empirical financial economics current approaches to performance measurement

Empirical Financial Economics

Current Approaches to Performance Measurement

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

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%


AS&P 500

BCD

Treasury Bills

Manager A best

Manager D worst

Average Return


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%


^

AS&P 500

BC

D

Treasury Bills

rf = 1.08%

σS&P = 20.0%

Average Return

Standard Deviation

RS&P = 13.68%


^

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%


^

AS&P 500

BC

D

Treasury Bills

13

rf = 1.08%

βS&P = 1.0

Average Return

Beta

RS&P = 13.68%

Treynor Measure comparison

A

S&P 500

BCD

Treasury Bills

14

rf = 1.08%

Average ReturnRS&P =

13.68%

Treynor Measure comparison

A

S&P 500

BCD

Treasury Bills

Manager B worstManager C best

Treynor measure =Average return – rf

Beta

βS&P = 1.0Beta

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

Other pricing kernels:

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

( )Mm a b r c

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 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 jtr j J

,jT Jt

Jt

J

Jt

Basis Assets

GSC (1997) approach …

vary through time but fixed for risk class

Allocate equities to risk classes directly using

Style benchmarks are given by


jt Jt jtr j J

,jT Jt

Jt

J

Jt

Brown, Stephen J. and William N. Goetzmann, 1997 Mutual Fund Styles. Journal ofFinancial Economics 43:3, 373-399.

Switching Regression

Quandt (1958)

If regimes not observed

1 1 1

2 2 2

3 3 3

1 0

0 0

t t t t

t t t t

t t t

tt

t

y X I

y X I

Z X

if ZI

if Z

= 1 (Regime 1)

= 0 (Regime2)

K means procedure

Hartigan (1975)

If regimes not observed, use iterative algorithm to determine regime membership

1 1

2 2

t t t

t t t

t Kt Kt

y

y

y K

(Regime 1)

(Regime2)

(Regime )

Switching Regression

Quandt and Ramsey (1978)

Method of moments ...

2 21 2

2 21 2

2 21 1 2 2

( ) ( )

2 21 2

1 2

2 2 2 21 2 1 2

( ) ( , ) (1 ) ( , )

1

2 2

(1 )

(1 ) 2 (1 )( )

.

y y

y

y

f y N N

e e

etc

Eight style decomposition

GSC1 GSC2 GSC3 GSC4 GSC5 GSC6 GSC7 GSC80%

20%

40%

60%

80%

100%

US Equity HedgeNon-US Equity HedgeEvent DrivenNon Directional/Relative ValueGlobal MacroPure Leveraged Currency

Five style decomposition

GSC1 GSC2 GSC3 GSC4 GSC50%

20%

40%

60%

80%

100%

US Equity HedgeNon-US Equity HedgeEvent DrivenNon Directional/Relative ValueGlobal MacroPure Leveraged Currency

Style classifications

GSC1 Event driven internationalGSC2 Property/Fixed IncomeGSC3 US Equity focusGSC4 Non-directional/relative valueGSC5 Event driven domesticGSC6 International focusGSC7 Emerging marketsGSC8 Global macro

Regressing returns on classifications: Adjusted R2

Year N GSC 8

classifications

GSC 5 classificati

ons

TASS 17 classificati

ons

1992 149 0.3827 0.1713 0.4441

1993 212 0.2224 0.132 0.1186

1994 288 0.1662 0.104 0.0986

1995 405 0.0576 0.0548 0.0446

1996 524 0.1554 0.0769 0.1523

1997 616 0.3066 0.1886 0.2538

1998 668 0.2813 0.2019 0.1998

Average 0.2246 0.1328 0.1874

“Informationless” investing

Analytic Optioned

Our fund purchases a stock and simultaneously sells a call option against the stock. By doing this, the fund receives both dividend income from the stock and a cash premium from the sale of the option. This strategy is designed for the longer term investor who wants to reduce risk. It is particularly suited for pension plans IRAs and Keoghs. Our defensive buy/write strategy is designed to put greater emphasis on risk reduction by focusing on “in-the-money” call options. The results speak for themselves. Over a twelve year period, of 153 institutional portfolios in the Frank Russell Co. universe, no other portfolio had a higher return with less risk than our All Buy/Write Accounts Index. In the terminology of modern portfolio theory, our clients’ portfolios dominated the market averages.

