Deriving the Combined Pure Premium Trend

Why Combine Frequency and Severity Trends?

A projected loss cost can change because the number of claims per exposure changes, because the average cost per claim changes, or both. Combining these effects correctly is important: the combined change is generally not just the sum of the two percentage changes. The exact result includes an interaction between frequency and severity.

This tutorial derives that combined result and shows how to quantify the interaction. It assumes that frequency and severity are measured for the same claims, exposures, coverage basis, and time interval. The result is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Definition: Let $f$ denote claim frequency, measured as claims per exposure, and let $s$ denote average severity, measured as dollars per claim. The pure premium $P$, measured as expected loss dollars per exposure, is $P=fs$, when frequency and severity refer to a consistent claim and exposure basis. A frequency trend factor $T_f$ is the ratio of projected frequency to starting frequency; a severity trend factor $T_s$ is the corresponding ratio for average severity.

For example, if a hypothetical portfolio has frequency $0.08$ claims per exposure and severity of $\$12{,}500$ per claim, its pure premium is $0.08 \times \$12{,}500=\$1{,}000$ per exposure. The units cancel appropriately: claims per exposure multiplied by dollars per claim gives dollars per exposure.

When using observed totals, the identity can also be seen by dividing total losses by exposure: total losses per exposure equals claims per exposure multiplied by total losses per claim. For projected values, that same decomposition is useful only when the components are defined consistently and represent the same projected claim population. A changed mix of claim types, exposures, or coverage terms can make component comparisons misleading even if the multiplication itself is correct.

Deriving the Combined Trend Factor

Suppose frequency and severity are projected from a starting period to a future period. If their respective trend factors are $T_f$ and $T_s$, then projected frequency is starting frequency multiplied by $T_f$, and projected severity is starting severity multiplied by $T_s$. The projected pure premium follows by multiplying those projected components.

P_0=f_0s_0,\qquad P_1=(f_0T_f)(s_0T_s)=P_0T_fT_s.

Therefore, the combined pure premium trend factor is the product of the frequency and severity trend factors. The combined percentage trend is that factor minus one. To express the two component trends as rates, write $T_f=1+g_f$ and $T_s=1+g_s$, where $g_f$ and $g_s$ are the frequency and severity percentage changes over the same interval.

T_P=T_fT_s=(1+g_f)(1+g_s), \qquad g_P=T_P-1=g_f+g_s+g_fg_s.

The term $g_fg_s$ is the interaction term. It arises because the severity change applies to the frequency-changed claim volume as well as the starting volume. The additive approximation $g_f+g_s$ leaves out this product. It may be close when both rates are small, but the multiplicative calculation gives the exact combined change under the stated component assumptions.

Key result: Over a common interval, combine frequency and severity percentage trends using $g_P=g_f+g_s+g_fg_s$, not merely $g_f+g_s$. If both component trends are positive, the interaction is positive. If one is positive and the other negative, it is negative. If either component trend is zero, the interaction is zero.

Worked Applications

Worked Example: Two Positive Trends

A hypothetical small-equipment portfolio has starting frequency $0.08$ claims per exposure and starting severity of $\$12{,}500$ per claim. Over the same projection interval, frequency is expected to rise by $4\%$ and severity by $6\%$. Calculate the combined pure premium trend and projected pure premium. Use the multiplicative method and identify the additive-approximation error.

Method. Calculate the starting pure premium, apply each component factor, and compare the exact combined rate with the sum of the component rates. This avoids treating the two percentage changes as if they acted only on the starting pure premium.

The starting pure premium is $P_0=0.08\times \$12{,}500=\$1{,}000$ per exposure. The component factors are $T_f=1.04$ and $T_s=1.06$. Thus, $T_P=1.04\times1.06=1.1024$, so the combined trend is $1.1024-1=0.1024$, or $10.24\%$.

For a component check, projected frequency is $0.08\times1.04=0.0832$ claims per exposure, and projected severity is $\$12{,}500\times1.06=\$13{,}250$ per claim. Their product is $0.0832\times \$13{,}250=\$1{,}102.40$ per exposure. The dollar increase is $\$1{,}102.40-\$1{,}000=\$102.40$, which is $10.24\%$ of $\$1{,}000$.

