Reinsurance and Solvency Capital Relief

Introduction

Reinsurance can reduce an insurer’s exposure to claims, but the treaty’s stated coverage is not automatically equal to solvency capital relief. Recoveries may be limited by attachment points, caps, collateral, or the reinsurer’s capacity to pay. The relevant comparison is between the insurer’s liabilities before and after reinsurance, both at the valuation date and under the specified stress.

This tutorial develops a transparent stress-based calculation. It treats the capital requirement as the increase in net liabilities under a stated set of adverse scenarios. This is a useful calculation framework, but it is not a universal regulatory formula: a particular solvency regime may prescribe other stresses, aggregation methods, or counterparty adjustments. The examples are hypothetical and aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: Let $L^G(s)$ be the insurer’s gross liability in scenario $s$, before reinsurance. Let $R(s)$ be the recoverable recognized after applying the treaty terms, counterparty limits, and the stated collectability assumptions. The net liability is $N(s)=L^G(s)-R(s)$. In a simplified scenario framework, the capital requirement is the largest positive increase in net liability from the baseline scenario $0$: $\max_s\{N(s)-N(0),0\}$.

Contractual Recovery and Recognized Recoverable

A contractual recovery is the amount payable under the reinsurance terms for the insurer’s covered claims. It is calculated from the contract before considering whether the reinsurer can pay. For example, a quota-share treaty may reimburse a fixed proportion of covered liabilities, while excess-of-loss cover may pay only the amount above an attachment point and up to a limit.

The recognized recoverable is the amount included in the insurer’s solvency valuation after considering collection risk. It can be less than the contractual recovery. Depending on the stated model, the adjustment may be represented by a collection percentage, a counterparty exposure cap, an explicit default scenario, or collateral. These are alternative ways of representing credit risk; they should not be applied twice to the same exposure.

For one simplified collateral model, let $C$ be collateral available to meet the recovery, $U$ the contractual recovery not covered by that collateral, $K$ a cap on eligible unsecured exposure, and $q$ the fraction of that exposure expected to be collected in the specified scenario. Then a stylized recognized recovery is:

R=C+q\min(U,K)

This expression assumes collateral is fully available and collectible. If the collateral itself is impaired, legally unavailable, or subject to a haircut, its recognized amount must also be adjusted. The expression is not appropriate when the treaty excludes the underlying claim or when the specified counterparty cap is defined on a different basis; in those cases, apply the actual contract and valuation rules instead.

Why Capital Relief Depends on the Change in Recoverables

The important quantity is not the total recovery in the stressed scenario by itself. Capital measures the change from the baseline net position. Subtracting the baseline net liability from the stressed net liability gives:

\begin{aligned} N(s)-N(0) &=\bigl(L^G(s)-R(s)\bigr)-\bigl(L^G(0)-R(0)\bigr)\\ &=\bigl(L^G(s)-L^G(0)\bigr)-\bigl(R(s)-R(0)\bigr). \end{aligned}

Thus, the stressed increase in gross liabilities is reduced by the increase in recognized recoverables. A recovery already reflected in the baseline liability is not new relief under the stress. If the recognized recovery falls in the adverse scenario, then $R(s)-R(0)$ is negative and the counterparty deterioration increases the net capital requirement.

Value the baseline.
Determine gross liabilities and recognized recoverables at the valuation date, using the same valuation basis for each.

Apply the stress to the gross liabilities.
Calculate the gross liability in each specified adverse scenario before applying reinsurance.

Recalculate recoveries under the scenario.
Apply treaty coverage, attachments, limits, aggregate caps, counterparty constraints, and any specified default or collateral assumptions.

Compare net liabilities.
For each scenario, subtract the baseline net liability from the stressed net liability, then take the largest positive increase if the framework defines the charge that way.

Worked Applications

Worked Example: Quota Share with a Collection Adjustment

A hypothetical insurer has a baseline gross liability of $10.00 million for a covered block. A quota-share treaty reinsures 40% of that liability. For this example, the recognized recovery is 95% of the treaty recovery in both the baseline and stress; there are no other limits. A prescribed liability stress increases the gross liability by 25%. Calculate the baseline and stressed net liabilities, the stress-based capital requirement, and the reduction relative to no reinsurance.

