Choosing Attachment Points From Loss Exceedance Curves

Why Attachment Points Should Be Read Against a Loss Curve

A catastrophe reinsurance layer trades premium for protection at specified loss levels. Its attachment point determines when recoveries begin; its limit determines how much protection is available above that point. A loss exceedance curve helps connect those contract terms to the insurer’s objective: for example, keeping the probability of a retained event loss above a chosen amount below a target.

The curve is not a contract recommendation by itself. It describes the modeled frequency or probability of losses exceeding levels; selecting a layer also requires considering its cost, wording, reinstatements, and the insurer’s risk appetite. The examples here focus on comparing attachment points and limits against stated loss-retention objectives.

Definition: Let $M$ be the maximum gross catastrophe event loss in one year. The occurrence exceedance probability at loss level $x$ is $P(M>x)$. An occurrence exceedance curve plots this probability against $x$, or reports the loss level associated with selected probabilities. A return period of $T$ years is sometimes used to describe an exceedance probability of $1/T$ per year; it is a probability summary, not a schedule of events.

A curve based on $M$ answers a question about the largest event in a year. It does not give the probability distribution of total annual catastrophe losses or total annual reinsurance recoveries. Those are different quantities. The distinction matters whenever an insurer may have more than one event in a year.

From Attachment and Limit to Retained Event Loss

Use $A$ for the attachment point and $L$ for the layer limit. For a single gross event loss $x$, the layer recovery is the part of the loss above $A$, capped at $L$. The insurer’s retained event loss is the gross loss less that recovery.

Definition: For a layer attaching at $A$ with limit $L$, the occurrence recovery and retained loss at gross event loss $x$ are $$ R_{A,L}(x)=\min\bigl((x-A)^+,L\bigr), \qquad N_{A,L}(x)=x-R_{A,L}(x), $$ where $(z)^+=\max(z,0)$. The layer is exhausted at gross loss $A+L$. These formulas describe one event and do not by themselves calculate annual aggregate recoveries.

The retained-loss function has three regions. Below attachment, there is no recovery. Between attachment and exhaustion, the insurer retains the attachment amount. Above exhaustion, the insurer retains the gross loss less the full layer limit.

N_{A,L}(x)= \begin{cases} x, & x\leq A,\\ A, & A<x\leq A+L,\\ x-L, & x>A+L. \end{cases}

This piecewise form explains why an attachment point cannot be selected from the curve without first stating a retention objective. A layer may reduce the retained severity of losses in its covered band, yet leave losses above exhaustion increasing with gross loss. It also cannot change a retained-loss threshold below the attachment: losses already above such a threshold but below attachment remain unreinsured.

Relating a Retention Objective to the Exceedance Curve

For a chosen retained-loss threshold $c$, the probability of exceeding it can be obtained from the gross-loss curve. The inverse relationship follows directly from the three regions of $N_{A,L}(x)$. If $c<A$, retained losses exceed $c$ exactly when gross losses exceed $c$. If $c=A$, the retained loss is strictly greater than $A$ only after the layer is exhausted. If $c>A$, exceedance occurs only when the gross loss exceeds $c+L$.

P\bigl(N_{A,L}(M)>c\bigr)= \begin{cases} P(M>c), & c<A,\\ P(M>A+L), & c=A,\\ P(M>c+L), & c>A. \end{cases}

The strict inequalities are important. For example, an event loss exactly equal to $A+L$ leaves the insurer with retention $A$, not a retention greater than $A$. The curve convention $P(M>x)$ must therefore match the wording of the objective. If the objective instead asks for $P(N_{A,L}(M)\geq c)$, the boundary cases can differ.

State the risk measure.
Identify whether the objective concerns the annual maximum event loss, a particular event scenario, or total annual loss. Do not substitute one for another.

State the retention target.
Write the retained-loss threshold and probability target precisely, including whether the inequality is strict.

Test each candidate layer.
Calculate its retained event losses or use the piecewise relationship to read the relevant probability from the exceedance curve.

Compare protection and cost.
Identify which candidates meet the objective, then compare their limits, attachment points, and any supplied prices or constraints.

Worked Layer Calculations

Worked Example: Checking Recoveries at and Above Exhaustion

A hypothetical insurer evaluates an occurrence layer attaching at $A=\$100{,}000$ with limit $L=\$200{,}000$. Consider gross event losses of $\$0$, $\$100{,}000$, $\$200{,}000$, $\$300{,}000$, and $\$600{,}000$. Calculate the amount of loss above attachment before the cap, the actual recovery, and retained loss for each amount.

