Combining Developed Frequency and Severity

Why Combine Developed Frequency and Severity?

A frequency-severity estimate expresses expected loss per exposure as the product of two components: claims per exposure and dollars per claim. When the experience is not fully developed, both components may need adjustment. Developing them separately makes the projection transparent, but only if the count, loss, and exposure bases match.

For example, a count development factor estimates how many claims will ultimately be associated with the experience. A severity development factor estimates the ultimate average loss for those claims. Multiplying the resulting ultimate frequency by ultimate severity gives estimated ultimate loss per exposure. A mismatch—such as developing counts for one coverage while measuring losses for another—breaks that interpretation.

Definition: Let $E$ be the exposure for a defined experience period, $N$ the observed claim count, and $L$ the observed losses on those claims. The observed frequency is $N/E$, and observed severity is $L/N$. If $D_N$ is a claim-count development factor and $D_S$ is a severity development factor, then estimated ultimate frequency is $ND_N/E$, estimated ultimate severity is $(L/N)D_S$, and estimated ultimate pure premium (loss per exposure) is their product.

Here, “severity” means average loss per claim on a clearly specified loss basis. “Exposure” is the denominator appropriate to the product, such as vehicle-years or machine-years. If exposure is recorded in months but the desired frequency is per year, convert it before calculating the rate. The development factors must be appropriate to the same experience period, coverage, claim definition, and loss measure used in the component estimates.

The Combination and Its Algebraic Check

For positive observed claim count, the component calculation is

\text{Ultimate frequency} = \frac{ND_N}{E}, \qquad \text{Ultimate severity} = \frac{L}{N}D_S, \qquad \text{Ultimate pure premium} = \frac{ND_N}{E}\times\frac{L}{N}D_S.

The observed count cancels in the product, giving

\text{Ultimate pure premium} = \frac{L}{E}D_ND_S, \qquad \text{Ultimate losses} = ND_N\times\frac{L}{N}D_S = LD_ND_S.

This is a useful reconciliation: the product of the count and severity development factors acts as an aggregate loss development factor when both components refer to compatible claim and loss bases. It does not mean any two factors can be multiplied without checking their definitions.

Assumptions for the component calculation: The count development factor projects the defined observed claims to the ultimate count for the same exposure period. The severity development factor projects the observed average loss to ultimate average loss for the corresponding claim population and loss basis. Exposure is measured consistently and is not inadvertently developed as if it were claims or losses. The selected factors are treated as applicable to the experience being projected.

The identity follows directly from the definitions: estimated ultimate losses equal estimated ultimate claim count times estimated ultimate dollars per claim. Dividing that product by exposure gives loss per exposure. The cancellation of $N$ is algebraic, not a reason to disregard the claim count; the count is still needed to calculate and interpret frequency and severity separately.

Worked Calculations

Worked Example: Converting Vehicle-Months to Vehicle-Years

A hypothetical fleet has 9,600 vehicle-months of exposure, 16 reported claims, and $80,000 of losses on those claims. The selected count development factor is 1.125 and the selected severity development factor is 1.10. Calculate ultimate frequency per vehicle-year, ultimate severity, ultimate pure premium per vehicle-year, and total ultimate losses. Use the count and severity factors on the stated claim and loss bases.

Method and exposure conversion. Convert vehicle-months to vehicle-years before computing annual frequency:

E=\frac{9{,}600}{12}=800\text{ vehicle-years}.

Develop the claim count.

N_U=16(1.125)=18\text{ claims}.

Thus ultimate frequency is $18/800=0.0225$ claims per vehicle-year.

Develop severity. Observed severity is $80{,}000/16=\$5{,}000$ per claim. Therefore,

S_U=\$5{,}000(1.10)=\$5{,}500\text{ per claim}.

Combine and check. The estimated ultimate pure premium is $0.0225(\$5{,}500)=\$123.75$ per vehicle-year. Ultimate losses are $18(\$5{,}500)=\$99{,}000$. Independently, the aggregate check gives $80{,}000(1.125)(1.10)=\$99{,}000$, and dividing by 800 vehicle-years gives $\$123.75$ per vehicle-year. The results agree.

Common mistake avoided: Using 9,600 as if it were vehicle-years would understate the annual frequency and pure premium by a factor of 12. The exposure must be converted before reporting a per-vehicle-year result.

Worked Example: Separate Count and Severity Development

A hypothetical boat insurer has 2,400 boat-years, 30 claims, and $210,000 of losses on the defined coverage. Suppose the count development factor is 1.20 and the severity development factor is 1.05. Calculate ultimate claim count, frequency, severity, pure premium, and total ultimate losses.

Develop the count.

N_U=30(1.20)=36\text{ claims}, \qquad f_U=\frac{36}{2{,}400}=0.015\text{ claims per boat-year}.

Develop severity. Observed severity is $210{,}000/30=\$7{,}000$ per claim. Its developed value is $7{,}000(1.05)=\$7{,}350$ per claim.

Calculate the pure premium.

0.015(\$7{,}350)=\$110.25\text{ per boat-year}.

Ultimate losses are $36(\$7{,}350)=\$264{,}600$. As a check, the observed loss per exposure is $210{,}000/2{,}400=\$87.50$. The combined factor is $1.20(1.05)=1.26$, so $87.50(1.26)=\$110.25$ per boat-year. Equivalently, $210{,}000(1.26)=\$264{,}600$. Both checks agree with the component calculation.

