Jakša Cvitanić, 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
500
Assets (in US$billions)
450
400
350
300
250
200
150
100
50
0
1990 1991 1992 1993 1994 1995 1996 1997 1998 1999
Source: Managing of Hedge Fund Risk, Risk Waters Group, 2000.
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.
Yale University: Asset Allocation (2000)
Bonds
9%
U.S. Stocks
14%
Real Estate
15%
Foreign stocks
9%
Hedge funds
19%
Other
8%
Private Equity
26%
Sources: New York Times, Pensions and Investments, Financial Times, IHT
Motivation
Optimal 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))
Motivation
Risk and Return Trade-Off
Potential Risk and Return Tradeoff
Annual Return
16%-20%
Global Asset
Allocators
14%-16%
Equity Hedge Funds
Equities
10%-12%
8%-10%
Event
Driven
High Yield
Corporate
Corporate
Bonds
Relative
Value
Government
Bonds
4%-6%
Short Term
Gov't Bonds
Annual Standard Deviation
8%-10%
10%-12%
14%-16%
Source: Schneeweis, Spurgin (1999)
16%-20%
Motivation
In-Sample Efficient Frontiers
R isk a nd R e turn o f S to c k, B o nd a nd H e dg e F unds:
J a nua ry, 1 9 9 0 - A pril, 2 0 0 0
P ortfolio Annualized
Return
2 0 .0 0 %
1 8 .0 0 %
1 6 .0 0 %
100% S&P 500
1 0 0 % EA C M 1 0 0
1 4 .0 0 %
1 2 .0 0 %
5 0 % S to c k a n d
1 0 .0 0 %
8 .0 0 %
50%
Bond
100% Leman Bros .
G o v t/C o r p . B o n d
6 .0 0 %
4 .0 0 %
2 .0 0 %
0 .0 0 %
0 .0 0 %
5 .0 0 %
1 0 .0 0 %
P o rtfo lio A nnua lize d S ta nda rd D e v ia tio n
Source: Schneeweis, Spurgin (1999)
1 5 .0 0 %
Motivation
Alpha 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!
Contribution
Empirical 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
Data
2000
Optimal hedge
fund position in
2000
Empirical Testing
Data
• 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 Testing
Asset 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 Testing
HF 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 Testing
HF Styles
• Event driven (distressed sec. and risk
arbitrage)
• Market neutral (arbitrage and long/short)
• Short-sales
• Fund of fund (niche and diversified)
Empirical Testing
Summary Statistics
Strategy
Ev. Dist.
Ev. Risk
Ev. Driven
FoF Div.
FoF Niche
FoF
Mkt Neutr. Arb
Mkt Neutr. L/S
Mkt Neutr.
Short Sale
Average
Beta
0.23
0.14
0.16
0.24
0.15
0.22
0.06
0.04
0.02
-0.91
0.03
Mean Return
10.94
13.14
12.28
12.31
11.87
11.22
16.62
12.01
11.02
6.37
11.78
Volat.
6.56
3.98
4.71
6.26
4.36
5.60
10.58
2.08
1.42
20.71
6.63
• 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 Testing
Alphas
Strategy
Ev. Dist.
Ev. Risk
Ev. Driven
FoF Div.
FoF Niche
FoF
Mkt Neutr. Arb
Mkt Neutr. L/S
Mkt Neutr.
Short Sale
Average
Meth 0
10.42
12.62
11.76
11.80
11.35
10.70
16.10
11.50
10.51
5.85
11.26
Meth 1
2.83
6.26
5.05
4.10
4.83
3.26
10.82
6.46
5.64
13.34
6.26
Meth 2
-0.68
4.67
2.77
0.93
2.32
0.13
9.66
6.45
4.60
13.90
4.48
Meth 3
2.20
5.84
4.55
3.73
4.42
2.86
10.47
6.45
5.53
14.19
6.02
Meth 4
1.53
7.82
5.74
-0.82
5.70
-0.20
7.16
9.50
9.12
1.59
4.72
Meth 5 Mean Alpha St. Dev. Alpha
-0.14
2.69
4.02
6.67
7.31
2.80
4.07
5.66
3.15
-2.01
2.96
4.96
3.26
5.31
3.19
-3.06
2.28
4.72
12.04
11.04
2.96
9.41
8.30
2.15
8.61
7.33
2.39
31.57
13.41
10.27
7.04
6.63
2.47
• Large deviation around alpha estimate
• This is a measure of model risk
Empirical Testing
Cross-Section of Average Alphas
average alpha
30
26
22
18
14
10
6
2
-2
-6
-1
0
-1
4
-1
8
-2
2
-2
6
50
40
30
20
10
0
-3
0
number of funds
Distribution of Average Alpha Across
Hedge Funds
Empirical Testing
Cross-Section of Standard Deviation of Alphas
150
100
50
dispersion of alpha across models
30
27
24
21
18
15
12
9
6
3
0
0
number of funds
Distribution of Standard Deviation of
Alpha Across Hedge Funds
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 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 Risk
Base Case – FOF Niche
FOF Niche
Sharpe ratio
No HF, no uncertainty
No uncertainty
Model uncertainty
SP500
0.82
31.83
-2.56
27.25
hedge fund
1.56
0.00
229.31
30.57
risk free
68.17
-126.75
42.19
Focusing on Model Risk
Base Case – Ev. Distr
Ev. Dist.
