From Candidate Prices to a Rollout Decision
A renewal recommendation is not complete when a model identifies a price with attractive expected contribution. The price must also satisfy customer-level and portfolio-level limits, perform acceptably under uncertain renewal outcomes, and survive a controlled test. This case follows a hypothetical renewal cohort through those decisions. The calculations illustrate a practical workflow aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.
For a policy offered price \(p_i\), let \(R_i\) equal 1 if the customer renews and 0 otherwise, and let \(C_i\) be the variable cost incurred if the customer renews. Write \(q_i(p_i)=P(R_i=1\mid p_i)\) for the renewal probability at the offered price. The expected variable cost conditional on renewal is \(c_i=E[C_i\mid R_i=1,p_i]\). The fixed-price expected contribution per offered policy is then determined by both the probability of renewal and the margin if renewal occurs.
The formula uses an expected conditional cost, not an assertion that every renewing policy has exactly that cost. To see why it holds, condition on renewal status. When there is no renewal, the contribution is zero under this one-period definition. When there is a renewal, the expected contribution conditional on renewal is the offered premium less expected conditional cost.
If renewal probability or conditional expected cost changes with price, use the value corresponding to each candidate price. In the worked cohort below, costs are treated as deterministic amounts conditional on renewal to make the simulation transparent. In practice, a simulation may also sample claim costs and other uncertain components.
Worked Case: Screen Prices Before Simulating
The hypothetical insurer is considering renewal prices for three segments. For each segment, the table gives the baseline price and renewal probability, an initially proposed price, and a constrained alternative where needed. Expected cost is per policy conditional on renewal. The insurer’s rules limit any individual price increase to 8% and the cohort’s premium-weighted average increase to 6.5%.
| Segment | Policies | Baseline price | Baseline renewal probability | Cost per renewal | Proposed price | Constrained price and renewal probability |
|---|---|---|---|---|---|---|
| A | 100 | $1,000 | 0.92 | $650 | $1,080 | $1,080; 0.90 |
| B | 80 | $800 | 0.86 | $520 | $880 | $840; 0.85 |
| C | 50 | $1,200 | 0.95 | $780 | $1,260 | $1,260; 0.94 |
Worked Example: Applying Customer and Portfolio Limits
Calculate expected contribution at baseline and at the constrained prices. Check the individual price cap and portfolio cap, then determine the expected contribution change.
Step 1: Apply the customer-level cap. Segment B’s initial proposed increase is \(($880-$800)/$800=10\%\), which exceeds the 8% limit. Its maximum allowed price is \($800(1.08)=$864\). The selected constrained price of $840 is within that maximum, with an increase of \(($840-$800)/$800=5\%\). Segment A increases by 8%, and segment C by 5%, so both also satisfy the customer-level rule.
Step 2: Calculate baseline expected contribution. For segment A, the margin upon renewal is \($1,000-$650=$350\), giving \(100(0.92)(350)=$32,200\). For B, it is \($800-$520=$280\), giving \(80(0.86)(280)=$19,264\). For C, it is \($1,200-$780=$420\), giving \(50(0.95)(420)=$19,950\). Total baseline expected contribution is \($32,200+$19,264+$19,950=$71,414\).
Step 3: Calculate constrained-price expected contribution. For A, the renewal margin is \($1,080-$650=$430\), giving \(100(0.90)(430)=$38,700\). For B, it is \($840-$520=$320\), giving \(80(0.85)(320)=$21,760\). For C, it is \($1,260-$780=$480\), giving \(50(0.94)(480)=$22,560\). The constrained plan totals \($38,700+$21,760+$22,560=$83,020\), an expected increase of \($83,020-$71,414=$11,606\).
Step 4: Check the portfolio premium limit. Baseline premium volume is \(100($1,000)+80($800)+50($1,200)=$224,000\). The proposed premium increase is \(100($80)+80($40)+50($60)=$14,200\). The premium-weighted average increase is \($14,200/$224,000=0.063392857\), or 6.3393%. It is below the 6.5% portfolio cap, so the constrained plan passes both stated limits.
Method and mistake avoided. Screen each customer price before calculating the portfolio result, then calculate the portfolio percentage using baseline premium as the denominator. A common error is to approve the 10% increase for B because the total portfolio increase is under its cap; a portfolio limit does not override an individual price limit.
