Excess Loss and Stop-Loss Reinsurance for Exam STAM

Excess of Loss Reinsurance

Per-occurrence excess of loss reinsurance covers individual claims above a retention (attachment point) M, up to a limit u. The reinsurer pays min(max(X - M, 0), u) for each claim. When a claim is below M, the reinsurer pays nothing. When a claim is between M and M+u, the reinsurer pays the excess above M. When a claim exceeds M+u, the reinsurer pays exactly u and the cedant retains the additional excess.

The expected reinsurance cost per claim is E[min(max(X - M, 0), u)] = E[X wedge (M+u)] - E[X wedge M], where E[X wedge c] is the limited expected value at c. This formula is the difference of two limited expected values, and computing it requires the severity distribution's limited expected value function. For common distributions, these have closed-form expressions that you should memorize for Exam STAM.

Pricing Excess of Loss Layers

An excess of loss layer is defined by its attachment point M and its width u. For example, a "500 excess of 1000" layer covers the portion of each claim between 1000 and 1500. The expected cost per claim for this layer is E[X wedge 1500] - E[X wedge 1000].

For a Pareto distribution with parameters alpha and theta, the limited expected value is E[X wedge c] = (theta/(alpha-1)) * [1 - (theta/(theta+c))^(alpha-1)] for alpha > 1. Suppose alpha = 2 and theta = 3000. Then E[X wedge 1500] = 3000 * [1 - (3000/4500)^1] = 3000 * [1 - 0.6667] = 1000, and E[X wedge 1000] = 3000 * [1 - (3000/4000)^1] = 3000 * [1 - 0.75] = 750. The expected cost per claim for the layer is 1000 - 750 = 250.

For an exponential distribution with mean theta, E[X wedge c] = theta * (1 - exp(-c/theta)). If theta = 2000, then E[X wedge 1500] = 2000*(1 - exp(-0.75)) = 2000*0.5276 = 1055.2 and E[X wedge 1000] = 2000*(1 - exp(-0.5)) = 2000*0.3935 = 787.0. The layer cost is 1055.2 - 787.0 = 268.2 per claim.

Reinsurance Premium Calculation

The total expected reinsurance cost for a portfolio is the expected cost per claim multiplied by the expected number of claims: E[N] * (E[X wedge (M+u)] - E[X wedge M]). The actual reinsurance premium adds a loading for expenses and profit. Common loading methods include a flat percentage loading, an expected value loading (adding a proportion of the expected cost), and a standard deviation loading (adding a multiple of the standard deviation of the reinsured losses).

The variance of the reinsurer's payment per claim is E[(min(max(X-M,0),u))^2] - (E[min(max(X-M,0),u)])^2. Computing the second moment requires the second limited moment: E[min(X,c)^2] = 2 * integral from 0 to c of x * S(x) dx, which equals 2 * integral from 0 to c of x * (1 - F(x)) dx. These integrals have closed forms for standard distributions.

Stop-Loss (Aggregate Excess) Reinsurance

Stop-loss reinsurance operates on the aggregate losses rather than individual claims. If S is the total aggregate losses for the portfolio, the reinsurer pays max(S - d, 0), where d is the aggregate retention (attachment point). The reinsurer may also have a limit, in which case the payment is min(max(S - d, 0), u_agg).

The expected stop-loss premium is E[max(S - d, 0)] = E[S] - E[S wedge d] = E[S] - integral from 0 to d of S_S(x) dx, where S_S is the survival function of S. Computing this requires knowledge of the entire aggregate loss distribution, not just the severity distribution. The aggregate distribution can be obtained through several methods: convolution (exact but computationally intensive), the Panjer recursion (efficient for discretized severities with (a,b,0) class frequency), normal approximation (quick but approximate), or Monte Carlo simulation.

Normal Approximation for Stop-Loss Premiums

A practical approximation uses the normal distribution for S. If E[S] = mu_S and Var(S) = sigma_S^2, the stop-loss premium is approximately sigma_S * [phi(z) - z*(1 - Phi(z))], where z = (d - mu_S)/sigma_S, phi is the standard normal PDF, and Phi is the standard normal CDF. This formula is derived from the properties of the truncated normal distribution.

For example, suppose a portfolio has E[S] = 100,000 and SD(S) = 20,000. The expected stop-loss premium with retention d = 120,000 uses z = (120,000 - 100,000)/20,000 = 1.0. Then phi(1.0) = 0.2420 and 1 - Phi(1.0) = 0.1587, giving the stop-loss premium as 20,000 * (0.2420 - 1.0*0.1587) = 20,000 * 0.0833 = 1,666. This approximation works well when S is approximately normal, which holds for large portfolios by the central limit theorem.

Comparing Reinsurance Structures

Per-occurrence excess of loss and aggregate stop-loss reinsurance serve different purposes. Per-occurrence excess protects against individual large claims and is useful for lines with heavy-tailed severity distributions like liability and property catastrophe. Stop-loss reinsurance protects against adverse aggregate experience from any combination of frequency and severity, including scenarios where many moderate claims accumulate to an unexpectedly large total.

The cost of each type depends on different features of the loss distribution. Per-occurrence excess cost depends primarily on the tail of the severity distribution. Stop-loss cost depends on the variability of aggregate losses, which is driven by both frequency and severity variation. A portfolio with volatile frequency but moderate severity might benefit more from stop-loss coverage, while a portfolio with stable frequency but heavy-tailed severity might prefer per-occurrence excess.

Common Pitfalls

A frequent error is confusing the attachment point with the limit. The attachment point M is where the reinsurer starts paying; the limit u is the maximum the reinsurer pays per occurrence (or in aggregate for stop-loss). The reinsurer's payment range is from M to M+u, not from 0 to u. Another common mistake is forgetting that stop-loss premiums require the aggregate distribution, not just the severity distribution. You cannot compute a stop-loss premium from limited expected values of individual claims alone.

When using the normal approximation for stop-loss, students sometimes forget the formula and try to reconstruct it from first principles during the exam. Memorize the formula sigma * [phi(z) - z*(1-Phi(z))] and practice computing it quickly with standard normal tables.

Study Tips

For Exam STAM, practice computing limited expected values for each major distribution family, as these are the building blocks for pricing excess of loss layers. Be comfortable with both per-occurrence and aggregate reinsurance structures. Work through problems that combine the two, such as computing the cedant's retained risk after both per-occurrence and stop-loss reinsurance. Know the normal approximation for stop-loss premiums and when it is appropriate to use. Time yourself on reinsurance problems to ensure efficiency during the exam.