What Premium Stability Measures
A renewal premium can change because the experience modification changed, because the premium base changed, or because both changed. Measuring stability therefore requires more than comparing two modification values: the comparison must identify what is held constant and what has changed. This is especially important when a renewal swing could disrupt a policyholder’s budgeting or influence a coverage decision.
For a simple comparison, write the premium before taxes and fees as the product of a premium base and the experience modification. The base represents the premium that would apply at the selected exposure and rating basis before applying the modification. If other multiplicative rating adjustments apply, include them in the base or hold them constant for the comparison.
These measures describe movement, not whether a premium is adequate, fair, or predictive. They also do not determine what size of movement is acceptable. A carrier, regulator, or insured may define a budgeting threshold for a particular purpose, but the threshold should be stated rather than treated as universal.
Choose a Comparable Renewal Basis
A stability comparison is most informative when the premiums are on a consistent basis. If exposure, insured operations, coverage, limits, or rating rules change, the observed premium difference may not reflect movement in the experience modification alone. For a modification-only comparison, hold the base fixed and calculate \(B M_{t-1}\) and \(B M_t\). For an actual billed-premium comparison, use the relevant base and modification at each renewal and report both.
The relative premium change uses the prior renewal premium as its denominator. This makes the measure directional: a rise from \(100\) to \(110\) is a 10% increase, while a fall from \(110\) to \(100\) is a 9.09% decrease. The dollar change is the same magnitude, but the relative changes are not equal because the starting premiums differ.
This ratio identifies the two multiplicative sources of change. In particular, when the base is constant, the premium ratio is exactly the modification ratio. When the base changes, attributing the full premium movement to the modification is incorrect.
Decide whether the goal is to isolate the modification effect or measure the actual renewal premium movement.
Hold exposure and coverage constant for an isolated comparison, or explicitly retain their changes in the actual premium comparison.
Use the previous premium as the denominator for a relative change, and state the direction of movement.
Count or describe movements exceeding the selected threshold; do not imply that the threshold is a universal standard.
Why Log Changes Help Separate Effects
The multiplicative premium relationship has a useful additive form after taking logarithms. This is a mathematical identity, not an assumption about a probability distribution. It is useful when both the base and the modification change, because it displays each component’s contribution on the same log-ratio scale.
The first equality follows from \(P_t=B_tM_t\) and \(P_{t-1}=B_{t-1}M_{t-1}\). The second follows from the logarithm rule \(\log(ab)=\log(a)+\log(b)\), for positive premium bases and modifications. Thus, the log premium movement is exactly the sum of the log base movement and the log modification movement. Ordinary percentage changes do not generally add this way.
Across several consecutive renewals, one concise stability diagnostic is the average absolute log change. For \(n\) renewals, there are \(n-1\) adjacent changes. Taking absolute values prevents increases and decreases from cancelling; exponentiating the average converts the result back to a multiplicative movement scale. This diagnostic summarizes typical movement, while a threshold exceedance rate highlights potentially disruptive individual changes. Neither one replaces inspection of the underlying renewal sequence.
When the purpose is to evaluate modification stability alone, calculate the same measure using \(M_t/M_{t-1}\), or use a constant base so premium ratios match modification ratios. A threshold diagnostic can be defined as the proportion of adjacent changes whose absolute relative premium change exceeds a chosen value \(c\). State whether the test is strictly greater than \(c\) or greater than or equal to \(c\).
Worked Renewal Comparisons
Worked Example: A Modification Change at a Fixed Base
A hypothetical commercial policy has a constant premium base of \( \$48{,}000 \). Its finalized modification changes from \(0.92\) to \(1.08\). Calculate the premium at each renewal, the dollar and relative premium changes, and whether the increase exceeds a stated 15% budgeting threshold.
Method and assumptions. Hold exposure, coverage, and all non-modification rating factors fixed, so \(B=\$48{,}000\) at both renewals. This is a deterministic illustration; no loss distribution, interest rate, or mortality assumption is used.
First renewal premium. \(P_1=\$48{,}000(0.92)=\$44{,}160.00\).
