Coinsurance With an Ordinary Deductible

How Coinsurance and an Ordinary Deductible Work Together

An ordinary deductible leaves the insured responsible for the first $d$ dollars of each loss. Coinsurance then specifies what fraction of the amount above that deductible the insurer pays. The order matters: the deductible is applied first, and the insurer’s percentage applies to the remaining excess.

This tutorial considers a nonnegative ground-up loss $X$, an ordinary deductible $d\geq 0$, and a coinsurance fraction $c$ with $0\leq c\leq 1$. There is no separate payment limit. The contract is hypothetical; no interest or mortality assumptions are needed because the calculations concern the payment on a single loss, without discounting.

Definition: The insurer’s payment per loss under coinsurance with an ordinary deductible is $$ Y^L=c(X-d)_+=c\max(X-d,0). $$ Thus, if $X\leq d$, the payment is zero; if $X>d$, the payment is $c(X-d)$. The insured’s retained amount is $X-Y^L$.

The notation $Y^L$ denotes the insurer’s payment for a loss. The amount above the deductible before coinsurance is applied is $(X-d)_+$. The contract therefore scales the ordinary deductible payment by $c$. For $c=1$, the insurer pays the full excess over the deductible; for $c=0$, the insurer pays nothing.

Why the Payment Formula Has This Form

The contract has two sequential steps. First, the ordinary deductible removes the initial $d$ dollars from the loss, leaving $(X-d)_+$. Second, the insurer pays the agreed fraction $c$ of that amount. Multiplying the excess by $c$ gives $c(X-d)_+$, including the zero-payment case when the loss does not exceed $d$.

For a positive excess, the insurer and insured divide that excess in proportions $c$ and $1-c$. The insured also retains the deductible. In particular, when $X>d$, the retained loss is $d+(1-c)(X-d)$, which is equivalent to $X-c(X-d)$.

X-Y^L= \begin{cases} X, & X\leq d,\\ d+(1-c)(X-d), & X>d. \end{cases}

This check helps interpret the contract: above the deductible, the insurer and insured share only the excess layer. The insured does not retain $(1-c)X$, because that would apply the coinsurance percentage to the deductible portion too.

Worked Example: Payments From a Discrete Loss Distribution

A hypothetical policy has deductible $d=\$1{,}000$ and coinsurance fraction $c=0.80$. The loss $X$ has the following discrete distribution. Find the insurer’s payment at each loss value, the expected insurer payment, and the expected amount retained by the insured.

Method and assumptions. Use the stated discrete distribution and calculate $Y^L=0.80(X-1{,}000)_+$ at each possible loss. The probabilities sum to $0.10+0.20+0.25+0.30+0.15=1.00$. No interest or mortality assumption is relevant.

Loss $X$	Probability	Excess $(X-d)_+$	Payment $Y^L$
$\$0$	0.10	$\$0$	$\$0$
$\$500$	0.20	$\$0$	$\$0$
$\$1{,}000$	0.25	$\$0$	$\$0$
$\$2{,}000$	0.30	$\$1{,}000$	$\$800$
$\$4{,}000$	0.15	$\$3{,}000$	$\$2{,}400$

Expected insurer payment. Weight each payment by its probability:

E[Y^L] =0.10(0)+0.20(0)+0.25(0)+0.30(800)+0.15(2{,}400) =240+360 =\$600.

As a check, the expected ordinary-deductible excess is $0.30(1{,}000)+0.15(3{,}000)=750$, and $0.80(750)=600$. The two calculations agree.

Expected amount retained. The expected ground-up loss is

E[X]=0.20(500)+0.25(1{,}000)+0.30(2{,}000)+0.15(4{,}000) =100+250+600+600 =\$1{,}550.

Since the insured’s retained loss is $X-Y^L$, its expectation is $E[X]-E[Y^L]=1{,}550-600=\$950$. Directly, the expected retained amount is $0.20(500)+0.25(1{,}000)+0.30(1{,}200)+0.15(1{,}600)=100+250+360+240=\$950$, confirming the result.

Common mistake avoided. A loss of $\$2{,}000$ produces a payment of $\$800$, not $0.80(2{,}000)-1{,}000=\$600$. The deductible is taken from the loss before the coinsurance fraction is applied.

