Which expression represents the expected payment per payment in terms of the expected payment per loss and S(d)?

• A. E(Payment per loss) / S(d)
• B. E(Payment per loss) * S(d)
• C. E(Payment per loss) + S(d)
• D. E(Payment per loss) - S(d)

Answer: (A)

The key idea is conditioning on the event that a payment actually occurs. If you have an average payment amount per loss, but only a fraction S(d) of losses result in a payment (the insured survives to the payment date with probability S(d)), then the average payment observed per actual payment must be higher to account for those losses that don’t trigger a payment. Mathematically, the per-payment expectation is the per-loss expectation divided by the probability that a loss leads to a payment: E(Payment per loss) / S(d). For example, if the average payment per loss is 100 and only 0.8 of losses produce a payment (S(d) = 0.8), then the average payment per actual payment is 100 / 0.8 = 125. The other operations would not reflect conditioning on the payment actually occurring.