A Worked Frequency-Severity Projection

Projecting Aggregate Losses from Components

A frequency-severity projection estimates aggregate losses by projecting the number of claims and the average cost per claim, then combining those components. This approach is useful when claim counts and severities have different development patterns or trends. The central discipline is to keep each component on a consistent basis: develop and trend frequency as frequency, develop and trend severity as severity, and apply each factor once.

The projection can be thought of as a sequence: determine projected exposure, select a frequency per exposure unit, estimate the expected claim count, select severity per claim, and multiply. If historical data are immature, the frequency and severity selections must first be brought to ultimate using their respective development assumptions.

Definition: Let $E$ denote exposure measured in a stated exposure unit, $N$ the claim count, and $X_j$ the loss amount for claim $j$. Claim frequency is expected claims per exposure unit, and claim severity is expected loss per claim. Aggregate loss is $S=\sum_{j=1}^{N}X_j$. Under a model in which the expected claim severity is $\mu$ and is the same across claim counts, $E[S]=E[N]\mu$. If expected frequency is $f$, then $E[N]=Ef$, so $E[S]=Ef\mu$.

The aggregate formula follows from conditioning on the number of claims. Given $N=n$, the aggregate is the sum of $n$ claim amounts, each with expected value $\mu$. Thus $E[S\mid N=n]=n\mu$. Taking expectations over $N$ gives $E[S]=E[N]\mu$. This result concerns the expected aggregate; it does not require claim amounts to be independent of one another. It does require the expected severity, conditional on the claim count, to be $\mu$. If severity changes systematically with claim count or another portfolio characteristic, the model must reflect that relationship rather than use one unadjusted average.

Separate Development and Trend by Component

A frequency estimate and a severity estimate can each be immature. A count development factor converts observed counts to estimated ultimate counts; a severity development factor converts the observed average severity to an estimated ultimate average severity. These factors should be defined on the data basis used in the estimate. In particular, do not apply a count development factor to severity or a severity development factor to claim counts.

N_{\mathrm{ult}}=N_{\mathrm{obs}}\,D_N, \qquad \mu_{\mathrm{ult}}=\mu_{\mathrm{obs}}\,D_\mu

Here $D_N$ is the selected count development factor and $D_\mu$ is the selected severity development factor. If the historical exposure is $E_{\mathrm{hist}}$, the developed historical frequency is $N_{\mathrm{ult}}/E_{\mathrm{hist}}$. A future projection then uses projected exposure and selected trend factors. For $t$ years of trend at constant annual rates $r_f$ for frequency and $r_s$ for severity, the factors are $(1+r_f)^t$ and $(1+r_s)^t$, respectively. Rates may be negative; for example, a frequency trend of $-2\%$ uses an annual factor of $0.98$.

E[N_{\mathrm{future}}] = E_{\mathrm{future}} \left(\frac{N_{\mathrm{ult}}}{E_{\mathrm{hist}}}\right) (1+r_f)^t, \qquad \mu_{\mathrm{future}} = \mu_{\mathrm{ult}}(1+r_s)^t

These expressions assume the selected rates are annual compound rates applicable over the stated period, exposure is measured consistently in the historical and future periods, and the selected frequency and severity remain appropriate for the future population. If trends vary by year, use the product of the applicable annual factors rather than a single repeated factor.

Worked Example: Projecting Two Years of Frequency and Severity Trend

A hypothetical insurer has 12,000 projected policy-years. Its selected starting frequency is 0.08 claims per policy-year and its selected starting severity is $4,500 per claim. Assume the frequency decreases by 2% each year and severity increases by 5% each year for two years. Assume the selected starting values are already on the appropriate development basis, exposure is unchanged in definition, and the annual trends compound. Estimate the future expected aggregate loss.

Method. Project the claim count and claim severity separately, then multiply them. This avoids treating the two distinct trend rates as one rate.

Step 1: Claim count before trend. The projected exposure times the starting frequency gives $12{,}000 \times 0.08=960$ expected claims before frequency trend.

Step 2: Frequency trend. The two-year factor is $1-0.02=0.98$ per year, so $0.98^2=0.9604$. The projected expected count is $960\times0.9604=921.984$ claims.

