Ordinary Versus Franchise Deductible Distributions

Why the Deductible Type Changes the Payment

A deductible determines when an insurer begins to pay, but it does not by itself determine how much the insurer pays once the threshold is crossed. With an ordinary deductible, the insured bears the first $d$ dollars of a covered loss, and the insurer pays only the amount above $d$. With a franchise deductible, the insurer pays nothing at or below $d$, but pays the entire loss when the loss exceeds $d$. The distinction matters especially for losses near the threshold.

Throughout this tutorial, assume the loss $X$ is nonnegative and the deductible $d$ is positive. For the franchise deductible, use the strict threshold rule: payment begins when $X>d$. A policy that specifies a different rule at equality must be modeled according to its wording.

Definition: The payment per loss under an ordinary deductible is $Y_O^L=(X-d)_+=\max(X-d,0)$. Under a franchise deductible with the strict threshold rule, $Y_F^L=0$ when $X\le d$, and $Y_F^L=X$ when $X>d$. Thus, at $X=d$, both payments are zero; above $d$, the ordinary payment is $X-d$, while the franchise payment is $X$.

The superscript $L$ indicates payment per loss. The difference is not just whether a payment occurs: above the threshold, the two contracts pay different amounts. For instance, for a loss of $d+12$, an ordinary deductible pays 12, whereas a franchise deductible pays $d+12$.

Deriving the Payment Distributions

Let $F_X(x)=P(X\le x)$ be the loss cumulative distribution function (CDF). A payment CDF is $F_Y(y)=P(Y\le y)$. Because both deductible payments are nonnegative, their CDFs are zero for $y<0$. The event that produces a payment no greater than $y$ differs between the contracts.

Theorem: For $y\ge0$, the CDFs of the deductible payments are $$ F_{Y_O^L}(y)=F_X(d+y) $$ and $$ F_{Y_F^L}(y)= \begin{cases} F_X(d), & 0\le y\le d,\\ F_X(y), & y>d. \end{cases} $$ For $y<0$, both CDFs are zero. In particular, both contracts have probability $F_X(d)$ of a zero payment.

To justify the ordinary-deductible formula, consider $y\ge0$. The event $Y_O^L\le y$ consists of losses at most $d$, which produce zero, together with losses above $d$ whose excess over $d$ is at most $y$. Combining these cases gives $X\le d+y$, so $P(Y_O^L\le y)=F_X(d+y)$. This reasoning includes losses exactly equal to $d$.

For the franchise deductible, if $0\le y\le d$, every positive payment is greater than $d$, and therefore greater than $y$. Only the zero-payment outcomes count toward $Y_F^L\le y$, giving $F_X(d)$. If $y>d$, the event $Y_F^L\le y$ includes all losses $X\le d$ and the triggering losses $d<X\le y$. Together these are exactly $X\le y$, giving $F_X(y)$.

These formulas apply to discrete, continuous, and mixed loss distributions. The CDF $F_X$ automatically accounts for any probability mass at the deductible. If $P(X=d)>0$, that mass produces zero payment under both strict-threshold contracts.

Identify the contract rule.
For an ordinary deductible, subtract $d$ after the loss exceeds it. For a strict franchise deductible, pay the full loss only when $X>d$.

Map each loss to a payment.
Check losses below, at, and above $d$; do not assume the two contracts share the same positive payment amount.

Build the payment CDF.
Translate $Y\le y$ into an event about $X$, then evaluate it using $F_X$, including any atom at the threshold.

Worked Distributions

Worked Example: A Discrete Loss With Mass at the Deductible

A hypothetical contract has a deductible $d=40$. Its loss $X$, in dollars, equals 10 with probability 0.15, 40 with probability 0.25, 60 with probability 0.35, and 100 with probability 0.25. Find both payment distributions and compare their CDFs at payments of 20 and 60 dollars.

Method. Apply each payment function to each possible loss, then combine probabilities for losses that produce the same payment. The probabilities sum to $0.15+0.25+0.35+0.25=1.00$.

