Jakša Cvitanić, Ali Lazrak, Lionel Martellini and Fernando Zapatero

Dynamic Portfolio Choice with Parameter Uncertainty

Growth of Hedge Fund Investing

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1990 1991 1992 1993 1994 1995 1996 1997 1998 1999

Ass

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Source: Managing of Hedge Fund Risk, Risk Waters Group, 2000.

Motivation The Growth of Hedge Fund Investing

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

Yale University: Asset Allocation (2000)

Foreign stocks9%

Hedge funds19%

Other8%

Private Equity26%

Real Estate15%

U.S. Stocks14%

Bonds9%

Motivation Hedge Fund in Institutional Portfolios

• 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))

MotivationOptimal HF Allocation

Potential Risk and Return Tradeoff

Annual Return

16%-20% Global AssetAllocators

14%-16% Equity Hedge Funds

Equities10%-12% Event

Driven High YieldCorporate

Corporate8%-10% Relative Bonds

ValueGovernmentBonds

4%-6% Short Term Gov't Bonds

Annual Standard Deviation8%-10% 10%-12% 14%-16% 16%-20%

Motivation Risk and Return Trade-Off

Source: Schneeweis, Spurgin (1999)

Risk and Return of Stock, Bond and Hedge Funds: January, 1990 - April, 2000

100% EACM 100

50% Stock and 50% Bond

100% Leman Bros. Govt/Corp. Bond

100% S&P 500

0.00%2.00%4.00%6.00%8.00%

10.00%12.00%14.00%16.00%18.00%20.00%

0.00% 5.00% 10.00% 15.00%

Portfolio Annualized Standard Deviation

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Motivation In-Sample Efficient Frontiers

Source: Schneeweis, Spurgin (1999)

• 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!

MotivationAlpha Uncertainty

• 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

Contribution Empirical Contribution

Calibration Data based prior

1996 2000

2000-prior parameters calibration

Optimal hedge fund position in 2000

Data

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

Strategy Beta Mean Return Volat.Ev. Dist. 0.23 10.94 6.56Ev. Risk 0.14 13.14 3.98

Ev. Driven 0.16 12.28 4.71FoF Div. 0.24 12.31 6.26

FoF Niche 0.15 11.87 4.36FoF 0.22 11.22 5.60

Mkt Neutr. Arb 0.06 16.62 10.58Mkt Neutr. L/S 0.04 12.01 2.08

Mkt Neutr. 0.02 11.02 1.42Short Sale -0.91 6.37 20.71Average 0.03 11.78 6.63

Empirical TestingAlphas

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

Strategy Meth 0 Meth 1 Meth 2 Meth 3 Meth 4 Meth 5 Mean Alpha St. Dev. AlphaEv. Dist. 10.42 2.83 -0.68 2.20 1.53 -0.14 2.69 4.02Ev. Risk 12.62 6.26 4.67 5.84 7.82 6.67 7.31 2.80

Ev. Driven 11.76 5.05 2.77 4.55 5.74 4.07 5.66 3.15FoF Div. 11.80 4.10 0.93 3.73 -0.82 -2.01 2.96 4.96

FoF Niche 11.35 4.83 2.32 4.42 5.70 3.26 5.31 3.19FoF 10.70 3.26 0.13 2.86 -0.20 -3.06 2.28 4.72

Mkt Neutr. Arb 16.10 10.82 9.66 10.47 7.16 12.04 11.04 2.96Mkt Neutr. L/S 11.50 6.46 6.45 6.45 9.50 9.41 8.30 2.15

Mkt Neutr. 10.51 5.64 4.60 5.53 9.12 8.61 7.33 2.39Short Sale 5.85 13.34 13.90 14.19 1.59 31.57 13.41 10.27Average 11.26 6.26 4.48 6.02 4.72 7.04 6.63 2.47

Empirical TestingCross-Section of Average Alphas

Distribution of Average Alpha Across Hedge Funds

01020304050

-30

-26

-22

-18

-14

-10 -6 -2 2 6 10 14 18 22 26 30

average alpha

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Empirical Testing Cross-Section of Standard Deviation of Alphas

Distribution of Standard Deviation of Alpha Across Hedge Funds

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0 3 6 9 12 15 18 21 24 27 30

dispersion of alpha across models

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Focusing on Model Risk Base Case - Parameter Values

