Extending Premium Through Multiple Rate Changes

Why Multiple Rate Changes Require Segmentation

When earned premium spans several rate changes, the premium in each part of the experience may reflect a different rate level. A single adjustment to the total can therefore be misleading: earlier premium portions and later portions do not necessarily need the same adjustment to reach the selected target level. The extension method handles this by tracking the premium and rate level for each earned portion.

The calculation has two distinct tasks. First, determine the cumulative rate level after each change; successive percentage changes compound. Second, extend each earned portion from its own applicable level to the target level, then sum the results. This is a policy-level application of on-leveling, aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Definition: Let $L_j$ denote the rate-level index after the $j$th rate change, measured relative to a chosen base level $L_0=1$. If the percentage change at step $j$ is $r_j$, then $L_j=L_{j-1}(1+r_j)$. For policy $i$, let $EP_{ij}$ be earned premium in a portion earned at level $L_j$. To extend that portion to target level $L_T$, multiply it by $L_T/L_j$. The portfolio extension-method factor is total extended earned premium divided by total recorded earned premium.

Build the Rate-Level Sequence First

A rate change is applied to the rate level immediately before it, not to the original level in isolation. For example, a 10% increase followed by a 5% decrease gives $1.00(1.10)(0.95)=1.045$, a cumulative level 4.5% above the base. Adding the percentages would give 5%, which is not the correct cumulative change.

For this method, the premium records must identify the earned premium associated with each rate level. If policy $i$ has earned-premium portions $EP_{i1},EP_{i2},\ldots$, its extended premium is:

EP_i^{\text{extended}} = \sum_j EP_{ij}\frac{L_T}{L_j}.

For the portfolio, add the extended amounts across policies and divide by the total recorded earned premium:

F_{\text{ext}} = \frac{\displaystyle\sum_i\sum_j EP_{ij}(L_T/L_j)} {\displaystyle\sum_i\sum_j EP_{ij}}.

This calculation assumes that, for the exposure being compared, premium is proportional to the rate-level index and that the rate changes represented by the indices are applicable to the premium portions in the records. It also assumes the premium portions have been earned consistently with the stated policy terms and dates. If the records do not identify the level associated with each portion, the extension cannot be performed accurately without additional information. The treatment of changes that apply only at renewal or during a policy term requires careful interpretation of the underlying records.

Why the Level-Ratio Adjustment Works

The level ratio follows from proportional premium. Suppose a fixed exposure would produce premium $B$ at the base level. At level $L_j$, its premium is $B L_j$. At target level $L_T$, the same exposure would produce $B L_T$. Their ratio is:

\frac{B L_T}{B L_j}=\frac{L_T}{L_j}.

Thus an earned premium portion recorded at level $L_j$ is converted to its target-level equivalent by multiplying by $L_T/L_j$. Adding the converted portions is valid because each represents premium for a portion of the exposure at the same target level. Dividing by recorded premium then gives the portfolio factor: the multiplier that converts the total recorded earned premium to its extended equivalent.

Worked Example: Three Earned Portions and Two Changes

A hypothetical annual policy has a base annual premium of $1,200. Its 18-month term is divided into three six-month portions. A 10% increase takes effect at the start of the second portion, and a 5% decrease takes effect at the start of the third. Find the earned premium at each level, then extend the total to the final rate level.

Method. Build the cumulative rate levels, earn premium for each six-month portion, and multiply each portion by the target-to-applicable-level ratio.

