Why Discrete Losses Need Careful Boundary Treatment
When loss amounts come from a finite set of possible values, a deductible and a policy payment limit can map several different losses to the same insurer payment. Losses at or below the deductible produce no payment; sufficiently large losses produce the maximum payment. The exact losses equal to the deductible and to the point where the payment reaches its limit must be assigned correctly. This matters especially when the severity model assigns positive probability to those exact values.
This tutorial considers a nonnegative, discrete loss \(X\), a per-loss deductible \(d\), and a per-loss payment limit \(u\). The limit \(u\) is the maximum insurer payment after applying the deductible, not a cap on the loss amount. All scenarios and data below are hypothetical.
The equalities in this definition are deliberate. A loss exactly equal to \(d\) produces no payment: the deductible has absorbed the whole loss. A loss exactly equal to \(d+u\) produces a payment of exactly \(u\), so it is already at the limit. Losses larger than \(d+u\) also produce payment \(u\). In a discrete model, any one of these boundary losses can carry positive probability.
From Loss Probabilities to Payment Probabilities
Suppose \(X\) takes values \(x_1,x_2,\ldots\) with probabilities \(p_j=P(X=x_j)\), where the probabilities sum to 1. For each possible loss, calculate its payment using the contract formula. Then add the probabilities of every loss value that maps to the same payment. This aggregation is the key step: the payment distribution is not obtained by simply relabelling each loss with a unique payment.
The intermediate-payment rule follows because, when \(0<y<u\), the loss must be in the uncapped part of the payment function. There \(Y^L=X-d\), so \(Y^L=y\) exactly when \(X=d+y\). By contrast, every loss at or above \(d+u\) maps to the same payment \(u\), and every loss at or below \(d\) maps to zero. Those two regions generally combine multiple loss probabilities into single payment probabilities.
Expected Payment for a Discrete Severity
For a finite or countable discrete severity distribution with finite expected payment, the expected insurer payment is the probability-weighted sum of the payment at each possible loss. This direct calculation is often the clearest method when a probability mass function is given as a table.
Grouping terms by the payment they produce gives an equivalent expression: add the probability-weighted intermediate payments and then add \(u\) times the probability of reaching the limit. Zero payments make no contribution. This grouping can reduce the number of terms, but it must preserve all the mass at the relevant boundary values.
The sum is over the distinct intermediate payment values in the discrete distribution; values with zero probability can be omitted. The formula follows by partitioning the possible outcomes of \(Y^L\) into zero, intermediate payments, and the limit payment. The zero-payment term contributes \(0\cdot P(Y^L=0)=0\); each other outcome contributes its payment times its probability. This also explains why the limit probability is \(P(X\ge d+u)\), not \(P(X>d+u)\).
Calculate \(d\) and \(d+u\). These divide losses into zero-payment, uncapped-payment, and capped-payment regions.
Apply \(Y^L=\min\{\max(X-d,0),u\}\) to every support point, including values equal to either threshold.
Add probabilities for equal payments, calculate the expectation, and verify that the payment probabilities sum to 1.
Worked Example: Aggregating Payments at the Deductible and Limit
A hypothetical contract has deductible \(d=\$1{,}000\) and per-loss payment limit \(u=\$1{,}500\). The loss \(X\) has the following distribution. Calculate the probability of no payment, the probability of payment at the limit, and \(E[Y^L]\).
| Loss \(x_j\) | \(P(X=x_j)\) |
|---|---|
| \$0 | 0.1000 |
| \$500 | 0.2000 |
| \$1,000 | 0.3000 |
| \$2,000 | 0.2500 |
| \$3,500 | 0.1500 |
Step 1: Check the probabilities and thresholds. The probabilities sum to \(0.1000+0.2000+0.3000+0.2500+0.1500=1.0000\). The payment reaches its limit at \(d+u=\$1{,}000+\$1{,}500=\$2{,}500\).
Step 2: Map the losses. Losses of \$0 and \$500 are below the deductible, and the loss of exactly \$1,000 equals the deductible. All three produce payment \$0. The \$2,000 loss produces \(\$2{,}000-\$1{,}000=\$1{,}000\). The \$3,500 loss exceeds \$2,500, so its payment is capped at \$1,500.
Step 3: Aggregate the requested probabilities. The probability of no payment is \(0.1000+0.2000+0.3000=0.6000\). Only the \$3,500 loss produces payment at the limit, so \(P(Y^L=\$1{,}500)=0.1500\).
Step 4: Calculate the expectation. \(E[Y^L]=\$0(0.6000)+\$1{,}000(0.2500)+\$1{,}500(0.1500)=\$0+\$250+\$225=\$475.00\).
Check. The payment probabilities are 0.6000 at \$0, 0.2500 at \$1,000, and 0.1500 at \$1,500; they sum to 1.0000. The expectation is between \$0 and the maximum payment of \$1,500. The method is direct payment mapping and aggregation; it avoids treating a loss equal to the deductible as a positive payment.
Worked Examples With Exact Boundary Losses
Worked Example: A Loss Exactly at the Payment-Limit Threshold
A hypothetical equipment policy has deductible \(d=\$800\) and payment limit \(u=\$1{,}000\). The possible losses are shown below. Find \(P(Y^L=0)\), \(P(Y^L=\$1{,}000)\), and the expected payment.
| Loss \(x_j\) | \(P(X=x_j)\) |
|---|---|
| \$400 | 0.1000 |
| \$800 | 0.1500 |
| \$1,200 | 0.2000 |
| \$1,800 | 0.2000 |
| \$2,400 | 0.2000 |
| \$3,000 | 0.1500 |
Step 1: Establish the threshold. The probabilities sum to \(1.0000\). The limit threshold is \(d+u=\$800+\$1{,}000=\$1{,}800\).
