Value at Risk and Expected Shortfall Improved Calculation Based on the Power Transformation Method

Abstract

Value-at-risk (VaR), expected shortfall (ES), and similar risk measures are based on knowledge of the underlying probability distribution of portfolio value, and in particular its lower tail. The theory is well developed for the familiar normal/lognormal case, but it is well known that the normal does not match the actual returns observed on portfolios of stocks and other risky assets. Empirical distributions tend to have fatter tails and negative skewness. Two standard ways to deal with this problem are either to assume the returns are generated by a probability law with more flexibility about tail shape than the Gaussian, such as one of the Johnson family of distributions, or else to develop an empirical fit to the unknown density using a technique such as Gram–Charlier or Cornish–Fisher approximation. In both approaches, the density is chosen to match the moments of the empirical density from the data. In this article, the author follows the second approach but suggests that a better approximation can be obtained using a power transformation of a standard normal variable whose coefficients are selected to match the first four moments of observed returns. Applying the technique on S&P 500 Index returns, the power transformation consistently produces more accurate estimates of VaR and ES than the other methods.