Integrated Reinsurance Program Design Workshop

Why Program Design Requires an Integrated View

A reinsurance program is not selected by finding the cheapest quoted layer or the largest available limit. A useful recommendation must connect several questions: how much expected loss is transferred, what premium is paid, how much economic capital is released, how much recovery may be lost to counterparty default, and whether the remaining exposure is acceptable. Those quantities can point in different directions. More protection can reduce retained loss and capital while still increasing total cost.

This workshop uses a hypothetical portfolio and an annual decision horizon. All monetary amounts in the portfolio case are in millions of dollars. Premiums, expected losses, and credit shortfalls are annual amounts; no discounting is applied. A capital charge is an annual cost for holding capital, not a claim payment. The approach is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

This comparison is a decision model, not a universal accounting identity. It assumes the capital measure and capital charge are applied consistently across alternatives, and that expected credit shortfall is included separately from the modeled insurance loss. If counterparty default risk is already included in the net-loss capital model, adding a separate capital adjustment for that same risk would double count it.

Start with Layer Payments and Capacity

For a per-occurrence layer attaching at \(a\) with limit \(u\), the contractual recovery for a loss \(X\) is \(\min((X-a)_+,u)\), where \((z)_+=\max(z,0)\). The layer pays nothing below attachment, pays the amount above attachment while the loss remains within the layer, and is capped at its limit. A tower’s capacity is the sum of its limits, but the actual payment depends on how the loss falls across the attachments.

Expected recovery and available capacity answer different questions. Expected recovery is a frequency-and-severity-weighted amount; capacity is the contractual maximum. For a peak event, map the loss through each layer and identify any gaps, exhausted limits, and retained amounts above the tower. Capacity should be allocated to the exposures the program is intended to protect, rather than treated as interchangeable across attachments.

Connect Reinsurance to Economic Capital

The key capital calculation starts with the same loss distribution before and after reinsurance. The expected loss changes when recoveries reduce retained claims; the tail measure also changes. Economic capital is the difference between the chosen tail measure and the expected loss, so the relief in capital is not generally equal to the expected recovery.

The identity follows directly from the definition: capital is the loss at the selected tail quantile in excess of the average loss. If a program changes the loss to \(L_{\text{net}}\), then \(EC_{\alpha,\text{net}}=\operatorname{VaR}_\alpha(L_{\text{net}})-E[L_{\text{net}}]\). The change in capital is therefore affected by both the change in the tail quantile and the change in expected loss.

Integrated Portfolio Case

Consider a hypothetical insurer comparing no reinsurance with two annual programs. Its portfolio model is an empirical distribution of 100,000 equally weighted simulated annual outcomes. The model estimates gross expected claims of \(41.2\) million and a 99% VaR of \(220.0\) million. Program A has expected contractual recoveries of \(12.0\) million, a premium of \(14.0\) million, an expected credit shortfall of \(0.3\) million, and a modeled net 99% VaR of \(180.0\) million. Program B has corresponding values of \(14.8\), \(18.0\), \(0.5\), and \(166.0\) million. The modeled net VaRs are outputs of the portfolio model, not gross VaR minus an expected recovery. The annual capital charge is 10% of economic capital.

Evaluate an Additional Layer Marginally

Once a base program is established, an additional layer can be assessed by its incremental economics. Let \(P\) be the layer premium, \(R\) its expected contractual recovery, \(C\) its expected credit shortfall, \(\Delta EC\) the reduction in economic capital, and \(k\) the annual capital charge rate. If the layer reduces expected retained claims by \(R\), while expected credit shortfall is added separately, its approximate change in total cost is:

The formula is obtained by comparing costs with and without the layer. Premium adds \(P\). The expected claim payment received reduces retained claims by \(R\), while the expected amount not collected adds \(C\). Finally, capital falls by \(\Delta EC\), reducing the annual charge by \(k\Delta EC\). A negative result is a modeled cost reduction; a positive result is an increase. The result alone does not settle the decision if the layer is needed to satisfy a peak-exposure or risk-appetite constraint.

Defend the Recommendation, Not Just the Arithmetic

A cost ranking is only one part of a program recommendation. The decision maker should also state what the program protects, how much capacity is available at the relevant attachment points, and what material risk remains with the insurer. For example, if the board requires at least a specified reduction in a peak-event net loss, the cheapest option is unacceptable if it fails that constraint. Conversely, buying more limit than the modeled risk appetite requires may not be justified by a small capital benefit.

Counterparty effects also require care. Expected credit shortfall is an expected-value adjustment, not a guarantee that losses will be small in a severe event. Concentration in one reinsurer, collateral quality, and default dependence with the underlying catastrophe may make tail collectability materially worse than its average suggests. The case calculation treats expected shortfall as a separate cost and excludes counterparty risk from capital; a practical recommendation should disclose that assumption and test whether a further capital allowance is needed.

A defensible final recommendation therefore reports the preferred option, the cost difference, the change in expected retained claims and economic capital, the peak-event retained amount, and the key assumptions. In the portfolio case, Program A is preferred on the stated cost measure, while Program B provides greater tail reduction. If the extra protection in B is required by risk appetite, capacity constraints, or management objectives, that benefit can justify its \(0.28\)-million higher modeled cost compared with A.