Jakša Cvitani ć , Ali Lazrak, Lionel Martellini and Fernando Zapatero

1. Jakša Cvitanić, Ali Lazrak, Lionel Martellini and Fernando Zapatero Dynamic Portfolio Choice with Parameter Uncertainty

2. MotivationThe Growth of Hedge Fund Investing Growth of Hedge Fund Investing Assets (in US$billions) Source: Managing of Hedge Fund Risk, Risk Waters Group, 2000.

3. MotivationHedge Fund in Institutional Portfolios Recently, a substantial number of large U.S. and non-U.S. institutions California Public Employees Retirement System, Northeastern University, Nestlé and UK Coal Pension and Yale University have indicated their continued interest in hedge fund investment. Sources: New York Times, Pensions and Investments, Financial Times, IHT

4. MotivationOptimal HF Allocation • Question: is 19% a reasonable number? • Positive answer: most people would argue for a 10 to 20% allocation to hedge funds • Normative answer: only available through static in-sample mean-variance analysis • Problems • Theoretical problems: • Static • In-sample results • Mean-variance • Empirical problems: tangent portfolio (highest Sharpe ratio) is close to 100% in HFs • Do we believe this? • Expected returns and volatility do not tell the whole story • Huge uncertainty on estimates of expected returns (Merton (1980))

5. MotivationRisk and Return Trade-Off Source: Schneeweis, Spurgin (1999)

6. MotivationIn-Sample Efficient Frontiers Source: Schneeweis, Spurgin (1999)

7. MotivationAlpha Uncertainty • Academic consensus that traditional active strategies under-perform passive investment strategies • Jensen (1968), Brown and Goeztman (1995) or Carhart (1997), among many others • Evidence more contrasted for hedge fund returns • Agarwal and Naik (2000a, 2000b, 2001), Brown and Goetzmann (1997, 2001), Fung and Hsieh (1997a, 1997b, 2000), • If positive alphas exist (risk adjusted performance), they are certainly difficult to estimate!

8. ContributionEmpirical Contribution • The uncertainty is coming from three sources : • Model risk : Investor’s have not a dogmatic beliefs in one particular risk adjusted performance measure • Estimation risk : Investor’s are aware that their estimator’s are not perfect • Selection risk : The persistence issue… • We calibrate and test the model by using a proprietary data base • Individual hedge fund monthly returns • We focus on indexes (until now) • Preliminary results: For “reasonable” values of the parameters, our results show • When incorporating Bayesian portfolio performance evaluation, allocation to hedge funds typically decreases substantially an approaches more acceptable values. • Overall, hedge fund allocation appears as a good substitute for a fraction of the investment in risk-free asset

9. Calibration Data based prior 2000-prior parameters calibration 1996 2000 Data Optimal hedge fund position in 2000

10. Empirical TestingData • Use a proprietary data base of individual hedge fund managers, the MAR database. • The MAR database contains more than 1,500 funds re-grouped in 9 categories (“medians”) • We focus on the sub-set of 581 hedge funds + 8 indices funds in the MAR database that have performance data as early as 1996

11. Empirical TestingAsset Pricing Models • We use 5 different pricing models to compute a fund abnormal return • Meth 1: CAPM. • Meth 2: CAPM with stale prices. • Meth 3: CAPM with non-linearities • Meth 4: Explicit single-index factor model. • Meth 5: Explicit multi-index factor model. • We also consider Meth 0: alpha = excess mean return • This is the common practice for HF managers who use risk-free rate as a benchmark. • OK only if CAPM is the true model and beta is zero.

12. Empirical TestingHF Indices • We apply these 6 models to hedge fund indices (as opposed to individual hedge funds) on the period 1996-2000 to estimate the alpha • These indices represent the return on an equally-weighted portfolio of hedge funds pursuing different styles • We also consider an “average” fund, with characteristics equal to the average of the characteristics of these indices (preliminary construction)

13. Empirical TestingHF Styles • Event driven (distressed sec. and risk arbitrage) • Market neutral (arbitrage and long/short) • Short-sales • Fund of fund (niche and diversified)

14. Empirical TestingSummary Statistics • Note the negative beta on short-sales, and the zero beta on market neutral • Risk-return trade-off on market-neutral looks very good

15. Empirical TestingAlphas • Large deviation around alpha estimate • This is a measure of model risk

16. Empirical TestingCross-Section of Average Alphas

17. Empirical TestingCross-Section of Standard Deviation of Alphas

18. Focusing on Model RiskBase Case - Parameter Values • Use variance of alphas across models as an estimate of dAxs22 • Base case parameter values • Risk-free rate: r = 5.06% • Expected return on the S&P500: mP =18.23% • S&P500 volatility: sP= 16.08% • Assume away sample risk: dP = 0 • Time-horizon: T=10 • Risk-aversion: a = -15 • This is consistent with a (68.2%,31.8%) Merton allocation to the risk-free versus risky asset

19. Focusing on Model RiskBase Case – FOF Niche

20. Focusing on Model RiskBase Case – Ev. Distr

21. Focusing on Model RiskBase Case – Mkt Neutral Arbitrage

22. Focusing on Model RiskBase Case – Mkt Neutral Long/Short

23. Focusing on Model RiskBase Case – FOF Div

24. Focusing on Model RiskBase Case – Short Sale

25. Focusing on Model RiskBase Case - Results • We find an optimal 16.86% allocation to alternative investments when the average hedge fund is considered • Substitute as a fraction of the risk-free asset to the hedge fund

26. Focusing on Model RiskImpact of Risk-Aversion: a=-30 • This value is consistent with a (83.6%,16.4%) Merton allocation to the risk-free versus risky asset • We find that the average fund generates a 8.48% to hedge funds (versus 16.86% for the base case) • Again, money is taken away from risk-free asset

27. Focusing on Model RiskImpact of Biases: Mean Alpha – 4.5% • This is a reasonable estimate of magnitude of data base biases • We find that the average fund generates a 5.42% to hedge funds (versus 16.86% for the base case) • Again, money is taken away from risk-free asset

28. ConclusionRecap • We obtain data based predictions on optimal allocation to alternative investments incorporating uncertainty on risk adjusted performance measure (a proxy for managerial skill) • That fraction • Is much larger for a short-term investor • Decreases with risk-aversion • Decreases when biases are accounted for • It is not dramatically affected by introduction of estimation risk and the model risk effect is more important • Overall, hedge fund allocation appears as a good substitute for a fraction of the investment in risk-free asset

29. ConclusionFurther Research • This paper is only a preliminary step toward modeling active vs passive portfolio management with the nice continuous time analytical tool • In particular, the analysis could be more realistic and • accounts for the presence of various kinds of frictions, such as lockup periods and liquidity constraints, • accounts for the presence of various kinds of constraints such as tracking error or VaR constraints • Finally, it would be interesting to address the following related issues: 1)model the active management process 2) analyze the passive and active investment problem in an equilibrium setting