Journée « Gestion alternative et Imperfections de marché » Hedge Fund Risk Profiling: A non-linear approach to assess the risk and optimise Funds of Hedge Funds allocation. Université d’Evry, 1-2 Avril 2004 Raphaël Douady Research Director, Riskdata ®   

2 Evry, April 1, 2004 Journée Gestion Alternative Hedge Fund Modelling The Investor Problem What is the most likely Hedge Fund behaviour under the various market conditions? What factor or event can put the Hedge Fund at risk? Is the risk of a portfolio well diversified across the funds Goal Build and Rebalance portfolio of Hedge Funds Select new Hedge Funds to invest in

3 Evry, April 1, 2004 Journée Gestion Alternative Hedge Fund Modelling Hedge Funds form asset class different from others Apparent Statistical Instability Structural Non-linearity stemming from Dynamic Trading Usual market factors inefficient to explain returns Seldom and imprecise information: Net Asset Value (weekly or monthly, delayed in all cases) Exposure and sensitivity report Position transparency only in some cases

4 Evry, April 1, 2004 Journée Gestion Alternative Hedge Fund Modelling Methodology Determine a set of Factors that define the “Market” Identify, for each Hedge Fund, the Factors that do impact the returns Build a Proxy of the fund, as a function of each Selected Factor, or of the subset of them HF return = Proxy + Prediction error Proxy t = E(HF return t | Factor t U  t-1 )

5 Evry, April 1, 2004 Journée Gestion Alternative What Statistical Model for H.F. Single factor vs. Multi-factor Factor choice? Linear vs. Non-linear Non-linear modelling? Instantaneous info vs. Lagged Number of periods for the Fund? For the Factors? Return series vs. Integrated series Extreme moves modelling

6 Evry, April 1, 2004 Journée Gestion Alternative Evaluation Criteria Explanatory Power In-sample modelling error Fund(t) = f  (Factor 1 (t), …, Factor n (t)) +  (t)  calibrated on the whole analysis period Predictive Power Out-of-sample modelling error Fund(t) = f  (t-1) (Factor 1 (t), …, Factor n (t)) +  (t)  calibrated on [t 0, t - 1]

7 Evry, April 1, 2004 Journée Gestion Alternative Explanation Power R-square obtained with a Set of 25 Factors – Linear Reg. TUNA Hedge Fund Indices Selection of best combination of 5 factors Factor set: S&P500, size/style indices Corp. Bond and HY indices US Libor, bond curve, swap curve MSCI World, Emerging markets Fama-French FX Basket Commodity index, Gold, Oil S&P options S&P historical and implied Vol US T-bond historical vol

8 Evry, April 1, 2004 Journée Gestion Alternative Risk Profiling: Pairwise Analysis Question: Does the Risk Factor significantly impacts the fund returns? Statistical Inference Sensitivity (Beta) Convexity (Gamma) Directional Sensitivities Conditional Statistics under Up and Down hypotheses  = 0  > 0 Beta = 0 does not imply no exposure to Risk Factor

9 Evry, April 1, 2004 Journée Gestion Alternative Evaluation Criteria Prediction Power Correlation between Predicted Series and Actual Returns Direction Match Probability Biased if the the Fund average return is ≠ 0 Unbiased measure: Correlation of Sign Series Prediction Power P2 P2 = 1 – Var(Error) / Var(Return) Negatively biased because of Spurious Selections Var(Error) = Var(Specific) + Var(Spurious)

10 Evry, April 1, 2004 Journée Gestion Alternative State of the Art Maximum Correlation Select, in a set of market factors, the factor that is the most correlated to the fund Proxy the fund by linear regression with respect to this factor Factor Model / Style Analysis Determine a fixed factor set Size limited to the number of data points Multi-dimensional regression of the Fund returns on this set Constrain by positive weights for stability (only with directional funds) Stepwise Regression Factor set Not Limited Exposed to Spurious Selections Still Linear

