Franchise Deductibles and Ordinary Deductibles

Why the Deductible Definition Matters

A deductible threshold can describe two different payment arrangements. Under an ordinary deductible, the insurer pays only the part of a loss above the deductible. Under a franchise deductible, the insurer pays nothing unless the loss exceeds the threshold, but then pays the full loss. These contracts can have the same threshold and the same underlying losses while producing substantially different expected insurer payments.

The key distinction is what happens to a loss that qualifies for payment. An ordinary deductible subtracts the deductible from that loss; a franchise deductible uses the threshold as a qualification test and does not subtract it. This comparison is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Definition: Let $X$ be a nonnegative loss and let $d \ge 0$ be a deductible threshold. For an ordinary deductible, the insurer payment per loss is $Y^L_o=(X-d)^+=\max(X-d,0)$. For a franchise deductible with the strict-threshold convention, the insurer payment per loss is $Y^L_f=X\mathbf{1}_{\{X>d\}}$, where $\mathbf{1}_{\{X>d\}}$ equals 1 when $X>d$ and 0 otherwise. Thus, a loss at or below $d$ receives no payment under either contract; a loss above $d$ receives the excess over $d$ under the ordinary deductible and the full loss under the franchise deductible.

Payment Functions and the Threshold

The payment functions make the difference explicit. Below the threshold, both are zero. At the threshold, both are zero under the strict franchise convention used here. Above the threshold, ordinary payment rises from zero at a slope of one, while franchise payment starts at the full loss amount: it jumps from zero to $d$ immediately after the threshold, then also rises at a slope of one.

Loss amount	Ordinary payment $Y^L_o$	Franchise payment $Y^L_f$
$X<d$	0	0
$X=d$	0	0 under the strict-threshold convention
$X>d$	$X-d$	$X$

The equality case deserves attention when losses can take discrete values. Some contract language defines franchise coverage as applying when the loss is at least the threshold, rather than when it exceeds it. Under that alternative wording, a loss exactly equal to $d$ receives a payment of $d$. For a continuous severity distribution, the probability of a loss exactly equal to a particular threshold is zero, so the choice of strict versus inclusive wording does not change expected payment. For a discrete or empirical distribution, it can change the result and must be resolved from the contract wording.

Worked Example: Discrete Losses and an Exact Threshold

A hypothetical equipment policy has a deductible threshold of $500. A covered loss $X$ is $200 with probability 0.20, $500 with probability 0.30, $800 with probability 0.30, and $1,300 with probability 0.20. Calculate expected insurer payment per loss under the ordinary deductible and under a strict-threshold franchise deductible.

Method. Apply each payment function separately to every possible loss, then weight each payment by its probability. This method avoids treating the franchise threshold as an amount subtracted from a qualifying loss.

Ordinary deductible. The payments are $(200-500)^+=0$, $(500-500)^+=0$, $(800-500)^+=300$, and $(1{,}300-500)^+=800$. The expected payment is

E[Y^L_o]=0.20(0)+0.30(0)+0.30(300)+0.20(800)=0+0+90+160=\$250.

Franchise deductible. The $200 and $500 losses do not exceed the threshold, so their payments are zero. The $800 and $1,300 losses qualify and are paid in full. Therefore,

E[Y^L_f]=0.20(0)+0.30(0)+0.30(800)+0.20(1{,}300)=0+0+240+260=\$500.

Check. The probability of exceeding $500 is $0.30+0.20=0.50$. The franchise expected payment should exceed the ordinary expected payment by $dP(X>d)=500(0.50)=\$250$. Indeed, $\$500-\$250=\$250$, matching the ordinary result. The loss exactly at $500 receives no payment under the stated strict-threshold convention.

Relating the Expected Payments

For every individual loss amount, the franchise payment can be written as the ordinary payment plus the deductible amount when the loss exceeds the threshold. If $X\le d$, both sides of the identity below are zero. If $X>d$, the ordinary payment is $X-d$, and adding $d$ gives the full loss $X$. Taking expectations gives a convenient relationship between the two expected payments.

