Why Constrain an Indicated Change?
A rate indication estimates the change supported by the selected data, assumptions, and method. Applying it without modification can produce abrupt changes for some segments, especially when experience is volatile or a segment has limited data. An insurer may therefore use a floor, a cap, or a phased transition when implementing rates. These are implementation choices: they do not, by themselves, alter the underlying actuarial indication.
The calculations require a clear comparison basis. In this tutorial, each segment’s current premium is held fixed as the weight for measuring premium impact. We assume the implemented premium is proportional to the implemented rate factor, with no changes in exposure, coverage, mix, or other rating variables. Under these assumptions, we can calculate the mechanical premium impact. Actual future premium can differ if the portfolio or customer behavior changes.
A positive change means an increase; a negative change means a decrease. A floor and cap can be stated as percentage changes, such as \(-8\%\) and \(+15\%\), or as premium factors, such as 0.92 and 1.15. Convert factor bounds to percentage changes before applying the formulas, or keep the entire calculation in factor form consistently.
The portfolio premium after implementation, under the fixed-exposure assumptions above, is \(\sum_j P_j(1+h_j)\). The dollar change is \(\sum_jP_jh_j\), and the portfolio percentage change is that dollar change divided by \(\sum_jP_j\). Segment changes must be weighted by their current premiums; a simple average of segment percentages generally does not give the portfolio impact.
Applying Floors, Caps, and Phasing
The order of operations matters. First calculate the indicated change. Next apply the applicable floor and cap to determine the constrained target. Then apply the phase fraction to the constrained change. In particular, a phase fraction is not applied to the unconstrained indication when a cap or floor has already changed the target.
Express the actuarial target relative to current rates as a factor \(q_j\), or as a percentage change \(r_j=q_j-1\).
Limit the change to the stated interval: \(c_j=\min(U_j,\max(L_j,r_j))\). If \(r_j\) is already within the interval, it is unchanged.
For a first phase that implements fraction \(\alpha_j\) of the constrained change, use \(h_j=\alpha_jc_j\). The first-phase factor is \(1+h_j\).
Multiply each segment’s current premium by its applicable factor, then sum. Divide the dollar change by total current premium to get the portfolio percentage change.
If the transition is completed in several phases, specify whether each phase is a fraction of the original constrained change or a percentage change applied to the then-current rate. Those conventions are not interchangeable. Here, a phase fraction is a fraction of the original constrained change, so the completed factor is \(1+c_j\). The year-over-year rate change in a later phase is calculated relative to the prior phase’s rate.
Worked Example: Applying Segment Caps and Floors
Method and mistake avoided. Clip each indicated percentage change to the stated interval, then calculate the premium effect using current-premium weights. This avoids averaging segment changes without regard to segment size.
A hypothetical portfolio has three segments with current annual premiums of \$600,000, \$300,000, and \$100,000. Their indicated changes are \(+24\%\), \(-18\%\), and \(+6\%\), respectively. Each segment is subject to a floor of \(-10\%\) and a cap of \(+15\%\). The full constrained change is implemented immediately.
Step 1: Constrain each indication. Segment A’s \(+24\%\) indication exceeds the \(+15\%\) cap, so its constrained change is \(+15\%\). Segment B’s \(-18\%\) indication is below the \(-10\%\) floor, so its constrained change is \(-10\%\). Segment C’s \(+6\%\) indication is within the interval, so its constrained change remains \(+6\%\).
Step 2: Calculate premiums after implementation. Segment A: \$600,000 \(\times 1.15=\$690,000\). Segment B: \$300,000 \(\times 0.90=\$270,000\). Segment C: \$100,000 \(\times 1.06=\$106,000\).
Step 3: Calculate the portfolio effect. Current total premium is \$600,000+\$300,000+\$100,000=\$1,000,000. Implemented total premium is \$690,000+\$270,000+\$106,000=\$1,066,000. The dollar increase is \$66,000, so the portfolio increase is \$66,000/\$1,000,000=\(6.6\%\).
Check against the unconstrained indication. Applying the raw indications would produce \$600,000(1.24)+\$300,000(0.82)+\$100,000(1.06)=\$744,000+\$246,000+\$106,000=\$1,096,000, or a \(9.6\%\) increase. The implemented \(6.6\%\) is lower because the cap and floor limit the two outlying changes. The raw \(9.6\%\) remains the portfolio indication under the given inputs; it is not replaced as an actuarial result by the implementation constraint.
