Center for Computational Finance Hedge Fund Risk Profiling: A non-linear approach to assess the risk and optimise Funds of Hedge Funds allocation. Carnegie Mellon University, March 21, 2005 Raphaël Douady Research Director, Riskdata ®    

2 CMU, March 21, 2005 Computational Finance Seminar INVESTOR’S PROBLEM 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 CMU, March 21, 2005 Computational Finance Seminar INVESTOR’S PROBLEM Risk Transparency Beyond past performance, can we anticipate market situations which can kill us? What information can we derive from past returns?

4 CMU, March 21, 2005 Computational Finance Seminar 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

5 CMU, March 21, 2005 Computational Finance Seminar Correlation of Long-Short Equity Funds to TUNA LS Index 24M slipping period (end indicated)

6 CMU, March 21, 2005 Computational Finance Seminar Modelling vs. Index  = 0  > 0 Beta = 0 does not imply no exposure to Risk Factor  2002  2003

7 CMU, March 21, 2005 Computational Finance Seminar Hedge Fund Modelling General 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 )

8 CMU, March 21, 2005 Computational Finance Seminar What Statistical Model for H.F. Single factor vs. Multi-factor Question: Which factor set? Linear vs. Non-linear Question: What type of non-linear modelling? Instantaneous info vs. Lagged Question: Number of periods for the Fund? For the Factors? Return series vs. Integrated series Extreme moves modelling Question: Which criterion for "extreme"

9 CMU, March 21, 2005 Computational Finance Seminar What Statistical Model for H.F. Single FactorMulti-Factor Linear CAPM: Poor explanation + Misleading Max correlation: Stable and Robust, but can miss explanation + no aggregation Factor set for each Class of strategy  no aggregation General Factor Set  Spurius analysis Stepwise Regression  Still spurious! Non- linear Collection of Pairwise Non-linear Models: Optimal trade-off Explanation vs. Complexity General Non-linear Multi- factor Representation  Too many parameters  Spurious Analysis

10 CMU, March 21, 2005 Computational Finance Seminar 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 CMU, March 21, 2005 Computational Finance Seminar Stepwise Regression Start from large Factor base Equity indices (country, sector, style…), Fixed income, etc. Select a small number of factors F 1 … F n such that R2 is maximum Start with most correlated factor Include factor that increases R2 the most, etc. Stop when increase is too small. Remove factors that decrease R2 the less. Stop when decrease is too large Continue until we can neither include nor remove factors. Set R2 threshold so that n be in chosen range (3 - 6 factors)

12 CMU, March 21, 2005 Computational Finance Seminar 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]

13 CMU, March 21, 2005 Computational Finance Seminar Explanation Power Evaluation Criteria R2 R2 = Var(Explained) / Var(Return) Other formula for R2 Var(Return) = Var(Explained) + Var(Error) R2 = 1 – Var(Error) / Var(Return) Spurious Selections act Positively Var(Explained) = Var(Really Explained) + Var(Spurious) R2 = Real R2 + Var(Spurious) / Var(Return)

14 CMU, March 21, 2005 Computational Finance Seminar 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

15 CMU, March 21, 2005 Computational Finance Seminar Prediction Power Correlation between Predicted Series and Actual Returns Not influenced by Spurious Selections Prediction Power P2 P2 = 1 – Var(Error) / Var(Return) Spurious Selections act Negatively Var(Error) = Var(Specific) + Var(Spurious) P2 = Real R2 – Var(Spurious) / Var(Return) Direction Match Probability Probability that Actual Return has the same sign as the Prediction Biased if the the Fund average return is ≠ 0 Unbiased measure: Correlation of Sign Series

16 CMU, March 21, 2005 Computational Finance Seminar 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…

17 CMU, March 21, 2005 Computational Finance Seminar 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 No Hedge Fund Indices

18 CMU, March 21, 2005 Computational Finance Seminar Stepwise Regression

19 CMU, March 21, 2005 Computational Finance Seminar Maximum Correlation Select, in each time period, the factor that is the most correlated to the fund Eliminate periods with a correlation below some threshold (positive or negative) Regress returns on the selected factor Compute Return Prediction

20 CMU, March 21, 2005 Computational Finance Seminar Max Correlation Threshold

21 CMU, March 21, 2005 Computational Finance Seminar Maximum Correlation

22 CMU, March 21, 2005 Computational Finance Seminar Maximum Correlation

23 CMU, March 21, 2005 Computational Finance Seminar Stepwise Regression Max Correlation

24 CMU, March 21, 2005 Computational Finance Seminar 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

25 CMU, March 21, 2005 Computational Finance Seminar Multiple Pairwise Analysis Select a factor is at least one of the statistical tests is positive Compute a different prediction for each factor Measure the prediction uncertainty Compute the MLE estimate of the fund return, knowing Each single-factor prediction + uncertainty Factor correlation structure Compare to actual Fund return

26 CMU, March 21, 2005 Computational Finance Seminar OUT OF SAMPLE TEST

27 CMU, March 21, 2005 Computational Finance Seminar YES: RISK PROFILING

28 CMU, March 21, 2005 Computational Finance Seminar Findings Classical Linear methods are either often spurious (stepwise regression) or miss essential factors (correlation) Non linear modelling is necessary Statistical factors, such as Hist. Vol., Correl Index, etc. explain 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.

29 CMU, March 21, 2005 Computational Finance Seminar 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