Worked Example of an Empirical Increased Limits Factor

From Ground-Up Claims to an Empirical ILF

An increased limits factor (ILF) compares the expected loss cost at one limit with the expected loss cost at a lower limit. An empirical estimate can be built directly from claim-level ground-up losses: calculate what each claim would cost under each limit, sum the amounts, and take their ratio. The essential operation is to cap each claim separately before summing.

For a ground-up claim amount $x_j$ and a per-occurrence limit $u$, the limited claim amount is $\min(x_j,u)$. For $n$ observed claims, the empirical limited-loss total is the sum of those capped amounts. This tutorial uses a common set of claims at both limits and assumes the losses have been placed on the same valuation and coverage basis.

Definition: For ground-up claim amounts $x_1,\ldots,x_n$, define the empirical limited-loss total at limit $u$ as $L(u)=\sum_{j=1}^{n}\min(x_j,u)$. If $L(u_0)>0$, the empirical ILF at limit $u$ relative to base limit $u_0$ is $L(u)/L(u_0)$. The same claims must be used in both totals.

Dividing each limited-loss total by the claim count would give the corresponding empirical average limited severity. The factor is unchanged, because the common divisor cancels:

\frac{L(u)/n}{L(u_0)/n}=\frac{L(u)}{L(u_0)}.

This is a claim-weighted calculation: each observed claim contributes one capped amount. It is not automatically a policy-weighted estimate. It is most directly interpreted as an estimate of a per-claim severity ratio when the observed claims are representative of the severity distribution relevant to the rating use. A different mix of policy exposures, claim counts, or coverage terms can affect the observed claim sample. Do not treat the ratio as a complete premium indication; it does not by itself include expenses, profit provisions, or other rating adjustments.

Why the Capped-Loss Ratio Is the Empirical Factor

For a nonnegative ground-up loss $X$, the limited amount at limit $u$ is $\min(X,u)$. Its expected value is the expected loss cost per claim at that limit, before any frequency or expense adjustment. The ratio of two such expected values is therefore the severity ILF.

The identity underlying the empirical calculation follows by writing a capped amount as the length of the interval from zero to the smaller of the loss and the limit. For any fixed nonnegative $x$,

\min(x,u)=\int_0^u \mathbf{1}(x>t)\,dt.

Taking expectations gives the standard limited-mean expression. For a continuous loss distribution it can be written using the survival function $S_X(t)=P(X>t)$; the integral form also holds for discrete losses, since changing the indicator at isolated points does not change the integral.

E[\min(X,u)]=\int_0^u S_X(t)\,dt.

Replacing the population survival probability at each threshold with the proportion of observed claims above that threshold yields an empirical estimate of the limited mean. Equivalently, each observed claim contributes $\min(x_j,u)$, so the estimated limited mean is $L(u)/n$. Taking the ratio at $u$ and $u_0$ cancels $n$, giving the empirical ILF definition above. This argument assumes ground-up loss amounts and a common, comparable sample; observed payments already censored by a lower policy limit cannot simply be treated as ground-up amounts.

Worked Calculations

Worked Example: Adding a Large Claim With a Separate Sublimit

A hypothetical claim sample has ground-up losses of $8, $19, $44, $73, and $160. Calculate the empirical factor at a $100 limit relative to a $40 base limit. Then add a ground-up claim of $500. For this added claim, a separate contractual sublimit caps the payable amount at $80, regardless of whether the per-occurrence limit is $40 or $100. Recalculate the factor using the actual covered amounts.

Method. Cap each original claim at each limit, sum the amounts, and divide the higher-limit total by the base-limit total. For the added claim, apply both the per-occurrence limit and its $80 sublimit. This example avoids the mistake of assuming that a ground-up amount of $500 necessarily contributes $100 at the higher limit when a separate sublimit applies.

Before adding the claim: At $100, the five covered amounts are $8, $19, $44, $73, and $\min(160,100)=100$. Thus $L(100)=8+19+44+73+100=244$. At $40, they are $8, $19, $40, $40, and $40, so $L(40)=8+19+40+40+40=147$.

The initial factor is $244/147=1.6598639\ldots$, or 1.6599 rounded to four decimal places.

After adding the claim: Its $500 ground-up loss is subject to an $80 sublimit. At the $100 per-occurrence limit, its covered amount is $\min(500,100,80)=80$. At the $40 base limit, it is $\min(500,40,80)=40$. The new totals are $L(100)=244+80=324$ and $L(40)=147+40=187$.

The revised factor is $324/187=1.7326203\ldots$, or 1.733 to three decimal places (1.7326 to four decimal places). The result is checked by using the same six claims and their covered amounts at both limits. The added claim increases the factor because its ratio of covered amounts, $80/40=2$, exceeds the original factor of about 1.6599. Without the separate sublimit, its amounts would instead be $100 and $40, and that would be a different calculation.

Worked Example: Calculating an ILF From Uncensored Ground-Up Losses

A hypothetical liability portfolio has five ground-up claims of $12, $28, $55, $95, and $240. Estimate the empirical ILF at a $75 limit relative to a $25 base limit. Assume the amounts are ground-up losses, with no separate sublimits or coinsurance.

