Risk Adjusted Performance Analysis

8/7/2019 Risk Adjusted Performance Analysis

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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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Slide 2Date: 31.03.2001Produced by: Andreas Steiner

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

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Risk Adjusted Performance Analysis

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