Coinsurance and Reinsurance Provisions for Exam STAM

Coinsurance Provisions

Under coinsurance, the insured retains a proportion (1 - c) of each loss above the deductible, and the insurer pays proportion c, where 0 < c <= 1. With deductible d and coinsurance factor c, the payment per loss is Y = c * (X - d)+ = c * max(X - d, 0). The expected payment per loss is c * E[(X - d)+] = c * (E[X] - E[X ^ d]).

Coinsurance simply scales the payment by a constant factor. This means E[Y] = c * E[(X-d)+], Var(Y) = c^2 * Var((X-d)+), and more generally all moments of Y are proportional to the corresponding moments of (X-d)+. The distribution of Y/c (the unscaled payment) has the same shape as the excess loss distribution.

Example: X ~ Pareto(3, 1000), deductible d = 200, coinsurance c = 0.80. E[X] = 1000/(3-1) = 500. E[X ^ 200] = 500 * (1 - (1000/1200)^2) = 500 * (1 - 0.6944) = 152.78. E[(X-200)+] = 500 - 152.78 = 347.22. Expected insurer payment = 0.80 * 347.22 = $277.78. The insured's expected out-of-pocket cost is 152.78 + 0.20 * 347.22 = $222.22.

Combining Deductible, Coinsurance, and Limit

A complete insurance policy typically combines all three provisions. The insurer pays Y = c * min((X - d)+, u), where d is the deductible, c is the coinsurance factor, and u is the maximum benefit (the policy limit). The expected payment is E[Y] = c * (E[X ^ (d + u/c)] - E[X ^ d]) when the limit applies to the insurer's share, or c * (E[X ^ (d + u)] - E[X ^ d]) when the limit applies to the ground-up loss.

The exact formulation depends on how the policy provisions interact. On Exam STAM, read carefully to determine whether the limit applies to the insurer's payment (maximum insurer outlay is u) or to the total loss above the deductible (maximum loss covered is u, of which the insurer pays proportion c). The payment formulas differ depending on this interpretation.

A common setup on Exam STAM: the insured pays a deductible of d, the insurer pays c of the excess up to a maximum insurer payment of m. The payment is Y = c * min((X-d)+, m/c) = min(c*(X-d)+, m). Here the effective coverage limit from the ground up is d + m/c.

Excess-of-Loss Reinsurance

In excess-of-loss (XOL) reinsurance, the reinsurer covers the portion of each individual claim that exceeds the direct insurer's retention r, up to a maximum reinsurer payment of m. The reinsurer pays min((X - r)+, m) per claim. The expected reinsurer payment per loss is E[X ^ (r + m)] - E[X ^ r].

This is mathematically identical to an insurance policy with deductible r and limit m, viewed from the reinsurer's perspective. All the formulas for deductibles and limits carry over directly. The direct insurer retains the first r of each claim and anything above r + m. The reinsurer covers the "layer" from r to r + m.

Example: Direct insurer has retention r = 10,000 and the reinsurer covers up to m = 40,000 above the retention. For a claim of $35,000, the reinsurer pays min(35000 - 10000, 40000) = $25,000. For a claim of $60,000, the reinsurer pays min(60000 - 10000, 40000) = $40,000, and the direct insurer retains 10,000 + (60,000 - 50,000) = $20,000.

Proportional Reinsurance

Under quota share reinsurance, the reinsurer pays a fixed proportion c of every claim, and the direct insurer retains proportion (1 - c). The reinsurer's payment per claim is c * X. This is the simplest form of reinsurance and is mathematically equivalent to coinsurance with no deductible.

Under surplus share reinsurance, the reinsurer's share depends on the relationship between the sum insured (policy face amount) and the direct insurer's retention. If the sum insured exceeds the retention, the reinsurer covers the excess proportion. The reinsurer's share of any claim on that policy is (sum insured - retention) / sum insured. Surplus share provides proportional coverage that varies by policy based on the size of the exposure.

Proportional reinsurance reduces the variability of the direct insurer's results but does not protect against individual large claims (unless combined with excess-of-loss coverage). Excess-of-loss reinsurance specifically targets large claims and is more efficient at reducing the tail risk.

Stop-Loss and Aggregate Reinsurance

Stop-loss (aggregate excess-of-loss) reinsurance applies to the total aggregate losses S rather than individual claims. The reinsurer pays (S - r)+ where r is the aggregate retention. The stop-loss premium is E[(S - r)+] = E[S] - E[S ^ r] = E[S] - integral from 0 to r of (1 - G(s)) ds, where G is the CDF of S.

Computing stop-loss premiums requires knowledge of the full aggregate loss distribution, which can be obtained via Panjer recursion, simulation, or the normal/translated gamma approximation. The stop-loss premium is a decreasing function of the retention r and satisfies E[(S-r)+] <= E[S] for any r >= 0.

A stop-loss premium with an aggregate limit caps the reinsurer's total payment: the reinsurer pays min((S-r)+, m). The expected payment is E[S ^ (r+m)] - E[S ^ r]. This combines aggregate deductible and aggregate limit concepts, paralleling the individual claim formulations.

Layered Reinsurance Programs

In practice, reinsurance programs often involve multiple layers. For example, the direct insurer retains the first $100,000 of each claim. Layer 1 covers $100,000 excess of $100,000 (from $100K to $200K). Layer 2 covers $300,000 excess of $200,000 (from $200K to $500K). Each layer has its own expected loss, which can be computed using the limited expected value formula applied to successive retention points.

The expected loss for each layer is E[X ^ (r_upper)] - E[X ^ (r_lower)], where r_lower and r_upper are the attachment point and exhaustion point of the layer. The sum of expected losses across all layers plus the retained loss equals E[X]. This additive decomposition is a fundamental property that provides a consistency check on calculations.

Study Tips and Common Pitfalls

The most common mistake is confusing the retention (deductible) with the limit. The retention is the amount below which the reinsurer does not pay; the limit is the maximum the reinsurer pays above the retention. The layer from r to r + m has retention r and limit m, covering losses in the interval [r, r + m].

Another frequent error is computing the stop-loss premium using the wrong aggregate distribution. Always verify that the aggregate distribution correctly reflects the frequency model, severity model, and any policy modifications (deductibles, limits) before computing stop-loss premiums. Practice at least 5 problems involving layered reinsurance calculations and 3 problems involving stop-loss premiums with different aggregate approximation methods.