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Exam ASTAM · Reinsurance and Risk Sharing · Tutorial 195 of 979

Integrated Reinsurance Allocation Workshop

Learn to apply multiple treaties in the stated order, calculate retained and ceded amounts, and verify that the allocations and moments reconcile.

Intermediate 11 min read

What You'll Learn

  • Allocate an individual claim between the insurer and successive treaties
  • Separate claim-level recoveries from annual aggregate recoveries
  • Calculate retained-loss moments for a discrete severity distribution
  • Derive compound Poisson aggregate means, variances, and covariances
  • Reconcile gross loss with total ceded and retained loss
  • Identify errors caused by applying treaty terms in the wrong order

Why an Allocation Needs an Order

When a portfolio is protected by more than one treaty, the main task is not simply to calculate each treaty’s recovery in isolation. The analyst must determine the amount each treaty applies to, apply the contracts in their specified order, and check that the final amounts add back to gross loss. A recovery calculated on the wrong base can produce a plausible-looking number while assigning the same loss dollar twice.

This workshop uses hypothetical contracts to connect claim-level allocations with an annual aggregate recovery and with aggregate loss moments. The amounts and treaty terms are invented for illustration; the key practice is to keep the bases and the order of application explicit.

Definition: For a gross claim \(X\), let \(C^Q\) denote the quota-share cession, \(C^X\) a per-claim excess-of-loss recovery, and \(R\) the amount retained after those claim-level treaties. For an annual aggregate treaty, let \(S_R\) be the sum of claim-level retained amounts, \(C^A\) the annual aggregate recovery, and \(R^A\) the final annual retained loss. Each amount is defined by the treaty terms and application order; the basic allocation check is \(X=C^Q+C^X+R\) for each claim.

Claim-Level Allocation: A Quota Share Followed by a Layer

Suppose a quota-share treaty cedes proportion \(q\) of each gross claim. The insurer’s amount after quota share is \((1-q)X\). If the next treaty is an excess-of-loss layer with attachment \(a\) and payment limit \(L\), and it applies to that post-quota amount \(Z=(1-q)X\), its recovery is \(\min((Z-a)_+,L)\), where \((w)_+=\max(w,0)\). The amount remaining after both treaties is \(Z-C^X\).

This definition makes the sequence unambiguous: first calculate the quota-share cession; then calculate the layer recovery using the amount remaining after quota share. Applying the layer to gross loss instead would describe a different treaty arrangement.

Worked Example: Allocating Three Possible Claim Sizes

A hypothetical insurer has a 20% quota-share treaty. After that treaty, a per-claim excess-of-loss treaty pays the amount above $1,600, up to a limit of $1,600. A hypothetical claim severity \(X\) is $1,000 with probability 0.5, $3,000 with probability 0.3, and $5,000 with probability 0.2. Find each claim-level allocation, and calculate the mean and variance of the insurer’s retained amount per claim. The layer applies to the post-quota amount.

Method and common mistake avoided. Apply the treaties in order and use the post-quota amount as the layer’s base. A common mistake is to calculate the layer recovery from gross \(X\), even though the stated contract applies after quota share.

For a $1,000 claim, the quota-share cession is \(0.20(1{,}000)=\$200\). The post-quota amount is \(\$1{,}000-\$200=\$800\). Since \((800-1{,}600)_+=0\), the layer recovery is $0 and the retained amount is $800. The check is \(\$200+\$0+\$800=\$1{,}000\).

For a $3,000 claim, the quota-share cession is \(0.20(3{,}000)=\$600\), leaving \(\$2{,}400\). The layer recovery is \(\min(2{,}400-1{,}600,1{,}600)=\$800\). The retained amount is \(\$2{,}400-\$800=\$1{,}600\). The check is \(\$600+\$800+\$1{,}600=\$3{,}000\).

For a $5,000 claim, the quota-share cession is \(\$1{,}000\), leaving \(\$4{,}000\). The layer recovery is \(\min(4{,}000-1{,}600,1{,}600)=\$1{,}600\). The retained amount is \(\$4{,}000-\$1{,}600=\$2{,}400\). The check is \(\$1{,}000+\$1{,}600+\$2{,}400=\$5{,}000\).

The retained amount \(R\) therefore takes values $800, $1,600, and $2,400 with probabilities 0.5, 0.3, and 0.2. Its mean is \(E[R]=0.5(800)+0.3(1{,}600)+0.2(2{,}400)=\$1{,}360\). Its second moment is \(E[R^2]=0.5(800^2)+0.3(1{,}600^2)+0.2(2{,}400^2)=\$2{,}240{,}000\). Thus \(\operatorname{Var}(R)=2{,}240{,}000-1{,}360^2=\$390{,}400\). The calculation is checked by using the variance definition \(E[R^2]-(E[R])^2\); the variance is in dollars squared.