Step 3: Severity trend. The two-year factor is $1+0.05=1.05$ per year, so $1.05^2=1.1025$. Projected severity is $\$4{,}500\times1.1025=\$4{,}961.25$ per claim.

Step 4: Aggregate loss. The expected aggregate is $921.984\times\$4{,}961.25=\$4{,}574{,}193.12$.

Check. Before trend, expected aggregate loss is $12{,}000\times0.08\times\$4{,}500=\$4{,}320{,}000$. The combined trend factor is $0.9604\times1.1025=1.058841$, and $\$4{,}320{,}000\times1.058841=\$4{,}574{,}193.12$, matching the component calculation. Money is rounded to the nearest cent only at the final step.

Worked Projection with Immature Components

When the starting experience is immature, development must be completed before the projected frequency and severity are formed. The following example assumes that observed counts are 80% developed and that the observed average severity requires a separate selected development factor. It also assumes those development factors and the subsequent trends apply to the stated data without overlap.

Worked Example: Developing, Trending, and Projecting Components

A hypothetical book has 10,000 historical exposure units, 680 reported claims, and an observed average severity of $3,600 per reported claim. The selected count development factor is 1.25, and the selected severity development factor is 1.10. A future period is expected to have 11,000 exposure units. Apply a 3% annual frequency trend and a 4% annual severity trend for two years. Estimate its future expected aggregate loss.

Method. Develop the two historical components separately, convert developed count to frequency using historical exposure, and then apply future exposure and component-specific trend factors. The common mistake to avoid is applying the severity development factor to counts, or applying a total loss development factor in addition to these component factors.

Step 1: Develop claim count. $680\times1.25=850$ estimated ultimate claims.

Step 2: Calculate developed frequency. $850/10{,}000=0.085$ claims per historical exposure unit. Before trend, the future exposure implies $11{,}000\times0.085=935$ expected claims.

Step 3: Trend frequency. The two-year factor is $1.03^2=1.0609$, so projected count is $935\times1.0609=991.9415$.

Step 4: Develop and trend severity. Ultimate severity is $\$3{,}600\times1.10=\$3{,}960$. The two-year severity trend factor is $1.04^2=1.0816$, giving projected severity of $\$3{,}960\times1.0816=\$4{,}283.136$ per claim.

Step 5: Combine. The projected expected aggregate is $991.9415\times\$4{,}283.136=\$4{,}248{,}620.348544$, or $4,248,620.35 to the nearest cent.

Check by direct aggregate calculation. The observed aggregate is $680\times\$3{,}600=\$2{,}448{,}000$. Applying the component development factors gives $\$2{,}448{,}000\times1.25\times1.10=\$3{,}366{,}000$ on an ultimate basis. The future-to-historical exposure ratio is $11{,}000/10{,}000=1.10$. Applying the exposure ratio and both trend factors gives $\$3{,}366{,}000\times1.10\times1.0609\times1.0816=\$4{,}248{,}620.348544$. The result agrees with the component projection.

Reconciliation to a Direct Aggregate Projection

A direct aggregate calculation is a useful check, but it is equivalent to the component calculation only when both use consistent bases and assumptions. In a portfolio with multiple groups, each group’s projected count and severity should be combined within that group before summing. A single portfolio-average frequency and severity can also reconcile, provided the averages are exposure- and claim-weighted appropriately and the same trend assumptions apply.

The following example makes the reconciliation explicit. Both groups use the same frequency and severity trend rates, so the common trend factors can be applied to the sum of their starting expected losses.

Worked Example: Reconciling Group Projections to Aggregate Loss

A hypothetical portfolio has two groups. Group A has 6,000 exposure units, selected frequency 0.06, and selected severity $5,000. Group B has 4,000 exposure units, selected frequency 0.10, and selected severity $3,000. Both groups have a 2% annual frequency trend and a 3% annual severity trend over two years. Assume all starting selections are on-level and on an ultimate basis. Estimate each group’s projected aggregate and reconcile the total to a direct calculation.

Method. Project each group’s count and severity using the common component trends. Then check the sum using the starting aggregate and the product of the two trend factors. This check would not be valid if the groups had different trends and the direct calculation ignored those differences.