Second renewal premium. \(P_2=\$48{,}000(1.08)=\$51{,}840.00\).
Dollar change. \(P_2-P_1=\$51{,}840-\$44{,}160=\$7{,}680.00\).
Relative change. \((\$7{,}680/\$44{,}160)\times100=17.3913\%\), or \(17.39\%\) to two decimal places. Equivalently, the premium ratio is \(1.08/0.92=1.173913\), so the increase is \(17.3913\%\).
The increase exceeds the stated 15% threshold. The modification rose by \(1.08-0.92=0.16\) points, but the premium rose by \(17.39\%\). Common mistake avoided: reporting the 0.16-point modification change as a 16% premium increase. The percentage premium change must be calculated relative to the starting modification or premium.
Check. The premium difference also equals \( \$48{,}000(1.08-0.92)=\$7{,}680\), matching the direct subtraction.
Worked Example: Separating Base and Modification Movement
A hypothetical account renews with its premium base increasing from \( \$60{,}000 \) to \( \$63{,}000 \), while its modification decreases from \(1.10\) to \(1.00\). Find the actual premium change and distinguish the base movement from the modification movement.
Method and assumptions. Use the stated renewal bases and finalized modifications. Treat the premiums as deterministic amounts before taxes and fees; no interest rate, mortality assumption, or claim distribution is needed.
Prior premium. \(P_1=\$60{,}000(1.10)=\$66{,}000.00\).
Renewal premium. \(P_2=\$63{,}000(1.00)=\$63{,}000.00\).
Actual change. The dollar change is \(\$63{,}000-\$66{,}000=-\$3{,}000.00\). Relative to the prior premium, the change is \(-\$3{,}000/\$66{,}000=-0.0454545\), or a \(4.55\%\) decrease to two decimal places.
Multiplicative decomposition. The base ratio is \(63{,}000/60{,}000=1.05\), and the modification ratio is \(1.00/1.10=0.909091\). Their product is \(1.05(0.909091)=0.954545\), which is the actual premium ratio \(63{,}000/66{,}000\). On the log scale, the contributions are \(\log(1.05)=0.048790\) and \(\log(0.909091)=-0.095310\); their sum is \(-0.046520\), consistent with the premium decrease.
A dollar attribution can also be made in sequence. First apply the base increase at the old modification: \((\$63{,}000-\$60{,}000)(1.10)=+\$3{,}300\). Then apply the modification decrease at the new base: \(\$63{,}000(1.00-1.10)=-\$6{,}300\). The net is \(+\$3{,}300-\$6{,}300=-\$3{,}000\), as required. This dollar allocation depends on the order chosen; the log-ratio decomposition does not.
Common mistake avoided: adding the base increase of 5% to the modification decrease of 9.09% and calling the result the exact premium percentage change. Multiplicative factors combine by multiplication, not by adding their ordinary percentage changes. The calculated \(4.55\%\) decrease is the correct net movement.
Worked Example: Measuring Repeated Renewal Movement
A hypothetical account has a constant annual premium base of \( \$40{,}000 \) and finalized modifications of \(0.90,\ 1.00,\ 0.95,\ 1.10,\ 1.05\) over five renewals. Calculate the premiums, the average absolute log movement, and the fraction of renewal changes exceeding a 15% absolute relative-change threshold.
Method and assumptions. The base, exposure, coverage, and other rating factors remain fixed. No interest rate, mortality assumption, or claim distribution is modeled. Calculate the threshold using the absolute relative change from each immediately preceding premium.
Premium sequence. Multiplying each modification by \( \$40{,}000 \) gives \( \$36{,}000,\ \$40{,}000,\ \$38{,}000,\ \$44{,}000,\ \$42{,}000 \).
Adjacent relative changes. The four changes are \((40{,}000-36{,}000)/36{,}000=11.1111\%\), \((38{,}000-40{,}000)/40{,}000=-5.0000\%\), \((44{,}000-38{,}000)/38{,}000=15.7895\%\), and \((42{,}000-44{,}000)/44{,}000=-4.5455\%\). Only one of the four absolute changes exceeds 15%, so the exceedance rate is \(1/4=25\%\).
