Measuring Economic Capital Relief from Reinsurance

Why Measure Capital Relief?

Reinsurance changes the insurer’s retained loss distribution, but the amount of ceded loss is not itself a measure of capital relief. Capital depends on both the level of retained losses in the tail and the expected loss used as the capital baseline. A treaty can reduce a high-quantile loss and still produce little capital relief if its retention is high, its limit is exhausted, or recoveries are not fully collectible.

A valid comparison uses the same loss horizon, confidence level, risk measure, and treatment of expenses before and after reinsurance. The calculation below uses Value at Risk (VaR) and defines economic capital as VaR less expected loss. Other capital frameworks may use a different risk measure, but the comparison must remain consistent.

Definition: For a loss \(L\) and confidence level \(p\), define \(\operatorname{VaR}_p(L)=\inf\{z:P(L\le z)\ge p\}\). Under the VaR-based convention used here, economic capital is \(\operatorname{EC}_p(L)=\operatorname{VaR}_p(L)-E[L]\). If \(X\) is gross loss and \(R(X)\) is the collectible reinsurance recovery, retained loss is \(N=X-R(X)\). Capital relief is \(\operatorname{EC}_p(X)-\operatorname{EC}_p(N)\); positive relief means lower modeled capital after reinsurance.

Set the Contract and Solvency Basis

For a per-occurrence excess-of-loss contract with retention \(a\) and limit \(u\), assume \(X\ge 0\) is the covered gross loss and the reinsurer pays the part above the retention, subject to the limit. The limit is the maximum recovery per occurrence, not the maximum retained loss.

R(X)=\min\bigl((X-a)^+,u\bigr),\qquad N=X-R(X),\qquad (z)^+=\max(z,0).

This specification assumes the contract responds to the modeled loss, all amounts use the same currency and time basis, and there are no exclusions, aggregate limits, reinstatement effects, or other contract features. If those features apply, the recovery function must represent them. For a portfolio aggregate-loss model, \(X\) and \(R(X)\) must both be defined at the same aggregate level; an occurrence limit should not be applied as though it were an aggregate limit.

The solvency basis must also be explicit. In particular, select one horizon and one confidence level \(p\), and use the same definition of loss before and after reinsurance. If a deterministic premium \(P\) is included in the post-reinsurance loss, that loss is \(N+P\). Under the stated economic-capital definition, a fixed premium shifts both VaR and the mean by \(P\), so it does not change economic capital. It can still materially affect expected cost or profit.

Why Expected Recovery Is Not Capital Relief

The difference between gross and net economic capital follows directly from the definition. Since \(N=X-R\), \(E[N]=E[X]-E[R]\), and therefore:

\begin{aligned} \operatorname{EC}_p(X)-\operatorname{EC}_p(N) &=\operatorname{VaR}_p(X)-E[X] -\left(\operatorname{VaR}_p(N)-E[N]\right)\\ &=\operatorname{VaR}_p(X)-\operatorname{VaR}_p(N)-E[R]. \end{aligned}

Thus, capital relief equals the reduction in VaR minus the expected recovery. The reason is that expected recovery reduces the mean loss as well as the tail loss; economic capital measures tail loss relative to that mean. Consequently, neither the expected recovery nor the decrease in VaR alone is the capital relief.

Key result: Under the same VaR confidence level and fully collectible recoveries, \(\operatorname{EC}_p(X)-\operatorname{EC}_p(N)=\operatorname{VaR}_p(X)-\operatorname{VaR}_p(N)-E[R]\). The result does not imply that relief must be positive. A treaty that has little impact at the selected quantile can reduce expected loss more than it reduces VaR, producing negative relief under this measure.

Worked Capital Calculations

Worked Example: Capital Relief from a Capped Layer

A hypothetical insurer models one-year gross loss \(X\) in thousands of dollars as follows: \(X=0\) with probability \(0.80\), \(X=100\) with probability \(0.15\), and \(X=500\) with probability \(0.05\). It buys fully collectible per-occurrence reinsurance of \(100\) excess of \(50\), with no other limits or costs. Calculate gross and net economic capital at \(p=0.90\), and the capital relief.

Method. Calculate the recovery and retained loss for each outcome, then obtain their means and 90% quantiles. All amounts in the calculations are thousands of dollars.

