Calculating Insurer Loss from LER

From the Loss Elimination Ratio to Insurer Payment

An ordinary deductible reduces the amount an insurer pays on each covered loss. The loss elimination ratio (LER) summarizes the expected amount removed by that deductible as a proportion of the expected ground-up loss. Once the ground-up mean and LER are known on the same basis, the expected insurer payment follows directly.

The calculation is useful in both directions. An actuary may know the ground-up expected loss and LER and need the insurer’s expected payment. Alternatively, the insurer payment and one of the other quantities may be known, and the missing mean or ratio must be recovered. In each case, the main task is to identify which proportion the LER represents.

Definition: Let $X$ be a nonnegative loss per claim with finite mean $E[X]$, and let $d\geq 0$ be an ordinary deductible. The insurer’s payment per loss is $Y^L=(X-d)_+=\max(X-d,0)$. For $E[X]>0$, the loss elimination ratio is $$ \operatorname{LER}(d)=\frac{E[\min(X,d)]}{E[X]}. $$ The numerator is the expected loss amount absorbed by the deductible, and the denominator is the ground-up expected loss per claim.

The LER is not the proportion of claims that receive no payment. It measures a proportion of expected dollars. A claim can be only partly eliminated: if its ground-up amount exceeds the deductible, the deductible absorbs $d$ dollars and the insurer pays the excess.

The Core Relationship

For each possible loss amount $X$, the ground-up loss separates into the portion absorbed by the deductible and the portion paid by the insurer. This equality holds whether the loss is below or above the deductible:

X=\min(X,d)+(X-d)_+.

If $X\leq d$, the two terms on the right are $X$ and zero. If $X>d$, they are $d$ and $X-d$. Taking expectations therefore gives $E[X]=E[\min(X,d)]+E[(X-d)_+]$. Substituting the definition of LER and rearranging yields the insurer-payment formula.

\begin{aligned} E[(X-d)_+] &=E[X]-E[\min(X,d)]\\ &=E[X]\left(1-\frac{E[\min(X,d)]}{E[X]}\right)\\ &=E[X]\bigl(1-\operatorname{LER}(d)\bigr). \end{aligned}

Theorem: For a nonnegative claim severity with $0<E[X]<\infty$ and an ordinary deductible $d\geq 0$, $$ \boxed{E[Y^L]=E[(X-d)_+]=E[X]\bigl(1-\operatorname{LER}(d)\bigr).} $$ Equivalently, $E[X]=E[Y^L]/(1-\operatorname{LER}(d))$ when $\operatorname{LER}(d)<1$, and $\operatorname{LER}(d)=1-E[Y^L]/E[X]$ when $E[X]>0$.

The relationship requires the same loss definition, valuation basis, and deductible in the LER and expected payment. It is a per-claim severity relationship; it does not by itself provide an annual or portfolio total. To convert a per-claim mean to a total, information about claim counts would also be needed.

Identify the quantities.
Determine whether the given amount is the ground-up mean, the insurer’s expected payment, or the expected amount eliminated. Keep the units and price basis explicit.

Translate the LER.
The insurer retains the fraction $1-\operatorname{LER}$ of the ground-up expected loss. The LER itself is the eliminated fraction.

Calculate and check.
Use the appropriate rearrangement, then verify that expected insurer payment plus expected loss eliminated equals the ground-up mean.

Worked Calculations

Worked Example: Calculate Payment from a Ground-Up Mean and LER

A hypothetical insurer is reviewing a policy with an ordinary deductible. On the policy’s current valuation basis, the ground-up expected loss per claim is $2,500, and the LER for this deductible is 0.25. Calculate the expected insurer payment per claim and the expected amount eliminated. No parametric severity distribution is specified; assume only a nonnegative loss with the stated finite mean and LER.

Method. Use the retained fraction $1-\operatorname{LER}$ to calculate insurer payment. The LER is a fraction of expected loss eliminated, not a claim-count probability.

First calculate the retained fraction:

$$ 1-0.25=0.75. $$

The expected insurer payment per claim is:

E[Y^L]=2{,}500(0.75)=\$1{,}875.

The expected amount eliminated is the LER times the ground-up mean:

E[\min(X,d)]=2{,}500(0.25)=\$625.

Check. The two expected amounts add to the ground-up mean: $\$1{,}875+\$625=\$2{,}500$. The common mistake avoided here is multiplying the mean by the LER to find insurer payment; that would calculate the amount eliminated instead.

Worked Example: Recover the LER from Expected Payment

For a hypothetical ordinary-deductible coverage, the ground-up expected loss is $3,600 per claim and the expected insurer payment is $2,700 per claim. Find the LER and the expected loss amount absorbed by the deductible. Assume the two expected values refer to the same severity basis and coverage.

Method. First subtract the insurer payment from the ground-up mean to find the expected amount eliminated. Then divide that amount by the ground-up mean to obtain the LER.

The expected amount eliminated is:

\$3{,}600-\$2{,}700=\$900.

