Portfolio LER as a Loss-Weighted Average

Why Portfolio LER Depends on Loss, Not Just Policy Count

A portfolio can contain risk classes with very different expected losses. A class may account for many policies but only a modest share of the portfolio’s ground-up losses; another may contain fewer policies but contribute much more expected loss. Since a loss elimination ratio measures the eliminated portion of ground-up loss, combining class ratios requires weighting by expected ground-up loss, not simply by the number or proportion of policies in each class.

This distinction matters whenever coverage terms or severity distributions vary across classes. The correct portfolio result follows from adding the classes’ expected losses and their expected eliminated amounts separately, then taking their ratio.

Definition: For a risk in class $j$, let $X_j$ be the ground-up loss and $Y_j$ the insurer’s payment, under the coverage being analysed. Assume $0 \leq Y_j \leq X_j$, and that the expectations are finite. The eliminated amount is $X_j-Y_j$. If $E[X_j]>0$, the class loss elimination ratio is $$ \operatorname{LER}_j=\frac{E[X_j-Y_j]}{E[X_j]} =1-\frac{E[Y_j]}{E[X_j]}. $$ For a portfolio with class probabilities $p_j$, where the classes are exhaustive and mutually exclusive, the portfolio LER is the expected eliminated loss divided by expected ground-up loss for a randomly selected exposure.

The class probability $p_j$ describes the share of exposures in class $j$, when the portfolio is represented by a randomly selected exposure. It is not, in general, the correct weight to apply directly to the class LERs. The expected loss per exposure also matters.

Deriving the Loss-Weighted Average

Write $\mu_j=E[X_j]$ for the class’s expected ground-up loss per exposure. The expected eliminated amount per exposure in that class is $\mu_j\operatorname{LER}_j$, by the definition of the class LER. Conditioning on class and adding across the mutually exclusive classes gives the portfolio result.

\begin{aligned} \operatorname{LER}_{\mathrm{portfolio}} &=\frac{\sum_j p_j E[X_j-Y_j]}{\sum_j p_j E[X_j]}\\ &=\frac{\sum_j p_j\mu_j\operatorname{LER}_j}{\sum_j p_j\mu_j}\\ &=\sum_j \alpha_j\operatorname{LER}_j, \qquad \alpha_j=\frac{p_j\mu_j}{\sum_k p_k\mu_k}. \end{aligned}

The first equality follows by partitioning expected loss and expected eliminated loss according to class. The second substitutes $E[X_j-Y_j]=\mu_j\operatorname{LER}_j$, which is just the class LER definition rearranged. The final equality divides each class’s expected-loss contribution by total portfolio expected loss. The weights satisfy $\alpha_j\geq 0$ and $\sum_j\alpha_j=1$, so the portfolio LER is an expected-loss-weighted average of the class LERs.

Theorem: For classes with positive expected ground-up loss, the portfolio weight for class $j$ is its share of total expected ground-up loss: $$ \alpha_j=\frac{\text{class }j\text{ expected ground-up loss}}{\text{total expected ground-up loss}}. $$ Equivalently, if there are $n_j$ exposures in class $j$, each with the same class-level expectations, then $\alpha_j=n_j\mu_j/\sum_k n_k\mu_k$. A class with zero expected ground-up loss contributes no expected loss and may be omitted from the ratio.

This argument uses additivity of expectation and does not require losses from different exposures to be independent. The essential requirements are that the classes form a valid partition, that the same portfolio basis is used in the numerator and denominator, and that the expected losses being combined are finite. If class expected losses are quoted per policy year, for example, combine them with policy counts for that same period.

Worked Examples

Worked Example: Two Classes With Different Expected Losses

A hypothetical insurer has a portfolio in which 80% of exposures belong to Class A and 20% belong to Class B. Class A has expected ground-up loss of $1,000 per exposure and class LER of 20%. Class B has expected ground-up loss of $4,000 per exposure and class LER of 5%. Calculate the portfolio LER. Use the expected-loss-weighting method, and compare it with weighting by exposure share.

