With a franchise deductible D and deductible amount d, the expected value of Y under a super policy is E[X] - E[min(X, D)] + d*S(d).

• A. E[X] - E[min(X, D)] + d*S(d)
• B. E[X] - E[min(X, D)]
• C. E[X] + d*S(d)
• D. E[X] - d*S(d)

Answer: (A)

The key idea is how the payout under this super policy breaks down and how to take expectations of those parts.

The policy pays the amount by which a claim exceeds the franchise D, whenever X > D. That part is (X − D)+, where (·)+ means max with 0. It also pays an extra fixed amount d whenever the claim exceeds d, which contributes d times the probability that X > d (i.e., d·S(d)).

So the total expected payout is E[(X − D)+] + d·P(X > d). Use the identity (X − D)+ = X − min(X, D) to get E[(X − D)+] = E[X] − E[min(X, D)]. Let S(d) denote P(X > d). Put these together:

E[Y] = E[X] − E[min(X, D)] + d·S(d).

This matches the given expression, since E[X] is the mean claim, subtracting E[min(X, D)] removes the part up to the franchise, and the term d·S(d) adds the extra fixed payment when the claim exceeds d.