Tail VaR / Conditional Tail Expectation

Value at Risk tells you where the bad zone begins but stays silent about how deep it goes. Imagine you ask not 'how cold is it on the coldest day I should plan for?' but 'when it really is one of those bad days, how cold is it on average?' That shift — from the edge of the disaster region to the average severity inside it — is exactly what Tail VaR captures. It looks past the cut-off and asks: given that we are in the worst slice of outcomes, what is the typical loss?

Precisely: fix a confidence level, say 95%. The 95% VaR is the loss at the 95th-percentile cut-off. The 95% Tail VaR (also called Conditional Tail Expectation, CTE) is the average of all the losses that fall in the worst 5% — it conditions on being beyond the VaR point and takes the mean there. Numerically, suppose the worst 5% of outcomes are losses of 100, 120, 150, 200 and 400 (equally likely). VaR would report roughly 100 (the entrance to the tail), but Tail VaR reports their average, (100+120+150+200+400)/5 = 194 — a number that feels the 400 disaster that VaR shrugged off. Tail VaR is therefore always at least as large as VaR at the same level, and it grows when the tail is fat.

This is why North American actuaries adopted CTE for variable-annuity and segregated-fund capital, and why the Swiss Solvency Test uses a tail measure rather than a plain quantile. Tail VaR is a coherent risk measure (it never punishes diversification) and it actually responds to extreme scenarios. The honest caveat: averaging the tail requires you to model the tail, and the tail is precisely where data is thinnest and assumptions matter most — a Tail VaR built on an optimistic tail model can be just as misleading as VaR, only with more false comfort.