Decode Contract Terms into Loss Random Variables

Why Contract Wording Must Become a Payment Function

A loss model starts with a ground-up loss $X$, the amount of covered damage before contract terms are applied. A deductible, a limit, coinsurance, or reinsurance changes who pays each part of that loss. Translating the wording into a function makes those allocations explicit and helps prevent errors about the order or scope of a contract term.

This tutorial focuses on per-claim payment functions. Unless a worked example says otherwise, assume each loss is nonnegative, the contract terms apply separately to each claim, and there are no expenses, taxes, interest adjustments, or other payments. The examples use hypothetical contracts and hypothetical loss distributions. No mortality assumption or interest rate is involved. The terminology and calculations are aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.

Definition: Let $X\geq 0$ be the ground-up loss on one claim. The insurer’s payment per loss is denoted $Y^L$. For a claim that produces a positive payment, the payment amount is described on a per-payment basis by $Y^P$. A deductible $d$ is the amount of loss the policyholder bears before the insurer pays; a limit specifies a cap, whose precise meaning must be read from the contract.

The random variable $X$ describes the loss before coverage changes. The payment random variable is a function of $X$ and the contract terms. A claim can therefore have $X>0$ but $Y^L=0$, for example when the loss does not exceed the deductible. The per-loss variable $Y^L$ includes zero-payment claims; $Y^P$ refers to the amount of a payment when a payment occurs. That distinction matters when reading data or describing a covered claim.

Deductibles and Limits

For any real number $z$, define the positive part $z_+=\max(z,0)$. Under an ordinary deductible $d$, the loss remaining after the deductible is $(X-d)_+$. If $u$ is a cap on the insurer’s payment per claim, the insurer’s payment is

Y^L=\min\bigl((X-d)_+,u\bigr).

This function is zero when $X\leq d$, increases dollar for dollar between $d$ and $d+u$, and is capped at $u$ when $X\geq d+u$. In particular, “a limit of $u$ on the insurer’s payment” means that the maximum insurer payment is $u$, not that the ground-up loss is capped at $u$.

Important distinction: If the contract instead limits the ground-up loss to $u$ before applying the deductible, the payment is $(\min(X,u)-d)_+$. When $u>d$, this is $\min((X-d)_+,u-d)$, not $\min((X-d)_+,u)$. State which interpretation the contract specifies before calculating.

The policyholder’s amount not paid by the insurer is $X-Y^L$, under the simplifying assumption that the only payment is the insurer’s covered payment. This amount includes the deductible and any loss above the insurer’s limit. It is not always equal to the deductible: for a large claim, the part above the cap can be substantial.

Worked Example: Deductible and a Limit on Insurer Benefits

A hypothetical policy has a deductible of $1,000 and a maximum insurer payment of $7,000 per claim. The ground-up loss $X$ has this discrete distribution: $500 with probability 0.20, $3,500 with probability 0.30, $9,000 with probability 0.30, and $15,000 with probability 0.20. Find the insurer payment and policyholder amount for each possible loss.

Method. Since the $7,000 limit is on insurer benefits, use $Y^L=\min((X-d)_+,u)$, with $d=1{,}000$ and $u=7{,}000$. No interest rate or mortality assumption applies; the stated discrete distribution is the full loss model needed here.

For $X=500$, $(500-1{,}000)_+=0$, so $Y^L=\min(0,7{,}000)=0$. The policyholder pays $500-0=\$500$.

For $X=3{,}500$, $(3{,}500-1{,}000)_+=2{,}500$, so $Y^L=\min(2{,}500,7{,}000)=\$2{,}500$. The policyholder pays $3{,}500-2{,}500=\$1{,}000$.

For $X=9{,}000$, the amount above the deductible is $9{,}000-1{,}000=8{,}000$, so $Y^L=\min(8{,}000,7{,}000)=\$7{,}000$. The policyholder pays $9{,}000-7{,}000=\$2{,}000$.

For $X=15{,}000$, the amount above the deductible is $14{,}000$, and the insurer payment remains capped at $Y^L=\$7{,}000$. The policyholder pays $15{,}000-7{,}000=\$8{,}000$.

The check is that each row adds back to the ground-up loss: $0+500=500$, $2{,}500+1{,}000=3{,}500$, $7{,}000+2{,}000=9{,}000$, and $7{,}000+8{,}000=15{,}000$. The common mistake avoided is treating the $7,000 benefit limit as a cap on $X$; it caps the insurer payment, not the loss.

Coinsurance and the Policyholder’s Share

Suppose the insurer pays a fraction $c$ of the loss above the deductible, with $0\leq c\leq 1$, subject to a maximum insurer payment $u$. Under this stated order of operations, the per-loss payment is

Y^L=\min\bigl(c(X-d)_+,u\bigr).