Adding an Annual Aggregate Treaty

An annual aggregate treaty applies to an annual total, not separately to each claim. For an annual deductible \(D\) and a maximum recovery \(M\), the recovery on the claim-level retained aggregate \(S_R\) is \(C^A=\min((S_R-D)_+,M)\). The final annual retained amount is \(R^A=S_R-C^A\). This treaty does not change the earlier claim-level allocation; it transfers part of the resulting annual total.

The distinction between a per-claim layer and an annual aggregate layer matters even if both have an attachment and a limit. A claim-level layer is evaluated claim by claim. The annual layer is evaluated once using the sum of the relevant retained claims for the year.

Worked Example: Reconciling Two Claims and an Annual Recovery

A hypothetical year has two gross claims, $1,500 and $4,000. The insurer first cedes 20% of each claim under quota share. A per-claim layer then applies to the post-quota amount, with attachment $1,000 and limit $1,200. Finally, an annual aggregate treaty applies to the sum retained after those claim-level treaties, with deductible $2,500 and maximum recovery $1,000. Find each allocation and reconcile the annual amounts.

Method and common mistake avoided. Allocate both claims through the quota share and per-claim layer first. Only then calculate the annual recovery. Do not apply the annual deductible to gross loss or subtract it separately from each claim.

For the $1,500 claim, quota share cedes \(0.20(1{,}500)=\$300\), leaving \(\$1{,}200\). The per-claim layer pays \(\min((1{,}200-1{,}000)_+,1{,}200)=\$200\). The insurer retains \(\$1{,}200-\$200=\$1{,}000\). The claim check is \(\$300+\$200+\$1{,}000=\$1{,}500\).

For the $4,000 claim, quota share cedes \(0.20(4{,}000)=\$800\), leaving \(\$3{,}200\). The per-claim layer pays \(\min(3{,}200-1{,}000,1{,}200)=\$1{,}200\), so the insurer retains \(\$3{,}200-\$1{,}200=\$2{,}000\). The claim check is \(\$800+\$1{,}200+\$2{,}000=\$4{,}000\).

The claim-level retained aggregate is \(S_R=1{,}000+2{,}000=\$3{,}000\). The annual aggregate recovery is \(\min(3{,}000-2{,}500,1{,}000)=\$500\), leaving final annual retention \(R^A=3{,}000-500=\$2{,}500\).

Across both claims, quota-share recoveries total \(\$300+\$800=\$1{,}100\), and per-claim layer recoveries total \(\$200+\$1{,}200=\$1{,}400\). Including the annual recovery, total ceded loss is \(\$1{,}100+\$1{,}400+\$500=\$3{,}000\). Gross loss is \(\$1{,}500+\$4{,}000=\$5{,}500\), and total ceded loss plus final retention is \(\$3{,}000+\$2{,}500=\$5{,}500\). The allocation reconciles.

Compound Poisson Moments After Claim-Level Reinsurance

Claim-level allocations also make it possible to calculate aggregate means and variances before any annual aggregate treaty. Assume \(N\) has a Poisson distribution with mean \(\lambda\), and that claim severities are independent and identically distributed, independent of \(N\). If \(Z\) is the per-claim amount of interest and \(T=\sum_{j=1}^{N}Z_j\), then \(E[T]=\lambda E[Z]\) and \(\operatorname{Var}(T)=\lambda E[Z^2]\). The \(Z_j\) must all represent the same quantity, such as gross loss, ceded loss, or retained loss.

Justification: Conditional on \(N\), the sum \(T\) has conditional mean \(N E[Z]\) and conditional variance \(N\operatorname{Var}(Z)\). The law of total expectation gives \(E[T]=E[N]E[Z]=\lambda E[Z]\). The law of total variance gives \(\operatorname{Var}(T)=E[N]\operatorname{Var}(Z)+\operatorname{Var}(N)(E[Z])^2\). For a Poisson variable, \(E[N]=\operatorname{Var}(N)=\lambda\), so \(\operatorname{Var}(T)=\lambda(\operatorname{Var}(Z)+(E[Z])^2)=\lambda E[Z^2]\). This result assumes the stated independent Poisson frequency and independent severities; it does not automatically apply to a fixed claim count or dependent claim amounts.

Worked Example: Comparing Gross, Ceded, and Retained Aggregate Moments

Use the severity distribution and the quota-share and per-claim layer terms from the first worked example. Suppose the annual claim count is Poisson with mean \(\lambda=2\), independent of severities. Find the means and variances of gross aggregate loss, total claim-level ceded loss, and claim-level retained aggregate loss. Do not apply an annual aggregate treaty.

Method and common mistake avoided. First calculate the per-claim first and second moments for each allocation. Then use the compound Poisson formulas. A common mistake is to use \(\lambda\operatorname{Var}(Z)\) as the aggregate variance; for Poisson frequency the correct expression is \(\lambda E[Z^2]\).

