1. Robust Bayesian Portfolio Construction
Josh Davis, PIMCO
Jan 12, 2009
UC Santa Barbara
Seminar on Statistics and Applied Probability

2. Introduction Modern portfolio theory 2 Main Ingredients
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

3. Markowitz Mean-Variance Efficiency
Original representation of portfolio problem
Investor maximizes following utility function:
Subject to:
Where:
Investor indifferent to higher order moments!
Gaussian distribution carries all information relevant to investor’s problem!

4. Markowitz Solution Theoretical Result: Diversification!
“Don’t put all your eggs in one basket”
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

5. 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:

6. 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

7. 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

8. 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!

9. 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
Subject to:

10. The Space of Plausible Alternatives

11. Characterizing Uncertainty
GUW take a ‘statistical approach’ based on confidence intervals
I modify this for the BL framework
Parameter ‘e’ determined by investor’s ‘confidence’ in the expected return

12. Determination of Uncertainty Parameter ‘e’

13. Solution
The inner minimization can be removed via the following adjustment:
Where the adjustment puts the expected return on the boundary of the plausible region

14. Example Posterior Moments
Commodities: % (12%)
US Bonds: % (14%)
US Large Cap: % (22%)
US Small Cap: % (25%)
Sovereign Bonds: % (18%)
EM Equity: % (28%)
Real Estate: % (16%)

15. Correlations from… (Monthly Jan ’96-Dec ’08)
Commodities: GSCI
US Bonds: LBAG
US Large Cap: Russell 200
US Small Cap: Russell 2000
Global Bonds: Citi Sovereign Index
EM Equity: MSCI Em Index
Real Estate: MSCI US Reit Index
Also, added constraint of weights b/w 0 and 1

16. Definitions
Reference Model:
Plausible Worst Case Model
Where:

17. ‘Optimal’ Weights Confidence in Model 1% 25% 50% 75% 95% 100%
Commodities 2
US Bonds 41 14
US Large Cap 10 20 25 27 26 24
US Small Cap 6 32 35
Sovereign Bonds 13 12 7
EM Equities 3 33 39
Real Estate 8 1
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

18. Optimal Weights

19. Endogenous Worst Case Returns

20. Robust Portfolios under Reference Model

21. Endogenous Worst Case Comparison

22. Historical Performance

24. 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 worst-case 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

25. Appendices

26. Example Derivation of Prior
In BL views take the following form:
Which can be represented as:
The investor’s updating distribution is

27. Posterior Derivation
The prior and updating distributions take the form
The posterior is Gaussian