Journ

1
Journée  Gestion alternative et Imperfections de
marché Hedge Fund Risk ProfilingA
non-linear approach to assess the risk and
optimiseFunds of Hedge Funds allocation.Univer
sité dEvry, 1-2 Avril 2004
? raphael.douady_at_riskdata.com ?
www.riskdata.com ? 33 1 44 54 35 00
Raphaël Douady Research Director, Riskdata
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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

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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

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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
• Proxyt E(HF returnt Factort U ?t-1)

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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

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Evaluation Criteria

• Explanatory Power
• In-sample modelling error
• Fund(t) fa(Factor1(t), , Factorn(t)) e(t)
• a calibrated on the whole analysis period
• Predictive Power
• Out-of-sample modelling error
• Fund(t) fa(t-1)(Factor1(t), , Factorn(t))
  e(t)
• a calibrated on t0, t - 1

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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
• SP500, 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
• SP options
• SP historical and implied Vol
• US T-bond historical vol

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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

Beta 0 does not imply no exposure to Risk Factor
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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)

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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

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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

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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

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Max Correlation
Threshold
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Max Correlation Prediction Correlation
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Max Correlation Selection Rate
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Stepwise Regression Prediction Correlation
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Stepwise Regression P2
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Other Selection Methods

• Non linear regression F-test, Log-likelihood
• Causality (non linear VARMA) F-test
• Cointegration. Non linear factor ? Factt² dt
• P2
• Direction Match
• Joint occurrence of Extreme Moves

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Selection Method Comparison Prediction
Correlation (1 fact)
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Selection Method Comparison Prediction
Correlation
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Selection Method Comparison Selection Rate
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Selection Method Comparison Direction Match
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Missed Selections
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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.

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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

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Correlation of Long-Short Equity Funds to TUNA LS
Index24M slipping period (end indicated)
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Riskdata FoFIX