Projecting Frequency and Severity Separately

Why Project Frequency and Severity Separately?

A pure premium can change because the number of claims per exposure changes, because the average cost per claim changes, or because both change. Separating these components makes the source of a projected change visible. It also gives a useful check: when the definitions and periods match, the frequency-and-severity projection must agree with a direct projection of pure premium using the combined trend factor.

The calculation depends on a consistent basis. Frequency must use the same claim count and exposure definition throughout; severity must use losses corresponding to those claims; and both components must be projected from the same historical level to the same prospective level. The component trend assumptions are hypothetical selections for the examples below, not industry estimates.

Definition: Claim frequency is the number of claims divided by the corresponding exposure measure. Claim severity is the corresponding loss amount divided by the number of claims. Pure premium is loss per exposure. When the claims, losses, and exposure basis are consistent, pure premium equals frequency multiplied by severity.

The Component Projection

Let $f_0$ be historical claim frequency per exposure and $s_0$ be historical severity per claim. Let $r_f$ and $r_s$ be the selected effective annual trend rates for frequency and severity, and let $t$ be the number of years from the historical level to the prospective level. Under constant annual compound trends, the component projections are $f_t=f_0(1+r_f)^t$ and $s_t=s_0(1+r_s)^t$. The resulting pure premium is $P_t=f_ts_t$.

P_t=f_ts_t =f_0s_0(1+r_f)^t(1+r_s)^t =P_0\left[(1+r_f)(1+r_s)\right]^t

Here $P_0=f_0s_0$ is historical pure premium. The rates can differ, and either may be negative if that component is expected to decline. The formula assumes each selected rate applies consistently throughout the stated interval. If the trend varies over time, use the product of the applicable period-specific factors instead of treating one annual rate as constant.

Key result: With constant annual component trends over the same $t$-year interval, the annual pure premium trend factor is $(1+r_f)(1+r_s)$. Thus, the equivalent annual pure premium trend rate is $r_f+r_s+r_fr_s$, not simply $r_f+r_s$.

Why the Factors Multiply

The identity follows directly from the definitions. If there are $N$ claims, total losses are $L$, and exposure is $E$, then frequency is $N/E$, severity is $L/N$, and pure premium is $L/E$. Provided $N>0$ and all three quantities refer to the same claims and exposure, the product cancels the claim count:

\frac{N}{E}\times\frac{L}{N}=\frac{L}{E}

Trending each component changes the historical product by the product of its two trend factors. For one year, that factor is $(1+r_f)(1+r_s)$. Expanding it gives $1+r_f+r_s+r_fr_s$; the cross-product $r_fr_s$ is why adding the two rates is only an approximation. Over multiple years, compound the combined factor for each year.

Set a common basis.
Identify the historical frequency and severity and confirm they use matching claims, losses, and exposures.

Align the interval.
Determine the number of years from the historical cost level to the prospective level. Apply that same interval to both components.

Project both components.
Apply the selected frequency and severity factors separately.

Combine and check.
Multiply projected frequency by projected severity, then compare with historical pure premium multiplied by the combined factor.

Worked Example: Two-Year Component Projection

A hypothetical coverage has historical frequency of $0.12$ claims per exposure and historical severity of $4,800 per claim. The selected annual frequency trend is 3% and the selected annual severity trend is 5%. Project both components and pure premium two years forward. Assume constant annual compounding and an unchanged coverage and exposure basis.

Historical pure premium. Multiply the historical frequency by historical severity: $0.12\times \$4{,}800=\$576.00$ per exposure.

Project frequency. The two-year frequency factor is $(1.03)^2=1.0609$. Therefore, projected frequency is $0.12\times1.0609=0.127308$ claims per exposure.

Project severity. The two-year severity factor is $(1.05)^2=1.1025$. Therefore, projected severity is $\$4{,}800\times1.1025=\$5{,}292.00$ per claim.

Combine the projections. The pure premium is $0.127308\times \$5{,}292.00=\$673.713936$ per exposure, or $673.71 per exposure to the nearest cent.

Check by direct projection. The combined two-year factor is $1.0609\times1.1025=1.16964225$. Thus, $\$576.00\times1.16964225=\$673.713936$, again $673.71 per exposure to the nearest cent. The method avoids adding the two component trends and treating that sum as the exact annual pure premium trend.

