Rate Level Change Versus Premium Change

Why Rate Level and Written Premium Can Move Differently

A rate change describes how the price charged for a comparable set of risks changes. Total written premium describes the dollars produced by the business actually written. Those quantities can move by different percentages: premium depends not only on rates, but also on how much business is written and which classes make up the portfolio. Keeping the comparison basis clear is essential when interpreting an indicated or implemented rate change. This material is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

The simplest case is a uniform rate change applied to an unchanged set of exposures. If the rate level rises by 6% and exposures and mix stay fixed, premium also rises by 6%. But if exposure volume grows, shrinks, or shifts among classes with different rates, the observed change in total premium is not the rate change alone.

Definition: Let $E_j$ be the number of exposure units in class $j$, and let $r_j$ be the rate per exposure unit for that class. Under a simple proportional rating model, total premium is $P=\sum_j E_jr_j$. To compare two rate schedules, a rate level change is the percentage change in premium for a fixed set of exposure weights: $$ \text{Rate level change} = \frac{\sum_j E_j r_{j,1}}{\sum_j E_j r_{j,0}}-1. $$ Here $r_{j,0}$ and $r_{j,1}$ are the old and new rates, and the same exposure counts $E_j$ are used in both totals. The result measures the change in rates on that reference portfolio; it is not necessarily the change in premium actually written.

An exposure unit is the rating basis used in the calculation—for example, a vehicle-year or a specified amount of insured value. In practice, premium may include fixed charges, minimum premiums, or other features that do not scale in direct proportion to the rate. The simple formulas here assume premium equals exposure multiplied by rate. If that assumption does not hold, those additional features must be handled explicitly rather than treating all premium as proportional.

Fixed Weights Define the Rate Level Comparison

The exposure weights matter because classes can have different rates and different rate changes. A portfolio with many exposures in a low-rate class will generally produce a different overall rate level result from one with many exposures in a high-rate class. For a specified reference portfolio, the old and new average rates per exposure are $A_0=\sum_j w_jr_{j,0}$ and $A_1=\sum_j w_jr_{j,1}$, where $w_j=E_j/\sum_k E_k$. These weights sum to 1, so the rate level change is $A_1/A_0-1$.

Holding the weights fixed is what makes the result a rate comparison rather than a comparison of two portfolios that differ in both rates and business composition. A different reasonable reference portfolio can produce a different weighted average when class changes are not uniform. Therefore, the exposure basis should be stated whenever the rate level result is reported.

Worked Example: An 8% Rate Change and a Larger Premium Increase

A hypothetical insurer writes 1,000 exposure units at a rate of $900 per unit. It then applies a uniform 8% rate increase. In the next period it writes 1,300 exposure units, all with the same risk mix. Find the rate level change and the change in total written premium. Method: calculate the rate change separately, then calculate each period's premium using its own exposure volume.

Step 1: Find the new rate. The old rate is $900 per exposure unit. The new rate is $900(1.08)=\$972$ per exposure unit. Thus the rate level change is $(972/900)-1=0.08$, or 8.00%.

Step 2: Calculate old written premium. $1{,}000\times \$900=\$900{,}000$.

Step 3: Calculate new written premium. $1{,}300\times \$972=\$1{,}263{,}600$.

Step 4: Compare premiums. The dollar increase is $\$1{,}263{,}600-\$900{,}000=\$363{,}600$. The percentage increase is $\$363{,}600/\$900{,}000=0.404$, or 40.40%.

Check and trap avoided. Exposure volume increased by $1{,}300/1{,}000-1=30\%$. The premium ratio is $1.08\times1.30=1.404$, confirming a 40.40% increase. Calling 40.40% the rate level change would incorrectly attribute the growth in exposure volume to rates.

How Mix Changes Affect the Observed Premium

For period $t$, define total exposure $E_t=\sum_j E_{j,t}$, class mix $w_{j,t}=E_{j,t}/E_t$, and average premium per exposure $A_t=\sum_j w_{j,t}r_{j,t}$. Then written premium is $P_t=E_tA_t$. This identity separates the total exposure volume from the average premium per exposure, but that average can itself change because of either rates or mix.

