Deductibles and Policy Limits in Loss Models for STAM

The Ground-Up Loss and Insurance Payment

The ground-up loss X represents the total loss before any insurance modifications. The insurer's payment depends on the policy provisions: deductible, policy limit, and coinsurance. Understanding how each provision transforms the ground-up loss into the insurer's payment is fundamental to loss modeling on Exam STAM.

There are two perspectives for analyzing insurance payments. The "per-loss" perspective considers all losses, including those below the deductible (which produce zero payment). The "per-payment" perspective considers only losses that exceed the deductible and produce a positive payment. Both perspectives are tested on Exam STAM, and it is essential to know which one the problem is asking for.

Per-Loss Deductible

With an ordinary (per-loss) deductible d, the insurer pays Y^L = (X - d)+ = max(X - d, 0) per loss. When X <= d, the policyholder absorbs the entire loss and the insurer pays nothing. When X > d, the insurer pays X - d. 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.

The limited expected value E[X ^ d] = integral from 0 to d of S(x) dx, where S(x) = 1 - F(x) is the survival function. This formula is one of the most important on Exam STAM and should be memorized. For common distributions: exponential with mean theta gives E[X ^ d] = theta*(1 - e^(-d/theta)); Pareto with parameters alpha and theta gives E[X ^ d] = theta/(alpha-1) * (1 - (theta/(d+theta))^(alpha-1)) for alpha not equal to 1.

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. This quantity is central to pricing insurance layers. For the exponential, e(d) = theta (constant, due to the memoryless property). For the Pareto, e(d) = (theta + d)/(alpha - 1), which increases with d.

Policy Limits

A policy limit caps the insurer's payment. The limit can be expressed in two ways, and Exam STAM problems use both. A maximum covered loss of u (measured from the ground up) means the insurer covers losses up to u. With a deductible d, the maximum payment is u - d. Alternatively, the limit may be stated directly as the maximum payment amount m, so the insurer pays min((X - d)+, m).

The expected payment per loss with both deductible d and maximum covered loss u is E[Y] = E[X ^ u] - E[X ^ d] = integral from d to u of S(x) dx. This elegant formula shows that the expected payment is simply the integral of the survival function over the coverage interval.

Example: X ~ Exponential(1000), deductible d = 200, maximum covered loss u = 5000. E[X ^ 200] = 1000*(1 - e^(-0.2)) = 181.27. E[X ^ 5000] = 1000*(1 - e^(-5)) = 993.26. Expected payment per loss = 993.26 - 181.27 = $811.99.

Effect on Claim Frequency

With a deductible, only losses exceeding d produce payments. The frequency of payments N^P differs from the frequency of losses N. The probability that a loss produces a payment is P(X > d) = S(d). If N ~ Poisson(lambda), then N^P ~ Poisson(lambda * S(d)). This "thinning" property of the Poisson is exact and preserves the Poisson form.

For negative binomial frequency with parameters r and beta, the thinned frequency is negative binomial with parameters r and beta*S(d). For binomial frequency with parameters n and q, the thinned frequency is binomial with parameters n and q*S(d). These results follow from the general theory of compound distributions applied to Bernoulli thinning.

The aggregate payment distribution uses the thinned frequency and the per-payment severity distribution (the excess loss distribution, conditional on X > d). The expected aggregate payment is E[S^P] = E[N^P] * E[Y^P | payment], where Y^P is the payment amount given a payment occurs.

Franchise Deductibles and Disappearing Deductibles

A franchise deductible differs from an ordinary deductible: when X <= d, the insurer pays nothing (same as ordinary), but when X > d, the insurer pays the full amount X (not X - d). The expected payment per loss is E[X] * P(X > d) + integral from d to infinity of 0 dx... more precisely, E[Y^franchise] = E[X * I(X > d)] = E[X] - E[X ^ d] + d * S(d). This exceeds the ordinary deductible payment by d * S(d).

A disappearing deductible linearly reduces the deductible as the loss increases, eventually providing full coverage above a certain loss level. These are less common on Exam STAM but may appear as a specialized problem. The key is to write down the payment function Y as an explicit function of X and then compute E[Y].

Inflation and Loss Elimination

When losses are subject to uniform inflation at rate r, each loss X becomes X*(1+r). With a fixed deductible d, the expected payment per loss changes in two ways: larger losses increase payments, and some losses that were previously below the deductible now exceed it. The loss elimination ratio (LER) measures the proportion of expected losses eliminated by the deductible: LER = E[X ^ d] / E[X]. After inflation, LER decreases because the fixed deductible eliminates a smaller proportion of the inflated losses.

The expected payment per loss after inflation with deductible d is E[(X*(1+r) - d)+] = (1+r) * E[(X - d/(1+r))+] = (1+r) * (E[X] - E[X ^ d/(1+r)]). This formula shows that uniform inflation with a fixed deductible is equivalent to reducing the effective deductible from d to d/(1+r) and then scaling the result by (1+r). This leveraging effect causes insurer payments to increase faster than the inflation rate.

Study Tips and Common Pitfalls

The most critical distinction on Exam STAM is between "per loss" and "per payment." Per-loss calculations use the full loss distribution and include zero payments. Per-payment calculations condition on X > d and use the excess loss distribution. Confusing the two is the single most common error on deductible problems.

Another frequent mistake is confusing the maximum covered loss u (from the ground up) with the maximum payment m. If the problem says "maximum payment of $5,000" with a deductible of $500, the maximum covered loss is u = $5,500, and the payment is min((X - 500)+, 5000). If it says "coverage up to $5,000" with a $500 deductible, u = 5000 and the maximum payment is $4,500. Read the problem statement carefully. Practice at least 6 problems combining deductibles and limits with different severity distributions.