Robust Portfolio Optimization with
Multiple Experts
Frank Lutgens and Peter Schotman
(Maastricht University)
Mean Variance Portfolios
• Bad input parameters (means, covariances) lead
to poor performing portfolios
• Symptoms
– Portfolios often extreme: large arbitrage
positions
– Error maximization: Michaud (1989)
• Performance
– Huge overestimation of real ex-post
performance
– Overestimation of Sharpe ratio: bias order N/T.
Example 1
• Hoevenaars, Molenaar, Schotman and Steenkamp
(2005) consider optimal mean-variance portfolios
of long-term investors with risky liabilities
• Optimal portfolios are a linear combination of
hedge portfolio and a speculative portfolio
1 1 1 1
w (1 ) c0
• Asset classes: stocks, bonds, real estate, credits,
commodities, hedge funds
Speculative portfolio
( =10, 5 year horizon)
600%
400%
200%
0%
te
its
s
es
s
ds
s
k
ill
nd
a
iti
oc
d
on
B
st
-200%
re
Fu
od
T-
St
lE
B
C
m
ge
ea
om
ed
R
C
H
-400%
-600%
-800%
Ridiculously leveraged, extreme weights
Example 2
• Dutch market index:
– 1975-2004, 30 years
– average returns: 12%
– standard deviation: 20%
• Standard error of the mean:
– s = /T = 20 / 30 = 3.5%
– 95% confidence interval: (5% - 19%)
• Much uncertainty about expected returns
Example 3
• Country selection correlation matrix
NL SWE FR GER IT UK ESP • Exclusively substantial
Netherlands 1 0.57 0.66 0.76 0.45 0.66 0.49 positive correlations
Sweden 1 0.47 0.56 0.43 0.46 0.51
France 1 0.66 0.52 0.57 0.50
Germany 1 0.47 0.51 0.50
Italy 1 0.42 0.49
UK 1 0.43
Spain 1
• And the inverse covariance matrix ...
• Positive diagonal and
NL SWE FR GER IT UK ESP negative off-diagonal
Netherlands 1.18 -0.10 -0.11 -0.48 0.00 -0.30 -0.02 • Arbitrage own expected
Sweden -0.10 0.38 0.01 -0.09 -0.04 -0.04 -0.09
France -0.11 0.01 0.53 -0.18 -0.08 -0.10 -0.06
return against all other
Germany -0.48 -0.09 -0.18 0.80 -0.03 0.07 -0.04 expected returns in –1
Italy 0.00 -0.04 -0.08 -0.03 0.27 -0.03 -0.07 • The higher the
UK -0.30 -0.04 -0.10 0.07 -0.03 0.46 -0.02 correlations, the more
Spain -0.02 -0.09 -0.06 -0.04 -0.07 -0.02 0.38
extreme the negative
versus positive entries
Solutions
• Restrict portfolio weights:
– Jagannathan and Ma (2003)
– Shortsale constraints
– Underdiversification
– Improved performance
• Bayesian estimation (using informative priors)
– Black and Litterman (FAJ 1992),
– Pastor and Stambaugh (JF 2000)
• Robust estimation and/or portfolio optimization
Robust decision rules
• Not the best possible solution in ideal
circumstances, but reasonably good under many
scenarios
• Portfolio weights should not change dramatically in
response to slight changes in the environment
(estimation error, model specification)
• Robust rules guarantee a minimum performance
• Two ways of introducing robustness
– In the estimation: correct for outliers, etc.
– In the portfolio optimization
Literature
• Behavioral Finance
– Ambiguity aversion: Gilboa & Schmeidler (JMathE,
1989)
– Underdiversification: Uppal & Wang (JF, 2003),
Boyle, Uppal & Wang (2004)
• Operations Research and Econometrics
– Better ex-post decisions
– Shrinkage: Jorion (JFQA, 1986), Kan & Zhou (2004),
Garlappi, Uppal & Wang (2004)
– Computational: Goldfarb & Iyengar (MOR, 2003),
Rustem, Becker & Marty (JEDC, 2000)
Multiple Priors
• N securities and riskfree rate
• Investor with MV utility:
Q(w) = w’ – ½ w’ w
• J experts with advice (j, j)
• Robust mean-variance problem
1
maxw min j w' j w' jw
2
Robust solution
1 1
J J
w S m S l j j m l j j
j 1 j 1
J
0 lj 1 lj 1
j 1
• Robust solution has same form as a Bayesian
solution, but with endogenous weights
(“probabilities”)
• No explicit analytic solution for lj.
