Worst-case robust decisions for multi-period mean–variance portfolio optimization

Abstract

In this paper, we extend the multi-period mean–variance optimization framework to worst-case design with multiple rival return and risk scenarios. Our approach involves a min–max algorithm and a multi-period mean–variance optimization framework for the stochastic aspects of the scenario tree. Multi-period portfolio optimization entails the construction of a scenario tree representing a discretised estimate of uncertainties and associated probabilities in future stages. The expected value of the portfolio return is maximized simultaneously with the minimization of its variance. There are two sources of further uncertainty that might require a strengthening of the robustness of the decision. The first is that some rival uncertainty scenarios may be too critical to consider in terms of probabilities. The second is that the return variance estimate is usually inaccurate and there are different rival estimates, or scenarios. In either case, the best decision has the additional property that, in terms of risk and return, performance is guaranteed in view of all the rival scenarios. The ex-ante performance of min–max models is tested using historical data and backtesting results are presented.

Introduction

In financial portfolio management, the maximization of return for a level of risk is the accepted approach to decision making. A classical example is the single-period mean–variance optimization model in which expected portfolio return is maximized and risk measured by the variance of portfolio return is minimized (Markowitz, 1952). Consider n risky assets with random rates of return r₁, r₂, … , r_n. Their expected values are denoted by E(r_i) or $\overline{r_i}$, for i = 1, … , n. The single period model of Markowitz considers a portfolio of n assets defined in terms of a set of weights wⁱ for i = 1, … , n, which sum to unity. Given an expected rate of portfolio return ${\overline{r}}_p$, the optimal portfolio is defined in terms of the solution of the following quadratic programming problem:

$$\underset{\mathbf{w}}{\min }\{〈\mathbf{w}\text{,}\Lambda \mathbf{w}〉|{\mathbf{w}}^{\prime }\overline{\mathbf{r}}⩾{\overline{r}}_p\text{,}{\mathbf{1}}^{\prime }\mathbf{w}=1\text{,}\mathbf{w}⩾0\}\text{,}$$

where Λ is the covariance matrix of asset returns. The quadratic program yields the minimum variance portfolio. Note that the classical stochastic linear programming formulation maximizes the expected return but takes no account of risk.

A multi-period framework to reformulate the single stage asset allocation problem as an adaptive multi-period decision process has been developed by using multi-period stochastic programming, see for example Birge and Louveaux, 1997, Kall, 1976, Kall and Wallace, 1994, Prekopa, 1995. In the multi-stage case, the investor decides based on expectations and/or scenarios up to some intermediate times prior to the horizon. These intermediate times correspond to rebalancing or restructuring periods. The mean or the variance of total wealth at the end of the investment horizon is modelled by either linear stochastic programming or quadratic stochastic programming in Gülpınar et al., 2002, Gülpınar et al., 2003.

Multi-period portfolio optimization entails the construction of a scenario tree representing a discretised estimate of uncertainties and associated probabilities in future stages. The multi-period stochastic mean–variance approach takes account of the approximate nature of the discrete set of scenarios by considering a variance term around the return scenarios. Hence, uncertainty on return values of instruments is represented by a discrete approximation of a multivariate continuous distribution as well as the variability due to the discrete approximation. The mean–variance framework is based on a single forecast of return and risk. In reality, however, it is often difficult or impossible to rely on a single forecast; there are different rival risk and return estimates, or scenarios. Two sources of further uncertainty might require a strengthening of the robustness of the mean–variance decision. The first is that some rival uncertainty scenarios may be too critical to consider in terms of probabilities. A worst-case optimal strategy would yield the best decision determined simultaneously with the worst-case scenario. The second is that the return variance estimate is usually inaccurate and there are different rival estimates, or scenarios. A worst-case optimal strategy protects against risk of adopting the investment strategy based on the wrong scenario. In either case, the best decision has the additional property that, in terms of risk and return, performance is guaranteed in view of all the rival risk and return scenarios. The min–max optimal performance will improve for any scenario other than the worst-case. This guaranteed performance and noninferiority property of min–max are discussed further in Section 3.

In this paper, multi-period mean–variance optimization framework is extended to the robust worst-case design problem with multiple return and risk scenarios. Our approach involves a min–max algorithm and a multi-period mean–variance optimization framework for the stochastic aspects of the scenario tree, (Gülpınar and Rustem, 2004). Since optimal investment decision is based on the min–max strategy, the robustness is ensured by the non-inferiority of min–max. The optimal portfolio is constructed (relative to benchmark) simultaneously with the worst-case to take account of all rival scenarios. The portfolio is balanced at each time period incorporating scalable (not fixed) transaction cost and its relative performance is measured in terms of returns and the volatility of returns.

The rest of the paper is organized as follows. In Section 2, the multi-period mean variance optimization problem is described. In Section 3, we introduce multi-period discrete min–max formulations of multi-period mean–variance optimization problem for robust, optimal investment strategies in view of rival return and risk scenarios (which are input scenarios in the min–max formulation). Section 4 focuses on the generation of scenario tree and forecasting rival risk and return scenarios. In Section 5, we present our computational results which are based on worst-case risk-return frontiers and backtesting (out-of-sample).