\frac{P_1}{P_0} = \frac{E_1}{E_0}\times\frac{A_1}{A_0}, \qquad A_t=\sum_j w_{j,t}r_{j,t}.

When class rates change by different percentages, a useful way to report the rate level change is to hold the starting mix fixed. The resulting benchmark is $A_{\text{rate}}=\sum_j w_{j,0}r_{j,1}$. The rate level change on that reference mix is $A_{\text{rate}}/A_0-1$. Comparing this benchmark with the actual later average premium per exposure shows the effect of mix under the later rates.

This separation has a sequence: first apply the rate changes to the starting mix, then change the mix at the new rates, then change total exposure volume. The mix and rate contributions can depend on the chosen sequence when rates vary by class. The total premium comparison does not depend on that sequence; it is the reference used to describe the components that must be stated.

Worked Example: Class Rate Changes With a Shift in Policy Mix

A hypothetical insurer has 80 exposure units in Class A and 20 in Class B. Old rates are $500 and $1,500, respectively. New rates are $550 in Class A and $1,560 in Class B. The next-period portfolio still has 100 exposure units, but now has 60 in Class A and 40 in Class B. Calculate the rate level change on the starting mix and the actual written premium change. Method: use the starting class counts to compare rates; use each period's actual counts to calculate its written premium.

Step 1: Old premium on the starting mix. Class A contributes $80\times\$500=\$40{,}000$, and Class B contributes $20\times\$1{,}500=\$30{,}000$. Total old premium is $\$70{,}000$.

Step 2: Premium at new rates but the same starting mix. Class A contributes $80\times\$550=\$44{,}000$, and Class B contributes $20\times\$1{,}560=\$31{,}200$. The benchmark is $\$75{,}200$. The rate level change is $(\$75{,}200/\$70{,}000)-1=0.0742857$, or 7.43% to two decimals.

Step 3: Actual next-period premium. With the new mix, Class A contributes $60\times\$550=\$33{,}000$, while Class B contributes $40\times\$1{,}560=\$62{,}400$. Actual next-period premium is $\$95{,}400$.

Step 4: Actual premium change. The increase is $\$95{,}400-\$70{,}000=\$25{,}400$, or $\$25{,}400/\$70{,}000=0.362857$, which is 36.29% to two decimals.

Check and trap avoided. At the new rates, the starting-mix average premium is $\$75{,}200/100=\$752$ per exposure, while the actual new-mix average is $\$95{,}400/100=\$954$. The shift toward the higher-rate Class B raises actual premium beyond the rate-only benchmark. Treating the 36.29% actual premium increase as the rate level change would confuse mix with rates.

Volume, Mix, and Rate: A Full Reconciliation

A sequential reconciliation is useful when both class mix and total exposure volume change. Let $P_0=E_0A_0$ be old premium. First calculate the rate effect on the old portfolio, then the mix effect at the new rates while holding total exposure at its old level, and finally the volume effect using the new mix and rates. Algebraically:

P_1-P_0 = E_0(A_{\text{rate}}-A_0) + E_0(A_1-A_{\text{rate}}) + (E_1-E_0)A_1.

The first two terms together equal $E_0(A_1-A_0)$, and adding the final term gives $E_1A_1-E_0A_0=P_1-P_0$. The identity therefore reconciles exactly. Its interpretation follows the stated order: rate changes are applied first to the old mix, then the mix changes at new rates, then exposure volume changes.

Worked Example: Separating Rate, Mix, and Volume Effects

A hypothetical portfolio starts with 80 exposure units in Class A and 20 in Class B. Old rates are $600 and $1,200. New rates are $630 and $1,320. In the next period, total exposure grows to 120 units, split equally between the two classes. Calculate the fixed-starting-mix rate level change, actual premium change, and dollar contributions under the sequence above. Method: reconcile rates, mix, then volume.

Step 1: Old premium and average. Old premium is $80\times\$600+20\times\$1{,}200=\$48{,}000+\$24{,}000=\$72{,}000$. With 100 exposures, $A_0=\$720$.