Method. Calculate the contractual quota share in each position, apply the 95% recognition factor, and compare the two net liabilities. The common mistake is to call 40% of the stressed gross liability the capital relief without deducting the baseline recoverable.

Baseline recovery. The contractual recovery is $0.40\times \$10.00\text{ million}=\$4.00\text{ million}$. The recognized recovery is $0.95\times \$4.00\text{ million}=\$3.80\text{ million}$. Therefore, the baseline net liability is $\$10.00-\$3.80=\$6.20$ million.

Stressed gross liability and recovery. The stressed gross liability is $\$10.00\times1.25=\$12.50$ million. Its contractual recovery is $0.40\times\$12.50=\$5.00$ million, and its recognized recovery is $0.95\times\$5.00=\$4.75$ million. The stressed net liability is $\$12.50-\$4.75=\$7.75$ million.

Capital calculation and check. The stress-based requirement is $\$7.75-\$6.20=\$1.55$ million. Without reinsurance, the gross liability increase would be $\$12.50-\$10.00=\$2.50$ million. The reduction is $\$2.50-\$1.55=\$0.95$ million. This equals the increase in recognized recoverables, $\$4.75-\$3.80=\$0.95$ million, confirming the calculation. Amounts are stated to the nearest cent where applicable; here they are exact to $0.01 million.

Worked Example: Excess-of-Loss Cover with a Counterparty Cap

A hypothetical block has a baseline gross liability of $4.00 million and a stressed gross liability of $12.00 million. An excess-of-loss treaty produces a contractual baseline recovery of $0.50 million and a contractual stressed recovery of $3.00 million after the treaty’s claim and aggregate limits have been applied. The counterparty adjustment recognizes 90% of contractual recovery, but the recognized amount cannot exceed $2.40 million in either position. Calculate the net-liability stress requirement and the capital reduction relative to no reinsurance.

Method. First apply the 90% collection factor, then enforce the cap on the recognized amount. The cap is on recognized recovery, so it must be checked after the collection adjustment.

Baseline. The uncapped recognized recovery is $0.90\times\$0.50=\$0.45$ million. This is below the $2.40 million cap, so $R(0)=\$0.45$ million. Baseline net liability is $\$4.00-\$0.45=\$3.55$ million.

Stress. The uncapped recognized recovery is $0.90\times\$3.00=\$2.70$ million. The cap reduces it to $R(s)=\$2.40$ million. Stressed net liability is $\$12.00-\$2.40=\$9.60$ million.

Capital calculation and check. The requirement is $\$9.60-\$3.55=\$6.05$ million. Without reinsurance, the gross increase is $\$12.00-\$4.00=\$8.00$ million, so the reduction is $\$8.00-\$6.05=\$1.95$ million. The increase in recognized recovery is also $\$2.40-\$0.45=\$1.95$ million. Ignoring the cap would incorrectly use a stressed recovery of $2.70 million and overstate capital relief by $0.30 million. Amounts are in millions.

Worked Example: A Combined Liability and Counterparty Stress

A hypothetical insurer has a baseline gross liability of $20.00 million and a contractual baseline recovery of $2.00 million. Of that recovery, $1.00 million is fully collateralized and the remaining $1.00 million is unsecured. In ordinary conditions, 95% of the unsecured amount is recognized. Under a combined adverse scenario, gross liabilities rise to $26.00 million and the treaty recovery rises to $4.00 million; $1.00 million remains collateralized and only 40% of the $3.00 million unsecured recovery is collectible. Calculate the capital requirement when the insurer considers the baseline and combined scenario. Also compare it with the liability stress assuming the baseline collection percentage continues to apply.

Method. Separate collateralized and unsecured recovery in each position. The common mistake is to subtract the full $4.00 million contractual recovery in the combined scenario even though the scenario specifies reduced collection on unsecured amounts.

Baseline recognized recovery. The collateral contributes $1.00 million, and the unsecured amount contributes $0.95\times\$1.00=\$0.95$ million. Thus, $R(0)=\$1.95$ million and the baseline net liability is $\$20.00-\$1.95=\$18.05$ million.

Combined-scenario recognized recovery. The collateral contributes $1.00 million. The unsecured recovery is $\$4.00-\$1.00=\$3.00$ million, of which $0.40\times\$3.00=\$1.20$ million is collectible. Hence $R(s)=\$1.00+\$1.20=\$2.20$ million. The stressed net liability is $\$26.00-\$2.20=\$23.80$ million, giving a capital requirement of $\$23.80-\$18.05=\$5.75$ million.

