Composite measures (Treynor’s, Sharpe’s, Jensen’s,Recall that in the Single index model R i =...

Road map and readings

Measures of returns: more in detail

Measures of risk: a brief summary

Composite measures (Treynor’s, Sharpe’s, Jensen’s,Information ratio)

Benchmark selection and Style analysis

Readings

BKM, Sections 24.1 and 24.5Sharpe, W. (1992), “Asset Allocation: Management Style andPerformance Measurement”, Journal of Portfolio Management,7-19. [Optional]

Suresh Mutuswami EC7092 Investment Management

The problem

How do we measure the performance of a portfolio manager?

by comparing with other portfolio managers?against some agreed benchmark?

How do we compute the average return achieved by aportfolio manager over a period of time?

complicated by intermediate inflows and outflows of cashoutside the portfolio manager’s control

Which risk measure to employ?

How do we adjust portfolio return for risk? (total risk ornon-diversifiable risk?)


Single period example

Consider a stock paying a dividend of $2 annually thatcurrently sells at Pt = $50. Assume that the stock is sold atthe end of the next period at Pt+1 = $53.

The rate of return is

HPRt,t+1 =53− 50 + 2

50= 0.10

Alternatively, we calculate the rate of return which equatesthe present value of all cash in-flows with the (initial) cashout-flow

50 =53 + 2

1 + IRR=⇒ IRR = 0.10


Intermediate cash flows

Calculating returns is more complicated if cash is added to, orwithdrawn from, the portfolio before the end of the holdingperiod.

Example

at the beginning of Year 1, an investor purchases a shares in astock at $50.at the end of Year 1, the stock is trading at $53, and theinvestor buys an additional share of the stock.at the end of Year 2, the stock price is $54, and the investorsells his complete holding.the stock pays a dividend of $2/share at the end of each of thetwo years.

What is the investors annual return?


Value-weighted returns (IRR)

The rate which solve the following equation is called theInternal rate of return (IRR), or dollar-weighted return:

50 +53

1 + IRR=

2

1 + IRR+

108 + 4

(1 + IRR)2

One can compute, in this case, that IRR = 0.071 or 7.1%.


Time-weighted returns

Alternatively, we can use a time-weighted measure which doesnot take the size of investment in each period.

r1 =53− 50 + 2

50= 0.10, r2 =

54− 53 + 2

50= 0.057

Time-weighted return is the average of the returns (r1, r2)over the two years. What kind of average?


Arithmetic versus geometric average

The arithmetic average of N returns is defined as

rA =

∑Ni=1 riN

It is an unbiased estimate of expected future returns onportfolio (but only if the expected returns do not change overtime).

The geometric average of N returns:

rG =[ΠNi=1ri

]1/NIt is the fixed return the portfolio would need to have earnedeach year in order to match actual performance.

The geometric mean is a good measure of past performance.

One can show that rG ≤ rA. Hence, one can regard thegeometric average as a downward biased estimate of futureperformance.


Arithmetic versus geometric average (continued)

In many cases, we can use the approximation rG = rA − 1/2σ2

where σ is the standard deviation of returns.

The rule holds exactly when returns are normally distributed.

Historical comparisons, US data, 1926-2002


Arithmetic versus geometric average (continued)

Which one should we use then?

for past performance, use the geometric average.

for future performance, we should amend the value of thearithmetic average (since it is biased when future returns arechanging over time).

rmod = rA ×[

1− H

T

]+ rG ×

H

T

Here, H is the forecasting horizon (i.e. H periods ahead) andT is the sample period used in the estimation.


And what about risk?

Average returns are not enough.

Returns must be adjusted for risk in order to compare themmeaningfully.

Where should we look at? Early practice stated that weshould compare investment with similar risk characteristics(such as high-yield bonds, growth stock equities etc.) orcomparison universe

When investments are grouped, then evaluate themindividually (for example ’the best out of 100 managers’, the’10th performing manager’ etc.)


And what about risk? (continued)


And what about risk? (continued)

Problems:

within the same universe we are not really looking at theeffective investment strategies (portfolio managers may focuson selected sub-groups). The assumption of same risk amongsecurities may be too restrictive.It does not tell us if portfolio managers have accomplishedindividual objectives and satisfied investment constraints.

Solution: composite portfolio performance measures.


Treynor performance measure

Treynor (1965) developed the first measure of portfolioperformance that included risk.

The measure applies to all investors regardless of their riskpreferences.

it is based on the definition of the SML (see notes on CAPM).it applies to fully diversified portfolios.

Treynor showed that each rational investor would like toachieve the largest the slope of SML or in turn maximize:

Tp =rp − rfβp

where rp and rf are the average return on portfolio P and therisk-free rate during a specified period of time and βp is thesystematic risk associated with portfolio P.


Treynor performance measure (continued)

ExamplePortfolio Returns Beta Treynor performance measure

M 0.14 1 TM = 0.14−0.081 = 0.06

W 0.12 0.9 TW = 0.12−0.080.9 = 0.044

X 0.16 1.05 TX = 0.16−0.081.5 = 0.076

Y 0.18 1.20 TY = 0.18−0.081.20 = 0.083

Assume that the average risk-free rate during the period was8%



In essence the Treynor measure is a risk premium per unit ofsystematic risk (beta).

Question Why is beta a measure of systematic risk?

Recall that in the Single index model

Ri = αi + βiRM + ei

σ2i = β2i σ2M + σ2(e)

The first component is the systemic risk and the second is thespecific risk.

Since the measure applies to fully diversified portfolios thesecond component is close to zero, therefore the relevantmeasure of risk is only a function of beta.


Sharpe performance measure

Sharpe (1966) developed another measure of portfolioperformance to evaluate mutual funds.

