Assuming a utility of consumption with constant relative risk aversion andconstant interest rates, we find that the decision to exercise the option at timeT and the decision to purchase a policy embedding a guaranteed annuity optionreduce to compare the guaranteed rate h and the interest rate r. It turns outthat the indifference valuation is based on two quantities: the actual value of theguaranteed continuous life annuity, discounted by the implicit guaranteed rate,and the actual value of a perpetuity discounted by the market interest rate.

1.4 Organization of the paper

The remainder of the paper is organized as follows. Section 2 describes thefinancial and actuarial setting where the model is defined. We characterize theoptimal exercise when the policy matures and the strategies at the initial time,when the policy is purchased. The end of this section gives the definition and theexplicit formula for the indifference valuation of the guaranteed annuity option.Section 3 show numerical examples for the equivalent valuation depending ondifferent scenarios for the interest rate. It also present a discrete-time version forthe numerical simulations.


2 The model

2.1 The financial market

We assume a policyholder who invests dynamically in a market consisting of arisky asset with price given by

dSt = µStdt + σStdWt

with initial condition S0 > 0, where Wt is a standard Brownian motion on afiltered probability space (Ω, F,Ft, P ) satisfying the usual conditions of com-pleteness and right-continuity, and µ and σ > 0 are constants. Furthermore, thepolicyholder can invest in a risk–free bank account described by

dBt = rBtdt

with initial condition B0 = 1, where r is a constant representing the continuouslycompounded interest rate.

The policyholder is assumed to consume at a instantaneous rate ct > 0 peryear, self-financing her position using the market gains she is able to realize.To this end, assume the agent is initially endowed by a positive wealth x andthe process Xt will denote the wealth process for t > 0. At each time t > 0,the policyholder chooses dynamically the amount πt to invest in the risky assetand, consequently, the amount Xt − πt to be invested in the risk free asset.The processes ct and πt need to satisfy some admissibility conditions, which wespecify in the next sections.

The assumed financial market follows the lines of Merton [27], [28], [29] andcan be generalized, at cost of less analytical tractability, following the contribu-tions provided, for example, by Trigeorgis [56], Kim and Omberg [21], Koo [22],Sørensen [54] and Wachter [57]. For example, in Grasselli & Silla [18] a shortnote with a non-stochastic labor income is considered.

2.2 The annuity market

Consider an individual aged χ at time 0. We shall denote by s−tpSχ+t the subjec-

tive conditional probability that an individual aged χ+t believes she will surviveat least s − t years (i.e. to age χ + s). We recall that s−tp

Sχ+t can be defined

through the force of mortality. Let Fχ+t(s) denotes the cumulative distributionfunction of the time of death of an individual aged χ + t. Assuming that Fχ+t

has the probability density fχ+t, the force of mortality at age χ + t + η is defiedby

λSχ+t(η) :=

fχ+t(η)

1 − Fχ+t(η),

which leads to

s−tpSχ+t = e−

∫

s−t

0λS

χ+t(η) dη


Since it is possible to obtain λSχ+t(η) = λS

χ+t+η(0) (see Gerber [17]), it is

useful to denote λSχ+t(η) by the symbol λS

χ+t+η. This leads to write

s−tpSχ+t = e−

∫

s

tλS

χ+η dη (1)

For the numerical simulations below, we will assume a Gompertz’s specifica-tion for the force of mortality λS :

λSχ+η :=

1

ςexp

(χ + η − m

ς

)

Similar formulas are given for both the objective conditional probability ofsurvival s−tp

Oχ+t and the objective hazard function λO. Employing the method

proposed by Carriere [13], we estimate the parameters m and ς in Table 1 usingthe Human Mortality Database for the province of Ontario, Canada, for a femaleand a male both aged 35 in the years 1970 and 2004.

Table 1: Estimated Gompertz’s parameters for a female and a male from the provinceof Ontario, Canada, conditional on survival to age 35. Source: Canadian Human Mor-tality Database available for year 1970 and 2004.

Female MaleYear m ς m ς

1970 85.3758 10.5098 79.1089 11.58902004 89.7615 9.3216 85.8651 10.1379

In a continuous compounding setting with a constant interest rate r, the(present) actuarial value of a life annuity that pays at unit rate per year for anindividual who is age χ + t at time t, is given by

aχ+t :=

∫ +∞

t

e−r(s−t)s−tp

Oχ+tds

where the survival probability is determined considering the objective mortalityassessment from the insurer’s point of view.

For a given aχ+t, an individual endowed by a wealth x > 0 is able to buyx/ aχ+t unit rate annuities, corresponding to a cash–flow stream at a nominalinstantaneous rate H := x/ aχ+t. This defines a conversion rate h := 1/ aχ+t, atwhich an amount x can be turned into a life long annuity with an income streamof H = xh per annum.

Notice that the conversion rate h imply a technical nominal instantaneousrate rh defined by the following expression:

1

h= a

(h)χ+t :=

∫ +∞

t

e−rh(s−t)s−tp

Oχ+tds


which depends on the mortality assumptions summarized by pO.

Returning to our g.a.o.policy, in order to offer a given conversion rate h tobe used by the policy holder at time T , the insurer considers the interest andmortality rates based on information available at time t0. However, improve-ments in mortality rates and the decline in market interest rates may representan important source of liabilities for the insurer. For instance, if at time T , theinterest rates will be below the technical rate rh and the policyholder decidesto exercise the guaranteed annuity option, the insurer has to make up the dif-ference between the two rates. Figure 1 plots the implicit rate rh with respectto different values for the conversion rate of h. The same figure also shows theimpact of the so called longevity risk : taking h = 1/9 (very common in 1970’sand 1980’s) a rate of rh = 0.0754 is implicitly determined, assuming a mortal-ity specification which was available from estimations in 1970. However rh risesto 0.0867, when the estimation of pO is made by the mortality tables availablein 2004. Hence, as remarked by Boyle & Hardy [7], if mortality rates improveso that policyholders live systematically longer, the interest rate at which theguarantee becomes effective will increase.

Fig. 1: Simulated implicit rate rh, assured by the an insurer in 1970 with respect todifferent values for the guaranteed conversion rate h. The policyholder is supposed tobe a 35 years old female, from the province of Ontario, who will be 65 at the time T

of retirement. Values for rh are compared (with an approximation of 1E-04) using aGompertz’s mortality function with (objective) parameters driven by survival tablesavailable in 1970 (solid line) and in 2004 (dashed line).

1/9 1/10 1/11 1/12 1/13 1/14 1/150

0.02

0.04

0.06

0.08

0.10

Guaranteed conversion rate

Impl

ied

disc

ount

rat

e


2.3 The valuation method

As outlined in the introduction, our valuation method will be based on two steps,motivated by the following two remarks:

1. provided the guaranteed annuity option has been purchased at time t0 = 0,the policyholder needs to decide on whether or not to exercise it at time T ;

2. assuming an optimal exercise decision at time T , the policyholder needs todecide how much she would pay at time t0 to embed such an option in herpolicy.

In order to obtain a well-posed valuation for this option, we need to assumethat the purchased insurance policy does not include any other guarantees orrights. We also assume that the conversion period (i.e. the time interval in whichthe agent is asked to take a decision on whether to exercise the option) reducesto the instant T .

2.4 Optimal exercise at time T

At time T , if the policyholder owns a guaranteed annuity option, she will be askedto take the decision to convert the accumulated funds in a life long annuity at theguaranteed rate, or to withdraw the money and invest in the market. Therefore,we need to compare the following two strategies:

i) If she decides to convert her accumulated funds A > 0 at the pre-specifiedconversion rate h, she will receive a cash flow stream at a rate H = A · hper annum. In this case, we assume that, from time T , she will be allowedto trade in the financial market. Henceforth, her instantaneous income willbe given by the rate H and by the gains she is able to realize by trading inthe financial market.

ii) On the other hand, if the policyholder decides not to convert her funds intothe guaranteed annuity, we assume she can just withdraw funds A and go inthe financial market. In this case, from time T , her income will be representedjust by the market gains she can realize. Her total endowment at the futuretime T will be then increased by the amount A.

Since the accumulation phase regards the period [t0, T ), we assume thatjust before the time T the policyholder’s wealth is given by XT

−

= xT > 0. Ifshe decides to convert her accumulated funds exercising the guaranteed annuityoption, the problem she will seek to solve is to maximize the present value ofthe expected reward represented by value function V defined as follows:

V (xT , T ) := supcs, πs

E

[∫ +∞

T

e−r(s−T )s−T pS

χ+T · u(cs) ds

∣∣∣∣ XT = xT

]

where the function u is the policyholder’s utility of consumption, which is as-sumed to be twice differentiable, strictly increasing and concave, and the wealth


process Xt satisfies, for all t > T , the dynamics

dXt = r(Xt − πt)dt + πt(dSt/St) + (H − ct)dt,

= [rXt + (µ − r)πt + H − ct] dt + σπtdWt, (2)

with initial condition XT = xT . On the contrary, if the policyholder decides notto exercise the option, she withdraws the accumulated funds A at time T andto solve a standard Merton’s problem given by:

U(xT + A, T ) := supcs, πs

E

[∫ +∞

T

e−r(s−T )s−T pS

χ+T u(cs) ds

∣∣∣∣ XT = xT + A

]

with initial condition XT = xT + A and subject to the following dynamics:

dXt = r(Xt − πt)dt + πt(dSt/St) − ctdt,

= [rXt + (µ − r)πt − ct] dt + σπtdWt, (3)

We assume the control processes ct and πt are admissible, in the sense thatthey are both progressively measurable with respect to the filtration Ft. Also,the following conditions hold a.s. for every t > T :

ct > 0,

∫ t

T

csds < ∞ and

∫ t

T

π2sds < ∞ (4)

At time T the policyholder compares the two strategies described above andthe respective expected rewards. We postulate she will decide to exercise theguaranteed annuity as long as

U(xT + A, T ) 6 V (xT , T )

The previous analysis, regarding the function U , considers a policyholderthat holds a policy embedding a guaranteed annuity option. A third strategyneeds to be considered in order to describe the case in which the policyholderholds a policy with no guaranteed annuity option embedded in it:

iii) If the policy does not embed a guaranteed annuity option, the policyholderdoes not have the right to convert the accumulated funds A into a lifelongannuity. In this sense, we assume that value function U will represent theexpected reward if at time t0 the policyholder purchased a plan without theguaranteed annuity option.

2.5 Optimal strategies at time t0

After defining the optimal exercise at time T , we can formalize the policyholder’sanalysis at the initial time t0, when the guaranteed annuity option may be em-bedded in her policy. We can summarize the two strategies that the policyholderfaces at time t0 as follows:


i) the policyholder purchases a policy without embedding a guaranteed annuityoption in it. In this case, she will pay a continuous premium at an annualrate P to accumulate funds A up to time T ;

ii) the policyholder decides to embed the guaranteed annuity option in herpolicy. She will pay a lump sum L0 for this extra benefit immediately andthe continuous premium P for the period [t0, T ), as in previous case.

In either case, the value of the accumulated funds is given by

A =

∫ T

t0

er(T−s)Pds

Assuming the policyholder’s income is given by the gains she can realizetrading in the stock market, between time t0 and time T , her wealth needs toobey to the following dynamics:

dXt = r(Xt − πt)dt + πt(dSt/St) − (P + ct)dt

= [rXt + (µ − r)πt − P − ct]dt + σπtdWt

(5)

with initial condition Xt0 = x0 > 0. Therefore, if the agent decides not to embedthe guaranteed annuity option in her policy she will seek to solve the followingoptimization problem

U(x0, t0) := supcs, πs

E

[∫ T

t0

e−r(s−t0)s−t0p

Sχ+t0 · u(cs)ds +

+ e−r(T−t0)T−t0p

Sχ+t0 · U(XT + A, T )

∣∣∣∣∣ Xt0 = w0

]

On the contrary, if she decides to embed a g.a.o.in her policy, paying thelump sum L0 at time t0, her wealth is still given by dynamics (5), but themaximization problem will be different, namely for w0 − L0 > 0:

V(x0 − L0, t0) := supcs, πs

E

[∫ T

t0

e−r(s−t0)s−t0p

Sχ+t0 · u(cs) ds +

+ e−r(T−t0)T−t0p

Sχ+t0 max

U(XT + A, T ; r), V (XT , T )

∣∣∣∣∣ Xt0 = x0 − L0

]

Notice that at time T the policyholder can an either exercise the option, remain-ing with the wealth XT plus the lifelong annuity obtained from converting A atthe rate h, or decide not to exercise the option and withdraw the accumulatedfunds A.

The same admissibility conditions are required for the control processes ct

and πt during the accumulation period, namely they are both progressively mea-surable with respect to Ft and satisfy (4)

We postulate that the agent will decide to embed a guaranteed annuity optionin her policy as long as the following inequality holds:

U(x0, t0) 6 V(x0 − L0, t0)


2.6 The inequality U(XT + A, T ) 6 V (XT , T )

Given a wealth XT at time T , Grasselli & Silla [18, App. A] find an explicit so-lution for a class of problems regarding value functions U , V , U and V, assuminga constant relative risk aversion (CRRA) utility function defined by

u(c) =c1−γ

1 − γ, γ > 0, γ 6= 1 (6)

and a constant interest rate satisfying the condition r > (1 − γ)δ where

δ := r + 1/(2γ) · (µ − r)2/σ2.

Namely, the value functions U and V are given by:

U(XT + A, T ) =1

1 − γ(XT + A)

1−γ· ϕγ(T ) (7)

V (XT , T ) =1

1 − γ

(XT +

H

r

)1−γ

· ϕγ(T ) (8)

where ϕ is is given by

ϕ(T ) =

∫ +∞

T

e−b(s−T ) · s−T pSχ+T ds (9)

for b := − [(1 − γ)δ − r] /γ. Notice that for every γ > 0, γ 6= 1, we have

U(XT + A, T ) 6 V (wT , T ) ⇔ r 6 h

From an economic point of view, the previous inequality tells us that, at timeof conversion T , the policyholder will find convenient to exercise the guaranteedannuity option if and only if the guaranteed rate h is greater than the prevailing

interest rate r. Moreover, recalling that 1/h = a(h)χ+T , the previous inequality can

be also written as follows:

U(XT + A, T ) 6 V (XT , T ) ⇔ a(h)χ+T 6 1/r,

which says that in order to come to a decision the policyholder compares theguaranteed cost of a unit rate lifelong annuity (assured by the insurance com-pany), whose the present value is given by a guaranteed implicit rate rh, with themarket cost of a unit rate perpetuity, whose present value is determined by the

market interest rate r. Notice that the indifference point is given by a(h)χ+T = 1/r,

highlighting the absence of bequest motives for the policyholder after time T .

2.7 A closed form for value functions U and V

Combining the results of the previous two sections, we have that the value func-tion U at time T needs to be equal to

g(XT ) =(XT + A)1−γ

1 − γϕγ(T ).


Using the change of variables technique proposed in Grasselli & Silla [18, App.B], we find that the value function U is given by

U(x0, t0) =1

1 − γ

(x0 − ξU(t0)

)1−γ

ϕγ(t0) (10)

where ξU is defined by

ξU(t) =P

r

(1 − er(t−T )

)− A · er(t−T ). (11)

Similarly, we have that the value function V, at time T , needs to be equal to

G(xT ) := maxU(xT + A, T ), V (xT , T )

=

(XT +A)1−γ

1−γ ϕγ(T ), if r > h

(XT +H/r)1−γ

1−γ ϕγ(T ), if r < h

Using the same change of variables technique, we arrive at the following expres-sion for the value function V:

V(w0, t0) =

U(w0, t0) if r > h

11−γ

(w0 − ξV(t0)

)1−γ

ϕγ(t0) if r < h

where ξV is given by

ξV(t) =P

r

(1 − er(t−T )

)−

H

r· er(t−T ) (12)

2.8 The indifference valuation for the guaranteed annuity option

Consider the policyholder that, at time t0, compares the two expected rewardsarising from the value functions U and V, and define the indifference value forthe guaranteed annuity option by

L∗0 := sup

L0 : U(w0, t0) 6 V(w0 − L0, t0), w0 − L0 > 0

If the indifference value exists, it is straightforward to deduce that it is given by

L∗0 =

(H

r− A

)e−r(T−t0)

2.9 Stochastic interest and mortality rates

As we mentioned in the introduction, the liabilities associated with guaranteedannuity options depends on the variations of interests rates and mortality ratesover the time. In this sense a richer model has to take into account and to


formalize these rates as stochastic processes. In present section we will just offera sketch for stochastic models for mortality intensity.

The debate over the stochastic mortality is very prolific and the literatureconcerning this problem is huge. For what concerns in particular mortality trendsand estimation procedure, we recall for example: Carriere [12] and [13], Frees,Carriere and Valdez [15], Stallard [55], Willets [59] and [60], Macdonald et al. [24]and Ruttermann [52]. In what concern stochastic diffusion processes to modelthe force of mortality, excellent contributions are offered by: Lee [23], Pitacco[51], [50], Olivieri and Pitacco [46], [47], [45], Olivieri [44], Dahl [14], Schrager[53], Cairns et al. [9], Marceau Gaillardetz [25], Milevsky, Promislow and Young[34], [35]. We also recall that the approach followed by Milevsky and Promislow[33] was the pioneering contribution that consider at one time both the stochasticmortality and a financial market model, in order to price the embedded optionto annuitise (what we call guarantee annuity option).

The contribution by Dahl [14], propose to model the mortality intensity by afairly general diffusion process, which include the mean reverting model proposedby Milevsky and Promislow [33]. Precisely the author consider a P dynamics forthe mortality intensity given by

dλχ+s = αλ (s, λχ+s) ds + σλ (s, λχ+s) dWs (13)

where αλ and σλ are non-negative and Ws is a standar Wiener process with

respect to the same filtration Fs, defined above, for s > t0. Ws is assumeduncorrelated with Ws.