The interaction is $0.04\times0.06=0.0024$, or $0.24$ percentage points. Adding the component rates alone gives $4\%+6\%=10\%$, which understates the exact combined rate by $0.24$ percentage points. Check: the exact factor calculation and the component-based projected pure premium both give $\$1{,}102.40$.

Worked Example: A Frequency Decline and a Severity Increase

A hypothetical repair-coverage portfolio has frequency of $0.05$ claims per exposure and severity of $\$8{,}000$ per claim. During the next common projection interval, frequency is expected to decline by $3\%$, while severity is expected to increase by $8\%$. Find the combined trend and projected pure premium. Explain the sign of the interaction.

Method. Convert each percentage change to a factor before multiplying. The frequency factor is $T_f=1-0.03=0.97$, and the severity factor is $T_s=1+0.08=1.08$. This avoids incorrectly treating a decline as a positive factor below one.

The starting pure premium is $P_0=0.05\times \$8{,}000=\$400$ per exposure. The combined factor is $T_P=0.97\times1.08=1.0476$, giving a combined trend of $4.76\%$. The interaction is $(-0.03)(0.08)=-0.0024$, or $-0.24$ percentage points. Thus the exact rate is $-3\%+8\%-0.24\%=4.76\%$, rather than the $5\%$ additive approximation.

Projected frequency is $0.05\times0.97=0.0485$, and projected severity is $\$8{,}000\times1.08=\$8{,}640$. The projected pure premium is $0.0485\times \$8{,}640=\$419.04$ per exposure. Equivalently, $\$400\times1.0476=\$419.04$. Check: the increase is $\$19.04$, and $\$19.04/\$400=0.0476=4.76\%$.

Common mistake avoided: adding $-3\%$ and $8\%$ gives $5\%$, but misses the fact that the increased severity applies to a smaller projected claim count. Here that interaction reduces the combined rate.

Worked Example: Separating the Dollar Effects

A hypothetical equipment policy has frequency $0.10$ claims per exposure and severity of $\$8{,}000$ per claim. For one projection interval, frequency rises by $5\%$ and severity rises by $3\%$. Calculate the pure premium increase in dollars and separate it into the frequency effect, severity effect, and interaction.

Method. Expand the product of the two changed components. This identifies the interaction in dollars and prevents counting only the two stand-alone effects.

The starting pure premium is $P_0=0.10\times \$8{,}000=\$800$ per exposure. Expanding the projection gives $P_1=P_0(1+0.05)(1+0.03)=P_0(1+0.05+0.03+0.05\times0.03)$. The frequency contribution is $\$800\times0.05=\$40.00$. The severity contribution is $\$800\times0.03=\$24.00$. The interaction contribution is $\$800\times0.0015=\$1.20$.

The total increase is $\$40.00+\$24.00+\$1.20=\$65.20$, so the projected pure premium is $\$865.20$ per exposure. Directly, projected frequency is $0.10\times1.05=0.105$, projected severity is $\$8{,}000\times1.03=\$8{,}240$, and their product is $0.105\times \$8{,}240=\$865.20$. Check: the direct and expanded calculations agree.

The combined rate is $5\%+3\%+0.15\%=8.15\%$. The common mistake is to report an $8\%$ increase and omit the $\$1.20$ interaction; the exact product includes it.

Worked Example: Compounding the Combined Trend Over Two Years

A hypothetical service-contract portfolio has starting frequency $0.10$ claims per exposure and severity of $\$8{,}000$ per claim. Frequency is projected to increase by $5\%$ each year and severity by $3\%$ each year. Assume those annual rates apply for two successive years. Find the annual combined trend, the two-year cumulative trend, and the pure premium at the end of year two.

Method. First combine the annual component factors, then compound that combined factor over two years. This avoids adding annual percentage rates across years without accounting for compounding.

For one year, the combined factor is $1.05\times1.03=1.0815$, so the annual combined trend is $8.15\%$. Over two years, the cumulative factor is $1.0815^2=1.16964225$, and the cumulative trend is $1.16964225-1=0.16964225$, or $16.964225\%$.

