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The Nonlinearities of Hedge Fund Returns

The Nonlinearities of Hedge Fund Returns. Joint work with E. Derman. raphael.douady@riskdata.com www.riskdata.com +33 1 44 54 35 00. Raphaël Douady Research Director, Riskdata ®. Traditional Equity Fund. FUND realized returns. y = 1,1051x - 0,0071. R. 2. = 0,8909.

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The Nonlinearities of Hedge Fund Returns

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  1. The Nonlinearities of Hedge Fund Returns Joint work with E. Derman • raphael.douady@riskdata.com • www.riskdata.com • +33 1 44 54 35 00 Raphaël Douady Research Director, Riskdata®

  2. Traditional Equity Fund FUND realized returns y = 1,1051x - 0,0071 R 2 = 0,8909 SP500 observed Returns Non Linear Modeling? • Do Hedge Funds support Linear Modeling? Fund Return = a + bx Index Return + e

  3. y = 0,4398x - 0,0215 R 2 = 0,0754 FUND realized returns SP500 observed Returns Non Linear Modeling? • Nonlinear Modeling • Missing nonlinearity  Erroneous a and b • Correlation is more than meaningless: misleading Fund Return = F(Index Return) + e

  4. Testing Nonlinearity • 1000 Hedge Funds • Distribution across Strategies similar to overall HF population • Including Dead Funds and their last return • Monthly Returns • Analysis Period • Jan 95 to Jun 05 (restricted to Fund existence) • Methodology • Select, among investable factors, the most explanatory • F-test of Quadratic and Cubic regression vs. Linear regression • Identify Funds that reject Linear model with Confidence 95%

  5. Hedge Fund Database

  6. Nonlinearity Test • Only ¼ of Hedge Funds support Linear Modeling

  7. 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% VALUE MACRO MISCELANEOUS LONG ONLY SHORT BIAS MORTGAGES COMMODITY DISTRESSED CURRENCIES MARKET TIMER REGULATION D EVENT DRIVEN OPPORTUNISTIC MULTI-STRATEGY FINANCE SECTOR OTHER ARBITRAGE SMALL/MICRO CAP COUNTRY SPECIFIC LONG/SHORT EQUITY EMERGING MARKETS SHORT:TERM TRADING SPECIAL SITUATIONS HEALTHCARE SECTOR OPTIONS STRATEGIES TECHNOLOGY SECTOR CTA/MANAGED FUTURES STATISTICAL ARBITRAGE MERGER/RISK ARBITRAGE MARKET NEUTRAL EQUITY CONVERTIBLE ARBITRAGE FIXED INCOME ARBITRAGE CAPITAL STRUCTURE ARBITRAGE FIXED INCOME (NON-ARBITRAGE) Nonlinearity Test by Strategy

  8. Linear and Nonlinear Strategies • Nonlinear Strategies • Short Bias, Market Timer, Currencies, Tech Sector, Stat Arb., Short Term Trading, CTA, L/S Equity • Dynamic Trading imply Optional Profile (Black-Scholes-Merton) • Convert. Arb., Option Strategies • Nonlinear Instruments • Merger/Risk Arb., Mortgages, Fixed Income Arb. • Correlation break under Liquidity Stress • Linear Strategies • Country specific, Emerging Markets, Commodities, Small Caps • Directional portfolio • Low turnover

  9. 3% 2% Period 30/06/2001 30/06/2005 1% 0% -3% -2% -1% 0% 1% 2% 3% -1% Fund Return -2% 2 y = 25.75x + 0.01x - 0.00 y = 0.07x + 0.00 -3% 2 2 R = 0.14 R = 0.01 -4% Sector Finance USA Nonlinear Profiles • Long/Short Equity with Quadratic Shape Strategy: “to achieve capital appreciation through the application of analytically and statistically based trading strategies, the beta neutral strategy is designed to return 12-14% annually with low drawdowns and low correlation to S&P 500”

  10. Simulation: Trend Follower • Quadratic shape • Negative outliers Strategy: Buy when Index Spot > 1M moving average Sell when Index Spot < 1M moving average

  11. Alternative Factors? • Create “nonlinear” factors • Returns of a Trend Following strategy • Trend Followers and Mean Reverters are now Linear with respect to this “Alternative Factor” • Literature • Naik-Agarval: Factor = returns of an Option Roll-over • Fung-Hsieh: Factor = returns of a Barrier Option • Industry practice: Hedge Fund Indices • Average of HF returns of a given strategy • HFR, EDHEC, etc.