Method. Calculate expected ground-up loss and expected eliminated loss across the exposure mix, then divide. The class expected eliminated amounts are:

\begin{aligned} E[X_A-Y_A]&=\$1{,}000(0.20)=\$200,\\ E[X_B-Y_B]&=\$4{,}000(0.05)=\$200. \end{aligned}

The portfolio expected ground-up loss per exposure is $0.80(\$1{,}000)+0.20(\$4{,}000)=\$800+\$800=\$1{,}600$. The expected eliminated loss per exposure is $0.80(\$200)+0.20(\$200)=\$160+\$40=\$200$. Therefore,

\operatorname{LER}_{\mathrm{portfolio}} =\frac{\$200}{\$1{,}600}=0.125=12.5\%.

As a check, the classes contribute $800 each to expected ground-up loss. Their loss weights are therefore $800/1{,}600=0.50$ each, giving $0.50(20\%)+0.50(5\%)=12.5\%$. Weighting by exposure share instead would give $0.80(20\%)+0.20(5\%)=17\%$, which is not the portfolio LER. That incorrect calculation gives too much influence to Class A’s ratio because it ignores the much higher expected loss per Class B exposure.

Worked Example: Calculate Class LERs Under Uniform Severities

A hypothetical portfolio has 80 exposures in Class A and 20 in Class B. In each class, ground-up loss $X$ for one exposure follows a uniform distribution from zero to $b$. Class A has $b=\$2{,}000$, Class B has $b=\$4{,}000$, and both classes have a per-loss deductible of $1,000 with no limit. Under this coverage, insurer payment is $Y=(X-d)^+$, where $d=\$1{,}000$. Find the portfolio LER.

Method. For a uniform loss on $[0,b]$, the ground-up mean is $E[X]=b/2$. Since $d<b$, the eliminated amount is $\min(X,d)$. Its expectation can be found by adding the expected loss below the deductible to the deductible applied to losses above it:

E[\min(X,d)] =\frac{1}{b}\int_0^d x\,dx+d\frac{b-d}{b} =\frac{d^2}{2b}+\frac{d(b-d)}{b}.

For Class A, $b=\$2{,}000$ and $d=\$1{,}000$. The expected ground-up loss is $\$2{,}000/2=\$1{,}000$. The expected eliminated loss is $\$1{,}000^2/(2\times \$2{,}000)+\$1{,}000(\$2{,}000-\$1{,}000)/\$2{,}000=\$250+\$500=\$750$. Thus $\operatorname{LER}_A=\$750/\$1{,}000=75\%$.

For Class B, $b=\$4{,}000$. The expected ground-up loss is $\$4{,}000/2=\$2{,}000$. The expected eliminated loss is $\$1{,}000^2/(2\times \$4{,}000)+\$1{,}000(\$4{,}000-\$1{,}000)/\$4{,}000=\$125+\$750=\$875$. Thus $\operatorname{LER}_B=\$875/\$2{,}000=43.75\%$.

Now multiply each per-exposure expectation by its class count. Class A contributes $80(\$1{,}000)=\$80{,}000$ of ground-up loss and $80(\$750)=\$60{,}000$ of eliminated loss. Class B contributes $20(\$2{,}000)=\$40{,}000$ of ground-up loss and $20(\$875)=\$17{,}500$ of eliminated loss. Total expected ground-up loss is $120,000, and total expected eliminated loss is $77,500. Therefore,

\operatorname{LER}_{\mathrm{portfolio}} =\frac{\$77{,}500}{\$120{,}000} =0.645833\ldots \approx 64.58\%.

The loss weights are $80/120=2/3$ for Class A and $40/120=1/3$ for Class B. The weighted result, $(2/3)(75\%)+(1/3)(43.75\%)=64.58\%$ to two decimal places, agrees with the direct ratio. The class proportions by count, 80% and 20%, are not the loss weights.

Worked Example: Combining Three Classes From Per-Exposure Results

A hypothetical property portfolio contains 120 policies in Class A, 50 in Class B, and 30 in Class C. Their expected ground-up losses per policy period are respectively $500, $1,600, and $3,000. The corresponding class LERs under their applicable coverage terms are 30%, 10%, and 4%. Calculate the portfolio LER and identify each class’s loss weight.

