## 1 Introduction

It is well known that asset returns probability distributions in general feature both skewness and excess kurtosis. Investor preferences of these higher moments are attracting increasing attention by portfolio choice researchers and the financial industry, often arriving at more complex utility functions than traditionally assumed. In full-scale optimization^{1} empirical return distributions are used in their entirety, and the choice of utility function is completely flexible. In this paper we assess the performance of FSO using a wide range of utility function specifications applied to equity indices. We show that when utility functions include loss aversion or prospect theory, even assets with only small deviations from normality are better selected in an FSO framework than with the traditional mean–variance (MV) approach.

The strength of FSO is that no analytical solution to the portfolio choice problem is pursued. This allows the distributional properties of returns to be left non-parameterized and the utility function to be specified to reflect investor preferences uncompromised by mathematical convenience. The absence of simplifying assumptions, yielding theoretical appeal, comes at the cost of computational burden, however. As the optimization problem is not convex, a grid search or a global search algorithm has to be used to find the utility-maximizing portfolio.

Cremers *et al.* (2005) show in a hedge fund selection problem that the performance of FSO (in terms of utility) is substantially better than Markowitz's (1952, 1959) MV approach when investor preferences are modeled to include loss aversion or prospect theory (based on Kahnemann and Tversky, 1979). The results are confirmed in an out-of-sample application by Adler and Kritzman (2007). Several other portfolio choice papers resembling FSO but not using the term have appeared lately. Statistical properties of the estimator have been explored by Gourieroux and Monfort (2005). Scenario-based approaches (using hypothesized outcomes with probabilities attached instead of empirical return distributions) have been dealt with by Grinold (1999) and Sharpe (2007). Higher moments properties of utility-maximizing portfolios have been investigated by Maringer (2008), who also proposes heuristic optimization methods to deal with the computational burden of the optimization problem.

The idea of utility maximization as a methodology for portfolio optimization problems, based on the utility theory founded by Von Neumann and Morgenstern (1947), can be traced back at least to Tobin (1958), and also appears in several assessments of the MV approach (e.g. Levy and Markowitz, 1979; Markowitz, 1987). As the latter studies showed that the performance difference between MV portfolios and utility-maximizing portfolios (using power utility) was negligible, the less burdensome MV approach became the model of choice in the financial industry, and the benchmark in academia.^{2}

The relevance of higher moments for investment decisions was pointed out by Levy (1969) and Samuelson (1970), and in realistic portfolio management situations, investors often express preferences that imply more complex utility functions than power or quadratic utility (Litterman, 2003, Ch. 2; Meucci, 2005, Ch. 5). It is when implementing such preferences (loss aversion and prospect theory) that the FSO has been proven superior to MV (Cremers *et al.*, 2005; Adler and Kritzman, 2007). In these assessments, however, assets with extremely non-normal returns (hedge funds) have been used, and the results have only been shown to hold for a few examples of utility function specifications. Our assessment of FSO features a selection of equity indices with returns much closer to a normal distribution (in general still non-normal though) and a much broader spectrum of utility function specifications. Finding that FSO performance is substantially better than MV when complex investor preferences (in particular prospect theory) hold, our results indicate that the earlier studies are robust with respect to utility function specification, as well as to other asset classes (equity indices).

This paper is organized as follows. Section 2 discusses the portfolio choice problem and the characteristics of investors' preferences. Section 3 presents data, utility functions and methodology used in our assessment of FSO. The results are presented and analyzed in Section 4, and Section 5 concludes.