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Mean-Variance-VaR Based Portfolio Optimization Authors: Wang J. Source: Working paper, Department of Mathematics and Computer Science, Valdosta State University Date: October 2000 |
In this paper, Wang compares mean-variance and mean-VaR approaches and develops a two step portfolio optimization procedure which is meant to benefit from the advantages of each approach and which appears to bypass most of their drawbacks. In fact, the use of variance (or standard deviation) to optimize portfolios not only reduces risk, but also reduces rewards. Even if VaR could be considered to be a new benchmark for managing risk, it does not constitute a coherent risk measure (it violates the subadditivity axiom which hinders diversification attempts) and it focuses on probability of loss and not on its magnitude. The first stage of his approach is to select efficient portfolios with a primary risk measure and then optimize those portfolios using a second risk measure. Analyzing simple examples, he finds that in general, a mean-variance efficient portfolio is not a mean-VaR efficient portfolio and vice-versa. The only case which gives the same result is the multivariate normal case (i.e under a normality assumption, the mean-VaR efficient portfolio is mean-variance efficient), where VaR is proportional to the variance. He also develops a mean-variance with a minimum conditional expected loss algorithm in order to deal with the magnitude of loss and not just its probability, as is the case with VaR. In the last part of his paper, Wang provides a global methodology based on both variance and VaR taken simultaneously as a double risk measure. The model he proposes leads to the conclusion that a mean-variance-VaR efficient portfolio may not be a mean-variance or a mean-VaR efficient portfolio. His last model includes the mean-variance and the mean-VaR models that can be seen to be nested in his mean-variance-VaR model.
