Weighting Frequency-Severity Estimates Across Groups

Why Group Weights Matter

Frequency-severity estimates are often prepared separately for groups such as vehicle types, territories, or coverage segments. To form a portfolio estimate, the group results must be combined in a way that reflects how much exposure and how many claims each group contributes. Averaging group estimates equally generally does not do this: a small group and a large group would receive the same weight, even if their exposure volumes differ substantially.

This tutorial develops the aggregation identities for observed experience and for selected group estimates. The key distinction is that frequency is exposure-weighted, while severity is claim-count-weighted. Pure premium is the expected loss per exposure, so it can also be aggregated directly by weighting each group’s expected loss by exposure.

Definition: For group $g$, let $E_g$ be exposure on a common basis, $N_g$ the number of claims, and $L_g$ the corresponding total loss. When $E_g>0$, observed frequency is $f_g=N_g/E_g$, and observed pure premium is $p_g=L_g/E_g$. When $N_g>0$, observed severity is $s_g=L_g/N_g$, so $p_g=f_gs_g$. Severity is undefined when $N_g=0$.

All group figures being combined must refer to compatible definitions: the same exposure unit, claim-count basis, loss basis, and evaluation period. For example, it would not be meaningful to combine one group’s developed losses with another group’s undeveloped losses without first putting them on a consistent basis. The earlier material on developing claim counts and severity addresses those component estimates; here the focus is their aggregation.

Pooling Observed Experience

The portfolio frequency is total claims divided by total exposure. The portfolio severity is total loss divided by total claims, provided there is at least one claim. The portfolio pure premium is total loss divided by total exposure. These are ratios of totals, not simple averages of group ratios.

f_{\mathrm{agg}}=\frac{\sum_g N_g}{\sum_g E_g} =\frac{\sum_g E_g f_g}{\sum_g E_g}, \qquad s_{\mathrm{agg}}=\frac{\sum_g L_g}{\sum_g N_g} =\frac{\sum_g N_g s_g}{\sum_g N_g}, \qquad p_{\mathrm{agg}}=\frac{\sum_g L_g}{\sum_g E_g}.

The group-frequency form assumes each group frequency is defined, which requires positive group exposure. The group-severity form assumes the group severity is defined for each included group, which requires positive group claim count; groups with zero claims can instead be included through their counts and losses in the ratio-of-totals form. The aggregate severity itself is defined only when $\sum_g N_g>0$. If every group has zero claims, aggregate severity is undefined, although aggregate observed pure premium is zero when the corresponding total loss is zero.

Key identity: If the total claim count is positive, $p_{\mathrm{agg}}=f_{\mathrm{agg}}s_{\mathrm{agg}}$. Equivalently, $p_{\mathrm{agg}}=\sum_g E_gp_g/\sum_g E_g$, when all group pure premiums are defined. Thus pure premium is an exposure-weighted average of group pure premiums, not generally their unweighted average.

Why the Identities Hold

The weighting rules follow from the definitions. A group frequency tells us claims per unit of exposure, so multiplying it by that group’s exposure recovers its claim count. A group severity tells us loss per claim, so multiplying it by the claim count recovers total loss. Adding those component totals across groups gives the portfolio ratios.

Recover each group’s count.
Since $f_g=N_g/E_g$, rearranging gives $N_g=E_gf_g$. Therefore total claims are $\sum_g E_gf_g$, and dividing by total exposure gives the exposure-weighted aggregate frequency.

Recover each group’s loss.
When $N_g>0$, $s_g=L_g/N_g$, so $L_g=N_gs_g$. Total loss is the sum of these group losses. Dividing total loss by total claims gives the claim-count-weighted aggregate severity.

Connect the aggregate components.
When total claims are positive, $f_{\mathrm{agg}}s_{\mathrm{agg}}=(\sum_gN_g/\sum_gE_g)(\sum_gL_g/\sum_gN_g)=\sum_gL_g/\sum_gE_g=p_{\mathrm{agg}}$. The total claim count cancels, proving the identity.

The proof also shows why an aggregate severity cannot be calculated from the product formula if there are no claims: its definition requires dividing total loss by total claim count. In that case, the direct pure-premium ratio remains the appropriate quantity if exposure is positive.

Worked Example: Pooling Observed Results

A hypothetical insurer reviews two groups of small commercial equipment policies. Group A has 1,000 exposure units, 50 claims, and $150,000 of loss. Group B has 400 exposure units, 40 claims, and $240,000 of loss. Calculate each group’s frequency, severity, and pure premium, then calculate the portfolio estimates. Use dollars per exposure unit and dollars per claim.

