Why the Choice of Measure Matters
Territorial experience can be summarized in more than one reasonable way. A loss ratio compares losses with premium; a loss cost compares losses with exposure; frequency and severity separate claim incidence from claim size. These measures answer related but different questions. A territory with a low loss ratio may have a high loss cost if its premium per exposure is also high, while a per-policy measure may conceal differences in insured property size.
The right measure depends on the purpose of the comparison and on whether the data bases are comparable. For territorial analysis, the losses, premiums, exposures, and claim counts should refer to the same territory and experience period, and should be on consistent valuation and coverage bases. Where adjustments have been made to the data, the adjustment basis should also be consistent across territories. This tutorial is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.
What Each Measure Says
A loss cost is average loss per exposure unit. It is useful when the question concerns losses associated with an exposure, and when comparable exposure counts are available. It does not, by itself, describe the premium charged for those exposures. A loss ratio is the proportion of earned premium represented by losses. It directly reflects the relationship between losses and the premium base used in the calculation, but it depends on that premium base: differences in premium per exposure can affect territorial comparisons.
Frequency and severity provide a decomposition of loss cost. Frequency measures how often claims occur per exposure; severity measures average loss per claim. Together, they can help explain why loss cost differs. Frequency is not a complete measure of expected loss when claim sizes differ, and severity should not be compared without considering claim count and the characteristics of claims.
This identity follows by multiplying the two ratios: \((N/E)(L/N)=L/E\), after cancelling \(N\). It requires \(N>0\) for the displayed severity ratio; if there are no claims, the observed loss cost and frequency are both zero, while average severity is undefined. The identity is an accounting decomposition, not a claim that frequency and severity are statistically independent.
The connection between loss ratio and loss cost is similarly direct. Since \(P/E\) is earned premium per exposure,
Thus, a loss ratio is loss cost divided by premium per exposure. Equal loss ratios do not imply equal loss costs unless premium per exposure is also equal. Conversely, equal loss costs can produce different loss ratios when premium per exposure differs.
Worked Comparisons
Worked Example: Equal Loss Ratios, Different Loss Costs
A hypothetical insurer compares two territories over the same period. Territory M has 840 earned exposure units, $252,000 of losses, and $280,000 of earned premium. Territory N has 840 earned exposure units, $315,000 of losses, and $350,000 of earned premium. Calculate and compare the loss ratios and loss costs. Assume each amount is on a consistent coverage and valuation basis.
Method. Calculate each territory’s loss ratio using losses divided by earned premium, and its loss cost using losses divided by earned exposures.
For M, the loss ratio is \(252{,}000/280{,}000=0.90\), or 90.00%. Its loss cost is \(252{,}000/840=\$300\) per exposure.
For N, the loss ratio is \(315{,}000/350{,}000=0.90\), or 90.00%. Its loss cost is \(315{,}000/840=\$375\) per exposure.
The loss ratios are equal, but the loss costs are not: M is $300 per exposure and N is $375 per exposure. The premium per exposure is $280,000/840=\$333.33 for M and $350,000/840=\$416.67 for N. Dividing each loss cost by its premium per exposure gives 90% in both cases.
Check. \(300/333.33\) and \(375/416.67\) are both approximately 0.90; using the unrounded premium-per-exposure values gives exactly 0.90. A common mistake is to infer equal loss costs from equal loss ratios. The correction is to calculate both denominators: equal loss ratios can coexist with different loss costs when premium per exposure differs.
Worked Example: Different Measures Give Different Rankings
A hypothetical insurer reviews two territories with 1,000 earned exposure units each. Territory A has $240,000 of losses and $300,000 of earned premium. Territory B has $300,000 of losses and $500,000 of earned premium. Calculate both measures, then explain their different rankings.
Method. Use \(L/E\) for loss cost and \(L/P\) for loss ratio. The shared exposure count makes the loss-cost comparison straightforward.
