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Comparison with Bayes-Stein Approach
I Bayes-Stein portfolio
wBS = φBS wM IN + (1 − φBS) wM V , (11)
where φBS =
N + (N +2)+T (ˆμ−μM IN 1 N )>Σ−^1 (ˆμ−μM IN 1 N )
I Multi-prior portfolio
wM P = φM P wM IN + (1 − φM P ) wM V , (12)
where φM P () =
( (^) √ ε γσ P∗ +
√ ε
)
=
√
(T −1)N T (T −N )
γσ∗ P +
√
(T −1)N T (T −N )
Example 3 - General Case
Uncertainty about expected returns estimated for subsets of
assets
I Let Jm = {i 1 ,... , iNm}, m = 1,... , M , be M subsets of { 1 ,... , N },
each representing a subset of assets.
I The constraint set for each subset of assets is
{ Tm(Tm − Nm)
(Tm − 1)Nm
(ˆμJm − μJm)
Σ
− 1 Jm (ˆμJm^ −^ μJm)^ ≤^ m
}
Example 4
Uncertainty about factor-generating model and expected returns
I N risky assets and K factors
I Return-generating model model (excess returns)
rat = α + βrf t + ut, cov(ut, u
t ) = Ω,^ (15)
I Hence, mean and variance of the returns can always be expressed as
μ =
( α + βμf
μf
)
( βΣf f β
Σf f β
Σf f
)
Multi-Prior Asset Allocation
max w
minμa,μf w
μ −
γ
w
Σw, (17)
subject to
(ˆμa − μa)
Σ
− 1
aa (ˆμa^ −^ μa)^ ≤^ a,^ (18)
(ˆμf − μf )
Σ
− 1
f f (ˆμf^ −^ μf^ )^ ≤^ f^.^ (19)
I If f = 0 and asset pricing model holds
- μˆa = βμf
- equation (18) is a multi-prior version of model uncertainty
- If a = 0 ⇒ investor believes dogmatically in the model
I If f > 0 , allocation depends on
- relative degree of uncertainty aversions: f vs. a
Empirical Application 1
Uncertainty about Expected Returns:
International Data
I Data
- MSCI month-end US$ value of equity index: Jan 1970–Jul 2001
- for 8 countries (CA, FR, GE, JP, IT, SW, UK, US)
I Rolling Windows
- Determine portfolio based on 60-month-window, and
- With these weights, calculate return in 61
st
month.
I Short-Selling Constraints Consider two cases:
- Short-selling is allowed, and
- Short-selling is not allowed.
US Portfolio (01/1975-07/2001) with shortselling allowed
Jan 75 Mar 79 May 83 Jul 87 Sep 91 Nov 95 Jan 99
−
−
0
500
1000
1500
2000
2500 MV ε= 1 ε= 3
Out-of-Sample Performance Analysis
I From the rolling-windows-portfolios, compute the out-of sample
1. Mean
2. Volatility
3. Ratio of mean to volatility
I Compare performance of multi-prior portfolio with
1. Mean-variance portfolio
2. Bayes-Diffuse-Prior (with μ = ˆμ and Σ =
(
1 +
1 T
)
3. Bayes-Stein (with μ = ωμmvp + (1 − ω)ˆμ)
Panel A: Short sales allowed
Strategy Mean Std.Dev.
Mean Std.Dev.
- Mean-Variance -0.517 42.085 -0.
- Bayesian approach
- Diffuse Prior -0.491 41.394 -0.
- Empirical Bayes-Stein -0.162 17.508 -0.
- Multi-prior approach
Empirical Application 2
Uncertainty on Expected Returns and Pricing Model:
US Domestic Data
I Data
- Assets. Monthly returns on HML and SMB: Jul 1926 to Dec 2002.
- Factor. Monthly excess return on MKT
NYSE, AMEX and NASDAQ stocks (from CRSP)
I Rolling Windows
- Determine portfolios based on 120-month window, and
- With these weights, calculate return in 121
st
month.
Portfolio weights
I Consider weights in SMB, HML, MKT
- When CAPM is used for estimating expected returns
(aversion to both parameter and model uncertainty)
- “Skeptical Bayesian” uses MLE (Maximum Likelihood Estimates).
- “Dogmatic Bayesian” uses CAPM (Capital Asset Pricing Model).
I Compare to Bayesian “Data and Model” approach of Pastor and
Stambaugh
Out-of-sample Sharpe Ratios (MLE reference estimate)
Strategy Sharpe Ratio
- Mean-Variance 0.
- Bayesian approach
- Bayes-Stein 0.
- Pastor-Stambaugh
with data only 0.
with ω = 0. 00 0.00 0.50 1.00 1.
% (0.0) (38.20) (68.07) (86.37)
a %
Out-of-sample Sharpe Ratios (with CAPM used for estimation)
Strategy Sharpe Ratio
- Mean-Variance 0.
- Bayesian approach
- Bayes-Stein 0.
- Pastor-Stambaugh
with model only 0.
with ω = 1. 00 0.0 0.5 1.0 2.0 2.5 3.
% (0.0) (38.20) (68.07) (95.22) (98.62) (99.67)
a %
