Empirical Increased Limits Factors

Why Estimate Increased Limits Factors from Data?

An increased limits factor (ILF) compares the expected insurer payment per claim at a selected limit with the expected payment per claim at a basic limit. When a suitable severity distribution has been specified, the factor can be calculated from that model. An empirical estimate instead uses observed ground-up claim severities directly: cap each severity at each limit, aggregate the capped amounts, and take their ratio.

This approach is useful when the data provide a credible picture of the relevant claim severity distribution but a parametric model is not being used. It is also transparent: each claim’s contribution to the basic-limit and selected-limit amounts can be inspected. The key is to use the same claim population in both parts of the ratio and to distinguish ground-up losses from amounts already censored by a policy limit.

Definition: Let $X_j$ be the ground-up severity of claim $j$, for $j=1,\ldots,n$, and let $B$ be the basic limit and $L$ a selected limit, with $0 < B \leq L$. The empirical increased limits factor is $$ \widehat{\mathrm{ILF}}(B,L) = \frac{\sum_{j=1}^{n}\min(X_j,L)} {\sum_{j=1}^{n}\min(X_j,B)}. $$ Each capped amount is the insurer’s payment per claim under a limit, assuming no deductible, coinsurance, or other payment adjustment. The denominator must be positive.

The factor is dimensionless. For example, a factor of 1.40 means that the aggregate capped severity at the selected limit is 1.40 times the aggregate capped severity at the basic limit for the sample. It does not mean that every claim costs 40% more at the selected limit. A factor is at least 1 when the selected limit is no lower than the basic limit, because $\min(X_j,L)\geq\min(X_j,B)$ for every claim.

What the Empirical Ratio Estimates

For a claim severity $X$, the limited payment at limit $u$ is $\min(X,u)$. Under a common severity distribution for claims in the population, the theoretical ILF is the expected limited payment at $L$ divided by the expected limited payment at $B$:

\mathrm{ILF}(B,L) = \frac{E[\min(X,L)]}{E[\min(X,B)]}.

The empirical formula replaces each expectation with the sample average. The sample averages are $\frac{1}{n}\sum_{j=1}^{n}\min(X_j,L)$ and $\frac{1}{n}\sum_{j=1}^{n}\min(X_j,B)$. Their common factor $1/n$ cancels, leaving the ratio of sums. This explains why the calculation uses the aggregate capped amounts rather than an unweighted average of claim-by-claim ratios.

\frac{\frac{1}{n}\sum_{j=1}^{n}\min(X_j,L)} {\frac{1}{n}\sum_{j=1}^{n}\min(X_j,B)} = \frac{\sum_{j=1}^{n}\min(X_j,L)} {\sum_{j=1}^{n}\min(X_j,B)}.

The empirical ratio estimates the population ratio when the observed claims are representative of the claim population to which the factor will be applied. A parametric severity distribution is not assumed. The data should be on a consistent basis, such as the same coverage, claim definition, and valuation basis. If claims have different weights representing multiple otherwise similar claims, each capped amount must be multiplied by its weight before summing.

Check the data basis.
Confirm that the severities are ground-up amounts and that the claims represent the population for which the factor is intended.

Apply each limit claim by claim.
For every severity, calculate $\min(X_j,B)$ and $\min(X_j,L)$. Do not cap the aggregate loss once at the limit.

Sum and divide.
Add the basic-limit payments, add the selected-limit payments, and divide the latter by the former.

Check the result.
When $L\geq B$, the selected-limit sum cannot be below the basic-limit sum. Check that the factor is at least 1 and that the units and claim set match.

Worked Calculations

In the examples below, dollar amounts are shown in thousands unless stated otherwise. These are hypothetical portfolios. No interest or mortality assumption is involved: the calculations use observed severities without discounting or survival contingencies. They are empirical rather than parametric, so no severity distribution or distribution parameters are assumed; the stated limits define the payment caps.

Worked Example: Calculating a Factor from Individual Claims

A hypothetical liability portfolio has ground-up claim severities of 40, 80, 140, 260, and 500 thousand dollars. Estimate the factor for a selected limit of 250 thousand dollars relative to a basic limit of 100 thousand dollars. Name the method and check the result.

Method. Use the empirical ratio of aggregate limited severities. For each claim, cap the severity separately at each limit.

At the 100-thousand-dollar basic limit, the capped amounts are:

$\min(40,100)=40$, $\min(80,100)=80$, $\min(140,100)=100$, $\min(260,100)=100$, and $\min(500,100)=100$.

The basic-limit total is $40+80+100+100+100=420$ thousand dollars.

At the 250-thousand-dollar selected limit, the capped amounts are:

$\min(40,250)=40$, $\min(80,250)=80$, $\min(140,250)=140$, $\min(260,250)=250$, and $\min(500,250)=250$.

The selected-limit total is $40+80+140+250+250=760$ thousand dollars. Therefore,

\widehat{\mathrm{ILF}}(100,250) = \frac{760}{420} = 1.809523\ldots

Rounded to four decimal places, the empirical factor is 1.8095. The selected-limit total exceeds the basic-limit total, as it must when $L\geq B$. This check confirms the direction of the ratio, though not the representativeness of the data.

Common mistake avoided. Averaging the five claim-level ratios would give $(1+1+1.4+2.5+2.5)/5=1.68$, which is not the empirical ratio of sample-average limited payments. The correct calculation is the ratio of sums, 760/420, because claims contribute different amounts to the basic-limit total.

