Separating Frequency and Severity Effects

Why Separate Frequency and Severity Effects?

A change in pure premium may reflect a change in how often claims occur, a change in the average cost of a claim, or both. Separating those effects helps explain an aggregate change and makes clear whether components reinforce or offset one another. The allocation is not always unique: if both components change, the amount attributed to each can depend on the order in which the changes are applied.

The calculations below compare two periods on a consistent exposure basis and use the same definition of loss in both periods. They isolate the arithmetic of frequency and severity decomposition; they do not by themselves establish why either component changed.

Definition: Claim frequency is the number of claims divided by the corresponding exposure measure. Claim severity is the average loss per claim, calculated using the same loss measure in the numerator as the pure-premium calculation. Pure premium is loss per exposure. Write the initial frequency and severity as $f_0$ and $s_0$, and the later values as $f_1$ and $s_1$. Under the standard compound-loss model, where claim severities have the same mean $s$ and are independent of claim count, expected pure premium is $P=fs$. For observed data on a consistent basis, aggregate loss per exposure also equals claims per exposure times average loss per claim.

Frequency may be expressed as a decimal, such as $0.08$ claims per exposure, or as a percentage, such as 8 claims per 100 exposures. Severity and pure premium are both measured in dollars, but their denominators differ: severity is dollars per claim, whereas pure premium is dollars per exposure.

Sequential Decomposition

A sequential decomposition changes one component at a time. If frequency is changed first, value its change at the initial severity, then value the severity change at the final frequency. Reversing the order values severity at the initial frequency and frequency at the final severity. Both routes reconcile exactly to the observed change in pure premium, but they need not assign the same amount to each component.

\Delta P=f_1s_1-f_0s_0

\begin{aligned} \text{Frequency first:}\quad &\Delta P_f^{(1)}=(f_1-f_0)s_0,\\ &\Delta P_s^{(1)}=f_1(s_1-s_0);\\[4pt] \text{Severity first:}\quad &\Delta P_s^{(2)}=f_0(s_1-s_0),\\ &\Delta P_f^{(2)}=(f_1-f_0)s_1. \end{aligned}

The superscripts distinguish the two orders, not different definitions of frequency or severity. In either sequence, the first component is evaluated while the other component remains at its initial value; the second is evaluated after the first change has been made. These are exact arithmetic decompositions, not estimates of causal effects.

Worked Example: Frequency Increases Before Severity

A hypothetical coverage has frequency $f_0=0.08$ claims per exposure and average severity $s_0=\$5{,}000$ in the initial period. In the later period, frequency is $f_1=0.09$ and severity is $s_1=\$5{,}500$. Calculate the pure-premium change and both sequential decompositions.

Method and initial calculation. Pure premium is frequency times severity. Initially, $P_0=0.08\times \$5{,}000=\$400$ per exposure. Later, $P_1=0.09\times \$5{,}500=\$495$ per exposure. Thus, $\Delta P=\$495-\$400=\$95$ per exposure.

Frequency first. Frequency increases by $0.09-0.08=0.01$. At the initial severity, its contribution is $0.01\times \$5{,}000=\$50$. Severity then increases by $\$5{,}500-\$5{,}000=\$500$. At the final frequency, its contribution is $0.09\times \$500=\$45$. The contributions sum to $\$50+\$45=\$95$.

Severity first. At the initial frequency, the severity change contributes $0.08\times \$500=\$40$. The frequency change, valued at the final severity, contributes $0.01\times \$5{,}500=\$55$. Again, $\$40+\$55=\$95$.

The frequency-first method assigns $\$50$ to frequency; the severity-first method assigns $\$55$. Both are valid paths from $\$400$ to $\$495$. Common mistake avoided: adding the two frequency-first contributions and then claiming that this makes the component allocation unique. Their sum is unique; the separate sequential attributions depend on order. The check is that both totals equal the directly calculated $\$95$ change.

