Why the Objective Matters
A price changes more than the premium paid by a customer. It can also change the probability that the customer accepts or renews, and therefore the number of policies that generate premium and costs. A high price may raise revenue per policy while reducing the number of policies; a low price may improve retention while leaving too little margin per policy. The preferred price depends on which outcome the decision maker is trying to optimize.
This tutorial compares four objectives using a common price, demand, and cost framework. The calculations are deliberately separated: a demand response is not itself a revenue or profit measure, and a price that maximizes one measure need not maximize another. The comparison is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.
For a simple comparison, suppose expected claim cost per bound policy is \(c\), and other price-sensitive operating expense per bound policy is \(e\). If these amounts are constant across the prices under consideration, expected contribution after claims is \(Nq(p)(p-c)\). Expected profit after claims and operating expense is \(Nq(p)(p-c-e)\), before any separate fixed expense. A fixed expense \(F\) that does not change with price gives expected profit \(\Pi(p)=Nq(p)(p-c-e)-F\).
These definitions make the cost treatment explicit. In practice, an organization may use “contribution” for a margin that includes some expenses, or may include additional provisions in its profit measure. The comparison remains valid if the chosen cost components are clearly defined and treated consistently. If expected cost per bound policy changes with price, use \(c(p)\) or \(e(p)\) rather than assuming a constant.
How the Objectives Relate
Revenue counts the premium from policies that are expected to bind, but does not subtract claims or expenses. Contribution subtracts a specified cost measure; profit subtracts all costs included in the definition of profit. Retention focuses on the proportion of eligible customers who stay, regardless of the premium or margin produced by those customers.
| Objective | Quantity to calculate | What it emphasizes |
|---|---|---|
| Premium revenue | \(Nq(p)p\) | Premium volume, without subtracting costs |
| Contribution after claims | \(Nq(p)(p-c)\) | Premium remaining after expected claims |
| Profit after claims and operating expense | \(Nq(p)(p-c-e)-F\) | Expected margin after the included costs |
| Retention | \(q(p)\) | Share of eligible customers who bind or renew |
The formulas follow from expected counts. For one eligible customer, let \(I\) equal 1 if the customer binds and 0 otherwise. Then \(E[I]=q(p)\). The premium received is \(pI\), so its expectation is \(E[pI]=pE[I]=pq(p)\). If a bound policy has expected claim cost \(c\), expected claim cost for that eligible customer is \(cq(p)\). Subtracting gives expected contribution \(pq(p)-cq(p)=q(p)(p-c)\). For \(N\) comparable customers, multiply the per-customer expectation by \(N\). This reasoning assumes the stated demand probability and conditional expected cost are appropriate for the price being evaluated.
One useful implication is that a fixed expense \(F\) that is the same at every price does not change which price maximizes profit. Subtracting the same number from every candidate profit changes the profit amounts but not their ordering. This is different from an expense incurred for each bound policy: its total expected amount is \(Nq(p)e\), which generally varies with price and demand.
Worked Comparison on a Discrete Price Menu
When the feasible choices are a specified set of prices, evaluate every objective at every choice using the same demand and cost assumptions. The selected price is the one with the largest value of the chosen objective on that menu; it is not necessarily the price with the largest per-policy margin.
Worked Example: Four Objectives, One Demand Curve
A hypothetical insurer is comparing annual prices for one homogeneous group of eligible customers. The estimated probability of binding is \(q(p)=1.2-0.005p\) for the listed prices from $100 to $200. Expected claims are $70 per bound policy, and operating expense is $40 per bound policy. Compare revenue, contribution after claims, profit after claims and operating expense, and retention. There are no additional fixed expenses.
Step 1: Calculate demand at each price. For example, at \(p=100\), \(q(100)=1.2-0.005(100)=0.70\). At \(p=160\), \(q(160)=1.2-0.005(160)=0.40\). Applying the same calculation to each price gives the demand column below.
