Why Combine Frequency and Severity Trends?
A projected claim cost can change because the number of claims per exposure changes, because the average cost per claim changes, or because both change. When frequency and severity are projected separately, their effects must be combined on the same time horizon and measurement basis. Simply adding their percentage changes is generally not exact.
The key relationship is that pure premium is frequency multiplied by severity. If the frequency trend factor is below 1 while the severity trend factor is above 1, the overall effect depends on the product of those factors—not on treating the two changes as if they cancel. The calculations in this tutorial are deterministic projections of expected measures; no discounting, mortality model, or severity probability distribution is assumed.
A trend factor is a multiplier that moves a measure from one period to another. For example, a 4% increase corresponds to a factor of \(1.04\), while a 2% decrease corresponds to \(0.98\). When the annual changes vary, multiply the factors for each year. Apply this separately to frequency and severity, then combine the resulting component factors.
Here, \(F_f\) is the cumulative frequency trend factor over the projection period, and \(F_s\) is the cumulative severity trend factor over that same period. These factors are multiplicative: a component factor of \(1.06\) means the projected measure is 1.06 times its starting value.
How the Component Factors Combine
The reason for multiplying is the frequency-severity relationship. Let the starting frequency be \(f\) and the starting severity be \(s\). Starting pure premium is \(fs\). After applying component factors \(F_f\) and \(F_s\), projected frequency is \(fF_f\), projected severity is \(sF_s\), and projected pure premium is:
Since \(fs\) is the starting pure premium, the combined pure premium trend factor is \(F_fF_s\). If the overall factor is written as \(1+g\), where \(g\) is the total percentage change, and the two component percentage changes are \(g_f\) and \(g_s\), then:
Expanding gives \(g=g_f+g_s+g_fg_s\). Adding the component percentages omits the cross-product \(g_fg_s\). That omission may be small for modest changes, but it is not generally zero. The exact calculation is to multiply the factors.
Check that frequency, severity, and pure premium refer to the same exposure unit, claim definition, loss basis, and projection period.
Multiply the annual frequency factors together and, separately, multiply the annual severity factors together.
Multiply the cumulative component factors, then apply the result to current pure premium. Alternatively, project frequency and severity separately and multiply them.
Confirm that the projected pure premium agrees with projected frequency times projected severity, before rounding.
Worked Applications
The following hypothetical examples use consistent exposure and claim definitions. Dollar results are rounded to the nearest cent only at the final step unless an intermediate value is explicitly identified as unrounded.
Worked Example: Opposing Annual Trends Over Two Years
A hypothetical property portfolio has a current frequency of 0.05 claims per unit-year and a current severity of $4,000 per claim. For each of the next two years, frequency is projected to decrease by 2% and severity to increase by 4%. Project the pure premium two years from now. Then calculate expected claims and expected loss for 1,200 projected unit-years.
Method. Compound each component across the two years, multiply the cumulative factors, and verify the result by projecting frequency and severity separately. No interest discounting or probability distribution is used; the expected measures and trend factors are treated as given.
Step 1: Calculate the current pure premium. The starting frequency is \(0.05\) claims per unit-year, and starting severity is $4,000 per claim:
Step 2: Calculate the cumulative component factors. The annual frequency factor is \(0.98\), so the two-year factor is \(0.98^2=0.9604\). The annual severity factor is \(1.04\), so its two-year factor is \(1.04^2=1.0816\).
Step 3: Combine the factors and project pure premium.
Thus, the pure premium increases by \(1.03876864-1=0.03876864\), or \(3.876864\%\). The projected pure premium is:
Step 4: Verify by projecting frequency and severity. Projected frequency is \(0.05\times0.9604=0.04802\) claims per unit-year. Projected severity is \(\$4{,}000\times1.0816=\$4{,}326.40\) per claim. Their product is:
Step 5: Calculate expected claims and loss for 1,200 unit-years. Expected claims are \(0.04802\times1{,}200=57.624\). An expected claim count need not be a whole number. Expected loss, using the projected severity, is:
The pure-premium check gives \(\$207.753728\times1{,}200=\$249{,}304.4736\), which agrees exactly before rounding. Common mistake avoided: treating the 2% decrease and 4% increase as if their effects cancel, or using an incorrect product for the two cumulative factors. The exact combined factor is \(1.03876864\), not a factor obtained by subtracting or adding the annual rates directly.
Worked Example: Different Annual Changes for the Two Components
A hypothetical equipment insurer has a starting frequency of \(0.08\) claims per exposure unit and a starting severity of $2,500. Frequency is projected to increase by 6% in year one and decrease by 2% in year two. Severity is projected to increase by 3% in year one and 4% in year two. Find the two-year projected pure premium and compare the exact percentage change with the sum of the net component changes.
Method. Multiply the two annual factors for each component, then multiply the cumulative frequency and severity factors. Do not assume the two annual component changes are the same.
Step 1: Find the current pure premium.
Step 2: Find the cumulative factors. The frequency factor is \(1.06\times0.98=1.0388\), a net increase of 3.88%. The severity factor is \(1.03\times1.04=1.0712\), a net increase of 7.12%.
Step 3: Combine the cumulative factors.
