Policyholder Retentions and Insurer Payments

From Insurer Payment to Policyholder Retention

A coverage modification determines how much of a ground-up loss the insurer pays. The policyholder’s out-of-pocket loss is the part that remains. Calculating that amount requires tracking what the contract does to the insurer payment, rather than subtracting a deductible or coinsurance percentage from the loss without regard to the contract’s sequence.

Throughout this tutorial, $X$ is the nonnegative ground-up loss for one claim, and $Y^L$ is the insurer’s payment per loss. Define the policyholder retention $R$ as the portion of $X$ not paid by the insurer. The calculations exclude premiums and any expenses outside the claim payment.

Definition: For a covered loss $X$ and insurer payment $Y^L$, the policyholder retention is $R=X-Y^L$. When $0\leq Y^L\leq X$, this also gives $0\leq R\leq X$. The payment rule must be specified before calculating $R$.

The identity is a conservation-of-loss statement: the ground-up loss is divided between the insurer’s payment and the policyholder’s retained amount. It is useful both for calculating retention directly and for checking a result.

Retention Under Common Payment Rules

For a deductible $d$, with the insurer paying the loss above the deductible and no payment cap, the payment and retention are

Y^L=(X-d)_+,\qquad R=X-(X-d)_+=\min(X,d),

where $(z)_+=\max(z,0)$. A deductible therefore limits the policyholder’s retention to $d$ when the insurer pays the full amount above the deductible. If the loss is below the deductible, the policyholder retains the entire loss.

For coinsurance, let $\alpha$ be the fraction of the amount eligible for payment that the insurer pays, with $0\leq\alpha\leq1$. For coinsurance applied to the entire loss, $Y^L=\alpha X$, so $R=(1-\alpha)X$. If the insurer pays a fraction $\alpha$ of the amount above a deductible, the deductible remains entirely with the policyholder, and the insurer pays a fraction of the excess.

The following combined rule makes that sequence explicit: first subtract deductible $d$, then apply insurer coinsurance fraction $\alpha$, and finally apply a maximum insurer payment $u$. Here, $u$ is a payment cap, not a cap on the loss eligible for payment.

Payment rule: For $d\geq0$, $u\geq0$, and $0\leq\alpha\leq1$, suppose the insurer pays a fraction $\alpha$ of the loss above $d$, subject to a maximum payment of $u$. Then $$ Y^L=\min\bigl(\alpha(X-d)_+,u\bigr),\qquad R=X-\min\bigl(\alpha(X-d)_+,u\bigr). $$

When $\alpha>0$, the payment cap begins to bind at the loss level $d+u/\alpha$: at that point the uncapped payment $\alpha(X-d)$ reaches $u$. Below the deductible, retention is $X$. Above the deductible but before the cap binds, retention is the deductible plus the policyholder’s coinsurance share of the excess. Once the cap binds, retention is the loss less the maximum insurer payment.

R= \begin{cases} X, & 0\leq X\leq d,\\ d+(1-\alpha)(X-d), & d<X<d+u/\alpha,\\ X-u, & X\geq d+u/\alpha. \end{cases}

At the threshold, both payment descriptions give $Y^L=u$, so the retention is $X-u$. If $u=0$, the insurer pays nothing and $R=X$. If $\alpha=0$, the insurer also pays nothing; the threshold expression involving division by $\alpha$ is not used.

Why the Retention Formula Holds

The piecewise formula follows by applying the contract’s payment rule in each range of $X$, then subtracting that payment from the ground-up loss. This is not an additional assumption about the severity distribution; it holds pointwise for each loss under the stated contract.

Below the deductible.
If $X\leq d$, then $(X-d)_+=0$, so $Y^L=0$ and $R=X$.

Above the deductible and below the cap threshold.
If $d<X<d+u/\alpha$, then $\alpha(X-d)<u$. Thus $Y^L=\alpha(X-d)$, and $R=X-\alpha(X-d)=d+(1-\alpha)(X-d)$.

