Choosing And Normalizing The Base Limit

Why the Reference Limit Matters

An increased limits factor (ILF) expresses the expected limited loss at one policy limit relative to the expected limited loss at a selected reference limit. The reference limit is also called the base limit. It is a scaling choice: changing it changes the numerical values in the factor table, but it does not change the underlying expected losses or the ratio between any two limits.

That distinction matters in practice. A factor table is not fully specified by its limit values alone; the base limit must also be clear. A factor of 1.40 means different things if it is relative to a $100,000 base limit than if it is relative to a $250,000 base limit. Once the reference is known, however, factors can be converted exactly by dividing by the factor at the new base.

Definition: Let $L(u)$ be the expected payment per claim for ground-up loss $X$ subject to a per-claim limit $u$, with no deductible, so $L(u)=E[\min(X,u)]$. For a chosen reference limit $b$ with $L(b)>0$, the increased limits factor at limit $u$, normalized to $b$, is $$ \operatorname{ILF}_{b}(u)=\frac{L(u)}{L(b)}. $$ Thus $\operatorname{ILF}_{b}(b)=1$. The same loss basis and severity assumptions must be used in numerator and denominator.

If the quantity being compared is expected annual limited loss rather than expected payment per claim, the same definition applies provided claim frequency is the same across the limits. If annual claim frequency is $\lambda$, expected annual limited loss is $\lambda L(u)$, and the common frequency cancels in the ratio. This normalization does not by itself account for expenses, profit, or other premium components.

Choosing a Reference Limit

A useful base limit is one that is relevant to the rating or comparison task and for which the limited-loss estimate is reasonably supported. A commonly used or requested policy limit can make a table easier to interpret. A base near the middle of the limits under study can also avoid presenting most factors as either very small or very large. These are practical considerations, not mathematical requirements.

The mathematical requirements are more basic: $L(b)$ must be positive, and it must be calculated on the same basis as the other $L(u)$ values. For example, do not normalize a ground-up, no-deductible expected loss at the base limit by a value calculated after a deductible, unless the intended factor definition explicitly uses that different basis. Unsupported extrapolation to a limit far beyond the available data is not repaired by choosing it as the base.

The base is a denominator, not a new loss estimate. It does not alter claim severity, the policy terms, or the data. It only determines where the factor scale is anchored at 1. Consequently, a different base should not be selected merely to make a particular factor look more favorable.

Worked Example: Normalize Empirical Limited Losses to $250,000

A hypothetical portfolio analysis estimates the following expected limited losses per claim from uncapped ground-up loss data. Values are in thousands of dollars. The coverage has no deductible, and the same claim population and loss basis are used at every limit. Select $250,000 as the reference limit and calculate the normalized ILFs at the four listed limits.

Method. Divide each estimated limited loss by the estimate at the selected base limit. The base value is $L(250)=165$, in thousands.

\begin{aligned} \operatorname{ILF}_{250}(50)&=\frac{50}{165}=0.303030\ldots,\\ \operatorname{ILF}_{250}(100)&=\frac{87.5}{165}=0.530303\ldots,\\ \operatorname{ILF}_{250}(250)&=\frac{165}{165}=1,\\ \operatorname{ILF}_{250}(500)&=\frac{227.5}{165}=1.378787\ldots. \end{aligned}

Rounded to four decimal places, the factors are 0.3030 at $50,000, 0.5303 at $100,000, 1.0000 at $250,000, and 1.3788 at $500,000. For example, the $50,000 factor is $50{,}000/165{,}000=0.3030$, not 0.2727. The $100,000 factor is $87{,}500/165{,}000=0.5303$.

Check. The selected base has factor exactly 1 before rounding. The limited losses increase with the limit, and all three non-base factors equal their respective limited losses divided by the same $165,000 denominator. A common mistake is to use a factor from a different limit as the numerator; the calculation must use the limited loss at the limit being factored.

Converting Factors to a Different Base

Suppose a table is already normalized to a base $a$, but a comparison requires a new base $b$. The table can be converted without returning to the original loss data, as long as its factors are unrounded or sufficiently precise and all factors use the same loss basis.

