Redistribution Between Risk Groups

Why Measure Redistribution?

An experience plan can change how premium is allocated among groups even when the portfolio’s total premium stays fixed. To see that effect, compare each group’s share of premium with its share of expected losses. A group receiving a larger share of premium than its expected-loss share is allocated relatively more; a group receiving a smaller share is allocated relatively less.

This is an allocation comparison, not a complete test of whether rates are adequate or whether a group is treated fairly in every respect. The expected-loss benchmark and the premium allocation must describe the same groups, coverage, period, and loss basis. If they do not, the measured difference may reflect inconsistent data rather than redistribution.

Definition: For mutually exclusive and exhaustive risk groups $g=1,\ldots,G$, let $L_g$ be group $g$’s expected losses on a common basis and $P_g$ its allocated premium on the corresponding basis. Let $L=\sum_g L_g$ and $P=\sum_g P_g$. The expected-loss share is $q_g=L_g/L$, the premium share is $s_g=P_g/P$, and the share difference is $\Delta_g=s_g-q_g$. The dollar redistribution relative to a proportional-to-expected-loss allocation is $D_g=P_g-Pq_g=P\Delta_g$.

A positive $\Delta_g$ means group $g$ has a greater premium share than expected-loss share; a negative value means it has a smaller share. Report $\Delta_g$ in percentage points, not as a relative percentage change. For example, a change from a 20% expected-loss share to a 23% premium share is a difference of $+3$ percentage points.

The dollar measure $D_g$ answers a related question: compared with assigning the same total premium $P$ in proportion to expected losses, how many dollars more or less are assigned to the group? It is not generally $P_g-L_g$. That raw difference compares allocated premium with expected losses and therefore includes any overall difference between the portfolio’s premium and expected-loss totals.

Calculation and Interpretation

Use one common loss basis for all groups. For example, the expected losses should be consistent in their treatment of coverage, exposure period, development, trend, and any selected loss components. The group premium amounts must likewise use a consistent definition and sum to the portfolio premium being analyzed. If expected losses are estimated from exposure counts and expected loss costs, calculate each group’s expected loss on that same basis before finding its share.

Set the comparison basis.
Identify the groups, the common expected-loss basis, and the premium amounts allocated to those same groups.

Calculate each group’s shares.
Divide each group’s expected losses by total expected losses, and each group’s premium by total premium.

Measure the difference.
Subtract the expected-loss share from the premium share. Multiply that difference by total premium to obtain the dollar redistribution.

Check the portfolio totals.
Because the shares in each allocation sum to 1, the share differences and dollar redistributions must each sum to zero, apart from rounding.

\sum_{g=1}^{G}\Delta_g =\sum_{g=1}^{G}s_g-\sum_{g=1}^{G}q_g =1-1=0, \qquad \sum_{g=1}^{G}D_g =P\sum_{g=1}^{G}\Delta_g=0.

The zero-sum property is an important check, not evidence that the allocation is reasonable. It says that, relative to the proportional benchmark, premium allocated above that benchmark in some groups must be offset by premium allocated below it in others.

Worked Comparisons

Worked Example: Correcting for the Overall Premium Level

A hypothetical plan allocates a total of $1,000,000 among three groups. Their expected losses are $150,000, $250,000, and $600,000; their allocated premiums are $180,000, $270,000, and $550,000. Calculate the share differences and dollar redistributions. The method is to compare shares first and then scale by total premium.

Step 1: Check totals. Expected losses total $150{,}000+250{,}000+600{,}000=\$1{,}000{,}000$. Premiums total $180{,}000+270{,}000+550{,}000=\$1{,}000{,}000$.

Step 2: Calculate expected-loss shares. The three shares are $150{,}000/1{,}000{,}000=15\%$, $250{,}000/1{,}000{,}000=25\%$, and $600{,}000/1{,}000{,}000=60\%$.

