Chapter 9

Performance Measurement on Fixed-Income Portfolios

FIXED-INCOME

Outline

• Return Measures– Arithmetic Rate of Return– Geometric Rate of Return

• Risk-Adjusted Performance Evaluation– Adjusting for Risk– Relative Performance Evaluation– Performance Evaluation in a CAPM Framework– Factor Models– Style Analysis

Return MeasuresMeasuring Investment Returns

• For one-period horizon, return is comprised of cash distributions and capital gain/loss

• For multi-period horizon, cash is added and withdrawn from the fund and issue is whether to recognize timing of cash flows or not

• Two choices:• Dollar (or Value) -weighted Average Returns• Time-weighted Average Returns

Return Measures Value-Weighted Return

• Equivalent to finding the IRR• Consider the different cash-flows:

– Just compute r

• If we are using a different time unit, we have to annualize the return

n

0ii

i

r1F0

)(

Return Measures Time-weighted Return

• Compute the returns between withdrawals– Assume that inflows and outflows took place at the end of a given period– Say a week or a month

• Then you compound the returns (geometric average)• If the horizon considered is less than a year, annualize• Example: compute returns for the following investment

– Initial investment: 50 (flow: -50)– After six months, it falls to 25– Deposit 25 (flow: -25)– After 6 months it has gone up to 100 (flow: +100)

• Use both methods– Value-weighted– Time-weighted

Return Measures Example

• Value weighted:

– With a result:

– This, annualized, yields:

2SS r1

100r1

25500)(

%.618rS

%).( 4011861 2

• Compounded:

• Total return:

Return Measures Example (continued)

• Time weighted:

%

%.

100150

50100r

505050

5025r

2

1

1r1r1 21 ))((

%011

Return Measures Comparison

• Dollar-weighted greater in this case because +100% applies to a larger sum than -50%;

• Dollar-weighted is better measure – when manager has control over inflows/outflows of money – manager should be rewarded when makes good decisions

• In other situations (e.g., for a pension fund manager), time-weighted is used

Adjusting for Risk Performance Measurement is a 2-Dimension

Problem

• Portfolio performance evaluation seems a trivial problem at first sight– Best manager is the one with the highest return!– What do we do about risk?

• The main problem of mutual fund performance evaluation is the problem of risk-adjustment– Before 1960s, risk adjustment took form of asset-type

classifications, imprecise and not very analytical– Intuitive definition of risk: how much an investor might lose– Need more specific analytical definitions of risk

Adjusting for RiskHow to Measure Risk ?

Monthly Stock Returns, 1926-1998

-60%

-40%

-20%

0%

20%

40%

60%

80%

1Mon

thly

Ret

urns

(%)

U.S. Small Stk TR S&P 500 TR

• Natural measure of risk is volatility• Volatility is standard deviation of returns• First define average return as

• Volatility is given by (the square-root of) the average squared deviation with respect to mean return

T

ttT R

TR

1,0

1

T

tTtT RR

T1

2

,0,01

Adjusting for RiskVolatility

• Sharpe computes the excess return per unit of standard deviation

• Sharpe index penalizes managers for taking risk

• This index is heavily used to measure performance

)(

p

fpp

rRES

Adjusting for RiskSharpe Ratio

• Sharpe ratio based on volatility as a measure of risk• Is it a good measure of risk?

– Measures average risk in a symmetric way– How about extreme risk?

• There is large empirical evidence that returns are not normally distributed (especially hedge fund returns)

– Exemple : the 2 distribution functions below have the same Sharpe ratio– Would you be indifferent?

Adjusting for RiskProblems with Sharpe Ratio

0

0.1

0.2

0.3

0.4

0.5

0.6

-5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Mixture of 2Gaussians

NormalizedGaussian

Normalized Gaussian

Sp=0 (mean=0, volatility=1)

Skewness = 0

Kurtosis = 3

Mixture of 2 Gaussians

Sp=0 (mean=0, volatility=1)

Skewness = -0.75 < 0

Kurtosis = 6.06 (fat tail)

• Investors dislike negative returns: they need a measure of downside risk, e.g., semi-variance

• Can be generalized in terms of any Minimum Acceptable Return (MAR)

• Sortino introduced an analogue to the Sharpe ratio that penalizes managers for taking on negative risks

Adjusting for RiskSemi-Variance for Investors who care about Skewness

Tt RRt

Tt RRT

,0 s.t.

2

,01

1)(

s.t.

