Point Masses and Scaling Under Coinsurance

Introduction

A coverage modification can change both the amount paid for a claim and the shape of the payment distribution. With a deductible and a payment limit, many different losses may produce the same payment: losses at or below the deductible produce zero, while sufficiently large losses all reach the limit. Coinsurance then scales each resulting payment. Understanding which losses are grouped together—and which probabilities stay attached to the resulting payment values—is essential when calculating point masses.

This tutorial focuses on how multiplying deductible-limit payments by a coinsurance fraction affects their attainable values, their point masses, and the probability of a positive payment. The treatment is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: Let $X$ be a nonnegative ground-up loss, $d\geq 0$ an ordinary deductible, $u>0$ a maximum payment per loss before coinsurance, and $c$ a coinsurance fraction with $0<c\leq 1$. Define the payment before coinsurance by $Z=\min\{(X-d)_+,u\}$, where $(x)_+=\max(x,0)$. The coinsurance-scaled payment is $Y=cZ$. Thus, coinsurance multiplies the amount remaining after applying the deductible and payment limit.

How Scaling Changes Payment Values

For a fixed loss $x$, the payment before coinsurance is zero when $x\leq d$, increases as $x-d$ for $d<x<d+u$, and is capped at $u$ when $x\geq d+u$. Multiplying by $c>0$ scales each of these payment amounts. It does not change which losses produce zero or reach the cap.

Y= \begin{cases} 0, & X\leq d,\\ c(X-d), & d<X<d+u,\\ cu, & X\geq d+u. \end{cases}

Consequently, all payment values lie between zero and $cu$, inclusive. The upper bound $cu$ is attainable only if the loss distribution assigns positive probability to losses at or above $d+u$. When $c>0$, multiplying payment values by $c$ is one-to-one: two distinct pre-coinsurance payment amounts remain distinct after scaling. But losses in the deductible region still share the zero payment, and all losses at or above the cap threshold still share the maximum payment.

Theorem: For $c>0$, the zero-payment probability is $P(Y=0)=P(X\leq d)$, and the positive-payment probability is $P(Y>0)=P(X>d)$. The maximum-payment probability is $P(Y=cu)=P(X\geq d+u)$. For an interior payment value $0<y<cu$, $P(Y=y)=P(X=d+y/c)$. A point mass at an interior payment therefore comes from a point mass at the corresponding ground-up loss.

These formulas include their boundary conditions. A loss exactly equal to $d$ produces no payment and belongs in $P(Y=0)$. A loss exactly equal to $d+u$ reaches the cap and belongs in $P(Y=cu)$. The strict inequality $X>d$ in the positive-payment event and the weak inequality $X\geq d+u$ in the cap event are therefore important.

Why the Point-Mass Formulas Hold

The formulas follow by examining the sets of losses that map to each payment. For zero payment, $\min\{(X-d)_+,u\}=0$ exactly when $(X-d)_+=0$, which is equivalent to $X\leq d$. Since $c>0$, $Y=0$ exactly when $Z=0$; likewise, $Y>0$ exactly when $X>d$.

At an interior value $y$, the payment cannot have come from the capped region, so $Y=c(X-d)=y$. Solving this equation gives $X=d+y/c$, a single ground-up loss value. At the maximum $cu$, the pre-coinsurance payment must be $u$; the definition of $Z$ shows that this happens exactly when $X-d\geq u$. Hence the cap mass is the probability of the entire tail $X\geq d+u$, not merely the probability of a loss exactly at the cap threshold.

P(Y=cu)=P(Z=u)=P(X\geq d+u).

This explains an important feature of capped payments: even a continuous ground-up loss distribution can produce a point mass at the maximum payment. If positive probability lies above the cap threshold, every loss in that tail is mapped to the same payment $cu$. By contrast, for an interior $y$, a continuous loss distribution has no point mass at the single value $d+y/c$, so that interior payment has zero point mass.

Find the ground-up thresholds.
The deductible threshold is $d$; the cap threshold is $d+u$.

Group losses by pre-coinsurance payment.
Losses $X\leq d$ produce zero, losses $X\geq d+u$ reach the cap, and losses between those thresholds produce an interior payment.

Scale the payment amounts.
Multiply each pre-coinsurance payment by $c$. For $c>0$, retain the probability attached to each corresponding group of losses.

