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It is of great importance for those in charge of managing risk to understand how financial asset returns are distributed. Practitioners often assume for convenience that the distribution is normal. Since the 1960s, however, empirical evidence has led many to reject this assumption in favor of various heavy-tailed alternatives. In a heavy-tailed distribution the likelihood that one encounters significant deviations from the mean is much greater than in the case of the normal distribution. It is now commonly accepted that financial asset returns are, in fact, heavy-tailed. The goal of this survey is to examine how these heavy tails affect several aspects of financial portfolio theory and risk management. We describe some of the methods that one can use to deal with heavy tails and we illustrate them using the NASDAQ composite index.

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We investigate the portfolio construction problem for risk-averse investors seeking to minimize quantile based measures of risk. Using dependence measures from extreme value theory, we find that most international equity markets are asymptotically independent. We also find that the few cases of asymptotic dependence occur mostly in markets which are in close geographic proximity. We then examine how extremal dependence affects the asset allocation problem. Following the structure variable approach, we focus on the portfolio and model its tail in a manner consistent with extreme value theory. We then develop a methodology for asset allocation where the goal is to guard against catastrophic losses. The methodology is tested through simulations and applied to portfolios made up of two or more international equity markets. We analyze in detail three typical types of markets, one where the assets are asymptotically independent and the ratio of marginal risks is not constant, the second where the assets are asymptotically independent but the ratio of marginal risks are approximately constant and the third where the assets are asymptotically dependent and the ratio of marginal risks is not constant. The results are compared with the optimal portfolio under the assumption of normally distributed returns. Surprisingly, we find that the assumption of normality incurs only a modest amount of extra risk for all but the largest losses. We make the software written in support of this work freely available and describe its use in the appendix.