2

MARRtt

pp

t

MARRT

MARRESortino

• You need a measure of extreme risks• J.P. Morgan has introduced in the 90s a measure of

extreme risk, the Value-at-Risk

Adjusting for RiskValue-at-Risk for Investors who care about Kurtosis

0

10

20

30

40

50

60

70

80

-18% -14% -10% -6% -2% 2% 6% 10% 14% 18% More

Returns

Frequency

1% of the cases 99% of the cases

VaR = -9.6%

• VaR can be used to define an analogue to the Sharpe ratio that penalizes managers for taking on extreme risks, as opposed to average risk

• Ratio Return/VaR

• More on VaR– Several methods : Parametric, Monte-Carlo and non parametric VaR– Beyond VaR (see next slide)

Adjusting for RiskRatio Renta/VaR

)(

Return/VaR Ratiop

fp

VaRrRE

• Useful to look beyond VaR : very extreme losses• BVaR:

• Distribution beyond VaR

Adjusting for RiskBeyond VaR

-10% -5% 0% 5% 10%Level of returns

Prob

abili

ty d

ensi

ty fu

nctio

n

Gaussian distribution Stable Paretian distribution

Probability p = 99%

VaR = 2.33%

Distributions of returns

Distributions of the returnsexceeding the VaR

0.00

0.50

2.50

2.00

1.50

1.00

dxxf

dxxfx = VaR)R | E(R = BVaR

RVaR

-

RVaR

-

• Whatever the measure of risk used, previous ratios were assessing the performance in absolute terms

• Need for relative performance evaluation– If bonds perform better than stocks, Sharpe (or Sortino) ratio of poor bond

manager likely to be higher than Sharpe ratio of good stock manager – Absolute evaluation measures performance of asset class, not necessarily

of manager!

• Two ways of doing relative performance evaluation1. Peer groups (statistical approach)2. Factor models (economic approach)

• Manager performance is measured with respect to 1. performance of comparable managers2. the risk factors manager is exposed to

Relative Performance Evaluation Absolute versus Relative Performance Evaluation

• Use statistical (cluster) to form objective peer groups– Maximize between-group-distances and minimize intra-group distances– Normalization of returns (std dev. or mean abs. dev.)– Use some notion of distance, e.g., Minkowski distance

• q=2 => standard Euclidean distance• q=1 => Manhattan distance (more robust for peer grouping

(Schneeweis and Kazemi (2001))

Relative Performance Evaluation Peer Groups

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t-1

qqT

tktjt RRkjd

1

1

),(

• Peer grouping is not explaining much– Explaining returns by returns!– Black-Box: good statistics but no economics

• Another approach: comparable funds are funds with similar risk exposure

• Relevant risk factors– Market risk is obvious choice– Other, multiple, sources of risk: volatility, credit, liquidity, etc.

• Modern portfolio theory provide guidance (models) that allow us to understand better the trade-off between risk and return

Relative Performance Evaluation Factor Models

Relative Performance Evaluation Normal and Abnormal Returns

• These models allow us to decompose mutual fund (excess) returns into

Total (excess) return = Normal return + Abnormal return (skill) – Normal return is generated as a fair reward for the risk(s) taken by mutual fund managers– Abnormal return (a.k.a. alpha) is generated through managers’ unique ability to “beat the

market” in a risk-adjusted sense (market timing and/or security selection skills)

• Abnormal return is a good way to rank managers– Total return is readily observable– Challenge is to measure normal return (not observable)

• We need a model to measure normal return– CAPM

• CAPM is a reference model for measuring normal return– Multi-factor models

• CAPM was extended under the form of multi-factor models (see below)

Performance Evaluation in a CAPM Framework CAPM

• Markowitz’s optimal portfolio selection (1952)

• Sharpe’s Capital Asset Pricing Model (1964)

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ijji2

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rRErRE

Sharpe ratio = slope

Performance Evaluation in a CAPM Framework Example

• Two professional money managers are evaluated– One averaged 19% last year and the other managed only 16%– First manager’s beta was 1.5 and second manager had a beta of 1.0

• Question: which manager performed better?– Market risk premium is 8%– T-bills yield 6%

• Results– Manager 1 should have got a performance of 6%+(1.5)x8%=18% < 19% --

good !!– Manager 2 should have got a performance of 6%+(1)x8%=14% <16% --

even better !!

