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Jakša Cvitani ć , Ali Lazrak, Lionel Martellini and Fernando Zapatero. Dynamic Portfolio Choice with Parameter Uncertainty. Motivation The Growth of Hedge Fund Investing. Growth of Hedge Fund Investing. Assets (in US$billions). Source: Managing of Hedge Fund Risk, Risk Waters Group, 2000.
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Jakša Cvitanić, Ali Lazrak, Lionel Martellini and Fernando Zapatero Dynamic Portfolio Choice with Parameter Uncertainty
MotivationThe Growth of Hedge Fund Investing Growth of Hedge Fund Investing Assets (in US$billions) Source: Managing of Hedge Fund Risk, Risk Waters Group, 2000.
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
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))
MotivationRisk and Return Trade-Off Source: Schneeweis, Spurgin (1999)
MotivationIn-Sample Efficient Frontiers Source: Schneeweis, Spurgin (1999)
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!
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
Calibration Data based prior 2000-prior parameters calibration 1996 2000 Data Optimal hedge fund position in 2000
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
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.
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)
Empirical TestingHF Styles • Event driven (distressed sec. and risk arbitrage) • Market neutral (arbitrage and long/short) • Short-sales • Fund of fund (niche and diversified)
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
Empirical TestingAlphas • Large deviation around alpha estimate • This is a measure of model risk
Empirical TestingCross-Section of Standard Deviation of Alphas
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
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
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
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
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
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