Step 2: Assign every payment. The losses \$400 and exactly \$800 produce \$0. The \$1,200 loss produces \(\$1{,}200-\$800=\$400\). The \$1,800 loss is exactly at the limit threshold and produces \$1,000. The \$2,400 and \$3,000 losses also produce \$1,000.
Step 3: Calculate probabilities. \(P(Y^L=0)=0.1000+0.1500=0.2500\). The limit payment occurs for losses at least \$1,800, so \(P(Y^L=\$1{,}000)=0.2000+0.2000+0.1500=0.5500\).
Step 4: Calculate expected payment. \(E[Y^L]=\$0(0.2500)+\$400(0.2000)+\$1{,}000(0.5500)=\$80+\$550=\$630.00\).
Check. The remaining probability is \(1.0000-0.2500-0.5500=0.2000\), matching the intermediate \$400 payment. The expectation is \(\$630.00\). The method aggregates losses by payment amount; it avoids excluding the loss exactly equal to \(d+u\) from the capped-payment probability.
Worked Example: Building the Entire Payment Distribution
A hypothetical repair plan has deductible \(d=\$500\) and payment limit \(u=\$1{,}000\). A discrete loss \(X\) has the distribution below. Construct the distribution of \(Y^L\) and calculate its expectation.
| Loss \(x_j\) | \(P(X=x_j)\) |
|---|---|
| \$0 | 0.0500 |
| \$500 | 0.2000 |
| \$1,000 | 0.2500 |
| \$1,500 | 0.1000 |
| \$2,000 | 0.2500 |
| \$2,500 | 0.1500 |
Step 1: Find the boundaries. The probabilities sum to \(1.0000\), and \(d+u=\$500+\$1{,}000=\$1{,}500\).
Step 2: Map and group the losses. Losses \$0 and \$500 produce \$0, with combined probability \(0.0500+0.2000=0.2500\). The \$1,000 loss produces \(\$1{,}000-\$500=\$500\), with probability 0.2500. The \$1,500 loss is exactly at the limit threshold, and the \$2,000 and \$2,500 losses exceed it; all three produce \$1,000, with combined probability \(0.1000+0.2500+0.1500=0.5000\).
Step 3: State the payment distribution. \(P(Y^L=\$0)=0.2500\), \(P(Y^L=\$500)=0.2500\), and \(P(Y^L=\$1{,}000)=0.5000\). These probabilities sum to \(1.0000\).
Step 4: Calculate the expectation. \(E[Y^L]=\$0(0.2500)+\$500(0.2500)+\$1{,}000(0.5000)=\$125+\$500=\$625.00\).
Check. The maximum possible payment is \$1,000, and the expectation of \$625.00 lies within the payment range. The method is to group the mapped payment values before taking the weighted average; it avoids counting the several capped losses as different payment outcomes.
Common Exam Trap
A frequent error is to use strict inequalities at both thresholds. For example, writing \(P(Y^L=u)=P(X>d+u)\) wrongly omits the loss \(X=d+u\), even though that loss produces payment \(u\) exactly. Likewise, writing \(P(Y^L=0)=P(X<d)\) wrongly omits the loss \(X=d\), which produces no payment. The correct events are \(X\ge d+u\) for a limit payment and \(X\le d\) for zero payment.
Another error is to cap the loss at \(u\) before subtracting the deductible, using \(\min(X,u)-d\). That is not the stated contract: it can produce a negative payment and applies the payment cap to the wrong quantity. For a deductible followed by a payment limit, subtract the deductible first, floor the result at zero, and then cap the payment at \(u\).
Check Your Understanding
For each hypothetical discrete loss model, apply the deductible first and interpret the payment limit as a cap on insurer payment. Money answers are in dollars.
- A contract has \(d=\$400\), \(u=\$700\), and losses \$0, \$400, \$900, and \$1,400 with probabilities 0.2000, 0.3000, 0.3000, and 0.2000. Find the zero-payment probability, limit-payment probability, and expected payment. (Answer: \(P(Y^L=0)=0.5000\), \(P(Y^L=\$700)=0.2000\), \(E[Y^L]=\$290.00\).)
- A contract has \(d=\$600\), \(u=\$600\). Losses \$300, \$600, \$900, \$1,200, and \$1,800 have probabilities 0.1000, 0.2000, 0.2500, 0.2500, and 0.2000. Find the payment distribution and expected payment. (Answer: payments \$0, \$300, and \$600 have probabilities 0.3000, 0.2500, and 0.4500; \(E[Y^L]=\$345.00\).)
- A contract has \(d=\$500\), \(u=\$1{,}000\). Losses \$0, \$500, \$1,000, \$1,500, and \$2,500 have probabilities 0.1500, 0.2000, 0.2500, 0.2500, and 0.1500. Find the probability of a positive payment and the expected payment. (Answer: \(P(Y^L>0)=0.6500\), \(E[Y^L]=\$525.00\).)
- A contract has \(d=\$700\), \(u=\$1{,}000\). Losses \$200, \$700, \$1,100, \$1,700, and \$2,200 have probabilities 0.1000, 0.2000, 0.2500, 0.1500, and 0.3000. Find \(P(Y^L=0)\), \(P(Y^L=\$1{,}000)\), and \(E[Y^L]\). (Answer: \(P(Y^L=0)=0.3000\), \(P(Y^L=\$1{,}000)=0.4500\), \(E[Y^L]=\$550.00\).)