Why Separate Frequency and Severity?
A change in total claims dollars does not, by itself, explain what happened to the underlying risk. More dollars could result from more claims per exposure, a higher average cost per claim, or both. Separating these effects helps an actuary interpret experience and compare periods on a consistent basis.
The basic comparison uses three quantities: claim frequency per exposure, average severity per claim, and loss cost per exposure (also called pure premium). This tutorial focuses on decomposing observed differences in those quantities. A comparison is meaningful only when the exposure definition, claim-count rules, and loss measurement basis are consistent between periods.
An exposure is the unit used to measure the amount of risk insured, such as one vehicle-year or one building-year. The exposure unit must have the same meaning in each period. A claim count must also use the same definition throughout: for example, the comparison should not count reopened claims in one period but exclude them in the other. Likewise, losses need to be measured on a consistent basis.
The identity \(p=fs\) follows directly from the definitions: \((N/E)(L/N)=L/E\), provided \(N>0\). It links the components exactly, not approximately. If there are no claims, frequency and pure premium are still defined, but average severity \(L/N\) is not; a zero-claim period therefore cannot be decomposed using an observed average severity.
Exact Decomposition of a Change
Let subscripts 0 and 1 denote the earlier and later periods. The pure-premium change is the difference between the products \(f_1s_1\) and \(f_0s_0\). One exact decomposition changes frequency first, holding severity at its earlier value, and then changes severity at the later frequency.
The first term is the frequency contribution under the earlier-period severity. The second is the severity contribution under the later-period frequency. Adding them gives the full change, including the effect of both factors changing. This is a valid accounting decomposition, but the allocation depends on the order in which the factors are changed.
Alternatively, change severity first and frequency second:
The difference between the two allocations is the interaction \((f_1-f_0)(s_1-s_0)\). When both factors move in the same direction, the interaction has a positive sign; when they move in opposite directions, it is negative. If an order-neutral allocation is desired, split this interaction equally between frequency and severity. The resulting symmetric decomposition is:
To verify it, expand the right-hand side. Its cross-terms cancel, leaving \(f_1s_1-f_0s_0\). The symmetric method is useful when neither order is preferred; the sequential method is often easier to interpret as a specified “change one factor, then the other” calculation. State which method is being used rather than presenting one allocation as uniquely determined.
Confirm that the exposure unit, claim-count definition, and loss basis are consistent across periods.
For each period, calculate \(f=N/E\), \(s=L/N\), and \(p=L/E\). Verify that \(fs=p\).
Calculate changes or ratios in frequency and severity, then check that their product explains the pure-premium change.
Worked Comparisons
Worked Example: Both Frequency and Severity Increase
A hypothetical insurer compares two groups of policy-years with the same claim definition and loss basis. In Period 0, there are 10,000 exposures, 500 claims, and $600,000 of losses. In Period 1, there are 12,000 exposures, 720 claims, and $972,000 of losses. Calculate both periods’ frequency, severity, and pure premium, then decompose the pure-premium change by changing frequency first. State the method and one mistake it avoids.
Period 0. Frequency is \(f_0=500/10{,}000=0.05\) claims per exposure. Severity is \(s_0=\$600{,}000/500=\$1{,}200\) per claim. Thus \(p_0=0.05(\$1{,}200)=\$60\) per exposure; directly, \(\$600{,}000/10{,}000=\$60\).
Period 1. Frequency is \(f_1=720/12{,}000=0.06\). Severity is \(s_1=\$972{,}000/720=\$1{,}350\). Thus \(p_1=0.06(\$1{,}350)=\$81\) per exposure; directly, \(\$972{,}000/12{,}000=\$81\).
The pure premium increases by \(\$81-\$60=\$21\) per exposure, or \(21/60=35\%\). The frequency-first decomposition gives a frequency contribution of \((0.06-0.05)(\$1{,}200)=\$12\). The severity contribution is \(0.06(\$1{,}350-\$1{,}200)=\$9\). Their sum is \(\$12+\$9=\$21\), matching the observed change.
As a ratio check, frequency increases by \(0.06/0.05=1.20\), or 20%, and severity increases by \(1{,}350/1{,}200=1.125\), or 12.5%. Their product is \(1.20(1.125)=1.35\), the 35% pure-premium increase. The method is a frequency-first exact decomposition. It avoids the mistake of adding 20% and 12.5% to claim a 32.5% increase: percentage changes in multiplicative components combine by multiplication.
Worked Example: More Claims but Unchanged Frequency
For a hypothetical coverage, Period 0 has 6,000 exposures, 240 claims, and $360,000 in losses. Period 1 has 7,000 exposures, 280 claims, and $490,000 in losses. A reviewer says frequency rose because the later period has more claims. Check the claim using rates per exposure and identify what changed.
In Period 0, \(f_0=240/6{,}000=0.04\), \(s_0=\$360{,}000/240=\$1{,}500\), and \(p_0=\$360{,}000/6{,}000=\$60\). The identity check gives \(0.04(\$1{,}500)=\$60\).
In Period 1, \(f_1=280/7{,}000=0.04\), \(s_1=\$490{,}000/280=\$1{,}750\), and \(p_1=\$490{,}000/7{,}000=\$70\). Again, \(0.04(\$1{,}750)=\$70\).
Claim counts rose by \(280-240=40\), but exposure rose by \(7{,}000-6{,}000=1{,}000\), so frequency remained at 0.04 claims per exposure. Severity increased by \(\$250\), or \(\$250/\$1{,}500=16.6667\%\). Pure premium likewise increased by \(\$10\), or \(16.6667\%\), entirely accounted for by severity because frequency did not change. The method is a per-exposure comparison. It avoids treating a larger raw count as a higher claim rate without accounting for the larger exposure base.