Modern Portfolio Theory

Covered Call Strategy

Stock

Value

Profit to Covered Call

Payoff to Covered Call

Unoptioned Portfolio Return

Portfolio Return

Expected

Return

Optioned Return

Portfolio Return

Expected

Return

Optioned Return (incl. premium)

Portfolio Return

Expected

Return

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 tradersGoetzmann, William N., Ingersoll, Jonathan E., Spiegel, Matthew I. and Welch, Ivo, 2007 Portfolio Performance

Manipulation and Manipulation-proof Performance Measures, Review of Financial Studies 20(5) 1503-1546.

Sharpe Ratio of Benchmark

-50% 0% 50% 100%-2

-1.5

-1

-0.5

0

0.5

1

Benchmark

Sharpe ratio = .631

Maximum Sharpe Ratio

-50% 0% 50% 100%-2

-1.5

-1

-0.5

0

0.5

1

Benchmark

Maximum Sharpe Ra-tio Strategy

Sharpe ratio = .748

Short Volatility Strategy

-50% 0% 50% 100%-2

-1.5

-1

-0.5

0

0.5

1

Benchmark

Short volatility

Benchmark return

Port

foli

o r

etu

rn

Sharpe ratio = .743

Concave trading strategies

-50% 0% 50% 100%-2

-1.5

-1

-0.5

0

0.5

1

Benchmark

Loss Averse Trading (Median)

Maximum Sharpe Ra-tio Strategy

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”



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 returnsare returns concave to benchmark?

Patterns of security holdingsdo security holdings produce

concave payouts?Patterns of trading

does pattern of trading lead to concave payouts?

Manipulation proof measure

Criteria:Ranks portfolios based on investor

preferencesCannot reward informationless tradingShould be scale invariantShould be consistent with market

equilibrium models

1

1

1 1ˆ n (1 ) (1 ) .(1 )

T

t ftt

r rt T

1

1

1 1ˆ n (1 ) (1 ) .(1 )

T

t ftt

r rt T

1

1

1 1ˆ n (1 ) (1 ) .(1 )

T

t ftt

r rt T

1

1

1 1ˆ n (1 ) (1 ) .(1 )

T

t ftt

r rt T

Goetzmann, William N., Ingersoll, Jonathan E., Spiegel, Matthew I. and Welch, Ivo, 2007 Portfolio Performance Manipulation and Manipulation-proof Performance Measures, Review of Financial Studies 20(5) 1503-1546.

Manipulation proof measure

1

1

1 1ˆ n (1 ) (1 ) .(1 )

ˆ

T

t ftt

r rt T

T

t

Certainty equivalent of portfolio return

Number of observations

Length of time between observationsChosen to make holding benchmark optimal for an uninformed investor

Implied risk aversion parameter

ln[ (1 )] ln[(1 )]

(1 )m f

m

E r r

Var r

What implied risk aversion parameter makes the

market participant indifferent to holding the

market portfolio?