• Use variance of alphas across models as an estimate of Ax

2

• Base case parameter values– Risk-free rate: r = 5.06%

– Expected return on the S&P500: P =18.23%

– S&P500 volatility: P = 16.08%

– Assume away sample risk: P = 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 Risk Base Case – FOF Niche

FOF Niche SP500 hedge fund risk freeSharpe ratio 0.82 1.56

No HF, no uncertainty 31.83 0.00 68.17No uncertainty -2.56 229.31 -126.75

Model uncertainty 27.25 30.57 42.19

Focusing on Model Risk Base Case – Ev. Distr

Ev. Dist. SP500 hedge fund risk freeSharpe ratio 0.82 0.90

No HF, no uncertainty 31.83 0.00 68.17No uncertainty 17.83 60.88 21.29

Model uncertainty 29.70 9.3 61.00

Focusing on Model Risk Base Case – Mkt Neutral Arbitrage

Mkt Neut Arb SP500 hedge fund risk freeSharpe ratio 0.82 1.09

No HF, no uncertainty 31.83 0.00 68.17No uncertainty 28.20 60.59 11.21

Model uncertainty 29.69 35.71 34.60

Focusing on Model Risk Base Case – Mkt Neutral Long/Short

Mkt Neut Long/Short SP500 hedge fund risk freeSharpe ratio 0.82 3.34

No HF, no uncertainty 31.83 0.00 68.17No uncertainty -8.72 1013.74 -905.02

Model uncertainty 27.45 109.68 -37.13

Focusing on Model Risk Base Case – FOF Div

FOF Div SP500 hedge fund risk freeSharpe ratio 0.82 1.16

No HF, no uncertainty 31.83 0.00 68.17No uncertainty 6.80 104.31 -11.11

Model uncertainty 30.09 7.25 62.66

Focusing on Model Risk Base Case – Short Sale

Short sale SP500 hedge fund risk freeSharpe ratio 0.82 0.06

No HF, no uncertainty 31.83 0.00 68.17No uncertainty 66.92 38.56 -5.48

Model uncertainty 38.17 6.96 54.87

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

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

Strategy Holding in Passive Holding in Active Relative Holding A versus P Holding in Risk-Free Delta Passive Delta Risk-Free Risk-FreeEv. Dist. 15.36% 4.68% 23.36% 79.97% 7.39% -2.71%Ev. Risk 12.68% 27.23% 68.22% 60.08% 10.06% 17.17%

Ev. Driven 13.75% 16.37% 54.34% 69.88% 8.99% 7.38%FoF Div. 15.57% 3.63% 18.90% 80.80% 7.18% -3.55%

FoF Niche 14.13% 15.35% 52.07% 70.52% 8.62% 6.73%FoF 15.75% 3.14% 16.62% 81.12% 7.00% -3.86%

Mkt Neutr. Arb 15.42% 18.17% 54.09% 66.41% 7.33% 10.84%Mkt Neutr. L/S 14.37% 54.92% 79.26% 30.71% 8.38% 46.54%

Mkt Neutr. 15.42% 41.50% 72.91% 43.08% 7.33% 34.17%Short Sale 19.62% 3.50% 15.13% 76.89% 3.13% 0.37%Av. Fund 16.14% 8.48% 34.44% 75.38% 6.61% 1.87%

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

Strategy Holding in Passive Holding in Active Relative Holding A versus P Holding in Risk-Free Delta Passive Delta Risk-Free Risk-FreeEv. Dist. 33.28% -6.24% -23.10% 72.96% 10.79% -17.04%Ev. Risk 28.97% 20.87% 41.87% 50.16% 15.11% 5.76%

Ev. Driven 30.75% 6.66% 17.80% 62.59% 13.32% -6.66%FoF Div. 32.74% -3.78% -13.06% 71.04% 11.33% -15.11%

FoF Niche 31.14% 4.68% 13.06% 64.18% 12.93% -8.26%FoF 33.18% -6.08% -22.42% 72.90% 10.90% -16.97%

Mkt Neutr. Arb 30.66% 21.14% 40.81% 48.20% 13.41% 7.73%Mkt Neutr. L/S 29.96% 50.12% 62.59% 19.92% 14.12% 36.00%

Mkt Neutr. 31.06% 32.05% 50.78% 36.90% 13.02% 19.03%Short Sale 36.04% 4.62% 11.37% 59.33% 8.03% -3.41%Av. Fund 31.65% 5.42% 14.61% 62.93% 12.42% -7.01%

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

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