Why Clipping Gives the Nearest Permitted Change
When each segment has its own permitted interval and there is no additional portfolio-level constraint, clipping has a useful interpretation: it selects the permitted change closest to the actuarial indication. To see why, suppose the indication for one segment is \(r\), and an implementable change \(c\) must satisfy \(L\leq c\leq U\). Consider minimizing the squared difference \((c-r)^2\).
If \(L\leq r\leq U\), the squared difference is smallest at \(c=r\), which is already permitted. If \(r>U\), every permitted \(c\) is less than \(r\), and moving \(c\) upward toward \(U\) decreases \((c-r)^2\); the minimum is at \(U\). If \(r<L\), moving \(c\) downward toward \(L\) decreases \((c-r)^2\); the minimum is at \(L\). Thus the minimizing choice is \(c=\min(U,\max(L,r))\).
The same reasoning applies to a sum of squared deviations across segments when each segment’s constraint is separate and each segment has a positive weight: minimizing \(\sum_j w_j(c_j-r_j)^2\) subject only to \(L_j\leq c_j\leq U_j\) produces the same clipped value for each segment. But this result does not imply that clipping is optimal under every implementation rule. A required portfolio-average change, a budget constraint, or linked segment constraints couples the choices. In those cases, independent clipping may not satisfy the full set of conditions, and a constrained optimization method is needed.
Worked Example: Phasing a Constrained Target
Method and mistake avoided. Apply the phase fraction to the constrained target, then compare the first-phase and completed premiums. This avoids treating the first phase as a percentage of the raw indication when the cap or floor has already changed it.
A hypothetical portfolio has current premiums of \$400,000, \$400,000, and \$200,000. The indicated changes are \(+30\%\), \(-16\%\), and \(+10\%\). A floor of \(-8\%\) and cap of \(+20\%\) apply. The first phase implements \(50\%\) of each constrained change; the completed transition implements \(100\%\).
Step 1: Find constrained changes. The first indication is limited from \(+30\%\) to \(+20\%\). The second is raised from \(-16\%\) to the \(-8\%\) floor. The third \(+10\%\) indication is unchanged. The constrained changes are therefore \(+20\%\), \(-8\%\), and \(+10\%\).
Step 2: Calculate first-phase changes. Multiplying each constrained change by \(0.50\) gives \(+10\%\), \(-4\%\), and \(+5\%\). First-phase premiums are \$400,000(1.10)=\$440,000, \$400,000(0.96)=\$384,000, and \$200,000(1.05)=\$210,000. Their sum is \$1,034,000, a \$34,000 or \(3.4\%\) increase over current total premium of \$1,000,000.
Step 3: Calculate the completed transition. The completed factors are 1.20, 0.92, and 1.10. Premiums are \$480,000, \$368,000, and \$220,000, summing to \$1,068,000. The completed portfolio increase is \$68,000/\$1,000,000=\(6.8\%\).
Step 4: Interpret the second phase. The additional premium between the first phase and completion is \$1,068,000-\$1,034,000=\$34,000. Relative to the first-phase premium, this is \$34,000/\$1,034,000\(\approx 3.288\%\), not \(3.4\%\). The \(3.4\%\) figure is the additional amount as a share of the original \$1,000,000 base. The raw, unconstrained indications would produce \(400,000(1.30)+400,000(0.84)+200,000(1.10)=\$1,076,000\), or \(7.6\%\). Thus the \(6.8\%\) completed increase is the constrained implementation target, distinct from the raw indication.
Factor Bounds, Phase Fractions, and Premium Impact
A cap stated as a maximum premium factor should not be mistaken for a cap on the percentage change. For example, a maximum factor of 1.20 permits an increase of \(20\%\), not \(120\%\). A minimum factor of 0.95 permits a decrease of \(5\%\), not \(95\%\). When a factor constraint is used, the constrained factor is the raw factor limited to the permitted factor interval.
A phased change also depends on the chosen convention. Under the convention used here, implementing \(75\%\) of a constrained \(12\%\) increase means applying a \(9\%\) first-phase increase, because \(0.75\times 12\%=9\%\). It does not mean multiplying the current rate by 75%, nor does it mean increasing the rate by 75 percentage points. State the convention before calculating, particularly when a schedule specifies several years.
Worked Example: Factor Limits and a Partial First Phase
Method and mistake avoided. Convert the factor limits to their equivalent changes, constrain the raw changes, and only then apply the phase fraction. This prevents confusion between a factor and a percentage change.