Expected Payment From a Survival Function

For a general nonnegative loss, the payment formula also gives a useful expectation result. Let $F_X$ be the distribution function of $X$, and let $S_X(x)=P(X>x)$ be its survival function. The expected payment is proportional to the expected excess over the deductible:

Theorem: For a nonnegative loss $X$, a deductible $d\geq0$, and a coinsurance fraction $0\leq c\leq1$, $$ E[Y^L]=cE[(X-d)_+] =c\int_d^\infty S_X(x)\,dx, $$ provided the expectation exists. This result includes distributions with discrete or continuous components.

To justify the integral, express the excess as the total length of the levels between $d$ and $X$:

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

For any fixed loss $X$, the indicator equals one precisely for levels $x<X$, so the integral is the length of the interval from $d$ to $X$ when $X>d$, and zero otherwise. Taking expectations and interchanging the nonnegative integral and expectation gives

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

Multiplying by $c$ gives the expected insurer payment. The formula does not require $X$ to have a density. In particular, an atom at the deductible contributes no excess payment at $X=d$. When using a survival function, be consistent with the convention $S_X(x)=P(X>x)$.

Worked Example: Integrating a Specified Survival Function

A hypothetical continuous loss has the following survival function for $x\geq0$:

S_X(x)= \begin{cases} 1-\dfrac{x}{2{,}000}, & 0\leq x\leq1{,}000,\\[4pt] \dfrac{3{,}000-x}{4{,}000}, & 1{,}000<x\leq3{,}000,\\[4pt] 0, & x>3{,}000. \end{cases}

The pieces agree at $x=1{,}000$: both give $0.5$. The function starts at $1$, is nonincreasing, and reaches zero at $3{,}000$, as required for this specified continuous loss model. The contract has $d=\$600$ and $c=0.75$. Find the expected insurer payment.

Method. Integrate the survival function from the deductible to the end of the support, then multiply by the coinsurance fraction. No interest or mortality assumption is involved.

From $600$ to $1{,}000$, the first piece applies:

\int_{600}^{1{,}000}\left(1-\frac{x}{2{,}000}\right)dx =\left[x-\frac{x^2}{4{,}000}\right]_{600}^{1{,}000} =(1{,}000-250)-(600-90) =750-510 =240.

From $1{,}000$ to $3{,}000$, the second piece applies. The survival curve declines linearly from $0.5$ to $0$, so its area is a triangle; evaluating the integral gives the same result:

\int_{1{,}000}^{3{,}000}\frac{3{,}000-x}{4{,}000}dx =\left[\frac{3{,}000x}{4{,}000}-\frac{x^2}{8{,}000}\right]_{1{,}000}^{3{,}000} =500.

The expected excess before coinsurance is $240+500=740$. Therefore,

E[Y^L]=0.75(740)=\$555.

The expected payment cannot exceed the expected excess above the deductible because $c=0.75\leq1$; indeed, $\$555\leq\$740$. A common error would be to multiply only one part of the integrated tail by $0.75$, or to integrate from zero and include losses below the deductible. Neither matches the contract.

Recovering a Coinsurance Fraction

The expectation formula can also be rearranged when the expected excess is known and a target expected insurer payment is specified. If $E[(X-d)_+]>0$, then

c=\frac{E[Y^L]}{E[(X-d)_+]}.

This calculation is valid only when the target is compatible with the contract: the resulting $c$ must lie between zero and one. A target above the expected excess cannot be attained by coinsurance on that excess alone.

Worked Example: Finding the Coinsurance Rate From a Target

A hypothetical discrete loss $X$ equals $\$400$, $\$900$, $\$1{,}500$, or $\$2{,}500$, with respective probabilities $0.25$, $0.20$, $0.35$, and $0.20$. An ordinary deductible is $d=\$1{,}000$, and the desired expected insurer payment is $\$380$. Find the required coinsurance fraction, the payment at each loss value, and the probability that the payment exceeds $\$500$.

Method and assumptions. The probabilities sum to $1$, and the loss distribution is discrete as specified. First find the expected ordinary-deductible excess, then solve for $c$. No interest or mortality assumption is needed.

The excesses at the four loss values are $\$0$, $\$0$, $\$500$, and $\$1{,}500$. Thus,

E[(X-d)_+] =0.25(0)+0.20(0)+0.35(500)+0.20(1{,}500) =175+300 =\$475.

