Stochastic Dominance and Risk Measure: A Decision-Theoretic Foundation for VaR and C-VaR

Is it possible to obtain an objective and quantifiable measure of risk backed up by choices made by some specific groups of rational investors? To answer this question, in this paper we establish some behavior foundations for various types of VaR models, including VaR and conditional-VaR, as measures of downside risk. Though supported to some extent with unanimous choices by some specific groups of expected or non-expected utility investors, VaRs as profiles of risk measures at various levels of risk tolerance are not quantifiable – they can only provide partial and incomplete risk assessments for risky prospects. Also included in our discussion are the relevant VaRs and several alternative risk measures for investors; these alternatives use somewhat weaker assumptions about risk-averse behavior by incorporating a mean-preserving-spread. For this latter group of investors, we provide arguments for and against the standard deviation vs. VaR and conditional VaRs as objective and quantifiable measures of risk in the context of portfolio choice.