Measuring Profitability and Retention Together

Why Measure Profitability and Retention Together?

A price test can improve one outcome while worsening another. A higher renewal price may increase contribution from customers who renew, but it may also lead more customers to leave. A lower price may retain more customers while reducing contribution per policy. Neither a retention rate nor a contribution figure, viewed by itself, gives a complete account of the test outcome.

The comparison must also use a clearly defined population and observation period. For example, a retention rate might mean the fraction of eligible customers still insured on a specified date after renewal. Contribution might include earned premium and incurred losses over the same post-renewal period, with variable expenses treated consistently. If the two measures use different populations or time windows, their apparent relationship can be misleading.

The calculations below compare groups assigned to alternative price plans. They describe observed outcomes over the stated period; a causal interpretation depends on the test design and analysis supporting that comparison. This material is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.

Definition: For a specified observation period, define an account's contribution as earned premium minus incurred losses and the variable expenses included in the analysis. Define retention as the proportion of eligible accounts in the assigned cohort that meet a stated in-force or renewal condition on a stated date. Report both outcomes using the original eligible assigned cohort as the denominator where applicable.

Set the Measurement Rules Before Comparing Plans

Let $g$ denote a test group, such as a control price plan or a treatment price plan, and let $N_g$ be the number of eligible accounts assigned to that group. For account $i$, let $C_{ig}$ be its contribution over the chosen observation period. A consistent account-level definition is

C_{ig}=\text{earned premium}_{ig}-\text{incurred losses}_{ig}-\text{included variable expenses}_{ig}.

The group’s average contribution per assigned eligible account and its retention rate are

\bar{C}_g=\frac{\sum_{i=1}^{N_g} C_{ig}}{N_g}, \qquad \hat{r}_g=\frac{\sum_{i=1}^{N_g} R_{ig}}{N_g},

where $R_{ig}=1$ if account $i$ meets the prespecified retention condition and $R_{ig}=0$ otherwise. Comparing treatment with control gives an incremental contribution of $\bar{C}_T-\bar{C}_C$ per assigned eligible account and a retention difference of $\hat{r}_T-\hat{r}_C$. State the units: contribution is dollars per eligible account over the period; retention difference is usually expressed in percentage points.

The assigned cohort is important. If the denominator is changed to include only accounts that renewed, the comparison excludes the customers most directly affected by a price change. For contribution, define what happens to an account that cancels or does not renew: it may still have earned premium and losses before leaving during the observation period. Those amounts should not disappear merely because the account is no longer in force at the endpoint.

Worked Example: A Higher Price With Lower Retention

A hypothetical insurer assigns 1,000 eligible accounts to each of two renewal price plans. Over the same post-renewal period, 800 control accounts and 760 treatment accounts renew. Control accounts produce $960,000 of earned premium, $480,000 of incurred losses, and $96,000 of variable expenses. Treatment accounts produce $950,000 of earned premium, $437,000 of incurred losses, and $95,000 of variable expenses. Calculate contribution per assigned account and retention for each plan, then compare them.

Method. Calculate each group’s total contribution on the same premium, loss, and expense basis; divide by all 1,000 assigned accounts. Separately calculate retention using the same 1,000-account denominators.

Control contribution. Total contribution is $ \$960{,}000-\$480{,}000-\$96{,}000=\$384{,}000$. Per assigned account, this is $ \$384{,}000/1{,}000=\$384.00$.

Treatment contribution. Total contribution is $ \$950{,}000-\$437{,}000-\$95{,}000=\$418{,}000$. Per assigned account, this is $ \$418{,}000/1{,}000=\$418.00$.

Retention. Control retention is $800/1{,}000=80\%$; treatment retention is $760/1{,}000=76\%$. The treatment-control contribution difference is $ \$418.00-\$384.00=\$34.00$ per assigned account. The retention difference is $76\%-80\%=-4$ percentage points.

Interpretation and check. The treatment has higher measured contribution per eligible account over this period, but lower retention. The total contribution difference is $ \$418{,}000-\$384{,}000=\$34{,}000$, which also equals $1{,}000\times \$34.00$. The common mistake avoided is declaring the treatment unambiguously better based only on its contribution increase, or unambiguously worse based only on its retention decline. These are distinct outcomes that must be considered together.

