Common Errors In Pricing Objectives And Constraints

Why Pricing-Optimization Errors Matter

A price optimization can be mathematically correct and still answer the wrong business question. A model may maximize retention when the intended goal is profitable growth, compare a portfolio dollar amount with a per-policy threshold, or report a price that violates one of the stated limits. These are not minor presentation issues: each changes what decision the analysis supports.

A reliable check separates three things: the objective being maximized or minimized, the constraints that define acceptable decisions, and the feasible set of decisions that satisfy every constraint. The checks in this tutorial are aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.

Definition: A pricing decision is feasible if it satisfies every stated bound and constraint. An objective is the quantity the optimization is directed to maximize or minimize. A guardrail restricts the allowed decisions but is not, by itself, the objective.

Retention Is Not the Same Objective as Profitability

Suppose $p$ is premium per policy, $n(p)$ is the number of policies retained at that price, and $c$ is the expected loss and expense cost per retained policy. A retention objective seeks a price with a large $n(p)$, or a large retention ratio. A simple contribution objective instead seeks a large portfolio amount $n(p)(p-c)$. The two objectives need not select the same price: reducing premium can raise retention while reducing the contribution earned on each policy.

A retention threshold can still be useful as a guardrail on a profitability objective. But maximizing retention subject to some bounds is not equivalent to maximizing contribution subject to a retention minimum. The objective and the constraint have different roles and should be written separately.

Worked Example: A Retention Maximum That Is Not a Profit Maximum

A hypothetical insurer considers monthly premiums $p$ from $30 to $80. Its expected retained policy count is $n(p)=1000-10p$, and its expected loss and expense cost is $20 per retained policy. Compare the price that maximizes retention with the price that maximizes total contribution over the permitted range.

Method. Evaluate the retention objective separately, then maximize the contribution function. This avoids treating a high policy count as proof of high profitability.

The retention count decreases as premium increases because the slope in $n(p)=1000-10p$ is $-10$. Therefore retention is maximized at the lowest allowed price, $p=$30$. The count is $1000-10(30)=700$, so contribution at that price is $700(30-20)=$7{,}000$.

To find the contribution maximum, write $C(p)=(1000-10p)(p-20)$. Expanding gives $C(p)=1200p-10p^2-20{,}000$. Its derivative is $C'(p)=1200-20p$, so the stationary point is $p=$60$. The second derivative is $-20$, which is negative; thus the contribution function is concave and this point is a maximum. It is inside the permitted range.

At $60, $n(60)=1000-600=400$, and contribution is $400(60-20)=$16{,}000$. As a boundary check, at $80 the count is 200 and contribution is $200(80-20)=$12{,}000$. The $60 contribution exceeds both boundary values and the $30 retention-driven contribution.

Check. The retention-only choice produces 700 policies but $7,000 of contribution; the contribution-maximizing choice produces 400 policies but $16,000. The common mistake is to claim that the price with the most retained policies is therefore the most profitable. It is not.

Keep Per-Policy and Portfolio Quantities Consistent

A per-policy amount is measured in dollars per policy; portfolio profit is measured in total dollars. These quantities cannot be compared directly. When a per-policy target applies to a portfolio, calculate the portfolio average using policy counts as weights. For segments $j$, with $n_j$ policies and contribution $m_j$ per policy, total contribution is $\sum_j n_jm_j$, and average contribution per policy is $\sum_j n_jm_j/\sum_j n_j$.

Worked Example: A Portfolio Threshold With the Wrong Units

A hypothetical portfolio has 100 policies in Segment A, each with contribution of $60, and 900 policies in Segment B, each with contribution of $20. A proposed guardrail requires average contribution of at least $25 per policy. Does the portfolio satisfy it?

Method. Calculate the total contribution, then divide by the total number of policies before comparing with the per-policy threshold.

Segment A contributes $100(60)=$6{,}000$. Segment B contributes $900(20)=$18{,}000$. Total contribution is $6{,}000+18{,}000=$24{,}000$, across $100+900=1{,}000$ policies. Thus average contribution is $24{,}000/1{,}000=$24$ per policy.

Because $24 per policy is less than the required $25 per policy, the portfolio fails the guardrail. A common incorrect calculation compares total contribution of $24,000 directly with $25 and declares the guardrail satisfied. That compares dollars with dollars per policy, so it is dimensionally invalid. The correction is to use the stated quantity: either compare the per-policy average with $25, or convert the requirement to a portfolio total of $1{,}000(25)=$25{,}000$. Since $24,000 is below $25,000, the same conclusion follows.

Check. The weighted average is not the simple average of the two segment contributions: $(60+20)/2=$40$ would give equal weight to segments with very different policy counts. The correct portfolio average is $24 per policy.

Intersect Bounds Before Reporting a Price

A price can be subject to an absolute floor and ceiling as well as a rate-change limit relative to an indicated rate. Each rule creates an allowed interval. A price satisfying one interval may violate another, so first find their intersection. For intervals $[L_1,U_1]$ and $[L_2,U_2]$, the intersection is $[\max(L_1,L_2),\min(U_1,U_2)]$ only if $\max(L_1,L_2)\leq\min(U_1,U_2)$. Otherwise there is no allowed price.

Worked Example: A Floor That Conflicts With a Rate-Change Cap

A hypothetical policy has an indicated premium of $90. An absolute floor requires a premium of at least $100, while a rate-change rule limits the premium to within 5% of the indicated amount. An absolute ceiling is $120. Is there a permitted price?

Method. Convert each rule into an interval and intersect all intervals. Do not assume that an absolute floor takes priority unless the rules explicitly say so.

