Rate-Change Caps Relative To Indicated Rates

Why Cap a Price Relative to Its Indicated Rate?

An actuarial indicated rate is a benchmark for the premium supported by the selected rating analysis. An optimized customer price may differ from that benchmark because the pricing objective accounts for customer response or another business objective. A rate-change cap limits how far the selected price may move above or below the indicated rate. It therefore turns a percentage rule into a feasible dollar interval for each policy.

The reference point matters. A cap stated relative to an indicated rate is not automatically a cap relative to the customer’s current premium, a prior quote, or another policy’s rate. The calculations below treat the indicated rate as the explicit base. All products, policies, and values are hypothetical. This topic is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.

Definition: Let $R_i>0$ be the actuarially indicated rate for policy $i$, expressed on the same premium basis as the price being optimized. Let $p_i$ be the selected price. If the permitted decrease is $d_i$ and the permitted increase is $u_i$, expressed as decimal proportions, the rate-change cap requires $$ -d_i \leq \frac{p_i-R_i}{R_i} \leq u_i. $$ Equivalently, the feasible price interval is $$ R_i(1-d_i) \leq p_i \leq R_i(1+u_i). $$ The decrease and increase limits need not be equal.

For example, a 7% permitted decrease is represented by $d_i=0.07$, while a 12% permitted increase is represented by $u_i=0.12$. The lower and upper dollar bounds are calculated separately. A symmetric cap of 10% would use $d_i=u_i=0.10$, but symmetry should not be assumed unless the rule specifies it.

The indicated rate and optimized price must be comparable. If the indicated rate is an annual premium before a particular fee, but the candidate price includes that fee, comparing them directly can produce a meaningless percentage change. First put both amounts on the same basis—for example, annualized premium for the same coverage and policy term—then apply the cap. The indicated rate is the reference for this constraint even if it is not the customer’s current price.

Calculating a Policy’s Feasible Interval

The calculation is mechanical, but the interpretation is important: the cap defines prices the optimization is allowed to select; it does not prescribe one particular price. A price within the interval is feasible under the cap. A price outside it must be moved to the nearest permitted boundary if the objective favors movement in that direction and no other constraint changes the result.

Worked Example: Asymmetric Caps for Three Policies

A hypothetical insurer evaluates three annual policies. Their indicated rates, permitted decreases and permitted increases are shown below. A separate pricing model proposes the raw optimized prices. Calculate each feasible interval and the price after applying only the rate-change cap.

Policy A. The indicated rate is $R_A=\$800.00$, the maximum decrease is 10%, and the maximum increase is 15%. Thus, $$ L_A=800.00(1-0.10)=800.00(0.90)=\$720.00, $$ $$ U_A=800.00(1+0.15)=800.00(1.15)=\$920.00. $$ The raw optimized price is $\$950.00$, which exceeds the upper bound. The feasible price after applying the cap is $\$920.00$.

Policy B. The indicated rate is $R_B=\$1{,}200.00$, the maximum decrease is 5%, and the maximum increase is 8%. The bounds are $$ L_B=1{,}200.00(0.95)=\$1{,}140.00,\qquad U_B=1{,}200.00(1.08)=\$1{,}296.00. $$ The raw optimized price of $\$1{,}100.00$ is below the lower bound, so the cap-adjusted price is $\$1{,}140.00$.

Policy C. The indicated rate is $R_C=\$650.00$, the maximum decrease is 12%, and the maximum increase is 6%. The bounds are $$ L_C=650.00(0.88)=\$572.00,\qquad U_C=650.00(1.06)=\$689.00. $$ The raw price of $\$660.00$ lies within this interval, so no adjustment is needed.

Check. The selected prices are $\$920.00$, $\$1{,}140.00$, and $\$660.00$. Their changes from the indicated rates are respectively $15\%$, $-5\%$, and approximately $1.5385\%$, each within its policy’s stated limits. The method is to calculate both bounds first and then test the proposed price against them; a common mistake is to apply only the increase limit and overlook the permitted decrease.