\operatorname{ILF}_{b}(u) =\frac{L(u)}{L(b)} =\frac{L(u)/L(a)}{L(b)/L(a)} =\frac{\operatorname{ILF}_{a}(u)}{\operatorname{ILF}_{a}(b)}.

The denominator in the conversion is the old-base factor at the new base limit. Dividing by it makes the factor at $b$ equal to one. This is why a change of base is performed by division, not multiplication. Multiplying by the old factor at $b$ would move the table farther from the required normalization rather than anchoring it there.

Identify both bases.
Record the original base $a$ and the desired new base $b$. Keep their units and limit definitions consistent.

Find the old factor at the new base.
Read or calculate $\operatorname{ILF}_{a}(b)$. This is the divisor for every factor in the old table.

Divide each old factor by that divisor.
For each limit $u$, calculate $\operatorname{ILF}_{a}(u)/\operatorname{ILF}_{a}(b)$.

Check the new anchor.
The converted value at $u=b$ must equal 1, apart from any final rounding.

Worked Example: Rebase a Table from $100,000 to $250,000

Use the hypothetical limited-loss estimates from the preceding example. An existing table is normalized to $100,000. Convert it to a $250,000 base and verify the $50,000 and $100,000 factors. Work from the unrounded ratios where possible.

Step 1: Establish the old factors. Since $L(100)=87.5$, the original factors are:

\operatorname{ILF}_{100}(50)=\frac{50}{87.5}=0.571428\ldots,\quad \operatorname{ILF}_{100}(100)=1,\quad \operatorname{ILF}_{100}(250)=\frac{165}{87.5}=1.885714\ldots,\quad \operatorname{ILF}_{100}(500)=\frac{227.5}{87.5}=2.6.

Step 2: Use the old factor at the new base as the divisor. The new base is $250,000, so the divisor is $\operatorname{ILF}_{100}(250)=165/87.5$.

Step 3: Convert the factors. At $50,000, the converted factor is $(50/87.5)/(165/87.5)=50/165=0.3030$ to four decimal places. At $100,000, it is $1/(165/87.5)=87.5/165=0.5303$. At $250,000, it is $(165/87.5)/(165/87.5)=1.0000$. At $500,000, it is $(227.5/87.5)/(165/87.5)=227.5/165=1.3788$.

Check. The converted values match direct division of each limited loss by $L(250)=165$. The $250,000 factor is 1 as required. The method avoids a rounding discrepancy by using the underlying ratios rather than factors already rounded to four decimal places.

Why the Conversion Formula Holds

The conversion follows directly from the definition. Under one fixed severity distribution and policy basis, an ILF is a ratio of two limited expected losses. The old factor at $u$ is $L(u)/L(a)$, while the old factor at the new base $b$ is $L(b)/L(a)$. Their ratio cancels the old denominator $L(a)$, leaving the required ratio $L(u)/L(b)$.

\frac{\operatorname{ILF}_{a}(u)}{\operatorname{ILF}_{a}(b)} = \frac{L(u)/L(a)}{L(b)/L(a)} = \frac{L(u)}{L(b)} = \operatorname{ILF}_{b}(u).

This cancellation explains both the validity and the limitation of the conversion. It is valid when both old factors were calculated from the same loss basis. If limits use different deductibles, coverage definitions, claim populations, or severity assumptions, the $L(a)$ terms do not represent a common basis and a simple re-normalization may not be appropriate.

Worked Example: Change the Base Using a Published Factor Table

A hypothetical liability factor table is normalized to a $80,000 limit. Its factors at limits of $40,000, $80,000, $120,000, and $200,000 are 0.625, 1.000, 1.250, and 1.500, respectively. Convert the table to a $120,000 base. Assume all entries use the same claim and coverage basis.

