Simulating Renewal Cohorts Under Alternative Price Plans

From Price Choices to Renewal Projections

A price plan is not evaluated only by the premium charged to a customer who renews. The price may also change the probability that the policy renews. A useful projection therefore applies each candidate price’s renewal probability to the policies eligible to renew, then aggregates the resulting expected counts and financial amounts. This connects customer-level price decisions with projected cohort outcomes.

The calculations are expectations, not promises about the exact number of policies that will renew. A cohort may have a projected count such as 73.5 renewals; that is a valid expected value even though the realized count must be an integer. Keep the projection at its expected value through the calculations, and round money only when reporting the final result.

Definition: A renewal cohort is a group of policies eligible to renew that share the inputs used in a projection, such as candidate price and expected loss cost. For cohort $j$, let $N_j$ be the number of eligible policies, $r_j$ the annual renewal price under the plan, $q_j$ the probability of renewal at that price, and $L_j$ the expected loss cost for each policy that renews.

For the basic projection, assume each eligible policy has the cohort’s specified renewal probability and, conditional on renewal, the cohort’s specified expected loss cost. Treat the annual renewal price as fully written for each renewal. Ignore cancellations, midterm changes, taxes, and fixed expenses unless the problem explicitly includes them. Loss costs and expenses must be measured on a consistent basis; for example, do not subtract expenses twice if they are already included in a stated cost.

Key result: Under these assumptions, expected renewal count is $N_jq_j$, expected written premium is $N_jq_jr_j$, and expected losses are $N_jq_jL_j$. If there are multiple cohorts, add each cohort’s expected amount. With no expenses included, expected contribution is expected written premium less expected losses.

\begin{aligned} E[\text{renewals in cohort }j] &= N_jq_j,\\ E[\text{written premium in cohort }j] &= N_jq_jr_j,\\ E[\text{losses in cohort }j] &= N_jq_jL_j,\\ E[\text{contribution}] &= \sum_j N_jq_j(r_j-L_j). \end{aligned}

Why the Aggregation Formula Works

The calculation follows from indicator variables and linearity of expectation. For each eligible policy $k$ in cohort $j$, define $I_{jk}=1$ if it renews and $I_{jk}=0$ otherwise. Then $E[I_{jk}]=q_j$. The cohort’s renewal count is the sum of these indicators, so its expected count is the sum of their expectations. Independence among policy renewal decisions is not required for this expectation.

E\left[\sum_{k=1}^{N_j}I_{jk}\right] =\sum_{k=1}^{N_j}E[I_{jk}] =N_jq_j.

Each renewal contributes $r_j$ of written premium, so the cohort’s written premium is $r_j\sum_k I_{jk}$. If $X_{jk}$ is the loss for a policy that renews and the conditional expected loss is $E[X_{jk}\mid I_{jk}=1]=L_j$, then $E[I_{jk}X_{jk}]=q_jL_j$. Summing across policies gives expected losses $N_jq_jL_j$. This uses the conditional loss-cost assumption; if renewal changes the expected loss cost within a cohort, use the appropriate conditional expected loss rather than an unrelated overall average.

\begin{aligned} E\left[r_j\sum_{k=1}^{N_j}I_{jk}\right] &=r_jN_jq_j,\\ E[I_{jk}X_{jk}] &=P(I_{jk}=1)E[X_{jk}\mid I_{jk}=1] =q_jL_j. \end{aligned}

For a plan with a variable expense rate $a_j$ applied to written premium and a fixed expense $e_j$ for each renewed policy, expected contribution becomes $\sum_jN_jq_j[r_j(1-a_j)-L_j-e_j]$. This version assumes the variable expense is proportional to premium and the per-policy expense is incurred only on renewal. State the expense basis before using the formula.

E[\text{contribution}] =\sum_jN_jq_j\left[r_j(1-a_j)-L_j-e_j\right].

Projecting and Comparing Candidate Plans

To compare plans, calculate every outcome using the renewal probability associated with that plan’s candidate price. Do not carry a current-price retention rate over to a new price unless the scenario explicitly says that the probability is unchanged. Keep the eligible count and the cohort’s loss-cost assumption visible in the calculation, so a price-related change in renewal probability is not confused with a change in expected cost per renewal.

Worked Example: Comparing Two Plans Across Two Cohorts

A hypothetical insurer has 1,000 small-equipment policies and 600 specialty-equipment policies eligible to renew. Under Plan A, the respective prices are $1,200 and $1,500, with renewal probabilities 0.80 and 0.70. Expected loss costs conditional on renewal are $700 and $950. Under Plan B, prices are $1,260 and $1,440, with renewal probabilities 0.74 and 0.65. Assume loss costs are unchanged between plans and there are no expenses. Project renewals, written premium, losses, and contribution under each plan. Method: Apply each plan’s own retention probability by cohort. The mistake to avoid is using one plan’s renewal probabilities for both plans.