Sharpe proposed (jointly with his work on the CAPM):

Sp =rp − rfσp

where rp and rf are the average return on portfolio p and therisk-free rate during a specified period of time and σP is ameasure of total risk for portfolio p during the same period.

The Sharpe measure is an risk premium return earned per unitof total risk.


Sharpe performance measure (continued)


Treynor versus Sharpe

The difference is only in the measure of risk adopted.

When is it appropriate to use them?

use Treynor when you compare fully diversified portfolios.use Sharpe when you compare poorly diversified (concetratedor focused) portfolios.

Treynor = Sharpe (will provide you with the same ranking)when evaluating fully diversified portfolios (the only source ofrisk is the systematic risk).

Treynor is better than Sharpe when evaluating poorlydiversified portfolios (differences in rank come directly fromdifferences in diversification)


Jensen performance measure

Jensen’s (1968) measure (or Jensen’s alpha) shares the sametheoretical foundations with the previous measures (it is basedon the CAPM).

The measure is calculated as the average return over andabove the one predicted by the CAPM (or single index model):

αp = rp − [rf + βp(rm − rf )]

Superior portfolio managers who forecast market turns or pickconsistently undervalued securities can earn higher risk premiathan the market risk premium (active portfolio management).Hence,

α > 0 denotes an (average) out-performance of thebenchmark.α < 0 denotes an (average) under-performance of thebenchmark.


Jensen performance measure (continued)


Jensen performance measure (continued)

In the investment industry Jensen’s alpha are calculated onperiod-by-period basis. Therefore the formula becomes

ap,t = rp,t − [rf ,t + βp,t(rm,t − rf ,t)]

This formula does not provide you with the averageperformance but the current performance at the end of eachperiod (i.e. month, quarter, year etc.)

If we do not believe in the CAPM, then we can use APT orsomething else:

αp,t = rp,t−[rf ,t + b1p,t(rm,t − rf ,t) + b2p,tSMBt + b3p,tHMLt ]

¯αp,t = rp,t −

rf ,t +k∑

j=1

Fj ,t


Using Treynor and Jensen measures


Using Treynor and Jensen measures (continued)

Note that for the example, aP = 0.02 and aQ = 0.03.However, TP = 0.11/0.9 = 0.120 andTQ = 0.19/1.6 = 0.118. Hence we have aP < aQ butTP > TQ .

When comparing two portfolio managers with fully diversifiedportfolios Jensen’s alpha could provide us with misleadingresults. Treynor measures are more appropriate.


Jensen measure: caveats

Always bear in mind that Jensen’s alphas are estimates(therefore there is some uncertainty associated with theirvalues).

Example: Assume that we have to evaluate a portfoliomanagers and we have got this monthly data

α = 0.2%, β = 1.2, s(e) = 2%

When we estimate alphas we want to know whether or notthey are significant (conditional on the sample ofobservations). We have to compute t-statistics.

Statistical rules tell us

s(a) =s(e)

N, t(a) =

α

s(a)=α√N

s(e)

Hence,

t(a) =0.2√N

2


Jensen measure: caveats

In order to have our t-statistic significant at 5%, its valuemust exceed 1.96. Hence

0.2√N

2> 1.96 =⇒ N > 384

This implies that in order to prove his/her true skills, theportfolio manager will have to produce this performance (onaverage) over 384 months (or 32 years)!

Statistical inference makes performance evaluation extremelydifficult. It is complicated to assess the quality of ex-antedecisions using ex-post data.


Information ratio

Strictly related to the previous measures of performance is theinformation ratio (or appraisal ratio).

It is an average return in excess of any predefined benchmark(CAPM, FF or Multifactor model) divided by the portfoliosspecific risk (or tracking error):

IRp =ap

s(ep)

It is a ratio between the benefit of a non-perfectdiversification (α) and its cost (σ(e)).

Values of IR > 0.5 denote a good performance, values ofIR > 1.0 denote an exceptional performance. However, howmany portfolio managers achieve such values?


Information ratio: example

Manager A has returns of 13% and a tracking error of 8%while Manager B has returns of 8% and tracking error of 4.5%

The index has returns of -1.5%

Manager A’s IR = [13− (−1.5)]/8 = 1.81 while Manager B’sIR = [8− (−1.5)]/4.5 = 2.11

Manager B had lower returns but a better IR. A high ratiomeans a manager can achieve higher returns more efficientlythan one with a low ratio by taking on additional risk.


Information ratio (continued)


Measure for measure

Jensen’s alpha can be used to compare the performance of aportfolio with that of a risk-adjusted benchmark:

it accounts for systematic risk.

Treynor or Sharpe measures can be used to compare theperformance of two or more portfolios with one another.

Treynor measure gives the excess return per unit of systematicrisk (fully diversified portfolios).Sharpe measure gives the excess return per unit of total risk(poorly diversified portfolios).


. . . for good measure


Benchmark problem?

Problem: what is market portfolio (Roll, 1987)? How can wemeasure it? How do we select the proper benchmark?

In general market portfolios are proxied by market indices (e.g.FTSE 100, S & P 500 etc.)However, different market indices or different benchmarks mayyield different results.

One alternative: Style analysis (Sharpe, 1992)

it can be used for obtaining an estimate of the portfoliosinvestment style and exposures to certain asset classes.


Style analysis


Style analysis (continued)

US New Templeton VanguardEquities Perspective World

a -0.26 0.03 0.00 0.02

USA 89% 51% 74% 4%

Europe 1% 35% 17% 67%

Japan 0% 5% 2% 24%

T-Bill 10% 9% 7% 5%

R2 0.83 0.87 0.85 0.83


Style analysis (continued)


Composite measures (Treynor’s, Sharpe’s, Jensen’s,Recall that in the Single index model R i =...

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