Robust Bayesian Portfolio Construction

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Robust Bayesian Portfolio Construction

• Josh Davis, PIMCO
• Jan 12, 2009
• UC Santa Barbara
• Seminar on Statistics and Applied Probability

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Introduction

• Modern portfolio theory
• Markowitzs seminal work (1952, JoF)
• Sharpes CAPM (1964, JoF)
• Rosss APT (1976,JET)
• Pimcos approach to asset allocation
• 2 Main Ingredients
• Utility Function of Investor
• Distribution of Asset Returns

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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 investors problem!

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Markowitz Solution

• Theoretical Result Diversification!
• Dont 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

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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

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Caveats

• Bayesian approach naturally integrates observed
  data and opinion
• Does the Gaussian updating distribution represent
  the investors 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
  doesnt describe investor behavior accurately

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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

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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!
• Shouldnt portfolio construction be robust to
  model misspecification?
• Yes!

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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 Sargents book Robustness
• Critical modification max-min objective
• Subject to

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The Space of Plausible Alternatives
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Characterizing Uncertainty

• GUW take a statistical approach based on
  confidence intervals
• I modify this for the BL framework
• Parameter e determined by investors
  confidence in the expected return

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Determination of Uncertainty Parameter e
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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

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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)

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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

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Definitions

• Reference Model
• Plausible Worst Case Model
• Where

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Optimal Weights
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Optimal Weights
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Endogenous Worst Case Returns
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Robust Portfolios under Reference Model
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Endogenous Worst Case Comparison
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Historical Performance
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Conclusion

• Bayesian Portfolio Methods theoretically
  appealing
• Attempts to correct for misspecification by
  incorporating additional information
• Doesnt 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

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Appendices
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Example Derivation of Prior

• In BL views take the following form
• Which can be represented as
• The investors updating distribution is

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Posterior Derivation

• The prior and updating distributions take the
  form
• The posterior is Gaussian