For A, loss cost is \(240{,}000/1{,}000=\$240\) per exposure, and loss ratio is \(240{,}000/300{,}000=80.00\%\).
For B, loss cost is \(300{,}000/1{,}000=\$300\) per exposure, and loss ratio is \(300{,}000/500{,}000=60.00\%\).
B has the higher observed loss cost, while A has the higher loss ratio. Premium per exposure is $300 for A and $500 for B, so the higher premium base in B yields a lower loss ratio despite its higher losses per exposure. The measures do not contradict each other: they answer different questions. Loss cost compares losses per exposure; loss ratio compares losses with the premium actually included in the calculation.
Check. \(240/300=0.80\) and \(300/500=0.60\). A common mistake is to call the lower loss ratio proof that B has lower loss per exposure. That conclusion ignores the different premium per exposure.
Worked Example: Decomposing Loss Cost Into Frequency and Severity
For a hypothetical coverage, Territory R has 1,200 earned vehicle-years, 96 claims, and $288,000 in losses. Territory S has 900 earned vehicle-years, 54 claims, and $243,000 in losses. Calculate frequency, severity, and loss cost for each territory, and identify the source of the loss-cost difference.
Method. Calculate claims per vehicle-year, losses per claim, and losses per vehicle-year. The identity in the preceding section should reconcile the results.
For R, frequency is \(96/1{,}200=0.08\) claims per vehicle-year. Severity is \(288{,}000/96=\$3{,}000\) per claim. Loss cost is \(288{,}000/1{,}200=\$240\) per vehicle-year. The decomposition checks: \(0.08\times3{,}000=\$240\).
For S, frequency is \(54/900=0.06\) claims per vehicle-year. Severity is \(243{,}000/54=\$4{,}500\) per claim. Loss cost is \(243{,}000/900=\$270\) per vehicle-year. The decomposition checks: \(0.06\times4{,}500=\$270\).
S has lower observed frequency but higher severity, and its loss cost is $30 per vehicle-year higher. The decomposition shows why frequency alone would give an incomplete comparison: S’s lower claim count per exposure does not offset its higher average loss per claim.
Check. Each loss cost equals its frequency multiplied by its severity. A common mistake is to compare the raw claim counts, 96 and 54, without accounting for the different exposure volumes. Frequency uses claims divided by exposure, not claims alone.
Worked Example: Selecting an Exposure Unit for Property Experience
A hypothetical property insurer compares two territories. Territory C has 1,000 earned policy-years, $500 million of insured value, and $500,000 of losses. Territory D has 800 earned policy-years, $320 million of insured value, and $400,000 of losses. Calculate loss cost per policy-year and losses per $1,000 of insured value. Explain which comparison addresses property amount more directly.
Method. First use policy-years as the exposure denominator. Then express insured value in units of $1,000 and divide losses by those units.
For C, loss cost per policy-year is \(500{,}000/1{,}000=\$500\). Its insured value is $500,000,000/$1,000 = 500,000 units of $1,000, so losses per $1,000 of insured value are \(500{,}000/500{,}000=\$1.00\).
For D, loss cost per policy-year is \(400{,}000/800=\$500\). Its insured value is $320,000,000/$1,000 = 320,000 units of $1,000, so losses per $1,000 of insured value are \(400{,}000/320{,}000=\$1.25\).
The per-policy-year measure is equal, while losses per $1,000 of insured value are higher in D. The insured-value measure directly scales losses by the amount insured; the policy-year measure does not account for differences in average insured value. Neither ratio automatically captures every risk characteristic, so the denominator should match the exposure concept relevant to the product and comparison.
Check. Both territories have $500 of losses per policy-year, but D has $1.25 per $1,000 insured compared with C’s $1.00. A common mistake is to compare raw losses, which would make C appear worse because its total is larger. The correction is to use an exposure measure suited to the question.
