Sharpening Sharpe Ratios

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Sharpening Sharpe Ratios

• Will Goetzmann
• Jonathan Ingersoll
• Matthew Spiegel
• Ivo Welch

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Background

• Sharpe Ratio
• Performance evaluation in practice.
• Asset pricing research.
• Limitations
• Misleading when shape of distribution changes.
• Problematic in presence of derivatives.

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Example

• Perfect foresight timer btw. US. Stocks and U.S.
  bonds.
• Sharpe 1926-2003 1
• Throwing all returns over 30/year away Sharpe
  1926-2003 1.06
• Smoothing works even better.

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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?

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Optimal Sharpe Ratio Distribution

• Left-skewed.
• Fat-tailed.
• Very sensitive to small-sample.
• Hard to distinguish luck vs. skill.

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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.

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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.

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Hedge Fund Strategies

• Fung and Hsieh (1997)
• Brown and Goetzmann (1997)
• Agarawal and Naik (2001)
• Contract-related non-linearity

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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).

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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.

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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 1.0098 gives ratio of .731

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Two Strikes

• Sell 2.58 puts at strike .88
• Sell .77 calls at strike 1.12
• Maximum Sharpe ratio is .748
• 18 increase in Sharpe ratio over the market.

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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.

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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.

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Manipulation-Free

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

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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.

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MFM
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Risk-Aversion Parameter

• Representative investor holds mkt
• ? 0 Rank on Arithmetic Average
• ? 1 Rank on Geometric Mean
• ? gt2 Higher Risk Aversion

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Empirical Tests

• A test of the Sharpe ratio.
• Equity mutual fund returns 1993 2003.
• Hedge fund returns 1992 2002.
• Examine rank correlations of Sharpe and MFM.
• Does skewness affect ranking differences?
• Parameter and time-period sensitivity.

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Rank Correlation between Manipulation-Free Measure and Sharpe Ratio Markets Percentile Performance within Class of Funds Rank Correlation between Manipulation-Free Measure and Sharpe Ratio Markets Percentile Performance within Class of Funds Rank Correlation between Manipulation-Free Measure and Sharpe Ratio Markets Percentile Performance within Class of Funds Rank Correlation between Manipulation-Free Measure and Sharpe Ratio Markets Percentile Performance within Class of Funds Rank Correlation between Manipulation-Free Measure and Sharpe Ratio Markets Percentile Performance within Class of Funds Rank Correlation between Manipulation-Free Measure and Sharpe Ratio Markets Percentile Performance within Class of Funds Rank Correlation between Manipulation-Free Measure and Sharpe Ratio Markets Percentile Performance within Class of Funds
CRSP Mutual Funds Oct 1993 ? Sept 1998 CRSP Mutual Funds Oct 1993 ? Sept 1998 CRSP Mutual Funds Oct 1993 ? Sept 1998 CRSP Mutual Funds Oct 1993 ? Sept 1998 CRSP Mutual Funds Oct 1993 ? Sept 1998 CRSP Mutual Funds Oct 1993 ? Sept 1998 CRSP Mutual Funds Oct 1993 ? Sept 1998
Category Sharpe ? 0 ? 1 ? 2 ? 6 N
All Mutual Funds 0.951 0.973 0.986 0.966 1008
All Mutual Funds 82.6 82.4 83.3 83.2 82.2 1008

TASS Hedge Funds Oct 1992 ? Sept 1997 TASS Hedge Funds Oct 1992 ? Sept 1997 TASS Hedge Funds Oct 1992 ? Sept 1997 TASS Hedge Funds Oct 1992 ? Sept 1997 TASS Hedge Funds Oct 1992 ? Sept 1997 TASS Hedge Funds Oct 1992 ? Sept 1997 TASS Hedge Funds Oct 1992 ? Sept 1997
Sharpe ? 0 ? 1 ? 2 ? 6 N
All Hedge Funds 0.595 0.680 0.748 0.895 411
All Hedge Funds 77.6 74.5 76.6 78.3 84.7 411
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Rank Correlation between Manipulation-Free Measure and Sharpe Ratio Markets Percentile Performance within Class of Funds Rank Correlation between Manipulation-Free Measure and Sharpe Ratio Markets Percentile Performance within Class of Funds Rank Correlation between Manipulation-Free Measure and Sharpe Ratio Markets Percentile Performance within Class of Funds Rank Correlation between Manipulation-Free Measure and Sharpe Ratio Markets Percentile Performance within Class of Funds Rank Correlation between Manipulation-Free Measure and Sharpe Ratio Markets Percentile Performance within Class of Funds Rank Correlation between Manipulation-Free Measure and Sharpe Ratio Markets Percentile Performance within Class of Funds Rank Correlation between Manipulation-Free Measure and Sharpe Ratio Markets Percentile Performance within Class of Funds
CRSP Mutual Funds Oct 1998 ? Sept 2003 CRSP Mutual Funds Oct 1998 ? Sept 2003 CRSP Mutual Funds Oct 1998 ? Sept 2003 CRSP Mutual Funds Oct 1998 ? Sept 2003 CRSP Mutual Funds Oct 1998 ? Sept 2003 CRSP Mutual Funds Oct 1998 ? Sept 2003 CRSP Mutual Funds Oct 1998 ? Sept 2003
Category Sharpe ? 0 ? 1 ? 2 ? 6 N
All Mutual Funds 0.981 0.962 0.886 0.552 3248
All Mutual Funds 42.7 42.5 45.8 49.5 55.7 3248
TASS Hedge Funds Oct 1997 ? Sept 2002 TASS Hedge Funds Oct 1997 ? Sept 2002 TASS Hedge Funds Oct 1997 ? Sept 2002 TASS Hedge Funds Oct 1997 ? Sept 2002 TASS Hedge Funds Oct 1997 ? Sept 2002 TASS Hedge Funds Oct 1997 ? Sept 2002 TASS Hedge Funds Oct 1997 ? Sept 2002
Sharpe ? 0 ? 1 ? 2 ? 6 N
All Hedge Funds 0.765 0.848 0.881 0.856 799
All Hedge Funds 9.3 8.8 10.5 13.5 18.4 799
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Effect of Skewness on Relative Performance
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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.