In order to avoid analytical difficulties, we investigate the effect of varyingmortality rates by comparing different scenarios for different survival probabili-ties. In particular, the next section highlights the effect of different parameterizedfunctions describing different specifications concerning the force of mortality.

3 Numerical examples and insights

3.1 Valuation under different scenarios interest rate scenarios

Consider t0 = 0 and, at this time, a female aged χ = 35 who is willing to purchasea policy. Also, suppose that this plan will accumulate, until time T := 30 (i.e.when the policyholder will be aged χ + T = 65) an amount A : = $350, 000.In order to be concrete, we can think that T may coincide with her retirementtime and that the purchase takes place in 1975. In this context, the g.a.o.(if theagent decides to embed such an option in her policy) could be exercised in 2005.We would like to stress that these calendar dates are not necessary to implementa numerical experiment. However they give a stronger economic meaning for acontract designed as follows: we assume that the agent is asked to decide whetherto include a guaranteed annuity option with a conversion rate h := 1/9 (verycommon in 1980’s and 1970’s), implying a guaranteed cashflow stream at thenominal rate H ≈ $38, 888.89 per year. Notice that, in this situation, if we referto survival tables available in 1970 (see table 1), the implicit discount rate is


Fig. 2: Value function U (solid) and value function V (dashed), for an individual char-acterized by γ = 1.4, that observes a financial market described by r = 0.07, µ = 0.08,σ = 0.12. The value of r and µ are taken large enough to simulate the 1970’s financialmarket. In this setting we find L∗

0 = 25, 171. The price is given for a g.a.o.exercisablein 2005, for a female in year 1970, from the province of Ontario, assuming a (subjective)mortality specification given by the survival table available in 1970, see Table 1.

0 200 400 600 800 1000−25

−20

−15

−10

−5

Initial wealth

Exp

ecte

d ut

iliy

rh ≈ 0.0754 and such an option was considered to be far out-of-the-money forthe policyholder.

Under the previous hypothesis, the value functions U and V are plotted infigure 2, where we assume a Gompertz’s mortality specification. We estimatethe parameters ς and m, minimizing a loss function using the method proposedby Carriere [13]. We refer to the Human Mortality Database for the province ofOntario, Canada, for a female and a male both aged 35 using tables available in1970 or in 2004. The results of our estimations are summarized in Table 1.

For some values of the market interest rate r, Table 2 shows the premium Pand the equivalent valuation L∗

0 for this policy. Figure 3 depicts the dependencyof L∗

0 on both the guaranteed conversion rate h and the interest rate r. Asexpected, the greater the interest rate, the lower the policyholder’s indifferenceprice for the option. Also, the analysis remains consistent with respect to h: thelower the guaranteed rate, the lower the agent’s indifference price.

Depending on r, Table 2, shows the nominal instantaneous rate for thepremium P (that the policyholder needs to pay to in order to accumulateA = $350, 000) and the indifference valuation L∗

0 for the g.a.o.. Notice thatit is not immediately possible to compare L∗

0 and P since the former denotesa lump sum, while the latter refers to a nominal instantaneous rate to be paidover time.

In order to better understand the meaning of P and L∗0, it can be useful

to think of an auxiliary problem. This problem is independent of the previous


Table 2: Premium and indifference valuation associated to the policy, depending onthe current interest rate.

r P L∗

0 p12 l12 Total

0.035 $6,594 $266,342 $550 $419 $9690.050 $5,026 $ 95,450 $420 $115 $5350.085 $2,519 $ 8,395 $211 $ 5 $216

indifference model, but will offer a way to validate the previous results. To dothis, consider a premium to be payed monthly for a pension or an insuranceplan. We can ask two questions: what is the value p12 of a monthly paymentwhose the future value, after 30 years, is exactly A; and what is the value ofa monthly payment l12 necessary to amortize, after 30 years, the lump-sum L∗

0

payed at t0 = 0.In order to compute l12, consider a horizon of T × 12 months. Thus l12 is

given by the following relation:

L∗0 = l12 · a T × 12 i12

where i12 := er/12 − 1 is the effective interest rate compounded monthly withrespect to er, and where in general we define

an i

:=1 − (1 + i)−n

i

as the present value of an annuity that pays one dollar for n periods, discountedby the effective interest rate i compounded each period. Similarly, define p12

such thatA = p12 · s T × 12 i12

where

sn i

:=(1 + i)n − 1

i= (1 + i)n · a

n i

represents the future value after n periods, of an annuity that pays one dollarper period, under an effective interest rate i compounded each period.

Coming back to Table 2 it is interesting to see that for r = 0.035, a monthlycash flow of $550 and a monthly stream of $419 equivalently amortize L∗

0. Settingr = 0.085, we observe a similar situation for a monthly premium of $211 anda monthly stream of only $5. These intuitive results are consistent with theliterature concerning the guaranteed annuity option. As mentioned by Boyle &Hardy [8], these guarantees were popular in U.K. retirement savings contractsissued in the 1970’s and 1980’s, when long-term interest rates were high. Thesame authors also write that at that time, the options were very far out–of–the–money and insurance companies apparently assumed that interest rates wouldremain high and thus the guarantees would never become active. As a result,


Fig. 3: Indifference price L∗

0 depending on the guaranteed conversion rate h and themarket interest rate r. The valuation is given for a g.a.o.exercisable in 2005, for afemale in year 1970, from the province of Ontario, assuming a (subjective) mortalityspecification given by the survival table available in 1970, see Table 1.

1/9

1/12

1/15

0.0350.045

0.0550.065

0.075

0

50

100

150

200

250

300

Conversion rate.Interest rate

Indi

ffere

val

uatin

for

the

g.a.

o.

from the indifference model discussed in the present paper, when the interestrate is very high - as was the case in the 1970’s and 1980’s - the guaranteedannuity option’s value, from the point of view of the policyholder, is very small.Interestingly, in the same period, empirically it was observed that a very smallvaluation was also given by insurers.

These facts are confirmed by the extremely low value of L∗0 = $8, 395 (over

T − t0 = 30 years), against the yearly nominal premium P = $2, 519. This isbetter seen in terms of the auxiliary “monthly valuation problem”: the lump sumL∗

0 can be amortized by a monthly cash flow of $5, against a monthly equiva-lent premium of $211. Moreover, p12 and l12 by construction are homogeneousquantities. Their sum gives an idea of the equivalent monthly value associatedto the policy the agent is willing to buy at time t0. This sum is showed in thelast column of Table 2. It is interesting to note the large difference between thetotal value corresponding to r = 0.035 compared to r = 0.085.

3.2 Valuation under different mortality scenarios

Through our analysis we consider a deterministic process for the force of mortal-ity. Under this assumption, the indifference valuation – in line with the previousliterature – depends on the difference between the interest rate r and the guaran-teed rate h. However it is interesting to simulate the effect arising from different


mortality rates, because even if the indifference value (at time t0) is still givenby L∗

0, the value functions U and V change. Figure 4 show the effect of assumingdifferent mortality specifications.

Fig. 4: Value function U (solid) and value function V (dashed), for a guaranteed annu-ity option maturing in 2005, for a 35 years old female in year 1975, from the provinceof Ontario, comparing a (subjective) mortality specification given by the survival tableavailable in 1970 (light lines) and in 2004 (demi-bold lines), see Table 1. The policy-holder is characterized by γ = 1.4, observing a financial market described by r = 0.07,µ = 0.08, σ = 0.12. The value of r and µ are taken large enough to simulate the 1970’sfinancial market.

0 125 250-28

-19

-10

Initial wealth

Exp

ecte

d u

tilit

y

VU

(a) Value functions U and V.

35 45 55 65 75 85 95 1050

0.005

0.010

0.015

0.020

Age

Pro

ba

bili

ty d

en

sity o

f d

ath

(b) Density of death.

For instance, we compare the optimal expecter reward considering the samepolicyholder under a different subjective assessment of the survival probability– notice that in both cases we assume deterministic process for pS , consideringdifferent scenarios. To make things easy, we keep referring to Table 1, comparingthe estimations available in 1970 and in 2004. The same value functions plottedin figure 2 are compared with the ones calculated using data available in 2004.

In other words, if the policyholder could use a more optimistic assessmentfor the survival probability, the convenience to by the option remains the same(i.e. V > U) but the expected utility is affected in the change in the value of ϕand the negative value of 1 − γ. The coefficient γ expresses the policyholder’srisk aversion over a larger trading horizon. For instance, the gap between the“new” V and the “old” V (as well as the “new” U and the “old” U) reducesfor smaller values of γ. Eventually, for γ → 0, the policyholder does not sufferany impact from different mortality scenarios. Notice, however, that this claimis true because different scenarios do not change during the trading period, thatis, there is no stochasticity other than the perturbation considered just at theinitial time t0.


4 Conclusions

We value the guaranteed annuity option using an equivalent utility approach.The valuation is made from the policyholder’s point of view. In a setting whereinterest rates are constant, we find an explicit solution for the indifference prob-lem, where power–law utility of consumption is assumed. In this setting wecompare two strategies when the policy matures, and two strategies at the ini-tial time. For the former we assume that, if the annuitant does not exercise theoption, she first withdraws her accumulated funds and then she seeks to solve astandard Merton problem under an infinite time horizon case. At the time whenthe policy matures, we compare the policyholder’s expected reward associated toa policy embedding a guaranteed annuity option, and the one which arise froma policy that does not embed such an option. We find that the option’s indif-ference price depends on the difference between the market interest rate r andthe guaranteed conversion rate h. Numerical experiments reveals that in periodscharacterized by high market interest rates, the value of the g.a.o.turns out tobe very small. Finally, we also consider an auxiliary (and independent) problemin which we compare the pure premium asked by the insurance company (foraccumulating the funds up to the time of conversion) and the indifference pricefor the embedded option.

For future research, the present model can be generalized in several ways.First, the policyholder can be allowed to annuitize her wealth more than onceduring her retirement period. This fact leads us to consider an unrestrictedmarket where the policyholder can annuitize anything at anytime, as defined byMilevsky & Young [39]. Second, the financial market can be modeled consideringa richer setting: stochastic interest rates and stochastic labor income. To thisend, we recall the work of Koo [22]. Third, and most important, in the presentframework, the longevity risk is considered by comparing different scenarios,given by the survival tables available in 1970 and in 2004. For this, a more generalstochastic approach, as proposed by Dahl [14], can be considered instead.

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Mergers, acquisitions, and innovation⋆

Hidenobu Hirata1

Graduate School of Economics,

Doshisha University

Karasuma Imadegawa, Kamigyo-ku, Kyoto,

602-8580, Japan

[email protected]

Abstract. Evidence shows that the merger firms are more successful in R&D

than those that are not. The question that then arises is how many firms should

the merger firms invest in while attempting to acquire innovation. We derive a

unique and closed-formed firm-level profit maximizing number of start-up en-

trepreneurial firms that the merger firms take equity positions in while attempting

to acquire innovation. The model is mainly described by two stochastic differen-

tial equations. For each stochastic differential equation, we apply the Bayesian

inferences to the construction of expectation on R&D process and to the rise of

profit excluding acquisition-related costs resulting from marketing.

Keywords. Stochastic differential equations, Bayes’ rule, innovation.

M.S.C. classification. 60G15, 60G35, 91B38.J.E.L. classification. C15, L20, M21.

[T]here is little justification for monopoly in a world of Open Innovation. – [10,

p.194]

1 Introduction

Open Innovation is essentially a paradigm that generates ideas by several firms or ac-

quires innovation through acquisition or capital investment. Chesbrough ([10]) insists

that innovation performed by only one firm is inefficient. Evidence by Griliches ([18])

estimates R&D and productivity at the firm level and it reveals that R&D investments

of the merger firms are more successful than those of non-merger firms. Griliches [18,

chap.5, pp.113-117] states that

⋆ The author is greatly indebted to Clair Brown and Tadashi Yagi. The author appreciates Yoshio

Itaba, Nobutaka Kawai, Yoshiaki Shikano, Kazuyuki Tokuoka, and the valuable suggestions

made by the workshop participants at the Graduate School of Doshisha University, members of

the Study Group of Employment and Industrial Relations and participants at the International

Conference MAF 2008 in Venice and anonymous referees. The author also acknowledges the

contributions of Robert Cole, Asli Colpan, Yoshi-fumi Nakata, Eiichi Yamaguchi, and mem-

bers of the Study Group of Innovation. The author is solely responsible for all the remaining

errors in this paper.

82 Hidenobu Hirata

About one firm out of five in our “complete” sample. . . appeared to be af-

fected. . . by considerable and generally simultaneous “jumps”. . . in gross plant,

number of employees, and sales. We have been able to check and convince our-

selves that most of these jumps do, in fact, result from mergers, although some

may be the result of very rapid growth. . . . One way of dealing with this prob-

lem is simply to drop the offending firms. This results in what we have called

the “restricted” sample. . . . The within estimates are. . . very sensitive, and the

estimated γ collapses, declining from .11 to .05 and -.03 in the complete, in-

termediate, and restricted samples, respectively [where γ is the elasticity of

output with respect to R&D capital]. . . . It is clear. . . that the merger firms are

responsible for the difference. . . . In other words, R&D seems most effective

for firms growing rapidly through mergers, and both phenomena (mergers and

R&D growth) are apparently related. . . . Such a finding raises questions that de-

serve additional analysis: Who are these “merger” firms and why would their

R&D investment be more successful? What kind of selectivity is at work here?

[The phrase within square brackets is added by the author]

Our motivation of this paper is to derive answers to the questions posed by Griliches.

The purpose of this paper is to solve an optimal number of start-up entrepreneurial firms

that the merger firms take equity positions in so as to maximize its profits while attempt-

ing to acquire innovation within a specified period of time. In this paper the merger firm

is a firm that is planning to merger a start-up entrepreneurial firm or the firm itself after

the merger. An equity position is an equity investment made by the merger firms for

the purpose of acquiring 50 percent or more of the shares issued by the start-up en-

trepreneurial firms after their IPO, the success of their R&D projects, or innovation. We

assume that the merger firms employ equity positions to finance them obtained by the

payment that equals the value of the call-option price multiplied by more than 50 per-

cent of the shares issued per start-up entrepreneurial firm if the start-up entrepreneurial

firm has been published.

In our model, we consider the cases where only the merger firms and the start-up

entrepreneurial firms are involved. Since the start-up entrepreneurial firms are relatively

small (before the IPO), the shareholders of the start-up entrepreneurial firms are only

the merger firms or the owners of the start-up entrepreneurial firms. Hence, shareholder

disapprovals do not occur once the merger announcement (contract) has been made for

there is no one who opposes to the merger deal. And because the start-up entrepreneurial

firms are relatively much smaller than the merger firms, regulatory considerations such

as anti-monopoly are not required.

Models that analyze acquisition and innovation have begun by Aghion and Tirole

([5, 6]) using the framework of Grossman and Hart ([19]). It is second-best to pur-

chase the other firm when the firm’s investment is relatively larger than the other’s [19].

Aghion and Tirole ([5, 6]) study the integrated case and nonintegrated cases. The for-

mer is the case the customer owns the research unit. Herein, the customer owns and

freely uses the innovation developed by the research unit. We analyze the integrated

case when there are several start-up entrepreneurial firms (research units) to finance

prior to the success of R&D. The merger firms (the customers) freely use the innova-

tion developed by the research unit after the former own the latter. Aghion and Howitt

Mergers, acquisitions, and innovation 83

([3]) provide more complex considerations to the study of Aghion and Tirole ([5, 6])

by introducing rent-sharing between researchers and developers. By incorporating the

concept of researchers and developers (in their model, researchers after becoming start-

up entrepreneurial firms, hire developers for applied research that is necessary for the

product to be sold), they study the positive effect of competition on growth. Klepper

([24]) considers how entry, exit, market structure, and innovation - the ratio of product

to process innovation - vary from the birth of industries through maturity. Arguing that

firm innovation and firm growth, entry, exit, and size distribution deserve an integrated

treatment, Klette and Kortum ([25]) capture and analyze in their model heterogeneous

firms, simultaneous exit and entry, optimal investments in expansion, explicit individ-

ual firm dynamics, and a steady-state firm size distribution. Loury ([30]) investigates

the relationship between the market structure and innovation through a model of non-

cooperative game. He analyzes that the shape of relationship between the aggregate

success rate of rivals and firm’s optimal investment in R&D is inverted U. Lee and

Wilde ([27]), using a model similar to that of Loury ([30]) which assumes the reward to

be the first to introduce the new technology is a fixed sum, where in Loury ([30]) that is

a flow, study the positive relationship between the aggregate success rate of rivals and

the firm’s optimal investment in R&D.

Several papers, including Cowan ([12]), insist that the relationship between the

number of firms and the total industry R&D is an inverted-U shape. Using the model

that is a discrete version of the model proposed by Loury ([30]), Cowan ([12]) suggests

that an increase in the number of firms in an industry decreases the number of R&D

projects undertaken by per firm. This causes a decline in the knowledge that is generated

by those R&D projects. On the basis of this fact, he shows that the relationship between

the number of firms in an industry and the rate of technological development - the total

number of R&D projects undertaken in an industry - is an inverted-U shape. Aghion

et al. ([1]) argue that the relationship between product market competition (PMC) and

innovation is an inverted-U shape, implying that there exists an optimal competition for

the greatest innovation. In this model, they use ”escaping competition” as an incentive

for engaging in innovation.

