Choosing Attachments and Limits by Marginal Value

Choosing Terms by Their Incremental Contribution

An attachment and a limit determine which losses a reinsurance layer pays. The choice is not simply a matter of finding the layer with the largest expected recovery: a larger recovery generally costs more, and its value depends on how much it reduces the risk the insurer intends to retain. A sound comparison therefore asks what an additional change in terms contributes to expected recovery, premium, and a stated measure of risk.

This tutorial uses variance as the risk measure in its worked calculations. Variance is useful for demonstrating the mechanics of comparing retained losses, but a program objective may instead specify another measure. The selection rule must match the objective actually stated; expected recovery, premium, and variance reduction are distinct quantities.

Definition: Let \(X\) be the insurer's loss before reinsurance, \(a\) the layer attachment, and \(u\) its limit. The layer recovery is \(Y^L=\min((X-a)_+,u)\), where \((z)_+=\max(z,0)\). The retained loss is \(R=X-Y^L\). Under a simplified premium assumption with loading \(\theta\), the premium is \(C=(1+\theta)E[Y^L]\). The variance reduction is \(\operatorname{Var}(X)-\operatorname{Var}(R)\).

The premium formula above is a stipulated comparison rule, not a universal market-pricing formula. It assumes the same proportional loading applies to the expected recovery for each candidate. If quoted premiums are supplied, use those premiums instead. All examples below use hypothetical losses and pricing assumptions.

Expected Recovery and Incremental Limit Value

For a discrete loss distribution, calculate the recovery at each possible loss, multiply each recovery by its probability, and sum. If a layer's limit increases from \(u_1\) to \(u_2\), with the same attachment \(a\), the added recovery is \(\min((X-a-u_1)_+,u_2-u_1)\). This isolates the loss portion covered only by the increase in limit. Its expectation is the incremental expected recovery; the corresponding incremental premium follows from the applicable pricing rule or quote.

An increase in limit does not necessarily create the same amount of recovery for every loss. Losses below the layer do not reach it, and losses already exhausting the old limit contribute only the added limit. That is why each candidate should be evaluated across the loss distribution rather than by multiplying the limit change by the probability of any loss.

Worked Example: Measuring the Value of an Added Limit

A hypothetical annual loss \(X\) is \(0,100,250,\) or \(500\), in thousands of dollars, with probabilities \(0.4,0.3,0.2,\) and \(0.1\), respectively. Compare a layer attaching at \(100\) with limit \(150\) to the same layer with limit \(300\). The premium loading is \(40\%\) of expected recovery. Find the incremental expected recovery and premium, and the incremental variance reduction.

Method. Calculate recovery and retained loss under each limit. Compare the two results rather than treating the additional \(150\) of limit as an automatic payment.

With limit \(150\), recoveries at losses \(0,100,250,500\) are \(0,0,150,150\). Thus \(E[Y^L]=0.4(0)+0.3(0)+0.2(150)+0.1(150)=45\). The premium is \(1.40(45)=63\), in thousands of dollars. Retained losses are \(0,100,100,350\), so their mean is \(0.3(100)+0.2(100)+0.1(350)=85\), and their second moment is \(0.3(10{,}000)+0.2(10{,}000)+0.1(122{,}500)=17{,}250\). The retained variance is \(17{,}250-85^2=10{,}025\), in thousands-of-dollars-squared.

With limit \(300\), recoveries are \(0,0,150,300\). Therefore \(E[Y^L]=0.2(150)+0.1(300)=60\), and the premium is \(1.40(60)=84\). Retained losses are \(0,100,100,200\), with mean \(0.3(100)+0.2(100)+0.1(200)=70\), second moment \(0.3(10{,}000)+0.2(10{,}000)+0.1(40{,}000)=9{,}000\), and variance \(9{,}000-70^2=4{,}100\).

The limit increase adds expected recovery \(60-45=15\) and premium \(84-63=21\), both in thousands of dollars. Its incremental variance reduction is \(10{,}025-4{,}100=5{,}925\) thousand-dollars-squared. As a check, the added recoveries are \(0,0,0,150\), with expected value \(0.1(150)=15\), as calculated by the difference in expected recoveries.

Common mistake avoided. The additional limit is \(150\), but the expected incremental recovery is not \(150\): only the loss outcome of \(500\), with probability \(0.1\), receives the extra \(150\). Multiplying \(150\) by the probability of any loss would use the wrong event.

Why Variance Reduction Depends on the Loss Relationship

Expected recovery measures average dollars paid. It does not, by itself, say how much uncertainty remains with the insurer. The link between the covered amount and the original loss matters. The following identity gives a direct way to see that relationship.