For \(X=0\), \(R=\min((0-50)^+,100)=0\), so \(N=0\). For \(X=100\), \(R=\min(50,100)=50\), so \(N=50\). For \(X=500\), \(R=\min(450,100)=100\), so \(N=400\).

The gross expected loss is \(E[X]=0(0.80)+100(0.15)+500(0.05)=40\). The gross cumulative probability is \(0.80\) at loss \(0\) and \(0.95\) at loss \(100\), so \(\operatorname{VaR}_{0.90}(X)=100\). Thus gross economic capital is \(100-40=60\).

The net expected loss is \(E[N]=0(0.80)+50(0.15)+400(0.05)=7.5+20=27.5\). The net cumulative probability is also \(0.95\) at loss \(50\), so \(\operatorname{VaR}_{0.90}(N)=50\). Net economic capital is \(50-27.5=22.5\), giving capital relief of \(60-22.5=37.5\) thousand dollars.

Check. Expected recovery is \(E[R]=0(0.80)+50(0.15)+100(0.05)=12.5\). The formula gives \(100-50-12.5=37.5\), matching the direct calculation. The common mistake is to call the VaR reduction of \(50\) the capital relief; that overstates relief because it ignores the \(12.5\) expected recovery.

Worked Example: A Contract Limit Leaves Tail Loss

A hypothetical portfolio has one-year gross losses of \(0\), \(100\), \(400\), and \(1{,}000\) thousand dollars with probabilities \(0.90\), \(0.07\), \(0.02\), and \(0.01\), respectively. A fully collectible treaty pays \(100\) excess of \(100\), subject to a \(300\)-thousand-dollar limit. Calculate economic capital at \(p=0.99\) before and after reinsurance.

Method. Apply the retention and limit to each outcome separately. In particular, the \(300\) limit caps recovery, not the insurer’s remaining loss.

At gross loss \(0\), recovery is \(0\) and retained loss is \(0\). At \(100\), recovery is \(0\) and retained loss is \(100\). At \(400\), recovery is \(\min(400-100,300)=300\), leaving \(100\). At \(1{,}000\), recovery is also capped at \(300\), leaving \(700\).

Gross expected loss is \(E[X]=0(0.90)+100(0.07)+400(0.02)+1{,}000(0.01)=7+8+10=25\). The cumulative probability reaches \(0.99\) at \(400\) and \(1.00\) at \(1{,}000\). Under the stated quantile definition, \(\operatorname{VaR}_{0.99}(X)=400\), so gross economic capital is \(400-25=375\).

Net expected loss is \(E[N]=0(0.90)+100(0.07)+100(0.02)+700(0.01)=7+2+7=16\). The net cumulative probability reaches \(0.99\) at \(100\), so \(\operatorname{VaR}_{0.99}(N)=100\). Net economic capital is \(100-16=84\). Capital relief is \(375-84=291\) thousand dollars.

Check. Expected recovery is \(0.02(300)+0.01(300)=9\). The quantile reduction is \(400-100=300\), and \(300-9=291\), as above. At the largest loss, the insurer retains \(700\), not \(300\). Confusing the recovery cap with a cap on retained loss understates the tail remaining after reinsurance.

Worked Example: A Recovery Haircut and a Fixed Premium

Use a hypothetical one-year gross loss distribution with \(X=0,80,400\) thousand dollars having probabilities \(0.80,0.15,0.05\), respectively. A treaty pays \(80\) excess of \(40\), subject to a \(160\)-thousand-dollar limit. For this sensitivity calculation, assume only \(75\%\) of each contractual recovery is collectible, and the insurer pays a fixed premium of \(5\) thousand dollars. Calculate economic capital at \(p=0.90\) including the premium, and compare it with gross economic capital.

Method. First calculate contractual recoveries. Apply the stated collectability factor to those recoveries, then calculate retained losses. Treat the premium as a deterministic addition to the post-reinsurance loss.

For \(X=0\), contractual recovery is \(0\). For \(X=80\), it is \(\min(80-40,160)=40\). For \(X=400\), it is \(\min(400-40,160)=160\). At \(75\%\) collectability, actual modeled recoveries are \(0,30,120\), respectively. The retained claims are therefore \(0,50,280\).

Gross expected loss is \(E[X]=80(0.15)+400(0.05)=12+20=32\). Its 90% VaR is \(80\), since the cumulative probability is \(0.80\) at zero and \(0.95\) at \(80\). Gross economic capital is \(80-32=48\).

