Calculating Expected Contribution Across Candidate Prices

Why Expected Contribution Depends on Renewal Probability

A candidate renewal premium does not produce premium revenue for every eligible policy. Some customers renew and generate premium, losses, and renewal-related expenses; others do not renew. A price decision therefore requires more than subtracting expected losses and expenses from the quoted premium. It must also account for the probability that the policy renews.

The key distinction is between contribution conditional on renewal and expected contribution per eligible policy. The first describes the economics of a policy that renews. The second averages over all policies offered the price, including those that lapse. For a portfolio, expected contribution is then summed across eligible policies or groups.

The calculations below use a one-term renewal horizon. Premium, loss, and renewal-related expenses are measured on the same basis for that term. A lapse generates no renewal premium, renewal loss, or renewal expense. The treatment of costs incurred even when a policy lapses is handled separately.

Definition: For eligible policy $i$ offered candidate premium $p_{ij}$, let $q_{ij}$ be its predicted renewal probability, $L_{ij}$ its expected loss conditional on renewal at that candidate price, $e_{ij}$ its fixed expense per renewal, and $\alpha_{ij}$ its variable expense rate as a proportion of premium. Conditional contribution is $(1-\alpha_{ij})p_{ij}-L_{ij}-e_{ij}$. If no costs are incurred when the policy does not renew, expected contribution per eligible policy is $q_{ij}[(1-\alpha_{ij})p_{ij}-L_{ij}-e_{ij}]$.

Deriving the Expected Contribution Formula

The formula follows from averaging the renewal outcome and the lapse outcome. Let $R_{ij}$ equal 1 if policy $i$ renews at candidate price $j$, and 0 otherwise. Thus $P(R_{ij}=1)=q_{ij}$. Let $X_{ij}$ be the loss for the renewal term, with $E[X_{ij}\mid R_{ij}=1]=L_{ij}$. If the policy renews, its contribution is premium less loss and expenses; if it lapses, its renewal-term contribution is zero.

C_{ij}=R_{ij}\bigl(p_{ij}-X_{ij}-e_{ij}-\alpha_{ij}p_{ij}\bigr)

Taking expectations, condition on whether renewal occurs. The renewal outcome has probability $q_{ij}$, and the lapse outcome contributes zero. Therefore:

\begin{aligned} E[C_{ij}] &=q_{ij}E[p_{ij}-X_{ij}-e_{ij}-\alpha_{ij}p_{ij}\mid R_{ij}=1]\\ &=q_{ij}\bigl[(1-\alpha_{ij})p_{ij}-L_{ij}-e_{ij}\bigr]. \end{aligned}

This derivation requires the loss expectation to be conditional on renewal at the price being evaluated. If the customers who renew at a higher price differ in risk from those who renew at a lower price, their conditional expected losses may differ too. Use the loss estimate appropriate to that candidate price and renewal population; do not assume it is unchanged unless that assumption is justified.

If a cost $k_i$ is incurred for every eligible policy regardless of renewal, it is not multiplied by $q_{ij}$. In that case, expected contribution is $q_{ij}[(1-\alpha_{ij})p_{ij}-L_{ij}-e_{ij}]-k_i$. The expense treatment must reflect when the cost is actually incurred.

Worked Calculations

Worked Example: Comparing Two Candidate Prices for One Policy

A hypothetical insurer considers two one-term renewal prices for an eligible policy. At a premium of $800, predicted renewal probability is 0.70. At a premium of $760, it is 0.78. For either price, expected loss conditional on renewal is $460, fixed expense per renewal is $24, and variable expense is 6% of premium. Find expected contribution per eligible policy at each price and identify the larger result.

Method. For each price, first calculate the variable expense and conditional contribution. Then multiply that contribution by the renewal probability. All stated expenses occur only if the policy renews.

At $800, variable expense is $0.06(800)=\$48$. Conditional contribution is:

800-460-24-48=\$268

Expected contribution per eligible policy is $0.70(268)=\$187.60$. As a component check, expected premium is $0.70(800)=\$560$, expected loss is $0.70(460)=\$322$, fixed expense is $0.70(24)=\$16.80$, and variable expense is $0.70(48)=\$33.60$. Their difference is $560-322-16.80-33.60=\$187.60$.

