Separating Insurer And Policyholder Payments

Why Separate the Two Payments?

A loss is not itself the amount an insurer pays. A deductible, a limit on the insurer’s payment, and coinsurance can all change how a loss is divided. For probability calculations, it is useful to define both payments as functions of the same random loss. Doing so makes it possible to find their distributions and expectations and to check that the two payments account for the entire loss.

Throughout this tutorial, assume a nonnegative loss $X$, a deductible $d\geq 0$, a finite payment limit $u>0$, and an insurer coinsurance share $c$ with $0<c\leq 1$. The contract convention used here is that the insurer first applies the deductible, then limits the insurer-eligible amount to $u$, and then pays the fraction $c$ of that amount. In particular, $u$ is a limit on the amount eligible for insurer payment after the deductible, not a limit on the gross loss.

Definition: Write $z_+=\max(z,0)$. Under the stated contract convention, the insurer payment per loss is $Y^L=c\min((X-d)_+,u)$. The policyholder payment per loss is $H=X-Y^L$. Thus, for every realized loss, $X=Y^L+H$.

The identity separates the amount the insurer pays from the amount the policyholder bears. The latter includes any loss within the deductible, any amount above the insurer’s payment limit, and the policyholder’s coinsurance share of the eligible amount. If $c=1$, the policyholder has no coinsurance share, but may still pay a deductible and amounts above the limit.

Payment Distributions

The insurer pays zero whenever $X\leq d$. For losses above the deductible, the insurer payment increases linearly with $X$ until $X=d+u$, where it reaches its maximum $cu$. After that point the insurer payment stays at $cu$. This creates a probability mass at zero and, if the loss can reach or exceed $d+u$, a mass at the maximum payment.

Let $F_X(x)=P(X\leq x)$. For the insurer payment, the event $Y^L\leq y$, when $0\leq y<cu$, is equivalent to $X\leq d+y/c$. Once $y\geq cu$, every possible insurer payment is at most $y$. Therefore:

F_{Y^L}(y)= \begin{cases} 0, & y<0,\\ F_X(d+y/c), & 0\leq y<cu,\\ 1, & y\geq cu. \end{cases}

In particular, $P(Y^L=0)=F_X(d)$, and $P(Y^L=cu)=P(X\geq d+u)$. These formulas allow for discrete as well as continuous losses: if the loss has probability exactly at $d+u$, that probability is included in the maximum-payment mass.

The policyholder’s payment is also a function of $X$, but the function changes slope at the deductible and at the point where the insurer limit is reached. For $c<1$, it increases throughout, with a smaller slope while the insurer and policyholder share eligible losses. Its distribution can be obtained by solving for the loss value corresponding to a given policyholder payment $h$.

H= \begin{cases} X, & X\leq d,\\ d+(1-c)(X-d), & d<X<d+u,\\ X-cu, & X\geq d+u. \end{cases}

For $c<1$, the corresponding CDF is:

F_H(h)= \begin{cases} 0, & h<0,\\ F_X(h), & 0\leq h\leq d,\\ F_X\!\left(d+\dfrac{h-d}{1-c}\right), & d<h\leq d+(1-c)u,\\ F_X(h+cu), & h>d+(1-c)u. \end{cases}

At the boundary $h=d+(1-c)u$, either inverse mapping gives $X=d+u$, so the CDF expressions agree. When $c=1$, the middle part of the policyholder payment is a flat segment: $H=d$ for $d<X\leq d+u$. Thus, for $h<d$, $F_H(h)=F_X(h)$; at $h=d$, $F_H(d)=F_X(d+u)$; and for $h>d$, $F_H(h)=F_X(h+u)$. This special case matters because the flat segment can create a probability mass at $d$.

Expected Payments and a Useful Derivation

The loss-payment identity gives an immediate expectation check: if $E[X]$ is finite, then $E[H]=E[X]-E[Y^L]$. To calculate the insurer’s expected payment, a survival-function integral is convenient. It follows from expressing the capped amount as an integral of indicators:

\min((X-d)_+,u) =\int_0^u \mathbf{1}_{\{X>d+t\}}\,dt.

For a fixed loss $X=x$, the indicator is one over precisely the portion of $0\leq t\leq u$ for which $d+t<x$; the length of that portion is $\min((x-d)_+,u)$. Taking expectations and integrating the indicator probabilities gives:

E[Y^L] =c\int_0^u P(X>d+t)\,dt =c\int_d^{d+u} S_X(x)\,dx, \qquad S_X(x)=P(X>x).

