Suppose a deductible d is chosen so that p% of X are below d. Then TVaR_p(X) equals:

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Multiple Choice

Suppose a deductible d is chosen so that p% of X are below d. Then TVaR_p(X) equals:

Tail Value at Risk at level p looks at the average loss in the tail beyond the p-quantile, here with the deductible chosen so that p% are below d (VaR_p(X) = d). The tail contains two kinds of contribution to the insurer’s payment: a fixed deductible portion for the cases that don’t exceed d, and the excess beyond d when the loss surpasses the deductible. The fixed part averages to p d, since p% of outcomes are at or below d and pay up to the deductible. The variable part is the mean excess beyond d in the tail, which is E[(X - d) | X > d]. Adding these together gives the total tail measure: dp + E[(X - d) | X > d]. Choosing the other forms would miss one of those two pieces: E[(X - d)+] counts only the excess beyond d; E[(X - d) | X > d] counts the tail excess but not the deductible portion across all outcomes; and d p only accounts for the deductible portion.

Tail Value at Risk at level p looks at the average loss in the tail beyond the p-quantile, here with the deductible chosen so that p% are below d (VaR_p(X) = d). The tail contains two kinds of contribution to the insurer’s payment: a fixed deductible portion for the cases that don’t exceed d, and the excess beyond d when the loss surpasses the deductible. The fixed part averages to p d, since p% of outcomes are at or below d and pay up to the deductible. The variable part is the mean excess beyond d in the tail, which is E[(X - d) | X > d]. Adding these together gives the total tail measure: dp + E[(X - d) | X > d].

Choosing the other forms would miss one of those two pieces: E[(X - d)+] counts only the excess beyond d; E[(X - d) | X > d] counts the tail excess but not the deductible portion across all outcomes; and d p only accounts for the deductible portion.

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