Integrated Stop-Loss Pricing from an Aggregate Distribution

Pricing a Stop-Loss Layer from Aggregate Losses

A stop-loss premium depends on the whole distribution of aggregate loss, not just its mean. Two portfolios with the same expected loss can produce different expected payments above a specified attachment because their probabilities of large losses differ. When a distribution is supplied directly, the pricing task is to apply the contract payment to each possible aggregate outcome, weight by its probability, and add.

The calculations in this tutorial use hypothetical aggregate-loss distributions. Unless a problem states otherwise, losses and payments are measured in dollars, premiums are undiscounted, and a pure premium means the expected insurer payment with no expense or risk loading.

Definition: Let $X$ be a nonnegative aggregate loss, with a supplied probability distribution. For an attachment $d\geq 0$, an unlimited excess-loss contract pays $(X-d)^+=\max(X-d,0)$. For a limited layer of width $u\geq 0$, the contract pays $\min((X-d)^+,u)$. Its pure premium is the expected payment. Here, $u$ is the maximum payment by the layer, so the layer covers losses from $d$ to $d+u$.

For a discrete distribution with possible outcomes $x_1,\ldots,x_n$ and probabilities $p_1,\ldots,p_n$, where $\sum_{j=1}^n p_j=1$, the pure premium is calculated by evaluating the payment at every outcome:

E[(X-d)^+]=\sum_{j=1}^{n}p_j\max(x_j-d,0), \qquad E[\min((X-d)^+,u)]=\sum_{j=1}^{n}p_j\min(\max(x_j-d,0),u).

An outcome exactly equal to the attachment produces no payment. An outcome above the top of a limited layer produces the full layer payment $u$, not the amount by which the loss exceeds the attachment.

Why Limited and Unlimited Payments Are Related

The key calculation for a limited layer is the difference between two unlimited excess-loss payments. This relationship both simplifies pricing and provides a useful check on direct calculations.

Consider one aggregate outcome.
Set $z=(X-d)^+$, the amount of loss above the attachment. Since $z\geq 0$, the limited payment is $\min(z,u)$.

Subtract the excess above the layer.
If $z\leq u$, then $(z-u)^+=0$, so $z-(z-u)^+=z$. If $z>u$, then $z-(z-u)^+=z-(z-u)=u$.

Identify the limited payment and take expectations.
Both cases give $z-(z-u)^+=\min(z,u)$. Therefore, averaging over the distribution gives the limited-layer premium identity.

\min((X-d)^+,u)=(X-d)^+-(X-d-u)^+, $$ $$ E[\min((X-d)^+,u)]=E[(X-d)^+]-E[(X-(d+u))^+].

The second term is the unlimited excess-loss premium at attachment $d+u$. Thus, a limited layer’s pure premium is the difference between the unlimited premiums at the bottom and top of the layer. This identity is valid for discrete, continuous, or mixed distributions, provided the expectations exist.

A second check follows from the pointwise decomposition $X=\min(X,d)+(X-d)^+$ for $X\geq 0$. Consequently, an unlimited excess-loss premium can also be checked as the mean aggregate loss minus the expected part retained below the attachment.

Worked Pricing and Attachment Comparisons

Worked Example: Price Both Forms of Cover from a Discrete Distribution

A hypothetical portfolio has aggregate loss $X$ with the following distribution. Find the unlimited excess-loss pure premium at attachment $d=200$, and the limited-layer pure premium with the same attachment and width $u=250$. Verify the limited premium using the layer-difference identity.

Aggregate loss $x$	0	100	300	600
Probability	0.10	0.20	0.40	0.30

Method and assumptions. The distribution is the stated probability mass function, whose probabilities sum to $0.10+0.20+0.40+0.30=1.00$. The unlimited payment is $\max(X-200,0)$; the limited payment is capped at $250$. These are expected payments, with no loading.

Unlimited cover. The payments at losses $0,100,300,600$ are respectively $0,0,100,400$. Weighting by the probabilities gives

0.10(0)+0.20(0)+0.40(100)+0.30(400) =0+0+40+120=\$160.

Limited layer. The payments are $0,0,100,250$, since the excess at loss $600$ would be $400$ but the layer pays at most $250$. Therefore,

0.10(0)+0.20(0)+0.40(100)+0.30(250) =0+0+40+75=\$115.

Identity check. At attachment $d+u=450$, the unlimited payments are $0,0,0,150$, so the unlimited premium is $0.30(150)=\$45$. The difference is $\$160-\$45=\$115$, agreeing with the direct limited-layer calculation. The common mistake avoided here is treating the limited payment at loss $600$ as $400$, rather than applying the cap of $250$.

Worked Example: Compare Limited Layers at Two Attachments

Using the same hypothetical aggregate distribution as in the preceding example, compare limited layers of width $u=200$ at attachments $d=100$ and $d=300$. Calculate both premiums directly and verify each as a difference of unlimited premiums.

At attachment $100$. The limited payments at losses $0,100,300,600$ are $0,0,200,200$. In particular, a loss of $300$ reaches the top of this layer. The pure premium is

0.10(0)+0.20(0)+0.40(200)+0.30(200) =80+60=\$140.

