Value at Risk and Tail Value at Risk (VaR, TVaR)

Value at Risk (VaR) at level p is the p-quantile of the loss distribution over a given horizon, the loss exceeded only with probability 1 − p, such as the one-year 99.5% VaR on which Solvency II calibrates the SCR. Tail Value at Risk (TVaR) at the same level, also called expected shortfall, is the average loss in the 1 − p worst cases, that is beyond the VaR; the Swiss regime (SST) calibrates its capital on it, at 99%. North American actuaries also call it CTE, conditional tail expectation, and the three names coincide for a continuous distribution. Set against TVaR, VaR has two flaws. Two books with the same VaR can have tails that look nothing alike, since VaR does not look past its threshold, and it fails subadditivity, so it can penalize diversification. TVaR has the opposite flaw. It rests entirely on the handful of extreme scenarios the modeler managed to write, and in cyber, where those scenarios are rarely observed, it measures the model as much as the risk. In reinsurance, cat models report both off the OEP and AEP curves, and TVaR is the usual basis for allocating capital across treaties. Climate VaR, as asset managers use the term, is a different metric, a portfolio's loss in value under a climate scenario.