Pure Premiums and Loaded Stop-Loss Prices

From Expected Claim Payment to Quoted Price

A stop-loss contract pays when an aggregate loss exceeds an attachment. The expected payment is an important pricing input, but it is not necessarily the amount charged to the purchaser. A quoted price may also include a risk or profit loading, expenses, or both. Keeping these amounts separate is essential: applying a loading to the wrong base can produce a material pricing error.

Let $S$ be a nonnegative aggregate loss over the contract period, and let $d\geq 0$ be the stop-loss attachment. The contract payment is the amount by which $S$ exceeds $d$, if any. In this tutorial, a pure premium means the expected value of that payment. It does not, by itself, include expenses or a loading.

Definition: The stop-loss payment with attachment $d$ is $(S-d)^+=\max(S-d,0)$. Its pure premium is $$ \pi(d)=E[(S-d)^+]. $$ A quoted price is a separate amount determined by a specified pricing rule. Unless the rule says otherwise, do not treat the pure premium and the quote as interchangeable.

The attachment is applied to the aggregate $S$, not separately to each underlying loss, unless the contract specifically defines a per-loss deductible. To calculate a pure premium, first determine the possible aggregate outcomes and their probabilities, calculate the payment for each outcome, and then take the expectation.

Why the Tail-Probability Formula Works

There is a useful alternative expression for the pure premium. It shows that the expected payment depends on how much probability remains above each level beyond the attachment.

Theorem: For a nonnegative aggregate loss $S$ with a finite expected stop-loss payment, $$ \pi(d)=\int_d^\infty P(S>x)\,dx. $$ For a discrete distribution, the direct probability-weighted sum of payments is often the simplest calculation. The integral formula also applies to discrete and mixed distributions.

To justify the formula, fix a realized aggregate value $S=s$. The length of the interval from $d$ up to $s$, when $s>d$, is $s-d$; when $s\leq d$, that interval has length zero. Thus

(s-d)^+=\int_d^\infty \mathbf{1}_{\{s>x\}}\,dx.

Taking expectations gives the result. The integrand’s expectation is the probability that $S>x$. Since the integrand is nonnegative, interchanging the expectation and integral is valid, including when $S$ has point probabilities. A single endpoint does not change the value of an integral.

E[(S-d)^+] =E\left[\int_d^\infty \mathbf{1}_{\{S>x\}}\,dx\right] =\int_d^\infty E[\mathbf{1}_{\{S>x\}}]\,dx =\int_d^\infty P(S>x)\,dx.

Loading and Expense Conventions

A loading is meaningful only after its base is specified. A proportional loading $\theta$ on the pure premium gives a claim-cost amount of $(1+\theta)\pi(d)$. A fixed expense $e$ is added as a dollar amount. By contrast, if expenses are a proportion $\alpha$ of the gross quote $Q$, the expense amount is $\alpha Q$, not $\alpha\pi(d)$.

For example, if the pricing rule applies a proportional loading $\theta$, adds a fixed expense $e$, and charges expenses equal to a proportion $\alpha$ of the gross quote, then the quote must satisfy

Q=(1+\theta)\pi(d)+e+\alpha Q.

Provided $\alpha<1$, solve for the quote:

Q=\frac{(1+\theta)\pi(d)+e}{1-\alpha}.

This equation treats the loading as an addition to expected claim cost and the percentage expense as a share of the final quote. If a problem instead defines expenses as a percentage of the pure premium, use that stated base; do not substitute one convention for another.

Worked Calculations

Worked Example: A Discrete Aggregate and a Fixed Expense

A hypothetical stop-loss contract has attachment $d=\$20{,}000$. The aggregate loss $S$ is $\$0$, $\$12{,}000$, $\$28{,}000$, or $\$60{,}000$, with respective probabilities $0.40$, $0.30$, $0.20$, and $0.10$. The quote applies a 15% proportional loading to the pure premium and adds a fixed expense of $\$300$. Find the pure premium and quote.