Common mistake avoided: Applying only the severity factor to observed losses would give $220,500, omitting the increase in claim count. The separate components require both factors in the aggregate loss check.

Worked Example: Deriving a Severity Factor from an Aggregate Loss Factor

For a hypothetical delivery-vehicle portfolio, the experience has 5,000 vehicle-years, 40 claims, and $360,000 in losses. A selected count development factor is 1.10. An analyst also has an aggregate loss development factor of 1.32 and wants component estimates consistent with it. Derive the implied severity development factor, then calculate ultimate frequency, severity, pure premium, and total ultimate losses.

Derive the severity factor. For compatible bases, the aggregate loss factor equals the count factor times the severity factor. Therefore, the implied severity factor is $1.32/1.10=1.20$. This is a ratio, not the aggregate factor itself.

Develop frequency. Ultimate claims are $40(1.10)=44$. Ultimate frequency is $44/5{,}000=0.0088$ claims per vehicle-year.

Develop severity. Observed severity is $360{,}000/40=\$9{,}000$ per claim. Applying the implied severity factor gives $\$9{,}000(1.20)=\$10{,}800$ per claim.

Combine and reconcile. The pure premium is $0.0088(\$10{,}800)=\$95.04$ per vehicle-year. Ultimate losses are $44(\$10{,}800)=\$475{,}200$. The aggregate check is $360{,}000(1.32)=\$475{,}200$, and $475{,}200/5{,}000=\$95.04$ per vehicle-year.

Common mistake avoided: Multiplying observed severity by 1.32 would produce $11,880 per claim and would count the count-development effect in severity as well. The compatible severity factor is $1.32/1.10=1.20$.

Basis Alignment and Interpretation

A mathematically correct product can still be an inappropriate estimate if its components do not describe the same experience. Before multiplying, make the following checks.

Claim definition: The count in the frequency calculation must correspond to the claims represented in the severity denominator. If one component counts claims while the other uses a different unit, such as claimants or occurrences, the simple cancellation of $N$ does not describe the data.
Loss basis: State what is included in losses. For example, paid losses and incurred losses are different measures; losses net of recoveries and gross losses are also different bases. Do not combine a factor selected for one basis with severity measured on another without a justified adjustment.
Exposure basis: Use the same exposure period and unit for the frequency denominator and the reported pure premium. A frequency per vehicle-month is not a frequency per vehicle-year.
Development interpretation: The count factor develops claim numbers; the severity factor develops average loss per claim. If instead an available factor develops aggregate losses, it already reflects changes in both components. Do not apply it as a severity factor on top of count development.
Period and population: The counts, losses, exposures, and selected factors should refer to the same defined experience period and portfolio, or any differences must be explicitly accounted for.

The combined pure premium estimates loss per exposure on the specified basis. It is not, by itself, a full indicated rate: expenses, profit provisions, and other ratemaking adjustments are outside this component calculation. The method is useful because it shows whether an estimate comes from more claims per exposure, higher loss per claim, or both.

Key takeaway: Develop claim count and average severity on compatible bases, divide ultimate claims by the correctly stated exposure, and multiply ultimate frequency by ultimate severity. Reconcile the result to observed losses per exposure multiplied by the product of the two development factors. This topic is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Check Your Understanding

For each problem, keep the claim, loss, and exposure bases aligned. Answers are given for checking.

A hypothetical equipment insurer records 1,200 machine-years, 12 claims, and $54,000 of losses. The count development factor is 1.25 and the severity development factor is 1.10. Find ultimate frequency, severity, pure premium per machine-year, and total ultimate losses. (Answer: 0.0125 claims per machine-year; $4,950 per claim; $61.875, or $61.88, per machine-year; $74,250 total.)
A hypothetical camper portfolio has 3,600 camper-years, 27 claims, and $243,000 of losses. Use a count development factor of $4/3$ and a severity development factor of 0.90. Find ultimate claims, severity, pure premium, and ultimate losses. (Answer: 36 claims; $8,100 per claim; $81 per camper-year; $291,600 total.)
A hypothetical repair plan has 7,500 covered-unit-years, 50 claims, and $500,000 of losses. Its count factor is 1.08 and aggregate loss factor is 1.188. Derive the compatible severity factor and calculate ultimate frequency, severity, pure premium, and losses. (Answer: severity factor 1.10; 0.0072 claims per unit-year; $11,000 per claim; $79.20 per unit-year; $594,000 total.)
A hypothetical courier fleet has 2,000 vehicle-years, 20 claims, and $150,000 of losses. The count factor is 1.15 and aggregate loss factor is 1.38. Calculate the implied severity factor and the component results. (Answer: severity factor 1.20; 0.0115 claims per vehicle-year; $9,000 per claim; $103.50 per vehicle-year; $207,000 total.)
A hypothetical machine warranty portfolio has 18,000 machine-months, 25 claims, and $100,000 of losses. The count factor is 1.12 and severity factor is 1.05. Convert exposure to machine-years and find the ultimate pure premium and losses. (Answer: 1,500 machine-years; 0.01867 claims per machine-year; $4,200 per claim; $78.40 per machine-year; $117,600 total.)

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