Maximizing Expected Contribution

Expected Contribution and the Price Decision

Expected premium revenue depends on both the price offered and how many customers are expected to buy at that price. But revenue alone does not account for the costs that arise when a policy is sold. Expected contribution brings those two parts together: it measures the expected premium revenue from sold policies after deducting the variable costs associated with those policies.

This objective is useful when comparing candidate prices using a supplied demand curve. A higher price may increase revenue per sale while reducing the number of sales; it may also change a variable expense such as a percentage-based commission. The calculation must reflect both the demand response and the costs on the policies actually sold.

Definition: For a price $p$, let $N(p)$ be the random number of policies sold and let $C_j$ be the variable cost associated with sold policy $j$. Expected contribution is expected premium revenue less expected variable costs:

\operatorname{EC}(p) = E\left[pN(p)-\sum_{j=1}^{N(p)} C_j\right].

If every sold policy has the same expected variable cost $c(p)$, measured on the same coverage and time basis as the premium, then

\operatorname{EC}(p) = \bigl(p-c(p)\bigr)E[N(p)].

This simplified expression assumes the expected variable cost per sold policy is $c(p)$ at price $p$. It does not deduct fixed costs that do not depend on the number of policies sold.

In an actuarial application, the variable cost per policy may include expected claim cost and variable expenses, if those quantities are supplied or can be estimated. A commission equal to a percentage of premium also belongs in the variable cost and changes with price. The calculation is on a common basis: for example, an annual premium should be compared with expected costs for the same annual policy period.

Expected contribution is not the same as expected premium revenue. Nor should the label be taken to mean that every expense or financial item has been included: fixed overhead and any other excluded items remain outside the stated objective. The objective here is specifically revenue less variable costs for policies sold.

Why the Per-Policy Formula Works

The simplified formula follows from linearity of expectation. Suppose there are $M$ prospects, and let $I_k$ equal 1 if prospect $k$ buys at price $p$, and 0 otherwise. Then $N(p)=\sum_{k=1}^{M} I_k$. If each prospect has the same purchase probability $q(p)$, then $E[N(p)]=Mq(p)$; independence between prospects is not required for this expectation.

If a sold policy has expected variable cost $c(p)$, then the expected total variable cost is $c(p)E[N(p)]$, provided this is the conditional expected cost for each sale. Thus:

\begin{aligned} \operatorname{EC}(p) &=pE[N(p)]-c(p)E[N(p)]\\ &=\bigl(p-c(p)\bigr)E[N(p)]. \end{aligned}

The formula is not a claim that the realized number of sales equals its expectation, or that each realized policy costs exactly $c(p)$. It calculates an expectation. If expected costs differ by customer segment, calculate the expected contribution in each segment and add the results rather than applying one cost to every policy.

Calculating Contribution at Candidate Prices

A supplied demand curve may give expected policy counts directly, or it may give purchase probabilities that must be multiplied by the number of eligible prospects. Once expected sales are known, calculate the variable cost at each candidate price, then multiply the per-policy contribution by expected sales. Comparing a finite list of candidate prices identifies the best option within that list; it does not establish a global optimum over prices that were not evaluated.

Worked Example: Comparing Candidate Prices with Constant Cost

A hypothetical insurer is evaluating annual prices for a small commercial package. The supplied demand estimates give 800 expected policies sold at $600, 700 at $650, 580 at $700, and 450 at $750. Assume a variable cost of $420 per policy at every price, including the expected claim cost and variable expenses for the same annual period. Calculate expected contribution at each candidate price and identify the highest.

Method. Since the cost per sold policy is constant, use $(p-c)E[N(p)]$ at each price. Fixed costs are excluded.

At $600, the contribution per policy is $600-420=\$180$. Expected contribution is $180(800)=\$144{,}000$.

At $650, the contribution per policy is $650-420=\$230$. Expected contribution is $230(700)=\$161{,}000$.

At $700, the contribution per policy is $700-420=\$280$. Expected contribution is $280(580)=\$162{,}400$.

At $750, the contribution per policy is $750-420=\$330$. Expected contribution is $330(450)=\$148{,}500$.

The highest contribution among the candidates is $162,400 at a price of $700. As a check, at $700 expected premium revenue is $700(580)=\$406{,}000$, expected variable costs are $420(580)=\$243{,}600$, and their difference is $406{,}000-243{,}600=\$162{,}400$. A common mistake is to select the highest price because it has the largest per-policy margin; here, the $750 price has a larger margin but fewer expected sales, and its total contribution is lower.

Worked Example: A Price-Dependent Variable Cost

For a hypothetical personal-lines product, 1,200 eligible prospects are offered one of three annual prices. The supplied purchase probabilities are 0.75 at $500, 0.65 at $550, and 0.54 at $600. Assume the expected variable cost per sold policy is $300 of expected claim cost, $90 of other variable servicing cost, and a commission equal to 8% of premium. Calculate contribution at each price.

Method. First calculate expected policies sold as the number of prospects times the purchase probability. Then calculate the price-dependent variable cost and the contribution per policy.

At $500, expected sales are $1{,}200(0.75)=900$. The commission is $0.08(500)=\$40$, so total variable cost per policy is $300+90+40=\$430$. Contribution per policy is $500-430=\$70$, and total expected contribution is $70(900)=\$63{,}000$.

At $550, expected sales are $1{,}200(0.65)=780$. The commission is $0.08(550)=\$44$, so variable cost per policy is $300+90+44=\$434$. Contribution per policy is $550-434=\$116$, and expected contribution is $116(780)=\$90{,}480$.

