Deductibles and Policy Limits: Insurance Applications of Probability

Ordinary Deductibles

An ordinary (or per-loss) deductible d means the insurer pays (X - d) when X > d and nothing when X <= d. The payment random variable is Y = max(X - d, 0), also written (X - d)+. The expected payment (the limited expected value) is E[Y] = integral from d to infinity of (x - d) * f(x) dx = E[X] - E[min(X, d)], where E[min(X, d)] = integral from 0 to d of x*f(x) dx + d*(1 - F(d)).

For the exponential distribution with mean theta, E[(X - d)+] = theta * e^(-d/theta) due to the memoryless property. This result is frequently used on Exam P.

Policy Limits

A policy limit u caps the insurer's payment at u. If there is no deductible, the payment is min(X, u), and E[min(X, u)] = integral from 0 to u of (1 - F(x)) dx. With both a deductible d and a limit u, the payment is min(max(X - d, 0), u) = min(X, d + u) - min(X, d).

For example, if X ~ Exponential(1000), d = 200, and the maximum payment is u = 5000, then the insurer pays min(X - 200, 5000) when X > 200. The expected payment is E[min(X, 5200)] - E[min(X, 200)] = 1000(1 - e^(-5.2)) - 1000(1 - e^(-0.2)) = 1000(e^(-0.2) - e^(-5.2)).

Loss Elimination Ratio

The loss elimination ratio (LER) measures the proportion of expected losses eliminated by a deductible: LER = E[min(X, d)] / E[X]. It represents the fraction of losses borne by the insured. On Exam P, you may be asked to find the deductible that eliminates a given fraction of losses, which requires solving E[min(X, d)] / E[X] = target for d.