Step 3: Calculate premium shares. The shares are $180{,}000/1{,}000{,}000=18\%$, $270{,}000/1{,}000{,}000=27\%$, and $550{,}000/1{,}000{,}000=55\%$.

Step 4: Subtract expected-loss shares from premium shares. The differences are $18\%-15\%=+3$ percentage points, $27\%-25\%=+2$ percentage points, and $55\%-60\%=-5$ percentage points.

Step 5: Convert to dollars. Multiply each difference by the $1,000,000 total premium: $0.03(\$1{,}000{,}000)=+\$30{,}000$, $0.02(\$1{,}000{,}000)=+\$20{,}000$, and $-0.05(\$1{,}000{,}000)=-\$50{,}000$.

Check and common mistake. The share differences sum to $3+2-5=0$ percentage points, and the dollar differences sum to $\$30{,}000+\$20{,}000-\$50{,}000=0$. A tempting but incorrect answer is $+3,-1,-2$ percentage points: the second group’s premium share is 27%, which is 2 points above 25%, not 1 point below it. The method avoids this by subtracting each group’s shares directly and checking that the differences balance.

Worked Example: Separating Redistribution from an Overall Loading

A hypothetical portfolio has expected losses of $200,000, $300,000, and $500,000. The plan allocates total premium of $1,200,000 as $300,000, $360,000, and $540,000. Find the share differences and dollar redistributions, then explain why subtracting expected losses directly gives a different result.

Step 1: Find totals and shares. Expected losses total $1,000,000, so the expected-loss shares are 20%, 30%, and 50%. Premiums total $1,200,000, so the premium shares are $300{,}000/1{,}200{,}000=25\%$, $360{,}000/1{,}200{,}000=30\%$, and $540{,}000/1{,}200{,}000=45\%$.

Step 2: Calculate percentage-point differences. The differences are $25\%-20\%=+5$ points, $30\%-30\%=0$ points, and $45\%-50\%=-5$ points.

Step 3: Scale to total premium. The dollar redistributions are $0.05(\$1{,}200{,}000)=+\$60{,}000$, $0(\$1{,}200{,}000)=\$0$, and $-0.05(\$1{,}200{,}000)=-\$60{,}000$.

Step 4: Interpret the benchmark. Premium assigned in proportion to expected losses would be 20%, 30%, and 50% of $1,200,000: $240,000, $360,000, and $600,000. The actual allocations differ from these by $+\$60{,}000$, $\$0$, and $-\$60{,}000$.

Check and common mistake. The dollar redistributions sum to zero. By contrast, subtracting expected losses from allocated premiums gives $+\$100{,}000$, $+\$60{,}000$, and $+\$40{,}000$, which sum to the overall $200,000 difference between premium and expected losses. Those raw differences mix the portfolio-wide loading with group allocation; they are not the redistribution measure.

Worked Example: Building the Benchmark from Exposure and Loss Cost

A hypothetical commercial program has three groups. Group A has 400 exposure units and an expected loss cost of $500 per unit; Group B has 250 units at $1,200 per unit; Group C has 500 units at $1,000 per unit. The plan allocates $1,100,000 of premium as $242,000, $363,000, and $495,000. Calculate the expected-loss shares, premium shares, and dollar redistributions. Use the stated loss costs on the same coverage and period basis.

Step 1: Calculate group expected losses. Group A has $400(\$500)=\$200{,}000$. Group B has $250(\$1{,}200)=\$300{,}000$. Group C has $500(\$1{,}000)=\$500{,}000$. Total expected losses are $1,000,000.

Step 2: Calculate expected-loss shares. Dividing by $1,000,000 gives 20% for A, 30% for B, and 50% for C.

Step 3: Calculate premium shares. The premium total is $242{,}000+363{,}000+495{,}000=\$1{,}100{,}000$. The group shares are $242{,}000/1{,}100{,}000=22\%$, $363{,}000/1{,}100{,}000=33\%$, and $495{,}000/1{,}100{,}000=45\%$.

