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

Value at Risk and Tail Value at Risk are two central risk measures in insurance and finance, used notably to calibrate solvency capital. Value at Risk, at a given confidence level and over a given horizon, indicates the loss threshold that should be exceeded only with a low probability, for example the 99.5 percent one-year loss underlying the SCR. Its major weakness is that it says nothing about the magnitude of losses once that threshold is breached, two portfolios with the same VaR being able to display very different extreme losses. Tail Value at Risk, also called expected shortfall, corrects this flaw by measuring the average loss conditional on exceeding the threshold, that is, the expected loss in the cases where the VaR is actually exceeded. TVaR is therefore more prudent and better suited to fat-tailed distributions, and it has better mathematical properties, notably sub-additivity, which rewards diversification. This is why it tends to be favored by modern regulators for risks with strong catastrophic potential. In cyber, where losses follow fat-tailed laws, TVaR is markedly more informative than VaR for appraising actual exposure beyond the threshold.

Two portfolios show an identical VaR of 100 million at 99.5 percent, but one has a TVaR of 130 million and the other of 220 million, revealing a far heavier extreme exposure for the second.

VaR, TVaR, value at risk, tail value at risk, expected shortfall