Y^L_f=Y^L_o+d\mathbf{1}_{\{X>d\}}

$$ E[Y^L_f]=E[Y^L_o]+dP(X>d) $$

This identity assumes the same nonnegative loss distribution and threshold for both contracts, and the strict-threshold convention. It requires the relevant expectations to be finite. It also shows precisely why the franchise expected payment is at least as large as the ordinary expected payment: the franchise contract restores $d$ dollars for each loss that exceeds the threshold. If $P(X>d)=0$, the two expected payments are equal to zero. If the threshold is exceeded with positive probability and $d>0$, the franchise expected payment is strictly larger.

Calculating Payments Under a Continuous Severity Model

For a continuous loss with density $f_X(x)$, expected payment can also be calculated by integrating the payment function over possible losses. Under the strict-threshold convention, the ordinary payment is positive only above $d$, where it equals $x-d$; the franchise payment is positive over the same range, where it equals $x$. Thus,

E[Y^L_o]=\int_d^\infty (x-d)f_X(x)\,dx,\qquad E[Y^L_f]=\int_d^\infty xf_X(x)\,dx.

The expected-payment identity follows directly by subtracting the integrands: $xf_X(x)-(x-d)f_X(x)=df_X(x)$ for $x>d$. Integrating gives $d\int_d^\infty f_X(x)\,dx=dP(X>d)$. This integral derivation and the loss-by-loss identity express the same result.

Worked Example: Exponential Severity

Suppose a hypothetical insured loss has an exponential distribution with mean $\theta=\$1{,}000$, so its density is $f_X(x)=\frac{1}{1{,}000}e^{-x/1{,}000}$ for $x\ge0$. The deductible threshold is $d=\$400$. Calculate the expected ordinary and strict-threshold franchise payments, rounding only the final dollar amounts to the nearest cent.

Method. For an exponential loss with mean $\theta$, $P(X>d)=e^{-d/\theta}$. The memoryless property gives the conditional expected loss above the threshold as $E[X\mid X>d]=d+\theta$. Therefore, the ordinary payment averages $\theta$ among losses that exceed $d$, while the franchise payment averages $d+\theta$ among those losses. Multiplying each conditional average by the probability of exceeding the threshold gives the unconditional expected payment.

First calculate the probability of exceeding the threshold:

P(X>400)=e^{-400/1{,}000}=e^{-0.4}\approx0.670320046.

For the ordinary deductible, the amount paid conditional on exceeding $400 has mean $\theta=\$1{,}000$. Hence,

E[Y^L_o]=1{,}000e^{-0.4}\approx\$670.32.

For the franchise deductible, the conditional mean loss among qualifying losses is $d+\theta=\$400+\$1{,}000=\$1{,}400$. Hence,

E[Y^L_f]=1{,}400e^{-0.4}\approx\$938.45.

Check. The difference is approximately $\$938.45-\$670.32=\$268.13$. The identity gives $dP(X>d)=400e^{-0.4}\approx\$268.13$, as required. The calculations use the full loss above the threshold for the franchise payment, not merely the amount in excess of $400.

Using Observed Losses

When an empirical severity sample is used to estimate expected payment, calculate the payment for each observation and take the arithmetic average. This is the empirical mean of the payment per loss. For the franchise deductible, count only observations that strictly exceed the threshold, but include their full loss amounts in the average. For the ordinary deductible, subtract the threshold from each qualifying observation. An observation at the threshold is not a qualifying loss under the strict convention.

Worked Example: Empirical Payment Estimates

A hypothetical sample contains six ground-up losses of $250, $450, $600, $600, $1,000, and $1,500. Use a strict franchise threshold and an ordinary deductible of $d=\$600$. Estimate the average insurer payment per observed loss under each contract.