Step 4: Calculate and scale the differences. The share differences are $+2$, $+3$, and $-5$ percentage points. Multiplying by $1,100,000 gives $+\$22{,}000$, $+\$33{,}000$, and $-\$55{,}000$.

Check and common mistake. The share differences sum to zero and the dollar differences sum to zero. A common error is to treat exposure counts as the expected-loss shares. The groups have different loss costs, so the relevant expected losses are $200,000, $300,000, and $500,000—not the exposure counts of 400, 250, and 500.

Derivation and Practical Use

The dollar formula follows directly from the comparison being made. If premium were allocated in proportion to expected losses while holding total premium fixed at $P$, group $g$ would receive $Pq_g$. Its actual allocation is $P_g$. The difference between actual and benchmark allocation is therefore $P_g-Pq_g$. Since the premium share is $s_g=P_g/P$, this can also be written $P(s_g-q_g)=P\Delta_g$. This derivation explains why the scaling uses total premium rather than total expected losses.

The proportional-to-expected-loss allocation is a reference point, not necessarily a prescribed rate plan. A plan may intentionally allocate premium differently because of expense provisions, coverage features, practical constraints, or other rating objectives. This measure isolates how allocations compare with the stated expected-loss benchmark; it does not determine whether a difference is justified.

For an evaluation, state whether expected losses represent the same coverage and exposure period as the premiums, and whether the grouping is unchanged across the comparison. If one group’s expected losses include a component excluded from another group’s basis, or if premium totals omit a component included elsewhere, the resulting shares are not a like-for-like diagnostic. When comparing alternative plan allocations, hold the expected-loss benchmark and total premium basis consistent so changes in the differences are attributable to allocation rather than a changed denominator or loss definition.

Key takeaway: Measure redistribution with $\Delta_g=s_g-q_g$, and express its dollar equivalent as $D_g=P\Delta_g$. Confirm that group shares use the same basis and that the differences balance to zero. Do not confuse this allocation comparison with the portfolio-wide difference between premium and expected losses.

Check Your Understanding

For each problem, use a consistent expected-loss basis, calculate shares before dollar differences, and check that the group differences balance.

A hypothetical plan has expected losses of $120,000, $280,000, and $600,000 and allocated premiums of $150,000, $350,000, and $500,000. Find each share difference and dollar redistribution. (Answer: Share differences $+3,+7,-10$ percentage points; dollar redistributions $+\$30,000,+\$70,000,-\$100,000$.)
Expected losses are $250,000, $250,000, and $500,000. Total premium is $1,500,000, allocated as $450,000, $375,000, and $675,000. Find the share differences and dollar redistributions. (Answer: Share differences $+5,0,-5$ percentage points; dollar redistributions $+\$75,000,\$0,-\$75,000$.)
Three groups have 1,000, 500, and 300 exposure units, with expected loss costs of $300, $800, and $1,000 per unit. Premium allocations are $375,000, $500,000, and $375,000. Calculate the share differences. (Answer: Expected losses are $300,000, $400,000, and $300,000; expected-loss and premium shares are 30%, 40%, and 30%, so all share differences and dollar redistributions are zero.)
Expected losses for two groups are $400,000 and $600,000. Their premiums are $500,000 and $700,000. Calculate each difference in percentage points and dollars using total premium. (Answer: Share differences $+1.6667$ and $-1.6667$ percentage points; dollar redistributions $+\$20,000$ and $-\$20,000$.)
Expected losses are $100,000, $300,000, and $600,000. Total premium is $2,000,000, allocated as $300,000, $600,000, and $1,100,000. Find the dollar redistributions and explain why raw premium-minus-loss differences are not the answer. (Answer: Share differences $+5,0,-5$ percentage points; dollar redistributions $+\$100,000,\$0,-\$100,000$. Raw differences are $+\$200,000,+\$300,000,+\$500,000$, reflecting the overall $1,000,000 premium-minus-loss difference as well as allocation.)

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