The target payment implies

c=\frac{380}{475}=0.80.

This is feasible because $0\leq0.80\leq1$. Applying the rate to each excess gives payments of $\$0$, $\$0$, $0.80(500)=\$400$, and $0.80(1{,}500)=\$1{,}200$. Their expectation is $0.35(400)+0.20(1{,}200)=140+240=\$380$, which checks the target.

Only the $\$2{,}500$ loss produces a payment greater than $\$500$, so the required probability is $0.20$. The $\$1{,}500$ loss produces a payment of exactly $\$400$, not a payment above $\$500$. This distinction between the loss threshold and the payment threshold is important.

Common Contract-Interpretation Errors

The key is to apply the contract in the stated order and to keep the ground-up loss, deductible excess, and insurer payment distinct.

Applying coinsurance to the whole loss. The payment is $c(X-d)_+$, not $cX$ less the deductible.
Subtracting the deductible after scaling. In general, $c(X-d)_+$ differs from $(cX-d)_+$. For example, at $X=\$1{,}500$, $d=\$1{,}000$, and $c=0.80$, the contract pays $0.80(500)=\$400$; the incorrect alternative gives $(1{,}200-1{,}000)_+=\$200$.
Paying at the deductible. For an ordinary deductible, $X=d$ gives $(X-d)_+=0$. A loss must exceed the deductible to produce a positive payment.
Using the wrong expectation. The expected insurer payment is $cE[(X-d)_+]$, not $c(E[X]-d)$ in general. The latter expression ignores the zero payment for losses at or below $d$.
Forgetting the no-limit assumption. The payment formula here has no cap. A separate limit would change the payment transformation and the expectation calculation.

Key takeaway: For coinsurance applied to the excess above an ordinary deductible, calculate the excess first and then multiply by the insurer’s fraction: $Y^L=c(X-d)_+$. For expected payment, use $E[Y^L]=c\int_d^\infty S_X(x)\,dx$ when the expected loss is finite.

Check Your Understanding

For each problem, apply the deductible before the coinsurance percentage. Amounts are hypothetical; round any dollar answer to the nearest cent.

A loss is $\$600$, $\$1{,}200$, or $\$3{,}000$ with probabilities $0.20$, $0.50$, and $0.30$, respectively. The contract has $d=\$1{,}000$ and $c=0.70$. Find the expected insurer payment and the probability of a positive payment. (Answer: expected payment $\$490$; probability $0.8000$)
A loss is $\$500$, $\$1{,}500$, or $\$2{,}500$ with probabilities $0.20$, $0.50$, and $0.30$, respectively. If $d=\$700$ and $c=0.60$, find the expected insurer payment. (Answer: $\$564$)
A continuous loss has survival function $S_X(x)=1-x/1{,}600$ for $0\leq x\leq800$, $S_X(x)=(2{,}400-x)/3{,}200$ for $800<x\leq2{,}400$, and zero for $x>2{,}400$. If $d=\$400$ and $c=0.50$, find the expected insurer payment using the survival integral. (Answer: $\$325$)
A loss equals $\$300$, $\$1{,}300$, or $\$2{,}300$ with probabilities $0.40$, $0.40$, and $0.20$, respectively. The deductible is $\$800$, and the target expected insurer payment is $\$350$. Find the required coinsurance fraction. (Answer: $c=0.70$)
A loss equals $\$1{,}000$, $\$2{,}000$, or $\$5{,}000$ with probabilities $0.30$, $0.40$, and $0.30$, respectively. For $d=\$1{,}000$ and $c=0.60$, find the expected insurer payment and the expected insurer payment conditional on a positive payment. (Answer: $\$960$; $\$1{,}371.43$)

Loss \(X\)	Probability	Excess \((X-d)_+\)	Payment \(Y^L\)
\(\$0\)	0.10	\(\$0\)	\(\$0\)
\(\$500\)	0.20	\(\$0\)	\(\$0\)
\(\$1{,}000\)	0.25	\(\$0\)	\(\$0\)
\(\$2{,}000\)	0.30	\(\$1{,}000\)	\(\$800\)
\(\$4{,}000\)	0.15	\(\$3{,}000\)	\(\$2{,}400\)

What You'll Learn