Discrete Losses Under Deductibles and Limits

Why Discrete Losses Require Careful Transformation

When ground-up losses can take only particular values, modifying coverage means transforming each possible loss into an insurer payment. The transformed distribution may have fewer possible values than the original distribution: several losses can produce no payment, and losses above a policy limit can all produce the same maximum payment. The key task is to track the probability attached to every payment value, including values created by merging outcomes.

This tutorial uses $X$ for the ground-up loss and $Y^L$ for the insurer payment per loss. Unless stated otherwise, the policy limit $u$ is a maximum insurer payment, not a cap on the ground-up loss. All examples are hypothetical.

Definition: Under an ordinary deductible $d$ and a maximum payment $u$, the insurer pays the part of a loss above the deductible, up to $u$: $$ Y^L=\min\bigl((X-d)_+,u\bigr),\qquad (z)_+=\max(z,0). $$ Under a franchise deductible $d$, using the convention that the deductible is met when $X\ge d$, the insurer pays the full loss once the threshold is met, subject to the same maximum payment: $$ Y^L= \begin{cases} 0,&X<d,\\ \min(X,u),&X\ge d. \end{cases} $$

The two contracts treat losses at and just above the deductible differently. An ordinary deductible subtracts $d$ from a covered loss. A franchise deductible instead determines whether the loss qualifies for payment; after the threshold is met, it does not subtract $d$. Contract wording controls the boundary convention. Here, a loss exactly equal to the franchise deductible qualifies for payment.

Mapping Loss Values to Payments

Suppose a discrete ground-up loss has possible values $x_1,x_2,\ldots$, with $P(X=x_j)=p_j$. To find the payment distribution, calculate the payment $y_j$ corresponding to each loss $x_j$, then add the probabilities of all loss values that produce the same payment. In particular, probabilities are not lost when outcomes merge.

Key result: For any possible payment value $y$, $$ P(Y^L=y)=\sum_{j:\,y_j=y}p_j, $$ where $y_j$ is the payment produced by ground-up loss $x_j$. More generally, for any set $B$ of payment values, $$ P(Y^L\in B)=\sum_{j:\,y_j\in B}p_j. $$

For an ordinary deductible, all losses at or below $d$ produce a zero payment. For a franchise deductible under the stated convention, only losses strictly below $d$ produce zero. With either contract, all losses that reach or exceed the maximum-payment point can accumulate probability at $u$. The payment distribution may also have other coincident values, so each transformed value must be checked rather than assumed unique.

Why Probability Masses Must Be Added

The probability rule follows directly from the events that define a payment. The event $Y^L=y$ occurs precisely when $X$ takes one of the ground-up values that transforms to $y$. Those ground-up outcomes are mutually exclusive, so their probabilities add.

\{Y^L=y\}=\bigcup_{j:\,y_j=y}\{X=x_j\}, \qquad P(Y^L=y)=\sum_{j:\,y_j=y}P(X=x_j)

This also explains why the probability at zero is a sum over the losses that the contract leaves unpaid, while the probability at the limit is a sum over all losses that reach the cap. Once the payment distribution is assembled, probabilities and expected payments are calculated from that distribution in the usual discrete way:

E[Y^L]=\sum_y y\,P(Y^L=y)

The expectation can equivalently be calculated by summing $y_jp_j$ over the original loss values. Aggregating first is often clearer for checking probabilities at zero or at the limit; summing by original loss is a useful arithmetic check.

Ordinary Deductible with a Payment Limit

Worked Example: Building the Ordinary-Deductible Payment Distribution

A hypothetical contract has an ordinary deductible of $d=\$100$ and a maximum insurer payment of $u=\$150$. The ground-up loss distribution is $P(X=\$0)=0.10$, $P(X=\$80)=0.15$, $P(X=\$100)=0.20$, $P(X=\$160)=0.25$, $P(X=\$260)=0.20$, and $P(X=\$400)=0.10$. Find the full payment distribution, $P(Y^L>0)$, and $E[Y^L]$.

Method. Transform each loss using $Y^L=\min((X-100)_+,150)$. Then group losses with identical payments.

Ground-up loss $X$	Probability	Payment calculation	Payment $Y^L$
\$0	0.10	$\max(0-100,0)=0$	\$0
\$80	0.15	$\max(80-100,0)=0$	\$0
\$100	0.20	$\max(100-100,0)=0$	\$0
\$160	0.25	$\min(160-100,150)=60$	\$60
\$260	0.20	$\min(260-100,150)=150$	\$150
\$400	0.10	$\min(400-100,150)=150$	\$150

The zero-payment mass is $0.10+0.15+0.20=0.45$. The probability at $\$60$ is $0.25$. The probability at the limit is $0.20+0.10=0.30$. These sum to $0.45+0.25+0.30=1.00$.

