Expected Premium Revenue as a Pricing Objective
A higher premium brings in more dollars from each policy that is purchased, but it may also lead fewer customers to buy. To compare candidate prices, an insurer needs to consider both effects: the premium per purchase and the expected number of purchases at that premium. The resulting objective is expected premium revenue.
This objective is useful for making a clear comparison among specified prices when a demand curve or purchase probabilities are available. It is not the same as comparing premium amounts alone. Nor does it account for expected claims or expenses: it measures premium revenue, not the amount remaining after costs.
The formula \(R(p)=pMq(p)\) assumes that each prospect is offered the same price, that a purchase generates premium \(p\), and that the prospect count \(M\) is the relevant count for the pricing decision. The purchase outcomes do not have to be independent for the expected-count calculation: by linearity of expectation, the expected total is the sum of the individual purchase probabilities. Independence would matter for the variance or distribution of \(N(p)\), but not for its expected value.
When prospects have different purchase probabilities, the same logic applies by summing their probabilities. If prospect \(j\) has purchase probability \(q_j(p)\), then \(E[N(p)]=\sum_j q_j(p)\). If prospects are grouped into segments, this becomes a sum of segment-level expected counts. The price used to calculate revenue must still match the price actually charged in each segment.
How to Compare Candidate Prices
For each candidate price, calculate the expected number of purchases first, then multiply by that price. Keep the prospect base and the measurement period consistent across candidates. If purchase probabilities are supplied directly, use them at the stated price; do not assume they remain unchanged when the price changes.
- List each candidate price and its corresponding purchase probability or expected purchase count.
- Calculate expected purchases from the prospect count and purchase probability, or use the expected count given.
- Multiply the candidate price by its expected purchase count.
- Compare the resulting expected revenues. The largest value identifies the revenue-maximizing candidate among those evaluated.
That last qualification matters. If only a few prices are evaluated, the result identifies the best of those candidates; it does not establish that no untested price would produce greater revenue. A continuous demand curve may require a separate analysis to identify a global maximum. This tutorial focuses on calculating and comparing expected revenue for candidate prices.
Worked Example: Comparing Three Prices for One Prospect Pool
A hypothetical insurer has 1,200 eligible prospects for a one-year policy. At an annual premium of $900, the estimated purchase probability is 0.42; at $960, it is 0.38; and at $1,020, it is 0.34. Calculate expected annual premium revenue at each price and select the best candidate.
Method. For each price, multiply the number of prospects by the purchase probability to obtain expected purchases. Then multiply expected purchases by the annual premium.
At $900, expected purchases are \(1{,}200(0.42)=504\). Expected revenue is \(504(\$900)=\$453{,}600\).
At $960, expected purchases are \(1{,}200(0.38)=456\). Expected revenue is \(456(\$960)=\$437{,}760\).
At $1,020, expected purchases are \(1{,}200(0.34)=408\). Expected revenue is \(408(\$1{,}020)=\$416{,}160\).
The revenues in descending order are $453,600, $437,760, and $416,160. Thus, $900 is the revenue-maximizing candidate among these three prices. As a check, the expected number of purchases falls as the price rises, and the calculation has multiplied each expected count by its own candidate premium. The common mistake avoided here is selecting the highest premium without accounting for the decline in expected purchases.
Why the Expected-Revenue Formula Holds
The formula follows from representing each prospect's purchase as an indicator. Let \(I_j(p)=1\) if prospect \(j\) purchases at price \(p\), and \(I_j(p)=0\) otherwise. Then the number of purchases is \(N(p)=\sum_{j=1}^{M} I_j(p)\). If the probability that prospect \(j\) purchases is \(q_j(p)\), then \(E[I_j(p)]=q_j(p)\).
The second line uses linearity of expectation, which does not require independent purchase decisions. If every prospect has the same probability \(q(p)\), the sum is \(Mq(p)\). Since each purchase produces the same premium \(p\), total premium revenue is \(pN(p)\), and taking expectations gives \(E[pN(p)]=pE[N(p)]\). This justifies the calculation without treating an expected count as a guaranteed count.
Worked Example: Aggregating Expected Purchases Across Segments
A hypothetical insurer considers a uniform annual premium for two prospect segments. Segment A has 500 prospects and Segment B has 300. Estimated purchase probabilities are:
| Annual premium | Segment A probability | Segment B probability |
|---|---|---|
| $500 | 0.60 | 0.35 |
| $560 | 0.52 | 0.31 |
| $620 | 0.44 | 0.27 |
Calculate total expected premium revenue at each candidate price. The same premium is charged to both segments.
At $500: Segment A contributes \(500(0.60)=300\) expected purchases. Segment B contributes \(300(0.35)=105\). Total expected purchases are \(300+105=405\), so expected revenue is \(405(\$500)=\$202{,}500\).
At $560: Segment A contributes \(500(0.52)=260\) expected purchases. Segment B contributes \(300(0.31)=93\). Total expected purchases are \(260+93=353\), so expected revenue is \(353(\$560)=\$197{,}680\).