Average absolute log movement. The absolute log changes are \(\left|\log(40/36)\right|=0.105361\), \(\left|\log(38/40)\right|=0.051293\), \(\left|\log(44/38)\right|=0.146604\), and \(\left|\log(42/44)\right|=0.046520\). Their sum is \(0.349778\); dividing by four gives \(A=0.0874445\). The corresponding typical absolute multiplicative movement is \(e^{0.0874445}-1=0.0914\), or about \(9.14\%\).
The average summarizes movement across all four transitions, while the exceedance rate identifies how often the selected threshold was crossed. They answer different questions. Common mistake avoided: dividing by the five renewal observations instead of the four adjacent changes, or comparing dollar changes against a percentage threshold.
Check. The largest absolute relative change is \(15.7895\%\), so the threshold count is one; the average absolute log movement is positive even though the sequence contains both increases and decreases.
Interpreting Stability Carefully
A stable modification does not by itself establish that the premium is adequate, and a large modification movement does not by itself establish that the rating result is unreasonable. Stability is a separate diagnostic: it describes how much the indicated or charged premium changes between renewals on a specified comparison basis.
For a portfolio or account study, report the number of matched risks and renewal transitions, the comparison basis, the movement statistic, and any threshold. If only continuously renewed accounts are included, state that scope; the result describes those observed renewals, not necessarily every risk eligible to renew. Avoid mixing a modification-only comparison with an actual billed-premium comparison in one summary without labeling the difference.
| Measure | What it describes | Important qualification |
|---|---|---|
| Dollar change | Amount added to or subtracted from premium | Depends on the size of the starting premium |
| Relative change | Percentage movement from the prior premium | Uses the prior premium as denominator; direction matters |
| Absolute log change | Multiplicative movement on a symmetric scale | Does not show whether the premium rose or fell unless its sign is retained separately |
| Threshold exceedance rate | Share of transitions beyond a stated movement limit | Depends on the selected threshold and comparison basis |
A practical report often presents both direction and magnitude: for example, the median relative change, the average absolute log change, and the proportion of renewals exceeding a specified budgeting threshold. Whatever measures are selected, use the same denominator convention and renewal definition throughout. These results are aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.
Check Your Understanding
For each problem, state the relevant comparison basis and check whether the requested measure is in dollars, percentage points, or percent.
- A hypothetical account has a fixed base of \( \$32{,}000 \). Its modification rises from \(0.80\) to \(0.88\). Find both premiums and the relative premium increase. (Answer: \( \$25{,}600.00\) and \( \$28{,}160.00\); increase \(=\$2{,}560.00\), or \(10.00\%\).)
- A fixed-base policy has a \( \$50{,}000 \) base and a modification changing from \(1.25\) to \(1.10\). Find the dollar and relative premium change, and the modification-point change. (Answer: premium falls from \( \$62{,}500.00\) to \( \$55{,}000.00\), a \( \$7{,}500.00\) or \(12.00\%\) decrease; modification change \(=-0.15\) points.)
- A renewal’s base rises from \( \$72{,}000 \) to \( \$75{,}600 \), while its modification rises from \(0.95\) to \(1.00\). Find the old and new premiums and the actual relative change. (Answer: \( \$68{,}400.00\) to \( \$75{,}600.00\); increase \(=\$7{,}200.00\), or \(10.5263\%\).)
- With a constant base, a risk has four consecutive modifications \(0.96,\ 1.04,\ 1.00,\ 1.10\). What proportion of the three absolute premium changes exceeds 9%? (Answer: \(1/3\), since the changes are \(8.3333\%\), \(3.8462\%\), and \(10.0000\%\) in absolute value.)
- A constant-base policy has two consecutive renewal transitions, with modifications \(0.85,\ 0.95,\ 0.90\). Using a strict threshold of more than 10% absolute relative premium change, find the exceedance rate. (Answer: \(1/2=50\%\); the two absolute changes are \(11.7647\%\) and \(5.2632\%\).)