Payment Functions for Modified Losses

Why Payment Functions Matter

A severity model describes a ground-up loss: the amount of loss before the insurance contract changes what the insurer pays. A deductible, a limit, and a coinsurance share each transform that amount in a different way. Writing the transformation as a function of the ground-up loss makes the contract’s effect explicit and helps prevent errors about which party pays which portion.

This tutorial focuses on the payment functions themselves, not on the resulting probability distributions or expected payments. Those functions are a necessary starting point for calculations involving modified losses. The definitions and techniques here are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: Let $X\geq 0$ be the ground-up loss for one occurrence. The insurer payment per loss, denoted $Y^L$, is a function of $X$ determined by the contract. The payment per loss can be zero. Unless a contract says otherwise, deductible and limit amounts below are nonnegative, and a coinsurance share $c$ is between 0 and 1.

Ordinary and Franchise Deductibles

An ordinary deductible $d$ leaves the insured responsible for the first $d$ dollars of each loss. If the loss does not exceed $d$, the insurer pays nothing. If it exceeds $d$, the insurer pays only the excess. Thus, the ordinary-deductible payment is the positive part of $X-d$.

Y^L=(X-d)_+=\max(X-d,0)

A franchise deductible has a different threshold effect. When the loss is at or below $d$, the insurer pays nothing; once the loss is above $d$, the insurer pays the entire loss. For this tutorial, the contract wording is interpreted as a strict trigger, $X>d$. If the policy specifies a different boundary convention, that wording governs.

Y^L= \begin{cases} 0, & X\leq d,\\ X, & X>d. \end{cases}

These functions agree that the payment is zero at $X=d$, but they differ for losses above the threshold. Under the ordinary deductible, the insurer pays the excess over $d$; under the franchise deductible, the insurer pays the full loss after the trigger is met.

Worked Example: Comparing Ordinary and Franchise Deductibles

A hypothetical policy has a deductible of $500. For each of three possible ground-up losses, calculate the insurer payment under both an ordinary and a franchise deductible. Assume each loss is fixed at the stated amount; equivalently, for each calculation $P(X=x)=1$. There is no limit or coinsurance.

Method. Apply each payment function separately to each loss. The important distinction is whether the insurer pays only the excess or the full loss once the threshold is crossed.

For $X=$350$, the ordinary-deductible payment is $\max(350-500,0)=\max(-150,0)=$0$. The franchise payment is also $0 because $350\leq500$.

For $X=$500$, the ordinary-deductible payment is $\max(500-500,0)=$0$. The franchise payment is $0 because the stated trigger is $X>500$, which is not met at equality.

For $X=$1,250$, the ordinary-deductible payment is $\max(1,250-500,0)=$750$. The franchise payment is the full loss, $1,250, because $1,250>500$.

Check. The ordinary payment never exceeds the ground-up loss, and for the third loss the difference between the franchise and ordinary payments is exactly the $500 deductible. The common mistake is to calculate the franchise payment as $750; that is the ordinary-deductible payment, not the franchise payment.

Policy Limits and Combined Deductible-Limit Payments

A policy limit caps the insurer’s payment. With no deductible, a limit $u$ on the insurer payment gives $\min(X,u)$: the insurer pays the loss when it is below the limit and pays $u$ when the loss reaches or exceeds it.

Y^L=\min(X,u)

If an ordinary deductible applies as well, a common contract interpretation is that $u$ is the maximum amount the insurer pays after the deductible. The payment function is then the excess over the deductible, capped at $u$. This is distinct from a limit on the ground-up loss before the deductible. The contract wording must identify which amount the limit caps.

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

The equivalent piecewise form makes the regions clear: no insurer payment at or below the deductible, an uncapped payment between the deductible and deductible plus limit, and a capped payment above that point.

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

Worked Example: Applying a Deductible and a Payment Limit

A hypothetical policy has an ordinary deductible of $400 and a maximum insurer payment of $1,200 per loss. The limit applies to the insurer’s payment after the deductible. Evaluate the payment for ground-up losses of $250, $900, and $2,000. Treat each stated loss as fixed, so $P(X=x)=1$ for that calculation.

Method. First subtract the deductible, replacing any negative result with zero. Then cap that amount at $1,200.

For $X=$250$, $(250-400)_+=0$, so the payment is $\min(0,1,200)=$0$.

For $X=$900$, $(900-400)_+=500$, so the payment is $\min(500,1,200)=$500$.

For $X=$2,000$, $(2,000-400)_+=1,600$, so the payment is $\min(1,600,1,200)=$1,200$.

Check. Each payment is nonnegative, no payment exceeds the $1,200 insurer limit, and the $400 deductible is not subtracted again after the cap. A common error is to treat $1,200 as a limit on the ground-up loss, calculating $\min(X,1,200)-400$. That is a different contract interpretation and does not match the wording in this example.