Plan A, small-equipment cohort. Expected renewals are $1{,}000(0.80)=800$. Written premium is $800(\$1{,}200)=\$960{,}000$. Expected losses are $800(\$700)=\$560{,}000$, giving contribution of $\$960{,}000-\$560{,}000=\$400{,}000$.

Plan A, specialty-equipment cohort. Expected renewals are $600(0.70)=420$. Written premium is $420(\$1{,}500)=\$630{,}000$. Expected losses are $420(\$950)=\$399{,}000$, giving contribution of $\$630{,}000-\$399{,}000=\$231{,}000$.

Plan A total. Expected renewals are $800+420=1{,}220$. Written premium is $\$960{,}000+\$630{,}000=\$1{,}590{,}000$; expected losses are $\$560{,}000+\$399{,}000=\$959{,}000$. Contribution is $\$1{,}590{,}000-\$959{,}000=\$631{,}000$.

Plan B, small-equipment cohort. Expected renewals are $1{,}000(0.74)=740$. Written premium is $740(\$1{,}260)=\$932{,}400$. Expected losses are $740(\$700)=\$518{,}000$, giving contribution of $\$414{,}400$.

Plan B, specialty-equipment cohort. Expected renewals are $600(0.65)=390$. Written premium is $390(\$1{,}440)=\$561{,}600$. Expected losses are $390(\$950)=\$370{,}500$, giving contribution of $\$191{,}100$.

Plan B total and check. Expected renewals are $740+390=1{,}130$; written premium is $\$932{,}400+\$561{,}600=\$1{,}494{,}000$; expected losses are $\$518{,}000+\$370{,}500=\$888{,}500$. Thus contribution is $\$1{,}494{,}000-\$888{,}500=\$605{,}500$, which equals $\$414{,}400+\$191{,}100$. Plan A’s projected contribution is $\$25{,}500$ higher, despite Plan B’s higher small-equipment contribution.

Including Expenses in Contribution

Premium minus expected losses is not necessarily the contribution measure used for a pricing decision. If the scenario provides variable expenses or expenses per renewed policy, include each once and apply it to the correct base. A percentage expense applies to written premium, while a per-policy expense applies to expected renewed policies. Neither should be applied to every eligible policy if it is incurred only when a policy renews.

Worked Example: Applying Expenses to Expected Renewals

A hypothetical renewal block has 400 policies in Cohort A and 250 in Cohort B. Under the selected plan, A has a $1,500 price, renewal probability 0.80, and expected loss cost $900. B has a $1,900 price, renewal probability 0.68, and expected loss cost $1,300. For both cohorts, variable expense is 10% of written premium and additional service expense is $60 per renewed policy. Calculate expected contribution. Method: Calculate premium, losses, percentage expense, and per-renewal expense separately. The mistake to avoid is charging per-policy expenses on all eligible policies rather than expected renewals.

Cohort A. Expected renewals are $400(0.80)=320$. Written premium is $320(\$1{,}500)=\$480{,}000$, and expected losses are $320(\$900)=\$288{,}000$. Variable expense is $0.10(\$480{,}000)=\$48{,}000$, and service expense is $320(\$60)=\$19{,}200$. Contribution is $\$480{,}000-\$288{,}000-\$48{,}000-\$19{,}200=\$124{,}800$.

Cohort B. Expected renewals are $250(0.68)=170$. Written premium is $170(\$1{,}900)=\$323{,}000$, and expected losses are $170(\$1{,}300)=\$221{,}000$. Variable expense is $0.10(\$323{,}000)=\$32{,}300$, and service expense is $170(\$60)=\$10{,}200$. Contribution is $\$323{,}000-\$221{,}000-\$32{,}300-\$10{,}200=\$59{,}500$.

Total and check. Total expected contribution is $\$124{,}800+\$59{,}500=\$184{,}300$. Across both cohorts, premium is $\$803{,}000$, losses are $\$509{,}000$, variable expense is $\$80{,}300$, and service expense is $\$29{,}400$. Subtracting the latter three amounts from premium also gives $\$184{,}300$.

Use Cohort Weights, Not a Simple Average

Portfolio renewal probability is a count-weighted average of cohort probabilities when the eligible policy count is the weighting basis. A simple average gives a small cohort the same influence as a large cohort, which generally does not describe the projected portfolio. Likewise, portfolio premium and losses must be summed from cohort amounts; applying a portfolio-average price or loss cost without appropriate weights can misstate the result.

