Why a Portfolio Objective Is a Sum
A pricing decision is often evaluated across many quote opportunities rather than by the outcome of one policy. Some customers accept an offered price and others do not, so the portfolio result is uncertain even when the quote count and prices are known. The natural starting point is to calculate each policy or segment’s expected dollar outcome and add those expectations.
The objective must match the business question. Expected premium revenue counts premium from policies that are bound. Expected contribution subtracts the costs that vary with bound business. Expected profit may also subtract expenses that are fixed for the portfolio. These are different objectives: a price that produces the most premium dollars need not produce the greatest contribution or profit.
When quotes are grouped into segments, the same calculations can use segment counts. If segment \(s\) has \(n_s\) equivalent quote opportunities, purchase probability \(q_s\), premium \(p_s\), and expected cost per bound policy \(c_s\), then its expected bound count is \(n_sq_s\). Its expected revenue is \(n_sq_sp_s\), and its expected contribution is \(n_sq_s(p_s-c_s)\). Segment costs should reflect the expected costs of the policies that bind in that segment; if those costs differ within a segment, use a suitable weighted average or calculate at a finer level.
Why Expected Outcomes Add
For each quote, revenue is \(p_jI_j\): it equals the premium if the policy binds and zero otherwise. Since \(E[I_j]=P(I_j=1)=q_j\), expected revenue from that quote is \(p_jq_j\). Adding across quotes gives the portfolio expression.
The key step is linearity of expectation. It does not require purchase decisions to be independent. Customers’ decisions may be related, but the expected total is still the sum of the expected policy outcomes, provided each \(q_j\) is the appropriate purchase probability. Dependence does matter for the variance or distribution of portfolio results, which is a different question.
For contribution, let \(C_j\) be the cost incurred if quote \(j\) binds, with conditional expected cost \(c_j=E[C_j\mid I_j=1]\). The quote’s contribution is \(I_j(p_j-C_j)\), so its expectation is \(q_j(p_j-c_j)\). This form assumes that the stated \(c_j\) is conditional on binding and that no corresponding cost is incurred when the quote does not bind. If costs are incurred regardless of binding, those costs must instead be included in the objective in the way they actually arise.
Worked Example: Adding Revenue Across Segments
Worked Example: Adding Revenue Across Segments
A hypothetical insurer is comparing the revenue expected from two groups of quotes at their current prices. Group A has 120 eligible quotes, a premium of $600 per bound policy, and a purchase probability of 0.70. Group B has 80 eligible quotes, a premium of $900, and a purchase probability of 0.45. Calculate expected bound policies, expected premium revenue, average revenue per eligible quote, and the ratio of expected revenue to expected bound policies.
Method. For each group, multiply quote count by purchase probability to obtain expected bound policies. Multiply that expected count by premium to obtain expected revenue.
Group A. Expected bound count is \(120(0.70)=84\). Expected revenue is \(84(\$600)=\$50{,}400\).
Group B. Expected bound count is \(80(0.45)=36\). Expected revenue is \(36(\$900)=\$32{,}400\).
Portfolio total. Expected bound count is \(84+36=120\), and expected revenue is \(\$50{,}400+\$32{,}400=\$82{,}800\). There are \(120+80=200\) eligible quotes, so average expected revenue per eligible quote is \(\$82{,}800/200=\$414\). The ratio of expected revenue to expected bound count is \(\$82{,}800/120=\$690\).
Check and trap avoided. The segment revenues sum to $82,800, and the segment expected bound counts sum to 120. A common mistake is to average the two segment premiums, \((\$600+\$900)/2=\$750\), and treat that as the portfolio’s average premium. That gives each segment equal weight despite their different expected bound counts. The appropriate ratio is \(\$82{,}800/120=\$690\).
Revenue, Contribution, and Profit Are Not Interchangeable
The revenue objective counts premium dollars but does not deduct claim costs or expenses. Contribution deducts costs that arise with bound business, such as expected losses and variable expenses, according to the cost definition being used. Profit additionally deducts any relevant fixed expenses. A fixed expense changes the profit level even when it does not change with the pricing decision; if it is constant across all alternatives, it does not by itself change which alternative maximizes profit.
Worked Example: From Revenue to Portfolio Profit
A hypothetical insurer evaluates three segments. Segment A has 50 quotes, a premium of $800, a purchase probability of 0.60, and an expected cost of $500 per bound policy. Segment B has 40 quotes, a premium of $1,200, a purchase probability of 0.40, and an expected cost of $850 per bound policy. Segment C has 30 quotes, a premium of $1,000, a purchase probability of 0.50, and an expected cost of $700 per bound policy. A fixed portfolio expense of $4,000 is incurred regardless of the number of policies that bind. Calculate expected revenue, expected variable cost, contribution, and profit.
Segment A. Expected bound count is \(50(0.60)=30\). Revenue is \(30(\$800)=\$24{,}000\), and expected variable cost is \(30(\$500)=\$15{,}000\). Contribution is \(\$24{,}000-\$15{,}000=\$9{,}000\).
Segment B. Expected bound count is \(40(0.40)=16\). Revenue is \(16(\$1{,}200)=\$19{,}200\), and expected variable cost is \(16(\$850)=\$13{,}600\). Contribution is \(\$19{,}200-\$13{,}600=\$5{,}600\).
Segment C. Expected bound count is \(30(0.50)=15\). Revenue is \(15(\$1{,}000)=\$15{,}000\), and expected variable cost is \(15(\$700)=\$10{,}500\). Contribution is \(\$15{,}000-\$10{,}500=\$4{,}500\).