Step 2: Calculate the four objectives per eligible customer. Revenue is \(q(p)p\), contribution after claims is \(q(p)(p-70)\), profit is \(q(p)(p-70-40)=q(p)(p-110)\), and retention is \(q(p)\).
| Price \(p\) | Binding probability \(q(p)\) | Revenue \(pq(p)\) | Contribution \(q(p)(p-70)\) | Profit \(q(p)(p-110)\) |
|---|---|---|---|---|
| $100 | 0.70 | $70 | $21 | -$7 |
| $120 | 0.60 | $72 | $30 | $6 |
| $140 | 0.50 | $70 | $35 | $15 |
| $160 | 0.40 | $64 | $36 | $20 |
| $180 | 0.30 | $54 | $33 | $21 |
| $200 | 0.20 | $40 | $26 | $18 |
Step 3: Select the price for each objective. Revenue is highest at $120, with $72 per eligible customer. Contribution after claims is highest at $160, with $36. Profit is highest at $180, with $21. Retention is highest at $100, where the binding probability is 0.70.
Check. At $180, the expected profit per eligible customer is \(0.30(180-70-40)=0.30(70)=\$21\), matching the table. Each selected price is the maximum in its own column. The mistake this comparison avoids is choosing the highest price because it has the largest margin per bound policy: that ignores the reduction in the probability of binding.
When Contribution and Profit Select the Same Price
The relationship between contribution and profit depends on which costs are included and how those costs vary with price and volume. If profit equals contribution minus a fixed amount that is independent of price, the two objectives have the same maximizer. If profit also subtracts a cost for every bound policy, the amount subtracted varies with demand and can change the selected price.
Worked Example: A Price-Independent Fixed Expense
A hypothetical insurer considers prices of $120, $160, and $200. The binding probabilities are respectively 0.60, 0.40, and 0.20. Expected claims and all price-sensitive operating costs together are $80 per bound policy. A separate $10 fixed expense applies to the whole pricing period and is unchanged by the selected price. Find the contribution- and profit-maximizing prices.
Step 1: Calculate contribution per eligible customer. At $120, contribution is \(0.60(120-80)=0.60(40)=\$24\). At $160, it is \(0.40(160-80)=0.40(80)=\$32\). At $200, it is \(0.20(200-80)=0.20(120)=\$24\).
Step 2: Subtract the same fixed expense at each price. Expected profit is contribution minus $10: at $120 it is \(24-10=\$14\); at $160 it is \(32-10=\$22\); and at $200 it is \(24-10=\$14\).
Step 3: Compare the maxima. Both contribution and profit are maximized at $160. Subtracting $10 from every contribution value changes the levels but not the ranking. The calculation assumes the fixed expense truly does not change with price, policy count, or the decision itself. Treating an expense per bound policy as if it were a price-independent fixed amount would not be justified.
Retention Is Not Revenue or Profit
Retention uses the binding probability itself. A retention-maximizing price can be lower than the price that maximizes revenue or contribution because retention does not reward a higher premium per retained customer. When evaluating a book, the expected number retained is the initial eligible count times the retention probability, but that count alone still does not measure the premium or profit generated.
Worked Example: Retention Versus Financial Outcomes
A hypothetical renewal portfolio has 250 eligible policies. Three offered prices have retention probabilities of 0.74 at $130, 0.62 at $150, and 0.49 at $170. Expected claims are $82 per renewed policy, and servicing expense is $8 per renewed policy. Compare retention, expected premium revenue, contribution after claims, and profit after claims and servicing expense.
Step 1: Find expected renewals. At $130, the expected number renewed is \(250(0.74)=185\). At $150, it is \(250(0.62)=155\). At $170, it is \(250(0.49)=122.5\). The fractional value is an expected count, not a claim that half a policy is actually renewed.
Step 2: Calculate revenue. Expected premium revenue is price times expected renewals. The three values are \(130(185)=\$24{,}050\), \(150(155)=\$23{,}250\), and \(170(122.5)=\$20{,}825\). Revenue is highest at $130.
