Proceedings of the 2006 Winter Simulation Conference L. F. Perrone, F. P. Wieland, J. Liu, B. G. Lawson, D. M. Nicol, and R. M. Fujimoto, eds. JACKKNIFE ESTIMATORS FOR REDUCING BIAS IN ASSET ALLOCATION Amit Partani David P. Morton Graduate Program in Operations Research The University of Texas at Austin Austin, TX 78712, U.S.A. Ivilina Popova Albers School of Business and Economics Seattle University Seattle, WA 98122, U.S.A. ABSTRACT We use jackknife-based estimators to reduce bias when estimating the optimal value of a stochastic program. Our discussion focuses on an asset allocation model with a power utility function. As we will describe, estimating the optimal value of such a problem plays a key role in establishing the quality of a candidate solution, and reducing bias improves our ability to do so efficiently. We develop a jackknife estimator that is adaptive in that it does not assume the order of the bias is known a priori. 1 INTRODUCTION Monte Carlo simulation is used in assessing whether a candidate solution to a stochastic program is near opti- mal, when we cannot solve the stochastic program exactly. Optimizing a sample-mean estimator yields an optimistic bound, in expectation, on the original problem’s optimal value, say, z ∗ . Restated, maximizing a sample mean yields a positively-biased estimate of z ∗ . Such bounds are anal- ogous to optimistic bounds that arise, e.g., via Lagrangian or integrality relaxations in deterministic integer and non- linear programming. A weak relaxation bound yields a weak statement regarding the quality of a candidate solu- tion. Similarly, when our estimate of the optimal value has large bias, it can significantly degrade our ability to establish that a candidate solution is near optimal. So, in this paper we seek to tighten an optimized sample-mean bound, reducing its bias using jackknife-based estimators. Unlike existing jackknife estimators, we do not assume the order of the bias is known when forming the estimator. Our approach is illustrated on an asset allocation model with a power utility function. An overview of stochastic programming, and other ap- proaches to problems that arise in finance, including pricing single instruments, time-static asset allocation, and time- dynamic asset-liability management is given in Ziemba and Mulvey (1998). Monte Carlo methods have seen extensive application to pricing financial securities; see, e.g., Ander- sen and Broadie (2004), Fu et al. (2001), and Glasserman (2003). Asymptotic justification of replacing population means with sample-mean estimators in portfolio optimiza- tion is discussed in Jensen and King (1992). The type of simulation-based solution-quality assessments we make here have also been done in the context of portfolio opti- mization in Morton et al. (2003, 2006), but those papers do not include bias-reducing estimators. We consider a portfolio allocation problem with m as- sets. These assets have random returns ξ =(ξ 1 ,...,ξ m ) ≥ 0. The allocation problem selects the proportion, x = (x 1 ,...,x m ), to invest in each asset to maximize ex- pected utility. The random return of allocation x is ξx ≡ ∑ m j=1 ξ j x j . We assume that ξ ’s distribution is known, has finite second moments, does not depend on x, and that we can generate independent and identically distributed (i.i.d.) observations of ξ . If ξ 1 takes value 1.5 and ξ 2 takes value 0.75 then we have a 50% return on the first asset and a loss of 25% on the second asset. The asset allocation problem is z ∗ = max x∈X Eu(x, ξ ), (1) where X = {x : ∑ m j=1 x j =1,x j ≥ 0,j =1,...,m} and u(x, ξ)=(ξx) γ − c‖x − x t ‖ 2 2 . The constraint set X re- quires all proportions sum to one and disallows shortselling. Eu(x, ξ) is the expected utility obtained by allocation x. Our primary objective is to maximize the expected value of the power utility function of return, u p (·)=(·) γ , where γ ∈ (0, 1) captures the decision maker’s aversion to risk. This is augmented by a secondary penalty term with 0 ≤ c ≪ 1. This discourages deviation from the investor’s existing target portfolio, x t ∈ X, unless the power utility provides a reason to do so. If x t is unknown or the secondary objective is not desired we set c =0. The approach we describe applies to a wide variety of utility functions, but in this paper we restrict ourselves to this variant of the power utility. The power utility function exhibits increasing 783 1-4244-0501-7/06/$20.00 ©2006 IEEE