Why Portfolio Constraints Need Their Own Calculation
A price recommendation can satisfy every customer-level guardrail and still miss a portfolio target. The reason is aggregation: a percentage change for a small premium may matter less than the same percentage change for a large premium, while a high quoted price may produce less expected premium if it reduces renewal probability. Portfolio controls therefore need a clearly specified denominator, weight, and measurement basis.
This tutorial formulates three distinct controls: average rate change on the current portfolio, expected total renewal premium, and expected contribution. They are not interchangeable. In particular, a limit on quoted price changes does not automatically control expected premium after customer response. The methods here are aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.
The count \(n_i\) may be 1 when a record represents one customer, or larger when it represents a group whose members share the relevant inputs. Aggregation must use the actual portfolio weights; taking a simple average across records is correct only when the records represent equal exposure and the chosen measure calls for that average.
Three Different Portfolio Measures
For a quoted average rate change, compare candidate quoted premium with current premium across the in-force policies. A common portfolio measure is the change in total quoted premium divided by total current premium:
A requirement that the average rate change lie between \(L\) and \(U\) is \(L \le R \le U\). This is a current-premium-weighted measure: each record’s percentage change is weighted by its current premium volume \(n_i p_i\). It describes the quoted price movement for the in-force portfolio, not the premium expected after renewals.
Expected renewal premium uses a different weight. If \(r_{ij}\) is the renewal probability at choice \(j\), the expected renewal premium is:
A lower bound \(T\) on expected premium is imposed as \(P_{\mathrm{exp}}\ge T\); an upper bound or target interval can be written in the same way. This calculation counts only expected renewing policies, under the assumed renewal probabilities and prices.
For contribution, define \(k_i\) as expected loss and expense cost per renewed policy for record \(i\), on the same time and coverage basis as \(q_{ij}\). If \(k_i\) does not vary by price choice, expected contribution is:
A contribution floor \(C_{\min}\) requires \(C_{\mathrm{exp}}\ge C_{\min}\). This version treats \(q_{ij}-k_i\) as contribution conditional on renewal and multiplies by the probability of renewal. If costs vary with price choice or customer response, the appropriate choice-specific expected contribution must be used instead. Any fixed costs not included in \(k_i\) must be handled separately if the target is net of those costs.
Why the Average Rate Formula Is Weighted
The portfolio rate change is a ratio of aggregate dollars, not an average of percentages with each record automatically receiving equal weight. To see why, define the percentage change for one record as \(g_i=(q_i-p_i)/p_i\), for a selected price \(q_i\). Its premium change is \(p_i g_i\). Summing those changes over represented policies gives:
The weights \(n_i p_i/\sum_h n_h p_h\) sum to 1, so this is a weighted average of the record-level percentage changes. A record with greater current premium volume has a greater effect on the portfolio result. This derivation assumes the specified portfolio measure is based on current premium; a regulator, business plan, or internal control may define a different denominator, which must be followed explicitly.
Formulating the Choices and Constraints
In a discrete price-menu problem, the price decision is represented by the selected choice for each record. The same selection variables can be used in the objective and in multiple aggregate constraints. For example, an objective that maximizes expected contribution can be written as \(\max \sum_i n_i\sum_j x_{ij}r_{ij}(q_{ij}-k_i)\), subject to one selected choice per record and any portfolio limits. The objective answers which plan is preferred; the constraints state which plans are permitted.
Identify the policy count or exposure represented by each record. Confirm that current and candidate premiums use consistent coverage periods and units.
Decide whether the control applies to quoted price movement, expected renewal premium, or expected contribution. State the denominator and threshold.
Apply the selected price and, where relevant, its renewal probability to each record, then multiply by the record’s count.
For example, a premium floor requires expected premium to be at least the target; an average rate-change cap requires the calculated rate change not to exceed its limit.
Worked Example: Testing a Portfolio Average Rate Cap
A hypothetical insurer has three renewal groups. Group A has 100 policies with current premium $600 each and a proposed price of $660. Group B has 20 policies with current premium $1,500 each and a proposed price of $1,500. Group C has 10 policies with current premium $4,000 each and a proposed price of $3,800. The maximum permitted current-premium-weighted average rate change is 3.0%. Determine whether the proposed prices satisfy the cap, and calculate the result if Group A’s price is changed to $650.