This argument applies to nonnegative losses with a finite payment limit; the integral is finite even if $E[X]$ is not. If $E[X]$ is finite as well, the policyholder expectation follows from the identity. The integral calculates the capped eligible loss first, so the factor $c$ must still be applied to obtain the insurer’s payment.

Worked Example: Uniform Loss With a Limit and Coinsurance

A hypothetical contract covers a loss $X\sim\operatorname{Uniform}(0,1000)$ dollars. It has a deductible of $d=\$100$, a post-deductible eligible-amount limit of $u=\$400$, and an insurer coinsurance share of $c=0.80$. Find the insurer payment CDF, the expected payments, and the probability that the insurer pays its maximum. Use the method of transforming the loss and the survival integral.

Insurer payment and CDF. The payment is $Y^L=0.80\min((X-100)_+,400)$, so its maximum is $0.80(400)=\$320$. The probability of zero payment is $P(X\leq100)=100/1000=0.1000$. For $0\leq y<320$, the CDF is $F_X(100+y/0.80)$. Since $X$ is uniform on $[0,1000]$, this is $(100+y/0.80)/1000$. At $y\geq320$, the CDF is 1. The probability of the maximum payment is $P(X\geq100+400)=P(X\geq500)=500/1000=0.5000$.

Expected insurer payment. The survival function is $S_X(x)=1-x/1000$ for $0\leq x\leq1000$. Thus the expected capped eligible amount is $\int_{100}^{500}(1-x/1000)\,dx$. An antiderivative is $x-x^2/2000$, so the integral is $(500-500^2/2000)-(100-100^2/2000)=375-95=\$280$. Applying the insurer’s 0.80 share gives $E[Y^L]=0.80(280)=\$224.00$.

Expected policyholder payment and check. For a uniform distribution on $[0,1000]$, $E[X]=500$. Therefore $E[H]=500-224=\$276.00$, and the expected payments add to the expected loss: $224+276=500$. As a pointwise check, if $X=\$600$, then $Y^L=0.80(400)=\$320$ and $H=600-320=\$280$; the payments sum to $\$600$. The method avoids confusing the $\$280$ capped eligible amount with the insurer’s payment, which is only 80% of that amount.

Discrete Losses: Keep Each Payment Attached to Its Loss

For a discrete loss, the safest approach is to calculate $Y^L$ and $H$ for every possible loss value, then combine probabilities when different losses produce the same payment. The CDF formulas still apply, but a table often makes the payment masses and the identity easier to verify.

Worked Example: Combining Loss Outcomes at the Limit

A hypothetical equipment policy has loss distribution $P(X=0)=0.20$, $P(X=200)=0.30$, $P(X=600)=0.30$, and $P(X=1000)=0.20$, with amounts in dollars. The deductible is $d=\$100$, the eligible-amount limit is $u=\$300$, and the insurer share is $c=0.50$. Find both payment distributions and their expectations.

Calculate each outcome. For $X=0$, the insurer pays $0$ and the policyholder pays $0$. For $X=200$, the eligible amount is $\min(200-100,300)=100$, so the insurer pays $0.50(100)=\$50$ and the policyholder pays $\$150$. For $X=600$, the eligible amount is $\min(500,300)=300$, so the insurer pays $\$150$ and the policyholder pays $600-150=\$450$. For $X=1000$, the insurer again pays its maximum of $\$150$, and the policyholder pays $1000-150=\$850$.

Combine repeated payments. The insurer payment is $0$ with probability $0.20$, $50$ with probability $0.30$, and $150$ with probability $0.30+0.20=0.50$. The policyholder payment is $0,150,450,850$ with probabilities $0.20,0.30,0.30,0.20$, respectively.

Compute and check expectations. The insurer expectation is $0(0.20)+50(0.30)+150(0.50)=0+15+75=\$90$. The loss expectation is $0(0.20)+200(0.30)+600(0.30)+1000(0.20)=0+60+180+200=\$440$. Hence $E[H]=440-90=\$350$. Directly, the policyholder expectation is $0(0.20)+150(0.30)+450(0.30)+850(0.20)=0+45+135+170=\$350$. The two expected payments sum to $\$440$, as required. The key method is to combine the two losses that reach the insurer limit when constructing the insurer’s distribution.

A Continuous Loss With a Long Tail

For an unbounded loss model, a finite insurer limit still makes the insurer’s payment bounded. The policyholder payment, in contrast, can remain unbounded because losses beyond the point where the insurer reaches its limit are borne by the policyholder.

