Expected Layer Recoveries from Limited Expected Values

Why Limited Expected Values Simplify Layer Calculations

A per-claim excess-of-loss layer pays only for the portion of an individual loss above an attachment point, up to a stated limit. Its expected recovery can be calculated directly from the severity distribution, but the payment’s piecewise definition can make a direct calculation awkward. Limited expected values provide a compact alternative: evaluate the expected loss capped at the exhaustion point, subtract the expected loss capped at the attachment point, and the difference is the expected layer payment.

This method applies to both continuous and discrete severities. It also handles probability mass exactly at the attachment or exhaustion point without requiring special adjustments. The calculations below concern the expected recovery per claim, not an aggregate annual recovery. This material is aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.

Definition: Let $X$ be a nonnegative claim severity. For $t\geq 0$, its limited expected value at $t$ is $$ L_X(t)=E[\min(X,t)]. $$ It is the expected amount of a loss retained when the loss is capped at $t$. For a per-claim layer with attachment $a\geq 0$ and limit $u>0$, the exhaustion point is $a+u$, and the per-loss payment is $$ Y^L=\min\bigl((X-a)_+,u\bigr), \qquad (z)_+=\max(z,0). $$

The attachment determines where the layer begins to pay; the limit determines the greatest amount it pays on one loss. The exhaustion point is therefore the attachment plus the limit, not the limit by itself. For a loss no greater than $a$, the recovery is zero. Between $a$ and $a+u$, the recovery rises with the loss. At or above $a+u$, the recovery is $u$.

The Limited-Expected-Value Identity

The key calculation is the difference between two limited expected values. It works for any nonnegative severity distribution for which the expectations exist, including distributions with atoms. No continuity assumption is needed.

Theorem: For a per-claim layer with attachment $a$ and limit $u$, $$ E[Y^L]=L_X(a+u)-L_X(a). $$ The first limited expected value uses the exhaustion point $a+u$; the second uses the attachment point $a$.

To see why this identity holds, compare the two capped amounts for each possible loss $x$. If $x\leq a$, both $\min(x,a+u)$ and $\min(x,a)$ equal $x$, so their difference is zero. If $a<x<a+u$, the difference is $x-a$. If $x\geq a+u$, the difference is $u$. These are exactly the three possible values of the layer payment.

\min(x,a+u)-\min(x,a)=\min\bigl((x-a)_+,u\bigr)

Taking expectations on both sides gives the theorem. This pointwise argument explains why the identity remains valid for discrete severities and for losses exactly at either treaty threshold: at $x=a$ the difference is zero, and at $x=a+u$ it is $u$.

For a nonnegative severity, the limited expected value can also be calculated from the survival function:

L_X(t)=\int_0^t S_X(s)\,ds

Indeed, for a fixed nonnegative loss $x$, the capped amount $\min(x,t)$ is the length of the portion of the interval from $0$ to $t$ lying below $x$. Equivalently, it is the integral over that interval of the indicator that $x>s$. Taking expectations of this indicator gives $P(X>s)=S_X(s)$, and integrating gives the formula. The identity is useful for continuous models; for a discrete severity, summing the probability-weighted capped losses is often simpler.

Identify the layer points.
Record the attachment $a$, limit $u$, and exhaustion point $a+u$. Keep the distinction between limit and exhaustion explicit.

Choose a way to evaluate each limited expected value.
Use $E[\min(X,t)]$, a survival-function integral, or a probability-weighted sum, depending on the severity model.

Subtract in the correct order.
Calculate $L_X(a+u)-L_X(a)$. The result is the expected recovery per claim and must lie between 0 and $u$.

Worked Applications

Worked Example: Exponential Severity With a Two-Point Layer

A hypothetical claim severity $X$ is exponential with mean $2{,}000$ dollars, so its rate is $1/2{,}000$ per dollar. A per-claim layer attaches at $a=1{,}000$ dollars and has limit $u=1{,}000$ dollars. Find the expected recovery per claim to the nearest cent. This is a severity calculation; no interest rate or mortality assumption is involved.

Method. For an exponential severity with mean $\theta$, $S_X(s)=e^{-s/\theta}$, and thus $L_X(t)=\int_0^t e^{-s/\theta}\,ds=\theta(1-e^{-t/\theta})$. Here $\theta=2{,}000$, and the exhaustion point is $a+u=1{,}000+1{,}000=2{,}000$ dollars.

Evaluate the limited expected value at exhaustion:

L_X(2{,}000)=2{,}000(1-e^{-2{,}000/2{,}000}) =2{,}000(1-e^{-1}) =1{,}264.2411\text{ dollars}.

Evaluate it at attachment:

L_X(1{,}000)=2{,}000(1-e^{-1{,}000/2{,}000}) =2{,}000(1-e^{-0.5}) =786.9387\text{ dollars}.

Subtract the attachment value from the exhaustion value:

E[Y^L]=1{,}264.2411-786.9387 =477.3024\text{ dollars}.

Equivalently, $E[Y^L]=2{,}000(e^{-0.5}-e^{-1})$. Rounded only at the final step to the nearest cent, the expected recovery is $477.30 per claim. This is less than the $1,000 layer limit, as required.

Common mistake avoided: The exhaustion point is $2,000, but the formula’s difference is already the expected payment for the layer. Dividing that difference by two would give an incorrect result.

Worked Example: Discrete Severity With Mass at Both Thresholds

A hypothetical severity has the following distribution. A per-claim layer attaches at $a=1{,}000$ dollars and has limit $u=1{,}500$ dollars. Find the expected recovery per claim. No interest rate or mortality assumption is involved.

