Optimal Portfolio Choice Under Incomplete Information

Abstract

Models of asset pricing generally assume that the variables which characterize the state of the economy are observable. However, the distributional properties of asset prices that are relevant for portfolio decisions are in general not observable, and therefore must be estimated. The estimation of expected returns is a particularly difficult problem and estimation errors are likely to be substantial. In this light, it is reasonable to examine whether the assumption of observability of expected returns and other relevant state variables causes significant mis-specification in equilibrium models of asset prices. This paper has three main objectives: first, to derive optimal estimators for the unobservable expected instantaneous returns using observations of past realized returns; second, to establish that estimation and portfolio choice can be solved in two separate steps; third, to analyze the impact of estimation error on investment choices. The estimators of expected returns are in general not consistent, i.e., the estimation error does not tend to disappear asymptotically. The effects of the estimation error, therefore, cannot be ignored even if realized returns are observed continuously over an infinite time period. for helpful comments and substantive suggestions. I would also like to thank the members of the finance group at the Sloan School of Management and the participants of numerous seminars for their contribution. Support from the College Interuniversitaire pour les Sciences du Management is gratefully acknowledged. 734The Journal of Finance theoretical issues. When selecting their optimal portfolio, investors have to take into account the fact that they have only estimates of "true" expected returns. Since investors form their portfolios using estimators, equilibrium levels of investment, the term structure of interest rates and contingent claims prices will generally be functions of the "perceived" (or estimated) state variables.The analysis here is intended to improve upon the modest state of our current knowledge regarding portfolio choice and general equilibrium under incomplete information on the distribution of asset returns. With the exception of Klein and Bawa [7, 8], Williams [18], Feldman [4], Dothan and Feldman [3] and de Temple [2], the heuristic approach in previous research has been to assume implicitly that the portfolio choice problem can be solved in two steps: parameters are first estimated, and then portfolios are chosen conditional on these parameter estimates. This separation of the estimation and optimization steps is optimal when a property which Simon [16] and Theil [17] termed "certainty equivalence" applies.'

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“…which is easy to compute once Ψ F I l and Ψ F I h are known from (34). When π hh = 1 − π ll = π, each of the equations produces a FI pricing kernel identical to that in (28).…”

“…Since we work with power utility, results in Rubinstein (1974) imply that an aggregation result holds whereby a representative agent exists if all individuals populating the economy have identical time preference parameters {ρ i } I i=1 , identical coefficients of relative risk aversion {γ i } I i=1 , and identical beliefs. 34 These are strong restrictions, not very dissimilar from imposing the existence of a single agent. Moreover, even if one found these assumptions acceptable, it would not resolve issues such as the no-trade theorem which implies a zero trading volume in this type of model.…”

Optimal Portfolio Management for Individual Pension Plans

“…Various authors since Detemple (1986) and Gennotte (1986) have argued that young investors may face considerable uncertainty about it. As investors grow older, they observe stock returns, and they update their beliefs by using Bayes' rule.…”

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“…where the unconditional mean and variance of income are y = µ/ρ and σ 2 y / (2ρ), respectively; the persistence coefficient ρ governs the speed of convergence or divergence from the steady state; 14 B y,t is a standard Brownian motion on the real line R; and σ y is the unconditional volatility of the 14 If ρ > 0, the income process is stationary and deviations of income from the steady state are temporary; if ρ ≤ 0, income is non-stationary. The ρ = 0 case corresponds to a simple Brownian motion without drift.…”

“…In particular, when introducing state uncertainty due to RI, I derive the continuous-time version of the information-processing constraint (IPC) proposed in Sims (2003), and find the explicit expressions for the stochastic properties of the RI-induced noise and the Kalman filtering equation. This paper is therefore closely related to the literature on imperfect information, learning, asset allocation and asset pricing (see Gennotte 1986, Lundtofte 2008, and Wang 2009). 11 Second, after solving the models explicitly, we can exactly inspect the mechanism through which these two types of induced uncertainty interact and affect different types of demand for the risky asset (i.e., the standard speculation demand and the income-hedging demand), the precautionary saving demand, and consumption dynamics.…”