Jak

1
Jakša Cvitanic, Ali Lazrak, Lionel Martellini
and Fernando Zapatero

• Dynamic Portfolio Choice with Parameter
  Uncertainty

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Motivation The Growth of Hedge Fund Investing

• Growth of Hedge Fund Investing

Assets (in USbillions)
Source Managing of Hedge Fund Risk, Risk Waters
Group, 2000.
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Motivation Hedge 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
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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))

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Motivation Risk and Return Trade-Off
Source Schneeweis, Spurgin (1999)
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Motivation In-Sample Efficient Frontiers
Source Schneeweis, Spurgin (1999)
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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!

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Contribution Empirical Contribution

• The uncertainty is coming from three sources
• Model risk Investors have not a dogmatic
  beliefs in one particular risk adjusted
  performance measure
• Estimation risk Investors are aware that their
  estimators 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

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

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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.

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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)

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Empirical TestingHF Styles

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

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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

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Empirical TestingAlphas

• Large deviation around alpha estimate
• This is a measure of model risk

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Empirical TestingCross-Section of Average Alphas
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Empirical Testing Cross-Section of Standard
Deviation of Alphas
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Focusing on Model Risk Base 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 SP500 mP 18.23
• SP500 volatility sP 16.08
• Assume away sample risk dP 0
• Time-horizon T10
• Risk-aversion a -15
• This is consistent with a (68.2,31.8) Merton
  allocation to the risk-free versus risky asset

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Focusing on Model Risk Base Case FOF Niche
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Focusing on Model Risk Base Case Ev. Distr
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Focusing on Model Risk Base Case Mkt Neutral
Arbitrage
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Focusing on Model Risk Base Case Mkt Neutral
Long/Short
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Focusing on Model Risk Base Case FOF Div
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Focusing on Model Risk Base Case Short Sale
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Focusing on Model Risk Base 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

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Focusing on Model Risk Impact 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

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Focusing on Model Risk Impact 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

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Conclusion Recap

• 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

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Conclusion Further 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