11 Evry, April 1, 2004 Journée Gestion Alternative Testing Procedure Test Pannel (250 funds) Directional: 75 Non directional: 64 Arbitrage: 32 Special/Event: 24 Aggregates: 23 Other: 22 Random: 10 Hedge Fund Analysis 3Y slipping window Monthly returns [Jan 99 – Dec 01] to[Jan 01 – Dec 03] Factor set ~200 factors Equity, IR, Commodity, FX… Volatility, Correlation, Trend…

12 Evry, April 1, 2004 Journée Gestion Alternative Overview of Riskdata ® Factor Set Market Variables Equity Indices: main, sectors, size, style, individual equity Fixed Income: Interest rates, Gov. bond yields, swap rates, credit spreads, high yield return indices, etc. Commodities: energy, metals, food FX, FX baskets Emerging markets Implied volatilities, implied correlation indices Market Rolling Statistics Historical volatilities Historical volatility indices Historical correlations Historical correlation indices Combinations and Spreads Equity: Size/Style vs. Main index, Sector vs. Main index Fixed Income: YC slope/butterfly, Bonds vs. Swaps, Credit spreads, etc. Implied volatility vs. statistical Simulated Strategies Dynamic portfolios Trend/Revert strategy Strategies involving options Lagged Series Hedge Fund Indices

13 Evry, April 1, 2004 Journée Gestion Alternative Max Correlation Threshold

14 Evry, April 1, 2004 Journée Gestion Alternative Max Correlation: Prediction Correlation

15 Evry, April 1, 2004 Journée Gestion Alternative Max Correlation: Selection Rate

16 Evry, April 1, 2004 Journée Gestion Alternative Stepwise Regression: Prediction Correlation

17 Evry, April 1, 2004 Journée Gestion Alternative Stepwise Regression: P2

18 Evry, April 1, 2004 Journée Gestion Alternative Other Selection Methods Non linear regression: F-test, Log-likelihood Causality (non linear VARMA): F-test Cointegration. Non linear factor: ∫ Fact t ² dt P2 Direction Match Joint occurrence of Extreme Moves

19 Evry, April 1, 2004 Journée Gestion Alternative Selection Method Comparison: Prediction Correlation (1 fact)

20 Evry, April 1, 2004 Journée Gestion Alternative Selection Method Comparison: Prediction Correlation

21 Evry, April 1, 2004 Journée Gestion Alternative Selection Method Comparison: Selection Rate

22 Evry, April 1, 2004 Journée Gestion Alternative Selection Method Comparison: Direction Match

23 Evry, April 1, 2004 Journée Gestion Alternative Missed Selections

24 Evry, April 1, 2004 Journée Gestion Alternative Findings Classical Linear methods are either often spurious (stepwise regression) or miss essential afctors (correlation) Non linear modelling is necessary Statistical factors, such as Hist. Vol., Correl Index, etc. expalin a lot of hedge fund returns Causality is efficient because of Lagged series Co-integration is useful to find the “right” factor, but not for prediction capabilities. Dickey-Fuller mean reversion test worsen statistics Direction match probability test good for “event” type strategies Large factor shifts should be analysed differently: use the frequency of joint large move occurrence between the fund and the factor.

25 Evry, April 1, 2004 Journée Gestion Alternative Conclusion Performance Analysis + Correlations are insufficient for the construction of Portfolios of of Hedge Fund A Complete Set of Risk Factors contains Factors that replicate Dynamic Strategies Sensitive to Volatility and Correlation of Assets Include Non-linear Features Hedge Funds must be Proxied by Non-linear functions of Factors Building a Risk Profile is the only way to identify Market Conditions under which Funds over/under-perform This is also the only way to extract Stable information from Return series

26 Evry, April 1, 2004 Journée Gestion Alternative Correlation of Long-Short Equity Funds to TUNA LS Index 24M slipping period (end indicated)

27 Evry, April 1, 2004 Journée Gestion Alternative Riskdata FoFIX ®