Empirical Financial Economics

5. Current Approaches to Performance Measurement

Stephen Brown NYU Stern School of Business

UNSW PhD Seminar, June 19-21 2006

Overview of lecture

Standard approachesTheoretical foundationPractical implementationRelation to style analysisGaming performance metrics

Performance measurement

Leeson InvestmentManagement

Market (S&P 500) Benchmark

Short-term Government Benchmark

Average Return

.0065 .0050 .0036

Std. Deviation

.0106 .0359 .0015

Beta .0640 1.0 .0

Alpha .0025(1.92)

.0 .0

Sharpe Ratio

.2484 .0318 .0

Style: Index Arbitrage, 100% in cash at close of trading

Frequency distribution of monthly returns

0

5

10

15

20

25

30

35

-1.00

%

-0.50%

0.00

%

0.50

%1.0

0%1.5

0%

2.00

%

2.50

%

3.00

%

3.50

%

4.00

%

4.50

%

5.00

%

5.50

%

6.00

%

6.50

%

Universe Comparisons

5%

10%

15%

20%

25%

30%

35%

40%

Brownian ManagementS&P 500

One Quarter

1 Year 3 Years 5 Years

Periods ending Dec 31 2002

Average Return

Total Return comparison

A

BCD

rf = 1.08%

Average Return

RS&P = 13.68%

Total Return comparison

AS&P 500

BCD

Treasury Bills

Manager A best

Manager D worst

Average Return

Total Return comparison

A

BCD

Average Return

Standard Deviation

Sharpe ratio comparison

A

BC

D

rf = 1.08%

σS&P = 20.0%

Average Return

Standard Deviation

RS&P = 13.68%

Sharpe ratio comparison

^

AS&P 500

BC

D

Treasury Bills

rf = 1.08%

σS&P = 20.0%

Average Return

Standard Deviation

RS&P = 13.68%

Sharpe ratio comparison

^

AS&P 500

BC

D

Treasury Bills

Manager D bestManager C worstSharpe ratio =

Average return – rf

Standard Deviation

rf = 1.08%

σS&P = 20.0%

Average Return

Standard Deviation

RS&P = 13.68%

Sharpe ratio comparison

^

AS&P 500

BC

D

Treasury Bills

rf = 1.08%

Average Return

RS&P = 13.68%

Jensen’s Alpha comparison

AS&P 500

BCD

Treasury Bills

Manager B worstJensen’s alpha = Average return

–

{rf + β (RS&P - rf )}

βS&P = 1.0Beta

Manager C best

Intertemporal equilibrium model

Multiperiod problem:

First order conditions:

Stochastic discount factor interpretation:

“stochastic discount factor”, “pricing kernel”

0

Max ( )jt t j

j

E U c

,( ) (1 ) ( )jt t i t j t jU c E r U c

, , ,

( )1 (1 ) ,

( )t jj

t i t j t j t jt

U cE r m m

U c

,t jm

Value of Private Information

Investor has access to information

Value of is given by where and are returns on optimal portfolios given and

Under CAPM (Chen & Knez 1996)

Jensen’s alpha measures value of private information

1 0I I

1 0I I 1 0[( ) ]t tE R R m 1R 0R1I 0I

1 0 1 1 1[( ) ] ( )t t t ft t mt ftE R R m r r

The geometry of mean variance

a

b

a

b

E

2 1a

1 1

2

1/1/

0

bx b

22

2

2a bE cE

ac b

Note: returns are in excess of the risk free rate

fr

Informed portfolio strategy

Excess return on informed strategy where is the return on an optimal orthogonal portfolio (MacKinlay 1995)

Sharpe ratio squared of informed strategy

Assumes well diversified portfolios

1 0f fR r R r

2 1 1 2 2 21 0 0 0 0( ) ( )f fr r

Informed portfolio strategy

Excess return on informed strategy where is the return on an optimal orthogonal portfolio (MacKinlay 1995)

Sharpe ratio squared of informed strategy

Assumes well diversified portfolios

1 0f fR r R r

2 1 1 2 2 21 0 0 0 0( ) ( )f fr r

Used in tests of mean variance efficiency of benchmark

Practical issues

Sharpe ratio sensitive to diversification, but invariant to leverage

Risk premium and standard deviation proportionate to fraction of investment financed by borrowing

Jensen’s alpha invariant to diversification, but sensitive to leverage

In a complete market implies through borrowing (Goetzmann et al 2002)

2 0

Changes in Information Set

How do we measure alpha when information set is not constant?

Rolling regression, use subperiods to estimate (no t subscript) – Sharpe (1992)

Use macroeconomic variable controls – Ferson and Schadt(1996)

Use GSC procedure – Brown and Goetzmann (1997)

1 1 1 ( )t t ft t mt ftr r 1tI

1 1 1( )f m ftr r

Style management is crucial …

Economist, July 16, 1995

But who determines styles?

