Normalizing an Increased Limits Factor Table

Why Normalize an Increased Limits Factor Table?

An increased limits factor (ILF) expresses the expected claim cost at a selected policy limit relative to the expected claim cost at a basic limit. A table may be supplied relative to one basic limit while a rating plan uses another. Normalizing the table puts its entries on the basis the plan needs, without changing the relative claim costs represented by the table.

The essential operation is a division: divide every table entry by the entry for the limit selected as the new basic limit. This makes that selected entry exactly one. It also preserves the relative relationship between every pair of limits, an important check when translating the table.

Definition: Let $X$ be the ground-up severity of a claim, and let $L$ be a per-claim policy limit. The insurer’s payment for that loss is $Y^L_L=\min(X,L)$. On a consistent severity and coverage basis, the expected limited loss at limit $L$ is $m(L)=E[\min(X,L)]$. If $B$ is the basic limit, the increased limits factor at limit $L$, relative to $B$, is $$ \operatorname{ILF}_B(L)=\frac{m(L)}{m(B)}. $$ Thus, $\operatorname{ILF}_B(B)=1$, provided $m(B)>0$.

The formula compares expected payments for the same underlying loss distribution and policy conditions. In practical use, factors should be compared only when their underlying basis is consistent—for example, the same coverage, claim population, and treatment of deductibles. If claim frequency is the same at every limit, it cancels from the ratio of expected claim costs, so the ratio of expected limited severities also describes the ratio of expected loss costs per exposure. This tutorial concerns changing the table’s reference limit, not estimating a new severity distribution.

The Normalization Rule

Suppose a table has entries $T(L)$, and the table is currently expressed relative to a limit $B$. If a different limit $C$ is to become the basic limit, the converted factor at any listed limit $L$ is $$ \operatorname{ILF}_C(L)=\frac{\operatorname{ILF}_B(L)}{\operatorname{ILF}_B(C)}. $$ The denominator is the old-table factor at the new basic limit. The same rule applies if the supplied table has a common arbitrary scale rather than being explicitly normalized at $B$: divide every entry by the entry at $C$.

The operation has two immediate checks. First, the entry at $C$ must equal one. Second, the ratio between any two converted entries must equal the ratio between the corresponding original entries. The table changes its reference point, not its underlying pattern.

Why Division Gives the Correct Factors

The normalization formula follows directly from the definition. The original factor at $L$ is $m(L)/m(B)$, while the original factor at the new basic limit $C$ is $m(C)/m(B)$. Taking their ratio removes the old denominator:

\frac{\operatorname{ILF}_B(L)}{\operatorname{ILF}_B(C)} = \frac{m(L)/m(B)}{m(C)/m(B)} = \frac{m(L)}{m(C)} = \operatorname{ILF}_C(L).

This also explains why the original table’s overall scale does not matter. If each entry is multiplied by the same positive constant, that constant appears in both numerator and denominator and cancels. Normalization therefore works for factors reported on a common basis even when that basis is not the one ultimately required.

Identify the target basic limit.
Find the table entry for the limit that the rating plan intends to set equal to one.

Use that entry as the divisor.
Divide every listed factor by the target-limit factor, using unrounded values if available.

Verify the result.
Check that the target limit is one and that ratios between limits are preserved. Round only for presentation.

Worked Example: Selecting a New Basic Limit

A hypothetical property program has an ILF table relative to a $100,000 basic limit. Its factors are 0.72 at $50,000, 1.00 at $100,000, 1.46 at $250,000, 1.78 at $500,000, and 2.05 at $1,000,000. The rating plan is to use $250,000 as its basic limit. Normalize the table and check the result.

Method. Divide each original factor by the factor at $250,000, which is 1.46. For example, the new $50,000 factor is $0.72/1.46=0.4931507\ldots$. Applying the same calculation to every row gives the following table, rounded to four decimal places:

Limit	Original factor	Calculation	Factor relative to $250,000
$50,000	0.72	0.72 / 1.46	0.4932
$100,000	1.00	1.00 / 1.46	0.6849
$250,000	1.46	1.46 / 1.46	1.0000
$500,000	1.78	1.78 / 1.46	1.2192
$1,000,000	2.05	2.05 / 1.46	1.4041

Check. The $250,000 factor is exactly one before rounding. The converted $1,000,000-to-$250,000 factor is $2.05/1.46=1.4041096\ldots$, matching the table. The calculation avoids treating the old $100,000 factor of one as if it had to remain one after the basic limit changes.

Rebasing a Table When the New Basic Limit Is Not the Old One

A factor below one is possible after rebasing. If the selected basic limit is higher than a listed limit, the expected limited loss at that lower limit will ordinarily be less than at the basic limit. The converted factor is then less than one because it is a ratio to the higher-limit expected loss. It is not an error merely because a factor is below one.