At $760, variable expense is $0.06(760)=\$45.60$. Conditional contribution is $760-460-24-45.60=\$230.40$. Expected contribution per eligible policy is $0.78(230.40)=\$179.712$, or $179.71 to the nearest cent.

The $800 candidate has the higher expected contribution per eligible policy: $187.60 versus $179.71. The comparison is not based only on conditional contribution: the lower price has a higher renewal probability, but that increase does not offset its lower contribution per renewal.

Common mistake avoided. Subtracting expenses after weighting only premium and losses would give $0.70(800-460)-24-48=\$166$. That incorrectly treats the $24 and $48 renewal expenses as though they were incurred even when the policy lapses. Weighting all renewal-contingent amounts gives the correct $187.60.

Worked Example: Aggregating Contribution Across Two Portfolio Groups

A hypothetical renewal portfolio has 1,000 eligible policies in Group A and 600 in Group B. Group A is offered a $900 premium, has renewal probability 0.80, expected loss of $520 per renewal, fixed expense of $35 per renewal, and variable expense of 5% of premium. Group B is offered a $700 premium, has renewal probability 0.65, expected loss of $400 per renewal, fixed expense of $20 per renewal, and variable expense of 4% of premium. Calculate expected contribution for each group and for the portfolio.

Group A. Variable expense per renewal is $0.05(900)=\$45$. Conditional contribution is $900-520-35-45=\$300$. Expected contribution per eligible policy is $0.80(300)=\$240$. For 1,000 eligible policies, expected group contribution is $1{,}000(240)=\$240{,}000$.

Group B. Variable expense per renewal is $0.04(700)=\$28$. Conditional contribution is $700-400-20-28=\$252$. Expected contribution per eligible policy is $0.65(252)=\$163.80$. For 600 eligible policies, expected group contribution is $600(163.80)=\$98{,}280$.

Total expected portfolio contribution is $240{,}000+98{,}280=\$338{,}280$. There are 1,600 eligible policies, so the expected contribution per eligible policy across the portfolio is $338{,}280/1{,}600=\$211.425$, or $211.43 to the nearest cent. This is a count-weighted average of group contributions per eligible policy, not the simple average of $240 and $163.80.

A component reconciliation gives the same portfolio result. Expected renewals are 800 in Group A and 390 in Group B. Expected premium is $800(900)+390(700)=\$993{,}000$. Expected losses are $800(520)+390(400)=\$572{,}000$. Fixed expenses total $800(35)+390(20)=\$35{,}800$, and variable expenses total $800(45)+390(28)=\$46{,}920$. Thus $993{,}000-572{,}000-35{,}800-46{,}920=\$338{,}280$, which checks the group calculation.

Common mistake avoided. Applying one portfolio-wide renewal probability or averaging the two group probabilities without weights would ignore the different group sizes and economics. Calculate each group on its own assumptions, multiply by its eligible count, and then sum.

Worked Example: Separating Renewal Expenses From an Unconditional Cost

For one hypothetical eligible policy, an insurer compares a $1,000 price with renewal probability 0.60 against a $920 price with renewal probability 0.72. At either price, expected loss conditional on renewal is $620, fixed expense per renewal is $25, and variable expense is 8% of premium. A separate $10 contact cost is incurred for every eligible policy, whether or not it renews. Calculate expected contribution per eligible policy at each price.

At $1,000. Variable expense per renewal is $0.08(1{,}000)=\$80$. Conditional contribution is $1{,}000-620-25-80=\$275$. Expected renewal contribution is $0.60(275)=\$165$. Subtract the $10 unconditional contact cost to obtain expected contribution of $155.00 per eligible policy.

The same answer follows by components: expected premium is $0.60(1{,}000)=\$600$; expected loss is $0.60(620)=\$372$; expected renewal expenses are $0.60(25+80)=\$63$; and unconditional cost is $10. The net is $600-372-63-10=\$155$.

At $920. Variable expense per renewal is $0.08(920)=\$73.60$. Conditional contribution is $920-620-25-73.60=\$201.40$. Expected renewal contribution is $0.72(201.40)=\$145.008$. After subtracting the $10 contact cost, expected contribution is $135.008, or $135.01 per eligible policy to the nearest cent.

