Exponential Severity Layer Calculations

Why Exponential Severity Is Useful for Layer Calculations

An excess-of-loss treaty pays only when an individual claim exceeds an attachment point. The expected payment depends both on how often claims penetrate the layer and on how much severity remains above the attachment. When claim severity is exponential, its survival function and memoryless property make these calculations especially direct.

This tutorial considers hypothetical per-claim treaties. There is no interest discounting or mortality assumption: claim amounts are evaluated at their stated nominal values, and the only random quantity modeled here is claim severity. The calculations are aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.

Definition: Let claim severity $X$ have an exponential distribution with rate parameter $\lambda>0$. Then, for $x\geq 0$, $$ f_X(x)=\lambda e^{-\lambda x},\qquad S_X(x)=P(X>x)=e^{-\lambda x},\qquad E[X]=\frac{1}{\lambda}. $$ For a per-claim excess-of-loss layer with attachment $d\geq 0$ and payment limit (layer width) $u>0$, the ceded payment per loss is $$ Y^L=\min\bigl((X-d)_+,u\bigr),\qquad (z)_+=\max(z,0). $$ Thus the layer starts at $d$, can pay at most $u$, and is exhausted at $d+u$. An unlimited layer has no finite payment limit.

The distinction between attachment and width matters. A claim at or below $d$ produces no recovery. A claim between $d$ and $d+u$ produces the amount above $d$, while a claim at or above $d+u$ produces the full layer payment $u$. At the exhaustion point, the layer does not pay the part of the claim above $d+u$.

Expected Recovery for an Unlimited Layer

For an unlimited layer, the recovery per claim is $Y^L=(X-d)_+$. The probability that a claim penetrates the attachment is $P(X>d)=e^{-\lambda d}$. Conditional on penetration, the expected amount above the attachment is $1/\lambda$. Consequently, the expected recovery is the product of the penetration probability and the mean excess severity.

E[(X-d)_+]=\frac{e^{-\lambda d}}{\lambda}.

The conditional-mean argument follows from the exponential memoryless property. For $t\geq 0$,

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}.

Thus, given that a claim exceeds $d$, the excess $X-d$ has the same exponential distribution as $X$, with mean $1/\lambda$. Multiplying this conditional mean by $P(X>d)$ gives the unlimited-layer expectation.

Worked Example: Unlimited Recovery Above an Attachment

A hypothetical insurer models individual claim severity as exponential with rate $\lambda=0.02$ per dollar, so $E[X]=1/0.02=\$50$. An unlimited per-claim layer attaches at $d=\$50$. Find its expected recovery per claim. There is no discounting or mortality assumption.

Method. Use the unlimited-layer formula. The probability a claim exceeds the attachment is $e^{-\lambda d}=e^{-(0.02)(50)}=e^{-1}=0.3678794$.

The expected excess amount, conditional on penetration, is $1/\lambda=1/0.02=\$50$. Therefore, $$ E[Y^L]=\frac{e^{-\lambda d}}{\lambda} =\frac{e^{-1}}{0.02} =\frac{0.3678794}{0.02} =\$18.39397. $$

Rounded to the nearest cent, the expected recovery is \$18.39 per claim. As a check, it is below the conditional excess mean of \$50 because only about 36.7879% of claims penetrate the attachment. A common mistake is to report \$50 as the expected recovery; that omits the probability of no payment. The calculation here includes that probability.

Expected Recovery for a Finite Layer

For a finite layer, the recovery cannot exceed $u$. A useful general identity for a nonnegative random variable $W$ is $E[W]=\int_0^\infty P(W>t)\,dt$, when the integral exists. Applying the identity to a recovery capped at $u$ gives an integral only from $0$ to $u$. For each $t$ in that interval, the recovery exceeds $t$ exactly when $X>d+t$.

Result: Under the exponential severity model, the expected recovery per claim for a finite layer with attachment $d$ and width $u$ is $$ E[Y^L] =\int_0^u P(X>d+t)\,dt =\int_0^u e^{-\lambda(d+t)}\,dt =\frac{e^{-\lambda d}-e^{-\lambda(d+u)}}{\lambda} =\frac{e^{-\lambda d}(1-e^{-\lambda u})}{\lambda}. $$

The integral provides a direct justification: the layer's expected payment is the area under the probability that the recovery exceeds each possible payment amount $t$. The upper limit is $u$ because a finite layer cannot pay more than its width. Equivalently, a claim must exceed $d+t$ for the layer payment to exceed $t$. Integrating the exponential survival probability over the payment range yields the formula.

This result is also the difference between two unlimited stop-loss expectations: $E[(X-d)_+]-E[(X-(d+u))_+]$. The first includes all severity above the attachment; the second removes the part above the layer's exhaustion point. Subtracting leaves the expected payment within the layer.

Worked Example: A Finite Layer With a Nonzero Attachment

A hypothetical portfolio has exponential individual claim severity with rate $\lambda=0.01$ per dollar, giving mean severity $1/\lambda=\$100$. Its per-claim treaty has attachment $d=\$40$ and width $u=\$60$, so it exhausts at $\$100$. Find expected recovery per claim. No interest discounting or mortality assumption is used.

Method. Apply the finite-layer formula. The two survival probabilities at the attachment and exhaustion points are $e^{-(0.01)(40)}=e^{-0.4}=0.6703200$ and $e^{-(0.01)(100)}=e^{-1}=0.3678794$.

