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Exam FAM · Inflation and Loss Elimination · Tutorial 682 of 988

Estimating LER from Ground-Up Loss Data

Learn to estimate an ordinary-deductible LER from observed ground-up losses and interpret the result without assuming a parametric severity model.

Intermediate 10 min read

What You'll Learn

  • Define the empirical loss elimination ratio for a fixed ordinary deductible
  • Calculate eliminated loss and ground-up loss from individual or grouped claim data
  • Explain why the ratio of empirical means is the appropriate estimate
  • Distinguish an aggregate loss-weighted ratio from an unweighted average of claim percentages
  • Identify data and denominator conditions needed for a meaningful estimate

Estimating an Ordinary-Deductible LER From Claims

When ground-up claim amounts are available, an ordinary-deductible loss elimination ratio can be estimated directly from the observed losses. The calculation compares the amount eliminated by the deductible with the total ground-up loss in the sample. It does not require fitting a parametric severity distribution, but it does require using the claim amounts and deductible consistently.

Suppose a hypothetical insurer observes claim severities \(x_1,\ldots,x_n\), all measured in the same currency and on the same ground-up basis. For a fixed ordinary deductible \(d\), the amount eliminated from a claim of size \(x_i\) is \(\min(x_i,d)\). The insurer payment for that claim is \((x_i-d)_+\), where \(z_+=\max(z,0)\). The sample calculation uses the eliminated amounts and the original ground-up losses.

Definition: For ground-up loss \(X\) with \(0<E[X]<\infty\), the ordinary-deductible loss elimination ratio is $$ \operatorname{LER}(d)=\frac{E[\min(X,d)]}{E[X]}. $$ Given complete ground-up observations \(x_1,\ldots,x_n\), the empirical estimate is $$ \widehat{\operatorname{LER}}(d) =\frac{\frac{1}{n}\sum_{i=1}^n \min(x_i,d)} {\frac{1}{n}\sum_{i=1}^n x_i} =\frac{\sum_{i=1}^n \min(x_i,d)}{\sum_{i=1}^n x_i}, $$ provided the observed ground-up total is positive.

The numerator is the empirical mean amount eliminated per claim. The denominator is the empirical mean ground-up claim amount. Because both means use the same \(n\) observations, the factors \(1/n\) cancel. The result is a ratio of total amounts, not an average in which every claim’s percentage receives equal weight.

Why the Empirical Ratio Has This Form

The population LER is a ratio of two expectations. To estimate it from observed losses, first form the empirical distribution that assigns probability \(1/n\) to each observed claim amount. Under that distribution, the mean ground-up loss is the average of the observed losses, and the mean eliminated loss is the average of the observed amounts \(\min(x_i,d)\). Taking their ratio gives the empirical formula.

$$ \widehat{E[X]}=\frac{1}{n}\sum_{i=1}^n x_i,\qquad \widehat{E[\min(X,d)]}=\frac{1}{n}\sum_{i=1}^n\min(x_i,d), $$ $$ \frac{\widehat{E[\min(X,d)]}}{\widehat{E[X]}} =\frac{\sum_{i=1}^n\min(x_i,d)}{\sum_{i=1}^n x_i}. $$

This is also a natural plug-in estimate: replace each population expectation in the definition by its empirical counterpart. If the observations are independent and identically distributed from the target ground-up severity distribution, with finite positive mean, the sample means converge to their corresponding expectations as the sample size grows. Their ratio therefore converges to the population LER. For a finite sample, the ratio is not generally an exactly unbiased estimator.

No interest rate, mortality assumption, or parametric severity distribution is used in this empirical calculation. The data are treated as the observed sample from the target severity population. Its interpretation depends on the sample being appropriate for that population and on having complete, uncensored ground-up amounts. A sample that records only insurer payments after deductibles is not ground-up data and cannot be substituted into this calculation as if it were.

A Reliable Calculation Process

1
Confirm the data basis.
Check that each observation is a ground-up loss, that all amounts use the same currency and valuation basis, and that the deductible is expressed in that same basis.
2
Calculate eliminated loss claim by claim.
For each \(x_i\), use \(\min(x_i,d)\). A loss below the deductible is entirely eliminated; a loss at or above it contributes \(d\).
3
Sum both sets of amounts.
Add the eliminated amounts for the numerator and the original ground-up losses for the denominator. Retain claim counts when data are given in grouped form.
4
Divide and interpret.
Divide eliminated total by ground-up total. The result is the share of sample ground-up loss eliminated by the deductible; report it as a decimal or percentage as requested.

Worked Calculations

Worked Example: Individual Ground-Up Claims

A hypothetical insurer reviews five complete ground-up claims of $100, $300, $700, $900, and $1,500. Estimate the ordinary-deductible LER for a $500 deductible. No parametric severity distribution, interest assumption, or mortality assumption is specified or needed.

Method. Calculate the eliminated amount \(\min(x_i,500)\) for each observed claim, then divide the total eliminated amount by the total ground-up amount.

The eliminated amounts are $100, $300, $500, $500, and $500. Thus the eliminated total is

$$ 100+300+500+500+500=1{,}900. $$

The ground-up total is

$$ 100+300+700+900+1{,}500=3{,}500. $$

Therefore,

$$ \widehat{\operatorname{LER}}(500) =\frac{1{,}900}{3{,}500} =0.542857\ldots \approx 0.5429. $$

To check the calculation, the insurer payments are $0, $0, $200, $400, and $1,000, totaling $1,600. Eliminated loss plus insurer payment equals the ground-up total: \(1{,}900+1{,}600=3{,}500\). The estimated LER is \(0.5429\), or 54.29% when expressed as a percentage. The common mistake this method avoids is using the deductible itself as the eliminated amount even for claims smaller than the deductible.