• Numerical solution is efficiently computable
Loss function
• Performance under alternative true values
L(, w) max Q(v ) Q(w)
v
1
( w)' 1( w)
2
• How do portfolios behave under alternative and
?
– Robust w will never be worst
Stock and bond: two experts
M1
• R M2
• •
Utility Q(w)
Q(w)
Robust portfolio not always most
conservative
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
w
Investment in stock
Loss function • Robust portfolio dominates for
moderate true mean/variance
• Robust rule is never worst
L(,2|w)
L( , 2|w )
1.5 2 2.5 3 3.5
/2
2
/
Robust Portfolio Choice
• N securities and riskfree rate
• Investor with MV utility:
Q(w) = w’ – ½ w’ w
• J experts with advice (j, j)
• Robust mean-variance problem
1
maxw min j w' j w' jw
2
N assets, 2 experts, equal
1 1
• Optimal portfolio expert j: wj
ˆ j
• Performance wrt expert i: ji ( j i )' w j
ˆ
• Robust solution
w1
ˆ 12 0
21w1 12w2
ˆ ˆ
wR
ˆ if min(12 , 21 ) 0
21 12
w2
ˆ 21 0
• When advice is very different, combine
Performance
• Define:
– True expected returns 0
10
Expected return asset 2
9
8
– Difference in optimal portfolio:
7
w2 w1
ˆ ˆ
6
5
• Robust investor has lower loss if
4
1 2
3
2 0 2
1
0 0 0
0 2 4 6 8 10
Expected return asset 1 – Two linear inequalities in 0
Robust Efficient Frontier
10 Optimal portfolio choice
9 with riskfree rate
8
7
6
5 1 Robust
4 2
3 R
2
1
0
0 2 4 6 8 10 12 14 16 18 20
Factor models
• 2 experts, second expert uses projection on factor
space with loadings B.
2 B(B' 1B)1 B' 1
• Robust solution is optimal portfolio from factor
model (w2)
• With nested advice, follow the restricted model
Priors and data
• Expert j has prior pj(, ).
• Experts agree on the likelihood function L(, | Y).
– Discipline on dispersion of advice
• Experts report their posterior mean of and :
j E j Y , j E j Y
• Robustness:
– Here: model uncertainty
– Later: estimation uncertainty
Priors
• Dogmatic
– strict restrictions on and
– Examples: CAPM, equal means, factor structure
• Loose, but informative
– Pastor and Stambaugh (JF, 2000)
yt a Bft ut , var(ut ) D
ft et , var(et )
– Different priors on a, or on covariance structure
Experimental Design I
• Four models
– Unrestricted sample moments
– Equal means = minimum variance
– Fama-French 3 factor model
– CAPM
• Data
– 25 Size-Value Fama-French portfolios
– January 1964 – December 2002
• Full sample moments set as population true values
• Bootstrap samples of 60 or 120 months
– Construct portfolios from sample data
– Evaluate performance (mean, variance, utility, Sharpe)
– Repeat 10,000 times
Model Misspecification
'Population' moments
25 FF Size-Value portfolios, 1963 - 2002
returns in percent per month, risk aversion = 5
Optimal portfolios
True MinV FF 3 CAPM
Expected return 4.62 1.53 1.11 0.18
Std. dev. 9.61 5.53 4.66 1.82
Utility, risk adj. return 2.31 0.76 0.54 0.08
Sharpe 0.48 0.28 0.23 0.09
Leverage 1.64 1.64 0.73 0.39
• “True values” favor unstructured model
• CAPM badly misspecified
Performance statistics
• Portfolio wk, based on bootstrap sample
• True values (0, 0)
• Performance
Ex-ante Ex-post
Mean ˆ
E wk k
E wk 0
Variance ˆ
S wk k wk
S wk 0wk
Utility ˆ ˆ ˆ
Q E S / 2 Q E S / 2
Loss L Q0 Q
Performance (T = 60)
Ex-Ante Sample MinV FF 3 CAPM Robust