Method. Calculate the group ratios first, then pool counts, losses, and exposures. This avoids treating the groups as if they had equal volume.

For Group A, frequency is $50/1{,}000=0.05$ claims per exposure. Severity is $$150{,}000/50=$3{,}000$ per claim. Pure premium is $$150{,}000/1{,}000=$150$ per exposure, also $0.05\times$3{,}000=$150$.

For Group B, frequency is $40/400=0.10$ claims per exposure. Severity is $$240{,}000/40=$6{,}000$ per claim. Pure premium is $$240{,}000/400=$600$ per exposure, also $0.10\times$6{,}000=$600$.

Total exposure is $1{,}000+400=1{,}400$; total claims are $50+40=90$; and total loss is $$150{,}000+$240{,}000=$390{,}000$. Therefore portfolio frequency is $90/1{,}400=0.0642857$ claims per exposure. Portfolio severity is $$390{,}000/90=$4{,}333.33$ per claim. Portfolio pure premium is $$390{,}000/1{,}400=$278.57$ per exposure.

Check: $0.0642857\times$4{,}333.33\approx$278.57$, matching the direct pure-premium calculation. The figures are rounded to two decimal places for dollar amounts and six decimal places for frequency.

Common mistake avoided: The unweighted average of the two group pure premiums is $($150+$600)/2=$375$, not the portfolio pure premium. It incorrectly gives the 400-exposure group the same weight as the 1,000-exposure group.

Combining Selected Group Estimates

In a projection or indication, group frequency and severity may be selected estimates rather than raw observed ratios. Let $f_g$ and $s_g$ now denote the selected frequency and severity for group $g$. On a common exposure basis, the selected expected claim count and loss for that group are:

\widehat N_g=E_gf_g, \qquad \widehat L_g=E_gf_gs_g.

Summing these quantities gives the aggregate selected estimates. Frequency is the total expected count divided by total exposure. Severity is total expected loss divided by total expected claim count, and therefore is defined only when that total expected count is positive. Pure premium can be calculated directly from total expected loss per total exposure.

f_{\mathrm{agg}}=\frac{\sum_g E_gf_g}{\sum_g E_g}, \qquad s_{\mathrm{agg}}=\frac{\sum_g E_gf_gs_g}{\sum_g E_gf_g} \quad\text{if }\sum_g E_gf_g>0, \qquad p_{\mathrm{agg}}=\frac{\sum_g E_gf_gs_g}{\sum_g E_g}.

If $\sum_g E_gf_g=0$, aggregate severity is undefined, not zero: there is no expected claim count over which to average loss per claim. If all selected frequencies are zero and selected severities are finite, every expected loss is zero, so the aggregate pure premium is zero for positive total exposure. The component product $f_{\mathrm{agg}}s_{\mathrm{agg}}$ should not be used in that case because $s_{\mathrm{agg}}$ is undefined.

Worked Example: Aggregating Selected Estimates

A hypothetical insurer projects two groups of specialty tool coverage. Group A has 2,000 exposure units, selected frequency $0.04$, and selected severity $$2{,}500$. Group B has 500 exposure units, selected frequency $0.12$, and selected severity $$5{,}000$. Find expected claims and losses by group, then the aggregate frequency, severity, and pure premium.

Method. Convert each group’s selected frequency into expected claims using exposure, and then use expected claim counts as the severity weights.

Group A expected claims are $2{,}000(0.04)=80$. Its expected loss is $80($2{,}500)=$200{,}000$. Group B expected claims are $500(0.12)=60$. Its expected loss is $60($5{,}000)=$300{,}000$.

Total exposure is $2{,}000+500=2{,}500$, total expected claims are $80+60=140$, and total expected loss is $$200{,}000+$300{,}000=$500{,}000$. Aggregate frequency is $140/2{,}500=0.056$. Aggregate severity is $$500{,}000/140=$3{,}571.43$. Aggregate pure premium is $$500{,}000/2{,}500=$200$.

Check: $0.056\times$3{,}571.43\approx$200$. Dollar amounts are rounded to the nearest cent; frequency is shown to three decimal places.

Common mistake avoided: The simple average of the group pure premiums, $(0.04($2{,}500)+0.12($5{,}000))/2=($100+$600)/2=$350$, is not the portfolio estimate. The exposure-weighted result is $(2{,}000($100)+500($600))/2{,}500=$200$.

Worked Example: A Group With Zero Selected Frequency

A hypothetical portfolio has three groups. Group A has 600 exposure units, selected frequency $0.05$, and selected severity $$4{,}000$. Group B has 900 exposure units, selected frequency $0$, and a stated severity of $$7{,}000$. Group C has 500 exposure units, selected frequency $0.08$, and selected severity $$2{,}500$. Calculate aggregate frequency, severity, and pure premium. Explain how Group B enters the calculation.