Worked Example: Frequency and Severity Move in Opposite Directions
A hypothetical portfolio has 5,000 exposures, 200 claims, and $400,000 of losses in Period 0. In Period 1 it has 6,000 exposures, 300 claims, and $540,000 of losses. Calculate the component changes and use the frequency-first decomposition to explain why the direction of the pure-premium change cannot be read from severity alone.
In Period 0, \(f_0=200/5{,}000=0.04\), \(s_0=\$400{,}000/200=\$2{,}000\), and \(p_0=\$80\). In Period 1, \(f_1=300/6{,}000=0.05\), \(s_1=\$540{,}000/300=\$1{,}800\), and \(p_1=\$90\). The identity checks are \(0.04(\$2{,}000)=\$80\) and \(0.05(\$1{,}800)=\$90\).
Frequency increased by \((0.05-0.04)/0.04=25\%\), while severity decreased by \((\$1{,}800-\$2{,}000)/\$2{,}000=-10\%\). The frequency-first contributions are \((0.05-0.04)(\$2{,}000)=\$20\) and \(0.05(\$1{,}800-\$2{,}000)=-\$10\). The net increase is \(\$20-\$10=\$10\), which equals \(\$90-\$80\).
The ratio check is \(1.25(0.90)=1.125\), so pure premium increases by 12.5%. The method is an exact sequential decomposition. It avoids assuming that a reduction in average claim size necessarily lowers loss cost: here, the frequency increase more than offsets the severity decrease.
Interpreting the Results Carefully
A component decomposition describes the arithmetic of the observed experience; it does not, on its own, establish why the components changed. A difference in frequency or severity may reflect changes in the insured population or mix, claim reporting or counting, coverage terms, or the period over which losses are measured. Those possibilities should be checked before interpreting a component as a change in underlying risk.
A useful diagnostic is to check both absolute changes and relative changes. Absolute contributions have units of dollars per exposure and sum to the pure-premium difference. Relative changes are percentages and combine multiplicatively. They answer different questions, so avoid mixing an absolute frequency contribution with a percentage severity change in the same sum.
| Quantity | Calculation | Interpretation |
|---|---|---|
| Frequency | \(N/E\) | Claims per exposure |
| Severity | \(L/N\), if \(N>0\) | Average losses per claim |
| Pure premium | \(L/E=fs\) | Loss dollars per exposure |
| Total losses | \(E p\) | Depends on both exposure volume and pure premium |
Total losses can increase even when pure premium is unchanged if the number of exposures increases. Conversely, total losses can fall while pure premium rises if exposure volume falls sufficiently. Therefore, when the question concerns claim cost per unit of risk, compare pure premium and its frequency-severity components rather than relying on total loss dollars alone.
- Wrong: Comparing raw claim counts and concluding that frequency increased because the later count is larger. Correction: Divide each period’s count by its exposure.
- Wrong: Comparing average claim dollars when the claim-count definition or loss basis differs between periods. Correction: Confirm a consistent basis before calculating and interpreting severity.
- Wrong: Adding frequency and severity percentage changes. Correction: Multiply their change factors to obtain the pure-premium change factor.
- Wrong: Reporting one decomposition’s dollar contributions without stating its order. Correction: Name the sequential method, or use the symmetric method if an order-neutral allocation is intended.
Check Your Understanding
For each hypothetical comparison, calculate the requested components before checking the final answer.
- A coverage has 8,000 exposures, 360 claims, and $486,000 of losses in Period 0. Calculate frequency, severity, and pure premium. (Answer: frequency \(0.045\) claims per exposure; severity \(\$1{,}350\) per claim; pure premium \(\$60.75\) per exposure.)
- Period 0 has 4,000 exposures, 160 claims, and $288,000 of losses. Period 1 has 5,000 exposures, 250 claims, and $450,000 of losses. Identify which component changed and calculate the pure-premium percentage change. (Answer: frequency rises from \(0.04\) to \(0.05\), severity stays at \(\$1{,}800\), and pure premium rises from \(\$72\) to \(\$90\), or 25%.)
- Period 0 has 9,000 exposures, 450 claims, and $810,000 of losses. Period 1 has 10,000 exposures, 450 claims, and $900,000 of losses. Calculate the frequency and severity percentage changes and explain why pure premium is unchanged. (Answer: frequency falls 10%, severity rises by \(11.1111\%\), and pure premium is \(\$90\) in both periods because \(0.05(\$1{,}800)=0.045(\$2{,}000)=\$90\).)
- Frequency rises from \(0.03\) to \(0.04\) claims per exposure and severity rises from \(\$1{,}000\) to \(\$1{,}250\). Calculate the pure-premium change and the frequency-first contributions. (Answer: pure premium rises from \(\$30\) to \(\$50\), a \(\$20\) increase; frequency contributes \(\$10\) and severity contributes \(\$10\).)
- A hypothetical portfolio has 12,000 exposures, 480 claims, and $720,000 of losses in Period 0. Period 1 has 15,000 exposures, 660 claims, and $1,122,000 of losses. Calculate the frequency, severity, and pure-premium changes, and check the pure-premium change using component ratios. (Answer: frequency rises from \(0.04\) to \(0.044\), or 10%; severity rises from \(\$1{,}500\) to \(\$1{,}700\), or \(13.3333\%\); pure premium rises from \(\$60\) to \(\$74.80\), or \(24.6667\%\), matching \(1.10(1.133333\ldots)=1.246666\ldots\).)