Worked Example: Checking the Portfolio Retention Rate

A hypothetical insurer has three renewal cohorts: 500 policies with a $1,100 price, 300 with a $1,500 price, and 200 with a $2,200 price. Their respective renewal probabilities are 0.90, 0.70, and 0.50; conditional expected loss costs are $600, $900, and $1,400. With no expenses, calculate total expected renewals, the portfolio renewal rate, written premium, losses, and contribution. Method: Aggregate cohort expectations and divide expected renewals by total eligible policies. The mistake to avoid is using the unweighted average of the three retention rates as the portfolio rate.

Expected counts. The three expected renewal counts are $500(0.90)=450$, $300(0.70)=210$, and $200(0.50)=100$. Their sum is $450+210+100=760$. There are $500+300+200=1{,}000$ eligible policies, so the projected portfolio renewal rate is $760/1{,}000=0.76$, or 76%.

Financial totals. Written premium is $450(\$1{,}100)+210(\$1{,}500)+100(\$2{,}200)=\$495{,}000+\$315{,}000+\$220{,}000=\$1{,}030{,}000$. Expected losses are $450(\$600)+210(\$900)+100(\$1{,}400)=\$270{,}000+\$189{,}000+\$140{,}000=\$599{,}000$. Contribution is $\$1{,}030{,}000-\$599{,}000=\$431{,}000$.

Check. The unweighted average renewal probability is $(0.90+0.70+0.50)/3=0.70$, which would incorrectly imply 700 renewals. The count-weighted calculation gives 760, consistent with the larger cohort having the highest renewal probability.

Common Projection Errors

A cohort simulation is simple arithmetic, but its output can still be misleading if the inputs or bases are inconsistent. Use the following checks before comparing plans:

Match retention to price. Use the renewal probability associated with each candidate price and cohort, not a single portfolio-wide rate by default.
Keep the denominator explicit. Renewal probability applies to eligible policies. Written premium and expected losses apply to expected renewals.
Use conditional loss costs. $L_j$ is expected loss cost per renewing policy under the stated assumptions, not total cohort losses or a cost per eligible policy.
Separate expected values from realized counts. Do not round an expected count such as 73.5 to 74 before calculating premium or losses. Such rounding changes the projection.
Check the contribution definition. Confirm whether expenses are included, and whether a supplied loss-cost figure already contains any expense or adjustment being subtracted separately.
Reconcile the totals. Total premium, losses, and contribution should equal the sums of their cohort values; contribution should also equal premium less every included cost.

Key takeaway: For each candidate plan, apply its renewal probability to each cohort’s eligible count, then value the expected renewals using the applicable price, conditional loss cost, and stated expenses. Aggregate cohort amounts only after applying the cohort-specific inputs.

Check Your Understanding

For each hypothetical projection, use expected counts without rounding and assume no expenses unless specified.

A cohort has 240 eligible policies, a $900 renewal price, a 0.75 renewal probability, and a $520 expected loss cost per renewal. Find expected renewals, written premium, expected losses, and contribution. (Answer: 180 renewals; $162,000 premium; $93,600 losses; $68,400 contribution.)
Cohort A has 150 eligible policies, renewal probability 0.80, price $1,200, and expected loss cost $700. Cohort B has 100 eligible policies, renewal probability 0.60, price $1,600, and expected loss cost $1,000. Find total expected renewals, premium, losses, and contribution. (Answer: 180 renewals; $240,000 premium; $144,000 losses; $96,000 contribution.)
A cohort has 500 eligible policies, a $1,400 price, a 0.82 renewal probability, and expected losses of $850 per renewal. Variable expense is 8% of written premium. Find expected contribution after variable expense. (Answer: 410 renewals; $574,000 premium; $348,500 losses; $179,580 contribution.)
Cohort A has 300 eligible policies with renewal probability 0.90, price $1,000, and expected loss cost $500. Cohort B has 200 eligible policies with renewal probability 0.50, price $1,800, and expected loss cost $1,200. Find the weighted portfolio renewal rate and total contribution, and compare the weighted rate with the unweighted average. (Answer: 370 renewals; 74.00% weighted renewal rate versus 70.00% unweighted; $195,000 contribution.)
Under Plan X, Cohort A has 100 eligible policies, renewal probability 0.80, price $2,000, and expected loss cost $1,200; Cohort B has 200 eligible policies, renewal probability 0.60, price $1,500, and expected loss cost $900. Under Plan Y, A’s renewal probability and price are 0.70 and $2,200; B’s are 0.65 and $1,400. Loss costs do not change. Calculate each plan’s contribution and identify which is higher. (Answer: Plan X has $136,000 contribution; Plan Y has $135,000; Plan X is higher by $1,000.)

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