Severity $x$ (dollars)	Probability
0	0.25
1,000	0.20
2,500	0.35
4,000	0.20

The probabilities sum to $0.25+0.20+0.35+0.20=1.00$. The exhaustion point is $a+u=1{,}000+1{,}500=2{,}500$ dollars. First calculate the limited expected value at exhaustion:

\begin{aligned} L_X(2{,}500) &=0.25\min(0,2{,}500)+0.20\min(1{,}000,2{,}500)\\ &\quad+0.35\min(2{,}500,2{,}500)+0.20\min(4{,}000,2{,}500)\\ &=0+200+875+500\\ &=1{,}575\text{ dollars}. \end{aligned}

At the attachment:

\begin{aligned} L_X(1{,}000) &=0.25\min(0,1{,}000)+0.20\min(1{,}000,1{,}000)\\ &\quad+0.35\min(2{,}500,1{,}000)+0.20\min(4{,}000,1{,}000)\\ &=0+200+350+200\\ &=750\text{ dollars}. \end{aligned}

Therefore, $E[Y^L]=1{,}575-750=825$ dollars per claim.

Check by direct payments. The recoveries at severities $0$, $1{,}000$, $2{,}500$, and $4{,}000$ are respectively $0$, $0$, $1{,}500$, and $1{,}500$ dollars. Thus the expected recovery is $0.25(0)+0.20(0)+0.35(1{,}500)+0.20(1{,}500)=525+300=825$ dollars. The two methods agree. The atom at attachment produces no payment; the atom at exhaustion produces the full layer limit.

Worked Example: Uniform Severity Using Survival Integrals

A hypothetical severity $X$ is uniform on $[0,4{,}000]$ dollars. A per-claim layer attaches at $a=1{,}000$ dollars and has limit $u=2{,}000$ dollars. Find the expected recovery per claim. No interest rate or mortality assumption is involved.

The density is $1/4{,}000$ between 0 and 4,000 dollars, and the exhaustion point is $a+u=3{,}000$ dollars. For $0\leq t\leq4{,}000$, the survival function is $S_X(s)=1-s/4{,}000$ for $0\leq s\leq t$. Therefore:

\begin{aligned} L_X(t) &=\int_0^t\left(1-\frac{s}{4{,}000}\right)\,ds\\ &=t-\frac{t^2}{8{,}000}. \end{aligned}

At exhaustion:

L_X(3{,}000)=3{,}000-\frac{3{,}000^2}{8{,}000} =3{,}000-1{,}125 =1{,}875\text{ dollars}.

At attachment:

L_X(1{,}000)=1{,}000-\frac{1{,}000^2}{8{,}000} =1{,}000-125 =875\text{ dollars}.

Thus $E[Y^L]=1{,}875-875=1{,}000$ dollars per claim. As a check, losses above $3,000 occur with probability $1{,}000/4{,}000=0.25$ and pay the full $2,000, contributing $500. Losses between $1,000 and $3,000 contribute $\frac{1}{4{,}000}\int_{1{,}000}^{3{,}000}(x-1{,}000)\,dx=500$ dollars. Their total is $1,000, matching the limited-expected-value calculation and staying below the $2,000 limit.

Checks, Conditions, and Common Traps

The limited-expected-value identity depends on the layer payment being capped excess of a single loss above its attachment. It does not by itself calculate a recovery based on the sum of multiple claims in a year. The severity $X$ should be nonnegative, and the attachment and limit should be expressed in the same monetary units as $X$. If the contract has a special feature that changes the per-loss payment from $\min((X-a)_+,u)$, verify the payment definition before applying the identity.

A quick reasonableness check is $0\leq E[Y^L]\leq u$. The recovery is also no greater than $E[X]$, since the layer payment on every loss is no greater than the loss itself. These bounds can reveal arithmetic or setup errors, but they do not replace checking the attachment and exhaustion points.

Using the limit as the exhaustion point. With attachment $a$ and limit $u$, the upper evaluation point is $a+u$, not $u$.
Reversing the subtraction. Since $a+u\geq a$, $L_X(a+u)\geq L_X(a)$; the expected recovery is the exhaustion value minus the attachment value.
Subtracting the attachment from each capped loss incorrectly. The identity uses two expected capped amounts. Do not subtract $a$ from every severity, including losses below attachment.
Ignoring discrete threshold values. The pointwise identity already handles atoms at either threshold. A loss exactly at attachment pays zero; one exactly at exhaustion pays the full limit.
Confusing per-claim and aggregate recoveries. This calculation gives the expected payment for one claim. An expected annual recovery also requires the claim count and the contract’s basis.

Key takeaway: For a per-claim layer attaching at $a$ with limit $u$, compute $L_X(a+u)-L_X(a)$. Use $a+u$ for exhaustion, retain the same severity model at both evaluation points, and check that the result lies between zero and $u$.

Check Your Understanding

For each hypothetical per-claim severity, find the requested expected recovery. Answers are given so you can check your calculation.

A severity is fixed at $2,400. A layer attaches at $900 and has a $700 limit. What is the recovery on this loss? (Answer: $700)
A severity is exponential with mean $3,000. A layer attaches at $1,500 and has a $1,500 limit. Give the expected recovery per claim to the nearest cent. (Answer: $715.95)
A discrete severity is $500 with probability 0.20, $1,800 with probability 0.50, and $4,000 with probability 0.30. A layer attaches at $1,000 and has a $2,000 limit. Find the expected recovery per claim. (Answer: $1,000)
A severity is uniform on $[0,6{,}000]$. A layer attaches at $2,000 and has a $3,000 limit. Find the expected recovery per claim. (Answer: $1,250)
A severity is exponential with mean $1,000. A layer attaches at $500 and has a $1,000 limit. Find the expected recovery per claim to the nearest cent. (Answer: $383.40)

What You'll Learn