Accounting for Portfolio Mix Shifts in Price Simulations

Why the Retained Portfolio Can Have a Different Risk Mix

A renewal price plan affects more than the number of policies that remain. If customers in different risk groups respond differently to their offered prices, the retained portfolio can have a different composition from the starting portfolio. Its expected claims therefore need not equal the retained policy count multiplied by the starting portfolio’s average loss cost.

For a renewal simulation, the key calculation is group-specific: estimate retained exposure in each risk group, apply that group’s expected loss cost, and then add the group results. This preserves the relationship between who renews and the expected cost of their risk. A portfolio-level shortcut is safe only when its average loss cost matches the retained cohort’s weighted average loss cost.

Definition: A risk group is a category used to distinguish exposures with potentially different expected loss costs, such as vehicle-use classes or property construction groups. For group $j$, let $N_j$ be the starting number of renewal exposures, $r_j$ the probability an exposure renews under the price plan, $L_j$ the expected loss cost per retained exposure, and $P_j$ the offered premium per retained exposure. If the group-specific assumptions are applied to exposures with equal weight, the expected retained count is $N_jr_j$, expected losses are $N_jr_jL_j$, and expected premium is $N_jr_jP_j$.

The loss cost $L_j$ should correspond to the same coverage, exposure period, and cost basis across groups. It may reflect a stated projection or adjustment, but it should not be silently replaced by a portfolio average. Retention rates in the calculations below are treated as supplied inputs to the renewal simulation; this tutorial does not infer them from price.

From Group Retention to Portfolio Results

For a portfolio with $J$ groups, the expected retained exposure count and expected losses are obtained by summing the group-level values. If $N_jr_j$ is fractional, it is an expected count, not a claim that a fractional policy literally renews. Fractional expected counts are appropriate for expected portfolio totals.

N_{\mathrm{ret}}=\sum_{j=1}^{J}N_jr_j, \qquad L_{\mathrm{ret}}=\sum_{j=1}^{J}N_jr_jL_j.

When $N_{\mathrm{ret}}>0$, the retained cohort’s average loss cost is its expected losses divided by expected retained exposure:

\bar L_{\mathrm{ret}} = \frac{\sum_{j=1}^{J}N_jr_jL_j} {\sum_{j=1}^{J}N_jr_j}.

If the simulation also uses a premium and a variable expense amount $E_j$ per retained exposure, expected contribution before any fixed expenses is calculated by group. Here contribution means premium less expected losses and the stated variable expense; it is not necessarily accounting profit.

C_{\mathrm{ret}} = \sum_{j=1}^{J}N_jr_j(P_j-L_j-E_j).

This calculation assumes the group’s quoted premium, expected loss cost, and expense are applicable to the expected retained exposures in that group. If individual-level values vary materially within a group, a more detailed grouping or customer-level calculation may be needed. Also, this is an expected-value calculation: it does not describe the random variation in realized renewals or claims.

Worked Example: Recalculating Loss Cost After Differential Retention

A hypothetical renewal portfolio has 600 lower-risk exposures with an expected annual loss cost of $400 each and 400 higher-risk exposures with an expected annual loss cost of $1,000 each. Under a proposed plan, the renewal probabilities are 0.90 and 0.60, respectively. Calculate expected retained exposures, expected losses, and retained average loss cost. Compare the result with a fixed-mix calculation.

Method. Calculate expected retained count and expected losses within each group, then add across groups.

For the lower-risk group, expected retained exposures are $600(0.90)=540$, and expected losses are $540(\$400)=\$216{,}000$.

For the higher-risk group, expected retained exposures are $400(0.60)=240$, and expected losses are $240(\$1{,}000)=\$240{,}000$.

Thus, the expected retained count is $540+240=780$, and expected losses are $\$216{,}000+\$240{,}000=\$456{,}000$. The retained average loss cost is $\$456{,}000/780=\$584.615\ldots$, or $584.62 per exposure to the nearest cent.

The starting portfolio average loss cost is $(600(\$400)+400(\$1{,}000))/1{,}000=\$640$. Multiplying the retained count by that starting average gives $780(\$640)=\$499{,}200$, which is not the group-based expected loss of $456,000. The retained cohort has a lower expected loss cost because the higher-risk group has the lower renewal probability.

