Which of the following correctly defines TVaRp(x) in terms of X and VaR_p?

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Multiple Choice

Which of the following correctly defines TVaRp(x) in terms of X and VaR_p?

Tail Value at Risk at level p is the average outcome in the tail beyond the p-quantile. It is defined as the conditional expectation of X given that X is at least the VaR at level p, so TVaR_p(x) = E[X | X ≥ VaR_p(x)]. This means you’re averaging only the worst outcomes, the losses that exceed the p-th percentile, rather than looking at the overall mean. This makes TVaR_p higher than VaR_p unless the tail is degenerate. It is not the unconditional mean E[X], and it is not simply a multiple of VaR_p, nor VaR_p itself. In continuous cases, you can also write it as E[X | X > VaR_p(x)].

Tail Value at Risk at level p is the average outcome in the tail beyond the p-quantile. It is defined as the conditional expectation of X given that X is at least the VaR at level p, so TVaR_p(x) = E[X | X ≥ VaR_p(x)]. This means you’re averaging only the worst outcomes, the losses that exceed the p-th percentile, rather than looking at the overall mean. This makes TVaR_p higher than VaR_p unless the tail is degenerate. It is not the unconditional mean E[X], and it is not simply a multiple of VaR_p, nor VaR_p itself. In continuous cases, you can also write it as E[X | X > VaR_p(x)].

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