Worked Example: Exponential Loss and the Maximum-Payment Mass

A hypothetical policy covers $X\sim\operatorname{Exponential}(\lambda=0.001\text{ per dollar})$, so $E[X]=1/\lambda=\$1000$. It has $d=\$200$, $u=\$500$, and insurer share $c=0.90$. Find the probability of no insurer payment, the probability of maximum insurer payment, the insurer’s expected payment, and the policyholder’s expected payment.

Payment thresholds. The insurer pays zero when $X\leq200$, so $P(Y^L=0)=1-e^{-0.001(200)}=1-e^{-0.2}=0.1813$. The maximum insurer payment is $0.90(500)=\$450$, reached when $X\geq200+500=700$. Therefore $P(Y^L=450)=P(X\geq700)=e^{-0.7}=0.4966$.

Expected payments. The exponential survival function is $S_X(x)=e^{-0.001x}$. Applying the survival integral, $E[Y^L]=0.90\int_{200}^{700}e^{-0.001x}\,dx$. An antiderivative is $-1000e^{-0.001x}$, giving $E[Y^L]=900(e^{-0.2}-e^{-0.7})=900(0.8187308-0.4965853)=\$289.93$, rounded to the nearest cent. Then $E[H]=1000-289.93=\$710.07$, also rounded to the nearest cent. The check $289.93+710.07=\$1000.00$ agrees with $E[X]$.

For a further pointwise check, if $X=\$600$, the eligible amount is $\min(600-200,500)=\$400$. The insurer pays $0.90(400)=\$360$, and the policyholder pays $600-360=\$240$. These payments sum to the loss. The limit is reached only at a loss of $\$700$, not $\$500$, because the deductible is applied before the post-deductible limit.

Common Traps and a Reliable Check

A frequent error is to stop after calculating the capped eligible amount and report it as the insurer’s expected payment. In the uniform-loss example, the capped eligible amount has expectation $\$280$. Reporting $E[Y^L]=\$280$ omits the insurer’s 0.80 coinsurance share. The correct calculation is $E[Y^L]=0.80(\$280)=\$224$. The $\$56$ difference is the insurer share that was incorrectly assigned to the policyholder.

Another trap is to apply a limit to the gross loss when the contract specifies a limit on the post-deductible eligible amount. Under the convention used here, the eligible amount is $\min((X-d)_+,u)$, and the maximum insurer payment is $cu$. Always identify what the stated limit caps before substituting into a payment formula; contract wording can use different limit conventions.

For a final consistency check, verify the identity for representative loss values and verify expectations when $E[X]$ is finite. If the model or contract defines both payments as parts of the same loss, then $Y^L+H=X$ for every outcome, and consequently $E[Y^L]+E[H]=E[X]$. This check is especially useful when discrete outcomes combine at the limit or a coinsurance fraction is easy to omit. The methods here are aligned with the SOA Exam P (Probability) syllabus.

Check Your Understanding

For each hypothetical contract, identify the relevant payment amounts or probabilities before calculating. Amounts are in dollars unless stated otherwise.

A loss is uniform on $[0,800]$. The deductible is $80$, the post-deductible eligible-amount limit is $200$, and the insurer share is $0.75$. Find the probability of zero insurer payment and the expected insurer payment. (Answer: $P(Y^L=0)=0.1000$; $E[Y^L]=\$116.25$)
A loss takes values $0,150,400,900$ with probabilities $0.25,0.25,0.30,0.20$, respectively. The deductible is $100$, the eligible-amount limit is $250$, and the insurer share is $0.60$. Find the insurer payment distribution and both expected payments. (Answer: $Y^L=0,30,150$ with probabilities $0.25,0.25,0.50$; $E[Y^L]=\$82.50$, $E[H]=\$255.00$)
A loss is exponential with rate $0.002$ per dollar. The deductible is $100$, the eligible-amount limit is $300$, and the insurer share is $0.80$. Find the probability of zero insurer payment and the expected insurer payment, rounded to the nearest cent. (Answer: $P(Y^L=0)=0.1813$; $E[Y^L]=\$147.76$)
A loss is uniform on $[0,600]$. The deductible is $100$, the eligible-amount limit is $200$, and the insurer share is $1$. Find the probability that the policyholder payment equals $100$. (Answer: $P(H=100)=200/600=0.3333$)
A loss is uniform on $[0,1000]$. The deductible is $100$, the eligible-amount limit is $400$, and the insurer share is $0.80$. Find the policyholder payment when the loss is $X=700$, then check the loss-payment identity. (Answer: $Y^L=\$320$, $H=\$380$, and $\$320+\$380=\$700$)

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