Performance of beta ranked portfolios

Jan-00 Sep-02 Jun-05 Mar-08 Dec-100

0.5

1

1.5

2

2.5

3

lobethighbetmktCRSPvw

Performance of vol ranked portfolios

Jan-00 Sep-02 Jun-05 Mar-08 Dec-100

0.5

1

1.5

2

2.5

lowvolhivolCRSPvw

Descriptive statistics of beta ranked portfolios

Monthly returns: January 1980 - December 2011

lobeta hibeta mktbeta lovol hivol lobeta-hibeta lovol-hivol

Mean 1.00% 0.71% 0.95% 1.03% 0.32% 0.29% 0.71%

Std.Dev 2.78% 9.38% 3.71% 2.76% 8.31% 8.36% 7.10%

Skewness -1.741 -0.253 -1.146 -1.207 -0.239 0.036 -0.228

Kurtosis 14.619 4.199 9.208 7.933 5.601 4.564 7.393

Beta 0.339 1.791 0.667 0.464 1.450 -1.449 -0.983

Sharpe 0.2080 0.0310 0.1415 0.2223 -0.0123 -0.0159 0.0411

Alpha 0.391% -0.697% 0.159% 0.352% -0.903% 0.667% 0.833%

t-value 2.805 -3.022 1.331 3.556 -3.522 2.339 2.810

FF alpha 0.322% -0.489% 0.158% 0.218% -0.776% 0.391% 0.574%

t-value 2.483 -2.866 1.645 2.426 -4.628 1.507 2.668

MPPM -.00205 .00391 -.00148 -.00221 .00453

Descriptive statistics of beta ranked portfolios

Daily returns: January 1980 - December 2011

lobeta hibeta mktbeta lovol hivol lobeta-hibeta lovol-hivol

Mean 0.05% 0.03% 0.04% 0.05% 0.01% 0.01% 0.04%

Std.Dev 0.34% 1.96% 0.56% 0.49% 1.39% 1.92% 1.11%

Skewness -2.926 -0.133 -1.829 -1.171 -0.597 -0.172 0.385

Kurtosis 46.557 10.471 30.116 37.263 12.931 10.886 16.465

Beta 0.070 1.615 0.416 0.378 1.033 -1.545 -0.655

Sharpe 0.0764 0.0060 0.0411 0.0570 -0.0085 -0.0029 0.0182

Alpha 0.024% -0.032% 0.012% 0.018% -0.040% 0.037% 0.038%

t-value 5.907 -3.423 2.994 5.574 -4.180 3.564 3.707

FF alpha 0.025% -0.034% 0.013% 0.012% -0.041% 0.039% 0.034%

t-value 6.394 -4.984 4.018 4.637 -6.527 4.505 4.615

MPPM -.00009 .00012 -.00007 -.00009 .00013

Hedge funds follow concave strategies

R-rf = α + β (RS&P- rf) + γ (RS&P- rf)2



Concave strategies: tβ > 1.96 & tγ < -1.96


Concave NeutralConve

x N

Convertible Arbitrage

Dedicated Short Bias

Emerging Markets

Equity Market Neutral

Event Driven Fixed Income

Arbitrage Fund of Funds Global Macro Long/Short Equity

HedgeManaged Futures

Other

5.38%0.00%

21.89%1.18%

27.03%2.38%

16.38%4.60%

11.19%2.80%5.00%

94.62%100.00

%77.25%97.06%72.64%95.24%82.06%91.38%86.62%94.17%91.67%

0.00%0.00%0.86%1.76%0.34%2.38%1.57%4.02%2.18%3.03%3.33%

13027

233170296126574174

109942960

Grand Total 11.54% 86.53% 1.93% 3318


Source: TASS/Tremont

Standard deviation as a function of the number of funds in FoHFs

0.35 3.5 35 3500

0.01

0.02

0.03

0.04

0.05

Std. Deviation S&P500 Index

Std. Dev. (naïve 1/N)

Median FoHF Std. Deviation

Number of underlying funds

Std

.Dev

. o

f p

ort

foli

o r

etu

rn

Brown, Stephen J., Gregoriou, Greg N. and Pascalau, Razvan C., Diversification in Funds of Hedge Funds: Is it Possible to Overdiversify? Review of Asset Pricing Studies 2(1), 2012, pp.89-110 http://ssrn.com/abstract=1436468

0.35 3.5 35 350-2

-1.5

-1

-0.5

0

Skewness S&P500 Index

Skewness (naïve 1/N)

Median of FoHF Skewness


Ske

wn

es o

f p

ort

foli

o r

etu

rn

Skewness as a function of the number of funds in FoHFs

0.35 3.5 35 3500

1

2

3

4

5

6

7

8

9

Kurtosis S&P500 Index

Kurtosis (naïve 1/N)

Median FoHF Kurtosis


Ku

rto

sis

of

po

rtfo

lio

ret

urn

Kurtosis as a function of the number of funds in FoHFs

Conclusion

Value of information interpretation of standard performance measures

New procedures for style analysis

Return based performance measures only tell part of the story

empirical financial economics current approaches to performance measurement

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