Comparison and check. If the 95% collection percentage continued to apply in the liability stress, the recognized stressed recovery would be $\$1.00+0.95\times\$3.00=\$3.85$ million. The stressed net liability would then be $\$26.00-\$3.85=\$22.15$ million, an increase of $\$22.15-\$18.05=\$4.10$ million. The combined default assumption therefore adds $\$5.75-\$4.10=\$1.65$ million to the requirement. This also equals the decline in recognized recovery relative to continued ordinary collection: $\$3.85-\$2.20=\$1.65$ million. Amounts are in millions.

Reading the Result and Avoiding Common Traps

A treaty’s nominal share or limit does not state its capital relief. Relief depends on how much the arrangement changes recognized recoverables between the baseline and the adverse scenario. The following distinctions are especially important:

Use recoverables, not gross treaty amounts. Apply the stated counterparty adjustment and eligibility limits before subtracting a recovery from liabilities.
Apply the treaty in each position. Attachments, per-claim limits, and aggregate caps may bind differently at baseline and under stress.
Do not treat a baseline recovery as new stress relief. The capital change depends on $R(s)-R(0)$, not on $R(s)$ alone.
Keep liability risk and counterparty risk consistent. If a scenario combines claim stress with reinsurer default, calculate the recovery under that combined scenario. Do not also subtract a separate default loss for the same exposure unless the stated framework requires it.
Check whether a recovery limit is contractual or credit-related. A treaty limit restricts what is owed; a counterparty cap restricts what is recognized or collected. They are not interchangeable.

Reinsurance may even increase the net capital requirement if the recognized recoverable deteriorates sufficiently under stress. For example, a large baseline recovery followed by a severe counterparty haircut can make $R(s)-R(0)$ negative. In the identity $N(s)-N(0)=\bigl(L^G(s)-L^G(0)\bigr)-\bigl(R(s)-R(0)\bigr)$, subtracting a negative change increases the result. The appropriate answer is not to assume capital relief, but to follow the specified scenario and report the calculated net change.

Key takeaway: For each scenario, calculate gross liabilities and recognized recoverables separately, then compare stressed net liabilities with baseline net liabilities. Capital relief is governed by the change in recognized recoverables after treaty terms and counterparty constraints, not by the headline reinsurance percentage.

Check Your Understanding

Use the stated stress-based framework. Amounts are in millions unless otherwise noted.

A block has baseline gross liabilities of $8.00 and stressed gross liabilities of $10.00. A 20% quota share applies, and 90% of each contractual recovery is recognized. Find the baseline net liability, the stressed net liability, and the capital requirement. (Answer: baseline net liability $6.56 million; stressed net liability $8.20 million; requirement $1.64 million.)
Baseline gross liability is $3.00 and stressed gross liability is $7.00. An excess-of-loss treaty has an attachment of $4.00 and a limit of $2.00. The baseline contractual recovery is stipulated to be $0.20. Recognized recovery is the lesser of 90% of contractual recovery and a $1.50 cap. Find the capital requirement and its reduction from the gross liability stress. (Answer: baseline recognized recovery $0.18 million; stressed recognized recovery $1.50 million; requirement $2.68 million; reduction $1.32 million.)
Baseline gross liability is $15.00, with a contractual recovery of $2.00: $0.50 collateralized and $1.50 unsecured. Recognize 90% of unsecured recovery at baseline. Under stress, gross liability is $18.00 and contractual recovery is $3.00: $0.50 collateralized and $2.50 unsecured, of which 25% is collectible. Calculate the baseline net liability, stressed net liability, and capital requirement. (Answer: baseline net liability $13.15 million; stressed net liability $16.875 million; requirement $3.725 million.)
Baseline gross liability is $10.00 with recognized recovery of $1.00. In scenario A, gross liability is $12.00 and recognized recovery is $1.80. In scenario B, gross liability is $11.00 and recognized recovery is $0.40. If the requirement is the maximum positive increase in net liability across scenarios, calculate it. (Answer: scenario A increase $1.20 million; scenario B increase $1.60 million; requirement $1.60 million.)

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