Worked Examples

Worked Example: A Discrete Loss With a Deductible and Cap

A hypothetical equipment policy has ground-up loss $X$ taking values $0$, $\$1{,}200$, and $\$3{,}000$, with probabilities $0.20$, $0.30$, and $0.50$, respectively. The ordinary deductible is $d=\$1{,}000$, the maximum payment before coinsurance is $u=\$1{,}500$, and the insurer pays a fraction $c=0.40$ of the deductible-limit amount. Find every payment value and point mass, the zero-payment probability, the maximum payment and its mass, and $P(Y>0)$.

Method. First apply the deductible and payment limit to each loss to obtain $Z$; then multiply each value of $Z$ by $0.40$. This avoids mistakenly treating the zero loss as a positive claim payment.

For $X=\$0$, the loss is below the deductible, so $Z=\$0$ and $Y=0.40(\$0)=\$0$. This outcome has probability $0.20$.

For $X=\$1{,}200$, the amount above the deductible is $\$1{,}200-\$1{,}000=\$200$, below the $\$1{,}500$ cap. Thus $Z=\$200$, and $Y=0.40(\$200)=\$80$. This outcome has probability $0.30$.

For $X=\$3{,}000$, the amount above the deductible is $\$3{,}000-\$1{,}000=\$2{,}000$, which exceeds the cap. Thus $Z=\$1{,}500$, and $Y=0.40(\$1{,}500)=\$600$. This outcome has probability $0.50$.

The payment masses are therefore $P(Y=\$0)=0.20$, $P(Y=\$80)=0.30$, and $P(Y=\$600)=0.50$. The maximum payment is $\$600$, with mass $0.50$. The positive-payment probability is $0.30+0.50=0.80$, which also equals $P(X>\$1{,}000)$. As a check, the masses sum to $0.20+0.30+0.50=1.00$.

Worked Example: Several Distinct Positive Payment Masses

A hypothetical repair-cost loss $X$ takes values $\$800,\$1{,}400,\$2{,}400,\$5{,}000$ with probabilities $0.10,0.25,0.30,0.35$, respectively. The deductible is $d=\$1{,}000$, the maximum payment before coinsurance is $u=\$2{,}000$, and $c=0.70$. Find all payment point masses and the probability of a positive payment.

Method. Calculate the capped payment $Z$ for each loss, then scale each result by $0.70$. In particular, do not merge all positive payments into a single mass: only losses reaching the cap share the maximum payment.

At $X=\$800$, the loss is below the deductible, so $Z=\$0$ and $Y=\$0$, with probability $0.10$.

At $X=\$1{,}400$, the amount above the deductible is $\$1{,}400-\$1{,}000=\$400$, below the $\$2{,}000$ cap. The scaled payment is $Y=0.70(\$400)=\$280$, with probability $0.25$.

At $X=\$2{,}400$, the amount above the deductible is $\$2{,}400-\$1{,}000=\$1{,}400$, also below the cap. The scaled payment is $Y=0.70(\$1{,}400)=\$980$, with probability $0.30$.

At $X=\$5{,}000$, the amount above the deductible is $\$5{,}000-\$1{,}000=\$4{,}000$, above the cap. The pre-coinsurance payment is $\$2{,}000$, so $Y=0.70(\$2{,}000)=\$1{,}400$, with probability $0.35$.

Thus $P(Y=\$0)=0.10$, $P(Y=\$280)=0.25$, $P(Y=\$980)=0.30$, and $P(Y=\$1{,}400)=0.35$. The positive-payment probability is $0.25+0.30+0.35=0.90$, equivalently $P(X>\$1{,}000)=0.90$. The masses sum to $1.00$, confirming that no outcome has been omitted.

Worked Example: Identifying the Scaled Support and Cap Mass

A hypothetical cargo loss $X$ takes values $\$500,\$1{,}800,\$2{,}600,\$4{,}200$, with probabilities $0.15,0.20,0.25,0.40$, respectively. The deductible is $d=\$1{,}000$, the maximum payment before coinsurance is $u=\$2{,}000$, and $c=0.60$. List the attainable payment values and their masses, and find the positive-payment and maximum-payment probabilities.

Method. Map each loss through the deductible and cap, then scale. The cap threshold is $d+u=\$1{,}000+\$2{,}000=\$3{,}000$, so losses at or above $\$3{,}000$ share the capped payment.

At $X=\$500$, the loss is below the deductible: $Z=\$0$ and $Y=\$0$, with probability $0.15$.

At $X=\$1{,}800$, $Z=\$1{,}800-\$1{,}000=\$800$, so $Y=0.60(\$800)=\$480$, with probability $0.20$.

