Beyond the VaR

Abstract

“Value at Risk has become a well-known and widely used approach to evaluating risk exposure, even though what VaR measures, in an important sense, is really the value not at risk. The 5% VaR level, for example, gives a lower bound on portfolio value that holds 95% of the time. But the VaR does not tell how large the loss is likely to be on the 5% of the occasions that the VaR level is penetratedÑonly that it will be worse than the 5% VaR value. To address this problem, Longin discusses a related concept, the expected value of the loss conditional on its being greater than the VaR, known as BVaR. For a lognormal distribution, there is a simple relationship between VaR and BVaR. But actual return distributions typically differ considerably from the lognormal, particularly in the extreme tails that are the focus of the VaR calculation. BVaR involves both the VaR and the shape of the entire tail of the distribution function. It is also better than VaR in taking account of the presence of securities with non-linear payoffs in the portfolio. Longin presents an analysis with historical data to show how VaR and BVaR calculations would differ for linear and non-linear positions based on the S&P 500 stock index and its options, under different assumptions about the returns distribution (e.g., lognormal, extreme value).”