LER for Mixtures of Severity Classes

Why Severity Mixtures Need Careful Weighting

A portfolio may contain several types of risks with different claim severity patterns. For example, a hypothetical insurer might cover both small commercial vehicles and heavy construction vehicles. Even if the deductible is the same for both, their expected losses eliminated by that deductible can differ substantially. The portfolio loss elimination ratio (LER) must reflect both the severity distribution within each class and the proportion of claims belonging to each class.

The key is to combine expected dollar amounts before forming the ratio. A class with higher average severity contributes more to the portfolio’s ground-up loss total than a class with the same claim share but lower average severity. Simply averaging the class LERs by claim proportions generally does not give the portfolio LER.

Definition: Suppose a claim is drawn from one of $k$ severity classes. Let $w_j$ be the probability that a randomly selected claim belongs to class $j$, where $w_j \geq 0$ and $\sum_{j=1}^k w_j=1$. Conditional on class $j$, let the ground-up loss be $X_j$, with finite, positive mean $E[X_j]$. For an ordinary deductible $d \geq 0$, the amount eliminated from a loss is $\min(X_j,d)$, and the insurer payment is $(X_j-d)_+$, where $z_+=\max(z,0)$. The mixture LER is the expected eliminated amount divided by the expected ground-up loss.

The weights $w_j$ are claim proportions, not automatically the proportions of policies, exposures, or insured values in each class. If claim frequencies differ by class, the claim proportions must account for those differences. This distinction can change the answer even when the class severity distributions are unchanged.

The Mixture Formula

For class $j$, define its expected eliminated loss as $E[\min(X_j,d)]$. The mixture’s expected ground-up loss is the weighted mean of the class means, and its expected eliminated loss is the weighted mean of the class-specific eliminated amounts. Therefore,

E[X]=\sum_{j=1}^k w_j E[X_j], \qquad E[\min(X,d)]=\sum_{j=1}^k w_j E[\min(X_j,d)].

Provided $0<E[X]<\infty$, the portfolio LER is

\operatorname{LER}(d) = \frac{\displaystyle\sum_{j=1}^k w_j E[\min(X_j,d)]} {\displaystyle\sum_{j=1}^k w_j E[X_j]}.

For each class, the ground-up loss splits into the amount eliminated and the insurer payment: $X_j=\min(X_j,d)+(X_j-d)_+$. Taking expectations and combining across classes gives the equivalent insurer-payment calculation:

\operatorname{LER}(d) = 1- \frac{\displaystyle\sum_{j=1}^k w_j E[(X_j-d)_+]} {\displaystyle\sum_{j=1}^k w_j E[X_j]}.

This alternative is useful when expected insurer payments are easier to calculate than expected eliminated amounts. Both expressions use the same mixture weights and the same ground-up denominator.

Why the Survival-Function Integral Works

For any nonnegative loss $X_j$, the eliminated amount can be written as the length of the portion of the interval from 0 to $d$ that lies below the loss:

\min(X_j,d)=\int_0^d \mathbf{1}_{\{X_j>x\}}\,dx.

Taking expectations and interchanging the expectation and integral gives

E[\min(X_j,d)] = \int_0^d P(X_j>x)\,dx = \int_0^d S_{X_j}(x)\,dx.

The mixture survival function is $S_X(x)=\sum_{j=1}^k w_j S_{X_j}(x)$, because a claim exceeds $x$ if it belongs to a class and that class’s loss exceeds $x$. Thus,

E[\min(X,d)] = \int_0^d S_X(x)\,dx = \sum_{j=1}^k w_j\int_0^d S_{X_j}(x)\,dx.

This establishes why class-specific eliminated amounts can be combined with the class claim proportions. The result holds for continuous, discrete, or mixed severities; it does not require a density. The survival-integral form requires nonnegative losses and a nonnegative deductible. The LER ratio additionally requires a finite, positive mixture mean.

Worked Example: Two Exponential Severity Classes

A hypothetical portfolio has two claim classes. A proportion $0.70$ of claims belongs to class A, whose losses are exponential with mean $\$1{,}000$; the remaining $0.30$ belongs to class B, whose losses are exponential with mean $\$4{,}000$. Calculate the LER for a $\$500$ ordinary deductible. Give the final ratio to four decimal places.

