A Note On Portfolio Efficient Frontier Based On Gev Under Various Risk Measures

Modern protfolio theory is finance finance domain arouses the widespread interest and the thorough discussion on key topic in the recent years. In 1952, the American economists, financial scientist Harry·m·Markowitz published "Portfolio Selection"and it symbolized the beginning of modern equity portfolio theory. In this paper Markowitz proposed the variance as the the target risk measurement. But from the point of view of risk measurement the variance is not a satisfactory measure. However, in the real life, such many investers have different opinions of risk utility that applyint the theory in practice causing many mistakes and the distribution of financial assets datas is not usually symmetric, but is characterized by several stylized facts: volatility clustering and excess kurtosis. In this paper, when the returns'distribution is subjectedd to fat-tailed, we consider various risk measures, i.e. variance, VaR and ES and we study the efficient frontiers obtained by solvingthe portfolio selection problem under these measures.In this work,after introducing the distribtion of multivariate generalized extreme value we have analyzed the portfolio selection problem following the mean-risk approach. We have shown that under the assumption of multivariate generalized extreme value distributed returns, the set of efficient portfolios under VaR and ES is a subset of the set of efficient portfolios under the standard deviation: (μ,σ)-portfolio selection could be inefficient under VaR or ES, but the opposite never occurs. Moreover, the set of efficient portfolios under VaR is a proper subset of the set of efficient frontier under ES. We have also shown that(μ,VaRα) and (μ,ESα) efficient frontiers could be empty for value ofαgreater than a given level. This suggests that the choice of the levelαneed some precautions. In the presence of a risk-free asset, the set of efficient portfolios under the various risk measures are identical, unless one of these is empty. This allows an extension of Tobin separation in the case of(μ,VaRα) or (μ,ESα) portfolio selection.