For the ordinary deductible, the losses 10 and 40 are at or below 40, so both produce zero. The loss 60 produces $60-40=20$, and the loss 100 produces $100-40=60$. Therefore the payment distribution is $P(Y_O^L=0)=0.15+0.25=0.40$, $P(Y_O^L=20)=0.35$, and $P(Y_O^L=60)=0.25$.

For the franchise deductible, losses 10 and 40 produce zero, including the loss exactly equal to the deductible. The triggering losses 60 and 100 are paid in full. Thus $P(Y_F^L=0)=0.40$, $P(Y_F^L=60)=0.35$, and $P(Y_F^L=100)=0.25$.

At a payment of 20, $F_{Y_O^L}(20)=0.40+0.35=0.75$, while $F_{Y_F^L}(20)=0.40$. At a payment of 60, $F_{Y_O^L}(60)=1.00$, while $F_{Y_F^L}(60)=0.40+0.35=0.75$. Both CDFs reach 1.00 at a payment of 100.

Check. Each payment distribution has total probability 1.00. The common mistake is to treat a loss of 40 as triggering a franchise payment; under the specified strict rule it does not.

Worked Example: Comparing CDF Values for an Exponential Loss

A hypothetical portfolio has individual loss $X$ with an exponential distribution of rate $\lambda=0.02$ per dollar, so $F_X(x)=1-e^{-0.02x}$ for $x\ge0$. A contract has deductible $d=30$. Find the zero-payment probability under each deductible, the ordinary and franchise CDFs at payment 20, and the probability that the ordinary payment exceeds 20.

Method. Use the general CDF formulas with the stated exponential model and translate the exceedance event back to a loss event.

First, $F_X(30)=1-e^{-0.02(30)}=1-e^{-0.6}$. Since $e^{-0.6}\approx0.5488$, the probability of zero payment is $1-0.5488=0.4512$ for each contract.

For the ordinary deductible, $F_{Y_O^L}(20)=F_X(30+20)=F_X(50)=1-e^{-1}\approx0.6321$. For the franchise deductible, 20 is below $d=30$, so $F_{Y_F^L}(20)=F_X(30)\approx0.4512$. The franchise CDF remains at its zero-payment probability throughout the interval from 0 to 30.

The ordinary payment exceeds 20 exactly when $X-30>20$, or $X>50$. Thus $P(Y_O^L>20)=P(X>50)=e^{-0.02(50)}=e^{-1}\approx0.3679$. This complements the CDF value: $0.6321+0.3679=1.0000$.

Check. The zero-payment probabilities agree, and the CDF values are no greater than 1. The common mistake is to calculate the ordinary CDF at $X=20$; the payment threshold 20 corresponds to a loss threshold of $d+20=50$.

Worked Example: A Mixed Loss With an Atom at the Threshold

A hypothetical loss has probability 0.20 of being 25, probability 0.30 of being 50, and probability 0.50 of following a continuous uniform distribution on $(50,100)$. Take $d=50$ with a strict franchise threshold. Find the zero-payment probability for both contracts and each payment CDF at 25 and 75.

Method. Separate the point masses from the continuous component. For $50<x<100$, the continuous component contributes $0.50(x-50)/50$ to $F_X(x)$, because its total probability is 0.50 and its interval length is 50.

The loss probabilities at or below 50 are $0.20+0.30=0.50$, so $F_X(50)=0.50$. Both contracts therefore have zero-payment probability 0.50. In particular, the atom of 0.30 at $X=50$ pays zero under the strict franchise rule.

For the ordinary deductible, $F_{Y_O^L}(25)=F_X(75)$. At 75, the continuous component contributes $0.50(75-50)/50=0.25$, in addition to the 0.50 at or below 50. Hence $F_X(75)=0.75$, and $F_{Y_O^L}(25)=0.75$. At payment 75, $F_{Y_O^L}(75)=F_X(125)=1.00$, since all possible losses are at most 100.

For the franchise deductible, 25 is below 50, so $F_{Y_F^L}(25)=F_X(50)=0.50$. Since 75 exceeds 50, $F_{Y_F^L}(75)=F_X(75)=0.75$.