Therefore, the LER is:

\operatorname{LER}=\frac{\$900}{\$3{,}600}=0.25.

The retained fraction provides an independent check:

1-\operatorname{LER}=1-0.25=0.75, \qquad \$3{,}600(0.75)=\$2{,}700.

Check. The eliminated amount is 25% of $3,600, and the insurer’s $2,700 payment is the remaining 75%. A common error is to label $2{,}700/3{,}600=0.75$ as the LER. That ratio is the retained fraction; the LER is its complement, 0.25.

Worked Example: Recover the Ground-Up Mean After Trending

A hypothetical analysis starts with a historical ground-up mean of $2,000 per claim. A trend factor of 1.10 converts it to the current price basis. On that current basis, the ordinary deductible has an LER of 0.25. Calculate the current ground-up mean, the expected insurer payment, and the expected amount eliminated. Treat the supplied LER as applicable to the current severity and deductible; no parametric distribution is assumed.

Method. First convert the ground-up mean to the stated current basis. Then apply the current-basis LER and its complement to that trended mean. Applying the ratio before converting the mean would give amounts on the wrong price basis.

The current ground-up mean is:

\$2{,}000(1.10)=\$2{,}200.

The insurer retains $1-0.25=0.75$ of this expected loss. Thus the expected payment is:

\$2{,}200(0.75)=\$1{,}650.

The expected amount eliminated is:

\$2{,}200(0.25)=\$550.

Check. The current-basis amounts reconcile: $\$1{,}650+\$550=\$2{,}200$. The common mistake avoided is applying a current-basis LER to the historical mean and reporting the result as a current-dollar payment.

Conditions, Interpretation, and Exam Traps

For a positive finite ground-up mean, the LER lies between zero and one. With no deductible, the eliminated amount is zero and the LER is zero. At the other extreme, an LER of one means the expected insurer payment is zero. That boundary matters when reversing the formula: if the insurer payment is zero and the LER is one, the ground-up mean cannot be recovered by dividing by $1-\operatorname{LER}$. The equation has a zero denominator and does not identify a unique mean.

When $\operatorname{LER}<1$, the reverse calculation is valid:

E[X]=\frac{E[Y^L]}{1-\operatorname{LER}}.

For example, an insurer payment of $1,440 with an LER of 0.20 implies a ground-up mean of $\$1{,}440/0.80=\$1{,}800$, not $\$1{,}440/0.20$. The corresponding eliminated amount is $\$1{,}800(0.20)=\$360$, which checks because $\$1{,}440+\$360=\$1{,}800$.

Keep these distinctions in view when setting up a calculation:

LER versus retained fraction. The insurer’s expected payment uses $1-\operatorname{LER}$, not the LER itself.
Expected dollars versus claim counts. The LER is a ratio of expected loss amounts. It does not directly state the proportion of claims below the deductible.
Per-claim versus total loss. The formula calculates expected insurer payment per claim when $X$ is defined per claim. An annual total requires claim-count information as well.
Consistent valuation basis. Ground-up mean, LER, and insurer payment must refer to compatible loss values and coverage terms. If a mean is trended, do not silently treat a historical-basis amount as current.
Valid reversal. Recovering the ground-up mean requires $1-\operatorname{LER}>0$. If the retained fraction is zero, division cannot recover the mean.

The calculation matters because it lets an actuary move between a ground-up view of claims and an insurer-payment view without rebuilding the full severity calculation, provided the LER is already known for the relevant deductible and basis. The underlying distribution need not be specified for this conversion. However, a change in deductible or severity basis can change the applicable LER, so a ratio from one setting should not be carried into another without justification.

Key takeaway: Under an ordinary deductible, expected insurer payment equals ground-up expected loss multiplied by the fraction not eliminated: $E[Y^L]=E[X](1-\operatorname{LER})$. To recover the LER, compare the eliminated expected amount with the ground-up mean; to recover the mean, divide the insurer payment by the retained fraction.

Check Your Understanding

For each hypothetical ordinary-deductible situation, use the stated per-claim values and assume they are on a consistent valuation basis. No parametric severity distribution is required.

The ground-up expected loss is $1,500 and the LER is 0.20. Find the expected insurer payment per claim. (Answer: $1,200)
The ground-up mean is $3,200, and the expected amount eliminated by the deductible is $800. Find the LER and expected insurer payment. (Answer: LER = 0.25; expected payment = $2,400)
The expected insurer payment is $1,470 and the ground-up mean is $2,100. Find the LER and expected amount eliminated. (Answer: LER = 0.30; expected amount eliminated = $630)
The LER is 0.20 and the expected insurer payment is $1,440. Recover the ground-up mean and the expected amount eliminated. (Answer: ground-up mean = $1,800; expected amount eliminated = $360)
A historical ground-up mean is $1,500. A trend factor of 1.20 applies to the current basis, and the current-basis LER is 0.30. Find the current ground-up mean and expected insurer payment. (Answer: current ground-up mean = $1,800; expected payment = $1,260)

What You'll Learn