Portfolio Performance Measurement

Eric Zivot

December 8, 2009

1

Investment Styles

1.1

Passive Management• Believe that markets are in equilibrium— Assets are correctly priced• Hold securities for relatively long periods with small infrequent changes

• Hold surrogates for market portfolio known as index funds— Low cost diversified portfolios (e.g. Vanguard Index Funds)— motivated by portfolio theory and CAPM: eﬃcient portfolios are combinations of T-Bills and a market index portfolio• Do not try to create portfolios to “actively” beat the returns on indexfunds

1.2

Active Management• Markets are not always in equilibrium— Some securities are “mis-priced”• Buy under-priced (positive “alpha”) assets and sell over-priced (negative“alpha”) assets• Active managers often “tweak” a benchmark (index) portfolio
Security
MSFT
GM
..

weight inbenchmark
.05
.02
..

weight in
active port.10-.05..

activeposition+.05-.07..

• Active management strategies— individual stock selection— sector selection (e.g. utility, technology)— asset class selection (stocks, bonds, real estate)• Most mutual funds are actively managed.
— management fees can vary substantially from fund to fund— fee is often a percentage of assets under management

2Evaluating Investment Performance

Q: Is it worthwhile to “pay” for active management of portfolios?
Key Concepts• Actively managed portfolios should be compared with passive (index) benchmarks of a similar risk class• Superior past performance could be luck or could be skill• Often very little historical data to evaluate managed portfolios— Statistical analysis is diﬃcult

2.1

Risk Adjusted Measures of Performance

Observe returns on active portfolio and benchmark over some time horizon (e.g.
5 years of monthly data)• Does the managed portfolio exhibit superior performance adjusted for risk?
• How to rank diﬀerent actively managed portfolios?

Measures of risk
• Market risk (portfolio beta, β p, from SI model or CAPM)• Total risk (portfolio standard deviation, σ p)
Ex Post (Historical) measuresTT
1 X
1 X
μ̂p =
Rp,t, r̂f =rf,t
T t=1T t=1

⎛

⎞1/21
⎝
σ̂ p =(Rp,t − μ̂p)2⎠
T − 1 t=1
dcov(Rp,t, RM,t)β̂ p =vdar(RM,t)
TX

Types of Performance Measures• Average return diﬀerence adjusted for risk
ave return on active portfolio ave return on risk adjusted benchmark• Risk adjusted reward/risk ratio

average excess returnrisk measure2.1.1

Performance Measures Based on Market Risk

Idea: Under CAPM, market risk is captured by β and expected returns arecaptured by the Security Market Line (SML)
μp,CAP M = rf + β p(μM − rf )Jensen’s alphaRisk-adjusted return diﬀerenceα̂∗p = μ̂p − μ̂p,CAP M
Computation: use linear regression to estimate the excess returns SI model
Rp,t − rf = α∗p + β p(RMt − rf ) + εpt, εpt ∼ iid N (0, σ 2ε )Statistical evaluation:
H0 : α∗p = 0 (no superior performance) vs. H1 : α∗p 6= 0

Information Ratio
dp =IRα̂∗pσ̂ εStatistical evaluation: Use bootstrap to compute standard error and confidence
interval

2.1.2

Performance Measures Based on Total Risk

Idea: Eﬃcient portfolios are combination of T-bills and tangency portfolio.
Under CAPM, the tangency portfolio is the market portfolio
Sharpe ratio
SRp =μ̂p − r̂fσ̂ p= excess return per unit portfolio risk
Statistical evaluation:
H0 : SRp = SRM (no superior performance) vs H1 : SRp 6= SRM
Evaluate H0 using bootstrap

R Package for Performance Evaluation

PerformanceAnalytics