The mean excess loss function e(d) is defined as which of the following?

• A. e(d) = E[X - d | X > d]
• B. e(d) = E[X | X > d] - d
• C. e(d) = E[(X - d)^+]
• D. e(d) = E[X] - d

Answer: (A)

The mean excess loss function describes the average amount by which a loss exceeds the deductible, but only for losses that actually exceed the deductible. This is captured by the conditional expectation E[X − d | X > d], which directly measures the typical excess above the deductible when a claim is larger than d. This form is also the same as E[X | X > d] − d, since subtracting the deductible after taking the conditional expectation yields the average excess. Note that the unconditional average excess, E[(X − d)⁺], would mix in cases where the loss doesn’t exceed the deductible (giving zero in those cases) and is a different quantity.