Characteristics-based Styles

Traditional approach …

are changing characteristics (PER, Price/Book)

are returns to characteristics Style benchmarks are given by

jt Jt Jt t jtr I j J

jt Jt jtr j J

JttI

Jt

Returns-based Styles

Sharpe (1992) approach …

are a dynamic portfolio strategy are benchmark portfolio returns Style benchmarks are given by

jt Jt Jt t jtr I j J

jt Jt jtr j J

JttI

Jt

Returns-based Styles

GSC (1997) approach …

vary through time but are fixed for style

Allocate funds to styles directly using Style benchmarks are given by

jt Jt Jt t jtr I j J

jt Jt jtr j J

,jT Jt

Jt

J

Jt

Eight style decomposition

0%

20%

40%

60%

80%

100%

GSC1 GSC2 GSC3 GSC4 GSC5 GSC6 GSC7 GSC8Other Pure PropertyPure Emerging Market Pure Leveraged CurrencyGlobal Macro Non Directional/Relative ValueEvent Driven Non-US Equity HedgeUS Equity Hedge

Five style decomposition

0%

20%

40%

60%

80%

100%

GSC1 GSC2 GSC3 GSC4 GSC5Other Pure PropertyPure Emerging Market Pure Leveraged CurrencyGlobal Macro Non Directional/Relative ValueEvent Driven Non-US Equity HedgeUS Equity Hedge

Style classifications

GSC1 Event driven international

GSC2 Property/Fixed Income

GSC3 US Equity focus

GSC4 Non-directional/relative value

GSC5 Event driven domestic

GSC6 International focus

GSC7 Emerging markets

GSC8 Global macro

Regressing returns on classifications: Adjusted R2

Year N GSC 8

classifications GSC 5

classificationsTASS 17

classifications1992 149 0.3827 0.1713 0.44411993 212 0.2224 0.1320 0.11861994 288 0.1662 0.1040 0.09861995 405 0.0576 0.0548 0.04461996 524 0.1554 0.0769 0.15231997 616 0.3066 0.1886 0.25381998 668 0.2813 0.2019 0.1998

Average 0.2246 0.1328 0.1874

Variance explained by prior returns-based classifications

Year N8 GSC

Classifications8 Principal

Components 8 Benchmarks(predetermined)

1992 198 0.3622 0.0572 0.17691993 276 0.1779 0.0351 0.17481994 348 0.1590 0.0761 0.04811995 455 0.0611 0.0799 0.08621996 557 0.1543 0.0286 0.06911997 649 0.2969 0.0211 0.06421998 687 0.2824 0.2862 0.2030

Average 0.2134 0.0835 0.1175

Variance explained by prior factor loadings

Year N8 GSC

Classifications8 Principal

Components 8 Benchmarks

(predetermined)1992 198 0.2742 0.1607 0.25521993 276 0.2170 0.0928 0.09321994 348 0.1760 0.1577 0.07001995 455 0.0670 0.0783 0.08291996 557 0.1444 0.0888 0.03491997 649 0.3135 0.3069 0.08991998 687 0.2752 0.3744 0.3765

Average 0.2096 0.1799 0.1432

Percentage in cash (monthly)

0%

20%

40%

60%

80%

100%

120%

31-Dec-1989 15-May-1991 26-Sep-1992 8-Feb-1994

Examples of riskless index arbitrage …

Percentage in cash (daily)

-600%

-500%

-400%

-300%

-200%

-100%

0%

100%

200%

31-Dec-1989 15-May-1991 26-Sep-1992 8-Feb-1994

“Informationless” investing

Concave payout strategies

Zero net investment overlay strategy (Weisman 2002)

Uses only public informationDesigned to yield Sharpe ratio greater than benchmarkUsing strategies that are concave to benchmark

Concave payout strategies

Zero net investment overlay strategy (Weisman 2002)

Uses only public informationDesigned to yield Sharpe ratio greater than

benchmarkUsing strategies that are concave to benchmark

Why should we care?

Sharpe ratio obviously inappropriate hereBut is metric of choice of hedge funds and

derivatives traders

We should care!

Delegated fund managementFund flow, compensation based on

historical performanceLimited incentive to monitor high

Sharpe ratiosBehavioral issues

Prospect theory: lock in gains, gamble on loss

Are there incentives to control this behavior?

Sharpe Ratio of Benchmark

-200%

-150%

-100%

-50%

0%

50%

100%

-50% 0% 50% 100%

Benchmark

Sharpe ratio = .631

Maximum Sharpe Ratio

-200%

-150%

-100%

-50%

0%

50%

100%

-50% 0% 50% 100%

Benchmark

MaximumSharpe RatioStrategy

Sharpe ratio = .748

Concave trading strategies

-200%

-150%

-100%

-50%

0%

50%

100%

-50% 0% 50% 100%

Benchmark

Loss AverseTrading(Median)MaximumSharpe RatioStrategy

Examples of concave payout strategies

Long-term asset mix guidelines

Unhedged short volatilityWriting out of the money

calls and puts

Examples of concave payout strategies

Loss averse trading a.k.a. “Doubling”

Examples of concave payout strategies

Examples of concave payout strategies

Long-term asset mix guidelines

Unhedged short volatilityWriting out of the money calls

and puts

Loss averse trading a.k.a. “Doubling”

Forensic Finance

Implications of concave payoff strategies

Patterns of returns

Forensic Finance

Implications of Informationless investing

Patterns of returnsare returns concave to benchmark?

Forensic Finance

Implications of concave payoff strategies

Patterns of returnsare returns concave to benchmark?

Patterns of security holdings

Forensic Finance

Implications of concave payoff strategies

Patterns of returnsare returns concave to benchmark?

Patterns of security holdingsdo security holdings produce

concave payouts?

Forensic Finance

Implications of concave payoff strategies

Patterns of returnsare returns concave to benchmark?

Patterns of security holdingsdo security holdings produce concave

payouts?

Patterns of trading

Forensic Finance

Implications of concave payoff strategies

Patterns of returnsare returns concave to benchmark?

Patterns of security holdingsdo security holdings produce concave

payouts?

Patterns of tradingdoes pattern of trading lead to concave

payouts?

Conclusion

Value of information interpretation of standard performance measures

New procedures for style analysis

Return based performance measures only tell part of the story