Why the Calculation Order Matters

The order effect is the interaction between the two component changes. To see it, let $\Delta f=f_1-f_0$ and $\Delta s=s_1-s_0$. Expanding the final product gives the initial pure premium, the separate changes evaluated at initial values, and one cross-product:

\begin{aligned} \Delta P &=(f_0+\Delta f)(s_0+\Delta s)-f_0s_0\\ &=s_0\Delta f+f_0\Delta s+\Delta f\Delta s. \end{aligned}

If frequency changes first, its contribution is $s_0\Delta f$, and severity is then evaluated at $f_1=f_0+\Delta f$. That severity contribution is $f_1\Delta s=f_0\Delta s+\Delta f\Delta s$. If severity changes first, it receives the initial-frequency term $f_0\Delta s$, while frequency is evaluated at $s_1=s_0+\Delta s$, giving $s_1\Delta f=s_0\Delta f+\Delta f\Delta s$. In each case the entire interaction term goes to whichever component changes second.

Key result: The interaction in dollar pure-premium terms is $\Delta f\Delta s$. The frequency contribution differs between the two sequential orders by $\Delta f\Delta s$, and the severity contribution differs by the same amount in the opposite direction. If either frequency or severity is unchanged, the interaction is zero and the sequential allocations agree.

When both components increase, the interaction is positive: the second change is applied to a higher value of the first component. When one rises and the other falls, the interaction is negative. A negative interaction does not mean the total pure-premium change must be negative; it describes how the component changes combine.

Symmetric Contributions

A symmetric decomposition averages the two sequential orders. Equivalently, it values the frequency change at average severity and the severity change at average frequency. This treats the two possible calculation orders equally and splits the interaction equally between the components.

\begin{aligned} \Delta P_f^{(\mathrm{sym})} &=\Delta f\,\frac{s_0+s_1}{2} =s_0\Delta f+\frac{1}{2}\Delta f\Delta s,\\ \Delta P_s^{(\mathrm{sym})} &=\Delta s\,\frac{f_0+f_1}{2} =f_0\Delta s+\frac{1}{2}\Delta f\Delta s. \end{aligned}

Adding the symmetric contributions gives $\Delta f(s_0+s_1)/2+\Delta s(f_0+f_1)/2$. Using $s_1=s_0+\Delta s$ and $f_1=f_0+\Delta f$, this sum is $s_0\Delta f+f_0\Delta s+\Delta f\Delta s$, exactly the product expansion for $\Delta P$. Thus the symmetric method both splits the interaction evenly and reconciles to the total change.

Worked Example: Frequency Falls While Severity Rises

For a hypothetical portfolio, frequency falls from $f_0=0.12$ to $f_1=0.10$ claims per exposure, while severity rises from $s_0=\$4{,}000$ to $s_1=\$4{,}400$. Find the total pure-premium change, both sequential allocations, and the symmetric contributions.

Method and total change. The initial pure premium is $P_0=0.12\times \$4{,}000=\$480$ per exposure. The final pure premium is $P_1=0.10\times \$4{,}400=\$440$ per exposure. Therefore $\Delta P=\$440-\$480=-\$40$ per exposure.

Sequential allocations. Here $\Delta f=-0.02$ and $\Delta s=\$400$. If frequency changes first, its contribution is $-0.02\times \$4{,}000=-\$80$. The severity contribution at final frequency is $0.10\times \$400=\$40$, so the total is $-\$80+\$40=-\$40$.

If severity changes first, its contribution at initial frequency is $0.12\times \$400=\$48$. The frequency contribution at final severity is $-0.02\times \$4{,}400=-\$88$. Their sum is $\$48-\$88=-\$40$.

Symmetric allocation. Average severity is $(\$4{,}000+\$4{,}400)/2=\$4{,}200$, so frequency contributes $-0.02\times \$4{,}200=-\$84$. Average frequency is $(0.12+0.10)/2=0.11$, so severity contributes $0.11\times \$400=\$44$. These sum to $-\$84+\$44=-\$40$. The interaction is $(-0.02)(\$400)=-\$8$; half, $-\$4$, is included in each symmetric contribution. Common mistake avoided: treating the severity increase as proof that pure premium must increase. Frequency falls enough to more than offset it, and the final check agrees with the direct change from $\$480$ to $\$440$.

Worked Example: Reconcile Counts, Losses, and Contributions

Worked Example: Separating Effects From Portfolio Totals

A hypothetical book has 10,000 exposures in each of two periods. The first period has 640 claims and aggregate loss of $\$2{,}080{,}000$; the second has 720 claims and aggregate loss of $\$2{,}592{,}000$. Calculate frequency and severity in each period, then decompose the pure-premium change symmetrically.