Before the limit applies, the policyholder pays the deductible plus the fraction $1-c$ of the amount above the deductible. If the insurer’s payment cap binds, the policyholder also bears the amount that would otherwise have been paid above that cap. The safest way to obtain the policyholder’s total amount is to calculate $X-Y^L$.

Worked Example: Coinsurance with a Payment Cap

A hypothetical policy has a $500 deductible, 80% insurer coinsurance on the loss above the deductible, and a $6,000 maximum insurer payment per claim. The discrete ground-up loss $X$ is $2,000 with probability 0.25, $6,000 with probability 0.50, and $12,000 with probability 0.25. Calculate the insurer payment and policyholder amount for each possible loss.

Method. Apply the deductible first, multiply the remaining amount by $c=0.80$, and then apply the insurer-payment cap $u=6{,}000$: $Y^L=\min(0.80(X-500)_+,6{,}000)$. The distribution is the stated discrete distribution; there is no interest or mortality assumption.

At $X=\$2{,}000$, the loss above the deductible is $2{,}000-500=\$1{,}500$. The uncapped insurer share is $0.80(1{,}500)=\$1{,}200$, below the cap, so $Y^L=\$1{,}200$. The policyholder pays $2{,}000-1{,}200=\$800$.

At $X=\$6{,}000$, the loss above the deductible is $6{,}000-500=\$5{,}500$. The uncapped insurer share is $0.80(5{,}500)=\$4{,}400$, also below the cap. Thus $Y^L=\$4{,}400$, and the policyholder pays $6{,}000-4{,}400=\$1{,}600$.

At $X=\$12{,}000$, the loss above the deductible is $12{,}000-500=\$11{,}500$. The uncapped insurer share is $0.80(11{,}500)=\$9{,}200$, which exceeds the $6,000 cap. Therefore $Y^L=\$6{,}000$, and the policyholder pays $12{,}000-6{,}000=\$6{,}000$.

The checks are $1{,}200+800=2{,}000$, $4{,}400+1{,}600=6{,}000$, and $6{,}000+6{,}000=12{,}000$. For the final claim, the policyholder amount includes the $500 deductible, 20% of the $11,500 above the deductible, and the $3,200 of otherwise applicable insurer coinsurance blocked by the cap. The common mistake avoided is applying 80% to the entire loss, including the deductible.

Reinsurance: Gross Liability, Recovery, and Net Liability

Reinsurance changes how much of the insurer’s policy liability the insurer ultimately retains. It does not, by itself, change the payment promised to the policyholder. For per-claim excess-of-loss reinsurance attaching at $a$ with a recovery limit $h$, let $Y^L$ be the insurer’s gross policy payment. The reinsurer’s recovery is

R=\min\bigl((Y^L-a)_+,h\bigr), \qquad \text{insurer net payment}=Y^L-R.

Here the attachment and limit apply to the insurer’s gross payment, not automatically to the ground-up loss $X$. If a treaty instead defines its layer on ground-up losses, or has an aggregate rather than per-claim limit, its wording requires a different function. Always identify the loss measure to which the reinsurance layer applies.

Worked Example: Reconciling Policy and Reinsurance Payments

A hypothetical primary policy has a $1,000 deductible and a $10,000 maximum insurer payment per claim. A reinsurer covers the portion of the primary insurer’s payment above $6,000, up to a $3,000 recovery per claim. The discrete ground-up loss $X$ is $4,000, $9,000, or $18,000, each with probability one-third. Find the policyholder amount, gross insurer payment, reinsurance recovery, and insurer net payment for each outcome.

Method. First calculate the primary insurer’s gross payment $Y^L=\min((X-1{,}000)_+,10{,}000)$. Then calculate $R=\min((Y^L-6{,}000)_+,3{,}000)$, and subtract $R$ from $Y^L$. There is no interest rate or mortality assumption; $X$ has the stated discrete distribution.

For $X=\$4{,}000$, the primary payment is $\min(4{,}000-1{,}000,10{,}000)=\$3{,}000$. The recovery is $\min((3{,}000-6{,}000)_+,3{,}000)=\$0$. The insurer’s net payment is $3{,}000-0=\$3{,}000$, and the policyholder pays $4{,}000-3{,}000=\$1{,}000$.

For $X=\$9{,}000$, the primary payment is $\min(9{,}000-1{,}000,10{,}000)=\$8{,}000$. The recovery is $\min(8{,}000-6{,}000,3{,}000)=\$2{,}000$. The insurer’s net payment is $8{,}000-2{,}000=\$6{,}000$, and the policyholder pays $9{,}000-8{,}000=\$1{,}000$.