Check. The CDFs are nondecreasing: each value at 25 is no greater than its value at 75. The common mistake is to discard the atom at 50; it is part of $F_X(50)$ and produces zero payment under both contracts.

Distributional Comparison and Exam Traps

Under the same loss $X$, franchise payment is never smaller than ordinary payment. When $X\le d$, both are zero. When $X>d$, the franchise payment is $X$, whereas the ordinary payment is $X-d$, so $Y_F^L-Y_O^L=d>0$. Consequently, for every $y$, $F_{Y_F^L}(y)\le F_{Y_O^L}(y)$: the franchise payment is at least as large in this point-by-point comparison.

The distribution shapes make the difference visible. Both payments have an atom at zero of size $F_X(d)$. Ordinary payments above zero are amounts of excess over the deductible. Franchise payments above zero are full losses, all strictly greater than $d$; hence there can be no franchise payment in $(0,d]$. For a discrete loss, additional atoms occur wherever the loss distribution has atoms above $d$. For a continuous loss, the franchise payment distribution has a gap between zero and $d$, while the ordinary payment can take positive values arbitrarily close to zero if losses can be arbitrarily close above $d$.

A frequent exam trap is to use $X-d$ for both contracts. That gives the ordinary payment, not the franchise payment. For example, if $X=55$ and $d=50$, the incorrect franchise calculation $55-50=5$ should be replaced by the full franchise payment of 55. At $X=50$, the strict franchise rule gives zero, not 50. Always check the loss below, at, or above the threshold before using a distribution formula.

These comparisons are useful when a problem asks for the probability of a payment falling within an interval, a CDF value, or the chance of no payment. State whether the event is about the loss $X$ or the insurer payment $Y^L$, then translate it carefully. The two contracts share a zero-payment probability but generally do not share positive payment amounts or CDF values.

Key takeaway: For an ordinary deductible, $Y_O^L=(X-d)_+$. For a strict franchise deductible, $Y_F^L=X$ when $X>d$, and zero otherwise. Both have zero-payment probability $F_X(d)$, but the franchise CDF has a plateau through $d$, and its positive payments are full losses.

Check Your Understanding

For each hypothetical loss model, identify the applicable payment rule first, then calculate the requested payment probabilities or CDF values.

A loss equals 15, 30, or 45 dollars with probabilities 0.20, 0.30, and 0.50, respectively; $d=30$. Find both payment distributions and both CDFs at payment 15. (Answer: Ordinary: 0 with probability 0.50 and 15 with probability 0.50; franchise: 0 with probability 0.50 and 45 with probability 0.50; the CDFs at 15 are 1.0000 and 0.5000, respectively.)
A loss is uniform on $[0,80]$, with $d=20$. Find the ordinary zero-payment probability and the ordinary and franchise CDFs at payment 10. (Answer: $0.2500$, $0.3750$, and $0.2500$, respectively.)
A loss is exponential with rate $0.01$ per dollar and $d=50$. Find the probability that the ordinary payment exceeds 30 and the probability that the franchise payment exceeds 80. (Answer: Both equal $e^{-0.8}\approx0.4493$.)
A loss is 20 with probability 0.10, 40 with probability 0.40, and 70 with probability 0.50; $d=40$. Find each payment distribution and $F_{Y_F^L}(40)$. (Answer: Ordinary: 0 with probability 0.50 and 30 with probability 0.50; franchise: 0 with probability 0.50 and 70 with probability 0.50; $F_{Y_F^L}(40)=0.5000$.)
A loss has probability 0.25 of being 20, probability 0.25 of being 60, and probability 0.50 of being uniform on $(60,100)$; $d=60$. Find $F_X(80)$, $F_{Y_O^L}(20)$, and $F_{Y_F^L}(80)$. (Answer: $F_X(80)=0.5000$, $F_{Y_O^L}(20)=0.5000$, and $F_{Y_F^L}(80)=0.5000$.)

What You'll Learn