Method and component calculations. Frequency is claims divided by exposures. Thus $f_0=640/10{,}000=0.064$ and $f_1=720/10{,}000=0.072$ claims per exposure. Severity is aggregate loss divided by claims: $s_0=\$2{,}080{,}000/640=\$3{,}250$ per claim and $s_1=\$2{,}592{,}000/720=\$3{,}600$ per claim.

The initial pure premium is $0.064\times \$3{,}250=\$208$ per exposure. The final pure premium is $0.072\times \$3{,}600=\$259.20$ per exposure. Directly from the totals, these are also $\$2{,}080{,}000/10{,}000=\$208$ and $\$2{,}592{,}000/10{,}000=\$259.20$. The increase is $\$51.20$ per exposure.

Symmetric contributions. Frequency changes by $0.008$, and average severity is $(\$3{,}250+\$3{,}600)/2=\$3{,}425$. Frequency therefore contributes $0.008\times \$3{,}425=\$27.40$ per exposure. Severity changes by $\$350$, and average frequency is $(0.064+0.072)/2=0.068$. Severity contributes $0.068\times \$350=\$23.80$ per exposure. Their sum is $\$27.40+\$23.80=\$51.20$.

The interaction is $0.008\times \$350=\$2.80$, split as $\$1.40$ into each symmetric component. Common mistake avoided: dividing aggregate loss by exposures to get pure premium and also treating that amount as severity. The correct severity denominator is claims; the check is that frequency times severity reproduces the pure premium in both periods.

Interpreting the Decomposition

The symmetric allocation is useful when a single order-neutral attribution is required, but it is not a causal model. It answers an accounting question: how can the observed product change be divided between two changing factors under a stated rule? A causal explanation requires evidence beyond this arithmetic.

Component percentage changes also combine multiplicatively rather than by simple addition. If frequency changes by proportion $a$ and severity by proportion $b$, then the pure-premium factor is $(1+a)(1+b)$, so the total proportional change is $a+b+ab$. For example, a 10% frequency increase and an 8% severity increase produce a pure-premium change of $1.10\times1.08-1=0.188$, or 18.8%, not 18%. The 0.8 percentage-point difference is the relative interaction.

State the units and denominator for frequency, severity, and pure premium before calculating.
Use the same exposure basis and loss measure in both periods.
For a sequential decomposition, name the order and evaluate each change at the value applicable at that step.
For an order-neutral allocation, use average severity for frequency and average frequency for severity.
Check that component contributions sum to the direct change in pure premium.

Separating frequency and severity effects is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus. The central exam discipline is to distinguish the total change, which is fixed by the two pure premiums, from the component attribution, which must be defined by the chosen decomposition method.

Check Your Understanding

For each problem, calculate on the stated consistent exposure basis and check that the component contributions reconcile to the total pure-premium change.

A hypothetical product's frequency rises from 0.05 to 0.06 claims per exposure while severity rises from $\$2{,}400$ to $\$2{,}500$. Find the symmetric frequency and severity contributions. (Answer: frequency $+\$24.50$, severity $+\$5.50$, total $+\$30.00$ per exposure.)
Frequency falls from 0.15 to 0.12, and severity rises from $\$1{,}800$ to $\$2{,}100$. Find the initial and final pure premiums and symmetric contributions. (Answer: $\$270$ and $\$252$ per exposure; frequency $-\$58.50$, severity $+\$40.50$, total $-\$18.00$.)
Frequency rises by 4% and severity falls by 5%. Calculate the total percentage change in pure premium and the interaction in percentage points. (Answer: pure premium decreases by 1.2%; interaction is $-0.2$ percentage points.)
Frequency changes from 0.08 to 0.10 and severity changes from $\$5{,}000$ to $\$4{,}700$. Calculate the severity-first sequential contributions and the symmetric contributions. (Answer: severity-first severity $-\$24$, frequency $+\$94$; symmetric frequency $+\$97$, severity $-\$27$; total $+\$70$ per exposure.)
A hypothetical book has 10,000 exposures and 800 claims with $\$3{,}200{,}000$ total loss in the first period; it has 10,000 exposures and 900 claims with $\$3{,}960{,}000$ total loss in the second. Find both frequencies and severities, then the symmetric contributions. (Answer: frequencies 0.08 and 0.09; severities $\$4{,}000$ and $\$4{,}400$; frequency contribution $+\$42$, severity contribution $+\$34$, total $+\$76$ per exposure.)

What You'll Learn