Both Cowan ([12]) and Aghion et al. ([1]) assume that the total number of R&D

projects undertaken in an industry M is not given and that it is a function of the to-

tal number of firms in an industry. In this paper, we assume that M is constant in the

short term regardless of whether or not there is a change in the number of firms. This is

because there is an upper bound to the number of researchers that firms contract with.

Firms that attempt to innovate, contract only with energetic researchers who are grow-

ing. These researchers face time constraints. The time constraints of the researchers and

the upper bound of the number of researchers restrict M. We assume that these require-

ments of firms for researchers do not change in the short term. Therefore, we assume

that M is constant in the short term. Later, we refer to this in detail and in Section 2, we

refer to the longer term when there is a possibility of variation in M.

Section 2 presents our model that is mainly comprised by two stochastic differential

equations. The optimal number of start-up entrepreneurial firms that the merger firms

take equity positions in for profit maximization is also derived in this section. Section

3 simulates the derived optimal number of firms that the merger firms take equity posi-

84 Hidenobu Hirata

tions in. Section 4 presents a brief review of the results and the conclusions that discuss

the selectivity of the merger firms.

2 The model

2.1 Assumptions

For companies like Intel and Hewlett-Packard, it makes perfect sense to invest substan-

tially in options, such as equity investments in small entrepreneurial companies with

interesting technology [32].

In this paper, the merger firms take equity positions in n start-up entrepreneurial

firms while attempting to acquire innovation. The equity investment c for the each

start-up entrepreneurial firm i has the value of the call-option price multiplied by more

than 50 percent of the shares issued per start-up entrepreneurial firm if the start-up en-

trepreneurial firm is published. Thus, after the publication of the start-up entrepreneurial

firms, the merger firms enjoy equal footings with regard to the right to purchase the

start-up entrepreneurial firms based on the “exercise price” of call options. Profits ex-

cluding acquisition-related costs x(t) of the merger firms and the number of success-

ful R&D projects j(t) in time t are given by stochastic differential equations, where

acquisition-related costs are nc + C and C has the value of the call-option strike price

multiplied by more than 50 percent of the shares issued per start-up entrepreneurial

firm if the start-up entrepreneurial firm is published. The merger firms acquire only one

start-up entrepreneurial firm from n that has the best innovation capabilities. As stated

in Loury ([30]), there are no externalities in the R&D process (for example, no stealing

of trade secrets). As noted above, M is constant in the short term. Firms are risk neutral.

Demand varies stochastically. The merger firms are in the monopolistically competitive

product markets under complete information and seek for their short term profits as my-

opic firms. They can raise as much funds as they require from the financial sector at the

constant short-term interest rate r. In the financial sector, complete stock markets with

the Black-Scholes ([8]) economy also prevail. Thus, there are no budget constraints.

There are no gains or losses from the acquisition itself, owing to the no-arbitrage con-

dition. Therefore, we can focus our study on the profit results from innovation and not

from the arbitrage of the acquisition. The gained assets, including intangible and tangi-

ble assets (net assets), are arbitrage free owing to the complete stock markets. Hence,

acquisition-related costs have no effects on the stock prices S (t) of the merger firms.

If the merger firms have the right to purchase the stocks of start-up entrepreneurial

firms that have succeeded in their R&D projects prior to the success with the call op-

tions, the risk of investments can be minimized by setting the “exercise price” before-

hand.

2.2 Number of times of successful R&D projects

In this section, we present a drift derived from Bayes’s Rule founded on the Poisson

distributions in order to express the increase in j(t). The success rate of R&D projects

depends on the increase in j(t) by his/her past experiences in them. The researcher’s


past experiences of success in R&D positively affects j(t). Hence, they increase the

success rate of R&D projects. We use the Poisson arrival rate of an innovation. Using

Bayes’ Rule founded on the Poisson distributions we get

λ′′ =j′′

m′′=

j′ + j

m′ + m(1)

[38, pp.345-362],1 where λ′′ is the posterior distribution, λ is the Poisson arrival rate of

an innovation, and j′ and m′ are the prior distributions. Further, m = MN= M

n+k, where m

is the number of times of R&D projects that are evenly allocated for each i and t by the

merger firms, N = n+k is the total number of firms in an industry, and k is the number of

other firms that the merger firms did not take equity positions in including the number

of rivals of the merger firms. In other words, k is the sum of the number of the start-up

entrepreneurial firms that the merger firms have no technological interest in and that of

firms that are rivals to the merger firms. Using the update of Bayes’s Rule founded on

the normal distributions ([9, p.25]), the update of j and m by the prior distributions will

be

j′′′ = j′′ + j

= 2 j + j′,

where j′′′ is the posterior distribution after 2 updates. Assuming that j′ = j, and apply-

ing the same to m,

λ(t) =t j

tm

=j

m.

Since, m is evenly allocated for each t, the increase in the success rate of R&D projects

depends on j. That is, if j(t) grows, the expression of the positive effects of previous

successful R&D experiences on the number of times a researcher succeeds will be2

d j(t) = µ j(t)dt + j(t)

√

j(t)

ndV(t), µ ≥ g . (2)

In (2), µ is the growth rate of j(t), the know-how, the experience required to succeed

in R&D gained by the researchers and V(t) depicts the Brownian motion of j(t). We

1 Pratt, Raiffa, and Schlaifer ([38]) derive (1) by assuming that the prior distribution is the

gamma-1 distribution. The definition of the function is given in Pratt, Raiffa, and Schlaifer

([38, p.202]). See Loredo ([28]) and Rainwater and Wu ([40]) for the theoretical frameworks

that derive exactly the same equation as the gamma-1 distribution. Gregory ([17, pp.376-378])

derives an application of it based on Loredo ([29]) with some theoretical expansion. Both

Loredo ([28, 29]) and Gregory ([17]) employ the Jeffrey’s prior as the prior distribution for

deriving it. See Jaynes ([21]) for a general discussion on the Jeffrey’s prior.2 See the Appendix for proofs of the existence and uniqueness of a solution of (2).

86 Hidenobu Hirata

employ the Brownian motion since the researchers’ efforts are the ones that are con-

tinuous, engaging in m R&D projects in each t. Though the successful R&D seems to

be discontinuous process, for the researchers themselves, it is a thing that can only be

accomplished by conducting numerous R&D projects including the ones that fail. Nu-

merous R&D projects have the role as a path to the success. Thus, we assume that j(t) is

a continuous process and apply Brownian motion to it. And j(t) does not jump. To see

why, we use contradiction. Suppose that j(t) jumps. Then, from (A10) in the Appendix,

the stock price of the merger firm S (t) also jumps. Because “such a process allows for

a positive probability of stock price change of extraordinary magnitude ([34, pp.126-

127]),” there would be arbitrage. And thus, this is not Pareto optimal, contradicting the

conclusion of the first fundamental theorem of welfare economics or first welfare the-

orem.3 Therefore j(t), especially those that are in n, do not jump. This is because the

risk induced by the jumps cannot be hedged and correspondingly, the economic model

is not complete ([7, p.1834]), which contradicts with our assumption of complete stock

market.

Since, firms have requirements for the researchers to undertake R&D, they only

contract with the researchers with µ ≥ g, where a constant g in the short term is the lower

bound of µ. Firms only contract with the researchers who are within their bounds that

reflect their qualification criteria for researchers. However, when a depression occurs

in the longer term, their qualification criteria for researchers become severe and their

bounds become narrower, that is, g becomes larger. Thus, during depressions in the

longer term, M decreases. Therefore, M is a negative function of g in the long term.

Notice that in the short term, M is constant as M. The reasons for M = M are as follows.

One of the reason is that firms do not change their standards to ensure consistency in

contracts. Thus, firms’ g do not vary in the short term. The difference between the short

term and the long term, in this paper, is whether it is shorter or longer than a specified

period of time [0,T ]. That is, the term [0, t] is

[0, t] =

short term, t ≤ T

long term, t > T,

where T is the maturity date. Since researchers with µ < g are not considered for en-

gaging in R&D undertakings, there are upper bounds to the numbers of researchers that

a firm contracts with in the short term. And these researchers that suffice firms’ require-

ments face time constraints. The number of researchers that firms contract with and the

time constraints faced by researchers restrict M. We also assume that the economy em-

ploys all these resources, that is, the researchers’ labor markets clear. Thus, M = M in

the short term. And µ does not depend on n. The start-up entrepreneurial firms that the

merger firms took equity positions in do not share their researchers’ know-how µ with

each other because they are rivals in terms of receiving the reward (C) from the merger

firms after the success of R&D. Each start-up entrepreneurial firm that receives equity

finance from the merger firms do not have any incentive to share the information about

its µ with each other. Hence, µ is not a function of n.

Since the discoveries of the researchers are actual facts, they vary stochastically.

3 See Mas-Colell, Whinston, and Green ([33, p.694]).


This is expressed by

√

j(t)

ndV(t), where we are use the variance of j(t) is j(t) consider-

ing the variance of the Poisson distribution λ =j(t)

mand m allocated evenly for each t.

√

j(t)

nis the volatility.

In the next section, we present a drift of x(t) derived from the inverse of an equation

derived from Bayes’s Rule founded on the normal distributions.

2.3 Marketing

We assume that firms perform marketing activities for the new product depending on the

invention. In this paper, these marketing activities contribute to the largest x(t) through

the four Ps, i.e. Product, Price, Place, and Production [26, pp.32-33]. In our model,

demand varies stochastically. The price that maximizes sales depends on the price elas-

ticity of demand. Assume that the increment of the inverse of the equation derived from

Bayes’ Rule founded on the normal distributions per time to be the drift of x(t). This

is expressing the augmentation of precision per time with respect to the true values of

the four Ps that generates the largest x(t) through marketing activities. Then, we get the

incremental process of x(t) as4

dx(t) =mn

σ2x(t)dt + x(t)

σ√

ndV(t) , (3)

where m is the number of times that marketing activities that are evenly allocated for

each i and t by the merger firms and V(t), the Brownian motion of x(t). The second term

in the RHS of (3) is derived by assuming that the merger firms disperse variance by

investing equally in each i. Because, x(t) is an actual event that is subject to fluctuation

this is expressed by σ√ndV(t). σ√

nis the volatility. Further, V(t) and V(t) are independent

of each other, since the former Brownian motion is the fluctuation of j(t) and the lat-

ter, the fluctuation of x(t). (3) is expressing the increment of x(t) whose growth rate is

depicted by the augmentation of precision per time with respect to the true value of the

four Ps that generate the largest x(t).

In this paper, m and m are assumed to be proportional to each other. This is be-

cause one of the functions of marketing is to determine the directions of technologies

before it is too late to change the characteristics of them. The more R&D projects are

conducted, the more difficult it is to change the directions of new technologies. Several

authors emphasize the importance of communication between the marketing and R&D

departments. “[C]ompanies serious about competing via innovation must recognize that

the educational task is multidirectional; urging R&D to educate marketing but not vice

versa is a mistake[20, p.220].” Further, “[w]ithout close communication between mar-

keting and R&D, the successful new brand development rate will be even lower than the

pitiful national average of 10 to 30 percent[31, p.81].” Thus, we assume that increases

in m increase m. Therefore m and m are conducted at the same pace. While deriving

the optimization, we assume that m = m due to the above mentioned reasons and the

simplicity of calculations. And here lies the “selectivity” of the merger firms that we

4 See the Appendix. Also see the Appendix for proofs of the existence and uniqueness of a

solution of (3).

88 Hidenobu Hirata

will discuss more in Section 4. Moreover, m ≮ m since we adopt Poisson distribution.

We need m to be large enough to makej

msmall. We do not discuss the allocation of re-

sources between the marketing and R&D departments since this problem deviates from

our thesis.

2.4 Expected profits of the merger firms

From (1), the expected profits of the merger firms π becomes5

π = nj(t)

mx(t) − nc −C , (4)

where j(t) = 1n

∑ni=1 ji. We assume that the start-up entrepreneurial firms receiving

equity finance through selling c to the merger firms cover the costs of R&D projects

and marketing activities through it.

Before deriving the merger firms’ profit maximizing n, we state the following Propo-

sitions 1 and 2, for solving this problem.

Proposition 1. m, m, and n have no effects on f (t, j(t), x(t)) = log

S (t)S (0)

where f (t, j(t), x(t))

is the call-option price as in Black and Scholes ([8]).

Proof. A proof of Proposition 1 is provided in the Appendix. 2

The implication of this Proposition is that as soon as m, m, and n are determined, the

effects on S (t) and therefore on S (0) are formulated so that the purchase of assets are

arbitrage free. An economic rationale of this example is that as soon as the merger

firm makes an announcement about n, the information of these start-up entrepreneurs

would spread through the market and the stock price of the merger firm S (0) would

coincides S (t), so that there would be no arbitrage. We can plausibly expect that as soon

as the merger firm decides n, the announcement of it would be made and information

of each n would be published. This is because large firm such as those that are listed

on the New York Stock Exchange (NYSE) firms have many existing interim sources

of information, for example, interim financial reports, trade journals, security analysts’

forecasts, industry forecasts, litigation, prospectuses, etc ([16, p.255]).

Proposition 2. The expected production function of the merger firms with respect to n

is

E

[

nj(t)

mx(t)

]

= x(0) exp

(µ +mn

σ2)t

,

and is S-shaped when Mt > 4σ2.

Proof. The proof of Proposition 2 is provided through simple calculations.6

5 See Aghion and Howitt ([4, p.55]) for nj(t)

m.

6 See Elliott and Kopp ([15, p.124]) for reference, and employ Proposition 1 and Lemma 1 in

the Appendix.


Proposition 2 depicts that initially, the merger firms hasten to take equity positions in

other start-up entrepreneurial firms while attempting to acquire innovation. However, it

is optimal for the merger firms to subsequently slow down their speed of acquisitions.

Using Proposition 2, we can derive the optimal number of start-up entrepreneurial

firms that the merger firms take equity positions in when t = T as7

n∗ = −k − MkT

2σ2W

− exp[

−(µ−r)σ2+MT2σ2

]

2σ2

√

x(0)

Mcσ2kT

, (5)

where µ is the drift of j(t) and W is the Lambert W function.8

The merger firms perform Open Innovation when n∗ > 1. When n∗ = 1, a firm

conducts R&D on its own. When n∗ ≤ 0, the firms do not conduct any R&D projects,

and thus do not purchase any call options.

In (5), k is the sum of the number of firms that the merger firms have no technologi-

cal interest in and that of its rivals. On account of monopolistic competition, the merger

firms disregard the behavior of other firms and thus the movement of k. Therefore n∗

is unique. Because we are dealing with the new technology, the resulting market based

on this product would only become monopoly or monopolistic competition. If the mar-

ket becomes monopoly, i.e. k = 0, n∗ would also becomes naught. Thus, in a world of

monopoly, there is no Open Innovation. As an economic rationale of this proposition,

for example, when MP3 player was first introduced into the market, the supplier of it

did not have to worry about the behavior of other rivals that were selling other audio de-

vices. But k affects n∗ because they are also conducting R&D projects regarding audio

devices. This is because the merger firms are trying to get the advantage of their rivals,

and thus k has influence on their n∗. However, the technology used in MP3 player and

other audio devices are different so the supplier of MP3 player disregards the behavior

of other firms.

3 Simulations

We provide the numerical simulations of n∗ below. We use Maple for these simulations.

In Fig.1, σ is in the range of 90 to 250, where c is normalized to 1, and M = 10000,

x(0) = 100, µ = 0.01, r = 0.1, and T = 3 are substituted. The figure represents the

cases when there is no observation noise related to consumer behavior. In the following

subsection, we model and simulate the case wherein there is an observation noise.

3.1 Marketing with an observation noise

When consumer characteristics are not perfectly known or the marketing department

observes noise related to consumer behavior whose characteristics are perfectly known,

7 Use mathematical software such as Maple for this derivation.8 See Corless et al. ([11]) for the Lambert W function.

90 Hidenobu Hirata

the Bayesian update founded on the normal distributions that is applied in (3) requires

modification. According to Chamley ([9]), imperfect information on consumer behavior

is operationally equivalent to observation noise related to consumer behavior whose

characteristics are perfectly known.

Fig. 1. Perfect information on consumer be-

havior.

Fig. 2. Imperfect information on consumer

behavior.

Hereafter, we modify the update of the equation derived from Bayes’ Rule founded

on the normal distributions as given in Chamley ([9, pp.48-50, (3.10)]) so as to cope

with situations when consumer behavior is unknown. The precision of marketing activ-

ities based on the four Ps becomes different, and we remodel the stochastic differential

equation expressing the incremental process of x(t) as

dx(t) =mn

σ2 + σ2η(1 + ρσ

2)2x(t)dt + x(t)

√

σ2 + σ2η(1 + ρσ

2)2

ndV(t) , (6)

where ρ is the inverse of the variance of the prior distribution σ2θ

and σ2η is the variance

of an observation noise. Proposition 1 can also be applied to the production function

that uses (6) and a proposition similar to Proposition 2 can be derived as follows.

Proposition 3. When consumer behavior is not known, the expected production func-

tion of the merger firms with respect to n is

E

[

nj(t)

mx(t)

]

= x(0) exp

(µ +mn

σ2 + σ2η(1 + ρσ

2)2)t

,

and is S-shaped when Mt > 4[σ2 + σ2η(1 + ρσ

2)2].

Proof. The proof of Proposition 3 is similar to that of Proposition 2.


See Fig.10 for the examples of Proposition 3. Fig.10 illustrates the situation when Mt <

4[σ2 + σ2η(1 + ρσ

2)2].

