Yale School of Management Sharpening Sharpe Ratios Will Goetzmann Jonathan Ingersoll Matthew Spiegel Ivo Welch

Yale School of Management Background Sharpe Ratio  Performance evaluation in practice.  Asset pricing research. Limitations  Misleading when shape of distribution changes.  Problematic in presence of derivatives.

Yale School of Management Example Perfect foresight timer btw. US. Stocks and U.S. bonds. Sharpe = 1 Throwing all returns over 30%/year away Sharpe = 1.06 Smoothing works even better.

Yale School of Management Our Approach What strategy maximizes Sharpe ratio? How much can it matter? Implications for risk-control. Dynamic strategies. Are there any measures that cannot be manipulated?

Yale School of Management Optimal Sharpe Ratio Distribution Left-skewed. Fat-tailed. Very sensitive to small-sample. Hard to distinguish luck vs. skill.

Yale School of Management Manipulation-Free Statistic Exists only under specification of utility. Provides a method to test the efficacy of the Sharpe ratio. Sharpe ratio does well under “normal” conditions. New measure is useful under non-normal conditions.

Yale School of Management Hedge Fund Applications Hedge funds unconstrained from dynamic and derivative strategies. Hedge funds often evaluated by Sharpe Ratio. Absolute return benchmark: Libor or T-bills Hedge funds seem prone to occasional, spectacular disasters.

Yale School of Management Hedge Fund Strategies Fung and Hsieh (1997) Brown and Goetzmann (1997) Agarawal and Naik (2001) Contract-related non-linearity

Yale School of Management

Art Institute vs. Integral Integral boasted “The highest Sharpe Ratio in the business.” Options-based strategy. Performance-based contract. Guaranteed 1% to 2% in flat or rising markets. Losses possible only if stocks dropped more than 30% (which they did).

Yale School of Management Maximal Sharpe Ratio in a Complete Market MSR is linear in the likelihood ratio of the state price per unit probability. Sell high-priced, low probability payoffs. Leverage does not change shape. Possible to nearly match it with a limited liability portfolio. Any basis asset is possible.

Yale School of Management

Incomplete Market: 1 Strike Restriction to index, put and call. Parameter values:  r = 5%, mu = 15%, T= 1. Sharpe ratio for stock =.631 Sell.843 calls at gives ratio of.731

Yale School of Management Two Strikes Sell 2.58 puts at strike.88 Sell.77 calls at strike 1.12 Maximum Sharpe ratio is % increase in Sharpe ratio over the market.

Yale School of Management

Dynamic Strategies Conditioning on past performance.  Brown, Harlow and Starks, Chevalier and Ellison, Brown, Goetzmann and Park, Carpenter and others. Result: poor performance implies increasing leverage. Good performance, implies decreasing expected return towards market.

Yale School of Management Intuition Conditional return in the first period, you can minimized expected variance over the whole period by choosing an expected return equal to it. Dynamic strategy is like static option strategy in that it moves state payoffs from one period to another to improve Sharpe ratio.

Yale School of Management

Manipulation-Free Manipulation = rebalancing of the portfolio away from the benchmark even when there exists no informational reason to do so.

Yale School of Management Requirements Should provide a unique ranking of funds for a meaningful set of investors. Should be “memoryless” – no dynamic strategy should allow improvement. Implies time-separable, concave utility. Wealth-independent – power utility. Uninformed investor should hold market. Implies a single risk aversion parameter.

Yale School of Management MFM

Yale School of Management Risk-Aversion Parameter Representative investor holds mkt:  = 0: Rank on Arithmetic Average  = 1: Rank on Geometric Mean  >2: Higher Risk Aversion

Yale School of Management Empirical Tests A test of the Sharpe ratio. Equity mutual fund returns 1993 – Hedge fund returns 1992 – Examine rank correlations of Sharpe and MFM. Does skewness affect ranking differences? Parameter and time-period sensitivity.

Yale School of Management Rank Correlation between Manipulation-Free Measure and Sharpe Ratio Market’s Percentile Performance within Class of Funds CRSP Mutual Funds Oct 1993  Sept 1998 CategorySharpe  = 0  = 1  = 2  = 6 N All Mutual Funds %82.4%83.3%83.2%82.2% TASS Hedge Funds Oct 1992  Sept 1997 Sharpe  = 0  = 1  = 2  = 6 N All Hedge Funds %74.5%76.6%78.3%84.7%

Yale School of Management Rank Correlation between Manipulation-Free Measure and Sharpe Ratio Market’s Percentile Performance within Class of Funds CRSP Mutual Funds Oct 1998  Sept 2003 CategorySharpe  = 0  = 1  = 2  = 6 N All Mutual Funds %42.5%45.8%49.5%55.7% TASS Hedge Funds Oct 1997  Sept 2002 Sharpe  = 0  = 1  = 2  = 6 N All Hedge Funds %8.8%10.5%13.5%18.4%

Yale School of Management Effect of Skewness on Relative Performance

Yale School of Management Conclusions Maximal Sharpe Ratio is a mirrored log-normal. Optimal strategies sell out of money option in asymmetric proportion. Dynamic strategies also possible. Manipulation free measure proposed. Sharpe ratio tested, and works well normally. Negative skewness in hedge funds associated with Sharpe ratio rank improvement.