Method. The old factor at the new base is $\operatorname{ILF}_{80}(120)=1.250$. Divide every old factor by 1.250:

\begin{aligned} \operatorname{ILF}_{120}(40)&=\frac{0.625}{1.250}=0.500,\\ \operatorname{ILF}_{120}(80)&=\frac{1.000}{1.250}=0.800,\\ \operatorname{ILF}_{120}(120)&=\frac{1.250}{1.250}=1.000,\\ \operatorname{ILF}_{120}(200)&=\frac{1.500}{1.250}=1.200. \end{aligned}

Check. The $120,000 entry is now 1.000, as required. For a direct interpretation, the old factor at $40,000 is 0.625 relative to $80,000, while the old factor at $120,000 is 1.250 relative to $80,000. Their ratio is $0.625/1.250=0.500$, which is the $40,000 factor relative to $120,000.

Common mistake avoided. Multiplying $0.625$ by $1.250$ gives $0.78125$, which is not the new-base factor. The denominator must be the old factor at the desired new base. The correct value is $0.625/1.250=0.500$.

Practical Checks and Exam Traps

A correctly normalized factor table has a factor of 1 at its stated base, and every factor should be traceable to the same limited-loss basis. Before accepting a converted table, check the label and arithmetic, not just whether the results look plausible.

Misidentifying the base. A factor of 1 identifies the reference limit. Do not assume that the smallest or largest listed limit is the base.
Using the wrong numerator. To calculate the factor at $u$, use $L(u)$ in the numerator. For the example above, $L(50)=50$, so $L(50)/L(250)=50/165=0.3030$.
Dividing by the wrong old factor. When converting from base $a$ to base $b$, divide by $\operatorname{ILF}_{a}(b)$, not by a factor at some other limit. In particular, the old $50,000 factor of 0.5714 is not the old $100,000 factor; the latter is 1.0000 when $100,000 is the old base.
Multiplying instead of dividing. Division cancels the old denominator and leaves the new one. Check that the converted factor at the new base equals 1.
Rounding too early. If the old factors are rounded, converting them can introduce small inconsistencies. Retain underlying limited losses or unrounded factors through the calculation and round only the final table.
Changing more than the base. Re-normalization does not make factors comparable across changed deductibles, claim populations, or coverage definitions. Those changes affect the loss basis itself.

A base-limit conversion is therefore a controlled rescaling, not a new severity analysis. This distinction is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus: candidates should be able to identify the denominator, normalize a factor table, and verify that the selected base has factor one.

Key takeaway: Choose and label a reference limit $b$, then define every factor as $L(u)/L(b)$. To change from an old base $a$ to $b$, divide each old factor by $\operatorname{ILF}_{a}(b)$. Keep the loss basis fixed and check that the new-base factor is 1.

Check Your Understanding

For each problem, use the stated loss basis and keep intermediate values unrounded.

A hypothetical portfolio has limited expected losses of $36,000 at a $60,000 limit, $54,000 at a $120,000 limit, and $72,000 at a $240,000 limit. Normalize the three factors to a $120,000 base. (Answer: 0.6667, 1.0000, and 1.3333, respectively.)
For ground-up claim severity $X$ in thousands of dollars, $P(X=10)=0.4$, $P(X=50)=0.4$, and $P(X=150)=0.2$. With no deductible, calculate limited expected losses at limits 40, 80, and 160, then normalize to 80. (Answer: $L(40)=28$, $L(80)=40$, $L(160)=54$; factors 0.7000, 1.0000, and 1.3500.)
A table based at $100,000 gives factors 0.72 at $60,000, 1.00 at $100,000, and 1.40 at $200,000. Convert it to a $200,000 base. (Answer: 0.5143 at $60,000, 0.7143 at $100,000, and 1.0000 at $200,000.)
A hypothetical empirical analysis estimates limited expected losses of $45,000 at $50,000, $75,000 at $100,000, and $120,000 at $200,000. What are the factors at $50,000 and $100,000 when the base is $200,000? Verify the base factor. (Answer: 0.3750, 0.6250, and 1.0000, respectively.)
An old table is based at $80,000 and gives factors 0.60 at $40,000, 1.00 at $80,000, and 1.80 at $240,000. Rebase it to $240,000. State the divisor and each converted factor, and identify a plausible conversion error. (Answer: Divisor $=1.80$; factors are 0.3333, 0.5556, and 1.0000. Multiplying by 1.80 instead of dividing is an error.)

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