Portfolio credit risk is often concerned with the tail distribution of the total loss, defined to be the sum of default losses incurred from a collection of individual loans made out to the obligors. The default for an individual loan occurs when the assets of a company (or individual) fall...

We introduce a class of quantile-based risk measures that generalize Value at Risk (VaR) and, likewise Expected Shortfall (ES), take into account both the frequency and the severity of losses. Under VaR a single confidence level is assigned regardless of the size of potential losses. We allow...