The absolute bounds give $[$100,$120]$. The 5% rate-change limit gives a lower endpoint of $90(1-0.05)=$85.50$ and an upper endpoint of $90(1+0.05)=$94.50$, so its interval is $[$85.50,$94.50]$.

The combined lower endpoint would be $\max(100,85.50)=$100$, while the combined upper endpoint would be $\min(120,94.50)=$94.50$. Since $100 exceeds $94.50, the intersection is empty. No premium satisfies all three rules.

Check. Choosing the floor of $100 is not a valid workaround: its increase relative to $90 is $(100-90)/90=0.1111$, or 11.11%, which exceeds the 5% cap. The correction is to report the conflict and seek an authorized change to a rule or its inputs, not to label a violating price feasible.

Test Whether Targets Are Attainable

A target can be individually understandable and still be impossible under the permitted price range. Before optimizing, translate each target into a condition on the decision variable and check whether those conditions overlap with the price bounds. If no price meets all the requirements, the constrained problem is infeasible: it has no feasible solution and therefore no constrained optimum.

Worked Example: A Retention Target Outside the Permitted Range

A hypothetical insurer has 1,000 eligible policies. For monthly premium $p$, its expected retained count is $n(p)=1200-10p$, with $p$ restricted to $40 through $80. Management requires retention of at least 85% and portfolio contribution of at least $10,000. Expected loss and expense cost is $25 per retained policy. Determine whether a feasible premium exists.

Method. Convert the retention and contribution requirements into inequalities, then test them against the allowed price interval.

Retention of at least 85% requires at least $0.85(1000)=850$ retained policies. Thus $1200-10p\geq850$, which implies $-10p\geq-350$, or $p\leq$35$. But the price floor requires $p\geq$40$. These two requirements cannot hold together.

Check the contribution requirement at the lowest allowed price, $40. The retained count is $1200-10(40)=800$; contribution is $800(40-25)=$12{,}000$, above the $10,000 minimum. Thus the contribution threshold and price bounds can be met at $40, but retention is only $800/1000=80\%$, below 85%.

Check. The retention requirement demands a price no higher than $35, while the permitted price range starts at $40. No feasible price exists, even though the contribution requirement alone can be met. The common mistake is to report $40 as an “optimal” solution because it meets the floor and profit threshold. It violates the retention target; the correct conclusion is that the combined problem is infeasible.

A Practical Diagnostic Sequence

Use a consistent sequence before interpreting any proposed price. It catches errors in what is being optimized, how constraints are measured, and whether the stated conditions can be met at all.

Name the objective.
Write the exact quantity to maximize or minimize, including whether it is a per-policy or portfolio amount.

Separate guardrails.
Write each minimum, maximum, floor, ceiling, or rate-change rule as its own condition. Do not silently turn a guardrail into the objective.

Check units and aggregation.
Confirm that both sides of each comparison use the same unit and that portfolio averages use policy-count weights.

Check feasibility.
Intersect price intervals and test every target before claiming an optimum. If the feasible set is empty, report infeasibility rather than a violating price.

Verify the candidate.
Substitute the proposed price into every constraint and calculate the objective at the candidate and relevant boundaries.

Common Traps to Avoid

Equating retention with profitability. More retained policies do not necessarily produce more contribution. State the intended objective explicitly.
Comparing unlike units. Total portfolio dollars cannot be compared directly with a dollars-per-policy threshold. Convert one side before testing the condition.
Applying only the most visible bound. A price floor does not override a rate-change cap unless the governing rules explicitly establish that priority.
Assuming every target is achievable. Translate each target into a condition on price and check it against the permitted range.
Calling a near-feasible price feasible. A price that misses even one hard constraint is not a solution to the stated problem.

Key takeaway: A sound pricing decision requires the right objective, consistent units, mutually compatible bounds, and a nonempty feasible set. If a target conflicts with the stated restrictions, identify the conflict rather than presenting a price that violates them.

Check Your Understanding

For each problem, identify the relevant objective or constraint carefully and check the arithmetic before deciding.

A hypothetical portfolio has 500 eligible policies and retention $n(p)=650-5p$, for $p$ between $30 and $90. Cost is $20 per retained policy. Find the price selected by maximizing retention alone, its retained count and contribution, and compare its contribution with the contribution at $60. What does this show? (Answer: Retention alone selects $30; 500 policies and $5,000 contribution. At $60, 350 policies yield $14,000, so the retention maximum is not the contribution maximum.)
Segment A has 80 policies contributing $75 each; Segment B has 320 contributing $15 each. Does the portfolio meet a minimum average contribution of $30 per policy? State the total and average, and identify the unit error in comparing total with $30. (Answer: Total $10,800; average $27 per policy, so it fails. Comparing $10,800 directly with $30 mixes dollars and dollars per policy.)
A hypothetical indicated premium is $80, an absolute floor is $90, and a rate-change cap permits increases of no more than 5%. The absolute ceiling is $110. Find the combined allowed interval or state that none exists. (Answer: The rate-change interval is [$76, $84]; its intersection with [$90, $110] is empty.)
Expected retained count is $n(p)=900-12p$ from 900 eligible policies, with $p$ between $35 and $65. Is a retention target of at least 60% feasible? (Answer: The target requires 540 policies, so $900-12p\geq540$ implies $p\leq$30$; the $35 floor makes the target infeasible.)
Expected retained count is $n(p)=900-10p$, with $p$ between $40 and $70, cost of $30 per retained policy, and a retention requirement of at least 50% of 900 eligible policies. Find the price that maximizes contribution subject to the bounds and retention requirement. (Answer: Retention requires $900-10p\geq450$, hence $p\leq$45$. The unconstrained contribution maximum is $60, so the constrained maximum is at $45; 450 policies at $15 contribution each produce $6,750.)

What You'll Learn