Why Rounding Requires a Second Review
An optimizer can return prices with cents or other precision that a rating or billing system cannot issue. Converting those outputs to permitted prices is an implementation step, but it can also change customer-level price movements, modeled renewals, expected contribution, and aggregate premium measures. Therefore, passing a control before rounding does not establish that the implemented prices pass it.
The appropriate workflow is to round each recommended price under a stated operational rule, then rerun the relevant customer and portfolio calculations using those rounded prices. If a control fails, adjustments should be made through a documented procedure that preserves operational rules and explains the tradeoff between constraint relief and the modeled effect on contribution.
Rounding rules need not mean “nearest dollar.” Permitted prices may be multiples of $5 or $10, or may be drawn from a policy-specific menu. A rounding rule should also specify what happens at an exact midpoint. Different rules can produce different implemented prices and different results.
Recalculate Customer and Portfolio Results
Suppose policy \(j\) has modeled renewal probability \(p_j(P)\) at price \(P\). If \(I_j\) is an indicator equal to 1 when the customer renews and 0 otherwise, then \(E[I_j\mid P]=p_j(P)\). Expected retained premium from that policy is \(p_j(P)P\). If the expected loss conditional on renewal is \(L_j\) and the variable expense rate on premium is \(e_j\), expected contribution is \(p_j(P)((1-e_j)P-L_j)\). These calculations use the renewal probability at the price actually offered.
For a portfolio of policies, expected retained premium and expected contribution are sums of the corresponding policy-level expectations. Linearity of expectation makes this true whether or not renewals are independent. A written-premium constraint, however, commonly uses the full premium for every policy rather than weighting by renewal probability. The control's definition determines the appropriate calculation.
The justification is direct. For one policy, retained premium is \(I_jP_j^R\). Since the price is fixed once offered, \(E[I_jP_j^R]=P_j^RE[I_j]=P_j^Rp_j(P_j^R)\). Expected contribution conditional on renewal is \((1-e_j)P_j^R-L_j\), so multiplying by the renewal indicator and taking expectations gives the second expression. Summing those policy-level expectations gives the portfolio formulas.
Worked Example: Recomputing Modeled Results After Rounding
A hypothetical insurer has three policy groups. The permitted prices are multiples of $5, and the optimizer's outputs are rounded to the nearest $5, with ties rounded upward. Variable expense is 5% of premium. The table gives group size, optimizer output, modeled renewal probability at both prices, and expected loss per renewal. Calculate expected retained premium and contribution before and after rounding.
| Group | Policies | Optimizer price | Rounded price | Renewal probability: optimizer / rounded | Expected loss |
|---|---|---|---|---|---|
| A | 20 | $523.40 | $525 | 0.795 / 0.790 | $300 |
| B | 12 | $873.20 | $875 | 0.724 / 0.720 | $510 |
| C | 8 | $1,456 | $1,460 | 0.612 / 0.610 | $860 |
Method. For each group, multiply the price by the modeled renewal probability and policy count to obtain expected retained premium. For contribution, first calculate the premium after variable expense, subtract expected loss, and then multiply by expected renewing policies.
Before rounding. Group A has \(20(0.795)(523.40)=\$8,322.06\) expected retained premium. Its contribution is \(20(0.795)(0.95(523.40)-300)=20(0.795)(197.23)=\$3,135.957\). Group B has \(12(0.724)(873.20)=\$7,586.3616\) expected retained premium and contribution \(12(0.724)(0.95(873.20)-510)=12(0.724)(319.54)=\$2,776.16352\). Group C has \(8(0.612)(1,456)=\$7,128.576\) expected retained premium and contribution \(8(0.612)(0.95(1,456)-860)=8(0.612)(523.20)=\$2,561.5872\).
The pre-rounding totals are \(8,322.06+7,586.3616+7,128.576=\$23,036.9976\) expected retained premium, and \(3,135.957+2,776.16352+2,561.5872=\$8,473.70772\) expected contribution.
After rounding. Group A has \(20(0.790)(525)=\$8,295\) expected retained premium and contribution \(20(0.790)(0.95(525)-300)=20(0.790)(198.75)=\$3,140.25\). Group B has \(12(0.720)(875)=\$7,560\) expected retained premium and contribution \(12(0.720)(0.95(875)-510)=12(0.720)(321.25)=\$2,775.60\). Group C has \(8(0.610)(1,460)=\$7,124.80\) expected retained premium and contribution \(8(0.610)(0.95(1,460)-860)=8(0.610)(527)=\$2,571.76\).
The rounded totals are \(\$8,295+\$7,560+\$7,124.80=\$22,979.80\) expected retained premium and \(\$3,140.25+\$2,775.60+\$2,571.76=\$8,487.61\) expected contribution. Thus rounding decreases expected retained premium by \(\$57.1976\) and increases modeled contribution by \(\$13.90228\). To the nearest cent, those changes are a decrease of $57.20 and an increase of $13.90. As a check, summing the three rounded group results reproduces each portfolio total.