Method. Calculate expected counts and losses group by group. Do not let a stated severity create expected loss when the selected frequency is zero.

Group A has $600(0.05)=30$ expected claims and $30($4{,}000)=$120{,}000$ expected loss. Group B has $900(0)=0$ expected claims and $0($7{,}000)=$0$ expected loss. Group C has $500(0.08)=40$ expected claims and $40($2{,}500)=$100{,}000$ expected loss.

Total exposure is $600+900+500=2{,}000$, expected claims total $30+0+40=70$, and expected loss totals $$120{,}000+$0+$100{,}000=$220{,}000$. Aggregate frequency is $70/2{,}000=0.035$. Since expected claims are positive, aggregate severity is defined and equals $$220{,}000/70=$3{,}142.86$. Aggregate pure premium is $$220{,}000/2{,}000=$110$.

Check: $0.035\times$3{,}142.86\approx$110$. Dollar amounts are rounded to the nearest cent; frequency is shown to three decimal places. Group B contributes exposure to the frequency denominator but contributes neither expected claims nor expected loss. Its stated severity has no effect on these totals.

Common mistake avoided: Do not include Group B’s severity as if it represented expected loss or assign it a positive claim-count weight. If all groups instead had zero selected frequency, aggregate severity would be undefined, even though aggregate pure premium would be zero.

Common Errors and Practical Checks

Weighting errors can produce plausible-looking numbers, so a calculation should be checked against the underlying totals. For estimates, first reconstruct expected claims and expected losses by group. Then aggregate those amounts rather than averaging the ratios directly.

Using equal group weights. An arithmetic average of group frequencies ignores exposure volume. Use exposure weights, $E_g/\sum_g E_g$.
Weighting severity by exposure. Severity is loss per claim, so use claim counts or expected claim counts as weights, not exposure alone.
Averaging pure premiums equally. Pure premium is loss per exposure. Use exposure weights, or calculate aggregate expected loss divided by aggregate exposure.
Applying the product identity without checking its condition. The aggregate severity formula and $p_{\mathrm{agg}}=f_{\mathrm{agg}}s_{\mathrm{agg}}$ require a positive total (expected) claim count.
Mixing incompatible bases. Counts, losses, and exposure must correspond to the same groups and measurement basis. A unit mismatch can invalidate every ratio even when the arithmetic is correct.

The direct ratio-of-totals calculation is a reliable audit. For observed data, confirm that aggregate frequency times aggregate severity equals total loss divided by total exposure whenever claims exist. For selected estimates, confirm that the product agrees with total expected loss per total exposure whenever expected claims are positive. These checks are aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Check Your Understanding

For each problem, calculate the requested portfolio quantities from the group data. Treat all exposures, counts, and losses as being on compatible bases.

A hypothetical portfolio has Group A exposure 800, 32 claims, and $96,000 loss; Group B exposure 1,200, 96 claims, and $432,000 loss. Find aggregate frequency, severity, and pure premium. (Answer: frequency $0.064$, severity $$4{,}125$, pure premium $$264$.)
Two hypothetical groups have selected estimates: Group A has exposure 1,500, frequency $0.02$, and severity $$3{,}000$; Group B has exposure 500, frequency $0.10$, and severity $$4{,}500$. Find total expected claims, expected loss, and aggregate pure premium. (Answer: 80 expected claims, $$315{,}000$ expected loss, and $$157.50$ per exposure.)
A hypothetical portfolio has Group A exposure 400, selected frequency $0.10$, and severity $$2{,}000$; Group B exposure 1,600, selected frequency $0.025$, and severity $$4{,}000$. Find aggregate pure premium and compare it with the unweighted average of group pure premiums. (Answer: aggregate pure premium $$120$; unweighted average $$150$.)
Every group in a hypothetical portfolio has positive exposure and selected frequency zero, with finite selected severity. State aggregate frequency, pure premium, and whether aggregate severity is defined. (Answer: aggregate frequency $0$, pure premium $$0$ per exposure, and aggregate severity is undefined.)
A hypothetical portfolio has Group A exposure 1,000, selected frequency $0.03$, and severity $$5{,}000$; Group B exposure 3,000, selected frequency $0.06$, and severity $$2{,}500$. Find expected claims, expected losses, aggregate frequency, aggregate severity, and aggregate pure premium. (Answer: 210 expected claims, $$600{,}000$ expected loss, frequency $0.0525$, severity $$2{,}857.14$, and pure premium $$150$ per exposure.)

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