Why the Cohort Average Combines Retained and Nonretained Outcomes

The group average contribution includes every account in the assigned cohort. To see what it captures, divide the group into accounts that meet the retention condition and those that do not. Let $r$ be the group’s retention proportion, and let $\bar{C}_{R}$ and $\bar{C}_{NR}$ be average contribution for retained and nonretained accounts, respectively, over the same period. The average is

\bar{C}=r\bar{C}_{R}+(1-r)\bar{C}_{NR}.

This identity follows by adding contribution across the two disjoint sets. If there are $N_R$ retained and $N_{NR}$ nonretained accounts, their total contribution is $N_R\bar{C}_{R}+N_{NR}\bar{C}_{NR}$. Dividing by $N=N_R+N_{NR}$ gives the identity because $N_R/N=r$ and $N_{NR}/N=1-r$. It applies to a common cohort and measurement period, including any contribution generated by nonretained accounts during that period.

A frequently used simplification, $\bar{C}=r\bar{C}_{R}$, is valid only when nonretained accounts have zero contribution over the measured period. That may be a reasonable definition for a strictly post-renewal period in which nonrenewing accounts generate no post-renewal premium, losses, or included expenses. It is not automatically valid for a broader period that includes time before cancellation or expiration.

This distinction explains why multiplying retention by contribution among retained accounts can produce the wrong answer. The product omits nonretained contribution unless it is genuinely zero. In addition, contribution among retained accounts is conditional on an outcome that may itself respond to price; it is not the same measure as contribution averaged over everyone assigned to the plan.

Worked Example: Keeping the Observation Window Consistent

In a hypothetical test, 500 eligible accounts are assigned to each plan. Retention is measured at the end of month 12. During the full 12-month observation period, the treatment group has $420,000 of earned premium, $250,000 of incurred losses, and $42,000 of variable expenses. The control group has $400,000 of earned premium, $230,000 of incurred losses, and $40,000 of variable expenses. At month 12, 330 treatment accounts and 350 control accounts are in force. Calculate both outcomes on the assigned cohorts.

Method. Include period contribution for every assigned account, including any contribution before a cancellation. Use the original 500 accounts in each group as the denominator for both group averages and retention.

Treatment. Total contribution is $ \$420{,}000-\$250{,}000-\$42{,}000=\$128{,}000$, so contribution per assigned account is $ \$128{,}000/500=\$256.00$. Month-12 retention is $330/500=66\%$.

Control. Total contribution is $ \$400{,}000-\$230{,}000-\$40{,}000=\$130{,}000$, so contribution per assigned account is $ \$130{,}000/500=\$260.00$. Month-12 retention is $350/500=70\%$.

The treatment-control contribution difference is $ \$256.00-\$260.00=-\$4.00$ per assigned account, and its retention difference is $66\%-70\%=-4$ percentage points. As a check, the total contribution difference is $-\$2{,}000$, equal to $500\times(-\$4.00)$.

Common mistake avoided. Dividing treatment contribution by the 330 accounts in force gives $ \$128{,}000/330=\$387.88$ per retained account; dividing control contribution by 350 gives $ \$130{,}000/350=\$371.43$. That comparison is not contribution per eligible account, and it selects different subsets of the assigned cohorts. The correct assigned-cohort comparison is $256.00 versus $260.00. The retained-account figures may answer a separate conditional question, but they do not replace the cohort measure.

When a Retention-Weighted Comparison Is Appropriate

A retention-weighted calculation can be useful if its assumptions match the outcome definition. Suppose contribution is measured only during a post-renewal period, and nonrenewing accounts generate no contribution during that period. Then $\bar{C}_{NR}=0$, so the decomposition reduces to $\bar{C}=r\bar{C}_{R}$. This can clarify why both the likelihood of retaining an account and its contribution if retained matter.

Worked Example: Comparing Plans When Nonrenewers Have Zero Post-Renewal Contribution

Two hypothetical renewal plans each have 400 eligible accounts. For this example, the measurement window starts after renewal, and accounts that do not renew have zero contribution in that window. Under Plan A, 360 accounts renew and average $300 of contribution each during the window. Under Plan B, 300 accounts renew and average $400 each. Find each plan’s retention and contribution per eligible account, and compare the results.

Method. Because nonrenewers have zero contribution in the defined post-renewal window, multiply the retention proportion by average contribution among renewers. Also verify the result using total contribution divided by 400.

