Reinsurance Pricing: Excess of Loss and Aggregate

Per-Risk Excess of Loss

Per-risk excess of loss reinsurance covers individual losses above a retention (attachment point) M, up to a limit L. The reinsurer pays min(max(X minus M, 0), L) for each loss X. This type of treaty protects the cedant against large individual claims that could materially affect their results. The reinsurance pure premium per loss is E[min(max(X minus M, 0), L)] = E[X wedge (M + L)] minus E[X wedge M], where E[X wedge d] is the limited expected value at d.

Pricing per-risk excess layers requires accurate modeling of the severity distribution in the tail above the retention. If the retention is high (say, 1 million), then only the portion of the loss distribution above 1 million matters for pricing. Standard distributions fitted to the entire loss range may not adequately capture the tail. Extreme value theory (fitting a GPD to exceedances above a threshold) or specialized large-loss data analysis provides more accurate tail estimates.

Increased Limits Factors

The increased limits factor (ILF) approach provides a convenient framework for pricing excess layers. The ILF at limit d is defined as E[X wedge d] / E[X wedge B], where B is a basic limit (typically the minimum policy limit offered). The ILF represents the ratio of the expected loss at limit d to the expected loss at the basic limit. An excess layer from M to M + L has a pure premium proportional to ILF(M + L) minus ILF(M).

ILFs must be increasing and concave (the marginal cost of additional coverage decreases with increasing limits) for the pricing to be internally consistent. If the fitted severity distribution produces ILFs that violate concavity, the distribution may need adjustment or a different distributional family may be needed. In practice, ILFs are computed from fitted loss distributions (Pareto, lognormal, or mixed exponential) calibrated to both the body and tail of the loss data.

Per-Occurrence Excess of Loss

Per-occurrence (or catastrophe) excess of loss reinsurance covers aggregate losses from a single event (such as a hurricane, earthquake, or industrial explosion) above a retention. Unlike per-risk excess, which applies to individual claims, per-occurrence excess aggregates all claims arising from a single event. Pricing requires modeling the distribution of event-level aggregate losses across all affected policies.

Event-level aggregate losses depend on the frequency of events, the number of policies affected per event, and the severity of individual policy losses. Catastrophe models (combining hazard, vulnerability, exposure, and financial modules) are the primary tool for estimating these distributions. For non-catastrophe perils, historical event-level loss data, trended and developed to current conditions, provides the pricing basis.

Reinstatement Provisions

Excess of loss treaties typically have limited coverage: after the reinsurer pays a loss, the coverage is exhausted up to the amount paid. Reinstatement provisions restore the coverage in exchange for an additional premium. A treaty might specify "one reinstatement at 100%," meaning after the first recovery, the full limit is reinstated upon payment of an additional premium equal to 100% of the original premium (pro-rated by the proportion of the limit exhausted).

The reinstatement premium for a partial loss is typically calculated as: reinstatement premium = (loss amount / limit) * reinstatement rate * original premium. For example, if the limit is 10 million, the original premium is 500,000, a loss of 6 million occurs, and the reinstatement rate is 100%, the reinstatement premium is (6/10) * 1.00 * 500,000 = 300,000. The total cost to the cedant for this recovery is the original premium plus the reinstatement premium, while the reinsurer's net result is the recovery paid minus the reinstatement premium received.

Aggregate and Stop-Loss Reinsurance

Aggregate excess (or stop-loss) reinsurance covers total losses over a specified period (typically one year) that exceed a threshold, subject to a limit. The threshold may be expressed as a dollar amount or as a loss ratio. For example, a stop-loss treaty might cover aggregate losses in excess of a 75% loss ratio, up to a 95% loss ratio. The reinsurer pays the difference between actual aggregate losses and the threshold, capped at the limit.

Pricing aggregate reinsurance requires the aggregate loss distribution, which incorporates both the frequency and severity of individual losses and their potential correlations. Parameter uncertainty (uncertainty about the true values of frequency and severity parameters) widens the aggregate loss distribution and increases the reinsurance premium. Correlation between risks (from common exposures to economic conditions, weather, or legal trends) further increases aggregate volatility.

Pricing Methods

The burning cost method calculates the historical reinsurance cost by applying the treaty terms to trended and developed historical losses. For each historical year, compute the amount that would have been recovered under the proposed treaty, trend these recoveries to the prospective rating period, and average over the experience period. This approach is simple and transparent but can be unreliable when the experience period is short or when the losses are highly variable.

Exposure rating derives the reinsurance premium from the modeled loss distribution rather than from historical treaty recoveries. It uses the fitted aggregate loss distribution (from frequency-severity models, catastrophe models, or simulation) to compute the expected layer loss. Experience rating blends the burning cost estimate with the exposure rating estimate using credibility weights that reflect the volume and relevance of historical data. The final price adds loadings for expenses, profit, and risk margin.

Common Pitfalls

A common error is pricing a per-occurrence layer using per-risk loss data without properly aggregating individual losses into event totals. Per-occurrence excess requires event-level aggregate distributions, not individual claim distributions. Another frequent mistake is forgetting to account for loss development and trend when computing burning cost estimates. Historical losses must be developed to ultimate and trended to the prospective period before applying the treaty terms.

Students also sometimes confuse reinstatement premiums with deposit premiums. Reinstatement premiums are additional charges triggered by losses, while deposit premiums are advance payments that may be adjusted retrospectively. Understanding the interaction of these terms is important for computing the total cost of a reinsurance program and the net result for both the cedant and the reinsurer.