The derivation of n∗, in the case with an observation noise, is similar to the deriva-

tion of (5). Thus, n∗ becomes

n∗ = −k −MkTσ4

θ

2δW

− exp

−(µ−r)δ+Mσ4θT

2δ

2√

x(0)δ

Mcσ4θkT

, (7)

where

δ = σ2σ4θ + σ

2ησ

4θ + 2σ2

ησ2θσ

2 + σ2ησ

4 = σ2σ4θ + σ

2η(σ

2θ + σ

2)2.

Hereafter, we operate our simulations based on (7), where c is normalized to 1 and

M = 10000, x(0) = 100, k = 100, µ = 0.01, r = 0.1, T = 3, σ = 80, ση = 42.3,

σθ = 100, and C = 50 are substituted except those arguments that are presented in the

figures.

Fig. 3. k (the sum of the number of firms

that the merger firms has no technologi-

cal interest in and the number of its ri-

vals) and n∗ (the optimal number of start-up

entrepreneurial firms that the merger firms

take equity positions in while attempting to

acquire innovation).

Fig. 4. M (total number of R&D projects) in

the long term.

92 Hidenobu Hirata

Fig.2 represents the version of Fig.1 that corresponds to the case when there is an

observation noise related to consumer behavior. n∗ is much smaller when the marketing

department observes a noise than when it does not. As an economic rationale when

merger firms are uncertain about the demand for their new technology, they reduce

their investment in n.

In Fig.3, recognizing that the merger firms attempt to innovate significantly as k

begins to increase, we can observe an incentive to “escape competition” ([1]). However,

when the number of rivals in an industry is excessively high, the merger firms eventually

desist from conducting R&D and taking equity positions in start-up entrepreneurial

firms. This phenomenon is due to the “business stealing” effect [1, 2, 4]. In the words

of Aghion and Howitt ([2]), rivals do not internalize the loss to the incumbents (the

merger firms) by their entry. Considering the future loss evoked by the entries of rivals,

the merger firms decrease n∗.

Fig.4 demonstrates the long term transition. It shows the transition in the long term

from the stage of no innovation to the stage wherein the merger firms take equity po-

sitions in n while attempting to acquire innovation. When the number of researchers in

an industry is limited and M is low, because of depression or scarcity of researchers in

the field, the merger firms do not conduct R&D projects or do not take equity positions

in other start-up entrepreneurial firms. Fig.4 shows that in order to take equity positions

in other start-up entrepreneurial firms in attempting to acquire innovation, the merger

firms require larger M and therefore an abundant number of researchers that engages in

R&D.

Fig.5 reveals that the merger firms’ performances are considerably better than that

of non-merger firms. As shown here, according to the effective R&D projects and the

augmented x(t) through marketing activities, at the optimal point, the merger firms’

profits are approximately 7 times larger than those of non-merger firms. We can ob-

serve that the difference between the profits of the merger and non-merger firms is

“very sensitive [18].” Also we can see from “Tabel 5.7 Analysis of Merger Differences

(scientific firms, 1966-77)” γ is almost 7 times higher comparing −0.03 for the “Re-

stricted” sample and 0.65 for the “No-jump” period of the Merger firms, even though

the laor productivity growth rates are equal for both [18, pp.116-117]. This conclusion

would be supported by our results. The reason of this comes from well known fact that

small firms have (relatively strong) merits in successful R&D projects and large firms

have merits in marketing. Thus merger firms have the advantage of the two, and more-

over, competitive behavior of researchers, since their internal labor market becomes

competitive market for researchers do not know when the entry occurs by their firms’

acquisitions.9 Whereas by the Closed Innovation i.e. the opposite of Open Innovation,

firm performs innovation by itself so that the boundary of the firm has a role as an entry

barrier to the other researchers outside. By Open innovation, internal labor market of

researchers becomes competitive market and this makes innovation cheaper.

And as an ex-post validity, Intel Capital held more than 475 investmenst in port-

folio companies, with a market value of more than $1.4 billion ([10, p.126]). Cisco

systems, from August 1998 to August 1999, acquired 14 companies with a total market

9 See Shy [41, p.63] for example.


Fig. 5. π (the merger firms’ profits) and

n (the number of start-up entrepreneurial

firms that the merger firms take equity po-

sitions in while attempting to acquire inno-

vation).

Fig. 6. T (the maturity date of the call-

option) and n∗.

value of nearly $12 billion ([36, pp.20-21, Exhibit 3]). In Sweden most successful small

technology based firms are in due course acquired by large firms ([13, p.174]).

We have figures 6, 7, 8, 9, and 10 illustrate the relationships between T and n∗, σ

and n∗, µ and n∗, x(0) and n∗, and n and x(t) respectively.

4 Concluding remarks

The profit maximizing number of start-up entrepreneurial firms that the merger firms

take equity positions in while attempting to acquire innovation is provided and sim-

ulated for each case when the marketing departments observe and do not observe a

noise on consumer behavior. Further, the relationship between the number of start-up

entrepreneurial firms that the merger firms take equity positions in while attempting to

acquire innovation and profits excluding acquisition-related costs x(t) is proven to be S-

shaped whenever the number of an industry R&D projects are large enough to cover the

variance of x(t). We also observe that the merger firms’ expected profits are revealed to

be considerably higher than those of non-merger firms. What are the gains that increase

the expected profits of the merger firms? Owing to the complete stock markets, the as-

sets gained, including intangible and tangible assets (net assets), are arbitrage free. An

acquisition per se does not result in gains or losses due to the no-arbitrage condition.

Note that, in this model, invention by itself does not contribute to the profits or sales

94 Hidenobu Hirata

Fig. 7. σ (derived from σ2 of x(t)) and n∗. Fig. 8. µ (the drift of j(t)) and n∗.

of the merger firms. If the invented new technologies can not suffice the demands of

consumers, they do not contribute to anything besides adding to the R&D costs. It is

the integration of the marketing organized for the new technological products and the

efficient R&D projects that correspond to the outcome of marketing activities during

the period of R&D projects that raises x(t) of the merger firms. The reason for this

is well known fact that small firms have (relatively strong) merits in successful R&D

projects and large firms have merits in marketing. Thus merger firms have the advantage

of the two if these are combined frictionless, and moreover, competitive behavior of re-

searchers, since their internal labor market becomes competitive market for researchers

do not know when the entry occurs by their firms’ acquisitions. By Open innovation,

internal labor market of researchers becomes competitive market and this makes inno-

vation cheaper.

Appendix

Derivation of (3). The variance of the equation derived from Bayes’ Rule founded on

the normal distributions after being updated t times is10

σ2σ2θ

σ2 + σ2θt, (A1)

where σ2 is the variance of x(t) and σ2θ

is the variance of the prior distribution. The

inverse of this variance, the precision, is

σ2

n+σ2θ

nmt

σ2

n

σ2θ

n

(A2)

10 See Chamley ([9, p.25]) for example.


Fig. 9. x(0) (profits excluding acquisition-related costs in time 0) and n∗.

where we replaced σ2 by σ2

nand σ2

θby

σ2θ

nin order to take into consideration of the

reductions in the variances by taking equity positions in n start-up entrepreneurial firms

and evenly allocated m for each i and t. σ2

nand

σ2θ

ncan be justified by assuming that the

rational merger firms do not allow the start-up entrepreneurial firms that are receiving

equity finance from them, to have the same prior distributions or to undertake the same

marketing activities. Differentiating (A2) with respect to t yields

d

σ2

n+σ2θ

nmt

σ2

n

σ2θ

n

dt=

σ2θm

n

σ2

n

σ2θ

n

=mn

σ2. (A3)

We assume that this increment of precision per time to be the drift of x(t) through

marketing activities. Thus, a stochastic differential equation describing the incremental

process of x(t) is given by (3).

Proofs of the Existence and Uniqueness of Solutions of (2), (3), and (6). We first provide

proofs of the existence and uniqueness of a solution of (2) in the range of [0,T ]. The

following proof is based on Klebaner ([23, pp.17-18]), Minotani ([35]), and Φksendal

([37, chap 5]). (2) satisfies the following condition:

|µ( j, t)| +

∣

∣

∣

∣

∣

∣

∣

√

j( j, t)

n

∣

∣

∣

∣

∣

∣

∣

≤ B(1 + | j|); j ∈ Rn, t ∈ 0, T (A4)

for some constant B, and the following Lipschitz condition:

96 Hidenobu Hirata

Fig. 10. n and x(t) (profits excluding acquisition-related costs of the merger firms).

|µ( j, t) − µ(l, t)| +

∣

∣

∣

∣

∣

∣

∣

√

j( j, t)

n−√

j(l, t)

n

∣

∣

∣

∣

∣

∣

∣

≤ D| j − l|; j, l ∈ Rn, t ∈ 0, T (A5)

for some constant D. Since µ is the drift of j, µ < 1. Thus, (A4) is satisfied by an

adequately large B, since j is the number of times of successful R&D projects. We

prove the Lipschitz condition below. Here we assume that µ( j, t) = µ1 j(t) + b and√

j(l,t)

n=

√

1n1

√

j(t)+b0. Owing to µ < 1 and an adequately large D, µ1 and√

1n1

satisfy

the following conditions:

|µ1| <D

2and

√

1

n1

<D

2. (A6)

Therefore,

|µ( j, t) − µ(l, t)| = |µ1 j(t) − l(t)| = |µ1| | j(t) − l(t)| ≤ D

2| j(t) − l(t)| , (A7)

and


∣

∣

∣

∣

∣

∣

∣

√

j( j, t)

n−√

l(l, t)

n

∣

∣

∣

∣

∣

∣

∣

=

∣

∣

∣

∣

∣

∣

√

1

n1

√

j(t) −√

l(t)

∣

∣

∣

∣

∣

∣

=

∣

∣

∣

∣

∣

∣

√

1

n1

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

√

j(t) −√

l(t)∣

∣

∣

∣

≤

≤ D

2

∣

∣

∣

∣

√

j(t) −√

l(t)∣

∣

∣

∣

, (A8)

Then the Lipschitz condition is satisfied. Consequently, the existence and uniqueness

of a solution of (2) in the range of [0,T ] are proved, provided that the second moment

of initial value j(0) is

E[ j(0)2] < ∞ . (A9)

Moreover, j(0) must be independent of V(t) and t ≥ 0. These conditions are satisfied

considering that j(0) is the number of times of successful R&D projects in time 0 and

is thus constant. 2

Proofs of the existence and uniqueness of solutions of (3) and (6) in the range of

[0,T ] are similar as to the above proof.

Proof of Proposition 1. Due to the Black-Scholes ([8]) economy, the call option price is

f (t, j(t), x(t)) = log

S (t)

S (0)

= log

R × (nj(t)

mx(t))

Q × (x(0))

, (A10)

where R and Q are the multipliers or price/earnings ratios (PERs). In (A10) earnings

are profits and nc+C are not subtracted from nj(t)

mx(t), since there are no gains or losses

from the acquisition per se by the no-arbitrage condition. Thus, we can focus our study

on the profits resulted from innovation and not from the arbitrage of the acquisition.

The gained assets including intangible and tangible (net assets) are arbitrage free owing

to complete stock markets of the financial sector. Thus, acquisition-related costs have

no influence on S (t). Hence, though S (t) is a function of price/book-value ratio (PBR),

PBR is not included in (A10) as PERs are. And without loss of generality, we can

assume that the numbers of shares of the merger firms are constant through [0,T ]. In

the above equation, since R&D projects and marketing activities are conducted in [0,T ],

m, m, and n are predetermined by the contract made before time 0. Due to complete

stock markets, as soon as m, m, and n are determined the effects on S (t) and therefore

S (0) are formulated so that the purchase of assets are arbitrage free. For simplicity, let

r = 0. Owing to the Black-Scholes ([8]) economy, the stock pays no dividends or other

distributions. Therefore, in this case, the forward price of S (t) at time 0 equals S (0).

Since, the expected settlement amount of the forward, the forward price, and the future

price coincide [14, chap. 8, pp.166-168] , nj(t)

mx(t) and x(0) have the multipliers so that

S (t) = S (0) and thus, m, m, and n can not affect f (t, j(t), x(t)). This is nothing but a

Radner equilibrium [see Radner ([39])] under complete markets. Therefore, there are

no effects of m, m, and n on f (t, j(t), x(t)). That is,

f (t, j(t), x(t),m, m, n) = f (t, j(t), x(t)) . (A11)

⊓⊔

98 Hidenobu Hirata

Lemma 1. E[exp∫ t

0

√

j(s)

ndV(s)] = exp

∫ t

0

j(s)

2nds.

Proof. First we prove that exp∫ t

0

√

j(s)

ndV(s) is a martingale. We depend the following

proof on Karatzas and Shreve ([22]). This is since,

Zt(X) = 1 +

d∑

i

∫ t

0

Zs(X)X(i)s dW (i)

s , ∀t ∈ [0,T ]

which shows that Z(X) is a continuous local martingale with Z0(X) = 1. Solving for

Zt(X) we get

Zt(X) = exp

∫ t

0

XsdWs,

where we applied Z0(0) = 1 and omitted∑d

i=1 for d, dimension, is one in this case.

Thus, Jt = exp∫ t

0

√

j(s)

ndV(s) is a continuous local martingale. If

E[exp(1

2

∫ T

0

||Xs| |2ds)] < ∞; 0 ≤ T < ∞,

then Z(X) is a martingale ([22, Section 3.5.D]) and therefore, Jt is also a martingale.

This is because,∫ t

0

√

j(s)

ndV(s) is bounded in t (since there is the maturity date) and j

(since no researcher can attain successful R&D unboundedly and because of the lower

bound g of µ). Martingale increments are independent of each other because, “when

squaring sums of martingale increments and taking the expectation, one can neglect

the cross-product terms[22, p.32].” Thus, applying E[exp

√

j(s)

nV(s)] = exp

j(s)

2ns,

E[exp

∫ t

0

√

j(s)

ndV(s) = E[explim

l→∞

l∑

h=1

√

j(δh)

n(V(sh) − V(sh−1))]

= liml→∞

l∏

h=1

E[exp√

j(δh)

n(V(sh) − V(sh−1))]

= exp

∫ t

0

j(s)

2nds,

where δh ∈ [sh, sh−1] and max(sl − sl−1) → 0. In the above calculations we have em-

ployed that martingales are independent of each other, and thus, the expectations of

multiplications are equal to the multiplications of expectations. 2

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Risk indicators in equity markets⋆

Leonard MacLean1, Giorgio Consigli2, Yonggan Zhao1, and William Ziemba3

1 School of Business Administration, Dalhousie University, 6100 University Avenue,Halifax, Canada B3H 3J5, [email protected]

l.c.maclean, [email protected] Department of Mathematics, Statistics and Computer Science, University of

Bergamo, Via dei Caniana, 24127 Bergamo, [email protected]

3 Sauder School of Business, University of British Columbia, 2053 Main Mall,Vancouver, British Columbia V6T 1Z2, Canada

[email protected]

Abstract. The distribution of securities prices in financial markets isknown to exhibit heavy tails, and furthermore the time trajectory hasoccasional extreme swings or reversals in direction. The modelling ofheavy tails has been achieved with the addition of a homogeneous pointprocess to a diffusive process. However, the timing of the jumps in thepoint process should capture the price reversals. In this paper a non-homogeneous point process is introduced, so that the intensity and sizeof jumps are state dependent. The state is characterized by stress mea-sures, which are composed from combinations of risk factors. The factorsconsidered are the bond-stock yield differential and the volatility index.The parameters in the model are estimated from data on the US marketfrom 1990 - 2007. An out-of-sample test is performed for 2008 - 2009.The model captures the swings in equities prices and provides a basis foranticipating reversals from risk factors.

Keywords. Cox process, endogenous instability, stress factors, extremeequity risk, maximum likelihood.

M.S.C. classification. 62F25, 91B24, 91B30.

J.E.L. classification. C13, C52, G12.

1 Introduction

The short and medium term dynamics of equity markets in developed economieshave in recent years (e.g. last two decades) increasingly shown evidence of fun-damental imbalances associated with frequent volatility regime switching andoccasional prolonged periods of one-sided tail events. The development of new

⋆ Supported by Natural Sciences and Engineering Council of Canada and the CanadaResearch Chairs program.

102 Leonard MacLean et al.

modeling approaches capturing those complex dynamics in several equity mar-kets (e.g. US, Japan, UK) has been unprecedented with a remarkable impacton financial practice and investment decisions (see [6], [20], [23] as recent refer-ences). A range of possible modeling frameworks are now available to fit highlyskewed and fat tailed as well as canonical market distributions and adaptedto risk assessment and portfolio optimization applications. In early studies [1],[14], [4] we have presented an equity return generation model for the US marketcharacterised by a continuous diffusion process with random drift and a Coxprocess with state-dependent intensity and shocks. The associated stock priceprocess belongs to the class of right-continuous left-limited (RCLL) processes.A key role, in the proposed framework, is played by what we call the risk or in-stability process defined uniquely by the point process (see below equation (5)).Relative to earlier approaches the model sets a clear and statistically testablerelationship between equity market reversals and an underlying source of riskwhose behaviour can be inferred from market trends. The introduction of aninner source of instability driving the market volatility provides a clear general-ization of previous approaches with a potential to link the economics of financialinstability and its statistical characterization.

From a financial viewpoint the possibility to identify stress factors induc-ing large inflows and outflows in the equity market is of primary importancefor strategic allocation decisions and policy makers interventions. The formerFederal Reserve Chairman Alan Greenspan has indeed pushed forward, in hisfamous 1996 irrational exuberance speech, the idea that the bond-stock earningdifferential should have been considered to assess equity markets relative mis-valuation [1], [18] before the 1987 crisis. Ziemba and Schwartz [22] had alreadyconsidered a similar risk measure in 1991. The economic rationale being that inthe very long run the equity market is expected to fluctuate around a theoreticalvalue determined by the market earning expectations and a discount factor re-flecting the 10 year interest rate behavior. This measure has been shown in [4] tooccasionally underestimate the observable market dynamics over the 1980-2005period in the US. Furthermore a risk process uniquely driven by the yield differ-ential has been shown to be statistically significant and appropriate to capturemarket reversals, though neither necessary nor sufficient to anticipate observedmarket shocks.