Combining Rate-Change Caps With Other Bounds

A rate-change interval may coexist with a separate price floor or ceiling. These restrictions are combined by taking the intersection of their feasible sets. If a separate floor is $F_i$ and a separate ceiling is $C_i$, the resulting interval is $$ \left[\max\{R_i(1-d_i),F_i\},\ \min\{R_i(1+u_i),C_i\}\right]. $$ This interval is feasible only if its lower endpoint does not exceed its upper endpoint. If the intersection is empty, the stated restrictions cannot all be satisfied simultaneously; the conflict must be resolved rather than ignored.

This intersection rule does not change the meaning of the percentage cap. It simply identifies prices that satisfy both the cap and the additional bounds. In particular, a separate floor cannot be used to claim that a price satisfies the rate-change cap when it exceeds the cap’s upper bound.

Worked Example: Intersecting a Cap With a Floor and Ceiling

For a hypothetical policy, the indicated annual rate is $\$950.00$. The allowed decrease is 7% and the allowed increase is 12%. A separate pricing rule imposes a floor of $\$900.00$ and a ceiling of $\$1{,}030.00$. The raw optimized price is $\$1{,}080.00$. Find the combined feasible interval and the selected price when the raw price is preferred.

First calculate the rate-change bounds: $$ L_{\text{cap}}=950.00(1-0.07)=950.00(0.93)=\$883.50, $$ $$ U_{\text{cap}}=950.00(1+0.12)=950.00(1.12)=\$1{,}064.00. $$ The separate bounds give the interval $[\$900.00,\$1{,}030.00]$. The lower endpoint of the intersection is $$ \max\{\$883.50,\$900.00\}=\$900.00, $$ and the upper endpoint is $$ \min\{\$1{,}064.00,\$1{,}030.00\}=\$1{,}030.00. $$ The combined feasible interval is therefore $[\$900.00,\$1{,}030.00]$. Since the raw price $\$1{,}080.00$ exceeds the combined upper endpoint, the selected price is $\$1{,}030.00$. Its change from the indicated rate is $(1{,}030-950)/950=0.0842105$, or approximately 8.4211%, which is below the 12% cap and within the separate ceiling.

Check. The combined lower bound is above the cap’s lower bound, and the combined upper bound is below the cap’s upper bound. The interval is nonempty, and $\$1{,}030.00$ satisfies all four stated bounds. The common mistake is to check the cap but forget to intersect it with the separate floor and ceiling.

How a Cap Restricts an Optimized Price

Suppose the pricing objective for one policy is a function $f(p)$, and the unconstrained maximizing price is $p^*$. When $f$ is concave over the feasible range, the constrained maximizer on an interval $[L,U]$ is the unconstrained maximizer if it lies inside the interval; otherwise, it is the nearest endpoint in the direction of $p^*$. In compact form, this is

p_{\text{c}}=\min\{U,\max\{L,p^*\}\}.

Here $p_{\text{c}}$ is the constrained price. Concavity matters: it ensures that the objective does not rise again farther from its maximum. If the objective is not concave or has multiple local maxima, simply clipping one candidate optimum is not sufficient; the objective must be compared over all relevant feasible candidates, including the boundaries.

Justification: For a concave objective that increases up to its unconstrained maximum $p^*$ and decreases after it, three cases exhaust the possibilities. If $L \leq p^* \leq U$, the unconstrained maximum remains feasible. If $p^*<L$, the objective is decreasing throughout $[L,U]$, so its largest value there is at $L$. If $p^*>U$, the objective is increasing throughout $[L,U]$, so its largest value there is at $U$. These cases give $p_{\text{c}}=\min\{U,\max\{L,p^*\}\}$.

Worked Example: A Binding Cap and the Objective Cost

Consider two hypothetical policies priced independently. Their contribution objectives, in dollars, are stipulated to be concave quadratic functions. Policy A has indicated rate $\$1{,}000.00$, a 5% maximum decrease, an 8% maximum increase, and objective $f_A(p)=500-0.01(p-1{,}200)^2$. Policy B has indicated rate $\$900.00$, a 10% maximum decrease, a 5% maximum increase, and objective $f_B(p)=400-0.02(p-850)^2$. Find the constrained prices and total contribution, and compare with the unconstrained total.

