Presentation on theme: "Robust Bayesian Portfolio Construction"— Presentation transcript:
1Robust Bayesian Portfolio Construction Josh Davis, PIMCOJan 12, 2009UC Santa BarbaraSeminar on Statistics and Applied Probability
2Introduction 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 allocation2 Main IngredientsUtility Function of InvestorDistribution of Asset Returns
3Markowitz Mean-Variance Efficiency Original representation of portfolio problemInvestor maximizes following utility function:Subject to:Where:Investor indifferent to higher order moments!Gaussian distribution carries all information relevant to investor’s problem!
4Markowitz Solution Theoretical Result: Diversification! “Don’t put all your eggs in one basket”Practical Issues to Model Implementation“Good” estimates of first two momentsThese moments are state dependentThese moments are also ‘endogenous’General Equilibrium vs. Partial EquilibriumSolution: Assume Investor is infinitesimal
5Bayesian Portfolio Construction Black-Litterman popularized the approachCombine subjective investor ‘views’ with the sampling distribution in a consistent mannerOrigins in the economics literature: ‘Minnesotta Prior’See Doan, Litterman, Sims (1984) or Litterman (1986)See Jay Walters’ excellent outline for more detailsExploit conjugate priors and Bayes Rule:
6CaveatsBayesian approach naturally integrates observed data and opinionDoes the Gaussian updating distribution represent the investor’s beliefs accurately?Black/Litterman implementation very mechanical and unintuitiveInconsistent with bounded rationality, rational inattentionIs the sampling distribution (prior) accurately represented by a Gaussian?Quality of asymptotic approximation?Regime switch?Posterior moments a function of this Gaussian frameworkEfficient 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
7UncertaintyAs 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
8Robustness Two types of uncertainties Statistical uncertainty (Calculable Risk)Model uncertainty (‘Knightian’ uncertainty)Ellsberg Paradox provides empirical evidenceMulti-prior representation (Gilboa and Schmeidler)Also related to literature on error detection probabilitiesIs the investor 100% certain in the model inputs?No!Shouldn’t portfolio construction be robust to model misspecification?Yes!
9Incorporating 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 RobustnessCritical modification: max-min objectiveSubject to:
15Correlations from… (Monthly Jan ’96-Dec ’08) Commodities: GSCIUS Bonds: LBAGUS Large Cap: Russell 200US Small Cap: Russell 2000Global Bonds: Citi Sovereign IndexEM Equity: MSCI Em IndexReal Estate: MSCI US Reit IndexAlso, added constraint of weights b/w 0 and 1
16DefinitionsReference Model:Plausible Worst Case ModelWhere:
17‘Optimal’ Weights Confidence in Model 1% 25% 50% 75% 95% 100% Commodities2US Bonds4114US Large Cap102025272624US Small Cap63235Sovereign Bonds13127EM Equities33339Real Estate81Portfolio Expected Return6.337.798.418.869.129.17Portfolio Expected Volatility7.889.4511.6913.6615.0715.39
24Conclusion Bayesian Portfolio Methods theoretically appealing… Attempts to correct for misspecification by incorporating additional informationDoesn’t rule out misspecificationRobust methods insure against plausible worst-case scenariosAccounting for uncertainty leads to…Lower volatility under ‘reference model’Lower expected return under ‘reference model’Improved risk/return tradeoff under ‘worst-case’ scenarios