Worked Example: Grouped Claim Amounts

A hypothetical portfolio has three ground-up claims of $200 each, two claims of $600 each, and one claim of $1,000. Estimate the LER for an ordinary deductible of $400. Treat the listed counts as the numbers of individual claims; no parametric severity model is assumed.

Method. Apply \(\min(x,400)\) to each claim amount and multiply by its frequency. The counts must be included: a listed amount with three claims contributes three times.

The ground-up total is

$$ 3(200)+2(600)+1(1{,}000)=600+1{,}200+1{,}000=2{,}800. $$

At $200, each claim contributes $200 of eliminated loss, for \(3(200)=600\). At $600, each contributes $400, for \(2(400)=800\). At $1,000, the eliminated amount is $400. The eliminated total is therefore

$$ 600+800+400=1{,}800. $$

The estimate is

$$ \widehat{\operatorname{LER}}(400) =\frac{1{,}800}{2{,}800} =\frac{9}{14} =0.642857\ldots \approx 0.6429. $$

As a check, insurer payments total \(0+2(200)+600=1{,}000\), and \(1{,}800+1{,}000=2{,}800\). The result is 64.29%. Ignoring the frequencies would treat the three claim-size categories as three equally represented observations, which is not the sample described.

Worked Example: A Claim Exactly at the Deductible

A hypothetical sample contains ground-up claims of $50, $150, $250, $450, $850, and $1,250. Estimate the LER for a $250 ordinary deductible. The observations are complete ground-up losses; no interest, mortality, or parametric distribution assumption is involved.

Method. Use \(\min(x_i,250)\) for each claim, including the claim exactly equal to $250. At that amount the deductible eliminates the entire loss, leaving no insurer payment.

The eliminated amounts are $50, $150, $250, $250, $250, and $250, so

$$ 50+150+250+250+250+250=1{,}200. $$

The ground-up total is

$$ 50+150+250+450+850+1{,}250=3{,}000. $$

Hence,

$$ \widehat{\operatorname{LER}}(250) =\frac{1{,}200}{3{,}000} =0.4000. $$

The insurer payments are $0, $0, $0, $200, $600, and $1,000, totaling $1,800. The check \(1{,}200+1{,}800=3{,}000\) confirms that eliminated and insured amounts partition the ground-up total. The estimate is 40.00%. A claim equal to the deductible contributes its full amount to eliminated loss; it is not treated as a claim above the deductible with a positive insurer payment.

Interpretation, Data Conditions, and Common Traps

The empirical LER is a loss-weighted summary. A large claim can contribute more dollars to the total loss than a small claim, even though both are individual observations. For positive losses, the same fact can be seen by writing the estimate as a weighted average of claim-level eliminated proportions:

$$ \widehat{\operatorname{LER}}(d) =\sum_{i=1}^n \left(\frac{x_i}{\sum_{j=1}^n x_j}\right) \left(\frac{\min(x_i,d)}{x_i}\right). $$

The weights \(x_i/\sum_jx_j\) are the claims’ shares of the sample ground-up total. Thus, averaging the individual fractions \(\min(x_i,d)/x_i\) with equal weights generally gives a different answer and is not the empirical LER defined by the ratio of aggregate eliminated loss to aggregate ground-up loss.

A meaningful estimate also depends on the data representing the intended severity experience. If the sample omits claims below a reporting threshold, caps large claims, or contains only payments after a deductible, then it is not a complete sample of ground-up losses. The calculation may still be arithmetically possible, but it estimates a quantity based on the altered observations, not necessarily the target ground-up LER. Likewise, amounts recorded in different currencies or valuation bases must not be combined without putting them on a consistent basis.

  • Wrong denominator: dividing by the total insurer payment estimates a different ratio. Use the ground-up loss total in the denominator.
  • Wrong claim-level average: averaging each claim’s eliminated percentage gives equal weight to claims rather than weighting by ground-up dollars.
  • Wrong eliminated amount: for an ordinary deductible, the amount eliminated is \(\min(x_i,d)\), not \(d\) for every claim.
  • Missing frequencies: when observations are grouped, multiply each amount’s contribution by its claim count.
  • Zero observed total: if every observed loss is zero, the empirical ratio has a zero denominator and is undefined.
Key takeaway: For complete ground-up claim data and an ordinary deductible \(d\), add \(\min(x_i,d)\) across claims and divide by the sum of the original losses. This ratio estimates the fraction of aggregate ground-up loss eliminated by the deductible.

Check Your Understanding

For each hypothetical data set, calculate the empirical ordinary-deductible LER from the stated ground-up losses. Unless a question says otherwise, no parametric severity distribution is assumed.

  1. A sample has ground-up claims of $80, $220, $500, and $900. Find the estimated LER for a $300 deductible. (Answer: \(0.5294\), or 52.94%.)
  2. A grouped sample has four claims of $100, three claims of $400, and two claims of $700. Find the estimated LER for a $250 deductible. (Answer: \(0.5500\), or 55.00%.)
  3. Ground-up claims are $120, $300, $480, $960, and $1,440. Find the estimated LER for a $480 deductible, treating the claim equal to the deductible correctly. (Answer: \(0.5636\), or 56.36%.)
  4. A sample has two claims of $150, three of $450, and one of $1,050. Calculate the estimated LER for a $300 deductible, using the listed claim counts. (Answer: \(0.5556\), or 55.56%.)
  5. For ground-up claims of $100, $500, and $1,500 with a $500 deductible, calculate the empirical LER. Explain why the unweighted average of the three individual eliminated proportions is not the requested estimate. (Answer: \(0.5238\), or 52.38%; the required estimate is total eliminated loss divided by total ground-up loss, not the unweighted mean of claim-level proportions.)

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