Deductibles and Policy Limits in Loss Models for STAM

Per-Loss Deductible

With an ordinary (per-loss) deductible d, the insurer pays Y^L = (X - d)+ = max(X - d, 0) per loss. The expected payment per loss is E[Y^L] = E[X] - E[X ^ d], where X ^ d = min(X, d) is the limited loss variable. E[X ^ d] = integral from 0 to d of S(x) dx, where S(x) = 1 - F(x) is the survival function.

The expected payment per payment (given a payment occurs) is E[Y^L | X > d] = E[Y^L] / S(d) = e(d), the mean excess loss function. For the exponential with mean theta, e(d) = theta (constant). For the Pareto, e(d) = (theta + d)/(alpha - 1), increasing in d.

Policy Limit

With a maximum covered loss of u (policy limit measured from the ground up) and deductible d, the insurer's payment per loss is Y = min((X - d)+, u - d). The maximum payment per loss is u - d. E[Y] = E[X ^ u] - E[X ^ d] = integral from d to u of S(x) dx.

On Exam STAM, be careful about whether the limit is stated as a maximum payment (u - d) or as a maximum covered loss level (u from the ground up). The formulas differ accordingly.

Effect on Frequency

With a deductible, only losses exceeding d produce payments. The frequency of payments is N^P, where P(payment) = P(X > d) = S(d). If N ~ Poisson(lambda), then N^P ~ Poisson(lambda * S(d)). This "thinned" frequency combined with the payment-size distribution (the excess loss distribution) gives the aggregate payment distribution. On Exam STAM, these modifications are fundamental to pricing insurance policies with deductibles and limits.