Coinsurance and Reinsurance Provisions for Exam STAM

Coinsurance

Under coinsurance, the insured retains a proportion (1 - c) of each loss above the deductible, and the insurer pays proportion c. 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)+]. Coinsurance simply scales the payment by a constant factor, so all moments of the payment distribution are scaled accordingly.

The combination of deductible, coinsurance, and policy limit is: Y = c * min((X - d)+, u), where u is the maximum benefit. Expected payment: c * (E[X ^ (d + u)] - E[X ^ d]).

Excess-of-Loss Reinsurance

In excess-of-loss (XOL) reinsurance, the reinsurer pays the amount by which each claim exceeds a retention r, up to a limit. If the direct insurer has retention r and the reinsurer covers up to m above r, the reinsurer pays min((X - r)+, m). 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.

Aggregate Reinsurance and Stop-Loss

Stop-loss (or 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]. Computing stop-loss premiums requires knowledge of the aggregate loss distribution, which can be obtained via Panjer recursion, simulation, or approximation. For Exam STAM, stop-loss premium calculations tie together frequency, severity, and aggregate modeling.