Comparing Premiums Across Policy Limits

Why Premium Comparisons Need a Consistent Basis

A premium comparison across policy limits is useful only when the compared amounts have a clear basis. If the insured, exposure, coverage terms, and other rating inputs are unchanged, the premium difference can be attributed to the limit change under the rating method being used. If those inputs also change, the total premium difference combines multiple effects.

An increased limits factor (ILF) expresses the relative premium level for one limit compared with a designated base limit. When comparing two limits, use the ratio of their ILFs—not automatically the higher-limit ILF by itself. That distinction matters whenever the premium at the starting limit is not at the ILF base limit.

Definition: Let $L_0$ be the base limit, $L$ a policy limit, and $\operatorname{ILF}(L)$ the ILF at $L$, normalized so that $\operatorname{ILF}(L_0)=1$. If the same limit-responsive premium basis and all non-limit rating inputs apply at both limits, then $$ P(L)=P(L_0)\frac{\operatorname{ILF}(L)}{\operatorname{ILF}(L_0)} =P(L_0)\operatorname{ILF}(L). $$ For two limits $L_1$ and $L_2$, the corresponding comparison is $P(L_2)=P(L_1)\operatorname{ILF}(L_2)/\operatorname{ILF}(L_1)$.

The formula assumes the ILFs are applicable to the coverage and loss basis being rated, and that the compared premium amount is the component to which the ILF is intended to apply. A fixed charge, a different deductible, or a coverage change may not be scaled by the same ILF. Follow the applicable rating procedure and separate such items rather than treating the entire billed premium as limit-responsive.

How to Compare Two Limits

A reliable comparison begins by identifying the limit at which the quoted or indicated premium applies. Then use the ratio of the ILF at the desired limit to the ILF at that starting limit. This preserves the proper starting point even when the base-limit ILF is not 1.00.

Fix the comparison basis.
Confirm that exposure, coverage terms, deductible, classification, and other rating inputs are held constant, or list which ones change.

Identify the starting premium and its limit.
Do not assume that a premium is at the ILF base limit unless the problem states or establishes that fact.

Apply the ILF ratio.
Multiply the starting limit-responsive premium by the destination ILF divided by the starting-limit ILF.

Measure the change from the correct denominator.
For a percentage increase from the starting premium, divide the dollar increase by that starting premium.

The percentage change from $L_1$ to $L_2$, under the same-basis assumptions, is

\frac{P(L_2)-P(L_1)}{P(L_1)} = \frac{\operatorname{ILF}(L_2)}{\operatorname{ILF}(L_1)}-1.

This percentage does not depend on the dollar size of the starting premium if the entire compared amount scales by the ILF ratio. The dollar change does depend on that starting premium.

Why the ILF Ratio Scales the Premium

The ratio follows from the definition of an ILF and a proportionality assumption. Let $M(L)$ denote the expected loss measure used to construct the ILF at limit $L$, and suppose the limit-responsive premium on a fixed rating basis is proportional to $M(L)$. Under that assumption, write the premium as $P(L)=cM(L)$, where $c$ is unchanged between the limits.

By definition, $\operatorname{ILF}(L)=M(L)/M(L_0)$. Therefore,

\frac{P(L)}{P(L_0)} = \frac{cM(L)}{cM(L_0)} = \frac{M(L)}{M(L_0)} = \frac{\operatorname{ILF}(L)}{\operatorname{ILF}(L_0)}.

The constant $c$ cancels because the comparison holds the other rating basis fixed. If the premium includes amounts that do not vary with the expected loss measure, or if $c$ changes between the options, that cancellation does not apply to the total premium. In that case, scale only the appropriate component and add the other components according to the rating procedure.

Worked Example: Comparing Two Higher Limits

A hypothetical liability account has a limit-responsive premium of $2,400 at a $100,000 per-occurrence limit. The applicable normalized ILFs are 1.00 at $100,000, 1.18 at $250,000, and 1.42 at $500,000. Assume that exposure, coverage terms, and all other rating inputs are unchanged. Find the premium at each higher limit, then compare the $500,000 option with the $250,000 option.

Method. Apply the destination-to-starting ILF ratio. The comparison avoids the mistake of treating the $500,000 ILF as the factor for a premium that already reflects the $250,000 limit.

At $250,000, the starting premium is at the ILF base limit, so

P(250{,}000)=\$2{,}400\left(\frac{1.18}{1.00}\right)=\$2{,}832.00.