At $600, expected sales are $1{,}200(0.54)=648$. The commission is $0.08(600)=\$48$, so variable cost per policy is $300+90+48=\$438$. Contribution per policy is $600-438=\$162$, and expected contribution is $162(648)=\$104{,}976$.

Among these three prices, $600 gives the highest expected contribution, $104,976. Check by separating revenue and cost at $600: expected premium revenue is $600(648)=\$388{,}800$, while expected variable costs are $438(648)=\$283{,}824$; their difference is $104,976. The commission must be recalculated at each price. Holding it at $40 would understate costs at $550 and $600.

Combining Customer Segments

When segments have different demand responses or variable costs, a single average cost can conceal the source of contribution. For a uniform price offered to multiple segments, calculate each segment’s expected sales and per-policy contribution separately, then add the segment contributions. The common price does not imply that every segment has the same cost or purchase probability.

Worked Example: Aggregating Two Segments

A hypothetical insurer is considering one annual price for two groups of eligible customers. Segment A has 400 prospects and expected variable cost of $350 per sold policy. Segment B has 300 prospects and expected variable cost of $500 per sold policy. At $800, the purchase probabilities are 0.60 for A and 0.40 for B. At $850, they are 0.52 for A and 0.34 for B. Assume the stated costs apply at either price. Compare expected contribution at the two candidate prices.

Method. For each segment and price, multiply eligible prospects by the purchase probability, then multiply expected sales by price less that segment’s cost.

At $800, expected sales are $400(0.60)=240$ in A and $300(0.40)=120$ in B. Segment A contributes $(800-350)(240)=450(240)=\$108{,}000$. Segment B contributes $(800-500)(120)=300(120)=\$36{,}000$. Total expected contribution is $108{,}000+36{,}000=\$144{,}000$.

At $850, expected sales are $400(0.52)=208$ in A and $300(0.34)=102$ in B. Segment A contributes $(850-350)(208)=500(208)=\$104{,}000$. Segment B contributes $(850-500)(102)=350(102)=\$35{,}700$. Total expected contribution is $104{,}000+35{,}700=\$139{,}700$.

The $800 price has the greater expected contribution, by $144{,}000-139{,}700=\$4{,}300$. Check the calculation at $850 using revenue less costs: revenue is $850(208+102)=\$263{,}500$; costs are $350(208)+500(102)=72{,}800+51{,}000=\$123{,}800$. The difference is $139,700. The common mistake is to apply one segment’s cost to all expected sales; the segment calculation avoids that error.

Common Mistakes and Practical Checks

The core calculation is simple, but the inputs need careful interpretation. Expected demand is the expected number of policies sold at the stated price, not the number of prospects who receive an offer. If demand is supplied as a purchase probability, convert it to expected sales using the applicable number of prospects. If the probability or number of prospects differs across segments, do the conversion separately for each segment.

Also check which costs vary with sales and which vary with price. A fixed cost incurred whether or not a policy is sold is not a per-policy variable cost. A commission stated as a percentage of premium is price-dependent and should be recalculated at every candidate price. Do not deduct a variable cost just once from total revenue when it is incurred on every sold policy.

Common mistake: At $700, if 580 policies are expected to be sold and the variable cost is $420 per policy, an incorrect calculation is $700(580)-420=\$405{,}580$. This subtracts the cost of only one policy. The correct calculation is $700(580)-420(580)=406{,}000-243{,}600=\$162{,}400$, or equivalently $(700-420)(580)=\$162{,}400$.

Before comparing prices, confirm that each candidate uses the same policy period and units, that costs are deducted for sold policies rather than all prospects, and that segment-specific costs have not been blended incorrectly. State whether the result is contribution only or includes any additional items; do not silently treat excluded fixed costs as if they were included.

The objective is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus: candidate prices must be assessed using the demand response and the costs attached to the resulting sales, rather than by premium revenue alone.

Check Your Understanding

For each problem, calculate expected contribution at the stated candidate prices and identify the price with the greatest contribution when asked.

A hypothetical product has a variable cost of $275 per sold policy. Expected sales are 600 at $500, 500 at $550, and 390 at $600. Which candidate price has the greatest expected contribution? (Answer: $550; contributions are $135,000, $137,500, and $126,750.)
A hypothetical insurer has 1,000 prospects. Purchase probabilities at prices of $400, $450, and $500 are 0.70, 0.62, and 0.53. Variable cost per sale is $250 plus 5% of premium. Which candidate gives the greatest contribution? (Answer: $500; expected contributions are $91,000, $110,050, and $119,250.)
A uniform price is offered to two hypothetical segments. Segment X has 200 prospects, a purchase probability of 0.50 at $700 and 0.42 at $750, and variable cost of $260 per policy. Segment Y has 150 prospects, probabilities of 0.30 at $700 and 0.25 at $750, and variable cost of $400 per policy. Compare the two prices. (Answer: $700; expected contribution is $57,500 at $700 and $54,285 at $750.)
For a hypothetical product, expected sales are 280 at $650 and 250 at $700. Variable cost per sale is $300 plus 10% of premium. Calculate the contribution at each price and select the higher one. (Answer: $650 gives $79,800; $700 gives $82,500, so $700 is higher.)
A hypothetical insurer offers a price of $800 to 500 prospects, with a purchase probability of 0.40. Variable cost is $360 per sold policy. A separate fixed campaign cost is $5,000 regardless of sales. What is expected contribution under the definition in this tutorial? (Answer: Expected sales are 200 and contribution is $(800-360)(200)=\$88,000$; the fixed campaign cost is excluded.)

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