Method. Use total proposed premium change divided by total current premium. For Group A, current premium volume is \(100(600)=\$60{,}000\), and the proposed change is \(100(660-600)=\$6{,}000\). For Group B, current premium volume is \(20(1{,}500)=\$30{,}000\), and the change is \(20(1{,}500-1{,}500)=\$0\). For Group C, current premium volume is \(10(4{,}000)=\$40{,}000\), and the change is \(10(3{,}800-4{,}000)=-\$2{,}000\).
Total current premium is \(\$60{,}000+\$30{,}000+\$40{,}000=\$130{,}000\). Total proposed change is \(\$6{,}000+\$0-\$2{,}000=\$4{,}000\). Therefore the average rate change is \(\$4{,}000/\$130{,}000=0.030769\), or 3.0769%. The proposed plan exceeds the 3.0% cap.
If Group A’s price is $650, its change becomes \(100(650-600)=\$5{,}000\). The revised total change is \(\$5{,}000-\$2{,}000=\$3{,}000\), so the portfolio rate change is \(\$3{,}000/\$130{,}000=0.023077\), or 2.3077%. The revised prices satisfy the cap. The result is checked against the limit: 2.3077% is no greater than 3.0%.
Common mistake avoided. The unweighted average of the three original percentage changes is \((10\%+0\%-5\%)/3=1.6667\%\). That calculation gives the groups equal importance despite their very different premium volumes, so it does not measure the specified portfolio rate change.
Expected Premium Depends on Renewal Response
A premium target based on expected renewals incorporates both the offered price and the probability of accepting or renewing at that price. The following example treats the renewal probabilities as given estimates for the hypothetical choices. It does not assume that a quoted-price increase produces a particular response in every setting; the relevant probabilities must come from the pricing and retention analysis used for the decision.
Worked Example: Meeting an Expected Renewal-Premium Floor
A hypothetical portfolio contains 10 policies in group A and 20 in group B. For group A, the choices are $900 with renewal probability 0.90, or $1,000 with probability 0.82. For group B, the choices are $600 with probability 0.95, or $700 with probability 0.85. The required expected renewal premium is at least $20,000. Calculate expected premium for each choice combination and identify a combination that meets the floor.
Method. For each group and choice, multiply policy count by renewal probability and quoted price. Group A’s $900 choice gives \(10(0.90)(900)=\$8{,}100\); its $1,000 choice gives \(10(0.82)(1{,}000)=\$8{,}200\). Group B’s $600 choice gives \(20(0.95)(600)=\$11{,}400\); its $700 choice gives \(20(0.85)(700)=\$11{,}900\).
The four portfolio totals are:
- A at $900 and B at $600: \(\$8{,}100+\$11{,}400=\$19{,}500\).
- A at $1,000 and B at $600: \(\$8{,}200+\$11{,}400=\$19{,}600\).
- A at $900 and B at $700: \(\$8{,}100+\$11{,}900=\$20{,}000\).
- A at $1,000 and B at $700: \(\$8{,}200+\$11{,}900=\$20{,}100\).
The combination of $900 for A and $700 for B meets the floor exactly: expected premium is $20,000, which is at least $20,000. The combination with both lower prices falls short by $500. The calculation is checked by adding the two group expected premiums, not by applying renewal probability to the total quote after aggregation.
Common mistake avoided. Treating all 30 policies as certain to renew would produce quoted totals rather than expected renewal premium. For example, the selected combination’s quoted total is \(10(900)+20(700)=\$23{,}000\), but its expected renewal premium is $20,000 after applying the two different renewal probabilities.
Expected Contribution and a Portfolio Floor
Expected contribution is not simply the premium on policies that might renew. It also subtracts the cost associated with those renewals. The calculation below assumes the stated cost per renewed policy is a suitable expected loss-and-expense amount for that group and is constant across that group’s price choices. The expected contribution floor is a portfolio constraint, not a statement that each individual choice must independently meet the floor.
Worked Example: Checking a Contribution Requirement
A hypothetical portfolio has 10 policies in group A, 5 in group B, and 8 in group C. Expected cost per renewal is $500, $600, and $400, respectively. The selected prices and renewal probabilities are: A at $800 with probability 0.80; B at $1,000 with probability 0.70; and C at $700 with probability 0.85. The required expected contribution is at least $5,600. Does the selected plan meet it?
Method. For each group, calculate the conditional contribution per renewal as price minus expected cost, multiply by the renewal probability, and then multiply by the policy count.