Method. First calculate $(x-A)^+$, the uncapped excess above attachment. Then cap that amount at $L$ to obtain the recovery, and subtract the recovery from $x$ to obtain retention. The layer exhausts at $A+L=\$300{,}000$.

Gross loss $x$	Excess above attachment $(x-A)^+$	Recovery $R_{A,L}(x)$	Retained loss $N_{A,L}(x)$
\$0	\$0	\$0	\$0
\$100,000	\$0	\$0	\$100,000
\$200,000	\$100,000	\$100,000	\$100,000
\$300,000	\$200,000	\$200,000	\$100,000
\$600,000	\$500,000	\$200,000	\$400,000

At a gross loss of $\$600{,}000$, the amount above attachment is $\$600{,}000-\$100{,}000=\$500{,}000$. That is the uncapped excess; the actual recovery is limited to $\$200{,}000$. The retained loss is $\$600{,}000-\$200{,}000=\$400{,}000$. The check is that recovery never exceeds the limit and that recovery plus retention equals gross loss in every row.

Common mistake avoided: Confusing excess above attachment with actual recovery. The $\$500{,}000$ calculation is correct as an uncapped amount, but calling it the recovery would ignore the layer limit.

Selecting an Attachment From a Retention Target

Worked Example: Meeting a Probability Limit on Retained Event Loss

A hypothetical insurer models the annual maximum catastrophe event loss $M$, in thousands of dollars, as follows. Losses are assumed to take only the listed values, and the probabilities sum to one. The insurer considers occurrence layers with limit $L=200$ and attachments $A=50$, $100$, or $150$. It wants $P(N_{A,L}(M)>75)\leq 0.12$. Which candidate layers meet the objective?

Annual maximum loss $M$ ($000)	0	50	100	150	250	300	400
Probability	0.35	0.20	0.15	0.12	0.08	0.06	0.04

Step 1: Check the probability total. The probabilities sum to $0.35+0.20+0.15+0.12+0.08+0.06+0.04=1.00$.

Step 2: Test $A=50$. The layer exhausts at $50+200=250$. Retained losses at the listed gross losses are $0,50,50,50,50,100,200$. Retention exceeds $75$ only at gross losses $300$ and $400$, so $P(N_{50,200}(M)>75)=0.06+0.04=0.10$.

Step 3: Test $A=100$. The layer exhausts at $300$. Retained losses are $0,50,100,100,100,100,200$. Retention exceeds $75$ at gross losses $100,150,250,300,$ and $400$, giving $0.15+0.12+0.08+0.06+0.04=0.45$.

Step 4: Test $A=150$. The layer exhausts at $350$. Retained losses are $0,50,100,150,150,150,200$. Retention exceeds $75$ at gross losses $100,150,250,300,$ and $400$, again giving $0.45$.

Only the $A=50$, $L=200$ layer meets the target, since $0.10\leq0.12$, while $0.45>0.12$ for each other candidate. The curve shortcut gives the same answer for $A=50$: because $75>A$, the relevant gross-loss threshold is $75+200=275$, and $P(M>275)=0.06+0.04=0.10$.

Common mistake avoided: Reading $P(M>75)$ for all candidate layers. For $A=50$, the layer reduces some losses above $75$, so the correct curve threshold is $275$, not $75$. For attachments above $75$, by contrast, losses above $75$ but below attachment remain unreinsured.

Comparing Protection and Expected Recovery

Worked Example: Comparing Two Layers on the Same Occurrence Curve

A hypothetical portfolio has annual maximum event loss $M$, in thousands of dollars, with values $0,100,200,300,500$ and respective probabilities $0.50,0.20,0.15,0.10,0.05$. Compare layer 1, $A=100$, $L=200$, with layer 2, $A=150$, $L=250$. For each layer, calculate expected recovery and the probability that retained event loss exceeds $150$. Assume no premium information is available, so the comparison is limited to modeled protection.

Step 1: Calculate layer 1. Its exhaustion point is $100+200=300$. Recoveries at gross losses $0,100,200,300,500$ are $0,0,100,200,200$. Thus expected recovery, in thousands of dollars, is $0.50(0)+0.20(0)+0.15(100)+0.10(200)+0.05(200)=15+20+10=45$. The retained losses are $0,100,100,100,300$, so retention exceeds $150$ only when $M=500$. Therefore $P(N_{100,200}(M)>150)=0.05$.