The rate levels are $L_0=1.000$, $L_1=1.000(1.10)=1.100$, and $L_2=1.100(0.95)=1.045$. Each six-month portion earns one-half of the annual premium at its applicable level:

\begin{aligned} EP_0 &= 1{,}200(1/2)(1.000)=600,\\ EP_1 &= 1{,}200(1/2)(1.100)=660,\\ EP_2 &= 1{,}200(1/2)(1.045)=627. \end{aligned}

Recorded earned premium is $600+660+627=\$1{,}887$. The target is $L_T=1.045$. The first portion is multiplied by $1.045/1.000$, the second by $1.045/1.100=0.95$, and the third by $1.045/1.045=1$:

\begin{aligned} EP^{\text{extended}} &=600(1.045)+660(0.95)+627(1)\\ &=627+627+627\\ &=\$1{,}881,\\ F_{\text{ext}}&=\frac{1{,}881}{1{,}887}=0.9968. \end{aligned}

Check. Each six-month portion at the final level would earn $1{,}200(1/2)(1.045)=\$627$, so the three portions total $1,881 as calculated. The factor is rounded to four decimal places. The common mistake is to add $10\%-5\%=5\%$ and use 1.05 as the final level; the correct compounded level is 1.045.

Calculate a Portfolio Factor from Policy Records

Policy records can have different premium sizes and different distributions of earned premium across rate levels. The portfolio factor is therefore a premium-weighted result. It is not generally the arithmetic average of policy factors, nor is it the factor for one representative policy unless that policy correctly represents the portfolio’s premium weights.

Worked Example: Combining Two Policies at the Latest Level

In a hypothetical portfolio, the rate changes are a 10% increase followed by a 5% increase. Policy A has a $1,000 annual base premium and earns six months at the base level and six months at the level after the first change. Policy B has a $2,000 annual base premium and earns three months at the first changed level and nine months at the latest level. Find the portfolio extension factor to the latest level.

Method. The successive indices are $L_0=1.000$, $L_1=1.100$, and $L_2=1.100(1.05)=1.155$. Extend every portion to $L_T=1.155$.

For Policy A, the recorded earned portions are $1{,}000(6/12)(1.000)=\$500$ and $1{,}000(6/12)(1.100)=\$550$. Their extensions are:

500\left(\frac{1.155}{1.000}\right)=577.50, \qquad 550\left(\frac{1.155}{1.100}\right)=577.50.

Policy A has recorded earned premium of $500+550=\$1{,}050$ and extended earned premium of $1,155.

For Policy B, the portions are $2{,}000(3/12)(1.100)=\$550$ and $2{,}000(9/12)(1.155)=\$1{,}732.50$. Extending them gives:

550\left(\frac{1.155}{1.100}\right)=577.50, \qquad 1{,}732.50\left(\frac{1.155}{1.155}\right)=1{,}732.50.

Policy B has recorded earned premium of $550+1{,}732.50=\$2{,}282.50$ and extended earned premium of $2,310.

The portfolio totals are recorded earned premium $1{,}050+2{,}282.50=\$3{,}332.50$ and extended earned premium $1{,}155+2{,}310=\$3{,}465$. Thus:

F_{\text{ext}}=\frac{3{,}465}{3{,}332.50}=1.0398.

Check. Policy A’s six-month portions both equal the target-level half-year premium of $577.50; Policy B’s four-quarter-equivalent portions together equal one annual target-level premium of $2,310. The portfolio factor is rounded to four decimal places. Applying a single factor to each policy without respecting its level mix would ignore the different weights of premium at each level.

A Target Level Need Not Be the Latest Level

The target is determined by the analysis, not automatically by the most recent rate level in the records. When the target is an earlier level, portions earned at higher levels are scaled down. The same level-ratio formula applies; the direction of the adjustment is determined by whether $L_T$ is above or below the portion’s level.

Worked Example: Extending Back to an Intermediate Level

A hypothetical policy with a $900 annual base premium earns four months at level $L_0=1.00$, four months at $L_1=1.20$, and four months at $L_2=1.08$. These levels result from a 20% increase followed by a 10% decrease. A second policy has a $600 annual base premium and earns six months at $L_1$ and six months at $L_2$. Find the portfolio factor to $L_1=1.20$.

Method. Calculate each policy’s earned premium at its applicable level, then extend all portions to the intermediate target $L_T=1.20$.