Simulating Renewal Outcomes
Expected contribution describes an average over possible outcomes, not a guaranteed result for the cohort. A simulation makes the uncertainty visible by generating renewal indicators and calculating the contribution in each trial. For a simple model, assume each policy renews independently at its segment’s price-specific probability, and its cost if renewed is the deterministic amount in the table. Independence is an assumption for this calculation, not a universal property of renewal portfolios.
For \(n\) independent policies with the same renewal probability \(q\) and deterministic per-renewal margin \(m=p-c\), total contribution is \(m\) times the number of renewals. The number of renewals has a binomial distribution with mean \(nq\) and variance \(nq(1-q)\). Multiplying by \(m\) gives the contribution mean and variance below.
When segment renewal outcomes are independent of one another, cohort expected contributions and variances add. If outcomes are correlated, for example because a common event affects multiple segments, covariance terms must also be included; simply adding segment variances would understate or overstate total variability depending on those covariances.
Worked Example: Cohort Simulation and Contribution Variability
Use the constrained prices from the prior calculation. Find expected cohort contribution and its standard deviation under the stated independent-renewal assumptions. Then calculate one possible simulated outcome.
Step 1: State the model. Each of the 100 A policies renews independently with probability 0.90 and contributes $430 if renewed. Each of the 80 B policies renews with probability 0.85 and contributes $320 if renewed. Each of the 50 C policies renews with probability 0.94 and contributes $480 if renewed. A nonrenewal contributes zero.
Step 2: Compute expected contribution. The segment means are \(100(0.90)(430)=$38,700\), \(80(0.85)(320)=$21,760\), and \(50(0.94)(480)=$22,560\). Their sum is $83,020, agreeing with the constrained-plan calculation.
Step 3: Compute variance. For A, \(100(0.90)(0.10)(430^2)=100(0.09)(184,900)=$1,664,100\). For B, \(80(0.85)(0.15)(320^2)=10.2(102,400)=$1,044,480\). For C, \(50(0.94)(0.06)(480^2)=2.82(230,400)=$649,728\). Total variance is \($1,664,100+$1,044,480+$649,728=$3,358,308\), in squared dollars. The standard deviation is \(\sqrt{$3,358,308}=$1,832.57\), rounded to the nearest cent.
Step 4: Show a possible trial. Suppose one simulated trial produces 91 A renewals, 67 B renewals, and 47 C renewals. Its contribution is \(91($430)+67($320)+47($480)=$39,130+$21,440+$22,560=$83,130\). This is one possible result, not the expected value or a forecast that the cohort will produce exactly that amount.
Method and mistake avoided. Repeating the independent Bernoulli draws many times produces a distribution of simulated contributions that can be summarized by percentiles or the probability of meeting a target. Do not confuse a single simulated trial with the expected contribution. Also, the calculated standard deviation is for contribution under this model; it is not automatically the standard deviation of the difference versus baseline, which requires modeling the joint outcomes under the two price plans.
Use a Controlled Test to Check the Recommendation
A simulation is conditional on its assumptions, including the renewal response assigned to each price. A controlled test can provide evidence about actual response to a price change. The comparison should be based on randomized assignment, a clearly defined outcome per assigned policy, and consistent measurement in both groups. The test results remain estimates: sampling variation does not disappear just because assignment was randomized.
The following hypothetical test compares two renewal offers for similar policies. Assume random assignment produced 2,000 control offers at $1,000 and 2,000 treatment offers at $1,050. For a compact example, assume deterministic variable cost of $650 conditional on renewal and zero contribution on nonrenewal. The test records 1,720 control renewals and 1,740 treatment renewals.
Worked Example: Test Evidence and a Guardrailed Decision
Estimate renewal and contribution differences, quantify uncertainty, and decide whether the hypothetical results support a phased rollout. The required retention guardrail is that the treatment renewal rate must not be more than 2 percentage points below control, assessed using a one-sided 95% lower confidence bound. The profitability criterion is that the two-sided 95% confidence interval for incremental contribution per assigned policy must be above zero.
Step 1: Estimate renewal rates. The control rate is \(1,720/2,000=0.86\); the treatment rate is \(1,740/2,000=0.87\). The estimated treatment-minus-control difference is \(0.87-0.86=0.01\), or 1 percentage point.
Step 2: Estimate contribution per assigned policy. Control margin on renewal is \($1,000-$650=$350\), so estimated contribution is \(0.86($350)=$301\) per assigned policy. Treatment margin is \($1,050-$650=$400\), giving \(0.87($400)=$348\). Estimated incremental contribution is \($348-$301=$47\) per assigned policy.