Method. For an exponential loss with mean $\theta$, the survival function is $S_X(x)=e^{-x/\theta}$. Integrating it from 0 to $d$ gives $E[\min(X,d)]=\theta(1-e^{-d/\theta})$. Apply this separately to each class, then form the ratio of mixture expected eliminated loss to mixture expected ground-up loss.

Class A eliminated amount.

E[\min(X_A,500)] = 1{,}000(1-e^{-500/1{,}000}) = 1{,}000(1-e^{-0.5}) \approx \$393.4693.

Class B eliminated amount.

E[\min(X_B,500)] = 4{,}000(1-e^{-500/4{,}000}) = 4{,}000(1-e^{-0.125}) \approx \$470.0124.

The mixture expected eliminated loss is

0.70(393.4693)+0.30(470.0124) = 275.4285+141.0037 = \$416.4322.

The mixture ground-up expected loss is

0.70(1{,}000)+0.30(4{,}000) = 700+1{,}200 = \$1{,}900.

Therefore,

\operatorname{LER}(500)=\frac{416.4322}{1{,}900}\approx 0.2192.

Check. The expected insurer payments by class are $1{,}000-393.4693=\$606.5307$ and $4{,}000-470.0124=\$3{,}529.9876$. Their mixture mean is $0.70(606.5307)+0.30(3{,}529.9876)=\$1{,}483.5678$. The eliminated and insured amounts sum to $\$1{,}900$, and $1-1{,}483.5678/1{,}900=0.2192$. The common mistake avoided here is treating the class proportions as weights for the class LERs alone; the calculation instead weights expected dollars in numerator and denominator.

Worked Examples with Other Severity Forms

Worked Example: A Discrete Mixture with Losses at the Deductible

A hypothetical equipment portfolio has $60\%$ of claims in class A and $40\%$ in class B. In class A, a loss is $\$200$ or $\$800$, each with probability $0.50$. In class B, a loss is $\$400$ with probability $0.25$ and $\$1{,}200$ with probability $0.75$. Find the LER for a $\$500$ deductible.

Method. For each possible loss, the eliminated amount is the smaller of the loss and the deductible. In particular, a loss above the deductible contributes exactly $\$500$ to eliminated loss, not its full ground-up amount.

For class A, the ground-up mean is

E[X_A]=0.50(200)+0.50(800)=100+400=\$500.

Its expected eliminated amount is

E[\min(X_A,500)] = 0.50(200)+0.50(500) = 100+250 = \$350.

For class B, the ground-up mean and expected eliminated amount are

E[X_B]=0.25(400)+0.75(1{,}200)=100+900=\$1{,}000, $$ $$ E[\min(X_B,500)] = 0.25(400)+0.75(500) = 100+375 = \$475.

The mixture ground-up mean is $0.60(500)+0.40(1{,}000)=300+400=\$700$. The mixture eliminated amount is $0.60(350)+0.40(475)=210+190=\$400$. Hence

\operatorname{LER}(500)=\frac{400}{700}=\frac{4}{7}\approx 0.5714.

Check. The expected insurer payment is $\$700-\$400=\$300$, so $1-300/700=0.5714$. The loss of $\$1{,}200$ in class B contributes $\$500$, not $\$1{,}200$, to the eliminated amount.

Worked Example: Converting Exposure Shares to Claim Shares

A hypothetical portfolio has 1,000 exposures in class A and 500 in class B. During the period, the expected claim frequency is $0.10$ per exposure in A and $0.20$ per exposure in B. Class A severity is uniform on $[0,2{,}000]$; class B severity is exponential with mean $\$3{,}000$. Calculate the portfolio LER for a $\$1{,}000$ deductible.

Method. First calculate expected claims in each class to obtain the claim proportions. Then calculate each class’s mean severity and expected eliminated amount. The claim shares are not the exposure shares because the claim frequencies differ.