Thus the payment distribution is $P(Y^L=\$0)=0.45$, $P(Y^L=\$60)=0.25$, and $P(Y^L=\$150)=0.30$. Therefore, $P(Y^L>0)=0.25+0.30=0.55$, and $E[Y^L]=0(0.45)+60(0.25)+150(0.30)=15+45=\$60.00$.

Check and trap avoided. The probabilities total $1.00$, and the expectation is also $0.25(\$60)+0.30(\$150)=\$60.00$. The loss of $\$100$ produces no payment under an ordinary deductible; treating it as a positive payment would incorrectly include the deductible itself.

Franchise Deductibles and the Threshold Boundary

A franchise deductible creates a different payment map: a loss below the threshold is unpaid, but a qualifying loss is paid in full, subject to the limit. The exact threshold is therefore important. Under the convention in this tutorial, $X=d$ qualifies.

Worked Example: Applying a Franchise Threshold and Limit

A hypothetical policy has a franchise deductible of $d=\$100$ and a maximum insurer payment of $u=\$250$. The ground-up losses and probabilities are $P(X=\$50)=0.20$, $P(X=\$100)=0.15$, $P(X=\$120)=0.25$, $P(X=\$200)=0.20$, $P(X=\$250)=0.10$, and $P(X=\$400)=0.10$. Find the payment distribution, $P(Y^L\ge \$200)$, and $E[Y^L]$.

Method. Losses below $\$100$ produce zero. At or above $\$100$, payment is $\min(X,\$250)$.

Ground-up loss $X$	Probability	Payment $Y^L$
\$50	0.20	\$0, because $50<100$
\$100	0.15	\$100, because the threshold is met
\$120	0.25	\$120
\$200	0.20	\$200
\$250	0.10	\$250
\$400	0.10	\$250, because the limit applies

The $\$250$ payment has probability $0.10+0.10=0.20$, because both the $\$250$ loss and the $\$400$ loss produce that payment. The full distribution is $P(Y^L=\$0)=0.20$, $P(Y^L=\$100)=0.15$, $P(Y^L=\$120)=0.25$, $P(Y^L=\$200)=0.20$, and $P(Y^L=\$250)=0.20$. The masses total $1.00$.

For the requested tail probability, the payments at least $\$200$ are $\$200$ and $\$250$, so $P(Y^L\ge \$200)=0.20+0.20=0.40$. The expected payment is $0(0.20)+100(0.15)+120(0.25)+200(0.20)+250(0.20)=15+30+40+50=\$135.00$.

Check and trap avoided. Summing payment times probability by the original loss values gives the same $\$135.00$. A common error is to subtract the franchise deductible from each qualifying loss. That would apply an ordinary deductible, not the franchise contract defined here. The $\$100$ loss qualifies and pays $\$100$.

Comparing Two Coverage Transformations

When contract terms differ, write out each payment rule before calculating. In particular, the two contracts can have different zero-payment masses and different values at the limit even when they use the same ground-up distribution. The following comparison also illustrates how to check an expectation directly against the transformed outcomes.

Worked Example: Comparing Ordinary and Franchise Deductibles

A hypothetical portfolio has ground-up loss distribution $P(X=\$20)=0.10$, $P(X=\$50)=0.20$, $P(X=\$90)=0.30$, $P(X=\$140)=0.25$, and $P(X=\$200)=0.15$. Compare (A) an ordinary deductible $d=\$50$ with payment limit $u=\$120$, and (B) a franchise deductible $d=\$50$ with payment limit $u=\$120$. Find each payment distribution and expected payment.

Method for A. Apply $Y^L=\min((X-50)_+,120)$. The losses $\$20$ and $\$50$ both pay $\$0$; their combined probability is $0.10+0.20=0.30$. The $\$90$ loss pays $\$40$ with probability $0.30$. The $\$140$ loss pays $\$90$ with probability $0.25$. The $\$200$ loss would pay $\$150$ above the deductible, so it is capped at $\$120$, with probability $0.15$.

Thus contract A has payment masses $0.30$ at $\$0$, $0.30$ at $\$40$, $0.25$ at $\$90$, and $0.15$ at $\$120$. They sum to $1.00$, and $E[Y^L]=0(0.30)+40(0.30)+90(0.25)+120(0.15)=12+22.50+18=\$52.50$.

Method for B. Losses below $\$50$ pay zero; losses at or above $\$50$ pay $\min(X,120)$. The $\$20$ loss pays zero with probability $0.10$. The $\$50$ loss pays $\$50$ with probability $0.20$, the $\$90$ loss pays $\$90$ with probability $0.30$, and both $\$140$ and $\$200$ pay the limit of $\$120$, with combined probability $0.25+0.15=0.40$.

