Exponential Losses Under Deductibles and Limits

Why Exponential Losses Simplify Coverage Calculations

A coverage modification changes the amount paid on a claim, not the underlying ground-up loss. For an exponential loss model, the probability of crossing a deductible and the expected payment can often be calculated directly. The key is to define precisely whether the deductible is ordinary or franchise and whether the limit caps the insurer’s payment.

This tutorial assumes a continuous, nonnegative ground-up loss and uses $d$ for the deductible and $u$ for the maximum insurer payment. All hypothetical examples state the loss units and exponential rate. This treatment is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: A ground-up loss $X$ has an exponential distribution with rate $\lambda>0$ if, for $x\geq 0$, $S_X(x)=P(X>x)=e^{-\lambda x}$, $F_X(x)=1-e^{-\lambda x}$, and $f_X(x)=\lambda e^{-\lambda x}$. The rate $\lambda$ is measured per loss unit, so $\lambda x$ is dimensionless and $E[X]=1/\lambda$.

For example, if losses are measured in thousands of dollars and $\lambda=0.1$ per thousand dollars, the mean ground-up loss is $1/0.1=10$ thousand dollars. A deductible or limit stated in dollars must first be converted to the same units.

Two Deductible Conventions and a Payment Limit

With an ordinary deductible $d$, the insurer pays only the part of the loss above $d$. If the maximum insurer payment is $u$, the payment per loss is

Y^L=\min\bigl((X-d)_+,u\bigr), \qquad (z)_+=\max(z,0).

Thus, a loss at or below $d$ produces no payment. A loss between $d$ and $d+u$ produces payment $X-d$, and a loss at or above $d+u$ produces the capped payment $u$.

With a franchise deductible $d$, a loss at or below $d$ produces no payment, but a loss above $d$ is paid from its first dollar. With a maximum payment $u$, the convention used here is

Y^L=\mathbf{1}_{\{X>d\}}\min(X,u).

In particular, the franchise deductible is not subtracted from the loss. If $u>d$, a loss between $d$ and $u$ is paid in full, while a loss at or above $u$ produces payment $u$. Since the exponential model is continuous, equality at a fixed boundary has probability zero; the stated inequalities still make the coverage convention clear.

Coverage	Positive payment occurs when	Payment reaches $u$ when
Ordinary deductible with limit $u$	$X>d$	$X\geq d+u$
Franchise deductible with limit $u>d$	$X>d$	$X\geq u$

The Memoryless Property and Ordinary-Deductible Payments

The exponential distribution has the memoryless property: conditional on a loss exceeding $d$, the excess loss $X-d$ has the same exponential distribution as $X$. For $t\geq0$,

P(X-d>t\mid X>d) =\frac{P(X>d+t)}{P(X>d)} =\frac{e^{-\lambda(d+t)}}{e^{-\lambda d}} =e^{-\lambda t}.

This identity explains why an ordinary deductible affects the chance of a payment through the factor $e^{-\lambda d}$, while the excess amount conditional on a payment retains the original exponential form.

Theorem: For an exponential loss with rate $\lambda$, an ordinary deductible $d\geq0$, and a maximum payment $u>0$, the probability of a positive payment is $e^{-\lambda d}$, the probability of reaching the limit is $e^{-\lambda(d+u)}$, and $$ E[Y^L]=e^{-\lambda d}\frac{1-e^{-\lambda u}}{\lambda}. $$ Without a payment limit, $E[(X-d)_+]=e^{-\lambda d}/\lambda$.

To justify the limited expectation, condition on whether the loss exceeds the deductible. Given $X>d$, the excess $X-d$ is exponential with rate $\lambda$. For an exponential variable $Z$ with that rate,

E[\min(Z,u)] =\int_0^u P(Z>y)\,dy =\int_0^u e^{-\lambda y}\,dy =\frac{1-e^{-\lambda u}}{\lambda}.

The first equality follows by expressing a nonnegative payment as the integral of the indicators that it exceeds $y$. Multiplying this conditional mean by $P(X>d)=e^{-\lambda d}$ gives the stated expectation. The limit is reached precisely when $X-d\geq u$, which has probability $e^{-\lambda(d+u)}$.

Worked Example: Ordinary Deductible With a Payment Limit

A hypothetical policy covers a ground-up loss $X$, measured in thousands of dollars, where $X$ is exponential with rate $\lambda=0.1$ per thousand dollars. It has an ordinary deductible of $d=2$ thousand dollars and a maximum insurer payment of $u=8$ thousand dollars. Find the probability of a positive payment, the probability of reaching the limit, and the expected payment.

