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RISK-ADJUSTED
PERFORMANCE ANALYSIS
Andreas Steiner
Zurich
May 2001
Slide 101.05.98Andreas Steiner
Date:Produced by:
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CONTENTS
1. INTRODUCTION
2. FRAMEWORK
3. DEFINITION ANDAPPLICATION OFCLASSICAL MEASURES
1. Sharpe Ratio2. Treynor Ratio
3. Information Ratio
4. M MEASURES
1. M2
2. M3
5. RAPP
6. RELATIONSHIP BETWEENTHE MEASURES
7. DISCUSSION
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INTRODUCTION (1/2)
Common wisdom today: Performance is only a biasedand noisysignal for the quality of asset management.
BIAS: Risk-return trade off
NOISE: Skill versus luck
Risk-adjusted performance analysis is about quantifying andanalyzing unbiased performance. It can also be used to distinguishskill from luck.
This presentation wants to summarize the best practice concepts
and methods in risk-adjusted performance analysis. It is of adescriptive nature.
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INTRODUCTION (2/2)
CONSULTANTS SWITZERLAND
Quantitative performance analysis as a criterion for manager selection has been
practiced for about 5 years a new toy
Most often requested statistics: Sharpe and Information Ratio
STRUCTURED ALPHA (Watson Wyatt)
Alpha: Net Fund Return Net Benchmark Return.Net = Fees & switching costs
Sigma: Tracking Error = Standard Deviation of Alpha
Financial Factors summarized inInvestment Efficiency: Net Alpha / TE = IR. Used to rank managers
Theta: Non-financial factors are of importance to trustees: Sleep Well factors (lossaversion), Seems Good factors (brand names)
According to WW, the relative influence of financial and Theta factors is 50/50.Source: Global Industry Survey, WW, 1999.
There exists a trade off between financial and Theta factors. Thats why you needWWs consulting service
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FRAMEWORK
1. CLIENT PREFERENCES
Client likes return, dislikes risk*:
3. INDEX MODELS
++= )r(rfBfP
0U
0U
dU
dU
dU
),(UU
P
P
P
P
P
P
PP
+
=
=
2. BENCHMARKING
Client chooses benchmark andsets targets/limits for alpha, beta
a.s.o. atinception
Portfolio Mgt controls alpha, betaand beta afterinception
*risk is usually defined as thesecond moment of the returndistribution.
Validity of index models to analyze
performance largely depends on theimplementation of benchmarking!
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MEASUREMENT
Annualized portfolio return, portfoliovolatility
Annualized risk-free rate Choice is important because it can
change ranking
Problematic in an internationalcontext
Aggregation
No straight-forward adding-upbecause of covariance effectsbetween volatilities
Are negative values ambiguous?
INTERPRETATION
Summary of the first two moments ofthe portfolio excess returndistribution. Model-free
Suitable for comparisons acrossasset classes
Target in Mean-VarianceOptimization
Does not assume a benchmark.Implicit benchmark is risk-free rate.
Statistical hypothesis testing: test fornon-zero performance
t-Stat = S * sqrt(T)
SHARPE RATIO (2/2) - APPLICATION
P
fPr
+
P
fPr
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TREYNOR RATIO (1/2) - DEFINITION
Betaio...Portfol
Ratee...Riskfrer
Returnio...Portfol
rT
P
f
P
P
fP
=
tion...Correla
nce...Covaria
PB
PB
fr
P
P
B
P
PB2
B
PB
==
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TREYNOR RATIO (2/2) - APPLICATION
MEASUREMENT
Annualized portfolio return,
annualized risk-free rate
Estimation of beta can be distortedby market timing. Extensions:Squared regression, H/M regression
Aggregation: Straight-forward. Betaof aggregate is weighted sum ofconstituents betas
INTERPRETATION
Accounts for systematic and
unsystematic risk (CAPM-based):Only systematic risk isconsidered.
Comparison across different assetclasses problematic (beta is
dependent on benchmark)
Choice of benchmark affectsranking
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INFORMATION RATIO (1/3) - DEFINITION
ErrorTrackingo...PorfoliET
Alphaio...Portfol
TEIR
P
P
P
P
=
TE
P
PTE
Active Portfolio Return: Alpha
Average annual performance
Jensens Alpha
Choice should be consistent to choice of TE definition
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IR (2/3) - TRACKING ERROR DEFINITIONS
( )BPPrrVarET =
== 2
PBPP1ET
with
Standard deviation of performance
++=BP
22
B2
B
2
P2
PB
22
B
22
P
+=+=
B
P
PB2
B
PB
==
Residual risk = Risk uncorrelated with BM.
