Worked Comparison of Reinsurance Structures

Why Compare Reinsurance Structures on the Same Model?

Quota share, per-claim excess-of-loss, and annual aggregate excess-of-loss can all reduce an insurer’s retained loss, but they respond to different features of the loss experience. Quota share cedes a fixed proportion of every covered loss. Per-claim excess-of-loss responds to the size of each individual claim. Annual aggregate excess-of-loss responds to the total loss over the year. A fair numerical comparison must hold the underlying frequency and severity assumptions fixed.

The hypothetical insurer below has two possible claim opportunities during a year. Each opportunity produces a claim with probability one-half, independently of the other. If a claim occurs, its loss is $100 or $300, each with probability one-half, independently of frequency and of other claim severities. All amounts are in dollars; there is no interest adjustment or claim expense.

Model and treaty definitions: Let $N\sim\operatorname{Binomial}(2,0.5)$ be the number of claims, let $X_j$ be the loss on claim $j$, and let $S=\sum_{j=1}^{N}X_j$ be annual aggregate loss (with $S=0$ if $N=0$). The claim severities are independent with $P(X_j=100)=P(X_j=300)=0.5$. The quota-share treaty cedes 40% of every loss. The per-claim excess-of-loss treaty has attachment $150$ and limit $150$ per claim. The annual aggregate excess-of-loss treaty has attachment $250$ and limit $250$ for the year.

For a loss $x$, an excess-of-loss layer with attachment $a$ and limit $u$ pays $\min((x-a)_+,u)$, where $(z)_+=\max(z,0)$. For the per-claim treaty, this formula is applied to each claim and the payments are summed. For the annual aggregate treaty, it is applied once to $S$. These are different calculations even if the attachment and limit amounts happen to be the same.

Build the Aggregate Distribution First

The frequency model gives $P(N=0)=0.25$, $P(N=1)=0.50$, and $P(N=2)=0.25$. When $N=1$, the annual loss is $100 or $300 with equal conditional probability. When $N=2$, the two severities produce totals of $200, $400, or $600 with conditional probabilities $0.25$, $0.50$, and $0.25$, respectively. Combining these cases gives the following distribution.

Annual loss $S$	Probability $P(S=s)$
$0	0.2500
$100	0.2500
$200	0.0625
$300	0.2500
$400	0.1250
$600	0.0625
Total	1.0000

As a check, the probabilities sum to $0.25+0.25+0.0625+0.25+0.125+0.0625=1$. The mean claim severity is $E[X_j]=0.5(100)+0.5(300)=200$, and the mean claim count is $E[N]=2(0.5)=1$. Therefore $E[S]=E[N]E[X_j]=200$, which also agrees with the aggregate distribution.

Justification for the compound moment calculation: If $W=\sum_{j=1}^{N}V_j$, where the $V_j$ are independent and identically distributed, independent of $N$, then $E[W\mid N]=N E[V_j]$ and $\operatorname{Var}(W\mid N)=N\operatorname{Var}(V_j)$. The law of total variance gives $$ \operatorname{Var}(W)=E[N]\operatorname{Var}(V_j)+\operatorname{Var}(N)(E[V_j])^2. $$ The first term is the average variation in severities given the claim count; the second is variation caused by the claim count itself. Here $E[N]=1$, $\operatorname{Var}(N)=2(0.5)(0.5)=0.5$, $E[X_j]=200$, and $\operatorname{Var}(X_j)=0.5(100-200)^2+0.5(300-200)^2=10{,}000$. Thus $\operatorname{Var}(S)=10{,}000+0.5(200)^2=30{,}000$.

Worked Comparison of Expected Payments

For each treaty, calculate the ceded payment from the actual loss measure to which the treaty applies. The insurer’s retained annual loss is $R=S-Z$, where $Z$ denotes total ceded loss under that treaty.

Worked Example: Expected Payments Under Three Hypothetical Treaties

Using the model and treaty terms above, calculate the expected ceded and retained annual losses for each structure. Then verify that expected retained plus expected ceded equals expected gross loss.

Method. Apply each treaty’s payment rule to the severity for quota share and per-claim excess-of-loss, and to the annual total for aggregate excess-of-loss. The common trap is to apply the annual aggregate attachment separately to each claim.

Quota share. The ceded amount is $Z=0.40S$, so $E[Z]=0.40E[S]=0.40(200)=80$. The insurer retains $R=0.60S$, giving $E[R]=0.60(200)=120$. The check is $80+120=200=E[S]$.

Per-claim excess-of-loss. The ceded payment on a $100 claim is $\min((100-150)_+,150)=0$. On a $300 claim it is $\min(300-150,150)=150$. Thus the mean ceded payment per claim is $0.5(0)+0.5(150)=75$. Since $E[N]=1$, expected annual cession is $75$. The retained payment per claim is $100 or $150, so its mean is $0.5(100)+0.5(150)=125$, and expected annual retained loss is $1(125)=125$. The check is $75+125=200$.

Annual aggregate excess-of-loss. Apply the layer to each possible value of $S$. At $S=0,100,200$, the payment is zero. At $S=300,400,600$, the ceded payments are respectively $50,150,250$. Therefore, $$ E[Z]=0.2500(50)+0.1250(150)+0.0625(250)=12.50+18.75+15.625=46.875. $$ Expected retained loss is $200-46.875=153.125$. Rounded to the nearest cent, expected ceded and retained amounts are $46.88 and $153.13; their unrounded values sum to $200.

The expected annual cessions are $80.00 for quota share, $75.00 for per-claim excess-of-loss, and $46.88 for annual aggregate excess-of-loss. These amounts describe expected payments under this model; they do not, by themselves, show how much variability each treaty transfers.