Check. The retained group counts sum to 780, and their group losses sum to $456,000. The retained average of $584.62 is below the starting average of $640, consistent with the shift away from the higher-cost group. The common mistake avoided is applying the starting average loss cost without checking the retained mix.

Why the Group-Weighted Calculation Holds

The group calculation follows from conditional expected losses and additivity. For each group $j$, suppose each starting exposure renews with probability $r_j$, and a retained exposure has expected loss $L_j$. The expected number of retained exposures in that group is $N_jr_j$. Multiplying by the conditional expected loss per retained exposure gives expected group losses $N_jr_jL_j$. Summing those group expectations gives total expected losses.

E[\text{total losses}] = \sum_{j=1}^{J}E[\text{group }j\text{ losses}] = \sum_{j=1}^{J}N_jr_jL_j.

This argument does not require the groups’ renewal outcomes or claim amounts to be independent. Linearity of expectation is enough to add the expected group losses. The group-level formula does require that $L_j$ is the appropriate expected loss cost for retained exposures in group $j$. If renewal within a group selects materially different risks from one another, a single group loss cost may not adequately represent them.

A fixed-mix shortcut uses a chosen average loss cost $\bar L_{\mathrm{fixed}}$ and estimates losses as $N_{\mathrm{ret}}\bar L_{\mathrm{fixed}}$. For a positive retained count, the shortcut gives exactly the same total expected losses as the group calculation if and only if:

\bar L_{\mathrm{ret}}=\bar L_{\mathrm{fixed}}.

Matching the starting and retained group proportions is sufficient for this equality when the fixed average is the starting portfolio average. Equal group loss costs are also sufficient. Neither condition is necessary: different mixes can still have the same weighted average loss cost. What matters for the shortcut’s accuracy is equality of the two averages, not equality of the group proportions.

Worked Example: Different Mixes With the Same Average Loss Cost

A hypothetical portfolio starts with 100 exposures in each of three groups. Their expected loss costs are $400, $600, and $800. A renewal plan retains expected counts of 50, 100, and 50, respectively. Calculate the starting and retained average loss costs and determine whether the starting average can be used to estimate retained losses.

Method. Calculate each weighted average from its own exposure counts. Do not assume that different proportions necessarily imply different average loss costs.

Starting expected losses are $100(\$400)=\$40{,}000$, $100(\$600)=\$60{,}000$, and $100(\$800)=\$80{,}000$. Total starting losses are $180,000 across 300 exposures, so the starting average is $\$180{,}000/300=\$600$.

Retained expected losses are $50(\$400)=\$20{,}000$, $100(\$600)=\$60{,}000$, and $50(\$800)=\$40{,}000$. Total retained losses are $120,000 across 200 exposures, so the retained average is $\$120{,}000/200=\$600$.

The starting and retained proportions differ: the starting mix is one-third in each group, while the retained mix is one-quarter, one-half, and one-quarter. Nevertheless, both averages equal $600. The fixed-mix estimate is $200(\$600)=\$120{,}000$, exactly matching the group calculation.

Check. The retained average equals the fixed average, so the shortcut is exact even though the group proportions differ and the group loss costs are unequal. The common mistake avoided is treating a mix change alone as proof that the fixed-average result must be wrong.

Recalculating Contribution Under a Price Plan

Premium and loss cost should be combined using the same retained group counts. A price plan can change both the number of policies retained and the premium collected per retained policy. Reporting premium for the original count, or expected losses for the retained count, would mix incompatible bases.

Worked Example: Comparing Contribution Across Two Price Plans

A hypothetical portfolio has 300 lower-risk exposures with loss cost $500 and 200 higher-risk exposures with loss cost $1,200. Ignore expenses. Under Plan A, premiums are $700 and $1,600, and renewal probabilities are 0.80 and 0.50. Under Plan B, premiums are $760 and $1,700, and renewal probabilities are 0.70 and 0.40. For each plan, calculate expected retained count, premium, losses, and contribution before expenses.

Method. Apply each plan’s retention probability to each group’s starting count. Then multiply the retained count by that plan’s group premium and the group’s loss cost.