Portfolio calculation. Expected revenue is \(\$24{,}000+\$19{,}200+\$15{,}000=\$58{,}200\). Expected variable cost is \(\$15{,}000+\$13{,}600+\$10{,}500=\$39{,}100\). Therefore contribution is \(\$58{,}200-\$39{,}100=\$19{,}100\), which also equals \(\$9{,}000+\$5{,}600+\$4{,}500\). Profit is \(\$19{,}100-\$4{,}000=\$15{,}100\).
Check and trap avoided. Revenue minus variable cost gives the same $19,100 as summing the three segment contributions. Subtracting the $4,000 fixed expense once gives profit of $15,100. Subtracting it once per expected bound policy would be incorrect: the problem specifies one portfolio expense, not a per-policy expense.
When an Average Does—and Does Not—Preserve the Objective
An average per eligible quote is the portfolio total divided by the number of eligible quotes. If that quote count is fixed across the pricing alternatives being compared, maximizing total expected revenue and maximizing expected revenue per eligible quote give the same ranking: division by a positive constant does not change which value is largest. The same principle applies to contribution or profit divided by a fixed eligible count.
An average per bound policy is different. The number of bound policies can change with price, so the ratio of expected revenue to expected bound count has a decision-dependent denominator. Maximizing that ratio need not maximize total revenue. It is also important to name the quantity precisely: \(\frac{E[R]}{E[B]}\), where \(B\) is the number of bound policies, is a ratio of expectations. It is not generally the same as the expected value of a realized average, \(E[R/B]\), particularly when \(B\) is random or can be zero.
Worked Example: Total Revenue Versus Revenue per Bound Policy
A hypothetical insurer considers two portfolio pricing strategies for the same 100 eligible quotes. Under Strategy L, the premium is $500 and the purchase probability is 0.80. Under Strategy H, the premium is $700 and the purchase probability is 0.50. Compare total expected revenue, average expected revenue per eligible quote, and the ratio of expected revenue to expected bound count.
Strategy L. Expected bound count is \(100(0.80)=80\). Expected revenue is \(80(\$500)=\$40{,}000\). Revenue per eligible quote is \(\$40{,}000/100=\$400\), and the ratio of expected revenue to expected bound count is \(\$40{,}000/80=\$500\).
Strategy H. Expected bound count is \(100(0.50)=50\). Expected revenue is \(50(\$700)=\$35{,}000\). Revenue per eligible quote is \(\$35{,}000/100=\$350\), and the ratio of expected revenue to expected bound count is \(\$35{,}000/50=\$700\).
Comparison. Strategy L produces greater total expected revenue, $40,000 versus $35,000, and greater revenue per eligible quote, $400 versus $350. Strategy H produces the greater ratio of expected revenue to expected bound count, $700 versus $500. Thus a conclusion about “average revenue” is incomplete unless the denominator is stated.
Check and trap avoided. The 100 eligible quotes are fixed, so dividing total revenue by 100 preserves its ranking. Dividing by the expected bound count does not: that count is 80 under L and 50 under H. Treating those two denominators as interchangeable would incorrectly equate a total-dollar objective with a per-bound-policy average.
Practical Checks Before Optimizing
Before evaluating a portfolio objective, state what is being summed, over which population, and which costs are included. A clear objective prevents an apparently correct calculation from answering the wrong business question.
- Specify the population. Identify whether the units are eligible quotes, policies that bind, or another defined set of exposures. Use the same population when comparing alternatives unless the objective explicitly accounts for a changing population.
- Use the appropriate purchase probability. Each \(q_j\) must correspond to the quoted price and relevant customer or segment. A probability from one offer cannot automatically be reused at a different price.
- Match cost treatment to the objective. Revenue has no cost deduction; contribution subtracts the stated variable costs; profit also includes relevant fixed costs. Do not subtract costs twice.
- State the denominator of any average. An average per fixed eligible quote preserves the ranking of a total. An average per bound policy can change the ranking because expected bound count varies.
These distinctions matter because portfolio pricing is judged on an aggregate result, but different aggregate measures reward different outcomes. The appropriate objective depends on whether the decision is intended to maximize total dollars, a per-quote amount, or a per-bound-policy ratio. This aggregation framework is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.
Check Your Understanding
For each problem, state the requested aggregate measure and keep its denominator explicit.
- A hypothetical portfolio has 60 eligible quotes, each with a $400 premium and purchase probability 0.55. Find the expected bound count, total expected revenue, and expected revenue per eligible quote. (Answer: 33 expected bound policies; $13,200 total expected revenue; $220 per eligible quote.)
- A hypothetical portfolio has 25 quotes with premium $1,000, purchase probability 0.40, and cost $650 per bound policy; it also has 40 quotes with premium $700, purchase probability 0.50, and cost $480 per bound policy. A fixed $2,500 expense applies once to the portfolio. Find expected contribution and profit. (Answer: $7,900 expected contribution; $5,400 expected profit.)
- For 200 eligible quotes, Strategy A has premium $450 and purchase probability 0.60. Strategy B has premium $600 and purchase probability 0.45. Compare total expected revenue and the ratio of expected revenue to expected bound count under each strategy. (Answer: both have $54,000 expected revenue; A has 120 expected bound policies and a $450 ratio, while B has 90 expected bound policies and a $600 ratio.)
- A portfolio has 80 eligible quotes, premium $900, purchase probability 0.50, and expected cost $620 per bound policy. A fixed expense of $3,000 applies once. Find expected profit and profit per eligible quote. (Answer: $8,200 expected profit; $102.50 per eligible quote.)
- A hypothetical portfolio has 40 quotes at premium $500 with purchase probability 0.80, and 60 quotes at premium $800 with purchase probability 0.30. Find total expected revenue, expected bound count, and the ratio of expected revenue to expected bound count. (Answer: $30,400 expected revenue; 50 expected bound policies; $608 per expected bound policy.)