At or above the cap threshold.
If $X\geq d+u/\alpha$, then $\alpha(X-d)\geq u$, so $Y^L=u$ and $R=X-u$.

A useful expectation check follows from the same identity. If $X$ and $Y^L$ have finite expectations, then linearity of expectation gives

$$ E[R]=E[X-Y^L]=E[X]-E[Y^L]. $$

This check does not require $X$ and $Y^L$ to be independent; the payment is determined by the loss and the contract. For a discrete severity model, one can also calculate $E[R]$ directly by multiplying each possible retention by its probability.

Worked Example: Deductible and Payment Limit

A hypothetical equipment policy pays the amount of each loss above a deductible of $500, up to a maximum insurer payment of $2,500. The single-claim severity distribution is $P(X=0)=0.1000$, $P(X=600)=0.2000$, $P(X=1{,}800)=0.4000$, and $P(X=4{,}500)=0.3000$. Find the retention for each loss and the expected retention. All amounts are at the same valuation date; no interest or mortality assumption is involved.

Method. This is a full insurer payment above the deductible with a payment cap, so $Y^L=\min((X-500)_+,2500)$, and $R=X-Y^L$.

Loss $X$	Probability	Uncapped amount above deductible	Payment $Y^L$	Retention $R$
$0	0.1000	$0	$0	$0
$600	0.2000	$100	$100	$500
$1,800	0.4000	$1,300	$1,300	$500
$4,500	0.3000	$4,000	$2,500	$2,000

For example, at $X=\$4{,}500$, the amount above the deductible is $\$4{,}500-\$500=\$4{,}000$. The payment is capped at $\$2{,}500$, leaving retention $\$4{,}500-\$2{,}500=\$2{,}000$.

The expected retention, calculated directly, is $0.1000(\$0)+0.2000(\$500)+0.4000(\$500)+0.3000(\$2{,}000)=\$900$. As a check, $E[X]=0.1000(\$0)+0.2000(\$600)+0.4000(\$1{,}800)+0.3000(\$4{,}500)=\$2{,}190$, and $E[Y^L]=0.1000(\$0)+0.2000(\$100)+0.4000(\$1{,}300)+0.3000(\$2{,}500)=\$1{,}290$. Therefore $E[X]-E[Y^L]=\$2{,}190-\$1{,}290=\$900$, agreeing with the direct calculation.

Worked Example: Coinsurance With a Capped Insurer Payment

A hypothetical policy has a $800 deductible. The insurer pays 75% of the amount above the deductible, subject to a maximum payment of $3,000. A claim has severity distribution $P(X=\$2{,}000)=0.2000$, $P(X=\$4{,}800)=0.5000$, and $P(X=\$6{,}800)=0.3000$. Find the retention at each possible loss and the expected retention. No interest or mortality assumption is needed because the calculation concerns one claim at one valuation date.

Method. Use $Y^L=\min(0.75(X-800)_+,3000)$. The cap threshold is $800+3000/0.75=\$4{,}800$.

At $X=\$2{,}000$, the excess is $\$2{,}000-\$800=\$1{,}200$. The uncapped payment is $0.75(\$1{,}200)=\$900$, which is below the cap. Thus $R=\$2{,}000-\$900=\$1{,}100$.

At $X=\$4{,}800$, the excess is $\$4{,}000$, and the uncapped payment is $0.75(\$4{,}000)=\$3{,}000$. The cap is exactly reached, so $R=\$4{,}800-\$3{,}000=\$1{,}800$.

At $X=\$6{,}800$, the excess is $\$6{,}000$, and the uncapped payment would be $0.75(\$6{,}000)=\$4{,}500$. The insurer instead pays the capped amount of $\$3{,}000$, leaving $R=\$6{,}800-\$3{,}000=\$3{,}800$.

The expected retention is $0.2000(\$1{,}100)+0.5000(\$1{,}800)+0.3000(\$3{,}800)=\$220+\$900+\$1{,}140=\$2{,}260$. For a check, $E[X]=0.2000(\$2{,}000)+0.5000(\$4{,}800)+0.3000(\$6{,}800)=\$4{,}840$, while $E[Y^L]=0.2000(\$900)+0.5000(\$3{,}000)+0.3000(\$3{,}000)=\$2{,}580$. Their difference is $\$4{,}840-\$2{,}580=\$2{,}260$, as required.

