## I. Introduction

The optimum asset mix in modern portfolio theory is determined by maximizing either the expected excess return of a portfolio per unit of risk in a mean-variance (MV) framework or the expected value of some utility function approximated by the expected return and variance of the portfolio. In both cases, portfolio risk is defined as the variance (or standard deviation) of portfolio returns. Modeling portfolio risk with these traditional volatility measures implies that investors are not allowed to treat the negative and positive tails of the return distribution separately. The standard risk measures determine the volatility of unexpected outcomes under normal market conditions, which corresponds to the normal functioning of financial markets during ordinary periods. However, neither the variance nor the standard deviation can yield an accurate characterization of actual portfolio risk during highly volatile periods (see Bali, Demirtas, and Levy 2009). Therefore, the set of MV efficient portfolios may produce an inefficient strategy for maximizing expected return of the portfolio while minimizing its risk.

In this article we propose a new approach to optimal asset allocation in a downside risk framework with a constraint on shortfall probability. We assume that investors have in mind some disaster level of returns and that they behave so as to minimize the probability of disaster. While minimizing the shortfall probability that the return on the portfolio will not fall below some given level, investors are also assumed to maximize expected return of the portfolio. Hence, the new approach allocates financial assets by maximizing expected return of the portfolio subject to a constraint on shortfall probability.

There is a significant amount of prior work on safety-first investors minimizing the chance of a disaster, introduced by Roy (1952) and Levy and Sarnat (1972). A safety-first investor uses a downside risk measure that is a function of value-at-risk (VaR). VaR is defined as the expected maximum loss over a given interval at a given confidence level. For example, if the given period is one day and the given probability is 1%, the VaR measure would be an estimate of the decline in the portfolio value that could occur with a 1% probability over the next trading day. In other words, if the VaR measure is accurate, losses greater than the VaR measure should occur less than 1% of the time. Roy points out that safety-first investors try to reduce the probability and magnitude of extreme losses in their portfolios.^{1}

In addition, we believe optimal portfolio selection under limited downside risk to be a practical problem. Even if agents are endowed with standard concave utility functions such that to a first-order approximation they would be MV optimizers, practical circumstances such as short-sale and liquidity constraints as well as some loss constraints such as maximum drawdown commonly used by portfolio managers impose restrictions that elicit asymmetric treatment of upside potential and downside risk.

Moreover, the MV optimization holds if the multivariate distribution of risky assets follows a normal density. However, as discussed by Bali, Demirtas, and Levy (2009), the physical distribution of asset returns is generally skewed and has thicker tails than the normal distribution. Hence, the standard measures of portfolio risk are not appropriate to use in dynamic asset allocation, especially during large market moves.

Finally, earlier studies indicate that investors have preference for positive skewness and lower kurtosis. For example, Arditti and Levy (1975) and Harvey and Siddique (2000) provide theoretical and empirical evidence that investors prefer to hold stocks with positively skewed return distributions and they are averse to stocks with high kurtosis. Because the shortfall probability is a function of higher order moments, the extended asset pricing models introduced by the aforementioned studies suggest significant impact of the shortfall probability (or downside risk) constraint on optimal asset allocation.

Because of a wide variety of motivating points discussed previously, we investigate the effects of extreme returns and shortfall probability constraints on optimal asset allocation problem (we refer the reader to Basak and Shapiro 2001; Basak, Shapiro, and Tepla 2006 for theoretical work on optimal portfolio selection with a constraint on VaR or shortfall return). The shortfall constraint reflects the typical desire of an investor to limit downside risk by putting a probabilistic upper bound on the maximum loss. Put differently, the investor wants to determine an optimal asset allocation for a given level of maximum likely loss (or VaR). Our analysis reveals that the degree of the shortfall probability plays a crucial role in determining the effects of the choice among a symmetric fat-tailed, skewed fat-tailed, and normal distribution. These effects concern the composition of optimal asset allocations as well as the consequences of misspecification of the degree of skewness and kurtosis for the downside risk measure.

We consider VaR bounds and optimal portfolio selection in a unified framework. We concentrate on the interaction between different distributional assumptions made by the portfolio manager and the resulting optimal asset allocation decisions and downside risk measures. We compare the empirical performance of the MV approach with the newly proposed asset allocation model in terms of their power to generate higher cumulative and risk-adjusted returns. We first model the shortfall probability constraint using the normal distribution. Then, we use the symmetric fat-tailed and skewed fat-tailed distributions in modeling the tails of the portfolio return distribution to determine the effects of alternative distributions on optimal asset allocation.^{2}