Robust Bayesian Portfolio
Construction
Josh Davis, PIMCO
Jan 12, 2009
UC Santa Barbara
Seminar on Statistics and Applied Probability
Introduction
• Modern portfolio theory
– Markowitz’s seminal work (1952, JoF)
– Sharpe’s CAPM (1964, JoF)
– Ross’s APT (1976,JET)
• Pimco’s approach to asset allocation
• 2 Main Ingredients
– Utility Function of Investor
– Distribution of Asset Returns
Markowitz Mean-Variance Efficiency
• Original representation of portfolio problem
– Investor maximizes following utility function:


maxw w   2 w w
– Subject to:
w 1 N 1
– Where: E
r 
Cov
r
• Investor indifferent to higher order moments!
• Gaussian distribution carries all information relevant to investor’s
problem!
Markowitz Solution
• Theoretical Result: Diversification!
– “Don’t put all your eggs in one basket”

1
w  
1

1 1 N
1 N1 1 N
1N
Zero Beta E
r
• Practical Issues to Model Implementation
– “Good” estimates of first two moments
• These moments are state dependent
• These moments are also ‘endogenous’
– General Equilibrium vs. Partial Equilibrium
– Solution: Assume Investor is infinitesimal
Bayesian Portfolio Construction
• Black-Litterman popularized the approach
– Combine subjective investor ‘views’ with the sampling distribution in a
consistent manner
– Origins in the economics literature: ‘Minnesotta Prior’
• See Doan, Litterman, Sims (1984) or Litterman (1986)
• See Jay Walters’ excellent outline for more details
– Exploit conjugate priors and Bayes Rule:
P
B|A
P
A
P
A|B
Bayes Rule
P
B

P
A N
, 
Prior distribution
P
B|A N
0 , 0 

P
A|B N
, 
Updating distribution
Posterior distribution
Caveats
• Bayesian approach naturally integrates observed data and opinion
– Does the Gaussian updating distribution represent the investor’s beliefs
accurately?
• Black/Litterman implementation very mechanical and unintuitive
• Inconsistent with bounded rationality, rational inattention
– Is the sampling distribution (prior) accurately represented by a
Gaussian?
• Quality of asymptotic approximation?
• Regime switch?
• Posterior moments a function of this Gaussian framework
– Efficient Frontier particularly sensitive to the expected return inputs
(Merton, 1992)
• What about the utility function?
– A wealth of economic literature suggests it doesn’t describe investor
behavior accurately
Uncertainty
• As we know,
There are known knowns.
There are things we know we know.
We also know
There are known unknowns.
That is to say
We know there are some things
We do not know.
But there are also unknown unknowns,
The ones we don't know
We don't know.
—Donald Rumsfeld, Feb. 12, 2002, Department of Defense
news briefing
Robustness
• Two types of uncertainties
– Statistical uncertainty (Calculable Risk)
– Model uncertainty (‘Knightian’ uncertainty)
• Ellsberg Paradox provides empirical evidence
• Multi-prior representation (Gilboa and Schmeidler)
• Also related to literature on error detection probabilities
• Is the investor 100% certain in the model inputs?
– No!
• Shouldn’t portfolio construction be robust to
model misspecification?
– Yes!
Incorporating Uncertainty
• Today I will follow the statistical approach of
Garlappi, Uppal and Wang (RFS, 2007)
– For a complete and rigorous treatment see Hansen
and Sargent’s book Robustness
• Critical modification: max-min objective


maxw min w   2 w w
– Subject to:

d
, e
w 1 N 1
The Space of Plausible Alternatives

d
j , j e
Characterizing Uncertainty
• GUW take a ‘statistical approach’ based
on confidence intervals
• I modify this for the BL framework

d
j , j 
j 
j

j
• Parameter ‘e’ determined by investor’s
‘confidence’ in the expected return
P
j 
j

j
e
1 p
Determination of Uncertainty
Parameter ‘e’
p  ’Confidence in Model’
e |1 
p/2| (Gaussian Case)
Solution
• The inner minimization can be removed
via the following adjustment:
maxw



w 
 adj  2 w w
• Where the adjustment puts the expected
return on the boundary of the plausible
region
adj
j
sign
w j  j  e j
Example Posterior Moments
•
•
•
•
•
•
•
Commodities:
US Bonds:
US Large Cap:
US Small Cap:
Sovereign Bonds:
EM Equity:
Real Estate:
4%
(12%)
5.5% (14%)
8%
(22%)
9%
(25%)
6.5% (18%)
10% (28%)
6%
(16%)
Correlations from…
(Monthly Jan ’96-Dec ’08)
•
•
•
•
•
•
•
Commodities:
US Bonds:
US Large Cap:
US Small Cap:
Global Bonds:
EM Equity:
Real Estate:
GSCI
LBAG
Russell 200
Russell 2000
Citi Sovereign Index
MSCI Em Index
MSCI US Reit Index
• Also, added constraint of weights b/w 0 and 1
Definitions

r 
• Reference Model: E

Cov
r 
• Plausible Worst Case Model

E
r|Confidence sign
w   e

Cov
r|Confidence
– Where:



w  arg max w 
 sign
w  e  2 w w
w
‘Optimal’ Weights
Confidence in Model
1%
25%
50%
75%
95%
100%
2
0
0
0
0
0
US Bonds
41
14
2
0
0
0
US Large Cap
10
20
25
27
26
24
US Small Cap
6
20
27
32
35
35
13
12
12
7
0
0
EM Equities
3
20
27
33
39
41
Real Estate
25
13
8
1
0
0
Portfolio Expected Return
6.33
7.79
8.41
8.86
9.12
9.17
Portfolio Expected Volatility
7.88
9.45
11.69
13.66
15.07
15.39
Commodities
Sovereign Bonds
Optimal Weights
Endogenous Worst Case Returns
Robust Portfolios under Reference
Model
Endogenous Worst Case
Comparison
Historical Performance
Conclusion
• Bayesian Portfolio Methods theoretically
appealing…
– Attempts to correct for misspecification by
incorporating additional information
– Doesn’t rule out misspecification
• Robust methods insure against plausible worstcase scenarios
• Accounting for uncertainty leads to…
– Lower volatility under ‘reference model’
– Lower expected return under ‘reference model’
– Improved risk/return tradeoff under ‘worst-case’
scenarios
Appendices
Example Derivation of Prior
• In BL views take the following form:
View 1: E
r 1 r 2 3%, Confidence w 1
View 2:E
r 1 5%,
Confidence w 2
• Which can be represented as:
P
1 1
1 0
Q
0. 03
0. 05

w1
0
0
w2
• The investor’s updating distribution is
1
P
B|A N
P 1 Q, 
P P

 diag
P

P 
(He and Litterman ’99)
Posterior Derivation
• The prior and updating distributions take
the form
 
P
A N
, 
1
P
B|A N
P 1 Q, 
P P

• The posterior is Gaussian




1 
 1
 P  P
 

 

1
 1


1
P  P
1
1
P 1 Q