At $500,000, relative to the $100,000 base limit,

P(500{,}000)=\$2{,}400\left(\frac{1.42}{1.00}\right)=\$3{,}408.00.

The dollar increase from $250,000 to $500,000 is $\$3{,}408.00-\$2{,}832.00=\$576.00$. The percentage increase uses the $250,000 premium as its denominator:

\frac{\$576.00}{\$2{,}832.00}=0.2033898\ldots=20.34\%.

Check. The same percentage follows from the ILFs: $1.42/1.18-1=0.2033898\ldots$. Rounded only at the final step, the premiums are stated to the cent and the percentage to two decimal places.

Portfolio Comparisons Require Premium Weighting

For a group of accounts, calculate the higher-limit premium account by account when their base premiums or ILF ratios differ, then add the resulting premiums. Applying a simple average ILF to the combined base premium is generally incorrect unless that average is weighted appropriately.

For account $j$, let $P_{j,1}$ be its starting premium and $r_j=\operatorname{ILF}_j(L_2)/\operatorname{ILF}_j(L_1)$ its limit ratio. The combined premium at the destination limit is $\sum_j P_{j,1}r_j$. The portfolio percentage increase is the total dollar increase divided by the combined starting premium. It is a starting-premium-weighted result, not necessarily the arithmetic mean of the account percentage increases.

Worked Example: Combining Accounts With Different ILF Ratios

Two hypothetical accounts have limit-responsive premiums of $1,200 and $2,800 at their respective starting limits. For the selected higher limits, their ILF ratios are 1.25 and 1.15. All other rating inputs are unchanged. Find the combined higher-limit premium and the percentage increase over the combined starting premium.

Method. Apply each account’s own ratio before summing. This avoids using an unweighted average factor when the accounts have different starting premiums.

For Account 1, the higher-limit premium is $\$1{,}200(1.25)=\$1{,}500.00$. Its increase is $\$1{,}500.00-\$1{,}200.00=\$300.00$.

For Account 2, the higher-limit premium is $\$2{,}800(1.15)=\$3{,}220.00$. Its increase is $\$3{,}220.00-\$2{,}800.00=\$420.00$.

The combined starting premium is $\$1{,}200+\$2{,}800=\$4{,}000.00$, and the combined higher-limit premium is $\$1{,}500+\$3{,}220=\$4{,}720.00$. Thus the total increase is $\$4{,}720-\$4{,}000=\$720.00$, and

\frac{\$720.00}{\$4{,}000.00}=0.18=18.00\%.

Check. The account increases are $300.00 and $420.00, which sum to $720.00. The combined premium is $4,720.00. The result is rounded to the cent for dollar amounts and to two decimal places for the percentage.

Separating Limit Changes From Other Rating Changes

A practical comparison may involve more than a limit change. Exposure may grow, a classification or experience factor may change, or the coverage terms may be revised. The observed premium difference then cannot be described as the ILF effect alone. A useful analysis constructs a counterfactual: calculate the premium at the new limit while holding the old exposure and other rating inputs constant. The difference from the old premium is the isolated limit effect on that chosen basis.

For example, if the limit-responsive premium is represented as exposure multiplied by a base rate, an other-rating factor, and an ILF, then the total change can be expressed as the product of the corresponding factor changes. A sequential dollar bridge can also be prepared, but its attribution of intermediate dollar changes depends on the order in which factors are changed. State the order when reporting such a bridge.

Worked Example: Isolating the Limit Effect in a Renewal

A hypothetical account has 100 exposure units, a base rate of $40 per unit, an other-rating factor of 1.10, and an ILF of 1.20 at its current limit. At renewal, exposure is 120 units, the other-rating factor is 1.05, and the ILF at the selected higher limit is 1.35. The base rate remains $40. Calculate the old premium, the premium at the higher limit on the old basis, the renewal premium, and a sequential bridge that changes limit first, exposure second, and the other-rating factor third.

Method. Hold all non-limit inputs at their old values to isolate the limit effect, then apply the other changes in the stated sequence. This avoids attributing the entire renewal change to the limit.

The old premium is

100(\$40)(1.10)(1.20)=\$5{,}280.00.

Changing only the ILF from 1.20 to 1.35 gives the limit-only premium on the old basis:

100(\$40)(1.10)(1.35)=\$5{,}940.00.

The isolated limit increase is $\$5{,}940-\$5{,}280=\$660.00$. Its percentage relative to the old premium is $\$660/\$5{,}280=0.125=12.50\%$, which also equals $1.35/1.20-1$.