Expected retained claims are \(E[N]=50(0.15)+280(0.05)=7.5+14=21.5\). Including the premium, the post-reinsurance loss values are \(5,55,285\), with probabilities \(0.80,0.15,0.05\). Their mean is \(21.5+5=26.5\), and their 90% VaR is \(55\). Post-reinsurance economic capital is \(55-26.5=28.5\). Capital relief is \(48-28.5=19.5\) thousand dollars.

Check. Without the premium, retained-claims VaR is \(50\) and the mean is \(21.5\), so economic capital is \(28.5\) as well. The premium shifts both VaR and mean upward by \(5\), leaving economic capital unchanged. The \(75\%\) collectability assumption is a simplified sensitivity, not a stochastic model of counterparty default; a stochastic model would need to specify the dependence between recoveries and gross losses.

Interpretation, Limits, and Common Traps

Capital relief is conditional on the modeled portfolio and the chosen solvency basis. Recompute it if the confidence level, horizon, loss definition, or contract terms change. If the treaty has aggregate limits or reinstatements, model those features in \(R\); do not treat a per-occurrence recovery as an unlimited aggregate recovery. If recoveries can be delayed, discounted, disputed, or unavailable, incorporate the relevant assumptions into the modeled retained loss rather than silently assuming full and immediate payment.

A deterministic recovery haircut can help separate contractual risk reduction from a simple collectability adjustment, as in the third example. It does not measure the full tail effect of counterparty default. In particular, an average haircut can miss the possibility that recoveries fail precisely in severe loss scenarios. A detailed stochastic treatment of counterparty exposure requires an explicit joint model; it should not be inferred from an expected recovery alone.

Using inconsistent bases. Comparing gross VaR at one confidence level with net VaR at another does not measure a treaty’s effect on the same solvency requirement.
Calling expected recovery capital relief. Under VaR-based economic capital, subtract expected recovery from the VaR reduction.
Ignoring the retained tail. A treaty limit caps recovery. Losses above retention plus limit remain with the insurer.
Assuming relief must be positive. A treaty can reduce expected loss without materially changing VaR at a selected confidence level, so economic capital can increase under the definition used here.
Mixing capital and cost. A deterministic premium does not change this economic-capital measure, but it changes expected outlay. Capital relief is not a complete measure of program value.

Key takeaway: Estimate gross and net capital under identical assumptions, calculate recoveries from the actual contract terms, and compare VaR less expected loss on both sides. Report separately the risk reduction, the expected recovery, and any modeled collectability adjustment.

Check Your Understanding

For each problem, use the stated VaR quantile convention and calculate economic capital as VaR less expected loss. Amounts are in thousands of dollars.

A hypothetical loss is \(0\) with probability \(0.85\), \(100\) with probability \(0.10\), and \(300\) with probability \(0.05\). A fully collectible treaty pays \(50\) excess of \(50\), with a \(100\) limit. Find gross and net economic capital and relief at \(p=0.90\). (Answer: gross \(75\); net \(35\); relief \(40\).)
A hypothetical loss is \(0\), \(200\), or \(600\), with probabilities \(0.90\), \(0.08\), and \(0.02\). A fully collectible treaty pays \(100\) excess of \(100\), with a \(300\) limit. Find gross and net economic capital and relief at \(p=0.95\). (Answer: gross \(172\); net \(86\); relief \(86\).)
A hypothetical loss is \(0\), \(100\), or \(500\), with probabilities \(0.80\), \(0.15\), and \(0.05\). A treaty pays \(50\) excess of \(50\), with a \(200\) limit, but only \(90\%\) of each recovery is modeled as collectible. Find net economic capital and relief at \(p=0.90\). (Answer: net capital \(30.75\); gross capital \(60\); relief \(29.25\).)
A hypothetical loss is \(0\), \(100\), \(400\), or \(1{,}000\), with probabilities \(0.90\), \(0.07\), \(0.02\), and \(0.01\). A fully collectible treaty pays \(100\) excess of \(100\), with a \(300\) limit. Find gross and net economic capital and relief at \(p=0.95\). (Answer: gross \(75\); net \(84\); relief \(-9\), so capital increases by \(9\).)
For the loss distribution in problem 1, suppose a deterministic premium of \(12\) is added to the post-reinsurance loss. What is the change in post-reinsurance economic capital compared with excluding that premium? (Answer: \(0\); both VaR and expected loss increase by \(12\).)

What You'll Learn