Safety-First Portfolio Linear Program Using Sampled Tail Distribution

Abstract

In mean-variance model, the performances of assets or portfolios are evaluated based on the mean and the variance of the corresponding return rates and the portfolio selection problem can be formulated into a simple mathematical program which can be solved very efficiently. However, the performances evaluated by investors are in more variety aspects. Investors would look for more information, especially the downside risk (tail-probability from the worse case), of the returns. In this study, we model a safety-first portfolio selection problem considering the downside risks. For better estimations of the downside risks, we usually try to use the extreme value theory to estimate them, however, it is not efficient to obtain an optimal portfolio if we adopt extreme value procedure. Instead of using extreme value theory directly, we use extreme copula to estimate the dependency on the distributions of the returns, based on which we simulate the returns and sample the joint worse cases (rare tail events) to obtain the approximated conditional tail distributions used for our model. By further presenting the corresponding evaluations in the fashion of linear equations, the corresponding safety-first portfolio optimization problem can be formulated as linear program, which can be solved by simplex method. Test results of ours performances and the results from the safety-first model using original historical data as well as the market's are compared.