Worked Example: Rebase to a Limit Above the Original Basic Limit

A hypothetical equipment-liability table is normalized to a $100,000 basic limit. It lists factors of 1.00 at $100,000, 1.32 at $200,000, 1.72 at $400,000, and 2.05 at $800,000. A revised plan selects $200,000 as its basic limit. Find the revised factors, rounded to four decimal places.

Method. The factor at the new basic limit is 1.32, so divide each factor by 1.32:

\begin{aligned} \operatorname{ILF}_{200}(100) &= 1.00/1.32 = 0.7575758\ldots,\\ \operatorname{ILF}_{200}(200) &= 1.32/1.32 = 1.0000000,\\ \operatorname{ILF}_{200}(400) &= 1.72/1.32 = 1.3030303\ldots,\\ \operatorname{ILF}_{200}(800) &= 2.05/1.32 = 1.5530303\ldots. \end{aligned}

Here the limit labels in the calculations are in thousands of dollars. The revised factors are 0.7576, 1.0000, 1.3030, and 1.5530, respectively. As a check, the ratio of the revised $800,000 factor to the revised $100,000 factor is $1.5530303\ldots/0.7575758\ldots=2.05$, the same as the original ratio $2.05/1.00$. The below-one factor at $100,000 is appropriate because that limit is below the new basic limit.

Applying a Normalized Table to Expected Claim Cost

Once the reference limit is changed, the factors can be used with a base expected claim cost defined at that new basic limit. The cost and factor must refer to the same coverage and loss basis. For example, if an expected claim cost per exposure is given at the new basic limit, multiplying it by the factor for another limit produces the expected claim cost per exposure at that other limit, subject to the rating assumptions behind the table.

Worked Example: Translating a Base Claim Cost

A hypothetical insurer has an expected claim cost of $840 per exposure at a $300,000 limit. A supplied table, originally relative to a $50,000 limit, gives factors of 0.64 at $50,000, 0.91 at $100,000, 1.42 at $300,000, 1.73 at $600,000, and 1.96 at $1,200,000. Calculate the costs at $50,000 and $1,200,000, using the $300,000 limit as the new basic limit. Round money to the nearest cent.

Step 1: Rebase the factors. Divide each factor by 1.42, the supplied factor at $300,000. The converted factor at $50,000 is $0.64/1.42=0.4507042\ldots$; the factor at $1,200,000 is $1.96/1.42=1.3802817\ldots$. The $300,000 factor is $1.42/1.42=1$.

Step 2: Apply the converted factors. At $50,000, the expected claim cost is $840(0.64/1.42)=378.5915\ldots$, or $378.59 per exposure. At $1,200,000, it is $840(1.96/1.42)=1,159.4366\ldots$, or $1,159.44 per exposure.

Check. The $1,200,000 cost divided by the $300,000 base cost is $1,159.4366\ldots/840=1.96/1.42$, the converted ILF. The cost at the basic limit remains $840 because its factor is one. These are expected claim costs, not necessarily final rates: other rating components would have to be handled consistently.

Common Mistake: Multiplying by the New-Basis Entry

A frequent error is to multiply each old factor by the factor at the new basic limit. That does not make the new basic-limit entry one. In the first example, this incorrect operation would make the $250,000 factor $1.46\times1.46=2.1316$, rather than one. The correct calculation is $1.46/1.46=1$, and every other entry must use the same divisor.

A second source of error is to assume that a table’s old reference limit remains its basic limit after conversion. A factor of one identifies the basic limit only for the basis on which that table is currently normalized. After a change of basis, the factor of one belongs at the newly selected limit.

Key takeaway: To normalize an ILF table to a selected basic limit, divide every table entry by the entry at that limit. The new basic-limit factor is one, relative factors are preserved, and a factor below one can be valid for a limit below the selected basic limit.

Check Your Understanding

For each hypothetical table, rebase the factors as requested. Round factors to four decimal places and money to the nearest cent where applicable.

A table relative to $100,000 gives factors 1.00 at $100,000, 1.40 at $250,000, and 1.75 at $500,000. What are the factors relative to $250,000? (Answer: 0.7143, 1.0000, 1.2500.)
A table gives factors 0.80 at $50,000, 1.00 at $100,000, and 1.60 at $400,000. Rebase it to $400,000. (Answer: 0.5000, 0.6250, 1.0000.)
A table relative to $200,000 gives factors 1.00 at $200,000, 1.25 at $300,000, and 1.75 at $600,000. Find the $300,000 and $600,000 factors relative to $300,000. (Answer: 1.0000 and 1.4000.)
A hypothetical expected claim cost is $720 per exposure at a $250,000 basic limit. The table relative to $100,000 has factors 0.70 at $50,000, 1.00 at $100,000, 1.50 at $250,000, and 1.80 at $500,000. What is the expected cost at $500,000 after rebasing? (Answer: $864.00 per exposure.)
A table has factors 0.90 at $100,000, 1.35 at $300,000, and 1.80 at $600,000. If $300,000 is selected as the new basic limit, what is the factor at $100,000, and why is its value not an error? (Answer: 0.6667; it is below one because the $100,000 limit is below the new basic limit.)

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