At $620: Segment A contributes \(500(0.44)=220\) expected purchases. Segment B contributes \(300(0.27)=81\). Total expected purchases are \(220+81=301\), so expected revenue is \(301(\$620)=\$186{,}620\).
The best of the evaluated candidates is $500, with expected revenue of $202,500. The segment calculations check because the expected counts at each price sum to the total before the common premium is applied. A tempting error is to average the two segment probabilities without weighting for segment size. For example, at $500 the unweighted average probability, 0.475, would imply 380 purchases from 800 prospects. The correct weighted expected count is 405 because the segments have different sizes and different purchase probabilities.
Interpreting the Result and Its Limits
Expected revenue is an average over the possible purchase outcomes. If a candidate has 100 expected purchases, the realized number could be more or fewer than 100. The expected revenue is not a promise about the premium actually collected. It is a comparison measure based on the purchase probabilities and prospect count used in the calculation.
The comparison also depends on using consistent units. If \(p\) is an annual premium, then the revenue calculated is annual premium revenue for the policies purchased, and the prospects and purchase probabilities must refer to that same offer and time period. A monthly premium cannot be compared directly with an annual premium without converting to a common basis. Similarly, if a stated amount is a premium per exposure unit rather than a full policy premium, it must be multiplied by the appropriate purchased exposure, not automatically by the number of customers.
Revenue maximization answers a narrow question: which candidate produces the greatest expected premium dollars under the stated demand assumptions? It does not say which price is adequate for expected costs, nor does it incorporate constraints or other objectives. Those questions require distinct analyses. A price with the most expected revenue can have a lower number of purchasers than another candidate, and it need not be the preferred price under a different objective.
Worked Example: A Small Price Increase Can Still Reduce Revenue
A hypothetical insurer receives 800 eligible quote requests for a one-year coverage option. At a premium of $640, estimated purchase probability is 0.55. At $700, it is 0.49, and at $760, it is 0.43. Compare expected revenue and explain why the premium increase from $640 to $700 does not improve this measure.
At $640, expected purchases are \(800(0.55)=440\). Expected revenue is \(440(\$640)=\$281{,}600\).
At $700, expected purchases are \(800(0.49)=392\). Expected revenue is \(392(\$700)=\$274{,}400\).
At $760, expected purchases are \(800(0.43)=344\). Expected revenue is \(344(\$760)=\$261{,}440\).
The expected revenue decreases at each successive candidate price, so $640 is best among the three. From $640 to $700, the premium per policy rises by $60, but expected purchases fall by 48. The full comparison is the relevant check: \(440(\$640)=\$281{,}600\), while \(392(\$700)=\$274{,}400\). The common mistake is to treat the higher premium per policy as evidence of higher total revenue without recalculating expected purchases.
Common Mistakes
- Comparing premiums rather than revenue. The revenue calculation includes expected purchases at each candidate price; a higher premium alone is not enough.
- Reusing one purchase probability at every price. Use the probability associated with each candidate price, since demand is price-sensitive in the setup.
- Using an unweighted average across segments. Multiply each segment's probability by its own prospect count, then add the expected counts.
- Calling expected purchases actual purchases. Expected counts can be fractional in general and represent averages over possible outcomes, not guaranteed policy counts.
- Claiming a global optimum from a short candidate list. The largest calculated value is only the best among the prices evaluated.
- Mixing time bases or units. Match annual premiums with annual purchase opportunities, and distinguish policy counts from exposure units.
Check Your Understanding
For each problem, compare expected premium revenue using the stated candidate prices and demand assumptions.
- A hypothetical insurer has 400 prospects. At annual premiums of $1,200, $1,300, and $1,400, purchase probabilities are 0.30, 0.27, and 0.23. Which candidate has the greatest expected annual premium revenue? (Answer: $1,200, with expected revenue of $144,000; the other revenues are $140,400 and $128,800.)
- For 1,500 prospects, purchase probabilities at premiums of $875, $910, and $950 are 0.40, 0.37, and 0.35. Calculate the expected revenue for each and identify the best candidate. (Answer: $875; expected revenues are $525,000, $505,050, and $498,750, respectively.)
- A uniform annual price is offered to two hypothetical segments. Segment A has 200 prospects and Segment B has 300. At $400, their purchase probabilities are 0.50 and 0.20; at $450, they are 0.43 and 0.18; at $500, they are 0.36 and 0.15. Which price maximizes expected revenue among these options? (Answer: $400; the respective expected revenues are $64,000, $63,000, and $58,500.)
- A hypothetical insurer has 600 eligible prospects. At an annual premium of $1,500, purchase probability is 0.32; at $1,650, it is 0.29. Calculate both expected revenues and explain which candidate is better on this objective. (Answer: $1,500, with $288,000 versus $287,100 at $1,650.)
- There are 250 prospects in Segment A and 750 in Segment B. At a uniform annual premium of $720, purchase probabilities are 0.48 in A and 0.24 in B. At $780, they are 0.43 in A and 0.22 in B. Calculate expected revenues and state the winning candidate. (Answer: $720; expected purchases are 300 and revenue is $216,000, compared with 272.5 expected purchases and $212,550 at $780.)