Worked Example: Comparing Factors Across Selected Limits

A hypothetical portfolio has ground-up severities of 25, 75, 120, 180, and 360 thousand dollars. Calculate empirical factors relative to a basic limit of 100 thousand dollars for selected limits of 150, 250, and 400 thousand dollars. State whether the pattern is consistent with the data.

Method. Use the same five claims at every limit. First calculate the basic-limit denominator:

$\min(X_j,100)$ gives 25, 75, 100, 100, and 100, for a total of $25+75+100+100+100=400$ thousand dollars.

For the 150-thousand-dollar limit, the capped amounts are 25, 75, 120, 150, and 150. Their total is $25+75+120+150+150=520$, so the factor is $520/400=\mathbf{1.3000}$.

For the 250-thousand-dollar limit, the capped amounts are 25, 75, 120, 180, and 250. Their total is $25+75+120+180+250=650$, so the factor is $650/400=\mathbf{1.6250}$.

For the 400-thousand-dollar limit, the capped amounts are 25, 75, 120, 180, and 360. Their total is $25+75+120+180+360=760$, so the factor is $760/400=\mathbf{1.9000}$.

The factors increase from 1.3000 to 1.6250 to 1.9000 as the selected limit rises. This agrees with the claim-level fact that raising a limit cannot reduce any capped payment. The result does not establish that the same pattern will hold in a different portfolio; it describes this sample.

Common mistake avoided. The denominator must remain the total at the same basic limit for every comparison. Changing the claim set or the denominator between selected limits would make the factors incomparable.

Worked Example: Applying Claim Weights Correctly

A hypothetical data extract represents three severity groups: a 50-thousand-dollar claim with weight 2, a 100-thousand-dollar claim with weight 3, and a 240-thousand-dollar claim with weight 1. Estimate the factor for a 200-thousand-dollar selected limit relative to an 80-thousand-dollar basic limit.

Method. Treat each weight as the number of represented claims and multiply each claim’s capped payment by that weight before summing.

At the 80-thousand-dollar basic limit, the capped amounts before weighting are 50, 80, and 80. The weighted contributions are $2(50)=100$, $3(80)=240$, and $1(80)=80$, for a basic-limit total of $100+240+80=420$ thousand dollars.

At the 200-thousand-dollar selected limit, the capped amounts before weighting are 50, 100, and 200. The weighted contributions are $2(50)=100$, $3(100)=300$, and $1(200)=200$, for a selected-limit total of $100+300+200=600$ thousand dollars.

Thus,

\widehat{\mathrm{ILF}}(80,200) = \frac{600}{420} = 1.428571\ldots

Rounded to four decimal places, the factor is 1.4286. As a check, the selected-limit total is greater than the basic-limit total, and the ratio is greater than 1.

Common mistake avoided. Ignoring the weights would calculate a factor for three records treated as three equally represented claims. The supplied weights say that the first and second severities represent more claims, so they must affect both totals.

Data Limitations and Common Traps

The most important limitation is censoring. The formula requires ground-up severities, or enough information to determine what each ground-up severity would be. If the recorded amounts are already capped at the basic limit, all claims above that limit appear identical in the data. For a claim recorded as exactly $B$, the true severity could be only $B$, or it could be far larger. Those possibilities produce the same basic-limit payment but different payments at $L>B$. Therefore, basic-limit-capped payments alone do not identify an empirical factor for a higher limit.

A second trap is to cap an aggregate amount rather than each claim. If the losses are 60 and 160 and the limit is 100, the correct total payment is $\min(60,100)+\min(160,100)=160$, not $\min(60+160,100)=100$. Policy limits apply claim by claim in this setting.

A third issue is inconsistent data. Mixing claim records from different coverage definitions, including different expense components in some severities, or using different claim populations in the numerator and denominator can distort the comparison. The ratio is meaningful only when both totals refer to the same underlying set of claims and a consistent severity basis. If the basic-limit total is zero, the factor is undefined.

For observed claims with unequal weights, weighted sums are appropriate when the weights represent claim counts or other justified representation weights. They are not a general remedy for a biased or unrepresentative sample. Likewise, an empirical estimate can be unstable when only a few large claims drive much of the selected-limit total; the arithmetic may be exact while the estimate remains uncertain.

Key takeaway: An empirical ILF is the selected-limit total of per-claim capped ground-up severities divided by the corresponding basic-limit total, using the same claims and appropriate weights. Correct capping and a consistent, uncensored data basis are essential to interpreting the result.

Check Your Understanding

For each hypothetical problem, use the empirical ratio of aggregate capped severities. Amounts are in thousands of dollars unless stated otherwise.

A portfolio has ground-up severities 30, 90, 160, and 310. Find the factor for a 200 limit relative to a 100 basic limit. (Answer: 1.5000)
Three severity records are 40 with weight 2, 120 with weight 1, and 210 with weight 2. Find the factor for a 250 limit relative to a 100 basic limit. (Answer: 1.6316)
Ground-up severities are 60, 140, and 300. Calculate factors at selected limits 200 and 400, each relative to a 100 basic limit. (Answer: 1.5385 and 1.9231, respectively)
A data file contains basic-limit payments of 70, 100, and 100, with a basic limit of 100. Can these values alone determine the empirical factor at a 200 limit? Explain briefly. (Answer: No; the two payments of 100 may represent ground-up losses above the basic limit, and their uncapped severities are unknown)
Ground-up severities are 20, 80, 150, and 400. Calculate the factor for a 300 limit relative to a 100 basic limit, showing the two capped totals. (Answer: Basic total 300; selected total 550; factor 1.8333)

What You'll Learn