For A, conditional contribution is \(\$800-\$500=\$300\). Expected contribution per eligible policy is \(0.80(\$300)=\$240\), giving \(10(\$240)=\$2{,}400\) for the group. For B, conditional contribution is \(\$1{,}000-\$600=\$400\). Expected contribution per eligible policy is \(0.70(\$400)=\$280\), giving \(5(\$280)=\$1{,}400\). For C, conditional contribution is \(\$700-\$400=\$300\). Expected contribution per eligible policy is \(0.85(\$300)=\$255\), giving \(8(\$255)=\$2{,}040\).
Total expected contribution is \(\$2{,}400+\$1{,}400+\$2{,}040=\$5{,}840\). The plan meets the $5,600 floor with \(\$5{,}840-\$5{,}600=\$240\) of headroom. The group totals add to the stated portfolio result, providing a direct arithmetic check.
Common mistake avoided. Ignoring renewal probabilities would give \(10(\$300)+5(\$400)+8(\$300)=\$7{,}400\), which overstates expected contribution for this model. Each conditional margin must be multiplied by the probability that the policy renews.
Interpreting Several Controls Together
A portfolio can be subject to more than one aggregate control. For example, the selected prices might need to satisfy both a maximum average rate change and a minimum expected contribution. Each expression must be calculated on its own stated basis. A quoted-price rate cap uses current premium and does not include \(r_{ij}\); expected renewal premium and contribution do include renewal probabilities. Adding these controls to the same decision problem does not make their underlying measures interchangeable.
| Control | Measure | Example condition |
|---|---|---|
| Average rate change | Quoted premium change divided by current premium | \(L \le R \le U\) |
| Expected renewal premium | Expected renewing policies times their quoted prices | \(P_{\mathrm{exp}}\ge T\) |
| Expected contribution | Expected renewals times contribution per renewal | \(C_{\mathrm{exp}}\ge C_{\min}\) |
A sound implementation records the metric and its inputs alongside each constraint. In particular, it should be possible to trace a portfolio total back to customer-level choices, weights, and probabilities. Before interpreting a result, check that the same policy count is not accidentally applied twice, that dollars and thousands of dollars have not been mixed, and that all probabilities are applied to the correct choices.
Common Exam and Implementation Traps
- Averaging percentages without the required weights. For a current-premium-weighted rate measure, use premium change divided by total current premium, not a simple mean of group percentage changes.
- Using quoted premium when the target is expected premium. Expected renewal premium includes the choice-specific renewal probability and may be materially different from the sum of quotes.
- Applying retention to the wrong quantity. For contribution under the stated model, multiply renewal probability by the margin per renewal, not by premium alone and not by a margin that has already been probability-weighted.
- Confusing policy counts and customer counts. If one customer has several policies, use the unit specified by the portfolio measure and apply the corresponding weight consistently.
- Assuming every limit is a floor. A cap uses an upper-bound inequality; a minimum premium or contribution requirement uses a lower-bound inequality. State the direction before evaluating the result.
Check Your Understanding
Use the stated measure and weights in each problem; answers are rounded only where needed.
- A portfolio has 40 policies at current premium $500 each with a proposed increase of 8%, and 10 policies at current premium $2,000 each with a proposed decrease of 4%. What is the current-premium-weighted average rate change? (Answer: 2.0%)
- A hypothetical renewal portfolio has 12 policies offered $750 with renewal probability 0.80 and 8 policies offered $900 with renewal probability 0.75. What is expected renewal premium? (Answer: $12,600)
- There are 20 policies offered $650 with renewal probability 0.90 and expected cost per renewal of $500, and 10 policies offered $1,000 with renewal probability 0.70 and expected cost per renewal of $700. What is total expected contribution? (Answer: $4,800)
- A portfolio has 10 policies offered $800 with renewal probability 0.90 and 15 policies offered $600 with renewal probability 0.80. Does the plan meet an expected premium floor of $15,000? Calculate the amount above or below the floor. (Answer: No; expected premium is $14,400, which is $600 below the floor)
- A portfolio’s current premium is $200,000. A selected price plan increases total quoted premium by $7,000, while expected renewal premium under the modeled renewal probabilities is $180,000. What is the quoted average rate change, and why is it not the ratio of expected renewal premium to current premium minus one? (Answer: 3.5%; the rate change is based on quoted premium change over current premium, whereas expected renewal premium also reflects renewal probabilities)