LESSON 6.1 · ABOUT 20 MINUTES
Losses, deductibles & limits
Translate a policy into a payment function.
By the end of this lesson
- Distinguish loss from payment.
- Apply a deductible and a maximum payment.
Define the contract
X is the loss. With ordinary deductible d and maximum PAYMENT L, Y=min(max(X−d,0),L). A limit on covered LOSS is a different contract; read the words before choosing a formula.
Piecewise behavior
Below d the insurer pays zero. Between d and d+L the insurer pays X−d. Above d+L the payment is capped at L. Continuous losses can create point masses at payment zero and at the cap.
Per loss vs per payment
E[Y] averages over all losses, including zero payments. E[Y|Y>0] averages only positive payments. If P(Y>0)>0, E[Y|Y>0]=E[Y]/P(Y>0).
Y = min(max(X−d,0),L), where L is maximum payment
Worked example
A loss is $900, the ordinary deductible is $200, and the maximum payment is $500. Find payment.
The loss after deductible is $700. Capping the PAYMENT at $500 gives Y=$500. The insured bears the remaining $400.