Under Plan A, the lower-risk retained count is $300(0.80)=240$, and its premium and losses are $240(\$700)=\$168{,}000$ and $240(\$500)=\$120{,}000$. The higher-risk retained count is $200(0.50)=100$, and its premium and losses are $100(\$1{,}600)=\$160{,}000$ and $100(\$1{,}200)=\$120{,}000$. Therefore, Plan A has 340 expected retained exposures, $328,000 expected premium, and $240,000 expected losses. Its contribution before expenses is $\$328{,}000-\$240{,}000=\$88{,}000$.

Under Plan B, the lower-risk retained count is $300(0.70)=210$, producing $210(\$760)=\$159{,}600$ premium and $210(\$500)=\$105{,}000$ expected losses. The higher-risk retained count is $200(0.40)=80$, producing $80(\$1{,}700)=\$136{,}000$ premium and $80(\$1{,}200)=\$96{,}000$ expected losses. Therefore, Plan B has 290 expected retained exposures, $295,600 expected premium, and $201,000 expected losses. Its contribution before expenses is $\$295{,}600-\$201{,}000=\$94{,}600$.

Plan B has $6,600 more expected contribution before expenses, despite retaining 50 fewer exposures and producing less premium in total. This comparison is limited to the stated expected contribution definition and assumptions; it does not establish that Plan B is preferable under other business or regulatory objectives.

Check. For each plan, group premiums and losses add to the stated totals, and contribution equals premium less losses. The common mistake avoided is comparing premiums from one retained count with losses from another.

Common Mistakes and Practical Checks

A renewal simulation is most useful when its arithmetic and interpretation make the risk mix visible. Before using a portfolio summary, check the following:

Do not reuse the starting average automatically. Calculate the retained weighted average, or verify that it equals the fixed average used in the shortcut.
Do not confuse a changed mix with a necessarily changed average. Different group proportions can produce the same weighted average when group loss costs and weights offset appropriately.
Use retained exposure as the denominator for retained average loss cost. Dividing retained losses by starting exposures answers a different question.
Keep price, retention, and loss cost aligned by group and plan. Applying a premium from one plan or group to another retained count can distort contribution.
Label expected counts clearly. A non-integer count from $N_jr_j$ is an expectation, not a realized policy count.

Key takeaway: In a price simulation with group-specific renewal, calculate expected retained exposure and expected losses by group before aggregating. A fixed-mix shortcut is exact precisely when the retained cohort’s weighted average loss cost equals the average used by the shortcut; matching proportions and equal group loss costs are sufficient, but not necessary, ways for that equality to hold.

Check Your Understanding

For each problem, use group-specific retained counts before calculating portfolio results.

A hypothetical portfolio has 400 exposures with loss cost $300 and 100 exposures with loss cost $900. Renewal probabilities are 0.75 and 0.40. Find expected retained exposures, expected losses, and retained average loss cost. (Answer: 340 exposures; $126,000 expected losses; $370.59 average loss cost.)
A portfolio has 100 exposures in each of three groups, with loss costs $2,000, $5,000, and $8,000. Expected retained counts are 50, 100, and 50. Find retained losses and average loss cost, and state whether the starting average loss cost gives the same total. (Answer: $1,000,000; $5,000 per exposure; yes, the starting average is also $5,000.)
A hypothetical portfolio has 200 lower-risk exposures with loss cost $450 and 100 higher-risk exposures with loss cost $1,050. Renewal probabilities are 0.90 and 0.60. Find expected retained count, expected losses, and retained average loss cost. (Answer: 240 exposures; $144,000; $600 per exposure.)
A portfolio has 250 exposures with loss cost $400 and 150 with loss cost $900. Renewal probabilities are 0.80 and 0.50; premiums are $650 and $1,300, respectively. Ignore expenses. Find expected premium, expected losses, and contribution before expenses. (Answer: $227,500 premium; $147,500 losses; $80,000 contribution.)
The starting mix across three groups is 20%, 50%, and 30%, with loss costs $300, $500, and $1,000. A plan produces a retained mix of 50%, 20%, and 30%. For 500 expected retained exposures, compare the group-based expected losses with the estimate using the starting average loss cost. (Answer: retained average $550 and group-based losses $275,000; starting average $610 and shortcut losses $305,000, an overstatement of $30,000.)

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