For gross severity \(X\), \(E[X]=0.5(1{,}000)+0.3(3{,}000)+0.2(5{,}000)=\$2{,}400\). Also, \(E[X^2]=0.5(1{,}000^2)+0.3(3{,}000^2)+0.2(5{,}000^2)=\$8{,}200{,}000\). Thus gross aggregate mean is \(2(2{,}400)=\$4{,}800\) and variance is \(2(8{,}200{,}000)=\$16{,}400{,}000\).

The total claim-level ceded amount \(C=X-R\) takes values \(\$200,\$1{,}400,\$2{,}600\) with probabilities \(0.5,0.3,0.2\). Therefore, \(E[C]=0.5(200)+0.3(1{,}400)+0.2(2{,}600)=\$1{,}040\), and \(E[C^2]=0.5(200^2)+0.3(1{,}400^2)+0.2(2{,}600^2)=\$1{,}960{,}000\). Total claim-level ceded aggregate mean is \(2(1{,}040)=\$2{,}080\), and its variance is \(2(1{,}960{,}000)=\$3{,}920{,}000\).

For retained loss, the first example gave \(E[R]=\$1{,}360\) and \(E[R^2]=\$2{,}240{,}000\). Retained aggregate mean is \(2(1{,}360)=\$2{,}720\), and its variance is \(2(2{,}240{,}000)=\$4{,}480{,}000\).

The means reconcile: \(\$2{,}080+\$2{,}720=\$4{,}800\). Variances do not generally add without a covariance term, because ceded and retained amounts from the same claims are dependent. Here \(E[RC]=0.5(800\cdot200)+0.3(1{,}600\cdot1{,}400)+0.2(2{,}400\cdot2{,}600)=\$2{,}000{,}000\). The aggregate covariance is \(\lambda E[RC]=\$4{,}000{,}000\). Hence \(\$4{,}480{,}000+\$3{,}920{,}000+2(\$4{,}000{,}000)=\$16{,}400{,}000\), matching gross aggregate variance.

Reconciliation Checks and Common Traps

A dependable allocation is supported by both a treaty-by-treaty calculation and an independent reconciliation. For claim-level treaties, verify the allocation for each claim. For a later annual treaty, verify the annual total separately. If moments are calculated, check means by additivity and check variances using the relevant covariance terms rather than assuming they add.

  • Using the wrong treaty base. A layer stated to apply after quota share uses the post-quota amount, not gross loss.
  • Reversing treaty order. If the contract says one treaty applies before another, changing that order can change both recoveries and retention.
  • Applying an annual deductible claim by claim. Aggregate deductibles operate on the annual sum specified by the contract.
  • Ignoring a layer limit. A layer recovery cannot exceed its payment limit, even if the amount above attachment is larger.
  • Forgetting the covariance in a variance check. For gross, ceded, and retained totals, \(\operatorname{Var}(G)=\operatorname{Var}(C)+\operatorname{Var}(R)+2\operatorname{Cov}(C,R)\), where \(G=C+R\).

An annual aggregate treaty is nonlinear in the claim-level retained total. Consequently, the first two moments of an individual retained claim generally are not enough to determine the exact moments after that annual treaty. The annual aggregate distribution or an appropriate distributional calculation is needed. The compound Poisson moments above therefore describe the portfolio before applying annual aggregate protection.

A concise exam solution can make the allocation auditable: state the order, show the base for each recovery, calculate the final retention, and close with a dollar reconciliation. This approach is aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each hypothetical case, apply the stated treaty order and check the final allocation.

  1. A claim of $2,400 is subject to a 25% quota share only. Find the quota-share cession and retained amount. (Answer: ceded $600; retained $1,800)
  2. A per-claim layer pays loss above $1,000, up to $1,500. For a gross claim of $4,200, calculate the recovery and the retained amount. (Answer: recovery $1,500; retained $2,700)
  3. A $3,000 claim is first subject to a 10% quota share, then to a per-claim layer attaching at $1,800 with a $1,000 limit, applied to the post-quota amount. Find both cessions and retention. (Answer: quota-share cession $300; layer recovery $900; retained $1,800)
  4. After all claim-level treaties, a hypothetical year has retained aggregate loss \(S_R=\$6{,}100\). An annual aggregate treaty has deductible $3,500 and maximum recovery $2,000. Find the recovery and final retention. (Answer: recovery $2,000; final retention $4,100)
  5. A compound Poisson portfolio has \(N\sim\operatorname{Poisson}(3)\), independent of severities. The per-claim retained amount is $500 or $1,000, each with probability 0.5. Find the aggregate mean and variance. (Answer: mean $2,250; variance $1,875,000)

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