Step 2: Rate change on the old mix. At new rates and old counts, premium is $80\times\$630+20\times\$1{,}320=\$50{,}400+\$26{,}400=\$76{,}800$. Thus $A_{\text{rate}}=\$768$, and the rate level change is $(768/720)-1=0.0666667$, or 6.67%.

Step 3: Mix change at new rates, holding volume at 100. Equal shares of 100 exposures mean 50 in each class. Premium would be $50\times\$630+50\times\$1{,}320=\$31{,}500+\$66{,}000=\$97{,}500$. The mix contribution is $\$97{,}500-\$76{,}800=\$20{,}700$.

Step 4: Volume change at the new average. The new average is $\$97{,}500/100=\$975$ per exposure. Increasing exposure from 100 to 120 adds $20\times\$975=\$19{,}500$.

Step 5: Check against actual premium. Actual premium is $60\times\$630+60\times\$1{,}320=\$37{,}800+\$79{,}200=\$117{,}000$. The components reconcile: $\$72{,}000+\$4{,}800+\$20{,}700+\$19{,}500=\$117{,}000$. Actual premium increased by $\$45{,}000/\$72{,}000=62.50\%$, much more than the 6.67% rate level change. The difference reflects the mix and volume effects, not an arithmetic inconsistency.

Common Interpretation Errors

The key question is what is being held constant. A rate level comparison holds the exposure weights fixed. An actual written premium comparison uses the exposures and mix that were actually written in each period. Confusing those bases can make a sound rate indication appear inconsistent with premium growth or decline.

Wrong: treating actual premium growth as the rate change. For example, in the first worked example, $1{,}263{,}600/900{,}000-1=40.40\%$ is the premium change, not the 8.00% rate change. The correction is to compare old and new rates using the same 1,000 reference exposures.
Wrong: using updated class counts in only one side of a rate-level comparison. Doing so changes both the rates and the mix. Use identical class exposure weights in the old-rate and new-rate totals for a rate level comparison.
Wrong: assuming unchanged total exposure means unchanged mix. A portfolio can have the same number of exposure units but a different distribution across classes. Class-level rates make that shift financially significant.
Wrong: multiplying a premium ratio by the rate change without checking the basis. The simple product of a rate factor and a volume factor applies when mix and the proportional-premium assumption are unchanged. With mix changes, calculate the average premium per exposure for each actual mix as well.

Key takeaway: A rate level change compares rate schedules on a stated, fixed exposure basis. Written premium is exposure volume multiplied by average premium per exposure, and that average can change with both rates and policy mix. Always state which comparison you are reporting.

Check Your Understanding

For each question, distinguish the rate comparison from the actual premium comparison. Dollar amounts are hypothetical.

A portfolio has 200 exposure units at $400 each. The rate rises uniformly by 6%, and next-period volume is 230 units with the same mix. What are the rate level change and written premium change? (Answer: Rate level change 6.00%; premium rises from $80,000 to $97,520, a 21.90% increase.)
A starting portfolio has 30 units in Class A and 70 in Class B, with rates of $700 and $1,100. New rates are $735 and $1,122. Find the rate level change using the starting mix. (Answer: Old premium $98,000; new premium on the same mix $100,590; rate level change 2.643%.)
There are 100 exposures in both periods. The initial mix is 40 in Class A and 60 in Class B; the new mix is 70 and 30. Rates rise from $500 to $550 in A and from $1,000 to $1,050 in B. What is the actual written premium change? (Answer: Premium falls from $80,000 to $70,000, a 12.50% decrease, despite both class rates increasing.)
A starting portfolio has 50 units in each of two classes, with old rates $400 and $900. New rates are $420 and $990. The next period has 120 units, 30% in the first class and 70% in the second. Find the starting-mix rate level change and actual premium change. (Answer: Rate level change 8.462%; premium rises from $65,000 to $98,280, a 51.20% increase.)
Assume unchanged mix and proportional premium. A uniform rate increase of 4% accompanies a 13.12% increase in written premium. What is the exposure volume change? (Answer: Exposure volume rises by $1.1312/1.04-1=8.769\%$, approximately.)

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