Robust Bayesian Portfolio Construction

Josh Davis, PIMCOJan 12, 2009

UC Santa BarbaraSeminar 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:

– Subject to:

– Where:

• Investor indifferent to higher order moments!• Gaussian distribution carries all information relevant to investor’s

problem!

maxww 2w w

Er

Covr

w 1N 1

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

w 1 1

Zero Beta Er

11N

1N 11N1N

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:

PA|B PB|APAPB

Bayes Rule

PA N , Prior distribution

PB|A N 0 , 0 Updating distribution

PA|B N , 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

– Subject to:

maxw min w 2w w

d , e

w 1N 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

• Parameter ‘e’ determined by investor’s ‘confidence’ in the expected return

d j, j

j j j

P j j j

e 1 p

Determination of Uncertainty Parameter ‘e’

p ’Confidence in Model’

e | 1p/2 | (Gaussian Case)

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

maxw w adj 2w

w

jadj signw j j ej

Example Posterior Moments

• Commodities: 4% (12%)

• US Bonds: 5.5% (14%)

• US Large Cap: 8% (22%)

• US Small Cap: 9% (25%)

• Sovereign Bonds: 6.5% (18%)

• EM Equity: 10% (28%)

• Real Estate: 6% (16%)

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

Definitions

• Reference Model:

• Plausible Worst Case Model

– Where:

Er

Covr

w w

argmax w signw e 2w

w

Er|Confidence signw e

Covr|Confidence

‘Optimal’ Weights

Confidence in Model

1% 25% 50% 75% 95% 100%Commodities 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

Sovereign Bonds 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

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 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

Appendices

Example Derivation of Prior

• In BL views take the following form:

• Which can be represented as:

• The investor’s updating distribution is

View 1: Er1 r2 3%, Confidence w1View 2:Er1 5%, Confidence w2

P 1 1

1 0Q

0.03

0.05

w1 0

0 w2

PB|A NP 1Q, P P 1

diagP P (He and Litterman ’99)

Posterior Derivation

• The prior and updating distributions take the form

• The posterior is Gaussian

PA N ,

PB|A NP 1Q, P P 1

1 P 1P 1

1 P 1Q

1 P 1P

1