Compare Retained-Loss Variability

Variance is another useful comparison, but it must be calculated for the retained payment under each structure. Proportional reinsurance scales the whole annual loss. Per-claim excess-of-loss instead changes the per-claim payment distribution, while annual aggregate excess-of-loss can be evaluated directly from the distribution of $S$.

Worked Example: Variance of Retained Annual Loss

Calculate the variance of the insurer’s retained annual loss under all three treaties. State which treaty has the smallest retained variance in this hypothetical model.

Quota share. Since $R=0.60S$, $\operatorname{Var}(R)=0.60^2\operatorname{Var}(S)=0.36(30{,}000)=10{,}800$.

Per-claim excess-of-loss. Retained severity $V_j$ is $100 or $150 with equal probability. Its mean is $0.5(100)+0.5(150)=125$, and its variance is $0.5(100-125)^2+0.5(150-125)^2=625$. Applying the compound moment formula, $$ \operatorname{Var}(R)=E[N](625)+\operatorname{Var}(N)(125)^2 =1(625)+0.5(15{,}625)=8{,}437.50. $$

Annual aggregate excess-of-loss. The retained amount is $R=S-Z$. From the aggregate distribution, the possible retained amounts are $0, $100, $200, $250, and $350. The $250 amount occurs for both $S=300$ and $S=400$, with combined probability $0.2500+0.1250=0.3750$. The other probabilities are 0.2500 at $0, 0.2500 at $100, 0.0625 at $200, and 0.0625 at $350. Hence $$ E[R^2]=0.2500(0^2)+0.2500(100^2)+0.0625(200^2)+0.3750(250^2)+0.0625(350^2)=36{,}093.75. $$ The mean retained amount is $153.125$, so $$ \operatorname{Var}(R)=36{,}093.75-(153.125)^2=12{,}646.484375. $$ Rounded to the nearest cent, the variance is $12{,}646.48$ dollars squared.

The per-claim treaty has the smallest retained variance, $8{,}437.50$, among these three structures in this model. This result depends on the specified frequency, severity, attachment, and limit; it is not a general ranking of treaty types.

How the Treaties Respond to a Large Year

Expected payments and variances summarize different aspects of the distribution. A threshold probability offers another comparison: it asks how often the insurer’s retained annual loss exceeds a chosen amount. It is important to apply the threshold to retained loss, not to gross loss or ceded loss.

Worked Example: Probability Retained Loss Exceeds $250

For each treaty, calculate $P(R>250)$. Use a strict inequality: retained loss equal to $250 does not count as exceeding $250.

Quota share. Retained loss is $0.60S$, so $R>250$ requires $S>250/0.60=416.666\ldots$. The only aggregate loss in the distribution above this threshold is $600, with probability $0.0625$. Therefore $P(R>250)=0.0625$.

Per-claim excess-of-loss. Retained amounts for annual losses $S=0,100,200,300,400,600$ are respectively $0, $100, $200, $250, $300, and $300. Thus retained loss exceeds $250 for $S=400$ or $S=600$, and $$ P(R>250)=0.1250+0.0625=0.1875. $$

Annual aggregate excess-of-loss. Retained amounts for the same annual losses are $0, $100, $200, $250, $250, and $350. Only $S=600$ results in retained loss greater than $250, so $P(R>250)=0.0625$.

In this case the quota-share and annual aggregate treaties have the same probability of exceeding the threshold, while the per-claim treaty has the larger probability. Their retained amounts are not otherwise the same: for example, at $S=400$, quota share retains $240, per-claim excess-of-loss retains $300, and annual aggregate excess-of-loss retains $250.

Common Mistake: Applying an Annual Layer Claim by Claim

Suppose two claims of $300 occur in the year, so $S=600$. Under the per-claim treaty, each claim produces a cession of $\min(300-150,150)=150$, for a total cession of $300. Under the annual aggregate treaty, the cession is $\min(600-250,250)=250$. Applying the annual attachment of $250 separately to each claim would produce $2(300-250)=100$, which is wrong because the treaty attaches to the annual total, not to individual claims. The annual limit also caps the cession at $250.

A useful calculation order is to identify the treaty basis, determine the relevant loss amount, apply the attachment, and then enforce the limit. For quota share, the relevant basis is each covered loss and there is no attachment. For per-claim excess-of-loss, apply the layer separately to each claim and add the payments. For annual aggregate excess-of-loss, first form $S$, then apply the layer once.

Key takeaway: A meaningful comparison uses the same frequency and severity assumptions for every structure, but applies each treaty to its correct basis. Expected cession, retained variance, and threshold probabilities answer distinct questions, so check the measure that the problem asks for.

Check Your Understanding

Use the hypothetical model and treaty terms in this tutorial. Report dollar amounts to the nearest cent and probabilities to four decimals.

Calculate the expected gross annual loss from the frequency and severity assumptions. (Answer: $200.00)
Under the 40% quota-share treaty, calculate the expected annual cession and the expected retained loss. (Answer: $80.00 ceded and $120.00 retained)
Under the per-claim excess-of-loss treaty, what is the probability that a claim produces a positive ceded payment, conditional on a claim occurring? (Answer: 0.5000)
Under the annual aggregate excess-of-loss treaty, calculate the probability that the reinsurer makes a positive payment during the year. (Answer: 0.4375)
Calculate the variance of annual ceded loss under the per-claim excess-of-loss treaty. The per-claim cession is $0 or $150 with equal probability. (Answer: $8,437.50 dollars squared)

What You'll Learn