The unlimited premium at attachment $100$ is $0.40(200)+0.30(500)=80+150=\$230$. At attachment $300=d+u$, it is $0.30(300)=\$90$. Thus, the difference is $\$230-\$90=\$140$.

At attachment $300$. The limited payments are $0,0,0,200$, giving

0.10(0)+0.20(0)+0.40(0)+0.30(200)=\$60.

The unlimited premium at attachment $300$ is $0.30(300)=\$90$. At attachment $500$, the unlimited premium is $0.30(100)=\$30$. Their difference is $\$90-\$30=\$60$.

The lower-attachment layer costs $\$140-\$60=\$80$ more in expected payments. The check also makes the attachment convention explicit: the top of a width-$200$ layer attached at $300$ is $500$, not $200$. Confusing layer width with the top attachment would produce the wrong second unlimited premium.

Worked Example: Check Aggregate Decomposition and Apply a Loading

A hypothetical portfolio has the following distribution of aggregate loss. A reinsurer offers a layer attached at $250$, with width $300$, and quotes a premium equal to 112% of the pure premium. Calculate the unlimited and limited pure premiums, check the limited premium in two ways, and find the quoted premium.

Aggregate loss $x$	0	150	400	900
Probability	0.25	0.25	0.30	0.20

Unlimited pure premium. At attachment $250$, the payments are $0,0,150,650$. Hence,

E[(X-250)^+] =0.25(0)+0.25(0)+0.30(150)+0.20(650) =45+130=\$175.

Limited pure premium. The layer pays $0,0,150,300$ at the four outcomes. The last payment is capped at the width $300$. Thus,

E[\min((X-250)^+,300)] =0.30(150)+0.20(300) =45+60=\$105.

First check: subtract the upper excess. The top attachment is $250+300=550$. Only the loss $900$ exceeds it, by $350$; therefore $E[(X-550)^+]=0.20(350)=\$70$. The difference $\$175-\$70=\$105$ matches the direct calculation.

Second check: decompose the aggregate loss. The retained amounts $\min(X,250)$ at the four outcomes are $0,150,250,250$. Their expectation is $0.25(0)+0.25(150)+0.30(250)+0.20(250)=37.50+75+50=\$162.50$. The expected aggregate loss is $0.25(0)+0.25(150)+0.30(400)+0.20(900)=37.50+120+180=\$337.50$. Their difference is $\$337.50-\$162.50=\$175$, confirming the unlimited premium.

Loaded quote. A quote at 112% of pure premium is $1.12(\$105)=\$117.60$. The loading is applied to the limited pure premium, not to the unlimited premium or the total expected aggregate loss. The result is a quoted premium of \$117.60.

Interpreting the Calculation and Avoiding Traps

The expected-payment method is especially useful when the aggregate distribution is already supplied. It does not require reconstructing the individual claim frequency and severity distributions. It does require applying the contract terms to each aggregate outcome correctly. In particular, distinguish the attachment $d$, which determines when payment starts, from the width $u$, which determines the maximum payment for a limited layer.

Apply the attachment first. The unlimited payment is zero at and below $d$, and equals $X-d$ above $d$. Do not pay on the entire aggregate loss once the attachment is crossed.
Apply the cap to the payment. For a limited layer, cap $(X-d)^+$ at $u$. A loss beyond $d+u$ still produces payment $u$, not the full excess above $d$.
Keep probabilities attached to their outcomes. Calculate the payment for each outcome, multiply by its probability, and then sum. Check that the probabilities total one.
Keep pure and loaded prices distinct. The expected payment is the pure premium. Apply a stated loading only after calculating that expectation, and use the loading rule specified by the problem.

These checks are valuable because a plausible-looking total can still use the wrong layer top or omit the cap. The difference-of-unlimited-premiums identity is a particularly efficient independent check: its second attachment must be $d+u$, where $u$ is the layer width.

Key takeaway: For a supplied aggregate distribution, price a stop-loss layer by averaging its payment over all outcomes. Check a limited layer using $E[(X-d)^+]-E[(X-(d+u))^+]$, and check an unlimited premium using $E[X]-E[\min(X,d)]$.

Check Your Understanding

For each hypothetical distribution, calculate the requested expected payment. Answers are given in dollars, with no loading unless one is specified.

$X$ takes values $0,100,250,500$ with probabilities $0.20,0.30,0.30,0.20$, respectively. Find the unlimited excess-loss pure premium at attachment $150$. (Answer: \$100)
Use the distribution in question 1. Find the limited-layer pure premium with attachment $150$ and width $200$. (Answer: \$70)
$X$ takes values $0,200,500$ with probabilities $0.25,0.50,0.25$, respectively. Find the limited-layer pure premium at attachment $300$ with width $100$. (Answer: \$25)
$X$ takes values $0,100,300,700$ with probabilities $0.10,0.20,0.40,0.30$, respectively. Compare the limited-layer pure premiums for width $250$ attached at $150$ and $400$. (Answer: \$135 at attachment 150; \$75 at attachment 400)
$X$ takes values $0,100,400,800$ with probabilities $0.20,0.30,0.30,0.20$, respectively. A width-$300$ layer attached at $250$ is priced at 110% of pure premium. Find the quoted premium. (Answer: \$115.50)

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