  12. 20% 3 2 y = 66.81x + 2.86x + 0.03x - 0.00 2 R = 0.35 15% Period 31/03/2001 31/07/2005 10% 5% y = 0.50x + 0.00 2 R = 0.25 0% Fund Return -15% -10% -5% 0% 5% 10% 15% -5% -10% -15% COMEX GOLD INDEX Cubic Profile • CTA Strategy:“The primary objective of the Advisor is the capital appreciation of the Company’s assets through the speculation in commodity futures contracts and cash currencies (FX). The Advisor will attempt to meet the objective of capital appreciation by making trading decisions based upon a proprietary trading method. (…)It believes that future price movements in all markets may be more accurately anticipated by historical price movements within a quantitative or technical analysis than by fundamental economic analysis. Since non-directional and limited price directional trading strategies are employed, major long-term price movements are not necessarily needed for the program to be successful. Rather, diverse models that have yielded good risk/reward characteristics in the past are combined with other uncorrelated models to form a robust trading program that is less dependent on any one particular market characteristic”

  13. 20% 15% Period 31/03/2001 31/07/2005 10% 5% 0% Fund Return -15% -10% -5% 0% 5% 10% 15% -5% 2 y = 9.17x - 0.13x - 0.02 -10% 2 R = 0.35 -15% TREND STRATEGY ON COMEX GOLD INDEX Specific Hedge Fund Index • CTA with respect to Alternative Factor • Reverse engineer CTA signal: Trend follower on Trend Following Strategy • Quadratic on Quadratic  Cubic shape

  14. Modeling with Lags • Do Hedge Fund support Modeling without Lags? • Null Hypothesis: The past has no influence on Fund returns Fund Return = F(Index Return) + e • Alternative Hypothesis: Fund returns are sensitive to • Past Fund returns • Past Index returns Fund Rtn = F(Idx Rtn) + G(Past Idx Rtn) + H(Past Fund Rtn) + e • Function H  Serial correlation • Function G  Delayed market impact

  15. Test for Lags • Only ¼ of Hedge Funds support Modeling without Lags

  16. Average Serial Correlation

  17. Serial Correlation • Less liquid strategies display higher serial correlation • Literature • Andy Lo: Many reasons for Serial Correlation • Illiquid Securities are serially correlated • Delayed influence of Markets on Illiquid Securities • Time mismatch between Trades and Fund Reporting • Return smoothing and “Hard to price” securities

  18. Need for Nonlinear Dynamic Model • Hypothesis Testing for the need of Nonlinear functional form and Lagged Factor Values • Models: • LS : Linear Static (without lags) • LD : Linear Dynamic (with lags) • NLS : Nonlinear Static (without lags) • NLD : Nonlinear Dynamic (with lags)

  19. CONCLUSIONS • For 89% of Hedge Funds, the Linear Model without Lags is Rejected with 95% Confidence • Nonlinear Modeling is even more important than Lag Modeling • Nonlinear Model without Lags rejected for ~1/2 of Hedge Funds • Linear Model with Lags rejected for ~ 3/4 of Hedge Funds • Nonlinear Returns ≠ Nonlinear instruments • Not Captured by Holdings analysis • Due to Dynamic Trading and Liquidity Stress • Lagged influence ≠ Serial Correlation≠ Return smoothing • ~1/2 of Hedge Funds are influenced by Past Factor Returns • Smoothing is only one of the explanations for Serial Correlation

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