Coinsurance and the Order of Contract Terms

Coinsurance specifies what fraction of an otherwise covered amount the insurer pays. Here, $c$ denotes the insurer’s share, so the insured share of that amount is $1-c$. For example, if the insurer pays 80% of the covered amount, $c=0.80$. The payment function depends on what amount the contract says is subject to coinsurance and where any limit applies.

For a contract that first applies an ordinary deductible, then applies the insurer’s share $c$ to the excess, and finally caps the insurer payment at $u$, the function is:

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

This is not always the only possible ordering. For instance, a contract could cap the amount eligible for coinsurance and then apply the share, giving $c\min((X-d)_+,u)$. These expressions need not agree. The limit and coinsurance language must therefore be translated in the order specified by the contract, rather than combined from memory.

Worked Example: Coinsurance With a Limit on the Insurer Payment

A hypothetical policy has an ordinary deductible of $300. The insurer pays 80% of the amount above the deductible, subject to a maximum insurer payment of $1,000 per loss. Calculate the insurer payment for ground-up losses of $200, $1,000, and $2,000. Assume each loss is fixed at its stated amount, with $P(X=x)=1$.

Method. Apply the deductible, multiply the remaining amount by $c=0.80$, and then apply the $1,000 cap to the insurer payment.

For $X=$200$, the amount above the deductible is $(200-300)_+=0$. The insurer share is $0.80(0)=$0$, and the cap does not change it.

For $X=$1,000$, the amount above the deductible is $1,000-300=$700$. The insurer share is $0.80(700)=$560$. Since $560 is below $1,000, the insurer payment is $560.

For $X=$2,000$, the amount above the deductible is $2,000-300=$1,700$. The insurer share before the cap is $0.80(1,700)=$1,360$. Applying the insurer-payment cap gives $\min(1,360,1,000)=$1,000$.

Check. The payments are $0, $560, and $1,000; none exceeds the stated cap. A different wording—capping the amount eligible for coinsurance at $1,000 before applying the 80% share—would give $0.80\min(1,700,1,000)=$800$ for the last loss. The example’s wording places the cap on the insurer payment, so $1,000 is correct.

Why the Formulas Hold

The functions follow directly from the contract’s allocation of a loss. For an ordinary deductible, consider the two possible regions. If $X\leq d$, the loss does not exceed the amount retained by the insured, so the insurer payment is zero. If $X>d$, the amount left after the insured’s first $d$ dollars is $X-d$. Combining those cases gives the positive-part formula.

Y^L= \begin{cases} 0, & X\leq d,\\ X-d, & X>d \end{cases} =\max(X-d,0)

For a payment limit, the insurer cannot pay more than either the amount otherwise payable or the limit $u$. The smaller of those two amounts is the payment. This explains why a deductible followed by an insurer-payment limit has the form $\min((X-d)_+,u)$. If coinsurance then reduces the covered amount, multiply by $c$ at the stage specified by the contract; apply a cap at the stage stated in its wording.

Identify the ground-up loss.
Start with $X$, before any deductible, limit, or coinsurance adjustment.

Translate each clause.
Determine whether a deductible is ordinary or franchise, what amount the limit caps, and which party’s share the coinsurance percentage represents.

Apply the stated order.
Write the function for the contract’s sequence, then check boundary cases such as $X=0$, $X=d$, and a sufficiently large loss.

Common Mistakes and Checks

Confusing ordinary and franchise deductibles. An ordinary deductible pays only the amount above $d$; a franchise deductible pays the full loss once its trigger is met.
Ignoring equality at the threshold. With the strict franchise trigger used here, a loss exactly equal to $d$ produces no payment. Follow any different boundary wording in the contract.
Applying a limit to the wrong amount. A cap on insurer payment after a deductible is not the same as a cap on ground-up loss before the deductible.
Reversing the coinsurance share. In this tutorial, $c$ is the insurer’s share, not the insured’s share.
Assuming operations can be reordered. In general, $c\min(z,u)$ and $\min(cz,u)$ are different. State what the limit applies to before choosing a formula.

Key takeaway: A payment function is a precise translation of contract wording into an amount payable for each ground-up loss. Define the threshold rule, the amount subject to a limit, and the insurer’s coinsurance share before evaluating the function.

Check Your Understanding

For each problem, write or apply the payment function specified by the hypothetical contract. Unless stated otherwise, treat each loss as fixed at the stated amount.

A policy has an ordinary deductible of $250 and no limit. What is the insurer payment for a ground-up loss of $830? (Answer: $580)
A franchise deductible is $600, with payment triggered only when the loss is strictly greater than $600. What are the insurer payments for losses of $600 and $920? (Answer: $0 and $920)
A policy has a $350 ordinary deductible and a $900 limit on insurer payment after the deductible. Find the payment for a ground-up loss of $1,700. (Answer: $900)
A policy pays 70% of the amount above a $200 ordinary deductible, with no limit. Find the insurer payment for a ground-up loss of $1,200. (Answer: $700)
A contract applies a $100 ordinary deductible, then pays 60% of the excess, with a $500 cap on the insurer payment. Find the insurer payment for a ground-up loss of $1,200. (Answer: $500)

What You'll Learn