22
P
2
PB
2
P +=
( ) ++=++=BBBBP
1
( ) 22B
2
BP1)(Var +=
For beta 1, the Stdev(perf) is alwayslarger than residual risk
Stdev(perf) depends on benchmark volatility
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IR (3/3) - APPLICATION
MEASUREMENT
Best practice in CH: TE asannualized volatility of
performance. Alpha as averageannualized performance.
Measurement of Alpha & TE withindex or factor models makes IRdependent on model specification
errors.
INTERPRETATION
Summary statistic: Active return /active risk trade off, efficiency ratio
Fundamental Law of Active Mgt:
IR ex ante = IC x BR
IC: Information Coefficient
Corr(Forecast r, Actual r)BR: Breadth of strategy
# of independent bets taken
Statistical hypothesis testing: Non-
zero alpha signals
t-Stat = IR * sqrt(T)
Generally not consistent with MVO
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M MEASURES - M2 (1/2)
( )
Ratee...Riskfrer
Returnio...Portfol
Volatilityrk...Benchma
Volatilityio...Portfol
ReturnAdjusted-...Risk
rr
f
f
B
P
RAP
ffP
P
B
RAP
+=
fr
RAP
PB
B
P
( )fPRAP
P
B
rd1d
dFactore...Leverag
+=
Performance is volatility-adjusted by
leveraging the fund with risk-free-investments so that the resultingvolatility equals the benchmark volatility.
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M MEASURES - M2 (2/2) The difference between M2 can be interpreted intuitively: Unit of
measurement is % Risk expressed in units of return
M2 rankings are independent of the chosen benchmark (benchmark riskas a scaling factor)
The M2 measure is a transformed Sharpe Ratio and therefore consistentwith MPT
( )fBffP
P
B
RAPrSrr +=+=
M2 ranking equals Sharpe Ratio ranking
Drawback: Correlation risk (timing, selection) is neglected
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M MEASURES - M3 (1/2)
( )
)1(
)1(b
)1(
)1(a
2
TE1
rba1ba
2
PB
2
PBPBPB
2
PB
2
PB
P
B
2
B
2
PBPB
fBPCAP
=
=
=
++=
M3 cannot be illustrated graphicallyin an elegant way (three dimensions)
Performance is correlation-adjustedby leveraging the fund with active,
passive and risk-free funds so that (1)the resulting volatility equalsbenchmark volatility and (2) the TEequals the Target TE
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M MEASURES - M3 (2/2) M3 is volatility-risk- and-correlation-risk-adjusted-performance
M3 rankings differ from M2 and rankings
If no target tracking error exists, a = 0 and M3 will equal M2
M3 can be used in a forward looking sense: It can provide ex anteguidance how to structure portfolios with TE restrictions (given thestability of distributional characteristics in the future)
Drawback (of all RAP measures): Timing and selection activitiesare not decoupled.
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RAPP (1/2)
PB
B
P
BU
PU
dU
dU
),(U),(UBBPP
+
+
dTE
d
Aversion...RiskU
U
TEU
),(U),(URAPP BBPP
=
+
Risk-Adjusted Performance and Positioning Index
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RAPP (2/2) The RAPP concept is very flexible (TE targets, for example)
Utility functions are considered at least problematic by manyeconomists, especially in decision making under risk (HomoOeconomicus debate, Behavioral Finance)
To implement RAPP, the marginal utilities of parameters (riskaversion, for example) have to be quantified. RAPP ranking willdepend on these marginal utilities.
Aggregation across asset classes is achieved by measuringeverything in terms of utilities. A new aggregation problem isintroduced: aggregating client preferences.
Non-financial aspects are neglected. Considering the importance of
such factors: Is it worth developing and maintaining an internal RAPmeasure?
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RELATIONSHIP BETWEEN MEASURES
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Months 01.02.93 - 31.03.01
IndexofChain-LinkedTotalR
eturns
Sharpe/M2 Max
Treynor/AlphaMax
MinVar
TE Min
IR Max
M3 Max
RAPP Max
MSCI World AC
Markets: S&P 500, DJ Euro STOXX 50, SPI, MSCI Japan, FTSE 100
Observations:- RAP strategies are highly correlated- The ex ante / ex post choice of RAP targets creates significant incentives
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DISCUSSION
ITS YOURTURN...