Plan A. Retention is $360/400=90\%$. Total contribution is $360\times \$300=\$108{,}000$, giving $ \$108{,}000/400=\$270.00$ per eligible account. Equivalently, $0.90\times \$300=\$270.00$.

Plan B. Retention is $300/400=75\%$. Total contribution is $300\times \$400=\$120{,}000$, giving $ \$120{,}000/400=\$300.00$ per eligible account. Equivalently, $0.75\times \$400=\$300.00$.

Plan B has retention 15 percentage points lower but contribution per eligible account $30.00 higher over this defined window. The totals check: $ \$120{,}000-\$108{,}000=\$12{,}000$, and $400\times \$30.00=\$12{,}000$.

Common mistake avoided. Reporting only the $400 versus $300 contribution among renewers would hide the lower retention under Plan B. Conversely, using retention alone would hide its higher contribution conditional on renewal. The product calculation is justified here only by the stated zero-contribution assumption for nonrenewers during this window; it should not be carried over to a period in which they can generate contribution before leaving.

Practical Checks Before Interpreting Results

Before comparing plans, write down the measurement rules and check that both groups follow them. In particular, specify the start and end dates, what qualifies as retained, and how earned premium, incurred losses, and variable expenses are assigned when an account cancels or changes status. Include incremental costs in the contribution definition if they are part of the decision being assessed; do not mix a contribution measure that includes such costs in one group with one that omits them in another.

Keep the contribution and retention denominators visible. A retention rate generally uses the eligible assigned accounts in the group, not only accounts that received a final offer or remained insured through the end, unless the question explicitly defines a different population. Similarly, if the purpose is to assess the plan across the assigned cohort, divide cohort contribution by assigned eligible accounts rather than by the number retained.

Finally, a point estimate is not by itself evidence that a difference is reliable or repeatable. The comparisons here describe the calculated outcomes; uncertainty and test design also matter when deciding how much weight to place on observed differences. The practical value of reporting both measures is that it makes the central trade-off visible without treating one outcome as a substitute for the other.

Key takeaway: Compare contribution per assigned eligible account and retention over explicitly stated, consistent observation periods. Contribution captures financial outcomes across the cohort; retention captures how many accounts meet the stated in-force condition. Read the two measures together, and use retention-weighted contribution only when the nonretained-account contribution assumption is valid.

Check Your Understanding

For each problem, use the stated cohort and observation window. Assume the figures are hypothetical and that no additional costs should be included unless specified.

A test assigns 200 eligible accounts to each plan. Control has 150 renewals, with total earned premium of $135,000, losses of $49,500, and variable expenses of $13,500. Treatment has 140 renewals, with total earned premium of $137,200, losses of $49,000, and variable expenses of $13,720. Find contribution per assigned account and retention for both groups, plus the treatment-control differences. (Answer: Control contribution $360.00 and retention 75%; treatment contribution $372.40 and retention 70%; differences are +$12.40 per account and -5 percentage points.)
In a 12-month cohort of 300 eligible accounts per plan, treatment contribution totals $75,000 and 195 accounts are in force at month 12. Control contribution totals $78,000 and 210 accounts are in force. Calculate both outcomes per plan and the differences. (Answer: Treatment $250.00 per eligible account and 65% retention; control $260.00 and 70%; treatment-control differences are -$10.00 and -5 percentage points.)
In a post-renewal window, nonrenewers generate zero contribution. Plan A retains 240 of 300 accounts, and retained accounts average $275 contribution. Plan B retains 210 of 300 and retained accounts average $330. Calculate contribution per eligible account and retention, then state which plan has the greater measured contribution. (Answer: Plan A retention 80%, contribution $220.00 per eligible account; Plan B retention 70%, contribution $231.00; Plan B is higher by $11.00 per eligible account.)
A study reports total 12-month contribution of $96,000 for 400 assigned treatment accounts, of whom 280 are in force at month 12. What is the treatment’s contribution per assigned account and retention? Why is dividing contribution by 280 not the requested cohort measure? (Answer: $240.00 per assigned account and 70% retention; dividing by 280 gives a retained-account denominator rather than the assigned-cohort denominator.)
In a defined post-renewal window, 80% of eligible accounts are retained and average contribution among retained accounts is $250. Nonretained accounts also generate an average of $20 contribution during the window. Calculate average contribution per eligible account. (Answer: $0.80\times \$250+0.20\times \$20=\$204.00$; multiplying retention by retained-account contribution alone would incorrectly give $200.00.)

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