Empirical Financial Economics Current Approaches to Performance Measurement

Overview of lecture  Standard approaches  Theoretical foundation  Practical implementation  Relation to style analysis  Gaming performance metrics

Performance measurement Leeson Investment Managemen t Market (S&P 500) Benchmark Short-term Government Benchmark Average Return Std. Deviation Beta Alpha.0025 (1.92).0 Sharpe Ratio Style: Index Arbitrage, 100% in cash at close of trading

Frequency distribution of monthly returns

Universe Comparisons 5% 10% 15% 20% 25% 30% 35% 40% Brownian Management S&P 500 One Quarter 1 Year3 Years5 Years Periods ending Dec

Average Return Total Return comparison A B C D

r f = 1.08% Average Return R S&P = 13.68% Total Return comparison A S&P 500 B C D Treasury Bills Manager A best Manager D worst

Average Return Total Return comparison A B C D

Average Return Standard Deviation Sharpe ratio comparison A B C D

r f = 1.08% σ S&P = 20.0% Average Return Standard Deviation R S&P = 13.68% Sharpe ratio comparison ^ A S&P 500 B C D Treasury Bills

r f = 1.08% σ S&P = 20.0% Average Return Standard Deviation R S&P = 13.68% Sharpe ratio comparison ^ A S&P 500 B C D Treasury Bills Manager D best Manager C worst Sharpe ratio = Average return – r f Standard Deviation

r f = 1.08% σ S&P = 20.0% Average Return Standard Deviation R S&P = 13.68% Sharpe ratio comparison ^ A S&P 500 B C D Treasury Bills

13 r f = 1.08% β S&P = 1.0 Average Return Beta R S&P = 13.68% Treynor Measure comparison A S&P 500 B C D Treasury Bills

14 r f = 1.08% Average Return R S&P = 13.68% Treynor Measure comparison A S&P 500 B C D Treasury Bills Manager B worst Manager C best Treynor measure = Average return – r f Beta β S&P = 1.0 Beta

r f = 1.08% Average Return R S&P = 13.68% Jensen’s Alpha comparison A S&P 500 B C D Treasury Bills Manager B worst Jensen’s alpha = Average return – {r f + β ( R S&P - r f )} β S&P = 1.0 Beta Manager C best

Intertemporal equilibrium model  Multiperiod problem:  First order conditions:  Stochastic discount factor interpretation:  “stochastic discount factor”, “pricing kernel”

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:

The geometry of mean variance Note: returns are in excess of the risk free rate

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

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 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)

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)

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

Returns-based Styles  Sharpe (1992) approach …  are a dynamic portfolio strategy  are benchmark portfolio returns  Style benchmarks are given by

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

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 Brown, Stephen J. and William N. Goetzmann, 1997 Mutual Fund Styles. Journal of Financial Economics 43:3,

Switching Regression  Quandt (1958)  If regimes not observed

K means procedure  Hartigan (1975)  If regimes not observed, use iterative algorithm to determine regime membership

Switching Regression  Quandt and Ramsey (1978)  Method of moments...

Eight style decomposition

Five style decomposition

Style classifications GSC1Event driven international GSC2Property/Fixed Income GSC3US Equity focus GSC4Non-directional/relative value GSC5Event driven domestic GSC6International focus GSC7Emerging markets GSC8Global macro

Regressing returns on classifications: Adjusted R 2 YearN GSC 8 classifications GSC 5 classification s TASS 17 classification s Average

“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 information  Designed to yield Sharpe ratio greater than benchmark  Using strategies that are concave to benchmark

Concave payout strategies  Zero net investment overlay strategy (Weisman 2002)  Uses only public information  Designed to yield Sharpe ratio greater than benchmark  Using strategies that are concave to benchmark  Why should we care?  Sharpe ratio obviously inappropriate here  But is metric of choice of hedge funds and derivatives traders 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)

Sharpe Ratio of Benchmark Sharpe ratio =.631

Maximum Sharpe Ratio Sharpe ratio =.748

Short Volatility Strategy Sharpe ratio =.743

Concave trading strategies

Examples of concave payout strategies  Long-term asset mix guidelines

 Unhedged short volatility  Writing out of the money calls and puts Examples of concave payout strategies

 Loss averse trading  a.k.a. “Doubling” Examples of concave payout strategies

 Long-term asset mix guidelines  Unhedged short volatility  Writing out of the money calls and puts  Loss averse trading  a.k.a. “Doubling”

Forensic Finance  Implications of concave payoff strategies  Patterns of returns  are returns concave to benchmark?  Patterns of security holdings  do 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 preferences  Cannot reward informationless trading  Should be scale invariant  Should be consistent with market equilibrium models 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)

Manipulation proof measure Certainty equivalent of portfolio return Number of observations Length of time between observations Chosen to make holding benchmark optimal for an uninformed investor

Implied risk aversion parameter What implied risk aversion parameter makes the market participant indifferent to holding the market portfolio?

Performance of beta ranked portfolios

Performance of vol ranked portfolios

Descriptive statistics of beta ranked portfolios Monthly returns: January December 2011 lobetahibetamktbetalovolhivollobeta-hibetalovol-hivol Mean1.00%0.71%0.95%1.03%0.32%0.29%0.71% Std.Dev2.78%9.38%3.71%2.76%8.31%8.36%7.10% Skewness Kurtosis Beta Sharpe Alpha0.391%-0.697%0.159%0.352%-0.903%0.667%0.833% t-value FF alpha0.322%-0.489%0.158%0.218%-0.776%0.391%0.574% t-value MPPM

Descriptive statistics of beta ranked portfolios Daily returns: January December 2011 lobetahibetamktbetalovolhivollobeta-hibetalovol-hivol Mean0.05%0.03%0.04%0.05%0.01% 0.04% Std.Dev0.34%1.96%0.56%0.49%1.39%1.92%1.11% Skewness Kurtosis Beta Sharpe Alpha0.024%-0.032%0.012%0.018%-0.040%0.037%0.038% t-value FF alpha0.025%-0.034%0.013%0.012%-0.041%0.039%0.034% t-value MPPM

Hedge funds follow concave strategies R-r f = α + β (R S&P - r f ) + γ (R S&P - r f ) 2

Hedge funds follow concave strategies R-r f = α + β (R S&P - r f ) + γ (R S&P - r f ) 2 Concave strategies: t β > 1.96 & t γ <

Hedge funds follow concave strategies ConcaveNeutralConvexN Convertible Arbitrage Dedicated Short Bias Emerging Markets Equity Market Neutral Event Driven Fixed Income Arbitrage Fund of Funds Global Macro Long/Short Equity Hedge Managed 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% % 77.25% 97.06% 72.64% 95.24% 82.06% 91.38% 86.62% 94.17% 91.67% 0.00% 0.86% 1.76% 0.34% 2.38% 1.57% 4.02% 2.18% 3.03% 3.33% Grand Total11.54%86.53%1.93%3318 R-r f = α + β (R S&P - r f ) + γ (R S&P - r f ) 2 Source: TASS/Tremont

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

Skewness as a function of the number of funds in FoHFs

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