Method. Apply $\min(x_j,u)$ to every claim at each limit, then take the ratio of the totals. This avoids the common mistake of capping only the largest claim or comparing totals formed from different claim samples.

At $75, the capped amounts are $12, $28, $55, $75, and $75. Therefore, $L(75)=12+28+55+75+75=245$.

At $25, the amounts are $12, $25, $25, $25, and $25. Therefore, $L(25)=12+25+25+25+25=112$.

The empirical ILF is $245/112=2.1875$. As a check, the average limited severities are $245/5=49$ at $75 and $112/5=22.4$ at $25; $49/22.4=2.1875$, as required.

Worked Example: Seeing the Effect of One Very Large Claim

A hypothetical sample of ground-up claims is $20, $60, $140, and $300. Calculate the empirical factors at limits of $100 and $200, each relative to a $50 base. Then remove the $300 claim and recalculate the $200-to-$50 factor. All claims have the same coverage terms.

Method. Calculate three capped totals for the full sample, then repeat the relevant totals for the reduced sample. The trap is to assume that a claim above both limits contributes its full ground-up amount; at a limit it contributes only the capped amount.

For the full sample, the amounts at $50 are $20, $50, $50, and $50, so $L(50)=20+50+50+50=170$. At $100 they are $20, $60, $100, and $100, giving $L(100)=280$. At $200 they are $20, $60, $140, and $200, giving $L(200)=420$.

The factors are $ILF(100;50)=280/170=1.6470588\ldots$, or 1.6471, and $ILF(200;50)=420/170=2.4705882\ldots$, or 2.4706, both rounded to four decimal places.

Now remove the $300 claim. The $50 total becomes $20+50+50=120$, and the $200 total becomes $20+60+140=220$. The revised factor is $220/120=1.8333333\ldots$, or 1.8333. Removing this large claim lowers the factor because, at the $200 limit, it contributed $200 while at the $50 base it contributed only $50. The computation is checked by confirming that each total includes exactly three claims after removal.

Interpreting the Result and Avoiding Errors

A large ground-up claim does not contribute its entire amount to every limited-loss total. Once its amount exceeds a limit, its contribution to that limit's total is capped. However, it can still have a substantial effect on an ILF because its contribution at the higher limit may be much larger than its contribution at the base limit.

The effect of adding a claim can be described exactly. If the original totals are $L(u)$ and $L(u_0)$, and the new claim contributes $a=\min(x,u)$ and $b=\min(x,u_0)$, the revised factor is $(L(u)+a)/(L(u_0)+b)$. When $b>0$, this is a weighted average of the original factor and the new claim's own capped-loss ratio $a/b$, with weights proportional to the old and new base-limit amounts. Thus a new claim raises the factor when $a/b$ is above the original factor, and lowers it when $a/b$ is below it.

For an uncapped ground-up claim with $x\geq u\geq u_0$, its own ratio is $u/u_0$. A separate sublimit or other coverage feature changes the actual payment function, as in the first example; it must be applied explicitly and consistently rather than silently treating a payment-limited amount as a ground-up loss.

Use a common denominator. Calculate both limits using the same claims and the same valuation and coverage basis.
Cap every claim separately. Do not cap the aggregate loss total once; in general, $\sum_j\min(x_j,u)$ is not $\min(\sum_jx_j,u)$.
Identify censoring. A claim recorded only as the payment at an existing policy limit does not reveal its ground-up amount above that limit. Do not treat the observed payment as an uncensored ground-up loss without a justified adjustment.
Know what the factor estimates. A raw claim-level ratio weights observed claims, not necessarily policies or exposure units. Consider whether the sample and coverage terms are appropriate for the intended use.

Key takeaway: An empirical ILF is the ratio of higher-limit to base-limit totals after capping each claim separately at each limit. Keep the sample and loss basis consistent, and investigate how large claims and coverage-specific caps affect the two totals.

Check Your Understanding

For each problem, use the stated claim-level loss basis and round factors to four decimal places unless another precision is specified.

A hypothetical set of ground-up claims is $10, $30, $70, and $130. Find the empirical ILF at $80 relative to $40. (Answer: 1.5833)
Three ground-up claims are $18, $42, and $90. Find the empirical ILF at $90 relative to $60. (Answer: 1.2500)
A sample has claims of $25, $80, and $150. Calculate its ILF at $100 relative to $50. Then add an uncensored ground-up claim of $300 and recalculate. (Answer: 1.6400 before; 1.7429 after)
For ground-up claims of $5, $35, $85, and $170, calculate the ILF at $100 relative to $35. Show the capped-loss totals before dividing. (Answer: $225/110=2.0455$)
A two-claim sample has ground-up losses of $40 and $80. Find the ILF at $120 relative to $40, then add a ground-up claim of $500 and recalculate. State why the added claim changes the factor. (Answer: 1.5000 before; 2.0000 after; its capped-loss ratio is $120/40=3$, above the original factor.)

What You'll Learn