Sharpe ratio
No HF, no uncertainty
No uncertainty
Model uncertainty
SP500
0.82
31.83
17.83
29.70
hedge fund
0.90
0.00
60.88
9.3
risk free
68.17
21.29
61.00
Focusing on Model Risk
Base Case – Mkt Neutral Arbitrage
Mkt Neut Arb
Sharpe ratio
No HF, no uncertainty
No uncertainty
Model uncertainty
SP500
0.82
31.83
28.20
29.69
hedge fund
1.09
0.00
60.59
35.71
risk free
68.17
11.21
34.60
Focusing on Model Risk
Base Case – Mkt Neutral Long/Short
Mkt Neut Long/Short
Sharpe ratio
No HF, no uncertainty
No uncertainty
Model uncertainty
SP500
0.82
31.83
-8.72
27.45
hedge fund
3.34
0.00
1013.74
109.68
risk free
68.17
-905.02
-37.13
Focusing on Model Risk
Base Case – FOF Div
FOF Div
Sharpe ratio
No HF, no uncertainty
No uncertainty
Model uncertainty
SP500
0.82
31.83
6.80
30.09
hedge fund
1.16
0.00
104.31
7.25
risk free
68.17
-11.11
62.66
Focusing on Model Risk
Base Case – Short Sale
Short sale
Sharpe ratio
No HF, no uncertainty
No uncertainty
Model uncertainty
SP500
0.82
31.83
66.92
38.17
hedge fund
0.06
0.00
38.56
6.96
risk free
68.17
-5.48
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
Strategy
Ev. Dist.
Ev. Risk
Ev. Driven
FoF Div.
FoF Niche
FoF
Mkt Neutr. Arb
Mkt Neutr. L/S
Mkt Neutr.
Short Sale
Av. Fund
Holding in Passive
15.36%
12.68%
13.75%
15.57%
14.13%
15.75%
15.42%
14.37%
15.42%
19.62%
16.14%
Holding in Active
4.68%
27.23%
16.37%
3.63%
15.35%
3.14%
18.17%
54.92%
41.50%
3.50%
8.48%
Relative Holding A versus P
23.36%
68.22%
54.34%
18.90%
52.07%
16.62%
54.09%
79.26%
72.91%
15.13%
34.44%
Holding in Risk-Free
79.97%
60.08%
69.88%
80.80%
70.52%
81.12%
66.41%
30.71%
43.08%
76.89%
75.38%
Delta Passive
7.39%
10.06%
8.99%
7.18%
8.62%
7.00%
7.33%
8.38%
7.33%
3.13%
6.61%
Delta Risk-Free Risk-Free
-2.71%
17.17%
7.38%
-3.55%
6.73%
-3.86%
10.84%
46.54%
34.17%
0.37%
1.87%
• 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 Risk
Impact of Biases: Mean Alpha – 4.5%
Strategy
Ev. Dist.
Ev. Risk
Ev. Driven
FoF Div.
FoF Niche
FoF
Mkt Neutr. Arb
Mkt Neutr. L/S
Mkt Neutr.
Short Sale
Av. Fund
Holding in Passive
33.28%
28.97%
30.75%
32.74%
31.14%
33.18%
30.66%
29.96%
31.06%
36.04%
31.65%
Holding in Active
-6.24%
20.87%
6.66%
-3.78%
4.68%
-6.08%
21.14%
50.12%
32.05%
4.62%
5.42%
Relative Holding A versus P
-23.10%
41.87%
17.80%
-13.06%
13.06%
-22.42%
40.81%
62.59%
50.78%
11.37%
14.61%
Holding in Risk-Free
72.96%
50.16%
62.59%
71.04%
64.18%
72.90%
48.20%
19.92%
36.90%
59.33%
62.93%
Delta Passive
10.79%
15.11%
13.32%
11.33%
12.93%
10.90%
13.41%
14.12%
13.02%
8.03%
12.42%
Delta Risk-Free Risk-Free
-17.04%
5.76%
-6.66%
-15.11%
-8.26%
-16.97%
7.73%
36.00%
19.03%
-3.41%
-7.01%
• 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
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