Common mistake avoided: retaining the optimizer's renewal probabilities after changing the offered price. The rounded-price probabilities are the ones used here; an implemented-price analysis must use model outputs for the implemented prices.
Test Aggregate Controls on the Implemented Prices
A portfolio premium cap may be defined as a limit on the change in written premium relative to a specified base. If \(B_j\) is policy \(j\)'s base premium, the aggregate base is \(\sum_j B_j\), and the proposed written-premium change is \(\sum_j(P_j^R-B_j)\). A cap of \(c\) on the increase requires \(\sum_j(P_j^R-B_j)\leq c\sum_j B_j\). It is not a cap calculated separately against each policy unless the control explicitly says so.
Worked Example: Restoring an Aggregate Premium Cap
A hypothetical portfolio has three groups. Base premiums are $600, $1,000, and $1,600 per policy; group sizes are 40, 25, and 15. After rounding to permitted $5 amounts, proposed prices are $620, $1,020, and $1,640. The maximum aggregate written-premium increase is 2.5% of aggregate base premium. Prices can be lowered in $5 increments by group. Based on the model, lowering one group's price by $5 for all its policies reduces expected contribution by $140.77 in group A, $85.79375 in group B, or $119.325 in group C. Find a controlled adjustment and recalculate expected results.
Test the control. Aggregate base premium is \(40(600)+25(1,000)+15(1,600)=24,000+25,000+24,000=\$73,000\). The permitted increase is \(0.025(73,000)=\$1,825\). The proposed increase is \(40(620-600)+25(1,020-1,000)+15(1,640-1,600)=800+500+600=\$1,900\), exceeding the cap by \(\$1,900-\$1,825=\$75\).
Choose an adjustment. A $5 decrease across a group reduces written premium by $5 times its policy count: $200 in A, $125 in B, or $75 in C. Compare modeled contribution loss per dollar of premium relief. The ratios are \(140.77/200=0.70385\) for A, \(85.79375/125=0.68635\) for B, and \(119.325/75=1.591\) for C. Group B offers the lowest modeled contribution loss per dollar of relief. Lowering B from $1,020 to $1,015 reduces the increase by $125, more than the $75 needed.
Verify the adjusted control. The adjusted increase is \(800+25(1,015-1,000)+600=800+375+600=\$1,775\). Since \(\$1,775\leq\$1,825\), the cap is met with \(\$50\) of remaining capacity.
Recalculate expected results. At the adjusted prices, the supplied model outputs are: A has renewal probability 0.70 and expected loss $390; B has probability 0.649 and expected loss $620; C has probability 0.51 and expected loss $1,000. Expected retained premium is \(40(0.70)(620)+25(0.649)(1,015)+15(0.51)(1,640)=17,360+16,468.375+12,546=\$46,374.375\), or $46,374.38 to the nearest cent.
Expected contribution is \(40(0.70)(0.95(620)-390)+25(0.649)(0.95(1,015)-620)+15(0.51)(0.95(1,640)-1,000)\). The three terms are \(28(199)=\$5,572\), \(16.225(344.25)=\$5,585.45625\), and \(7.65(558)=\$4,268.70\). Their sum is \(\$15,426.15625\), or $15,426.16 to the nearest cent. The adjustment reduces expected contribution relative to the original rounded plan by $85.79375, matching the stated group B impact. Both the aggregate-control check and the contribution-impact check reconcile.
Common mistake avoided: applying the 2.5% cap to the wrong base or comparing it with an unweighted average rate change. The specified control uses the aggregate $73,000 base premium, so its dollar limit is $1,825.
A Controlled Adjustment Procedure
A controlled response to a failed post-rounding check should be reproducible. It should not be an undocumented series of manual price edits. The following sequence separates identifying the breach from choosing a remedy.
Record the permitted price set, the rounding method, and the midpoint rule. Confirm that each implemented price is operationally valid.
Obtain customer-level modeled renewal probabilities and recompute expected retained premium, contribution, and any other measure used in decision controls.
For example, compute dollars above a written-premium cap using the control's defined base. Do not substitute an expected retained-premium measure for a written-premium measure.
For each eligible price step, calculate constraint relief and modeled impact. When appropriate, compare contribution loss per dollar of relief, while respecting customer-level guardrails and any other applicable constraints.
After making an adjustment, rerun all affected customer and portfolio calculations. Confirm every control, document any remaining slack, and retain the adjusted prices and rationale.
A useful ranking metric is contribution loss divided by the amount of constraint relief. It is not a universal optimization rule: it is a decision aid when contribution is the relevant tradeoff and the candidate adjustments are otherwise permissible. If a control requires a minimum premium rather than a maximum, the direction of the adjustment changes. Multiple controls may also conflict, in which case each candidate change must be tested against all controls rather than only the one that prompted the adjustment.
