For per-loss super with deductible d, policy limit u, coinsurance alpha, and inflation rate r, the per-loss expected payment equals which expression? The formula uses m = (u/alpha) + d.

• A. alpha*(1+r)*( E[ X ∧ m/(1+r) ] - E[ X ∧ d/(1+r) ] ), where m = (u/alpha) + d
• B. alpha*(1+r)*( E[ X ∧ m ] - E[ X ∧ d ] )
• C. (alpha*(1+r))*E[X]
• D. (1/(alpha*(1+r))) * E[ X ∧ m/(1+r) ]

Answer: (A)

The main idea is that the insurer’s per-loss payment with a deductible and a coinsurance share up to a limit is the post-deductible amount, capped by the policy limit, and then adjusted for inflation or discounting when we move to present value. After the deductible d, the insurer covers a fraction α of the remaining loss, but cannot exceed the total limit u. So the payment is P = min{ α*(X − d)+, u }. This can be rewritten as α * min{ (X − d)+, u/α } which in turn equals α times [ min{ X, d + u/α } − min{ X, d } ], because the difference between those two mins captures exactly the portion of X that lies above the deductible but not beyond the cap when scaled by α.

Let m = d + u/α. Then the expected payment is E[P] = α * E[ min{ (X − d)+, u/α } ] = α * E[ min{ X, m } − min{ X, d } ]. To express this in present value terms with inflation rate r, use the identity min{ (1+r)Y, m } = (1+r) min{ Y, m/(1+r) }. If X represents the loss in present-value terms after adjusting for r, the present-value expectation becomes α(1+r) times the difference E[ min{ X, m/(1+r) } − min{ X, d/(1+r) } ]. Substituting m = (u/α) + d gives the stated expression, which is the best option. The other forms omit the proper discounting/discounted cap structure or collapse the cap incorrectly, so they do not represent the per-loss payment with these features.