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Performance Slides

This document discusses methods for evaluating investment performance, including: 1) Passive management strategies that track market indexes versus active management strategies that aim to beat the market. 2) Risk-adjusted measures like Jensen's alpha, the Information Ratio, and the Sharpe Ratio that compare returns of actively managed portfolios to benchmarks adjusted for risk. 3) The R package PerformanceAnalytics that can be used to statistically evaluate different performance measures and test if an active strategy exhibits superior performance to a passive benchmark.

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52 views14 pages

Performance Slides

This document discusses methods for evaluating investment performance, including: 1) Passive management strategies that track market indexes versus active management strategies that aim to beat the market. 2) Risk-adjusted measures like Jensen's alpha, the Information Ratio, and the Sharpe Ratio that compare returns of actively managed portfolios to benchmarks adjusted for risk. 3) The R package PerformanceAnalytics that can be used to statistically evaluate different performance measures and test if an active strategy exhibits superior performance to a passive benchmark.

Uploaded by

hatem
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© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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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 com-


binations of T-Bills and a market index portfolio

• Do not try to create portfolios to “actively” beat the returns on index


funds
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

weight in weight in active


Security
benchmark active port position
MSFT .05 .10 +.05
GM .02 -.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


2 Evaluating Investment Performance

Q: Is it worthwhile to “pay” for active management of portfolios?

Key Concepts

• Actively managed portfolios should be compared with passive (index) bench-


marks 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) measures

1 XT 1 XT
μ̂p = Rp,t, r̂f = rf,t
T t=1 T t=1
⎛ ⎞1/2
1 T
X
σ̂ p = ⎝ (Rp,t − μ̂p)2⎠
T − 1 t=1
d
cov(R p,t, RM,t)
β̂ p =
vd
ar(RM,t)
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 return
risk measure
2.1.1 Performance Measures Based on Market Risk

Idea: Under CAPM, market risk is captured by β and expected returns are
captured by the Security Market Line (SML)

μp,CAP M = rf + β p(μM − rf )
Jensen’s alpha

Risk-adjusted return difference

α̂∗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 =
α̂∗p
IR
σ̂ ε

Statistical 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
μ̂p − r̂f
SRp =
σ̂ 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

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