Derivation: For \(R=X-Y^L\), the variance identity \(\operatorname{Var}(A-B)=\operatorname{Var}(A)+\operatorname{Var}(B)-2\operatorname{Cov}(A,B)\) gives \(\operatorname{Var}(R)=\operatorname{Var}(X)+\operatorname{Var}(Y^L)-2\operatorname{Cov}(X,Y^L)\). Rearranging, \(\operatorname{Var}(X)-\operatorname{Var}(R)=2\operatorname{Cov}(X,Y^L)-\operatorname{Var}(Y^L)\). Thus variance reduction depends on both the covariance of the recovery with the original loss and the recovery's own variance. This identity applies when the relevant second moments exist; it does not imply that every cover reduces variance.

For an ordinary layer, recovery tends to be larger for larger losses, so its covariance with \(X\) is generally positive. But the identity still requires calculation: the recovery has its own variability, and the net variance reduction is not equal to the expected recovery or to twice the covariance. In a discrete example, the most direct method is to calculate the variance of \(X\) and of \(R\) separately.

Worked Example: Comparing Two Attachments

A hypothetical loss, in thousands of dollars, is \(0,100,\) or \(300\), with probabilities \(0.5,0.3,\) and \(0.2\). Compare two candidate layers: one attaches at \(100\) with limit \(200\); the other attaches at \(200\) with limit \(100\). Both have a \(25\%\) premium loading on expected recovery. Calculate the expected recovery, premium, and variance reduction for each, then compare their marginal contributions.

Method. For each candidate, calculate its recoveries and retained losses outcome by outcome. Use \(E[X^2]-(E[X])^2\) to calculate each variance.

The original loss has mean \(E[X]=0.3(100)+0.2(300)=90\) and second moment \(E[X^2]=0.3(10{,}000)+0.2(90{,}000)=21{,}000\). Thus \(\operatorname{Var}(X)=21{,}000-90^2=12{,}900\).

For the layer attaching at \(100\), the recoveries at \(0,100,300\) are \(0,0,200\). Expected recovery is \(0.2(200)=40\), and premium is \(1.25(40)=50\). Retained losses are \(0,100,100\), with mean \(0.3(100)+0.2(100)=50\) and second moment \(0.3(10{,}000)+0.2(10{,}000)=5{,}000\). Retained variance is \(5{,}000-50^2=2{,}500\), so variance reduction is \(12{,}900-2{,}500=10{,}400\).

For the layer attaching at \(200\), the recoveries are \(0,0,100\). Expected recovery is \(0.2(100)=20\), and premium is \(1.25(20)=25\). Retained losses are \(0,100,200\), with mean \(0.3(100)+0.2(200)=70\) and second moment \(0.3(10{,}000)+0.2(40{,}000)=11{,}000\). Retained variance is \(11{,}000-70^2=6{,}100\), so variance reduction is \(12{,}900-6{,}100=6{,}800\).

Moving to the lower attachment adds expected recovery \(40-20=20\), premium \(50-25=25\), and variance reduction \(10{,}400-6{,}800=3{,}600\). These differences describe the value and cost of choosing the first candidate instead of the second under this distribution and pricing assumption. The check is consistent with the original variance of \(12{,}900\): the retained variances are \(2{,}500\) and \(6{,}100\), respectively.

Common mistake avoided. A common error is to report the first layer's variance reduction as the variance of its recovery. The recovery has outcomes \(0,0,200\), so its variance is \(0.2(40{,}000)-40^2=6{,}400\), not \(10{,}400\). Variance reduction must be calculated from the original and retained loss variances, or from the covariance identity.

Selection Against Explicit Program Objectives

A comparison becomes a selection only after the objective and constraints are stated. For example, a program may require variance reduction of at least a specified amount while keeping premium within a budget. Calculate each candidate on the same loss basis and with the same premium convention, eliminate candidates that fail either constraint, and select among the feasible choices according to the stated preference. If no candidate is feasible, the correct conclusion is that none meets the objective—not that the closest one satisfies it.

Specify the decision criteria.
State the required risk reduction, premium limit, or other objective in measurable terms.

Calculate each candidate consistently.
Use the same loss distribution, recovery definition, and pricing basis for every attachment and limit.

Measure the incremental contribution.
Compare recovery, premium, and risk reduction against the relevant alternative, not just against zero.

Apply the constraints and verify.
Check every requirement separately and confirm that the selected terms meet all of them.

Worked Example: Selecting Terms Under a Budget and Risk Target

A hypothetical loss \(X\), in thousands of dollars, is \(0,200,\) or \(500\), with probabilities \(0.5,0.3,\) and \(0.2\). Choose between a layer attaching at \(100\) with limit \(200\), and one attaching at \(200\) with limit \(200\). Each premium equals \(1.20\) times expected recovery. The program objective is a premium no greater than \(60\) and a variance reduction of at least \(15{,}000\) thousand-dollars-squared.