Method. Calculate the stop-loss payment at each aggregate outcome, take its expectation, and then apply the stated pricing rule. This avoids applying the attachment to each component loss or applying the loading to the wrong amount.

At $S=\$0$, the payment is $\max(\$0-\$20{,}000,0)=\$0$. At $S=\$12{,}000$, it is $\max(\$12{,}000-\$20{,}000,0)=\$0$. At $S=\$28{,}000$, it is $\$8{,}000$. At $S=\$60{,}000$, it is $\$40{,}000$.

The expected payment is

\begin{aligned} \pi(20{,}000) &=0.40(0)+0.30(0)+0.20(8{,}000)+0.10(40{,}000)\\ &=0+0+1{,}600+4{,}000\\ &=\$5{,}600. \end{aligned}

The 15% loading is $0.15(\$5{,}600)=\$840$, so the claim-cost amount after loading is $\$5{,}600+\$840=\$6{,}440$. Adding the fixed expense gives

Q=\$6{,}440+\$300=\$6{,}740.

Check. The quote exceeds the pure premium by $\$840+\$300=\$1{,}140$, and $\$5{,}600+\$1{,}140=\$6{,}740$. The pure premium is $\$5{,}600$; the quote is $\$6{,}740$.

Worked Example: Percentage Loading and Expenses on the Quote

A hypothetical aggregate loss takes values $\$10{,}000$, $\$30{,}000$, and $\$70{,}000$ with probabilities $0.25$, $0.50$, and $0.25$. A stop-loss contract attaches at $d=\$25{,}000$. The price includes a 20% loading on the pure premium and expenses equal to 5% of the gross quote. Find the pure premium and quote to the nearest cent.

Method. First calculate expected layer payments. Then include the loading on the pure premium and solve an equation for the quote, because the expense amount depends on the quote itself. This avoids treating the 5% expense as 5% of the pure premium.

The payments at the three aggregate outcomes are

\max(10{,}000-25{,}000,0)=0,\quad \max(30{,}000-25{,}000,0)=5{,}000,\quad \max(70{,}000-25{,}000,0)=45{,}000.

Therefore, the pure premium is

\begin{aligned} \pi(25{,}000) &=0.25(0)+0.50(5{,}000)+0.25(45{,}000)\\ &=0+2{,}500+11{,}250\\ &=\$13{,}750. \end{aligned}

The claim-cost amount after the 20% loading is $1.20(\$13{,}750)=\$16{,}500$. Let $Q$ be the gross quote. Expenses are $0.05Q$, so

Q=\$16{,}500+0.05Q,\qquad 0.95Q=\$16{,}500,\qquad Q=\frac{\$16{,}500}{0.95}=\$17{,}368.421\ldots.

Rounded to the nearest cent, the quote is $\$17{,}368.42$. Check. Five percent of $\$17{,}368.42$ is approximately $\$868.42$, and $\$16{,}500+\$868.42=\$17{,}368.42$. The pure premium is $\$13{,}750$, not the loaded claim-cost amount.

Worked Example: Aggregating Two Independent Claims Before Applying the Attachment

Two independent hypothetical claims are each $\$5{,}000$ with probability $0.10$, and zero otherwise. A stop-loss contract attaches at $\$4{,}000$ to their aggregate $S$. The quote has a 25% loading on the pure premium and expenses equal to 4% of the gross quote. Find the pure premium and quote to the nearest cent.

Method. Find the aggregate distribution using independence, then apply the stop-loss attachment to each aggregate outcome. This avoids applying the attachment claim by claim or overlooking the probability of exactly one claim.