Method. Calculate the two payments observation by observation, including zero payments for losses that do not qualify, and divide each total by all six observations. Dividing only by the number of losses that qualify would estimate a conditional payment, not the average payment per observed loss.

Ordinary deductible. The payments are $0, $0, $0, $0, $400, and $900. Their total is $0+0+0+0+400+900=\$1{,}300$. The average per observed loss is

\widehat{E[Y^L_o]}=\frac{1{,}300}{6}=\$216.67.

Franchise deductible. The first four observations are at or below $600 and receive no payment. The last two observations receive their full amounts, $1,000 and $1,500. Their total is $\$2{,}500$, so the average per observed loss is

\widehat{E[Y^L_f]}=\frac{1{,}000+1{,}500}{6}=\frac{2{,}500}{6}=\$416.67.

Check. Two of six observations exceed $600. The estimated difference should be $600(2/6)=\$200$. The calculated difference is $\$416.67-\$216.67=\$200.00$, subject to cent rounding. Both observations exactly at $600 receive zero under the stated convention.

Common Mistake and Practical Interpretation

A common exam error is to calculate a franchise payment as $(X-d)^+$. That is the ordinary deductible payment function, not the franchise function. For example, if a loss is $900 and the threshold is $600, the ordinary payment is $\$900-\$600=\$300$, but the franchise payment is the full $\$900$. The incorrect franchise calculation understates that payment by $600.

Another error is to apply the threshold to claim frequency but then use the ordinary payment amount for losses that qualify. The franchise condition and the payment amount are separate parts of the contract: first determine whether $X>d$; if so, the insurer pays $X$. This is why threshold wording and payment function should be stated before any expected value is calculated.

These functions describe payment per loss. They do not, by themselves, account for how many losses occur in a policy period. If a problem asks for expected aggregate payments, frequency information is also needed; the expected payment per loss is not automatically the expected payment per policy. Keep the requested measure clear and use the same set of losses when comparing the two deductible forms.

Key takeaway: With a strict threshold, an ordinary deductible pays $(X-d)^+$, while a franchise deductible pays $X\mathbf{1}_{\{X>d\}}$. Their expected payments differ by $dP(X>d)$. Losses at or below the threshold receive no payment under either form, but a loss above the threshold is paid in full only under the franchise form.

Check Your Understanding

For each problem, use a strict franchise threshold unless the wording states otherwise. Treat each requested average as a payment per loss.

A hypothetical loss is $100 with probability 0.25, $300 with probability 0.25, and $800 with probability 0.50. At $d=\$300$, calculate the expected ordinary and franchise payments. (Answer: ordinary $250; franchise $400.)
A hypothetical exponential loss has mean $\$2{,}000$, and the threshold is $\$500$. Calculate $P(X>d)$, then calculate the ordinary and franchise expected payments to the nearest cent. (Answer: $P(X>d)=0.7788$; ordinary $1,557.60; franchise $1,947.00.)
Observed losses are $200, $500, $900, $1,200, and $2,000. At $d=\$500$, calculate the empirical average payment under each deductible form. (Answer: ordinary $520; franchise $820.)
For a loss distribution, the ordinary expected payment at $d=\$200$ is $360, and $P(X>200)=0.30$. Use the expected-payment relationship to find the franchise expected payment. (Answer: $420.)
A discrete loss is $400 with probability 0.40, $500 with probability 0.20, and $900 with probability 0.40. At $d=\$500$, calculate both expected payments under the strict-threshold convention. Then state how the franchise expected payment changes if the contract instead pays the full loss when $X\ge d$. (Answer: strict ordinary $160 and strict franchise $360; inclusive franchise $460.)

Loss amount	Ordinary payment \(Y^L_o\)	Franchise payment \(Y^L_f\)
\(X<d\)	0	0
\(X=d\)	0	0 under the strict-threshold convention
\(X>d\)	\(X-d\)	\(X\)

What You'll Learn