The expected payment amount beyond the deductible, conditional on a payment occurring, is which of the following?

• A. E[X] / d
• B. E[(d - X)+] / S(d)
• C. E[(X - d)^2] / S(d)
• D. E[(X - d)+] / S(d)

Answer: (D)

The key idea is the average amount paid beyond the deductible, given that a payment actually happens. The insurer pays the excess over the deductible, written as (X − d)+, which is zero when X ≤ d and equals X − d when X > d. A payment occurs precisely when X > d, with probability S(d) = P(X > d).

So the conditional expected excess is E[(X − d)+ | X > d]. By the definition of conditional expectation, this equals E[(X − d)+] / P(X > d) = E[(X − d)+] / S(d). This is exactly the expression given.

Intuitively, the numerator averages the excess over all claims, while the denominator weights only the cases where a payment occurs, yielding the average excess for those payments.

The other forms don’t match this quantity: they either concern the shortfall below the deductible, the squared excess, or an unconditional mean that doesn’t condition on a payment.