FAM: Payment per loss and payment per payment have different denominators

A deductible creates zero payments on smaller losses. Expected payment per loss includes those zeroes; expected payment conditional on a positive payment does not. Define the payment random variable before calculating its mean.

Worked example or practice scenario

For an exponential loss with mean 1,000 and an ordinary deductible of 200, Y=(X−200)+. E[Y]=1,000 e^−.2 ≈ 818.73. Conditional on X>200, E[Y|Y>0]=1,000 by memorylessness. The two expectations answer different questions.

Try this next

Divide the unconditional mean by P(Y>0)=e^−.2 to verify the conditional mean. Then replace the ordinary deductible with a franchise deductible and write a new payment variable; the formula must change.

Reading sources

• Official syllabus and resources

ActNet editorial guide · October 1, 2026 · Original illustrative examples.