Contract B therefore has payment masses $0.10$ at $\$0$, $0.20$ at $\$50$, $0.30$ at $\$90$, and $0.40$ at $\$120$. They sum to $1.00$, and $E[Y^L]=0(0.10)+50(0.20)+90(0.30)+120(0.40)=10+27+48=\$85.00$.

Check and trap avoided. The franchise expected payment exceeds the ordinary expected payment by $\$85.00-\$52.50=\$32.50$, consistent with the franchise contract paying the full loss once the threshold is met. Do not treat the $\$50$ loss as unpaid under the franchise contract: equality meets the threshold in this example.

Common Errors to Check Before Finalizing

Subtracting the deductible twice or under the wrong contract. The ordinary payment is the excess over $d$; the franchise payment, once its threshold is met, is the full loss up to the limit.
Misreading the equality case. Under the convention used here, an ordinary loss exactly equal to $d$ pays zero, whereas a franchise loss exactly equal to $d$ qualifies.
Leaving out merged probability mass. Add probabilities when several losses produce zero, the same intermediate payment, or the limit.
Applying a limit to the wrong amount. Here $u$ caps the insurer's payment. Under an ordinary deductible the cap applies after subtracting $d$.
Checking only the expected value. A plausible expectation does not establish that the full distribution is right. Confirm that all payment masses sum to one and that boundary and limit outcomes are included.

Key takeaway: Transform every possible ground-up loss using the contract's exact payment rule, then add the probabilities of all losses that map to the same payment. Explicitly check the zero-payment boundary, the franchise threshold, and the cap before calculating probabilities or expected payment.

Check Your Understanding

For each hypothetical contract, transform the stated discrete loss distribution and combine any coincident payment values. Dollar amounts are exact; no rounding is needed.

An ordinary deductible is $\$50$, and the maximum payment is $\$100$. Losses are $\$0,\$50,\$80,\$150,\$200$, with respective probabilities $0.10,0.20,0.30,0.25,0.15$. Find the payment distribution, the probability of a positive payment, and the expected payment. (Answer: masses $0.30$ at $\$0$, $0.30$ at $\$30$, and $0.40$ at $\$100$; $P(Y^L>0)=0.70$; $E[Y^L]=\$49.00$.)
A franchise deductible is $\$75$, and the maximum payment is $\$160$. Losses are $\$40,\$75,\$100,\$160,\$210$, with respective probabilities $0.20,0.10,0.25,0.20,0.25$. Find the payment distribution and $P(Y^L\ge \$100)$. (Answer: masses $0.20$ at $\$0$, $0.10$ at $\$75$, $0.25$ at $\$100$, and $0.45$ at $\$160$; $P(Y^L\ge \$100)=0.70$.)
An ordinary deductible is $\$30$, and the maximum payment is $\$80$. Losses are $\$25,\$30,\$50,\$90,\$110$, with respective probabilities $0.15,0.15,0.20,0.25,0.25$. Find the payment distribution, $P(Y^L\ge \$60)$, and the expected payment. (Answer: masses $0.30$ at $\$0$, $0.20$ at $\$20$, $0.25$ at $\$60$, and $0.25$ at $\$80$; $P(Y^L\ge \$60)=0.50$; $E[Y^L]=\$39.00$.)
A franchise deductible is $\$100$, and the maximum payment is $\$180$. Losses are $\$80,\$100,\$140,\$180,\$240$, with respective probabilities $0.20,0.20,0.25,0.15,0.20$. Find the expected payment and the probability that payment exceeds $\$100$. (Answer: $E[Y^L]=\$118.00$; $P(Y^L>\$100)=0.60$.)
For losses $\$20,\$50,\$90,\$140,\$200$ with probabilities $0.10,0.20,0.30,0.25,0.15$, compare an ordinary deductible and a franchise deductible, each with threshold $\$50$ and payment limit $\$120$. Find both expected payments and state which contract has the larger expected payment. (Answer: ordinary expected payment $=\$52.50$; franchise expected payment $=\$85.00$; the franchise contract is larger by $\$32.50$.)

Ground-up loss \(X\)	Probability	Payment calculation	Payment \(Y^L\)
\$0	0.10	\(\max(0-100,0)=0\)	\$0
\$80	0.15	\(\max(80-100,0)=0\)	\$0
\$100	0.20	\(\max(100-100,0)=0\)	\$0
\$160	0.25	\(\min(160-100,150)=60\)	\$60
\$260	0.20	\(\min(260-100,150)=150\)	\$150
\$400	0.10	\(\min(400-100,150)=150\)	\$150

Ground-up loss \(X\)	Probability	Payment \(Y^L\)
\$50	0.20	\$0, because \(50<100\)
\$100	0.15	\$100, because the threshold is met
\$120	0.25	\$120
\$200	0.20	\$200
\$250	0.10	\$250
\$400	0.10	\$250, because the limit applies

What You'll Learn