The starting pure premium is $0.10\times \$8{,}000=\$800$ per exposure. After two years, frequency is $0.10\times1.05^2=0.11025$ claims per exposure. Severity is $\$8{,}000\times1.03^2=\$8{,}487.20$ per claim. Their product is $0.11025\times \$8{,}487.20=\$935.7138$, which rounds to $\$935.71$ per exposure. The factor check gives $\$800\times1.16964225=\$935.7138$, also $\$935.71$ to the nearest cent.

The calculation keeps the annual time unit consistent: each year's frequency factor is paired with that year's severity factor. Common mistake avoided: doubling the annual combined rate to claim a $16.30\%$ two-year change ignores compounding; the cumulative change is $16.964225\%$.

Conditions, Interpretation, and Exam Traps

The product formula is an algebraic consequence of the pure premium decomposition, but its application to projections depends on suitable definitions and assumptions. Frequency must count the claims whose costs are represented in severity, and both components must use the same exposure basis and projection interval. For example, a frequency trend based on one claim definition cannot automatically be combined with severity for a differently limited claim definition.

The formula also assumes the projected component factors can be applied together to the relevant portfolio. If the mix of policyholders, territories, or claim types changes, the observed frequency and severity may change for mix reasons as well as underlying trend. The multiplication still gives the result for the specified factors, but it does not by itself establish that those factors represent a comparable population. Estimating or selecting the component trends is a separate task.

Another trap is confusing the interaction in percentage points with the interaction in dollars. The percentage interaction is $g_fg_s$; the dollar interaction for starting pure premium $P_0$ is $P_0g_fg_s$. For the two-positive-trend example, these are $0.24$ percentage points and $\$2.40$, respectively, because $\$1{,}000\times0.0024=\$2.40$. In the dollar-effects example, the interaction is $0.15$ percentage points and $\$1.20$, because $\$800\times0.0015=\$1.20$.

Check the basis.
Confirm that frequency, severity, and exposure refer to the same claim population, coverage basis, and interval.

Convert rates to factors.
For an increase $g$, use $1+g$; for a decline, use a negative $g$, so the factor is less than one.

Multiply and interpret.
Calculate $T_P=T_fT_s$, subtract one for the combined rate, and identify $g_fg_s$ as the interaction when using percentage changes.

Check the result.
Apply the component factors to the starting frequency and severity, multiply the projected components, and compare with the starting pure premium times $T_P$.

The combined pure premium trend is useful because it translates separate changes in claim counts and claim costs into one change in expected loss per exposure. The practical point is not that the interaction is always large; rather, it is part of the exact calculation and can be especially material when component changes are substantial or when a precise dollar projection is required.

Check Your Understanding

For each problem, use matching time intervals and component bases unless the scenario states otherwise. Answers are provided for checking.

A hypothetical product has frequency increasing by $2\%$ and severity increasing by $5\%$ over one year. What is the exact combined pure premium trend, and how many percentage points does it differ from adding the rates? (Answer: $7.10\%$; the interaction is $0.10$ percentage points.)
For a hypothetical coverage, frequency falls $4\%$ while severity rises $10\%$ over the same interval. Calculate the exact combined trend and the interaction term. (Answer: $5.60\%$; interaction $=-0.40$ percentage points.)
A portfolio has starting frequency $0.06$ claims per exposure and severity $\$15{,}000$ per claim. Frequency rises $3\%$, and severity rises $4\%$. Find the starting pure premium and projected pure premium. (Answer: starting $\$900.00$; projected $\$964.08$ per exposure.)
Annual frequency trend is $+2\%$, and annual severity trend is $-1\%$, each applied for two years. Find the annual combined trend and the cumulative two-year trend. (Answer: annual trend $0.98\%$; cumulative two-year trend $1.969604\%$.)
In a hypothetical matched comparison, frequency changes from 120 to 132 claims per fixed exposure base, while average severity rises from $\$5{,}000$ to $\$5{,}250$. Calculate both component trends, the interaction, and the exact pure premium trend; compare with simple addition. (Answer: frequency $+10\%$, severity $+5\%$, interaction $+0.50$ percentage points, exact trend $15.50\%$, versus $15.00\%$ by addition.)

What You'll Learn