We propose in this work an extension of the financial model to account foran additional risk factor directly generated by the options market. The ChicagoBoard Options Exchange (CBOE) defines an implied volatility measure calledthe VIX. The index is updated daily on the basis of 30 day ATM options tradedon the S&P 500 index. As such the index reflects investors’ expectations offorward market movements. According to the structural default model [16], [3],the aggregate implied volatility in the equity market, capturing the relationshipbetween leverage and equity returns, is a key variable in assessing the credit cycleand it is heavily correlated with the prevailing spreads in the credit markets.

If the rational for the inclusion of the bond yield differential as a potentialstress measure comes from the (continuously varying) expected impact of the

Risk indicators in equity markets 103

risk free interest rate and the market expectation over future earnings, the con-sideration of the VIX does reflect both an aggregate measure of an economy widecredit cycle (according to the structural approach to credit risk) and a directsignal of the market uncertainty over future corporate performance. An increas-ing implied volatility indicator would then reflect a generalized contraction ofinvestors planning horizons and a reduction of corporate earnings, resulting intoan equity market downturn.

The ultimate aim of our study is twofold: we introduce a general statisticalprocedure based on a recursive maximum likelihood estimator to characterizeunderlying risk processes determining equity returns, and we propose the intro-duction of two risk indicators with the potential to anticipate market reversals.The presented methodology can be generalized to many instability factors, andalternative indicators can be put forward to capture the equity market senti-ment. In this article we show that the introduction of market stress measures,which are combinations of risk factors (we just consider two such factors), jointlywith a canonical geometric Brownian motion model may in certain periods im-prove the fitting of market dynamics even during periods of severe instability.The study is conducted over the 1990-2009 period in the US market.

In Section 2 of this paper the risk indicators impacting the securities mar-ket are defined. The dynamics for securities prices are separated into a diffusionand a nonhomogeneous point process, referred to as the risk process. The risk ismodeled by Weibull processes, with jump size and intensity parameters depend-ing monotonically on the risk factors through stress measures. A conditionalmaximum likelihood estimation procedure is discussed in Section 3, where theidentification of jump times in the point process follows from the monotonicity inthe stress measures. In Section 4 the methodology is applied to market data onstocks and bonds in the US for the period 1990 - 2009. The market implicationsof the pricing model and in particular the risk process are discussed in Section5.

2 Risk indicators

The focus of our study is represented by the equity market (here specifically theUS equity market). To account for investment movements from and towards thefixed income market we present a stochastic model for the equity and the bondmarket.

Consider in particular: S(t) = stock price at time t; B(t) = bond price attime t. The stock and bond prices are random variables defined on a probabilityspace (Ω,F , P ), representing the uncertain dynamics of the market. With theprices on a log scale, let Y1(t) = ln(B(t)), Y2(t) = ln(S(t)).

It is assumed that the dynamics of price movements are defined by geometricBrownian motion for bonds, and geometric Brownian motion plus a marked pointprocess for stocks. The level I or conditional log-price dynamics, given parametervalues and initial conditions Y1(0) = y1, Y2(0) = y2, are defined by the equations


dY1(t) = [µ1(t)dt+ δ1(t)dW1(t)] (1)

dY2(t) = [µ2(t)dt+ δ2(t)dW2(t)] + [dR(t)] . (2)

In these equations, W1t and W2t are independent standard Wiener processes.It is assumed that the drift parameters α1(t) and α2(t) are random variables,whose distributions are affected by common market forces. Those common forcesalso generate the correlation structure between the prices on stocks and bonds.For the drift, the dependence on factors is implicit rather than explicit. Considerthe market factor, given by F (t) at time t, where F (t) is a standard Gaussianvariable. Then the random drift parameters are

µ1(t) = µ1 + γ1F (t) (3)

µ2(t) = µ2 + γ2F (t). (4)

These equations represent the effect of the market forces on the direction of assetprices. The log-prices for the assets are correlated, with the correlation capturedby the relationship to the common factor F , as defined by (γ1, γ2). Thereforecorr(dY1, dY2) = γ1γ2.

The idiosyncratic volatility parameters are assumed to be deterministic, soδ1(t) = δ1, and δ2(t) = δ2. The full set of parameters in the Brownian motioncomponent are represented as Θ = (µ1, µ2, γ1, γ2, δ1, δ2).

The risk component dR(t) = dR1(t)+dR2(t) , where dR1(t) and dR2(t) repre-sent up and down shocks respectively, is a marked point process with time/statedependent sizes and intensities. The separate point processes determine theshocks to the stock price. The components of the risk processes are assumedto be affected by multiple risk factors, X = (X1, . . . , XJ), So the size ϑi(X)and the intensity λi(X) of up, i = 1, and down, i = 2, shocks depend on X. Itis assumed that up and down shocks are mutually exclusive. The rationale forincluding separate processes for up and down shocks is the possible differencesin investor reactions to high and low values of the risk factors.

The risk process dynamics are

dR(t) = ϑ1(t)dN1(λ1(t)) + ϑ2(t)dN2(λ2(t)). (5)

The processes N1 and N2 characterize up shocks (E(ϑ1(t)) > 0) and down shocks(E(ϑ2(t)) < 0), respectively. There are two factors or risk indicators which areconsidered in this analysis: (i) the differential in yields on stocks and bonds; (ii)the implied volatility of stocks. To capture the stress from yields, consider thevariables: U(t) = the stock market implied yield at time t; r(t) = bond market

yield at time t. Let ν(t) = r(t)U(t) = ratio of bond yield to stock yield, and ν∗ = the

average or long term yield ratio.Then the following yield variables are proposedas risk factors.

1. Yield up

X11(t) = max

ν∗

ν(t), 1

. (6)


This factor is a force for an upward shock in equity prices based on theundervaluation of securities.

2. Yield down

X12(t) = max

ν(t)

ν∗, 1

. (7)

This factor is a force for a down shock in equity prices based on the overval-uation of securities.The Chicago Board Options Exchange (CBOE) defines an implied volatilitymeasure called the VIX. It represents the expected forward volatility of theequity index around the risk free rate over the upcoming 30 day period (onan annualized basis). The movement can be up or down. The VIX is reportedas a percentage. Since its introduction the indicator varied from historicallylows of 10% up to a maximum level of 50%. A large VIX value indicatesan expectation of a sharp increase of equities price volatility or shocks inthe terminology of this work. The risk factor based on implied volatility isdefined by the VIX:

X2(t) = 1 +V IX

100. (8)

The direction of a shock is not revealed by the VIX, so another indicator isrequired. The obvious direction indicator is the yield ratio.

3. Volatility: Up

X21(t) =

X2(t) if ν(t) < ν∗

1 if ν(t) ≥ ν∗ .(9)

4. Volatility: Down

X22(t) =

X2(t) if ν(t) < ν∗

1 if ν(t) ≥ ν∗ .(10)

It is anticipated that the risk factors impact investor decisions. If equities areovervalued, as determined by the yield ratio, then downward pressure on stockprices grows and the likelihood af a shock increases. An expectation of volatilityin the equities market, as measured by the VIX, is an indication of an impendingshock. It is possible that the risk factors act separately or in combination. Themechanism is stress or market discordance measures, with measure k defined as

πik(t) =

J∏

j=1

Xwjk

ji (t), (11)

where wjk > 0,∑J

j=1 wjk = 1. On the log scale

ψik(t) = ln(πik(t)) =J∑

j=1

wjk · ln(Xji).

The stress measures are analogous to principal components generated from therisk factors. Since the measures are constructed to explain price movements


rather than correlation between risk factors, the linear combinations will differfrom components.

The theory for the shock processes in this paper proposes that the shockintensities depend monotonically on the stresses generated by the risk factors,with increased stress implying a greater chance of a shock. Let πik, i = 1, 2, k =1, . . . ,K be the stress for an up and down shocks respectively. An increasingintensity implies a Weibull process, so that πik follows a Weibull distributionwith density for i = 1, 2

fik(πik) =βikφik

(πikφik

)βik−1

e

(πikφik

)βik.

(12)

The cumulative distribution is Fik(πik) = 1 − e−

(π

φik

)βik

. The cumulative canbe written in terms of the stress measure as

ln − ln [1− Fik(πik)] = −βik ln(φik) + βikψik(t). (13)

The intensity associated with stress measure k is

λik(t) =fik(πik)

1− Fik(πik). (14)

With the Weibull processes, where π has a Weibull distribution, it is knownthat ψik = ln(πik) has an extreme value distribution. To have the shock sizereflecting extreme returns, it is assumed that size depends linearly on ψik(t). Ifthere is a shock at time t, the size is assumed to be

ϑik(t) = θik0 + θik1ψik(t) + ηikZik(t) (15)

where Zik(t), are independent, standard Gaussian variables. So the expectedshock size is proportional to the stress. With the stress related intensity and sizedefined, the risk process dynamics are

R(t) =∑

k

(ϑ1k(t)dN(λ1k(t) + ϑ2k(t)dN(λ2k(t)) . (16)

The distinguishing feature of the asset pricing model is the risk process. Theparameters in the risk process for a stress measure are Ξ = (Ξ(1), . . . , Ξ(J)),where Ξ(k) = (θ1k0, θ1k1, θ2k0, θ2k1, η1k, η2k, φ1k, φ2k, β1k, β2k). It is hypothe-sized that the risk factors characterize market stress, which in turn affects shocksto equity prices through the model parameters: X → π → Ξ → R. In subsequentsections these relationships will be explored with price data on bonds and stocksin the US financial market.

3 Parameter estimation

The methods in this section provide estimates for the parameters (Θ,Ξ) in themodel for asset price dynamics. The parameters in Ξ depend on the risk factors,


and define the risk process. So a framework is in place to study the risk processand the risk factors.

Consider the set of observations at regular intervals in time (days) of log-prices for bonds and stocks

y1, . . . , yt ,where

y′s = (y1s, y2s), s = 1, . . . , t .

Let the observed daily changes in log-prices be

es = ys − ys−1, s = 1, . . . , t . (17)

The risk factors are needed for the period preceding a shock. The preferredmeasure of stock yield is the earnings - price ratio. For the yields on stocks, theearnings-price ratio is recorded: Us =

Es

Ss. With the rate on the 10-year bond rs,

the yield ratio is vs =rsUs. The VIX values are reported by the CBOE.

It is assumed there is an unobservable set of random shocks imbedded in theobserved increments. Let e = (e1, . . . , et) be the vector of observations and

I = (I11, . . . , I1t, I21, . . . , I2t)

be the associated shock indicators, where Iis = 1 for a shock, and Iis = 0otherwise and I1s · I2s = 0, s = 1, . . . t. The conditional likelihood is L(Θ,Ξ) =p(e, I) = p(e|I)×p(I), and then, the log of the likelihood is l(Θ,Ξ) = ln(p(e, I)) =ln(p(e|I)) + ln(p(I)). The diffusion (random walk) parameters and the jumpsize parameters can be estimated using the conditional log likelihood l(Θ|I∗) =ln(p(e|I∗)) for a given jump sequence I∗. Note that ln(p(e|I)) = ln

∏t

s=1 p(es|Is) =∑t

s=1 ln(p(eis|Is)). The intensity parameters (φ1k, β1k) and (φ2k, β2k) can be es-timated independently from ln(p(I)) using the Weibull distribution.

If the shock sequences can be identified, conditional maximum likelihoodcan be used to estimate parameters. The hypothesis in the model is that therisk process parameters depend monotonically on the risk factors through themarket stress functions. If there are thresholds such that the chance of a shock isalmost certain, then identifying the thresholds is critical. The approach is to setthreshold values for extreme stress, and times where the value is exceeded areidentified. If threshold values for high stress are (π∗

1k, π∗

2k), then the incrementswith π1k > π∗

1k identify up shock times and π2k > π∗

2k identify down shocktimes. Given those shock times, conditional maximum likelihood estimates formodel parameters are determined. The value of the parameter estimates andthe conditional likelihood are compared for combinations of (π∗

1k, π∗

2k) , andappropriate thresholds for up/down shocks are established. Since the expectationis that the risk/shock component of a price movement dominates the diffusion,the positive and negative price changes will be used as a secondary indicator ofa shock period.

In considering market stress and the risk factors, there are two issues. (i)the stress thresholds (π∗

1k, π∗

2k); (ii) the weights assigned to the risk factors


wjk, j = 1, . . . , J . It is important to identify thresholds where shocks are likelyto occur, but it is also useful to identify the relative importance of the factors.The method proceeds as follows:

1. Set the weights wjk, j = 1, . . . , J , and calculate the stress values πik(t), i =1, 2.

2. Calculate the empirical distribution F1k for stress values π1k, and F2k forstress values π2k over the study period (1, t).

3. Specify a grid size ω > 0, an initial tail area p for deviations, and a stepnumber mmax. Set m = 0.

4. With a grid point m and a tail area αm = p−mω, identify times/indices

T1m =s | [es ≥ 0] ∧ [F1(π1,s−1) ≥ 1− αk]

T2m =s | [es < 0] ∧ [F2(π2,s−1) ≥ 1− αk]

.

5. Assume there is an up shock at times s ∈ T1m,and a down shock for timess ∈ T2m. For this sequence of shocks, calculate the conditional maximumlikelihood estimates for model parameters.

6. Reset m and return to [3].7. Select the thresholds and shock times which provide the best fit - minimum

mean squared error.8. Reset the weights and repeat the process.

For given thresholds and the identified shock times, a conditional likelihood max-imization is required. The diffusion and the jump size parameters are estimatedby maximizing the log-likelihood, l(Θ|I) = ln(p(e|I)) for a given sequence I. TheWeibull parameters are estimated from l(Ξ|I) = ln(p(I)).

The conditional distribution for es given I and Θ is a bivariate normal distri-bution with mean vector ξs(I,Θ) and covariance matrix, Σs(I,Θ), respectively,

(ξ1sξ2s

)=

(µ1

µ2 + I1s∑

k(θ1k0 + ψ1ksθ1k1) + I2s∑

k(θ2k0 + ψ2ksθ2k1)

)(18)

(σ21s σ12σ12 σ

22s

)=

(γ21 + δ21 γ1γ2γ1γ2 γ22 + δ22 +

∑k η

21kI1s +

∑k η

22kI2s

). (19)

It is informative to note that the mean and volatility for stocks are time vary-ing. For the given values of the shocks indicators I = ((I11, . . . , I1t), (I21, . . . , I2t)),

yield deviations((ψ1,0, . . . , ψ1,t−1), (ψ2,0, . . . , ψ2,t−1)

)and data e = (e1, . . . , et),

the data can be split into sets based on times with shocks. Let Ai = t|Iit = 1,i = 1, 2, and A = s|Iis = 0, i = 1, 2.

Consider the statistics on increments e for the subsamples: (i) the number of

values nAi, i = 1, 2, nA; (ii) means - ξAi

, i = 1, 2, ξA; (iii) covariance matricesSAi

, i = 1, 2, SA. The subsample statistics are the basis of maximum likelihood


estimates for parameters. The conditional likelihood for the diffusion and shocksize parameters is

L(Θ,Ξ|I, ψ, e) =

= (2π)−t|t∏

s=1

Σs(I)|−1

2 exp

−1

2

t∑

s=1

(es − ξs(I))′Σ−1

s (es − ξs(I))

. (20)

The function l(Θ,Ξ|I, ψ, e) = ln(L(Θ,Ξ|I, ψ, e)

)is solved iteratively for

conditional maximum likelihood estimates. The structure in the covariance ma-trix is important for model fitting. Consider Γ ′ = (γ1, γ2) and

∆s(Is) =

[δ21 00 δ22 +

∑η21kI1s +

∑η22kI2s

].

ThenΣs(I) = ΓΓ ′+∆s(Is). This decomposition of the covariance matrix intoa matrix determined by common market factors and a matrix of specific varianceswill be important in the estimation of parameters. For a given covariance matrix,

Σ =

[σ21 σ12

σ12 σ22

], it is possible to write the solution to the structural equation:

Γ ′ =(√ρσ1,

√ρσ2

), ∆ = diag((1 − |ρ|)σ2

1 , (1 − |ρ|)σ22), where ρ = σ12

σ1σ2

. Soestimates for the covariance naturally lead to estimates for the parameters Γand ∆.

With a given sequence of shocks, we can estimate (φ1k, β1k, φ2k, β2k) usingthe Weibull distribution. The power law intensity (Weibull) implies that thestress measure is the driving force in the occurance of a shock.

Consider the data on stress measures πi at the actual shock times I,

xi =xi1, . . . , xinAi

, i = 1, 2.

With this data, consider the likelihood equations

G(βik|xi) = nAi

nAi∑

j=1

xβik

ij +

nAi∑

j=1

ln(xij)

nAi∑

j=1

βikxβik

ij −nAi

nAi∑

j=1

β2ikx

βik−1ij , i = 1, 2.

The conditional maximum likelihood estimates for the Weibull parameters

(φi, βi) are (φi, βi), where G(βik|xi) = 0 and φik =

∑nAij=1

xβikij

nAi

, i = 1, 2.

All estimation routines were programmed in Matlab.

4 Market fitting

Data on the stock and bond markets in the United States are now analyzedusing the proposed model for asset prices. The objectives are to determine: (i)if the risk process improves the fitting of a model to an actual price trajectory;(ii) if the components of the risk process depend on the risk factors: bond-stockyield differential and volatility index.