Method. Use the ordinary-deductible formulas. The limit is an insurer-payment cap, so it applies to $X-d$, not directly to $X$.

The probability of a positive payment is $P(X>2)=e^{-0.1(2)}=e^{-0.2}=0.818730753$, or 0.8187 to four decimal places.

The limit is reached when $X\geq d+u=2+8=10$ thousand dollars. Thus $P(Y^L=8)=P(X\geq10)=e^{-0.1(10)}=e^{-1}=0.367879441$, or 0.3679.

The expected payment, in thousands of dollars, is $e^{-0.2}(1-e^{-0.8})/0.1$. The intermediate values are $e^{-0.8}=0.449328964$, $1-e^{-0.8}=0.550671036$, and $0.818730753(0.550671036)=0.450851312$. Therefore

E[Y^L]=\frac{0.450851312}{0.1}=4.50851312 \quad\text{thousand dollars}.

Rounded only at the final step, the expected payment is $4,508.51 to the nearest cent. As a check, the same value is $10(e^{-0.2}-e^{-1})=10(0.818730753-0.367879441)=4.50851312$ thousand dollars.

Franchise-Deductible Payments

For a franchise deductible, the positive-payment probability is also $P(X>d)=e^{-\lambda d}$. But the payment after crossing the deductible is different: the insurer pays $X$, not $X-d$, until the limit is reached. Assume $u>d$. The probability of reaching the limit is $P(X\geq u)=e^{-\lambda u}$, and the expected payment is

E[Y^L] =\int_d^u x\lambda e^{-\lambda x}\,dx+u e^{-\lambda u} =(d+1/\lambda)e^{-\lambda d}-(1/\lambda)e^{-\lambda u}.

The integral accounts for losses $d<X<u$, which are paid in full. The final term accounts for losses $X\geq u$, all of which produce payment $u$. To evaluate the integral, an antiderivative of $x\lambda e^{-\lambda x}$ is $-(x+1/\lambda)e^{-\lambda x}$. The integral is therefore $(d+1/\lambda)e^{-\lambda d}-(u+1/\lambda)e^{-\lambda u}$; adding $u e^{-\lambda u}$ leaves the formula above.

If there is no payment limit, the same calculation over $d<X<\infty$ gives $E[X\mathbf{1}_{\{X>d\}}]=(d+1/\lambda)e^{-\lambda d}$. This is not the ordinary-deductible expectation: a franchise deductible does not subtract $d$ from a covered loss.

Worked Example: Franchise Deductible With a Limit

A hypothetical equipment policy has exponential ground-up losses measured in thousands of dollars, with rate $\lambda=0.1$ per thousand dollars. It uses a franchise deductible $d=4$ and a maximum insurer payment $u=10$. Find the probability of a positive payment, the probability of payment $10$, and the expected payment.

Method. Since $u>d$, a loss above $4$ but below $10$ is paid in full; a loss at least $10$ is capped at $10$.

The positive-payment probability is $e^{-0.1(4)}=e^{-0.4}=0.670320046$, or 0.6703. The probability of reaching the cap is $e^{-0.1(10)}=e^{-1}=0.367879441$, or 0.3679.

For the mean payment, the formula gives $(4+1/0.1)e^{-0.4}-(1/0.1)e^{-1}$. The first term is $14(0.670320046)=9.384480644$; the second is $10(0.367879441)=3.678794411$. Subtracting gives

E[Y^L]=9.384480644-3.678794411 =5.705686233 \quad\text{thousand dollars}.

The expected payment is therefore $5,705.69 to the nearest cent. It is less than the maximum payment of $10,000, as it must be. Also, the cap probability $0.3679$ is less than the positive-payment probability $0.6703$, consistent with every capped loss being a positive-payment loss.

Reading Payment Probabilities From the Coverage Definition

For an ordinary deductible, payments strictly between zero and the limit correspond to losses strictly between $d$ and $d+u$. For example, if $0<a<b<u$, then

P(a<Y^L<b) =P(d+a<X<d+b) =e^{-\lambda(d+a)}-e^{-\lambda(d+b)}.

The difference of survival probabilities is used because the event requires $X$ to exceed the lower boundary but not the upper one. The payment-limit atom has probability $e^{-\lambda(d+u)}$; the zero-payment atom has probability $1-e^{-\lambda d}$. Between zero and $u$, the payment has density $\lambda e^{-\lambda(d+y)}$.

