An Extreme Value Theory Approach to the Allocation of Multiple Assets

Abstract

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.

Tail dependence and diversification benefits in emerging market stocks: an extreme value theory approach

“…This result implies that the assumption of a multivariate normal distribution for asset returns leads to underestimation of the probability of extreme co-crashes. These results are consistent with Bradley and Taqqu (2004), Turgutlu and Ucer (2010) and Choletta et al (2012). Bradley and Taqqu (2004) studied extremal dependence for international stock markets and how it affects asset allocation.…”

“…These results are consistent with Bradley and Taqqu (2004), Turgutlu and Ucer (2010) and Choletta et al (2012). Bradley and Taqqu (2004) studied extremal dependence for international stock markets and how it affects asset allocation. In particular, they studied the impact of extremal dependence on the weight given to riskier asset in a two-asset portfolio optimization problem for risk-averse investors.…”

“…Following this idea, Li et al (2003) used the reduction in variance as a measure of the benefits of international diversification. In this study, following Bradley and Taqqu (2004), we assume that strongly risk-averse investors, in an attempt to guard against extreme losses, would seek to minimize VaR or ES rather than variance. Another motivation for this approach is that we are approaching the diversification problem from a regulatory perspective, in which the diversification benefit is primarily measured by the reduction in risk.…”

Extreme Value Theory in Finance: A Survey

“…In agreement with Poon et al . (), Bradley and Taqqu () find for 12 international equity markets that most of them are asymptotically independent and the few cases of asymptotic dependence are related by geographical proximity.…”

“…Bradley and Taqqu () improve on this algorithm by proposing a two‐step procedure, which uses, in the first step, Bensalah's sampling algorithm to generate a sensible starting point for the application of an incremental trade algorithm at the second step. Along the same lines as Bensalah (), Bradley and Taqqu () study the risk‐minimization problem comparing normal and extreme value distributions in the computation of VaR and ES as risk measures. Making both simulation and empirical analyses, they find that: the optimal allocation is quantile‐based, i.e.…”