It is proposed that GBM is supplemented with a risk process, which dependson yields and volatility. Figure 1 displays the path of the bond-stock yield ratioand the VIX. The time covered is from 1990 when VIX was first introduced. Inboth graphs the mean is plotted to give an indication of stable levels. Notice thediverging paths of the yield ratio and the VIX during the year 2007: rapidly de-creasing interest rates in the US have driven down the yield ratio while the equitymarket implied volatility was increasing and thus anticipating the instability tocome.

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

Date: January 02, 1990 −− December 31, 2007

Bon

d/S

tock

Yie

ld R

atio

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50001.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

Date: January 02, 1990 −− December 31, 2007

VIX

Fig. 1. Risk factors: January 1990 - December 2007

4.1 Single stress measure

The estimated parameters for the model using conditional maximum likelihoodwith single stress measures generated by separate combinations of the yield ratio


and VIX as risk factors are given in Table 1 and Table 2. The purpose in thisanalysis is to understand the individual effects of combinations of factors.

Table 1. Diffusion parameter estimates

w

Parameter 1.00 0.75 0.50 0.25 1.00

µ1 0.0000 0.0000 0.0000 0.0000 0.0000

µ2 0.0006 0.0005 0.0004 0.0003 0.0004

γ1 0.0004 0.0021 0.0037 0.0014 0.0016

γ2 -0.0011 -0.0002 -0.0001 0.0001 0.0016

δ1 0.0042 0.0036 0.0020 0.0039 0.0039

δ2 0.0094 0.0095 0.0097 0.0095 0.0063

Table 2. Parameter estimates for risk process

w

Shock Parameter 1.00 0.75 0.50 0.25 0.00

UP θ10 -0.0044 -0.0163 -0.0354 0.0117 -0.0072

θ11 0.0093 0.0446 0.1181 0.0086 0.0750

η1 0.0000 0.0000 0.0000 0.0069 0.0051

φ1 1.6115 1.4852 1.4561 1.3707 1.2831

β1 16.1342 18.7659 23.6526 47.7722 20.7511

DOWN θ20 -0.0008 0.0112 -0.0016 0.1305 0.0056

θ21 -0.0007 -0.0416 -0.0241 -0.0591 -0.0717

η2 0.0057 0.0057 0.0000 0.0035 0.0055

φ2 1.4539 1.4330 1.4202 1.3338 1.2738

β2 9.4330 17.9974 57.0399 94.5792 21.5550

The estimates are somewhat similar across the various combinations. In allcases the dependence of the shock intensity on the stress measure is strong(βik ≫ 1). The fitted accumulated increments (log-prices) for the models withthe combinations of yield ratio and VIX are given in Figure 2. These fits areactually predictions. That is, with the estimated parameters from in-sample data,a backcast/forecast was performed for the entire study 1990 - 2007. Starting fronJanuary 1, 1990, the predicted/expected increments were calculated as:

˜∆Y2(t) = µ2 + θ10 + θ11E(∆N1(λ(π1(t− 1)))) + θ20 + θ21E(∆N2(λ(π2(t− 1)))),

where E(∆Ni(λ(πi(t − 1)))) is the probability of an up/down shock in periodt calculated from the Poisson distribution with intensity based on the stress inperiod t.


0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

Date: January 02,1990 – December 31, 2007

Gro

ss C

umul

ativ

e R

etur

ns

Stocksalpha=1alpha=0.75alpha=0.25alpha=0

Fig. 2. Fitted prices for the in-sample daily data

In most cases the fits are good, with the risk process capturing the bubbleeffect of strong growth in stock prices followed by a collapse.

The weights which gave the best fit were: w = 0.0, 1−w = 1.0. These weightswere the same for up and down shocks. Although the VIX factor is best overall,the dominance is not uniform over the time period. In some time points, theBSYD (w = 1.0, 1− w = 0.0) is closer to the actual stock price trajectory.

A comparison of the mean squared errors for the models with the yield ratio,the VIX and combinations as risk factors is given in Table 3. The actual SP500index and the market dynamics implied by the bond yield differential, the VIXand the combinations of the two are reported in Figure 2.

Table 3. Performance comparison: BSYD and VIX combinations

Combination 1.00 0.75 0.50 0.25 0.00

RMSE 0.1897 0.2300 0.3391 0.3050 0.1770

LogLikelihood 34124 34147 34161 34205 35551

These calculations from the full trajectory indicate the dominance of the VIX.There are important points following from the analysis of stock prices with themodel containing a diffusion and a risk process based on a single stress measure:

(i) The inclusion of a marked point process (or risk process) to the diffusionprovides a closer match to the dynamics of prices in the US market.


(ii) When they are considered separately, the bond-stock yield ratio and thevolatility index are more significant as risk indicators of the extreme pricemovements associated with bubbles. The VIX is the single best stress mea-sure.

(iii) When the stress is defined by combination of the factors, the results areweaker due to an averaging effect, which produces fewer extreme values onthe stress measure.

4.2 Multiple stress measures

With the evidence that the factors can be considered individually as stress mea-sures, the multiple stress model is estimated. With both BSYD and VIX in-cluded, it is appropriate to work with weekly data, since the BSYD is reportedweekly. The best fitting multiple stress model in our analysis is pictured in Figure3.

0 100 200 300 400 500 600 700 800 900 1000−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Weekly Data: January 1990 − December 2007

Gro

ss C

umul

ativ

e R

etur

ns

ActualFitted

Fig. 3. Fitted prices from multiple stress model for weekly in-sample data

The fit is an improvement over the single stress measure models. The es-timated parameters in the fitted model are provided in table 4. For the sizeestimates, p-values are also reported: (estimate, p-value). The intensity esti-mates include a confidence interval: (estimate, 95% confidence interval). Testson standard deviations are not included.

The size parameter estimation included both risk factors. The estimates forthe intensity parameters were computed based on the VIX alone, as the sizecoefficient for the BYSD is negligible. The test results shown in the p-valuesindicate the statistical significance of the risk factors in the full model. Therefore,


Table 4. Parameter estimates for stress associated components

Market Scenarios Size Parameters Intensity Parameters

UP

θ10 : (−0.0096, 0.1208)φ1 : [1.3715, (1.3563, 1.3867)]

θ11 : (0.0136, 0.1448)θ12 : (0.1239, 0.0000)

β1 : [33.0178, (25.8816, 42.1152)]σ1 : (0.0163,*)

DOWN

θ20 : (0.0062, 0.0000)φ2 : [1.3160, (1.3058, 1.3262)]

θ21 : (0.0000, 0.9994)θ22 : (−0.1226, 0.0000)

β2 : [35.5460, (29.6528, 42.6104)]σ2 : (0.0182, ∗)

the traditional geometric Brownian motion model is rejected in favor of the riskprocess with jumps.

4.3 Prediction

The in-sample fitting for the model with the data from 1990 - 2007 is very good,and the obvious issue is the predictability of stock price movements using thefitted model. The estimated model was used to produce weekly forecasts of pricesfor the period January, 2008 to June 30, 2008. Weekly values for BSYD and VIXare used to produce one week ahead forecasts for shock sizes and probabilities,and the expected shock sizes are added to the Brownian motion increments. Theweekly price changes are combined to give the trajectory in Figure 4.

0 10 20 30 40 50 60 70−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

Weekly Data: Janauary 02, 2008 − Aprial 21, 2009

Cum

ulat

ive

Ret

urn

Actual

Fitted

Fig. 4. Out-of-sample forecast: January 02, 2008 - April 21, 2009


The forecast follows the pattern of the actual trajectory, although the pre-diction at the beginning of the period is a bit higher than the actual.

5 Market implications

We have presented a novel approach to study short and medium term equitymarket returns employing a general random process with a discontinuous com-ponent driven by a set of underlying risk sources. The model development restson the general view that observable market dynamics are to a certain extentmotivated by diverging expectations over forthcoming economic and corporateconditions and that a set of stressed factors may induce a sudden market reversalcharacterized by a change of the correlation structure in the market. As shownin the previous section this view is strongly supported by the achieved marketfits. Both instability factors employed in our study are driven by expectationson forward market movements. The approach does not assume the existence ofa theoretical equity market fair value but only a potential relative misvaluationgiven current equity and bond prices.

The bond-yield differential focuses on expected in-outflows from the equityinto the bond market, and vice versa, induced by diverging equity and bondyields, resulting in a measure of over-under valuation of the equity market. Itis worth remarking that according to the so called Fed model [12] the defaultfree 10 year rate is adopted as a perpetual discount factor of future earnings.Under these assumptions a market increase (decrease) is determined by increas-ing earnings expectations and decreasing interest rates and an expansion of thecredit available in the economy. Similarly a market adjustment will be inducedby a sudden revision of earnings expectations associated with an increase of theterm structure of interest rates. The above sequence of events is consistent witha possibly very serious market crisis if positive market returns contribute to im-prove conditional expectations and a restrictive monetary policy turns out tobe insufficient over a prolonged period of time to induce a revision of marketexpectations. This is exactly the sequence of events that anticipated the 1987market crisis as well as the 2000 dot com crisis.

The volatility index, generated by possibly the most liquid market in theworld, reflects expected changes in market forward volatility. This turns out tofollow closely the US market reversals. The evidence is in this case: low impliedvolatility in the option market, positive credit cycle, positive market expecta-tions and positive equity returns. High implied volatility with respect to a 15year average, unstable expectations, growing market uncertainty, investors’ hori-zon contractions and sudden market adjustments. The latter is also conditionalon recent market performance. Market uncertainty is better captured by thevolatility index and the stylized evidence of negative market turns after periodsof relatively high market volatility is confirmed. The volatility index appears alsoto provide an effective mapping between leverage-based growth strategies andfuture corporate performance. Notably, according to the structural approach tocredit risk, high implied volatility at an aggregate level implies increasing credit


spreads in the corporate market, reduced earning expectations and, in presenceof high leverage after a prolonged period of market growth, anticipates a marketreversal. The recent 2007 credit crisis was accurately captured by the index thatstarted around 11% in January 2007, reaching 25% in mid August 2007, where itstill is in June 2008! The correlation with the S&P 500 is remarkable. We have inthis case a joint signal of credit crunch and negative stock market performance.

In either case, the inclusion of the risk process with random intensity andshock measure facilitates the definition of a risk premium directly associatedwith such factors and supports the view that equity markets do on average offera higher expected return than fixed income instruments.

The risk process, driven by market instability factors, clearly captures theswings in stock price trajectories around the standard diffusive process. Thissupports the intuition about investor behavior when faced with risk. The addi-tion of the risk process to the price model has corresponding implications for thepremium on risk for securities. If the average daily returns on stocks and longbonds are calculated for the 1990 - 2006 period, the results are µ2 = 0.00038,and µ1 = 0.00036, which give annualized rates of 1.10 and 1.09, respectively.On that basis the premium for stocks is negligible. With the addition of the riskprocess, the diffusive returns are µ2 = 0.00059 and µ1 = 0.00036. The annualizedreturns are 1.16 and 1.09 for stocks and bonds respectively. The volatility in therisk process is a part of the premium and increases it to 7%.

In this respect national equity markets will possibly have associated differentinstability factors and more importantly different risk premia: the higher thecorrelation between equity markets, the more homogenous the risk structure ofequity premia.

6 Conclusions

This paper studies the nature of jumps in equity returns in financial markets.It is proposed that jumps significantly improve the fitting of equity returns, andfurthermore the essential features of jumps are affected by current market condi-tions. That is, the rate of jumps and the size of jumps are variables depending oncertain observable risk factors. The model is tested with data on equity returnsand the factors: bond stock yield differential, volatility index. From the resultsof this study the following conclusions are reached.

1. The addition of non-homogeneous point processes to a diffusion greatly im-proves the fitting to actual equity returns.

2. Both the intensity of jumps and the size of jumps depend on the risk factors- BSYD, VIX.

3. The VIX and BSYD are somewhat complimentary in that during some peri-ods the dependence on the VIX is more pronounced, while in other periodsthe dependence on the BSYD is clearer.

4. Because of the complementarity, risk measures based on combinations of thefactors average out or smooth the extremes and result in a low frequency ofshocks and a poorer fit.


A model which anticipates extreme price movements is a key ingredient in astochastic control model for decisions on investments in risky assets. As such,the risk process model in this paper is a reasonable foundation for a decisionmodel which controls the risk of large losses.

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High watermarks of market risks⋆

Bertrand Maillet1,3,4,5, Jean-Philippe Medecin2,4, and Thierry Michel6

1 University of Paris-1CES/CNRS, MSE

106 bd de l’Hopital F-75647 Paris cedex 13, [email protected]

2 Paris School of EconomicsUniversity of Paris 1 Pantheon-Sorbonne

[email protected] A.A.Advisors-QCG (ABN AMRO),

4 Variances5 Europlace Institute of Finance

6 [email protected]

Abstract. We present several estimates of measures of risk amongst themost well-known, using both high and low frequency data. The aim ofthe article is to show which lower frequency measures can be an accept-able substitute to the high precision measures, when transaction datais unavailable for a long history. We also study the distribution of thevolatility, focusing more precisely on the slope of the tail of the vari-ous risk measure distributions, in order to define the high watermarks ofmarket risks. Based on estimates of the tail index of a Generalized Ex-treme Value density backed-out from the high frequency CAC40 series inthe period 1997-2006, using both Maximum Likelihood and L-momentMethods, we, finally, find no evidence for the need of a specification withheavier tails than in the case of the traditional log-normal hypothesis.

Keywords. Financial crisis, volatility estimator distributions, range-based volatility, extreme value, high frequency data.

M.S.C. classification. 62F03, 62J10, 91G70.

J.E.L. classification. G.10, G.14.

⋆ We are grateful to Charles Corrado for help and encouragement in preparing thiswork. We also acknowledge Tim Bollerslev, Christian Bontemps, Thierry Chauveau,Thierry Foucault, Sylvain Friederich, Alan Hutson, Yannick Malevergne, Nour Med-dahi, Roel Oomen, Richard Payne, Didier Rulliere and Dick van Dijk for usefulcomments. The first author thanks the Europlace Institute of Finance for financialsupport. We thank the two anonymous referees as well as the editors for valuablecomments that contributed to improvements in the article. The usual disclaimersapply.

120 Bertrand Maillet et al.

1 Introduction

The measure of risk plays a central role in the theory and practice of finance. Themost used version by professional remains today the so-called Simple Volatility.However, following for instance Barndorff-Nielsen and Shephard [8], or Andersenet al. [6], volatility should be viewed as a latent factor (namely, the quadraticvariation affecting the Brownian motion in some representations, for instance)that can only be estimated using its signature on market prices. It is only whenthe process is known (and simulated), as in Andersen and Bollerslev ([3] and[4]), that we know what the true volatility is. As shown by Barndorff-Nielsenand Shephard [7], when the underlying process is more sophisticated, or whenobserved prices suffer from market microstructure distortion effects (see [11]),the results are less clear.

The Realized Volatility is considered, since its first use (see [4]), as the bestestimator for the latent factor of risk. The daily volatility obtained from transac-tion data is shown to be accurate when controlling for a microstructure effect andthus empirically supports the Clark’s Mixture of Distribution Hypothesis ([13]).Among the high-frequency estimators, the one using all the available transac-tions performs better than the Realized Volatilities that use a lower samplingrate (see [9]). Oomen [31] empirically also shows that estimating the volatilityin business-time (transaction time) is more efficient than using the traditionalcalendar-time, as it samples the process when it is most informative. Aıt-Sahaliaet al. [1] argue that the most precise estimator, so far, is the mean of the RealizedVolatilities chosen at the optimal frequency but measured at different phases.

However, when high-frequency data are unavailable, the best estimations ofthe unobservable risk factor are obtained through the Range-based (or ExtremeValue) estimators. The price range, defined as the difference between the highestand lowest market prices over a fixed sampling interval, is known for a longtime as a volatility estimator. Starting with Parkinson [32], there is a wealth ofliterature1 devoted to refinements of this measure (using various assumptionsabout the underlying process).

The aim of the present article then is to study the main properties of lowand high-frequency measures of volatility, in order to find if there are glaring dis-crepancies between the empirical evidence and the usual assumptions that thedistribution of volatility is Gaussian. We first recall definitions and properties ofsix main estimates of volatility, based on daily and intra-day data. We then com-pute them on the French CAC40 index over a ten-year high frequency sample.We secondly study their distributional properties by testing their Goodness-of-Fit against the Gaussian hypothesis. We thirdly focus on extreme volatilities.We fit a General Extreme Value distribution to the right-hand tail of daily riskmeasures, in order to get estimated frequencies of high watermarks of extrememarket events.

1 Relevant literature includes [32], [19], [38], [26] and [40].

High watermarks of market risks 121

2 From low to high frequency measures of risk via

extreme value estimators of volatility

It is well known that the amplitude of price changes is not constant, but fluctuat-ing with time in a somewhat predictable fashion. The Integrated Variance (i.e.,the variance of the instantaneous returns over a period) can be approximatedthrough estimators of the Quadratic Variation of prices. We present hereafter themain measures of daily volatility, computed from either daily data or intra-daydata.

2.1 Measures of risk and extreme value estimators of volatility

The usual indicator of risk is the variance obtained from the series of closingprices. Since this indicator is not constant over time, a way to diminish itsvariations in the computation is to use a rolling window with a fixed range.The general expression of the daily volatility is calculated with daily data in thefollowing manner:

σt =

1

(N − 1)

t∑

n=t−N+1

[

ln

(

Pn

Pn−1

)

− µt

]2

1

2

, (1)

where N is the estimation window expressed in a number of business days,n = [1, . . . , T ] and t = [N, . . . , T ] are daily dates, Pn is a sequence of closingprices, and µt is given by (with previous notations):

µt =1

N

t∑

n=t−N+1

[

ln

(

Pn

Pn−1

)]

,

which is an estimation of the mean log-return on the reference period.