Expected claims are $1{,}000(0.10)=100$ in class A and $500(0.20)=100$ in class B, for 200 expected claims total. Thus $w_A=100/200=0.50$ and $w_B=100/200=0.50$.

For a uniform loss on $[0,b]$, the mean is $b/2$. When $0\leq d\leq b$, integrating the survival function $1-x/b$ from 0 to $d$ gives $E[\min(X,d)]=d-d^2/(2b)$. Therefore, for class A,

E[X_A]=\frac{2{,}000}{2}=\$1{,}000, \qquad E[\min(X_A,1{,}000)] = 1{,}000-\frac{1{,}000^2}{2(2{,}000)} = \$750.

For the exponential class, with mean $\theta=3{,}000$,

E[X_B]=\$3{,}000, \qquad E[\min(X_B,1{,}000)] = 3{,}000(1-e^{-1{,}000/3{,}000}) \approx \$850.4061.

The mixture expected ground-up loss per claim is $0.50(1{,}000)+0.50(3{,}000)=\$2{,}000$. The mixture expected eliminated loss per claim is $0.50(750)+0.50(850.4061)=\$800.2031$. Consequently,

\operatorname{LER}(1{,}000) = \frac{800.2031}{2{,}000} \approx 0.4001.

Check. Over 200 expected claims, the expected ground-up amount is $100(1{,}000)+100(3{,}000)=\$400{,}000$, and the expected eliminated amount is $100(750)+100(850.4061)=\$160{,}040.61$. Their ratio is $160{,}040.61/400{,}000\approx0.4001$. Weighting by exposure shares instead would use $2/3$ and $1/3$, which is wrong because the claim rates differ.

Common Mistake: Averaging Class LERs Directly

A frequent error is to calculate each class LER and then take an ordinary claim-share-weighted average. That gives each class’s percentage the same kind of weight as a dollar amount, even though the classes can have very different ground-up means. The correct portfolio calculation combines expected eliminated dollars and expected ground-up dollars separately, then divides.

For example, in the exponential example above, the class LERs are $393.4693/1{,}000\approx0.3935$ and $470.0124/4{,}000\approx0.1175$. Their claim-share average is $0.70(0.3935)+0.30(0.1175)\approx0.3107$, which is not the portfolio LER of $0.2192$. The difference occurs because class B has a much larger mean loss, so its ground-up dollars have a substantial effect on the portfolio denominator.

A reliable exam method is to calculate the mixture’s expected eliminated loss and ground-up loss explicitly. Keep the class means and class eliminated amounts visible until the final division, and verify that the expected insurer payment plus the expected eliminated amount equals the expected ground-up amount.

Key takeaway: For a mixture of severity classes, use claim proportions to combine class-specific expected dollar losses. The portfolio LER is the resulting expected eliminated loss divided by the resulting expected ground-up loss; it is not generally the claim-share average of the class LERs. This calculation is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, identify the claim weights, combine the class-specific expected losses, and form the LER.

A portfolio has $40\%$ exponential claims with mean $\$500$ and $60\%$ exponential claims with mean $\$2{,}000$. Find the LER for a $\$500$ deductible. (Answer: $0.2799$)
Class A has $75\%$ of claims and is uniform on $[0,1{,}000]$; class B has $25\%$ and is uniform on $[0,3{,}000]$. Find the LER for a $\$600$ deductible. (Answer: $0.6000$)
Class A has $20\%$ of claims and every loss is $\$1{,}000$. Class B has $80\%$ of claims, with loss $\$500$ or $\$2{,}500$, each with probability $0.50$. Find the LER for an $\$800$ deductible. (Answer: $0.4857$)
A portfolio has 600 class A exposures with expected claim frequency $0.10$ and 400 class B exposures with expected claim frequency $0.30$. Class A severity is exponential with mean $\$1{,}000$; class B severity is exponential with mean $\$2{,}000$. Find the LER for a $\$1{,}000$ deductible. (Answer: $0.4412$)
A claim belongs to class A with probability $0.25$ and has a certain loss of $\$300$; otherwise it belongs to class B and has a certain loss of $\$900$. Find the LER for a $\$600$ deductible. (Answer: $0.7000$)

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