Worked Example: Inferring a Loss From Its Retention

A hypothetical contract has a $700 deductible, pays 70% of the amount above the deductible, and caps the insurer payment at $2,100. A claim’s retention is reported as $3,300. Infer the ground-up loss $X$, and verify that the inferred loss is consistent with the contract. Treat the reported retention as exact; no severity distribution, interest rate, or mortality assumption is needed.

Method. The cap threshold is $700+2100/0.70=\$3{,}700$. Below the threshold but above the deductible, the retention formula is $R=700+0.30(X-700)=0.30X+490$. If this branch applied, solving $\$3{,}300=0.30X+\$490$ would give $X=\$2{,}810/0.30=\$9{,}366.67$. This is above the threshold, so the uncapped branch is inconsistent.

In the capped branch, $R=X-2100$. Therefore $X=R+2100=\$3{,}300+\$2{,}100=\$5{,}400$. This loss is above the $\$3{,}700$ threshold. Verification: the uncapped insurer payment would be $0.70(\$5{,}400-\$700)=0.70(\$4{,}700)=\$3{,}290$, which exceeds the cap, so the actual payment is $\$2{,}100$. The resulting retention is $\$5{,}400-\$2{,}100=\$3{,}300$, matching the reported amount.

Common Calculation Trap

A common mistake is to apply the coinsurance percentage to the entire ground-up loss when the contract applies it only to the amount above the deductible. In the second example, an incorrect calculation might take 25% of $\$6{,}800$ as the retention, giving $\$1{,}700$. That calculation ignores both the deductible and the insurer payment cap.

The correct calculation first determines the insurer payment: the amount above the deductible is $\$6{,}800-\$800=\$6{,}000$; 75% of that is $\$4{,}500$; and the $\$3{,}000$ payment cap reduces the actual payment to $\$3{,}000$. Retention is then $\$6{,}800-\$3{,}000=\$3{,}800$. The reliable check is to add insurer payment and retention: $\$3{,}000+\$3{,}800=\$6{,}800$.

The payment cap also changes how retention grows with the loss. Before the cap binds, an additional dollar of loss increases retention by $1-\alpha$ dollars under the stated coinsurance rule. After the cap binds, the insurer payment stays at $u$, so each additional dollar of loss increases retention by one dollar. This distinction is important when a loss is near or above the cap threshold.

Key takeaway: State the payment rule, calculate $Y^L$ under that rule, and then use $R=X-Y^L$. For coinsurance with a payment cap, check whether the uncapped payment exceeds the cap before subtracting the insurer payment from the loss.

Check Your Understanding

For each problem, calculate the requested retention or expected retention under the stated contract.

A hypothetical policy has a $750 deductible and pays the full amount above it, with no payment cap. For a loss of $2,900, calculate the insurer payment and the policyholder retention. (Answer: insurer payment $2,150; retention $750.)
A hypothetical contract has no deductible and the insurer pays 65% of each loss, with no payment cap. For a loss of $4,200, calculate the retention. (Answer: $1,470.)
A hypothetical contract has a $600 deductible, pays 80% of the amount above it, and caps the insurer payment at $1,600. For a loss of $2,600, calculate the payment and retention. (Answer: payment $1,600; retention $1,000.)
A hypothetical policy pays each loss above a $500 deductible in full, with no cap. The severity distribution is $P(X=\$0)=0.2000$, $P(X=\$1{,}000)=0.3000$, and $P(X=\$3{,}000)=0.5000$. Calculate the expected retention. (Answer: $400.)
A hypothetical contract has a $400 deductible, pays 75% of the excess, and caps the insurer payment at $1,800. For a loss of $4,000, identify whether the cap binds and calculate the retention. (Answer: the cap binds; retention $2,200.)

What You'll Learn