Method. Evaluate both candidate layers against both constraints. Do not choose on the basis of recovery or risk reduction alone.

The original loss has mean \(0.3(200)+0.2(500)=160\), second moment \(0.3(40{,}000)+0.2(250{,}000)=62{,}000\), and variance \(62{,}000-160^2=36{,}400\).

For attachment \(100\), recoveries at \(0,200,500\) are \(0,100,200\). Expected recovery is \(0.3(100)+0.2(200)=70\), so premium is \(1.20(70)=84\). Retained losses are \(0,100,300\), with mean \(0.3(100)+0.2(300)=90\), second moment \(0.3(10{,}000)+0.2(90{,}000)=21{,}000\), and variance \(21{,}000-90^2=12{,}900\). Its variance reduction is \(36{,}400-12{,}900=23{,}500\). It meets the risk target but exceeds the premium budget.

For attachment \(200\), recoveries are \(0,0,200\). Expected recovery is \(0.2(200)=40\), and premium is \(1.20(40)=48\). Retained losses are \(0,200,300\), with mean \(0.3(200)+0.2(300)=120\), second moment \(0.3(40{,}000)+0.2(90{,}000)=30{,}000\), and variance \(30{,}000-120^2=15{,}600\). Its variance reduction is \(36{,}400-15{,}600=20{,}800\). It meets both constraints, so the second candidate is selected. As a check, \(48\leq60\) and \(20{,}800\geq15{,}000\).

Common mistake avoided. The attachment-\(200\) layer does not recover \(200\) on the loss of \(200\); it attaches at that amount, so recovery there is zero. Its only positive recovery is \(200\) when the loss is \(500\), giving expected recovery \(0.2(200)=40\), not \(60\).

Common Traps in Marginal Comparisons

Confusing limit with payment. The limit is the maximum recovery, not the expected recovery. Apply the attachment and limit to each loss first.
Comparing totals when the decision is incremental. If the question asks what an added limit contributes, subtract the current program's results from the proposed program's results.
Using expected recovery as a risk measure. Expected recovery is an average payment. It does not determine the reduction in retained variance by itself.
Mixing premium conventions. Apply the stated loading or quoted premium consistently. Do not apply a loading twice or use one candidate's pricing basis for another.
Ignoring one constraint. A candidate that meets the risk target but breaks the budget is infeasible when both conditions are mandatory.

Key takeaway: For every candidate, calculate the recovery distribution, expected recovery, premium, and retained-risk measure. Then compare changes against the current alternative and select only terms that satisfy the stated program objective.

Check Your Understanding

For each hypothetical problem, calculate the requested quantities before checking the final answer.

A loss \(X\), in thousands of dollars, is \(0,100,\) or \(400\), with probabilities \(0.6,0.25,\) and \(0.15\). A layer attaches at \(100\) with limit \(200\). Find expected recovery and variance reduction. (Answer: Expected recovery \(30\); variance reduction \(13{,}800\) thousand-dollars-squared.)
A loss is \(0,100,250,\) or \(500\), with probabilities \(0.4,0.3,0.2,\) and \(0.1\). At attachment \(100\), the limit increases from \(150\) to \(300\). Premium is \(40\%\) above expected recovery. Find the incremental expected recovery and premium. (Answer: Incremental expected recovery \(15\) and incremental premium \(21\), both in thousands of dollars.)
A loss is \(0,100,\) or \(300\), with probabilities \(0.5,0.3,\) and \(0.2\). Compare attachment \(100\), limit \(200\), with attachment \(200\), limit \(100\), at a \(25\%\) loading. Find the lower-attachment option's incremental expected recovery and variance reduction relative to the other option. (Answer: Incremental expected recovery \(20\), premium increase \(25\), and incremental variance reduction \(3{,}600\) thousand-dollars-squared.)
A loss is \(0,150,\) or \(350\), with probabilities \(0.5,0.3,\) and \(0.2\). Compare attachment \(100\) with limits \(100\) and \(200\), at a \(10\%\) loading. The objective is expected recovery of at least \(50\) and premium no greater than \(62\), in thousands of dollars. Which limit meets the objective? (Answer: Limit \(200\); its expected recovery is \(55\) and premium is \(60.50\).)
A loss is \(0,200,\) or \(500\), with probabilities \(0.5,0.3,\) and \(0.2\). A candidate layer attaches at \(200\) with limit \(200\), and premium is \(20\%\) above expected recovery. Does it meet a premium budget of \(50\) and variance-reduction target of \(20{,}000\) thousand-dollars-squared? (Answer: Yes; expected recovery is \(40\), premium is \(48\), and variance reduction is \(20{,}800\) thousand-dollars-squared.)

What You'll Learn