Step 1: Starting group counts and losses. Group A has $6{,}000\times0.06=360$ expected claims and starting expected loss $360\times\$5{,}000=\$1{,}800{,}000$. Group B has $4{,}000\times0.10=400$ expected claims and starting expected loss $400\times\$3{,}000=\$1{,}200{,}000$. Starting total expected loss is $\$1{,}800{,}000+\$1{,}200{,}000=\$3{,}000{,}000$.

Step 2: Apply component trend factors. The frequency factor is $1.02^2=1.0404$. The severity factor is $1.03^2=1.0609$. Their product is $1.0404\times1.0609=1.10376036$.

Step 3: Project Group A. Its projected count is $360\times1.0404=374.544$, and its projected severity is $\$5{,}000\times1.0609=\$5{,}304.50$. Its projected aggregate is $374.544\times\$5{,}304.50=\$1{,}986{,}768.648$.

Step 4: Project Group B. Its projected count is $400\times1.0404=416.16$, and its projected severity is $\$3{,}000\times1.0609=\$3{,}182.70$. Its projected aggregate is $416.16\times\$3{,}182.70=\$1{,}324{,}512.432$.

Step 5: Sum and reconcile. The component total is $\$1{,}986{,}768.648+\$1{,}324{,}512.432=\$3{,}311{,}281.08$. The direct aggregate calculation is $\$3{,}000{,}000\times1.0404\times1.0609=\$3{,}311{,}281.08$. The two methods agree. Reporting the final group amounts and total to the nearest cent does not change the total.

Common Traps and Checks

Frequency and severity trends compound separately. Adding their annual rates is not a substitute for multiplying their factors. For the two-year trends in the preceding example, an incorrect shortcut would use $1+2(0.02+0.03)=1.10$, producing $\$3{,}000{,}000\times1.10=\$3{,}300{,}000$. The correct combined factor is $1.02^2\times1.03^2=1.10376036$, producing $\$3{,}311{,}281.08$. The shortcut both omits compounding and omits the product term between the trends.

Other checks help locate projection errors:

Check units. Frequency must be claims per the same exposure unit used in the exposure projection. Do not multiply a per-policy frequency by policy-years unless the definitions are compatible.
Keep development and trend distinct. Development moves experience to the selected ultimate basis; trend moves the selected basis to the future period. Apply each only for its stated purpose and period.
Do not double count. If a projected aggregate already includes frequency and severity trends, do not apply those trends again to the aggregate.
Use the right severity weight. Overall severity is weighted by claim counts, not by exposures alone. A simple average of group severities generally will not equal portfolio average severity.
Reconcile on consistent assumptions. A direct aggregate check should use the same exposure, development, trends, and group mix as the component calculation. A difference may signal inconsistent inputs rather than an arithmetic error.

Key takeaway: A frequency-severity projection is an expected count multiplied by an expected cost per claim. Develop, trend, and weight those components on consistent bases; then use a direct aggregate calculation as a check only when it preserves the same assumptions.

Check Your Understanding

For each problem, use the stated assumptions and keep frequency and severity calculations separate. Round money to the nearest cent at the end.

A hypothetical program projects 5,000 exposure units at a selected frequency of 0.04 claims per unit and severity of $2,500. Frequency increases 5% and severity increases 8% over one year. Find the projected expected aggregate loss. (Answer: $567,000.00)
A hypothetical book has 420 reported claims, a count development factor of $1/0.70$, and 6,000 historical exposure units. Future exposure is 6,600 units. Apply a $-1\%$ annual frequency trend for two years. Ultimate starting severity is $2,800 and its annual trend is 6% for two years. Find the projected aggregate. (Answer: $2,035,092.19)
Two hypothetical groups have starting expected losses of $900,000 and $1,100,000. Both receive a 3% annual frequency trend and a 2% annual severity trend for three years. Find the total projected aggregate. (Answer: $2,319,221.27)
A hypothetical book has 300 reported claims, 8,000 historical exposure units, observed severity of $2,500, count development factor $1/0.75$, and severity development factor 1.20. Future exposure is 9,000 units. Apply one year of 4% frequency trend and $-2\%$ severity trend. Calculate the projected aggregate from the components and verify it using the developed historical aggregate. (Answer: $1,375,920.00 by either method)

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