A main critic of this daily estimator concerns the serial dependences. Indeed,the same return observations are used in the computation of many successivevolatilities, which is all the more true since N is large. Moreover, Poon andGranger [36] have noticed that the statistical properties of the sample meanmake it a very inaccurate estimate of the true mean. This is particularly true forsmall samples since taking deviations around zero - or around a very long periodmean - instead of the sample mean on a short window, increases the accuracy ofthe estimate (even if biased). If we consider this approach, the simplest measureof Simple Volatility should be defined by the squared return between only twoobservation dates (days), which is written:

σSt =

[

1

τln

(

Pt

Pt−τ

)2]1/2

, (2)


where τ is the periodicity (one day per default), and Pt is the series of theprice of the asset at time t.

In this case, there is no hypothesis about the mean return, and also noserial dependencies. Then, we will use it as our instantaneous low frequencyvolatility in the rest of the paper. Nevertheless, its time-variation is greatly noisy,and, therefore, this estimate is not recommended for practical applications. It ispossible to reduce some of the noise, affecting the previous daily-based estimates,by using an Exponential Moving Average2 (EMA). The EMA estimator is definedby induction with the following equation:

σEMAt =

ρ(σEMAt−1 )2 + (1 − ρ)

[

ln

(

Pt

Pt−1

)]2

1

2

, (3)

where ρ is the parameter governing the smoothness3.

Moreover, the counterpart of the simplicity of the previous volatility measurecomputations is that they do not take into account the information given by thepath of the price inside the period of reference. For example, even at the low(daily) frequency, supplementary information is often available in addition tothe closing price, such as the opening price and extremal prices within the day.Parkinson [32] proposes, then, an estimator of the volatility based on this typeof data, given by:

σPt =

1

θN

N∑

n=1

[

ln

(

Hn

Ln

)]2

1

2

, (4)

where θN = 4N ln(2) is a correction parameter, and:

Hn = MaxPt

Pt | t ∈ [n − 1, n] is the highest price on day n

Ln = MinPt

Pt | t ∈ [n − 1, n] is the lowest price on day n.

The efficiency of Parkinson’s Extreme Value Volatility estimator comes intu-itively from the fact that the range of intra-daily quotes gives more informationregarding the true volatility than two arbitrarily spaced points in these series(the closing prices), for the low cost of two data points per day. By this definition,Parkinson’s estimator implicitly assumes that log-stock prices follow a geomet-ric Brownian motion with no drift. Taking this assumption into consideration,Rogers and Satchell [38] propose an improvement of the volatility estimator.They add a drift term in the stochastic process that can be incorporated into avolatility estimator (with previous notations):

2 Sometimes called a RiskMetrics type of measure.3 It has been set to .5 (mild smoothing), corresponding to a half-life of one day or so.


σRSt =

1

N

t∑

n=t−N+1

ln (Hn/On) [ln (Hn/On) − ln (Cn/On)]

+ ln (Ln/On) [ln (Ln/On) − ln (Cn/On)]

1

2

, (5)

where On is the open price on day n.At last, Yang and Zhang [40] propose another improvement by presenting an

Extreme Value Volatility estimator that is unbiased, independent of any drift,and consistent in the presence of opening price jumps. Their estimator writes(with previous notations):

σY Zt =

1

(N − 1)

t∑

n=t−N+1

[

ln (On/Cn−1) − ln (On/Cn−1)]2

+κ

(N − 1)

t∑

n=t−N+1

[

ln (Cn/On) − ln (Cn/On)]2

(6)

+ (1 − κ)(

σRSt

)21/2

,

with:

κ =.34

[

1.34 + N+1(N−1)

] ,

and with Cn being the closing price on day n, the notation Xn standing forthe unconditional mean of the sequence of the variable Xn, and σRS

t being theRogers-Satchell estimator (see definition above).

Concerning the previous defined estimators, Alizadeh et al. [2] underline thatrange-based estimators have many interesting properties compared to low fre-quency estimators or even, in some cases, to high-frequency based volatility es-timators ([1], [30]). The range is a highly efficient volatility estimator as shownby Brandt and Diebold [11] in a multivariate setting. For example, when themarket is characterized by drops and recoveries in the same day, the classicalclose-to-close volatility can take low values while the daily range indicates thatthe volatility is truly high. Furthermore, the range is robust to microstructurebiases such as the bid-ask bounce. When one measures the ratio of the variance ofthe Extreme Value estimators over the Close-to-Close Simple Volatility, all pre-vious estimators provide very substantial improvements. For example, Corradoand Miller [14] report that Parkinson’s estimator allows a theoretical relativeefficiency gain comprised between 2.5 and 5.

Moreover, while the extreme estimators are still dealing with traditional mea-sures, the availability of tick-by-tick data led to a reframing of both theoretical


and empirical literature on volatility. Instead of considering a constant volatilityover a certain period of time (a day for instance), the continuous time modelassumes a continuously varying volatility. The risk over the considered period isthus no longer a constant value, but the so-called Integrated Volatility. In a con-tinuous framework, the most common stochastic equation of the price processis:

d log(Pt) = µt dt + σtdBt, (7)

with Pt the price at time t, µt the drift term, Bt the standard Brownian mo-tion and σt the instantaneous volatility. This leads to the Integrated Volatilityover the time interval τ , that is

∫ τ

0σ2

t dt. In this case, the empirical IntegratedVolatility is in fact the Realized Volatility defined as:

σRVt,τ =

t/τ∑

j=1

ln

(

Pj

Pj−1

)2

1

2

, (8)

where t is the time interval between two successive observations and Pj thesequence of high frequency (intraday) prices.

The next section is devoted to the study and the comparison of the sixpreviously defined volatility estimators, namely the Realized, the Parkinson, theRogers-Satchell, the Yang-Zhang, the Simple and the EMA Volatility estimators,by considering their time series and empirical distributions.

2.2 Descriptive statistics, correlations and distribution diagnoses of

the volatility

We represent in Figure 1 the various weekly estimates of daily volatility, usingCAC40 French stock index intraday quotes, resampled at a 30’ frequency inthe period 01-01-1997 to 12-31-2006. The peaks of the variance estimates areapproximately synchronous, but the general behavior of the series differs, bothin the range of variances and persistence phenomenon (see next section). Weremark also that Parkinson’s estimator is the closest to the Realized Volatilityin terms of similarity and general behavior.

Table 1 presents the four first moments of the empirical log-volatilities. Theasymmetry coefficient of skewness is mostly positive (with the exception ofRoger-Satchells volatility, exhibiting many very small values); the mass of prob-ability on the right side of the distribution appears slightly larger than on theleft side. The kurtosis differs across measures, with the Simple Volatility andthe Rogers-Satchell measures appearing leptokurtic (due in fact to the existenceof many observations close or equal to zero). Overall, as already seen in Fig-ure 1, estimators using intra-day data are less volatile (more accurate) than theclassical estimator.

Table 2 corresponds to the Pearson and Spearman correlation coefficients ofrisk log-estimations. It confirms once again that Parkinson’s volatility is veryclose to the Realized Volatility.


01/97 05/98 09/99 02/01 06/02 11/03 03/05 08/06

25%

50%

75%

100%Realized Volatility

01/97 05/98 09/99 02/01 06/02 11/03 03/05 08/06

25%

50%

75%

100%Parkinson Volatility

01/97 05/98 09/99 02/01 06/02 11/03 03/05 08/06

25%

50%

75%

100%Rogers−Satchell Volatility

01/97 05/98 09/99 02/01 06/02 11/03 03/05 08/06

25%

50%

75%

100%Yang−Zhang Volatility

01/97 05/98 09/99 02/01 06/02 11/03 03/05 08/06

25%

50%

75%

100%EMA Volatility

01/97 05/98 09/99 02/01 06/02 11/03 03/05 08/06

25%

50%

75%

100%Simple Volatility

Fig. 1. Daily Estimates of Annualized Volatilities (Source: Euronext, 30’ sampled in-traday CAC40 French stock index quotes from the period 01-01-1997/12-31-2006. Com-putations by the authors).

It is generally admitted, since the seminal paper by Cizeau et al. [12], thatthe log-volatility is approximately Gaussian for a daily integrated-horizon (see[5]), even if it is still discussed (see [37]) or can be generalized (see [10]).

The Probability-to-Probability plots reported on Figure 2 show the empiricalcumulative distributions of each volatility versus the Gaussian hypothesis. Allof the scale and shape parameters are estimated using the Maximum Likelihoodestimation method (e.g., [27]).

A simple eye-ball analysis confirms the diagnosis based on higher moments:Gaussianity cannot be rejected at traditional significance levels for the most ac-curate estimates (namely Realized and Range-based Volatilities). Nevertheless,the differences in the left hand tails or in the mode and, likewise, small localdifferences in the curve can be diluted in the whole sample. A specific diagno-sis of market volatility is relevant in turbulent periods and some inaccuracy inlow risk periods can be tolerated provided, the estimator performs better oth-erwise. However, when studying volatility distributions, the area of interest isthe right-hand tail where the highest volatilities are located. We have chosen tostudy more precisely these particular observations in the next section, by using


Table 1. Statistics of the (Log-)Volatilities

Mean Std Skewness Kurtosis Beta(% ) ( %) (log vol) (log vol)

Realized 16.31 9.82 .24 2.97 1.00Parkinson 14.53 9.37 .10 2.98 .95Rogers-Satchell 15.27 10.41 -1.28 7.29 .93Yang-Zhang 20.42 12.91 .02 2.97 .87Simple 16.65 16.16 -1.29 6.13 .80EMA 19.72 12.27 .04 2.93 .73

Source: Euronext, 30’ sampled intraday CAC40 French stock index quotes from theperiod 01-01-1997/12-31-2006. The Beta is computed for every estimator with respectto the Realized Volatility. Computations by the authors.

Table 2. Pearson’s and Spearman’s Correlations between Risk Measures

Realized Parkinson Rogers- Yang- Simple EMASatchell Zhang

Realized 1.00 .88 .65 .79 .34 .67Parkinson .87 1.00 .62 .75 .39 .66Rogers-Satchell .78 .77 1.00 .71 .09 .32Yang-Zhang .77 .74 .81 1.00 .42 .65Simple .39 .44 .18 .48 1.00 .62EMA .66 .64 .42 .64 .71 1.00

Source: Euronext, 5’ sampled intraday CAC40 French stock index quotes from theperiod 01-01-1997/12-31-2006. This table contains empirical Pearson (upper triangle)and Spearman (lower triangle) correlation coefficients between risk measures. Compu-tations by the authors.

the Parametric Block Maxima method for a Generalized Extreme Value (GEV)distribution of extrema.

3 Extreme values of the daily risk estimates

The Generalized Extreme Value distribution (Cf. [25]) is characterized by threeparameters: h ∈ IR, the location parameter, α ∈ IR+, the scale parameter andξ ∈ IR known as the shape parameter (which is the inverse of the tail index).The last one measures the rate of decrease of the probability in the tails. The


0 0.2 0.4 0.6 0.8 10

0.5

1Parkinson

0 0.2 0.4 0.6 0.8 10

0.5

1Rogers−Satchell

0 0.2 0.4 0.6 0.8 10

0.5

1Yang−Zhang

0 0.2 0.4 0.6 0.8 10

0.5

1Simple

0 0.2 0.4 0.6 0.8 10

0.5

1EMA

0 0.2 0.4 0.6 0.8 10

0.5

1Realized

Fig. 2. Goodness-of-Fit of a Gaussian Distribution (Source: Euronext; 5’ resampledintraday CAC40 French stock index quotes from the period 01-01-1997/12-31-2006.Computations by the authors. The P-P Plots show the cumulative distribution of eachvolatility - on the x-axis - versus the Gaussian hypothesis - on the y-axis).

GEV Cumulative Distribution Function is given by:

H(σ) =

exp

−[

1 + ξ (σ−h)α

]

−ξ−1

if ξ 6= 0,

exp

− exp[

− (σ−h)α

]

otherwise,

(9)

for every σ ∈ IR such that [1 + ξ(σ − h)α−1] > 0.For fat-tailed distributions, the shape parameter will be significantly positive.

We are then interested in the following to test the null hypothesis H0 of thepositivity of shape parameters of the noisy volatility estimators.

Among the multiple methods of estimation for the parameters of a GEVdistribution, the most common one is to use a direct numerical Maximizationof the Log-likelihood. However, a natural challenger is proposed in the case ofsmall samples (which is by definition the case when studying extreme events)and proved as one of the best methods for parameter estimations. This com-plementary method is based on the computation of the Probability WeightedMoments (see [21] and [24]). For the following empirical applications, we need touse the estimation of sample counterparts of L-moments for assessing the shapeparameter as underlined hereafter. The L-moments, which are linear functions


of the expectations of order statistics, were introduced by Sillitto [39]. One ofthe main advantages over conventional moments is that they suffer less from theeffects of sampling variability because they are linear functions of the ordereddata. They have been shown to provide more robust estimators of higher mo-ments than the traditional sample moments. They can also characterize a widerrange of distributions compared to the usual moments. Formally, the L-momentof order r is defined as:

λr =r

∑

k=1

p∗k−1,r−1 βk−1, (10)

with:

βk−1 = k−1 E(X[k:k]),

where p∗.,. are the shifted Legendre polynomials coefficients, and βk−1 are theProbability Weighted Moments of order k = [1, . . . , r].

They can be estimated without bias from the sample Probability WeightedMoments computed such as:

βk−1 =1

n

n∑

i=1

k−1∏

j=1

[

(i − j)

(n − j)

]

X[i:n]

, (11)

where X[i:n] is the i-th order statistic of a sample of n realizations (see Appendix).When the shape parameter is different from zero, the three first L-moments,

as a function of the characteristic parameters of a GEV Cumulative DensityFunction, are given by (see [23], [18], and the proof in the Appendix):

λ1 = h − αξ + α

ξ Γ (1 − ξ)

λ2 = αξ

(

2ξ − 1)

Γ (1 − ξ)

λ3 = αξ

[

1 − 3(2ξ) + 2(3ξ)]

Γ (1 − ξ)

(12)

where h ∈ IR is the location parameter, α ∈ IR+ the scale parameter, ξ ∈ IR∗

the shape parameter and Γ (a) =∫ +∞

0ta−1e−tdt is the Gamma function.

These estimates of the three first L-moments are sufficient to get an esti-mation of all three parameters characterizing a GEV distribution, using anyclassical numerical solving method. In order to check that the GEV density pro-vides a good approximation for the distribution of the maxima in our sample, weapply the Kolmogorov-Smirnov Goodness-of-Fit test to the resulting distribu-tions. This test never rejects the hypothesis that the GEV density fits the data,with the lowest P-value being .16 for the 5%-threshold maxima of the realizedvolatility.

The following Table 3 gives estimates of the shape parameter, based on max-ima of the daily log-volatilities, with the Maximum Likelihood and the L-momentmethods. We first notice that the shape parameter estimations obtained with thetwo methods are similar most of the time. Moreover, the shape parameter estima-tion of the Realized Volatility on a weekly basis (-.17) and the one of the EMA on


a quarterly basis (-.14) are the closest to zero. We also remark that, with longerwindows, estimations are smaller than with short windows. In other words, thelower the frequency for collecting the index, the more limited the presence offinancial krachs, marked by extreme volatilities. Finally, and more importantly,whatever the method, the frequency and the estimate, none of the estimatedshape parameters (and then tail indexes) is positive. This clearly proves thatnone of the underlying distributions can be considered as fat-tailed.

Table 3. Estimates of Shape Parameters of Generalized Extreme Value Distribu-tions of Daily Log-volatilities Period Maxima via Maximum Likelihood and ProbabilityWeighted Moment Methods

Method Frequency Realized Parkinson Rogers- Yang- Simple EMASatchell Zhang

Maximum Weekly -.17 -.23 -.24 -.22 -.31 -.24Likelihood Monthly -.24 -.21 -.35 -.24 -.24 -.22

Quarterly -.48 -.31 -.56 -.35 -.31 -.25

Weekly -.17 -.21 -.24 -.24 -.30 -.25L-Moments Monthly -.20 -.24 -.30 -.26 -.23 -.20

Quarterly -.30 -.27 -.48 -.37 -.18 -.14

Source: Euronext, 30’ sampled intraday CAC40 French stock index quotes from theperiod 01-01-1997/12-31-2006. Computations by the authors.

The previous shape parameters correspond to real market data series. In or-der to reinforce our previous conclusions and following Danielsson and de Vries[15], we renew the shape parameter estimation exercise using bootstrapped seriesof volatilities for assessing some inferences about the shape parameter estima-tions. More precisely, after computing the daily volatilities according to eachestimator, we draw with replacement volatility series from these estimates andcompute new weekly, monthly and quarterly maxima from these virtual samples.These new maxima being uncorrelated by construction, we can then fit a GEVdistribution these series and draw the empirical distribution of the resultingshape parameters. We present hereafter in Figure 3 (respectively, in Figure 4),the GEV shape parameters estimated using the Maximum Likelihood method(respectively, the L-Moment method) based on 500 bootstrapped series of weeklymaxima of daily volatilities. This frequency is chosen since the one for which theestimation of the shape parameter for the Realized Volatility (the benchmark)is the closest to zero4. Ranging from -.39 (Simple Volatility) to -.12 (RealizedVolatility), it appears that none of the shape parameters of volatility estimatorsexhibit a positive value on the rebuilt new artificial extreme volatility series.

4 Results taken into account at monthly and quarterly frequencies (not reported here)lead to the same kind of qualitative conclusions.