For Policy A, the interval is $$ [1{,}000(0.95),1{,}000(1.08)]=[\$950.00,\$1{,}080.00]. $$ The quadratic is maximized at $p_A^*=\$1{,}200.00$, outside the interval above its upper endpoint. Because the function is concave, its maximum over the feasible interval occurs at $\$1{,}080.00$. The constrained contribution is $$ f_A(1{,}080)=500-0.01(1{,}080-1{,}200)^2 =500-0.01(-120)^2 =500-144 =\$356.00. $$

For Policy B, the interval is $$ [900(0.90),900(1.05)]=[\$810.00,\$945.00]. $$ The unconstrained maximum, $p_B^*=\$850.00$, is inside the interval, so the constrained price is $\$850.00$ and its contribution is $$ f_B(850)=400-0.02(850-850)^2=\$400.00. $$ The total constrained contribution is $\$356.00+\$400.00=\$756.00$. Without caps, the two unconstrained maximum contributions would total $\$500.00+\$400.00=\$900.00$. The cap therefore reduces the total by $\$900.00-\$756.00=\$144.00$, entirely from Policy A. Policy A’s cap binds at its upper endpoint; Policy B’s cap does not bind.

Check. Policy A’s selected change is $(1{,}080-1{,}000)/1{,}000=8\%$, exactly its maximum increase. Policy B’s change is $(850-900)/900=-5.5556\%$, which is between $-10\%$ and $5\%$. The method uses the objective’s maximizing price and the feasible interval; the common mistake is to assume every policy must be priced at a cap boundary.

Common Mistake: Using the Wrong Reference Premium

A percentage change is meaningful only after its denominator has been identified. Suppose a policy has an indicated rate of $\$1{,}000.00$, a current premium of $\$900.00$, and a permitted range from 5% below to 10% above the indicated rate. The correct interval is $$ [1{,}000(0.95),1{,}000(1.10)]=[\$950.00,\$1{,}100.00]. $$ A mistaken calculation applies the percentages to the current premium, giving $[900(0.95),900(1.10)]=[\$855.00,\$990.00]$. That is the wrong interval because the cap is relative to the indicated rate. For example, a proposed price of $\$1{,}020.00$ is feasible under the stated cap: its increase from the indicated rate is 2%. The mistaken interval would incorrectly reject it as too high.

Other checks help prevent errors: convert percentages to proportions before multiplying; calculate lower and upper bounds separately when the percentages are asymmetric; retain cents through the calculation and round only the final reported premium; and check the selected price’s percentage change against the indicated rate. A cap does not, by itself, imply that a price should be raised or reduced. It only limits the permitted range.

Key takeaway: For each policy, identify the indicated-rate basis, convert each percentage limit into a dollar endpoint, and test the proposed price against the resulting interval. If other bounds apply, intersect the intervals before selecting a price.

Check Your Understanding

For each hypothetical case, calculate the requested bound or selected price using the indicated rate as the reference.

A policy has an indicated annual rate of $\$740.00$, a maximum decrease of 8%, and a maximum increase of 12%. Find the feasible interval. (Answer: $[\$680.80,\$828.80]$)
A policy’s indicated rate is $\$1{,}250.00$, with a 6% maximum decrease and a 9% maximum increase. A raw optimized price is $\$1{,}140.00$. Calculate the interval and the price after applying the cap. (Answer: interval $[\$1{,}175.00,\$1{,}362.50]$; selected price $\$1{,}175.00$)
An indicated rate is $\$960.00$, with a 4% maximum decrease and an 11% maximum increase. A separate floor is $\$950.00$, a separate ceiling is $\$1{,}040.00$, and the raw price is $\$1{,}070.00$. Find the combined interval and selected price. (Answer: interval $[\$950.00,\$1{,}040.00]$; selected price $\$1{,}040.00$)
A concave objective has its unconstrained maximum at $\$1{,}400.00$. The indicated rate is $\$1{,}280.00$, with a 5% maximum decrease and a 7% maximum increase. Find the constrained maximizing price and state whether the cap binds. (Answer: $\$1{,}369.60$; the upper cap binds)
A policy has an indicated rate of $\$1{,}200.00$ and a permitted range from 8% below to 5% above that rate. A proposed price is $\$1{,}176.00$. Calculate its percentage change from the indicated rate and determine whether it is feasible. (Answer: $-2\%$; feasible, since the change is between $-8\%$ and $5\%$)

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