Step 3: Calculate the contribution standard error. Under the deterministic-cost model, an assigned policy’s contribution is either zero or the renewal margin. Its variance is \(q(1-q)m^2\). For control this is \(0.86(0.14)(350^2)=0.1204(122,500)=$14,749\). For treatment it is \(0.87(0.13)(400^2)=0.1131(160,000)=$18,096\). With independent randomized groups of 2,000 each, the standard error of the difference in sample mean contributions is
The approximate 95% interval is \($47\pm1.96($4.0525)\), or about \($39.06\) to \($54.94\) per assigned policy. It is entirely above zero, meeting the stated profitability criterion under the model.
Step 4: Check the retention guardrail. The estimated standard error of the renewal-rate difference is \(\sqrt{0.86(0.14)/2,000+0.87(0.13)/2,000}=\sqrt{0.00011675}\approx0.01080\). The one-sided 95% lower bound for treatment minus control is \(0.01-1.645(0.01080)\approx-0.00777\), or about -0.78 percentage points. That is above -2 percentage points, so this test meets the specified noninferiority guardrail.
Decision. Under the stated assumptions and decision rules, the results support a phased rollout rather than an unrestricted immediate change. The point estimate is positive for both renewal and contribution, and both specified statistical criteria pass. Before expanding, verify that actual assignment and offer delivery were consistent with the test design, apply the customer and portfolio price limits, and monitor renewal and contribution by relevant groups. These checks matter because aggregate results can conceal weaker outcomes in a segment, and because the simplified test model treats cost as deterministic.
Method and mistake avoided. Evaluate contribution per policy assigned, not only among customers who renewed; conditioning on renewal would compare selected groups. Also, do not infer that the 1-point observed retention increase is certainly real: its uncertainty is why the guardrail uses a confidence bound.
Putting the Steps Together
A defensible renewal-price recommendation connects constraints, expected economics, uncertainty, and test evidence. Each stage answers a different question: Is the price permitted? What is its expected contribution under the response assumptions? How variable could the cohort result be? Does controlled evidence satisfy pre-specified profitability and customer-impact criteria?
Apply customer-level caps first, then calculate the aggregate portfolio measure using its defined denominator.
Use price-specific renewal probabilities and conditional expected costs; compare the candidate plan with the baseline on the same exposure basis.
Simulate renewal outcomes under explicit assumptions and report distributions or decision-relevant probabilities, not just a single trial.
Use randomized test outcomes, quantify uncertainty, apply the pre-set guardrails, and scale gradually with monitoring.
A recurring exam and practice trap is to treat an attractive expected portfolio result as sufficient evidence for rollout. It is not. A candidate price may breach a customer cap; a portfolio average may mask a poor segment result; a simulation may depend on unsupported independence assumptions; and a test may show a favorable point estimate without ruling out unacceptable harm. State the assumptions, calculate each constraint on its specified basis, and distinguish estimated outcomes from guaranteed outcomes.
Check Your Understanding
Work each problem using the stated assumptions. Amounts are hypothetical and contribution excludes expenses not specified.
- A segment has 60 offered policies, renewal probability 0.80, offered price $900, and expected cost conditional on renewal of $600. Find expected segment contribution. (Answer: $14,400)
- A policy’s baseline price is $1,200 and the customer-level increase cap is 7%. A candidate price is $1,296. Does it pass, and what is the maximum permitted price? (Answer: No; the candidate is an 8% increase, and the maximum permitted price is $1,284.00.)
- For 40 independent policies, renewal probability is 0.90 and deterministic contribution per renewal is $250. Find expected total contribution and its standard deviation. (Answer: Expected contribution = $9,000; standard deviation = \(\sqrt{225,000}\approx$474.34\).)
- A controlled test has 1,000 policies per group. Renewal rates are 0.86 for control and 0.84 for treatment. A noninferiority rule permits a treatment decline of at most 3 percentage points and uses a one-sided 95% lower confidence bound. Does the evidence pass? Use \(1.645\) as the normal critical value. (Answer: No; the difference is -0.02, the standard error is \(\sqrt{0.86(0.14)/1,000+0.84(0.16)/1,000}\approx0.01596\), and the lower bound is approximately \(-0.0463\), below \(-0.03\).)
- A candidate price is $700, conditional renewal cost is $450, and renewal probability is 0.92 for 80 policies. Baseline price is $680 with the same cost and renewal probability 0.94. Find the candidate’s expected contribution and its change from baseline. (Answer: Candidate contribution = $18,400; baseline contribution = $17,296; expected increase = $1,104.)