Step 3: Calculate contribution and profit. Contribution after claims is expected renewals times price less $82. At $130 it is \(185(130-82)=185(48)=\$8{,}880\). At $150 it is \(155(68)=\$10{,}540\). At $170 it is \(122.5(88)=\$10{,}780\). Profit after the additional $8 servicing expense is, respectively, \(185(130-82-8)=185(40)=\$7{,}400\), \(155(60)=\$9{,}300\), and \(122.5(80)=\$9{,}800\).
Step 4: State the choices. Retention is highest at $130, with probability 0.74. Revenue is also highest at $130, but for a different reason: the product of price and expected renewals is largest there. Contribution and profit are highest at $170. For a check, the $170 profit is \(122.5(\$80)=\$9{,}800\). The three objectives need not agree even when all calculations use the same renewal probabilities and costs.
Assumptions and Common Traps
A comparison is meaningful only if the demand and cost measures match the decision being assessed. If a price changes the composition of customers who bind, expected claims per bound policy may differ by price. In that case, applying one constant claim cost to every option can misrank contribution and profit. Calculate the appropriate conditional expected cost for each price, or state and justify the assumption that the cost is constant.
Similarly, determine whether the objective is per eligible customer, per bound policy, or for the entire book. For a homogeneous group with the same price and response, multiplying a per-customer objective by the same positive \(N\) does not change its maximizing price. For heterogeneous groups or multiple pricing decisions, that simplification may not hold; the demand and cost quantities must be aggregated consistently.
- Ignoring demand: choosing the largest price or largest per-policy margin does not account for how many customers bind.
- Calling revenue profit: \(pq(p)\) includes no deduction for claims or expenses.
- Subtracting a fixed expense incorrectly: a price-independent fixed amount shifts profits but does not change their ranking; a per-policy expense generally varies in total with demand.
- Confusing retention with volume or dollars: retention is a proportion. Expected retained counts and premium revenue require additional calculations.
- Assuming costs are constant without checking: if the expected cost of bound policies varies with price, use the price-specific cost in the objective.
Check Your Understanding
For each question, calculate the requested objective at every listed price before selecting the maximizing price.
- A hypothetical product has prices of $90, $110, and $130, with binding probabilities 0.80, 0.60, and 0.40. Expected claims are $55 per bound policy and operating expense is $10 per bound policy. Which price maximizes revenue, contribution after claims, profit after claims and expense, and retention? (Answer: Revenue—$90; contribution—$110; profit—$110; retention—$90.)
- At prices $80, $100, and $120, binding probabilities are 0.75, 0.55, and 0.35. Total price-sensitive cost is $60 per bound policy, and there is an additional fixed expense of $8 independent of price. Which price maximizes contribution and profit? (Answer: Both select $100; contribution values are $15, $22, and $21, while profit values are $7, $14, and $13.)
- A renewal offer has prices of $120, $150, and $180 with retention probabilities 0.70, 0.55, and 0.35. Expected claims are $105 per renewed policy, with no other expense. Which price maximizes revenue, contribution after claims, and retention? (Answer: Revenue—$120; contribution—$180; retention—$120.)
- At price $140, the binding probability is 0.50; at price $160, it is 0.40. Compare expected revenue per eligible customer at the two prices. Which produces more revenue, and how much lower is the binding probability at the higher price? (Answer: $140 produces $70 revenue versus $64 at $160; the higher price lowers binding probability by 0.10.)
- A hypothetical insurer considers prices of $100 and $140. The binding probabilities are 0.80 and 0.50, and expected claims conditional on binding are $60 and $90, respectively. Which price gives higher contribution after claims per eligible customer? Why should one not use a single common claim cost for both options? (Answer: At $100, contribution is \(0.80(100-60)=\$32\); at $140, it is \(0.50(140-90)=\$25\), so $100 is higher. The stated conditional claim costs differ by price, so one common cost would misstate at least one option.)