The $1,000 candidate has the higher expected contribution by $155.00-135.008=\$19.992$, or $19.99 to the nearest cent. The $10 contact cost does not affect the comparison here because it is the same at both prices, but it must be included when reporting the contribution level.

Common mistake avoided. Multiplying the unconditional cost by the renewal probability would charge only $0.60(10)$ or $0.72(10)$, despite the problem stating it is incurred for every eligible policy. Conversely, a cost incurred only upon renewal must be multiplied by renewal probability.

From Individual Calculations to a Portfolio Measure

For $n$ eligible policies, expected portfolio contribution is the sum of their expected contributions. If each policy has its own price, renewal probability, loss expectation, and expense assumptions, then:

E[C_{\mathrm{portfolio}}] =\sum_{i=1}^{n}q_{ij}\bigl[(1-\alpha_{ij})p_{ij}-L_{ij}-e_{ij}\bigr] -\sum_{i=1}^{n}k_i

Here $k_i$ represents any cost incurred for eligible policy $i$ regardless of renewal. The summation uses the candidate price assigned to each policy. Linearity of expectation means this total does not require renewal outcomes for different policies to be independent. It does require the renewal probability, conditional loss expectation, and expenses to be defined consistently for each policy and price.

If policies are grouped and all members of group $g$ share the same assumptions, the group contribution is the number eligible $n_g$ multiplied by group expected contribution per eligible policy. Summing group totals gives portfolio contribution. Be explicit about the denominator: contribution per renewed policy, per eligible policy, and total portfolio contribution answer different questions.

Checks and Practical Interpretation

A reliable calculation follows a fixed sequence: identify the candidate price and renewal probability; calculate expenses on the correct base; determine conditional contribution; apply the renewal probability to renewal-contingent amounts; then scale by the number of eligible policies if a portfolio total is required. Keep all quantities on the same time and exposure basis.

Check the cost trigger. A cost per renewal is weighted by renewal probability. A cost per eligible policy is not.
Check the loss basis. Use expected loss conditional on renewal at the candidate price, not a portfolio average that may represent a different risk mix.
Check the denominator. Dividing by eligible policies gives expected contribution per eligible policy; dividing by expected renewals gives a different conditional measure.
Check portfolio aggregation. Multiply each group's per-eligible result by its eligible count before adding groups.
Check units and rounding. Keep dollar amounts and expense rates distinct, retain precision through intermediate steps, and round the reported money only at the end.

Expected contribution is one economic measure of a candidate price, not a substitute for checking that the input predictions and assumptions are suitable. Renewal probabilities and loss estimates should correspond to the offered price and the population to which the calculation is applied. Correctly combining those inputs is a foundational implementation step aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.

Check Your Understanding

For each question, use the stated expense trigger and report dollar amounts to the nearest cent when needed.

A hypothetical policy is offered a $500 renewal premium. Renewal probability is 0.80, expected loss per renewal is $310, fixed expense per renewal is $15, and variable expense is 4% of premium. Find expected contribution per eligible policy. (Answer: $124.00)
A hypothetical policy has a $1,200 candidate premium, renewal probability 0.55, expected loss per renewal of $760, fixed expense per renewal of $40, and variable expense of 6% of premium. Calculate expected contribution per eligible policy. (Answer: $180.40)
A group of 250 hypothetical eligible policies is offered a $640 premium. Renewal probability is 0.75, expected loss per renewal is $390, fixed expense per renewal is $18, and variable expense is 5% of premium. Find expected contribution per eligible policy and for the group. (Answer: $150.00 per eligible policy; $37,500.00 for the group)
A hypothetical portfolio has 400 eligible policies in Group A and 300 in Group B. Group A has premium $800, renewal probability 0.70, expected loss $470 per renewal, fixed expense $25 per renewal, and variable expense 5%. Group B has premium $900, renewal probability 0.60, expected loss $500 per renewal, fixed expense $30 per renewal, and variable expense 4%. Find total expected portfolio contribution. (Answer: $134,320.00)
A hypothetical policy has a $850 premium, renewal probability 0.72, expected loss of $540 per renewal, fixed expense of $30 per renewal, and variable expense of 7% of premium. A separate $12 cost is incurred for every eligible policy, whether or not it renews. Find expected contribution per eligible policy and for 200 eligible policies with identical assumptions. (Answer: $146.76 per eligible policy; $29,352.00 total)

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