For $X=\$18{,}000$, the amount above the deductible is $17,000, but the primary payment is capped at $Y^L=\$10{,}000$. The recovery is $\min(10{,}000-6{,}000,3{,}000)=\$3{,}000$. The insurer’s net payment is $10{,}000-3{,}000=\$7{,}000$, and the policyholder pays $18{,}000-10{,}000=\$8{,}000$.

The allocation checks are $1{,}000+3{,}000+0=4{,}000$, $1{,}000+6{,}000+2{,}000=9{,}000$, and $8{,}000+7{,}000+3{,}000=18{,}000$. In each equation, the three amounts are policyholder payment, insurer net payment, and reinsurer recovery. The common mistake avoided is subtracting reinsurance from the policyholder’s payment: the recovery reimburses the insurer and leaves the primary policy payment unchanged.

Why the Payment Functions Work

The deductible-and-limit formula follows directly from the contract’s successive restrictions. A deductible removes the first $d$ dollars of each loss from the insurer’s responsibility. If $X\leq d$, nothing remains, and $(X-d)_+=0$. If $X>d$, the amount remaining is $X-d$. A limit on insurer benefits then caps that amount at $u$, giving the smaller of $(X-d)_+$ and $u$.

Y^L= \begin{cases} 0, & 0\leq X\leq d,\\ X-d, & d<X<d+u,\\ u, & X\geq d+u. \end{cases}

This piecewise form is equivalent to $\min((X-d)_+,u)$: below the deductible the positive part is zero; between the deductible and the cap threshold it is $X-d$, which is less than $u$; at or above the threshold it is at least $u$, so the minimum is $u$. The derivation also shows why a positive-part function must be defined for real inputs: $X-d$ can be negative.

Identify the starting amount.
Decide whether each term applies to ground-up loss $X$, loss above a deductible, the insurer’s payment, or an aggregate amount.

Translate each term in its stated order.
For example, subtract a deductible and take the positive part before applying a cap when the limit is on insurer benefits.

Separate the parties’ amounts.
Find the policyholder amount from $X-Y^L$; for reinsurance, find the recovery and then the insurer net payment.

Reconcile the claim.
Check that policyholder payment plus insurer net payment plus reinsurer recovery equals $X$, when these are the only parties and payments in the model.

Common Contract-Translation Traps

Using the wrong cap. A limit on insurer benefits is not the same as a limit on ground-up loss. Write the formula that matches the wording before inserting numbers.
Applying coinsurance to the deductible. If coinsurance applies to the amount above the deductible, calculate $(X-d)_+$ first and multiply that amount by $c$.
Forgetting loss above the cap. The policyholder amount is $X-Y^L$, not always just $d$ or $(1-c)(X-d)$.
Applying reinsurance to the wrong base. A treaty may apply to the primary insurer’s payment, to ground-up loss, or to an aggregate. Use only the base stated in the wording.
Confusing gross and net insurer amounts. Reinsurance recovery reduces the insurer’s retained cost; it does not reduce the primary insurer’s contractual payment to the policyholder.
Ignoring the unit of application. A per-claim limit is applied to each claim. An aggregate annual limit cannot be treated as a separate limit on every claim.

Key takeaway: Define the ground-up loss first, state exactly what amount each contract term acts on, and build the payment function in the specified order. Then reconcile the amounts paid by the policyholder, insurer, and reinsurer.

Check Your Understanding

For each hypothetical contract, write or evaluate the relevant per-claim amounts. Answers are provided for checking.

A policy has a $900 deductible and a $4,500 maximum insurer payment per claim. For a ground-up loss of $6,200, find the insurer payment and policyholder amount. (Answer: insurer $4,500; policyholder $1,700.)
A contract pays 75% of loss above a $400 deductible, subject to a $4,800 maximum insurer payment. For $X=\$8{,}000$, find the insurer payment and policyholder amount. (Answer: insurer $4,800; policyholder $3,200.)
A primary policy has a $1,000 deductible and an $8,000 maximum insurer payment. A per-claim reinsurer layer attaches at $5,000 of the primary payment and has a $2,000 limit. For $X=\$13{,}000$, find the policyholder payment, reinsurer recovery, and insurer net payment. (Answer: policyholder $5,000; recovery $2,000; insurer net $6,000.)
A contract limits ground-up loss to $6,000 before applying a $1,000 deductible. For $X=\$12{,}000$, find the insurer payment and compare it with a contract that instead caps insurer benefits at $6,000. (Answer: ground-up-loss cap payment $5,000; insurer-benefit cap payment $6,000.)
A policy pays 60% of loss above a $2,000 deductible, with a $5,000 insurer-payment cap. Reinsurance covers the primary insurer’s payment above $3,000, up to $1,500. For $X=\$15{,}000$, find the policyholder amount, gross insurer payment, recovery, and insurer net payment. (Answer: policyholder $10,000; gross insurer payment $5,000; recovery $1,500; insurer net payment $3,500.)

What You'll Learn