For a franchise deductible with $u>d$, payments in the interval $(d,u)$ equal the ground-up loss itself. Thus, for $d<a<b<u$, $P(a<Y^L<b)=e^{-\lambda a}-e^{-\lambda b}$, and the limit-payment probability is $e^{-\lambda u}$. These expressions follow from the coverage transformation and the ground-up survival function; the ordinary and franchise formulas cannot be interchanged.

Worked Example: Probability of an Intermediate Ordinary Payment

A hypothetical warranty has exponential ground-up losses in thousands of dollars with rate $\lambda=0.25$ per thousand dollars. It applies an ordinary deductible of $d=2$ and a maximum payment $u=4$. Find the probability that the payment is strictly between $1$ and $3$ thousand dollars, and find the expected payment.

Method. A payment between $1$ and $3$ is below the cap, so $Y^L=X-2$ on this event. Consequently, $1<Y^L<3$ is equivalent to $3<X<5$.

The probability is $e^{-0.25(3)}-e^{-0.25(5)}=e^{-0.75}-e^{-1.25}$. Using $e^{-0.75}=0.472366553$ and $e^{-1.25}=0.286504797$, the result is $0.185861756$, or 0.1859 to four decimal places.

For the expected payment, first evaluate the ordinary deductible and limit formula:

E[Y^L] =e^{-0.5}\frac{1-e^{-1}}{0.25} =0.606530660\frac{0.632120559}{0.25} =1.533602 \quad\text{thousand dollars, approximately}.

The expected payment is $1,533.60 to the nearest cent. As a check, it is below the maximum payment of $4,000; the probability in the requested intermediate-payment range is also below the positive-payment probability $e^{-0.5}=0.6065$.

Common Exam Trap: Subtracting a Franchise Deductible

A common error is to use the ordinary-deductible formula when the policy has a franchise deductible. In the franchise example above, treating the coverage as ordinary would give $e^{-0.4}(1-e^{-1})/0.1=4.23723$ thousand dollars (approximately). That is the expected payment for an ordinary deductible of $4$ with a payment cap of $10$, not the stated franchise coverage.

The correction is to return to the payment definition: if $X>4$, a franchise policy pays $\min(X,10)$, not $\min(X-4,10)$. Its expected payment is $5.705686233$ thousand dollars. A second frequent error is to say the ordinary limit is reached at $X=u$; because the ordinary deductible is first subtracted, the correct ground-up threshold is $X=d+u$.

Key takeaway: Write the payment transformation before calculating. Under an ordinary deductible, the cap $u$ is reached at ground-up loss $d+u$, and the limited expected payment is $e^{-\lambda d}(1-e^{-\lambda u})/\lambda$. Under a franchise deductible with $u>d$, the cap is reached at $u$, and losses above the deductible are not reduced by $d$.

Check Your Understanding

For each hypothetical model, use the stated loss units and round probabilities to four decimal places and money to the nearest cent where applicable.

A loss in thousands of dollars is exponential with rate $0.2$ per thousand dollars. An ordinary deductible is $1.5$ thousand dollars, with no payment limit. Find the positive-payment probability and expected payment. (Answer: $0.7408$; $3.7041$ thousand dollars)
A loss in thousands of dollars is exponential with rate $0.1$ per thousand dollars. An ordinary deductible is $3$ and the maximum payment is $5$. Find the expected payment and the probability of reaching the limit. (Answer: approximately $2.9149$ thousand dollars; $0.4493$)
A loss in thousands of dollars is exponential with rate $0.2$ per thousand dollars. A franchise deductible is $2$ and the maximum payment is $7$. Find the probability of a positive payment, the probability of payment $7$, and the expected payment. (Answer: $0.6703$; $0.2466$; approximately $3.4593$ thousand dollars)
A loss in thousands of dollars is exponential with rate $0.25$ per thousand dollars. Under an ordinary deductible of $2$ and a maximum payment of $3$, find $P(1<Y^L<3)$ and $P(Y^L=3)$. (Answer: $0.1859$; $0.2865$)
A loss in thousands of dollars is exponential with rate $0.5$ per thousand dollars. Compare the expected payment under (i) an ordinary deductible of $2$ with maximum payment $4$, and (ii) a franchise deductible of $2$ with maximum payment $4$. Find the difference, franchise minus ordinary. (Answer: ordinary approximately $0.6362$ thousand dollars; franchise approximately $1.2008$ thousand dollars; difference approximately $0.5647$ thousand dollars)

Coverage	Positive payment occurs when	Payment reaches \(u\) when
Ordinary deductible with limit \(u\)	\(X>d\)	\(X\geq d+u\)
Franchise deductible with limit \(u>d\)	\(X>d\)	\(X\geq u\)

What You'll Learn