Subtract the probabilities and divide by the rate: $$ E[Y^L] =\frac{e^{-0.4}-e^{-1}}{0.01} =\frac{0.6703200-0.3678794}{0.01} =\frac{0.3024406}{0.01} =\$30.24406. $$

Rounded to the nearest cent, expected recovery is \$30.24 per claim. It is less than the \$60 maximum layer payment, as it must be. A common mistake is to treat \$100 as the layer width because that is the exhaustion point. The width is $100-40=\$60$; the correct formula uses $u=60$, not $u=100$.

Adjacent Layers and a Consistency Check

When a layer is split into adjacent sections, the expected payments from those sections should add to the expected payment over their combined range. This provides a useful check on both the layer definitions and the arithmetic. For a layer from $d$ to $d+u$, the formula uses the survival probabilities at those two endpoints; adjacent layers share an endpoint, so the middle survival terms cancel when the expectations are added.

\frac{e^{-\lambda d}-e^{-\lambda(d+u_1)}}{\lambda} + \frac{e^{-\lambda(d+u_1)}-e^{-\lambda(d+u_1+u_2)}}{\lambda} = \frac{e^{-\lambda d}-e^{-\lambda(d+u_1+u_2)}}{\lambda}.

Worked Example: Checking Two Adjacent Layers

For hypothetical exponential claim severity with rate $\lambda=0.01$ per dollar and mean $\$100$, calculate the expected payment in a first layer from $\$0$ to $\$50$ and a second layer from $\$50$ to $\$150$. Then check their sum against a single layer from $\$0$ to $\$150$. There is no discounting or mortality assumption.

First layer. Its attachment is $d=0$ and its width is $u=50$. Since $e^0=1$ and $e^{-0.5}=0.6065307$, $$ E[Y^L_1] =\frac{1-e^{-0.5}}{0.01} =\frac{1-0.6065307}{0.01} =\$39.34693. $$

Second layer. Its attachment is $d=50$ and its width is $u=100$, with exhaustion at $\$150$. Since $e^{-1.5}=0.2231302$, $$ E[Y^L_2] =\frac{e^{-0.5}-e^{-1.5}}{0.01} =\frac{0.6065307-0.2231302}{0.01} =\$38.34005. $$

The sum is $\$39.34693+\$38.34005=\$77.68698$. Directly calculating the combined layer gives $$ \frac{1-e^{-1.5}}{0.01} =\frac{1-0.2231302}{0.01} =\$77.68698. $$

The two methods agree, so the layer boundaries and calculations are consistent. Rounded to the nearest cent, the first layer has expected payment $\$39.35$, the second $\$38.34$, and the combined payment $\$77.69$. The method avoids a frequent allocation error: the second layer begins at \$50, not at zero, and therefore its expected payment must use $e^{-0.5}$ as its attachment survival probability.

Interpretation, Assumptions, and Exam Traps

The exponential formulas are per claim. They describe the expected payment conditional on modeling one claim severity as $X$; they do not by themselves specify how many claims occur. If a model also specifies a claim count $N$ with finite mean and claim severities having the stated distribution, linearity of expectation gives expected total ceded payments $E[N]E[Y^L]$ under the usual frequency-severity setup. That multiplication is an aggregate expectation, not a change to the per-claim layer formula.

The model assumes nonnegative severity with the exponential density and survival function stated above. It is appropriate to use these closed forms only when that assumption is specified or justified. For a different severity distribution, the general tail integral still applies when the expectation exists, but the exponential evaluations do not. A finite layer's maximum payment is $u$, so its expected recovery must lie between $0$ and $u$; an unlimited layer's expected recovery is nonnegative and no greater than $E[X]$.

Mixing up attachment and width. A layer attaching at $d$ with width $u$ exhausts at $d+u$, not at $u$.
Ignoring penetration probability. The conditional mean excess above $d$ is $1/\lambda$, but the unconditional expected recovery also accounts for $P(X>d)$.
Using the unlimited formula for a capped layer. If a layer has finite width, subtract the expected excess above $d+u$, or integrate only to $u$.
Interpreting the exponential parameter as the mean. Here $\lambda$ is the rate, so the mean is $1/\lambda$. If the problem gives the mean, take its reciprocal to obtain $\lambda$.
Using the formula for an annual aggregate treaty. These results apply to a layer on each individual claim; they do not model a cap applied to the total of all claims in a year.

As a final check, identify the layer endpoints before substituting values: $d$ is the attachment, $d+u$ is the exhaustion point, and the survival probabilities are evaluated at those endpoints. This simple check catches many errors before any arithmetic is done.

Check Your Understanding

For each hypothetical problem, assume no discounting or mortality. Round monetary answers to the nearest cent.

Claim severity is exponential with mean $\$250$. Find the expected per-claim recovery for an unlimited layer attaching at $\$100$. (Answer: $\$167.58$)
Claim severity is exponential with rate $\lambda=0.02$ per dollar. A finite layer attaches at $\$25$ and has width $\$75$. Find its expected recovery per claim. (Answer: $\$23.56$)
Claim severity is exponential with rate $\lambda=0.01$ per dollar. Find expected recovery for a layer attaching at $\$100$ with width $\$100$; state the exhaustion point as part of your setup. (Answer: Exhaustion at $\$200$; expected recovery $\$23.25$)
A hypothetical portfolio has a Poisson claim count with mean $2.5$ and independent exponential claim severities with mean $\$80$. A per-claim layer attaches at $\$40$ and has width $\$80$. Find the expected total ceded payment. (Answer: $\$76.68$)
Exponential claim severity has rate $\lambda=0.01$ per dollar. Find the attachment $d$ for which an unlimited layer has expected recovery $\$20$ per claim. (Answer: $d=\$160.94$, approximately)

What You'll Learn