Method. First calculate each class’s total expected ground-up loss. Class A contributes $120(\$500)=\$60{,}000$. Class B contributes $50(\$1{,}600)=\$80{,}000$. Class C contributes $30(\$3{,}000)=\$90{,}000$. The total is $\$60{,}000+\$80{,}000+\$90{,}000=\$230{,}000$.

The loss weights are $60/230=0.260869\ldots$ for Class A, $80/230=0.347826\ldots$ for Class B, and $90/230=0.391304\ldots$ for Class C. They sum to 1. The expected eliminated amounts by class are $\$60{,}000(0.30)=\$18{,}000$, $\$80{,}000(0.10)=\$8{,}000$, and $\$90{,}000(0.04)=\$3{,}600$. Total expected eliminated loss is $\$18{,}000+\$8{,}000+\$3{,}600=\$29{,}600$. Thus,

\operatorname{LER}_{\mathrm{portfolio}} =\frac{\$29{,}600}{\$230{,}000} =0.128695\ldots \approx 12.87\%.

As a check, applying the three loss weights to the class ratios gives the same result: $(60/230)(30\%)+(80/230)(10\%)+(90/230)(4\%)=12.87\%$ to two decimal places. The number of policies alone would not reveal these weights: Class C has only 30 policies, but it supplies the largest share of expected ground-up loss.

Interpretation and Common Exam Traps

A portfolio LER is bounded by the smallest and largest class LERs when all loss weights are nonnegative and sum to one. This follows because it is a weighted average. That property is a useful reasonableness check: if a calculated portfolio ratio falls outside the range of the class ratios, revisit the weighting or arithmetic.

Weighting by policy share. Policy counts determine exposure shares, but class expected loss per exposure can differ. Multiply each class’s per-exposure expected loss by its exposure count before finding its portfolio weight.
Averaging class ratios without weights. An ordinary average is justified only when the classes have equal expected ground-up loss contributions.
Mixing units or periods. Use consistent expected-loss measures, such as per policy period for every class, and combine them with counts for those same policy periods.
Using different totals in the two parts of the ratio. The portfolio numerator must include each class’s expected eliminated loss, and the denominator must include each class’s corresponding ground-up loss.

For the conventional indemnity setting used here, the eliminated amount is ground-up loss minus insurer payment. If a problem supplies class LERs directly, those ratios already summarize the class-level coverage effect; the remaining task is to weight them by expected ground-up loss. If instead it supplies severity distributions and coverage terms, calculate each class’s expected eliminated and ground-up losses first.

Key takeaway: For each class, multiply its class LER by its expected ground-up loss contribution. Add those eliminated-loss contributions, then divide by total expected ground-up loss. The resulting portfolio LER is a ground-up expected-loss-weighted average, not generally a policy-count-weighted average.

Check Your Understanding

For each hypothetical portfolio, calculate the portfolio LER using expected ground-up loss weights. Round percentages to two decimal places where needed.

A portfolio has equal numbers of policies in two classes. Their expected ground-up losses per policy are $800 and $2,400, and their class LERs are 25% and 10%, respectively. What is the portfolio LER? (Answer: 13.75%)
A portfolio has 40 policies in Class A and 60 in Class B. Expected ground-up loss per policy is $1,000 in A and $500 in B; the class LERs are 40% and 20%. Find the portfolio LER. (Answer: 31.43%)
A randomly selected exposure belongs to Class A with probability 0.70 and Class B with probability 0.30. The conditional expected ground-up losses are $600 and $3,000, with class LERs 50% and 10%, respectively. Find the portfolio LER. (Answer: 22.73%)
A hypothetical portfolio has 25 policies in Class A, 25 in Class B, and 50 in Class C. Expected ground-up loss per policy is $1,200, $2,400, and $600, respectively; class LERs are 20%, 15%, and 50%. Calculate the portfolio LER. (Answer: 25.00%)
Three classes contribute $20,000, $60,000, and $20,000 of expected ground-up loss to a portfolio. Their LERs are 20%, an unknown value, and 40%, respectively. If the portfolio LER is 25%, determine the unknown class LER. (Answer: 21.67%)

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