Performance Evaluation in a CAPM Framework Implementation

• Alphas and betas are measured statistically using historical returns on the portfolio and a market portfolio proxy, e.g. S&P 500

• Simple regression model, known as Market Model titftMiitfti rRrR ,,,,,

• Betas measure the sensitivity of changes in portfolio return to changes in market portfolio proxy’s return

• Alpha is the regression intercept (=0 if CAPM holds)

Performance Evaluation in a CAPM Framework Characteristic Line

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

0.60

-0.2 -0.1 0 0.1 0.2 0.3 0.4

Market Excess Return

Fund

's E

xces

s Re

turn

Beta=Slope of Regression Line

=Regression Intercept

Regression Line

• One is allowed to consider alpha = abnormal return– If alpha term is statistically significant (estimation risk)– If CAPM is the true model (model risk)

Performance Evaluation in a CAPM Framework Is CAPM a good Model? The Theoretical Answer

• Very restrictive assumptions– Equilibrium model– Investors are price takers– One-period investment horizon (“myopic”)– Fixed quantities of assets and all marketable– No taxes, transactions costs, regulations, etc.– Risk-averse utility-maximizing investors– All investors analyze securities in same way with same probabilistic

forecasts for each– Returns are normally distributed

• Roll’s critique (1977)– True market portfolio is not observable (S&P is only a poor proxy!)– All tests of CAPM are joint tests of CAPM and proxy for the market portfolio

Multi-Factor Models Is CAPM a Good Model? The Empirical Answer

• CAPM claims that (1) beta and (2) only beta explains cross-section of expected returns

• Empirical question: is the market portfolio the only rewarded factor affecting asset returns?

• Fama and French (1992) – Firm size and book-to-market equity capture the cross-sectional variation in

average stock returns– CAPM beta does not explain cross-sectional variation in average stock

returns

E(r) = 2.07 – 0.17 – 0.12(Size Factor) + 0.33(B/M Factor)

(6.55) (-0.62) (-2.52) (4.80)

Multi-Factor Models Multi-Factor Models

• Modern portfolio theory and practice is based upon multi-factor models

• The (excess) return on asset or fund i is Rit = i + i1F1t + ... + iKFKt + eit

– Fkt is factor k at date t (k = 1,…,K)

– eit is the asset specific return– ik measures the sensitivity of Ri to factor k, (k = 1,…,K)

• Generalization with respect to CAPM is two-fold– No unique factor– Market portfolio may not be a factor

• How many is not too many?– Easy to increase R-square (=only roughly 30% with CAPM!) by including more factors– To ensure fair comparaison, need to penalize (AIC, Schwartz, …)

• APT Line (hyper plan) is the natural generalization of the security market line and allows us to measure normal return

-0.1-0.05

00.05

0.10.15

0.20.25

0.3

-0.1 0 0.1 0.2-0.1

0.05

0.2

1F2F

E Ri( )

Kk

kikKk

fFikfi rRErREk

,...,1,...,1

)()(

Multi-Factor Models Factor Models and Portfolio Performance Evaluation

Multi-Factor Models Three Types of Factor Models

• Implicit factor models – Factors: principal components, i.e., uncorrelated linear combinations of asset returns– Industry application: Aptimum, Quantal– Advantage: no (low) model risk– Drawbacks: estimation risk, not easily interpretable (problem with Classement Le Monde)…

artificial artefact…

• Explicit factor models – macro factors– Factors (Chen, Roll, Ross (1986)): inflation rate, growth in industrial production, spread long-

short treasuries, etc.– Don’t work very well in practice

• Explicit factor models – micro “factors”– Factors (actually attributes): size, country, industry, etc.– Industry application: BARRA– Advantages: good interpretation, great marketing– Drawbacks: model risk; BARRA has failed during the tech stock bubble

Multi-Factor Models Sharpe’s Answer (26 years after the CAPM)

• I like index as a factor, you like multi-factor => how about multi-indices?

– Many indices collectively represent the market– This is a pragmatic way to address Roll’s critique: get closer to true market

portfolio with many asset classes

• Rather than directly looking for factors, let’s look for various asset classes, themselves exposed to unknown risk factors

– Closer to investor‘s interpretation– Borrow from explicit and implicit– Generalize Elton and Gruber by adding constraints

• Style analysis– Sharpe (1992): equity styles are as important as asset classes– Examples: growth/value, small cap/large cap, etc.

• Model: Rit = i1F1t + i2F2t + ... ikFkt + eit – Rit = (net of fees) excess return on a given portfolio or fund

– Fkt = excess return on index j for the period t– wik = style weight (add up to one)– eit = error term

• Style analysis of a given fund consists in explaining past returns of that fund with an (hypothetical) portfolio (style weights add up to 1) of indexes that mimics the returns of the fund

Style Analysis Return-Based Style Analysis

• Divide the fund return into two parts• Style: i1F1t + i2F2t + ... ikFkt

– Part attributable to asset classes movements

• Skill: eit – Part unique to the manager which emanates from 2 sources

• Manager’s active bets: active picking within classes and/or class timing• Manager’s exposure to other asset classes not included in the analysis

– If the deviation on returns is positive, the manager was successful– Otherwise, it would be better to follow the passive strategy

Style Analysis Return-Based Style Analysis