Abstract
The foundations of portfolio theory was established by Harry Markowitz in his pioneering work of 1952. In that work, Markowitz introduced the principle of diversification based on the minimal risk – the hallmark of modern portfolio optimization. The expected return (mean) and the variance, the foundation stones of his theory, lead to a popular name, mean – variance approach. The center of our criticism is association of the risk with the variance. True, the larger the variance, the more uncertainty in the return outcome. The variance (or standard deviation) is an integral symmetric measure but risk is asymmetric in nature. For example, the loss from a low return typically is more harmful than the gain from a high return. The goal of the present work is to extend the mean – variance approach to a separate analysis of downside and upside risk assuming that the investor specifies the lower and upper limits of the expected return. Our optimal portfolio allocation minimizes the risk to get the return outside of the interval. Specifically, the optimal portfolio weights are chosen to maximize the probability of obtaining the return within specified limits. We introduce the fundamental solution to portfolio optimization and show that many criteria can be derived from the interval probability approach as special cases. We believe that the interval return is better suited for expressing investors' financial goals and facilitates decision making.