At $X=\$2{,}600$, $Z=\$2{,}600-\$1{,}000=\$1{,}600$, below the cap. Thus $Y=0.60(\$1{,}600)=\$960$, with probability $0.25$.

At $X=\$4{,}200$, the amount above the deductible is $\$3{,}200$, which exceeds the cap. Therefore $Z=\$2{,}000$ and $Y=0.60(\$2{,}000)=\$1{,}200$, with probability $0.40$.

The attainable payment values are $\$0,\$480,\$960,\$1{,}200$, with respective masses $0.15,0.20,0.25,0.40$. The positive-payment probability is $0.20+0.25+0.40=0.85$, while the maximum-payment probability is $0.40$. These agree with $P(X>\$1{,}000)=0.85$ and $P(X\geq\$3{,}000)=0.40$.

Important Conditions and Common Traps

The positive-payment result depends on $c>0$. If $c=0$, then $Y=0$ for every loss, so $P(Y>0)=0$, regardless of the deductible or cap. In that case, all probability is concentrated at zero; the scaled payment support is not a positive rescaling of the original payment support.

For $c>0$, scaling preserves the probability of each pre-coinsurance payment amount, but it changes the dollar amount at which that probability appears. For example, a pre-coinsurance payment of $\$400$ becomes $\$280$ when $c=0.70$; its point mass does not change. The cap itself also scales: the maximum payment is $cu$, not $u$.

Common exam trap: Do not assign the zero-payment mass from only the loss value $X=0$. Every loss $X\leq d$, including a loss exactly equal to the deductible, produces zero payment. Similarly, the cap mass includes every loss $X\geq d+u$, not just losses exactly equal to that threshold. For $c>0$, coinsurance changes payment amounts but does not change either threshold event.

A reliable calculation keeps three questions separate: which losses produce zero, which losses reach the cap, and what amounts the remaining losses produce after scaling. This separation is useful when a loss distribution is discrete, mixed, or continuous. In all cases, the deductible determines the zero mass, the cap groups the upper tail into a maximum-payment mass, and positive coinsurance scales the payment axis without changing those group probabilities.

Check Your Understanding

For each problem, identify the relevant loss thresholds before calculating payment values. Money amounts are in dollars.

A hypothetical policy has losses $300,1{,}000,1{,}700,2{,}600$ with probabilities $0.10,0.20,0.30,0.40$. Let $d=700$, $u=1{,}200$, and $c=0.50$. Find all payment masses, the maximum payment and its mass, and $P(Y>0)$. (Answer: $P(Y=0)=0.10$, $P(Y=150)=0.20$, $P(Y=500)=0.30$, $P(Y=600)=0.40$; maximum $\$600$ with mass $0.40$; $P(Y>0)=0.90$.)
For a hypothetical loss distribution, suppose $P(X\leq \$500)=0.18$, $P(X\geq \$2{,}500)=0.12$, and $P(X>\$500)=0.82$. A contract has $d=\$500$, $u=\$2{,}000$, and $c=0.60$. Find the zero mass, maximum payment and its mass, and positive-payment probability. (Answer: $P(Y=0)=0.1800$; maximum payment $\$1{,}200$ with mass $0.1200$; $P(Y>0)=0.8200$.)
A hypothetical loss takes values $\$600,\$1{,}600,\$3{,}100$ with probabilities $0.25,0.45,0.30$. With $d=\$1{,}000$, $u=\$1{,}500$, and $c=0.80$, find the payment support and all point masses. (Answer: $P(Y=0)=0.25$, $P(Y=\$480)=0.45$, $P(Y=\$1{,}200)=0.30$.)
A contract has $d=\$900$, $u=\$1{,}600$, and $c=0$. The hypothetical loss exceeds $\$900$ with probability $0.73$, and reaches or exceeds $\$2{,}500$ with probability $0.21$. Find the zero-payment and positive-payment probabilities. (Answer: $P(Y=0)=1.0000$ and $P(Y>0)=0.0000$.)
A hypothetical loss takes values $0,\$900,\$1{,}400,\$1{,}900,\$4{,}000$ with probabilities $0.05,0.15,0.20,0.25,0.35$. For $d=\$1{,}000$, $u=\$1{,}000$, and $c=0.80$, find the payment masses, $P(Y>0)$, and the cap mass. (Answer: $P(Y=0)=0.20$, $P(Y=\$320)=0.20$, $P(Y=\$720)=0.25$, $P(Y=\$800)=0.35$; $P(Y>0)=0.80$; cap mass $0.35$.)

What You'll Learn