For the first policy, the recorded portions are $900(4/12)(1.00)=\$300$, $900(4/12)(1.20)=\$360$, and $900(4/12)(1.08)=\$324$. The target-level equivalents are:

300(1.20/1.00)=360,\qquad 360(1.20/1.20)=360,\qquad 324(1.20/1.08)=360.

Its recorded total is $300+360+324=\$984$; its extended total is $1,080.

For the second policy, recorded portions are $600(6/12)(1.20)=\$360$ and $600(6/12)(1.08)=\$324$. Extending them gives $360$ and $324(1.20/1.08)=\$360$, respectively. The second policy’s totals are $684 recorded and $720 extended.

The portfolio recorded total is $984+684=\$1{,}668$, and the extended total is $1{,}080+720=\$1{,}800$. Therefore $F_{\text{ext}}=1{,}800/1{,}668=1.0791$, rounded to four decimal places.

Check. Each policy contributes a full year of exposure, so at the target level their extended premiums are $900(1.20)=$1,080 and $600(1.20)=$720. Their sum matches the calculation. The common mistake is to use the latest level, 1.08, as the target despite the stated target of 1.20.

Common Traps and Practical Checks

Before accepting an extension calculation, verify the level sequence, the assignment of earned premium to levels, and the target. These checks catch most errors without requiring a separate model.

Adding changes instead of compounding. Calculate each new level as the prior level multiplied by one plus the next percentage change.
Using the wrong ratio. The multiplier is target level divided by the portion’s applicable level, $L_T/L_j$, not its reciprocal.
Applying one adjustment to all recorded premium. Separate the premium by rate level first; different portions may require different multipliers.
Confusing the latest level with the target. Use the target specified for the analysis, even if it is an intermediate level.
Ignoring policy term fractions. Earn only the portion of premium attributable to each stated part of the term before extending it.

A useful consistency check is to recompute the target-level premium for a policy directly from its base premium and exposure, when those are supplied. The result should match the sum of its extended portions under the proportional-premium assumption. At portfolio level, the extension factor should also equal extended total premium divided by recorded total premium; it is not a simple average of the individual multipliers.

Key takeaway: Track cumulative rate levels by compounding each change, segment earned premium by applicable level, and extend every segment using $L_T/L_j$. Sum the converted portions before calculating the portfolio factor.

Check Your Understanding

For each hypothetical problem, determine the requested extended premium or factor. Rate-level indices are relative to the stated base level.

A $1,500 annual policy earns four months at the base level, four months after a 8% increase, and four months after a further 4% increase. Find total recorded earned premium and the factor to the latest level. (Answer: recorded earned premium $1,601.60; extension factor 1.0519.)
A $2,000 annual policy earns 40% of its term at $L_0=1.00$, 30% at $L_1=1.12$, and 30% at $L_2=1.12(0.95)=1.064$. Find the factor to $L_1$. (Answer: extended earned premium $2,240; factor 1.0614.)
Two policies are extended to the latest level after successive 10% increases, so the levels are 1.00, 1.10, and 1.21. Policy A has a $1,200 annual base premium, earning three months at level 1.00 and nine months at 1.10. Policy B has an $800 annual base premium, earning six months at 1.10 and six months at 1.21. Find the portfolio factor. (Answer: recorded earned premium $2,214; extended earned premium $2,420; factor 1.0930.)
A $1,000 annual policy earns 25% at level 1.00, 25% after a 6% increase, and 50% after a further 4% decrease. Find the factor to the latest level after one more 3% increase. (Answer: latest level 1.048128; recorded earned premium $1,023.80; extended earned premium $1,048.13; factor 1.0238.)
Two policies are extended to the latest level after a 10% increase and then a 5% increase. Policy A has a $1,000 annual base premium, earning two months at level 1.00, four months at 1.10, and six months at 1.155. Policy B has a $1,500 annual base premium, earning four months at 1.10 and eight months at 1.155. Find the portfolio factor. (Answer: recorded earned premium $2,815.83; extended earned premium $2,887.50; factor 1.0255.)

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