These last results clearly indicate that the negative values of the shape param-eters observed with the real series are neither exceptional nor due to the specificcharacteristics of the volatility time series; the sign remains the same whetherthese characteristics are or are not accounted for. To sum up, the resamplingresults allow us to assert the significance of the negativeness of the shape pa-rameter of the log-volatilities: there is no need to use fat-tailed distributions toaccount for the extremes of the log-volatilities. The log-normal approximationproves adequacy, at least for the asset (the CAC40 index), the frequency (30’quotes) and the sample (1997-2006) considered and by using our methodology(Maximum Likelihood and L-Moment methods), with the chosen density (GEVdistribution) and the horizon considered (daily, weekly and quarterly).

−0.4 −0.35 −0.3 −0.25 −0.20

0.5

1Simple

−0.35 −0.3 −0.25 −0.2 −0.15 −0.10

0.5

1Rogers−Satchell

−0.35 −0.3 −0.25 −0.2 −0.15 −0.10

0.5

1Yang−Zhang

−0.3 −0.25 −0.2 −0.15 −0.1 −0.050

0.5

1Realized

−0.3 −0.25 −0.2 −0.15 −0.1 −0.050

0.5

1Parkinson

−0.35 −0.3 −0.25 −0.2 −0.15 −0.10

0.5

1EMA

Fig. 3. Bootstrapped Values of the Maximum Likelihood GEV Shape Parameters ofthe Daily Volatility Weekly Maxima (Source: Euronext; 30’ resampled intraday CAC40French stock index quotes from the period 01-01-1997/12-31-2006. Computations by theauthors. The bootstrapped values of the shape parameters of the Realized Volatilitiesare plotted on the x-axis, the empirical cumulative distribution on the y-axis).

To confirm the previous observations, we present in Table 4 the shape param-eters estimated on bootstrapped series of volatilities and on series of volatilitiesobtained from bootstrapped series of returns. In order to reshuffle the series, weuse four methods of bootstrap: the simple one ([16]); the stationary one ([34]);the circular one ([33] and [35]); the accelerated one ([17] and [20]). We observethat the shape parameters obtained are similar whatever the bootstrap methodsand series, and are also equivalent to the estimates obtained from the original


−0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.20

0.5

1Realized

−0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.20

0.5

1Parkinson

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.40

0.5

1Rogers−Satchell

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.60

0.5

1Yang−Zhang

−0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.10

0.5

1Simple

−0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.10

0.5

1EMA

Fig. 4. Bootstrapped Values of the L-Moment method GEV Shape Parameters of theDaily Volatility Weekly Maxima (Source: Euronext; 30’ resampled intraday CAC40French stock index quotes from the period 01-01-1997/12-31-2006. Computations by theauthors. The bootstrapped values of the shape parameters of the Realized Volatilitiesare plotted on the x-axis, the empirical cumulative distribution on the y-axis).

series. This allows us to conclude that the shape parameters remain undoubtedlystrictly negative.

For finally validating furthermore these results, we introduce herein a finalsimple reality check: given the sample estimates of the parameters, it is nowpossible to compute the probability of observing the historical volatility peaksunder the various measures and hypotheses. The sample spans from January1997 to December 2006, with the highest volatilities occurring in most casesat the terrorist attack on the Twin Towers (September 2001). Using the shapeparameter from the GEV distribution, estimated from the Maximum LikelihoodMethod, we now compute the probability of these events and their associatedreturn-times. Table 5 presents these probabilities5.

Though this reality check has a very limited statistical significance, it allowsus to filter the results according to our subjective estimation of the likelihoodof a major event. Even if the shape parameter appears relatively stable overthe choice of the estimators, we obtain important different return-times, whichimplies large differences in the other parameters of the GEV distribution. Toillustrate this idea, we give in the following figure the density of each estimate,

5 For information, when the return distribution is estimated by the Maximum Like-lihood Method for a Normal distribution, the three largest probabilities of returnsare 3.88 10−6%, 5.67 10−6% and 6.10 10−5% which give respectively the followingreturn-times: 103,085 years, 70,538 years and 6,567 years.


Table 4. Comparison of Estimates of Shape Parameters of Generalized Extreme ValueDistributions of Daily Log-volatilities Weekly Maxima using Maximum Likelihood andBootstrap Methods

Series Methods Statistics Realized Parkinson Rogers- Yang- Simple EMASatchell Zhang

Shape Param. -.24 -.23 -.22 -.21 -.28 -.25Simple [5%;95%] [-.27;-.21] [-.27;-.19] [-.27;-.18] [-.26;-.18] [-.32;-.25] [-.28;-.22]

KS P-stat (.51) (.57) (.52) (.54) (.56) (.54)

Shape Param. -.15 -.21 -.24 -.19 -.31 -.27Return Stationary [5%;95%] [-.19;-.10] [-.26;-.17] [-.29;-.18] [-.24;-.14] [-.36;-.26] [-.32;-.22]

KS P-stat (.61) (.60) (.54) (.57) (.48) (.54)

Shape Param. -.19 -.21 -.22 -.20 -.29 -.25Circular [5%;95%] [-.23;-.15] [-.27;-.16] [-.27;-.16] [-.25;-.15] [-.34;-.25] [-.29;-.2]

KS P-stat (.61) (.56) (.52) (.58) (.51) (.55)Shape Param. -.22 -.20 -.22 -.22 -.31 -.24

Accelerated [5%;95%] [-.26;-.18] [-.23;-.16] [-.26;-.17] [-.25;-.17] [-.33;-.28] [-.27;-.20]KS P-stat (.47) (.45) (.50) (.49) (.53) (.46)

Shape Param. -.22 -.19 -.25 -.24 -.32 -.24Volatil. Stationary [5%;95%] [-.26;-.17] [-.24;-.14] [-.29;-.21] [-.29;-.20] [-.36;-.27] [-.29;-.19]

KS P-stat (.58) (.57) (.53) (.52) (.54) (.52)

Shape Param. -.22 -.21 -.22 -.22 -.30 -.24Circular [5%;95%] [-.31;-.11] [-.27;-.15] [-.32;-.11] [-.30;-.12] [-.35;-.25] [-.29;-.19]

KS P-stat (.35) (.51) (.44) (.41) (.51) (.46)

Source: Euronext; 30’ resampled intraday CAC40 French stock index quotes from theperiod 01-01-1997/12-31-2006. The shape parameters for the various volatility mea-sures are estimated by the Method of Maximum Likelihehood of a GEV density with aweekly frequency (the block maxima length) on 10,000 series obtained with bootstrapmethods (simple: [16]; stationary: [34]; circular: [33] and [35]; accelerated: [17] and[20]) on series of returns or on series of volatilities. The 90% confidence intervalsof shape parameters are reported in brackets, whilst P-statistics of Goodness-of-FitKolmogorov-Smirnov tests (denoted KS P-stat.) are between parentheses. Computa-tions by the authors.

obtained for a GEV distribution, and we compare them to the empirical densityfunctions. It appears the two curves are similar, which is a good sign regardingthe pertinence of probabilities and return-times we obtained.

Further within the tail, the variability increases and estimates can still differby a factor larger than two, so the choice of the measure is not insignificant.Overall, the estimates seem to give return-times which are more in line with thesize of the sample (about ten years), except for the Yang-Zhang one, which givesreturn-times slightly larger as the sample length. We also notice the return-timesdecrease quickly between the first and the third extreme values. From these es-timations of the extreme values, computed with several methods and for variousvolatility estimators, we can prudently infer with reasonable confidence that it isunlikely that a fat-tailed distribution is needed to fit the high volatilities. Indeed,we have obtained in most cases negative shape parameters, which correspondsto a reversed Weibull-kind of distribution. However, range-based and intra-day


Table 5. Probablity of Largest Negative Returns and Largest Daily Volatilities, andReturn-Times using Maximum Likelihood GEV Estimates

Weekly Monthly QuarterlyEstimator Crisis Values Prob. Return Prob. Return Prob. Return

Times Times Times

11/09/2001 -7.68% .05% 37.73* .54% 14.81* .78% 33.34*Returns 15/04/2000 -7.55% 1.22% 1.64 1.33% 6.02 1.91% 13.61*

14/03/2003 -7.00% 3.88% 0.51 4.20% 1.90 5.94% 4.38

12/09/2001 .91 .63% 3.15 2.01% 3.99 5.92% 4.40Realized 24/07/2002 .71 .68% 2.96 2.12% 3.77 6.27% 4.14

03/04/2000 .70 .88% 2.31 2.69% 2.98 7.99% 3.2511/09/2001 .91 .64% 3.11 2.21% 3.42 4.01% 6.85

Parkinson 15/03/2003 .71 .93% 2.03 2.32% 3.24 6.59% 3.9225/07/2002 .70 .97% 1.96 2.81% 2.78 7.52% 3.4105/04/2000 .70 .68% 2.92 2.39% 3.35 4.16% 6.26

Rogers-Satchell 24/07/2002 .64 1.06% 1.88 3.47% 2.31 7.22% 3.6021/09/2001 .63 1.10% 1.82 3.57% 2.24 7.50% 3.4717/04/2000 1.20 .17% 12.08* .57% 14.07* 1.87% 13.93*

Yang-Zhang 05/01/2000 1.00 .51% 3.93 1.67% 4.79 5.65% 4.6021/09/2001 .90 .89% 2.26 2.79% 2.86 9.09% 2.8611/09/2001 1.17 .19% 10.39* 1.15% 6.94 4.61% 5.64

Simple 14/03/2003 1.14 .23% 8.55 1.30% 6.14 5.05% 5.1529/07/2002 1.11 .31% 6.38 1.57% 5.08 5.83% 4.4614/03/2003 .98 .15% 13.01* .79% 10.13* 3.59% 7.25

EMA 15/10/2002 .85 .50% 4.01 1.84% 4.35 6.58% 3.9511/09/2001 .84 .54% 3.69 1.96% 4.08 6.89% 3.78

Source: Euronext, 30’ sampled intraday CAC40 French stock index quotes on the period01-01-1997/12-31-2006. Return-times are expressed in years and they are marked withan asterisk * when they are larger than the size of the sample, i.e. 10 years. Computa-tions by the authors.

volatilities are not incompatible with the log-normal hypothesis and thus thestandard approximation is not significantly flawed.

4 Conclusion

The Realized Volatility, despite its known shortcomings, remains a benchmarkto which measures of risk should be compared. We show here that, among thelow-frequency volatility measures, Parkinson’s volatility was the closest to thehigh-frequency benchmark measure. This estimator should thus be the one usedwhen trying to get long-horizon historical estimates, or to complement series ofRealized Volatilities. Generally speaking, estimations of the whole distributionof the empirical volatilities cannot help to easily distinguish between the can-didate functional forms. Given the rationale for estimating these distributions -retrieving possible risk - and the main differences between them - in the tails -it seems natural to try instead to use the Extreme Value Theory and concen-trate on estimating the asymptotic distribution for the extreme measures of risk.The estimations for the Generalized Extreme Value indicate that the fat-taileddistribution is not needed to fit the sample volatilities. A log-normal process,as in the traditional stochastic volatility model, seems sufficient to reproduce


−1.2 −1.1 −1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3

0.8

0.85

0.9

0.95

1Realized

EstimatedActual

−1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2

0.8

0.85

0.9

0.95

1Parkinson

−1.4 −1.2 −1 −0.8 −0.6 −0.4

0.8

0.85

0.9

0.95

1Rogers−Satchell

−1 −0.8 −0.6 −0.4 −0.2 0

0.8

0.85

0.9

0.95

1Yang−Zhang

−0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1

0.8

0.9

1Simple

−1 −0.8 −0.6 −0.4 −0.2

0.8

0.9

1EMA

Fig. 5. Empirical and Estimated Density Functions of the Volatility Estimates (Source:Euronext; 30’ resampled intraday CAC40 French stock index quotes from the period01-01-1997/12-31-2006. Are represented in this Figure, on the y-axis, the empiricalcumulative functions of the log-volatilities (dots), altogether with their GEV best fits(thin lines). Annualized Daily Volatilities are represented on the x-axis. Computationsby the authors.

the extreme empirical volatilities observed in the ten year ultra-high frequencystudied sample.

However, we can think about confirming these results in line with others(see [5]) with a complementary analysis including additional measures (e.g., [26];[19]), other samples (containing this time individual stocks), more recent observa-tions (highlighting recent market turmoils and credit linked events in 2007, 2008and 2009), different methodologies (Parametric Block Maxima and ParametricPeaks-over-the-Threshold), complementary estimation methods (other types ofMoment Estimations versus the Maximum Likelihood method), using variousdistributions (Generalized Pareto Distribution versus GEV distribution), otherrealistic sampling schemes (from one-minute to one-hour quote frequency) andother horizons (30” to a quarter). One may finally think about a Reality CheckTest based on the various estimators, assets and methods of estimation (see [41]),for reinforcing our preliminary results on the best specification for the probabilitymodel for volatility. In particular, the final conclusion for a thin tail distributionfor the volatility is deeply related to the choice of the estimation period. Therecent turmoil on the financial markets, with its scope and it persistence, shouldhave an important impact on the distribution of the volatility and then shouldquestions its extreme behavior.


A practical application of these results will be to plug the appropriate esti-mates and distributions of volatilities in the Index of Market Shocks (IMS, see[28] and [29]) in order to get a clear ranking of the historical crises and an accu-rate estimation of the return-times of extreme scenarii, to ultimately preciselydefine the high watermarks of market risks.

Appendix

Let Xj, with j = [1, . . . , n], be a sequence of n independent and identicallydistributed non-degenerated random variables with a cumulative continuous dis-tribution function F (x) and with a quantile function Q(u) = F−1(u).

We recall that the r-th L-moment is defined, for every r = [1, . . . , n], by:

λr =1

r

r−1∑

j=0

(−1)j(

r−1j

)

E(X[r−j:r]). (13)

where X[i:r] is the i-th order statistic of a sample of r random variables.

Now since:

E(X[i:r]) =r!

(i − 1)! (r − i)!

∫ 1

0

Q(u)ui−1(1 − u)r−idu, (14)

we obtain:

λr =1

r

∫ 1

0

Q(u)

r−1∑

j=0

(−1)j(

r−1j

) r!

(r − j − 1)!j!ur−j−1(1 − u)j

du. (15)

It is often useful to express the r-th L-moment as a linear function of the Prob-ability Weighted Moments, i.e.:

λr =

r∑

k=1

p∗k−1,r−1 βk−1, (16)

where p∗.,. are the Legendre polynomials coefficients, and βk = k−1E(X[k:k]) arethe Probability Weighted Moments of order k = [1, . . . , r].

They can be estimated without bias by the following estimations:

βk−1 =1

n

n∑

i=1

k−1∏

j=1

[

(i − j)

(n − j)

]

X[i:n]

, (17)

where X[i:n] is the i-th order statistic of a sample of n realizations.


The three first sample L-moments can then be written in the following man-ner:

λ1 =1

n

n∑

i=1

X[i:n]

λ2 =1

n(n − 1)

n∑

i=1

(2i − 1 − n)X[i:n]

λ3 =1

n(n − 1)(n − 2)

n∑

i=1

[6(i − 1)(i − 2) − 6(i − 1)(n − 2)

+(n − 1)(n − 2)]X[i:n].

(18)

Based on these previous definitions, we can now give a specific result on theL-moments for a GEV distribution.

Proposition 1. (see [23] and [18]). The three first L-moments, as a functionof the three characteristic parameters of a GEV Cumulative Density Function,are given by:

λ1 = h − αξ + α

ξ Γ (1 − ξ)

λ2 = αξ

(

2ξ − 1)

Γ (1 − ξ)

λ3 = αξ

[

1 − 3(2ξ) + 2(3ξ)]

Γ (1 − ξ)

(19)

where λr is the r-th L-moment, h ∈ IR the location parameter, α ∈ IR+ the

scale parameter, ξ ∈ IR∗ the shape parameter and Γ (a) =∫ +∞

0 ta−1e−tdt is theGamma function.

Proof. We get the following intermediate result, for every k ∈ N :

∫ 1

0

uk[− ln(u)]−ξdu =

∫ +∞

0

x−ξe−(k+1)xdx

=1

(1 + k)1−ξ

∫ +∞

0

y−ξe−ydy (20)

=1

(1 + k)1−ξΓ (1 − ξ),

thanks to simple changes of variable.Given the quantile function of the GEV distribution such as:

Q(u) = h −α

ξ+

α

ξ[− ln(u)]−ξ, (21)

we can then compute the first L-moment:

λ1 =

∫ 1

0

Q(u)du

=

∫ 1

0

(h −α

ξ)du +

α

ξ

∫ 1

0

[− ln(u)]−ξdu. (22)


Using result (20), we obtain the final expression for the first L-moment so that:

λ1 = h −α

ξ+

α

ξΓ (1 − ξ). (23)

Similarly, the second L-moment is given by:

λ2 =

∫ 1

0

h −α

ξ+

α

ξ[− ln(u)]−ξ

[2u − 1]du

=

∫ 1

0

2uα

ξ[− ln(u)]−ξdu −

∫ 1

0

α

ξ[− ln(u)]−ξdu (24)

and applying result (20) once again is leading to the following expression of thesecond L-moment:

λ2 =α

ξ

(

2ξ − 1)

Γ (1 − ξ). (25)

Finally, the straightforward expression of the third L-moment is:

λ3 =

∫ 1

0

h −α

ξ+

α

ξ[− ln(u)]−ξ

[1 − 6u + 6u2]du (26)

which is leading, still using the intermediate result (20), to the expression of thethird L-moment:

λ3 =α

ξ

[

1 − 3(2ξ) + 2(3ξ)]

Γ (1 − ξ). (27)

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