The combined increase is \(1.11276256-1=0.11276256\), or \(11.276256\%\). Simply adding the two net component changes gives \(3.88\%+7.12\%=11.00\%\), which is not the exact combined change. The difference is \(0.276256\) percentage points.
Step 4: Project frequency, severity, and pure premium. Projected frequency is \(0.08\times1.0388=0.083104\) claims per exposure unit. Projected severity is \(\$2{,}500\times1.0712=\$2{,}678.00\) per claim. Therefore:
The direct pure-premium calculation gives the same result: \(\$200.00\times1.11276256=\$222.552512\). Common mistake avoided: adding net component percentages and reporting an 11.00% increase as exact. That shortcut would produce $222.00 per exposure unit rather than the correctly projected $222.55.
Worked Example: A Decrease in Severity Partly Offsets a Frequency Increase
A hypothetical delivery-vehicle portfolio has a frequency of \(0.12\) claims per vehicle-year and a severity of $4,000 per claim. Over a projection period, frequency is expected to rise by 10% and severity to fall by 8%. Calculate projected pure premium and expected loss for 750 vehicle-years.
Method. Convert the stated changes to factors and multiply them. Then verify the projected loss using both the frequency-severity route and the pure-premium route.
Step 1: Calculate the current pure premium.
Step 2: Calculate the combined factor. The frequency factor is \(1.10\), and the severity factor is \(0.92\):
The combined increase is \(1.2\%\), not zero. Projected frequency is \(0.12\times1.10=0.132\) claims per vehicle-year. Projected severity is \(\$4{,}000\times0.92=\$3{,}680.00\) per claim. The projected pure premium is:
Step 3: Calculate expected loss for 750 vehicle-years. Expected claims are \(0.132\times750=99\). Expected loss by the claim-count route is \(99\times\$3{,}680.00=\$364{,}320.00\). The pure-premium check is \(750\times\$485.76=\$364{,}320.00\), which agrees.
Common mistake avoided: treating a 10% increase and an 8% decrease as a net 2% increase. Their factors multiply to \(1.012\), so the exact increase is 1.2%. The difference arises because the severity decrease applies to the already changed frequency when the two factors are combined.
Conditions for a Useful Combined Projection
Multiplying component factors is straightforward, but a valid projection also requires the components to be comparable. Frequency and severity must use the same claim definition and relevant exposure basis. The severity measure must correspond to the same loss basis in both the starting period and the projected period. For example, a change in coverage limits or deductibles can change measured severity even if the underlying cost trend has not changed.
The calculation also assumes that the selected frequency and severity factors are appropriate for the population being projected. If the mix of risks, coverages, or claim types changes, the observed aggregate movement may not be represented by applying one frequency factor and one severity factor to the starting portfolio. In that case, the actuarial analysis must address the changed basis or composition rather than interpreting the product mechanically.
| Component change | Factor | Meaning |
|---|---|---|
| Increase of 5% | 1.05 | Multiply the starting measure by 1.05 |
| Decrease of 5% | 0.95 | Multiply the starting measure by 0.95 |
| Two annual increases of 5% | \(1.05^2=1.1025\) | Two years of compounding produce a 10.25% increase |
| Frequency factor \(F_f\), severity factor \(F_s\) | \(F_fF_s\) | Combined pure-premium factor when the bases are consistent |
Common Exam Traps
- Adding percentages instead of multiplying factors. The sum omits the cross-product of the component changes. Multiply the factors for the exact combined result.
- Using only one component. A frequency projection alone does not project pure premium if severity also changes, and vice versa.
- Using the wrong number of periods. Two annual changes require two annual factors for each component, even if one component has a decrease in one of those years.
- Mixing measurement bases. A severity trend on one coverage basis cannot automatically be combined with frequency measured under a different claim definition or exposure basis.
- Rounding too early. Retain unrounded cumulative factors and projected measures through the calculation; round only the final reported amount.
Check Your Understanding
For each hypothetical problem, calculate with unrounded factors until the final answer.
- A portfolio's frequency decreases by 3% per year for three years, while severity increases by 2% per year for three years. Find the combined pure-premium factor and percentage change. (Answer: factor \(0.968535888984\), a decrease of \(3.1464\%\).)
- Frequency increases by 4% in year one and decreases by 1% in year two. Severity increases by 5% in year one and 2% in year two. Find the combined two-year factor and percentage change. (Answer: factor \(1.1027016\), an increase of \(10.27016\%\).)
- Current frequency is \(0.10\) claims per unit and severity is $1,500. Frequency falls by 5% in each of two years, while severity rises by 6% in each of two years. Find the projected pure premium to the nearest cent. (Answer: $152.11 per unit.)
- Current frequency is \(0.025\) claims per unit and severity is $8,000. Over the projection period, frequency decreases by 4% and severity increases by 6%. Find the projected pure premium and expected loss for 400 units. (Answer: pure premium $203.52 per unit; expected loss $81,408.00.)
- A book's frequency rises by 8% while severity falls by 7%. If current pure premium is $320, calculate the exact projected pure premium and compare the exact percentage change with the sum of the component changes. (Answer: $321.41 per unit; exact increase \(0.44\%\), versus the additive estimate of \(1.00\%\).)