A hypothetical product has three segments with current premiums of \$250,000, \$500,000, and \$250,000. Raw indicated factors are 1.32, 0.91, and 1.08. The permitted factor interval for every segment is 0.95 to 1.20, and the first phase implements \(75\%\) of the constrained change.
Step 1: Constrain the factors. The first factor is capped at 1.20, the second is raised to the 0.95 floor, and the third remains 1.08. In change terms, the constrained targets are \(+20\%\), \(-5\%\), and \(+8\%\).
Step 2: Apply the first-phase fraction. The first-phase changes are \(0.75(20\%)=15\%\), \(0.75(-5\%)=-3.75\%\), and \(0.75(8\%)=6\%\). The corresponding factors are 1.15, 0.9625, and 1.06.
Step 3: Re-rate the segments. The resulting premiums are \$250,000(1.15)=\$287,500; \$500,000(0.9625)=\$481,250; and \$250,000(1.06)=\$265,000. Total premium is \$1,033,750, compared with current total premium of \$1,000,000. The first-phase dollar increase is \$33,750 and the portfolio increase is \(3.375\%\).
Check against the indication and constrained target. Raw indicated premiums are \$250,000(1.32)+\$500,000(0.91)+\$250,000(1.08)=\$330,000+\$455,000+\$270,000=\$1,055,000, a \(5.5\%\) indicated increase. Fully constrained premiums would total \$250,000(1.20)+\$500,000(0.95)+\$250,000(1.08)=\$300,000+\$475,000+\$270,000=\$1,045,000, a \(4.5\%\) target increase. The \(3.375\%\) first-phase result is lower because only \(75\%\) of the constrained changes is implemented.
Common Traps and Practical Interpretation
The most important distinction is between an indication and an implementation schedule. The indicated change is the model- and assumption-based result. Caps, floors, and phase limits describe how much of that result is put into rates under the stated plan. A constrained or phased portfolio effect should not be relabelled as the unconstrained actuarial indication.
- Using an unweighted average. Averaging segment changes gives each segment equal influence, regardless of premium. Use current premium weights for the mechanical portfolio premium impact.
- Applying the phase before the constraint. First determine the permitted target; then calculate the phase fraction of that target.
- Mixing factors and changes. A factor of 1.15 is a \(15\%\) increase. A factor floor of 0.95 is a \(-5\%\) change.
- Confusing cumulative and year-over-year changes. A later phase’s change relative to the original rate is not necessarily the same as its change relative to the prior phase’s rate.
- Assuming clipping solves a coupled problem. Independent clipping is appropriate for separate segment bounds, not necessarily when there is also a portfolio-level constraint.
In practice, the premium-impact calculation is conditional on its assumptions. If exposure counts, coverage limits, segment mix, or other rating variables change, then multiplying current premium by a rate factor is not a complete forecast of future written premium. The calculation remains useful as a controlled comparison of rate changes on a fixed base, provided that limitation is stated.
The method of distinguishing raw indications from constrained implementation rates, and of measuring their mechanical premium effects under stated assumptions, is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.
Check Your Understanding
For each hypothetical situation, calculate the requested constrained or phased premium effect. Assume current premiums are fixed weights and there are no other rate changes.
- A segment has current premium of \$200,000 and an indicated change of \(+28\%\). Its cap is \(+18\%\), and the full constrained change is implemented. What is its premium increase in dollars? (Answer: \$36,000)
- Two segments have current premiums of \$300,000 and \$700,000, with indicated changes of \(-14\%\) and \(+12\%\). A floor of \(-6\%\) and cap of \(+10\%\) apply. What is the portfolio percentage change after full implementation? (Answer: \(5.2\%\))
- Two segments have current premiums of \$400,000 and \$600,000, with raw indicated factors of 1.25 and 0.88. The permitted factors are 0.90 to 1.15, and the first phase implements \(40\%\) of each constrained change. What is the first-phase portfolio percentage change? (Answer: \(0.0\%\))
- Two segments have current premiums of \$150,000 and \$350,000, with indicated changes of \(+24\%\) and \(-12\%\). A cap of \(+15\%\) and floor of \(-9\%\) apply. The first phase implements one-third of each constrained change. What is the first-phase dollar and percentage portfolio change? (Answer: a decrease of \$3,000, or \(-0.6\%\))