Worked Example: A Small Rate Cap Breach With Group-Level Steps
A hypothetical portfolio has 24 policies with base premium $750 and rounded price $770, and 16 policies with base premium $1,250 and rounded price $1,280. Prices are permitted in $10 increments. A maximum written-premium increase of 2.5% applies. A $10 reduction across the first group is estimated to reduce contribution by $142.824; the same reduction across the second group is estimated to reduce contribution by $18.56. At the original rounded prices, renewal probabilities are 0.80 and 0.55, respectively. After reducing the second group's price to $1,270, its modeled renewal probability is 0.56. Variable expense is 5%; expected losses per renewal are $440 and $800.
Calculate the cap. Base premium is \(24(750)+16(1,250)=18,000+20,000=\$38,000\), so the maximum increase is \(0.025(38,000)=\$950\). The proposed increase is \(24(770-750)+16(1,280-1,250)=480+480=\$960\), which exceeds the cap by $10.
Select an allowed group-level adjustment. Reducing the first group by $10 provides $240 of written-premium relief; reducing the second group by $10 provides $160. Their contribution-loss-per-relief ratios are \(142.824/240=0.5951\) and \(18.56/160=0.1160\), respectively. Choose the second group. The adjusted increase is \(480+16(1,270-1,250)=480+320=\$800\), below the $950 cap.
Recompute modeled results. The first group has \(24(0.80)=19.2\) expected renewals and expected retained premium \(19.2(770)=\$14,784\). Its contribution is \(19.2(0.95(770)-440)=19.2(291.50)=\$5,596.80\). The adjusted second group has \(16(0.56)=8.96\) expected renewals, expected retained premium \(8.96(1,270)=\$11,379.20\), and contribution \(8.96(0.95(1,270)-800)=8.96(406.50)=\$3,642.24\). Thus expected retained premium is \(\$14,784+\$11,379.20=\$26,163.20\), and expected contribution is \(\$5,596.80+\$3,642.24=\$9,239.04\).
For comparison, before adjustment the second group's expected contribution was \(16(0.55)(0.95(1,280)-800)=8.8(416)=\$3,660.80\). The change is a reduction of \(\$3,660.80-\$3,642.24=\$18.56\), exactly the stated estimate. The adjusted plan therefore passes the written-premium cap and its calculated contribution change agrees with the adjustment estimate.
Common mistake avoided: assuming a small dollar breach permits a smaller-than-operational price step. Here a group-level step changes premium by $160 or $240, not $10. The selected adjustment is the smallest modeled-cost remedy among the stated eligible group steps, even though it leaves more cap slack than strictly necessary.
Common Rounding and Control Traps
- Using stale model outputs. A renewal probability at the optimizer's price is not automatically the probability at the rounded price. Re-query or recalculate the model at the implemented price.
- Applying a percentage to the wrong premium base. A portfolio cap stated as a percentage of aggregate base premium must use the sum of base premiums, not the number of policies, an unweighted rate average, or expected retained premium.
- Confusing written premium with expected retained premium. A written-premium control may count every offered policy price. Expected retained premium weights prices by renewal probabilities. They answer different questions.
- Rounding only a portfolio total. Rounding the total does not create valid customer-level prices. Round at the operational unit, then aggregate the resulting records.
- Failing to recheck after adjustment. An adjustment that fixes one constraint can affect others. Recalculate every affected control and model output after each adjustment.
Check Your Understanding
For each problem, calculate using the stated control definition and round money only at the final step.
- A group has 50 policies, base premium $800, and rounded price $835. The maximum aggregate written-premium increase is 3% of aggregate base premium. Does the group meet the cap, and by how much does the increase exceed or fall below it? (Answer: No; the increase is $1,750, exceeding the $1,200 cap by $550.)
- A portfolio has 12 policies with base premium $500 and rounded price $515, and 8 policies with base premium $900 and rounded price $930. Calculate aggregate written-premium increase as a percentage of base premium. (Answer: $420 / $13,200 = 3.1818%.)
- A price of $612.40 must be rounded to the nearest $5, with ties rounded upward. The base premium is $600 and the policy's maximum permitted increase is 2% of base premium. Find the implemented price, the dollar increase, and whether the movement cap is met. (Answer: $610; increase $10; yes, since the cap is $12.)
- A hypothetical portfolio has 30 policies priced at $700 with renewal probability 0.80 and 20 policies priced at $900 with renewal probability 0.60. Calculate expected retained premium. (Answer: \(30(0.80)(700)+20(0.60)(900)=\$27,600\).)
- A portfolio has 20 policies with base premium $1,000 and rounded price $1,030, and 10 policies with base premium $600 and rounded price $620. The written-premium increase cap is 2% of aggregate base premium. If each policy may be reduced by $10 independently, what is the minimum number of policies that must be reduced by one $10 step to meet the cap? (Answer: 28 policies; the initial $800 increase exceeds the $520 cap by $280.)