# 8.5 Coinsurance, deductibles, and limits

The final common coverage modification is coinsurance. In this case the insurance company pays a proportion, α, of the loss and the policyholder pays the remaining fraction. If coinsurance is the only modification, this changes the loss variable *X* to the payment variable, *Y* = α*X*. The effect of multiplication has already been covered. When all four items covered in this chapter are present (ordinary deductible, limit, coinsurance, and inflation), we create the following per-loss random variable:

For this definition, the quantities are applied in a particular order. In particular, the coinsurance is applied last. For the illustrated contract, the policy limit is α(*u − d*), the maximum amount payable. In this definition, *u* is the loss above which no additional benefits are paid and is called the *maximum covered loss*. For the per-payment variable, *Y*^{P} is undefined for *X < d*/(1 + *r*).

Previous results can be combined to produce the following theorem, given without proof.

**Theorem 8.7** *For the per-loss variable*,

*The expected value of the per-payment variable is obtained as*

Higher moments are more difficult. Theorem 8.8 gives the formula for the second ...

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