Next, change exposure from 100 to 120 units while keeping the new ILF and the old other-rating factor:

120(\$40)(1.10)(1.35)=\$7{,}128.00.

This exposure step adds $\$7{,}128-\$5{,}940=\$1{,}188.00$. Finally, change the other-rating factor from 1.10 to 1.05:

120(\$40)(1.05)(1.35)=\$6{,}804.00.

That final step reduces the premium by $\$6{,}804-\$7{,}128=-\$324.00$. The bridge reconciles: $\$5{,}280+\$660+\$1{,}188-\$324=\$6{,}804$. The total renewal increase is $1,524.00, or $\$1{,}524/\$5{,}280=28.86\%$ when rounded to two decimal places.

Check. The direct renewal calculation and the sequential bridge both give $6,804.00. The $660.00 increase is the limit-only effect on the old basis, not the full $1,524.00 renewal increase. The bridge’s intermediate dollar allocations use the order specified in the problem.

Coverage Changes and Non-ILF Premium Components

An ILF comparison is not automatically valid when the policy’s terms change along with its limit. A different deductible, attachment point, covered exposure, or limit structure may alter the loss basis to which the factor applies. Before comparing premiums, determine whether the stated ILFs correspond to each option’s actual terms. If they do not, multiplying by their ratio does not isolate a valid limit effect.

Similarly, a total billed premium can include charges that are not proportional to the ILF. If the rating procedure specifies a fixed policy fee, for instance, the fee remains separate unless the procedure explicitly says otherwise. Consider a hypothetical policy with $3,200 of limit-responsive premium and an $85 fixed fee at the starting limit. If the ILF ratio is 1.25, the new total is $\$3{,}200(1.25)+\$85=\$4{,}085$, not $(\$3{,}200+\$85)(1.25)$. The latter would incorrectly scale the fixed fee.

When comparing options, make the included premium components explicit. A clear comparison states whether it concerns the limit-responsive indicated premium, the total premium before fees, or the final amount including fees and other charges.

Common Exam Traps

Applying the destination ILF to a premium already at another limit. Use $\operatorname{ILF}(L_2)/\operatorname{ILF}(L_1)$ when moving from $L_1$ to $L_2$.
Using the wrong denominator for a percentage change. Divide the increase by the premium at the starting limit, not the destination premium.
Averaging account factors without weights. Calculate account premiums first, or use an average weighted by the relevant starting premiums.
Calling a total renewal change a limit effect. Hold other inputs constant to calculate a limit-only comparison, and separately account for exposure and rating changes.
Scaling every billed amount automatically. Confirm that fees and other components are intended to vary with the ILF before applying the factor.

Key takeaway: A sound premium comparison identifies the starting limit and premium basis, applies the correct ILF ratio, and holds other rating inputs constant when measuring the limit effect. If the inputs or premium components change, separate those changes explicitly.

Check Your Understanding

For each problem, identify the relevant starting basis before calculating. Dollar answers are stated to the cent, and percentages are rounded to two decimal places.

A hypothetical account has a $1,750 limit-responsive premium at a $100,000 limit. Its ILFs are 1.00 at $100,000, 1.12 at $200,000, and 1.36 at $500,000. Find both higher-limit premiums and the percentage increase from $200,000 to $500,000. (Answer: $1,960.00 and $2,380.00; 21.43% increase.)
Two hypothetical accounts have starting premiums of $900 and $2,100. Their ratios for their selected higher limits are 1.30 and 1.10, respectively. Find the combined higher-limit premium and the percentage increase over the combined starting premium. (Answer: $3,480.00; 16.00% increase.)
A limit-responsive premium of $2,400 applies at a $100,000 limit, where the ILF is 1.08. The ILF at $300,000 is 1.35. Assuming all other inputs are unchanged, calculate the $300,000 premium and its dollar increase. (Answer: $3,000.00 premium; $600.00 increase.)
A hypothetical account has 60 exposure units, a $75 base rate per unit, an other-rating factor of 1.20, and an ILF of 1.10. At renewal, exposure is 75 units, the other-rating factor is 1.10, and the selected-limit ILF is 1.32; the base rate is unchanged. Find the old premium, the limit-only premium on the old basis, and the final renewal premium. (Answer: $5,940.00; $7,128.00; $8,167.50.)
A hypothetical policy has $3,200 of limit-responsive premium and a separate $85 fixed fee at its starting limit. The ILF ratio to the selected higher limit is 1.25. Find the total premium at the higher limit and the percentage increase over the starting total premium. (Answer: $4,085.00; 24.35% increase.)

What You'll Learn