Expected return 24.8 4.32 2.31 0.56 0.49
Std. dev. 21.9 7.95 6.38 2.68 2.41
Utility 12.4 2.16 1.16 0.28 0.24
Sharpe 1.09 0.40 0.32 0.13 0.12
Leverage 3.22 3.27 0.86 0.45 0.55
Robust 14% 86%
Ex-Post Sample MinV FF 3 CAPM Robust
Expected return 9.50 3.33 1.27 0.20 0.31
Std. dev. 43.2 15.8 7.13 2.81 2.82
Expected Loss 44.7 8.20 2.56 2.43 2.35
Sharpe 0.22 0.18 0.17 0.05 0.10
Performance (T = 120)
Ex-Ante Sample MinV FF 3 CAPM Robust
Expected return 11.7 2.36 1.68 0.36 0.34
Std. dev. 15.2 6.38 5.53 2.20 2.11
Utility 5.85 1.18 0.84 0.18 0.17
Sharpe 0.76 0.32 0.28 0.11 0.11
Leverage 2.20 2.22 0.80 0.42 0.47
Robust 6% 94%
Ex-Post Sample MinV FF 3 CAPM Robust
Expected return 6.34 2.12 1.19 0.19 0.24
Std. dev. 20.6 8.60 5.88 2.27 2.21
Expected Loss 7.03 2.40 2.09 2.32 2.27
Sharpe 0.31 0.24 0.20 0.06 0.11
Experimental Design II
• Five models
– Unrestricted sample moments
– CAPM
– Fama-French 3 factor model
– International CAPM
– International FF
• Data
– 81 Size-Value portfolios for 9 European countries
– January 1975 – December 2001
• Bootstrap samples of 150 months
– Construct portfolios from sample data
– Evaluate performance (mean, variance, utility, Sharpe)
– Repeat 10,000 times
Pastor-Stambaugh priors
• Except for the unrestricted sample moments, all
models are misspecified
– Incorrect advice even in large samples
– Tradeoff between estimation error and model
misspecification
• Non-dogmatic priors
– Pastor-Stambaugh (2000)
y it a i Bi ft t
– Multiple priors on a and B for all assets
– Four models: strict or loose on a, CAPM or FF 3-fact
– Uninformative priors on B and factor premia
– Loose covariance prior around diagonal D.
Pastor-Stambaugh results
25 FF Size-Value portfolios, 1964 - 2002
T = 60
Fama-French CAPM
Ex-Ante Sample strict loose strict loose Robust
Expected return 24.7 2.14 18.2 0.54 11.3 0.54
Std. dev. 21.9 6.12 18.6 2.63 14.8 2.63
Utility 12.4 1.07 9.12 0.27 5.64 0.27
Sharpe 1.09 0.31 0.94 0.13 0.74 0.13
Leverage 3.26 0.81 2.59 0.44 1.94 0.44
Fama-French CAPM
Ex-Post Sample strict loose strict loose Robust
Expected return 9.52 1.18 7.36 0.20 5.08 0.21
Std. dev. 43.2 6.57 30.0 2.70 18.8 2.70
Expected Loss 44.4 2.43 19.0 2.41 6.61 2.40
Sharpe 0.22 0.17 0.24 0.05 0.27 0.05
Hoevenaars et al (2006)
(0) (1) (2) (3) (4) (5) (6) (7)
(0) OLS 0 2.1 5.7 21 33.8 4.9 3.1 0.9
(1) Flat 4.8 0 1.4 14 32.8 1 0.2 0.4
(2) Pessimist, uncertain 12.3 1.4 0 4.6 18.6 0 0.6 3.4
(3) Pessimist 33.4 10.4 4.5 0 7.5 5.3 7.9 15.5
(4) Pessimist, dogmatic 79.1 49.6 40.9 15 0 42.1 45.9 56.7
(5) Optimist, uncertain 9.7 0.9 0 5.3 18.5 0 0.3 2.7
(6) Optimist 5.1 0.2 0.5 7.4 18.7 0.3 0 1
(7) Optimist, dogmatic 2.3 0.4 3.5 23.3 48.5 2.7 1.2 0
• For each prior compute the optimal portfolio
• Evaluate each portfolio under all priors (columns
are portfolios, rows are priors)
• Robust portfolio is minimum over maximum loss
per column
Concluding remarks
• Robust mean-variance portfolios attain lower
expected loss compared to conditional mean-
variance portfolios
• Empirical example
– CAPM often most pessimistic
– CAPM severely misspecified
– Robust portfolio outperforms
– With strict prior need much data to increase performance
• Extensions
– Different objective function (Liabilities, Tracking error)
– Intertemporal choice