The probability of no claims is $0.90(0.90)=0.81$, and the aggregate is then $\$0$. The probability of exactly one claim is

$$ 0.10(0.90)+0.90(0.10)=0.09+0.09=0.18, $$

giving an aggregate of $\$5{,}000$. The probability of two claims is $0.10(0.10)=0.01$, giving an aggregate of $\$10{,}000$. The corresponding stop-loss payments are $\$0$, $\$1{,}000$, and $\$6{,}000$. Thus

\begin{aligned} \pi(4{,}000) &=0.81(0)+0.18(1{,}000)+0.01(6{,}000)\\ &=0+180+60\\ &=\$240. \end{aligned}

The claim-cost amount after the 25% loading is $1.25(\$240)=\$300$. Let $Q$ be the quote. Since the expense is 4% of $Q$,

Q=\$300+0.04Q,\qquad 0.96Q=\$300,\qquad Q=\frac{\$300}{0.96}=\$312.50.

Check. The expense is $0.04(\$312.50)=\$12.50$, and $\$300+\$12.50=\$312.50$. The pure premium is $\$240$; the quote is $\$312.50$.

Common Pricing Traps

The most frequent errors arise from mixing up the contract payment, its expected value, and the pricing additions. Before calculating, write down the payment formula and state the base for each percentage. These checks are especially useful when a question includes both a loading and an expense rate.

Using the aggregate mean as the pure premium. The expected aggregate loss $E[S]$ is generally not $E[(S-d)^+]$. The attachment changes the payment. Calculate the layer payment at each outcome or use the tail-probability formula.
Subtracting the attachment even when the aggregate is below it. The payment is $\max(S-d,0)$, never a negative amount.
Applying the attachment to individual claims. For an aggregate stop-loss contract, sum the claims first, then apply the attachment to $S$.
Applying a quote-based expense to the pure premium. If expenses are $\alpha$ of the gross quote, solve $Q=C+\alpha Q$, where $C$ is the specified claim-cost amount plus any fixed expense.
Confusing loading conventions. A 20% loading on the pure premium gives $1.20\pi(d)$. It does not mean adding 20% of the final quote unless the pricing rule explicitly defines it that way.

For instance, in the second worked example, adding 5% of the loaded claim cost would give $\$16{,}500(1.05)=\$17{,}325$, which is not the stated quote. The stated expense is 5% of the quote, so the correct equation is $Q=\$16{,}500+0.05Q$, yielding $\$17{,}368.42$.

Key takeaway: Calculate the expected stop-loss payment first: $\pi(d)=E[(S-d)^+]$. Then apply each loading and expense exactly to its stated base. In particular, an expense that is a percentage of the gross quote belongs on both sides of the quote equation.

Check Your Understanding

For each problem, distinguish the expected stop-loss payment from any added pricing amounts. Round only where requested.

A hypothetical aggregate loss is $\$0$, $\$10{,}000$, or $\$25{,}000$ with probabilities $0.50$, $0.30$, and $0.20$, respectively. The attachment is $\$8{,}000$. Find the pure premium. (Answer: $\$4{,}000$)
A stop-loss contract attaches at $\$10{,}000$. Its aggregate loss is $\$5{,}000$, $\$15{,}000$, or $\$35{,}000$ with probabilities $0.20$, $0.50$, and $0.30$. The quote adds a 10% loading to the pure premium and a fixed expense of $\$500$. Find the pure premium and quote. (Answer: Pure premium $\$10{,}000$; quote $\$11{,}500$)
A hypothetical aggregate loss is uniformly distributed from $\$0$ to $\$40{,}000$. The attachment is $\$10{,}000$. Find the pure premium, then find the quote if the claim cost is loaded by 20% and expenses are 5% of the gross quote. Round the quote to the nearest cent. (Answer: Pure premium $\$11{,}250$; quote $\$14{,}210.53$)
Two independent hypothetical claims are each $\$4{,}000$ with probability $0.20$ and zero otherwise. A stop-loss contract attaches at $\$5{,}000$ to the aggregate. The quote adds a 10% loading to the pure premium and a fixed expense of $\$30$. Find the pure premium and quote. (Answer: Pure premium $\$120$; quote $\$162$)
A contract has pure premium $\$2{,}000$. Its quote includes a 12% loading on the pure premium, a fixed expense of $\$100$, and expenses equal to 8% of the gross quote. Find the quote to the nearest cent. (Answer: $\$2{,}543.48$)

What You'll Learn