Direct Pure Premium Trending as a Check

A direct pure premium projection starts with $P_0$ and applies the combined trend factor. Under the constant-rate assumptions above, it is algebraically identical to projecting frequency and severity separately. This gives two calculation routes, not two different estimates: a discrepancy indicates a period, basis, or arithmetic inconsistency.

Worked Example: Comparing Component and Direct Projections

A hypothetical book has historical frequency $0.08$ claims per exposure and severity of $6,250 per claim. Its selected annual frequency trend is 2%, its selected annual severity trend is 6%, and the prospective interval is three years. Assume constant annual compounding and consistent claim and exposure definitions. Calculate projected pure premium by both routes.

Historical pure premium. $0.08\times \$6{,}250=\$500.00$ per exposure.

Component route. The frequency factor is $(1.02)^3=1.061208$, so projected frequency is $0.08\times1.061208=0.08489664$. The severity factor is $(1.06)^3=1.191016$, so projected severity is $\$6{,}250\times1.191016=\$7{,}443.85$. Multiplying gives $0.08489664\times \$7{,}443.85=\$631.957853664$, or $631.96 per exposure to the nearest cent.

Direct route. The combined three-year factor is $1.061208\times1.191016=1.263915707328$. Therefore, $\$500.00\times1.263915707328=\$631.957853664$, or $631.96 per exposure to the nearest cent. The agreement verifies that both components were trended over the same three-year interval.

Interpreting Component Selections

Separate projections are useful because frequency and severity need not move together. For example, a hypothetical portfolio could have fewer claims per exposure while repair costs per claim rise. A falling frequency factor does not by itself imply falling pure premium: the combined factor determines the direction and size of the projected pure premium change.

This decomposition is valid only when the component measures have compatible definitions and the selected trends can reasonably be applied to them. A change in the mix of risks, claim-counting practices, coverage terms, or exposure units can alter observed frequency or severity without representing a like-for-like trend. Component projections should not silently treat such a change as a pure trend. The earlier material on mix shifts and trend diagnostics provides context for assessing those issues; here, the calculation assumes the basis is consistent.

Projection approach	Calculation	Useful check
Separate components	Project frequency and severity, then multiply	Confirms each component's interval and basis
Direct pure premium	Project historical pure premium using the combined factor	Must match the component result under the same assumptions

Common Mistake: Adding Component Rates

Adding the rates omits the cross-product in the combined factor. As a numerical check, suppose historical pure premium is $960, with annual frequency and severity trends of 2% and 5% for two years. The exact component projection is:

\$960(1.02)^2(1.05)^2 =\$960(1.147041) =\$1{,}101.15936

Rounded to the nearest cent, this is $1,101.16. The incorrect shortcut adds 2% and 5%, then applies 7% annually: $\$960(1.07)^2=\$960(1.1449)=\$1{,}099.104$, or $1,099.10 to the nearest cent. It is not the component projection. The exact one-year combined rate is $1.02\times1.05-1=0.071$, or 7.1%, not 7%.

Another trap is applying different projection periods to the two components. Even correct factors will not produce the intended prospective pure premium if, for example, frequency is projected two years but severity three. Before multiplying, verify the endpoint and interval for each component.

Key takeaway: Match the historical and prospective bases, project frequency and severity over the same interval, and multiply their factors. Under those conditions, the result equals direct pure premium trending with the combined factor.

Check Your Understanding

For each hypothetical situation, use consistent component trends over the stated interval. Round dollar answers to the nearest cent.

A portfolio has historical frequency $0.10$ claims per exposure and severity of $5,200 per claim. Annual trends are 3% for frequency and 2% for severity. What is the one-year projected pure premium? (Answer: $546.31 per exposure)
Historical frequency is $0.15$ and severity is $4,000. Annual frequency and severity trends are 2% and 4%. Project two years and report pure premium. (Answer: $675.18 per exposure)
A hypothetical group has frequency $0.16$ claims per exposure and severity of $3,750. Annual trends are 4% for frequency and 2% for severity. Project three years. (Answer: $716.23 per exposure)
Historical pure premium is $800. Annual frequency trend is $-1\%$, and annual severity trend is 5%. What is projected pure premium after two years? (Answer: $864.45 per exposure)
The annual pure premium trend is 5%, while annual severity trend is 2%. Under the multiplicative component model, what annual frequency trend gives the same combined factor? (Answer: $1.05/1.02-1=0.0294118$, or approximately 2.9412%)

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