Deductible

Study guides: Exam P-1, Exam ASTAM, Exam MAS-I

A Deductible is the initial amount of a loss that the policyholder must pay before insurance coverage begins.

If the loss $X$ is less than or equal to the deductible $d$ , the insurer pays nothing; above $d$ it pays the excess $X - d$
Deductibles save the expense of processing small claims, lower the premium, and give the policyholder an economic incentive to prevent losses
On Exam 5, a deductible is stated either as a flat dollar amount or as a percentage of the coverage amount: a 5% deductible on a home insured for $500,000 is equivalent to a flat $25,000 deductible

$Y = (X - d)_+$

$= \max(X - d,\ 0)$

$d = \text{deductible}, \quad X = \text{ground-up loss}$

Expected Payment with an Ordinary Deductible Ground-up losses $X \sim \text{Exponential}(\theta = 1000)$ . An ordinary deductible of $d = 200$ applies. Find $E[Y]$ where $Y = (X - 200)_+$ .

Answer For an exponential distribution, the expected payment with deductible $d$ is: $E[Y] = E[(X-d)_+] = \theta\, e^{-d/\theta} = 1000\, e^{-200/1000} = 1000\, e^{-0.2} \approx 818.73$ The insurer expects to pay approximately $818.73 per loss (including the zero payments when $X \leq 200$ ).

Insurer Payment at Different Loss Levels Under a policy with a $ $500$ deductible, what does the insurer pay on losses of $ $300$ and $ $2{,}000$ ?

Answer

For a $ $300$ loss: $(300 - 500)^+ = 0$ . The insurer pays nothing.

For a $ $2{,}000$ loss: $(2000 - 500)^+ = 1500$ . The insurer pays $ $1{,}500$ .