Step 2: Calculate layer 2. Its exhaustion point is $150+250=400$. Recoveries are $0,0,50,150,250$. Expected recovery is $0.50(0)+0.20(0)+0.15(50)+0.10(150)+0.05(250)=7.5+15+12.5=35$. Retained losses are $0,100,150,150,250$, so retention exceeds $150$ only when $M=500$. Thus $P(N_{150,250}(M)>150)=0.05$.

Step 3: Interpret the comparison. Both layers produce the same modeled probability of retained loss above $150$, but layer 1 has expected recovery of $45$ thousand compared with $35$ thousand for layer 2. The expected gross loss is $0.20(100)+0.15(200)+0.10(300)+0.05(500)=105$ thousand. Hence expected retained event loss is $105-45=60$ thousand for layer 1 and $105-35=70$ thousand for layer 2. These results describe the annual maximum event under the stated model; they are not estimates of total annual recovery if multiple events occur.

Common mistake avoided: Treating equal exceedance probabilities at one threshold as equivalent protection. Here the layers have equal probability of retention above $150$, but different expected recoveries and different retained amounts at intermediate event losses.

How to Use the Curve Carefully

An occurrence curve can support a decision in two complementary ways. First, test a retention objective: translate the retained-loss threshold into the corresponding gross-loss threshold, then read the exceedance probability. Second, compare the full set of event losses or curve levels to see where each layer provides protection and where it is exhausted. A single probability point may conceal material differences elsewhere on the curve.

The derivation of the threshold relationship is a direct consequence of the piecewise retention function. For $c<A$, the covered layer cannot change whether a loss already exceeds $c$, because losses through attachment are retained in full and losses above attachment leave retention at least $A$. For $c=A$, retention equals $A$ throughout the covered band, so it is strictly greater only above exhaustion. For $c>A$, a retained loss greater than $c$ must occur above exhaustion, where retention is $M-L$; solving $M-L>c$ gives $M>c+L$.

Results depend on the modeled curve and on consistent loss definitions. Confirm whether values are gross or net of other protection, whether the curve represents annual maximum occurrence losses, and how event-loss estimates are treated. Where the curve is estimated from simulation or interpolation, uncertainty and model limitations should be considered rather than treating each plotted point as exact.

Common Exam Traps

Applying the limit to the wrong quantity. Calculate excess above attachment first, then cap the recovery at the limit. The uncapped excess is not necessarily the recovery.
Confusing the layer limit with its exhaustion point. For attachment $A$ and limit $L$, exhaustion occurs at $A+L$.
Using an aggregate-loss interpretation for an occurrence curve. The annual maximum event loss does not determine total annual loss or total annual recoveries.
Ignoring strict versus non-strict exceedance. At exact exhaustion, retention is $A$; whether that meets a target depends on whether the objective uses $>$ or $\geq$.
Choosing only by expected recovery. The layer with the larger expected recovery need not be the one that best meets a specific tail-retention objective, and premium or other contract terms may change the decision.

Key takeaway: Select an attachment point by first stating the retained event-loss objective, then mapping it through the layer’s attachment and limit to the appropriate point on the occurrence exceedance curve. Verify the result with the recovery formula and keep occurrence results distinct from aggregate annual results.

Check Your Understanding

For each problem, treat listed probabilities as probabilities for the annual maximum event loss unless stated otherwise. Money amounts are in thousands of dollars.

A layer attaches at $A=80$ with limit $L=120$. For a gross event loss of $310$, calculate uncapped excess above attachment, recovery, and retained loss. (Answer: uncapped excess $230$; recovery $120$; retention $190$.)
An annual maximum loss is $0,100,250,400$ with probabilities $0.60,0.20,0.15,0.05$. For $A=100,L=200$, find $P(N_{A,L}(M)>100)$. (Answer: $0.05$.)
A candidate layer has $A=60,L=150$, and the retained-loss target is $c=80$. The curve gives $P(M>230)=0.07$. Find the probability that retained event loss exceeds $80$. (Answer: $0.07$, since $c>A$ and the corresponding gross threshold is $c+L=230$.)
A layer attaches at $A=120$ with limit $L=180$. A modeled event loss is exactly $300$. Calculate recovery and retention, then state whether retention is strictly greater than $120$. (Answer: recovery $180$, retention $120$; no.)
Two layers have the same attachment $A=100$, with limits $L=100$ and $L=250$. The occurrence curve gives $P(M>350)=0.03$. For each layer, calculate $P(N_{A,L}(M)>100)$ and identify which has the lower probability. (Answer: for $L=100$, $P(M>200)$ is needed, so the stated information is insufficient; for $L=250$, the probability is $P(M>350)=0.03$. The second layer’s probability is $0.03$, but the first cannot be determined from the supplied curve point.)

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