Conditions That Favor Each Measure
The measure should be selected before interpreting a territorial ranking. A practical comparison begins by asking what the denominator represents, whether it is measured consistently, and whether it addresses the question of interest.
| Measure | Most informative when | Important limitation |
|---|---|---|
| Loss cost \(L/E\) | Comparable earned exposures are available and the question concerns losses per exposure unit. | Does not incorporate premium charged per exposure. |
| Loss ratio \(L/P\) | Comparable earned premium is available and the question concerns losses relative to that premium base. | Can reflect differences in premium per exposure as well as differences in losses. |
| Frequency \(N/E\) | Claim counts and exposure units are consistently defined, and claim incidence is of interest. | Ignores claim size; different frequencies need not imply different loss costs in the same order. |
| Severity \(L/N\) | Claim counts and losses are paired consistently, and average claim size is of interest. | Can be unstable with few claims and does not account for how often claims occur. |
| Losses per amount insured or another exposure base | The selected base better represents the volume of risk than a simple policy count. | Requires reliable, comparable measurement of that exposure base. |
A loss ratio is most interpretable when premium has been earned consistently and the premium base is appropriate for the comparison. If territories have different rates or premium composition, loss ratios can reflect those differences along with loss experience. A loss-cost comparison avoids using premium as its denominator, but it is only meaningful when the exposure units are comparable and measure relevant risk volume.
Similarly, a policy-year is not automatically the best exposure unit for every product. If one territory’s policies insure much larger risks on average, equal losses per policy may conceal a difference per unit of insured value. Conversely, a value-based denominator may not be appropriate if the intended comparison is cost per policy or if values are measured inconsistently. No denominator is universally correct; it must be defined in context.
Data Checks and Common Traps
Check that losses, premium, exposures, and claim counts refer to the same territories, periods, and coverage populations. A mismatch can create a ratio that is arithmetically valid but not comparable.
Distinguish, for example, earned policy-years from policies written, or vehicle-years from vehicle counts at a single date. Do not compare rates based on different exposure definitions as if they were the same.
Loss ratio uses premium, loss cost uses exposure, frequency uses claim count and exposure, and severity uses losses and claim count. One cannot be substituted for another without changing the question.
Check whether premium per exposure, risk size, or claim size could explain a difference in the metric. A single ratio is a summary, not a complete explanation of territorial experience.
A frequent exam trap is to treat the terms “loss ratio” and “loss cost” as interchangeable because both use losses in the numerator. They are not: the loss ratio denominator is premium, and the loss cost denominator is exposure. Another trap is to infer that fewer claims mean lower loss cost. Unless claim sizes are comparable, the frequency comparison alone cannot support that conclusion.
Check Your Understanding
For each problem, identify the relevant measure and keep its denominator explicit.
- A hypothetical territory has 1,500 earned exposure units, 105 claims, $420,000 of losses, and $500,000 of earned premium. Calculate frequency per exposure, severity per claim, loss cost per exposure, and loss ratio. (Answer: frequency 0.0700 claims per exposure; severity $4,000 per claim; loss cost $280 per exposure; loss ratio 84.00%.)
- Territory U has 800 earned exposures, $200,000 of losses, and $250,000 of premium. Territory V has 1,000 earned exposures, $270,000 of losses, and $300,000 of premium. Calculate each territory’s loss cost and loss ratio, and state which has the higher value on each measure. (Answer: U has $250 loss cost and 80.00% loss ratio; V has $270 loss cost and 90.00% loss ratio; V is higher on both measures.)
- A hypothetical insurer records 72 claims and $216,000 of losses for 900 earned vehicle-years in one territory. Calculate frequency, severity, and loss cost, and verify the decomposition. (Answer: frequency 0.0800 claims per vehicle-year; severity $3,000 per claim; loss cost $240 per vehicle-year; \(0.08\times\$3{,}000=\$240\).)
- Territory W has $360,000 of losses, 900 earned policy-years, and $450 million of insured value. Calculate loss cost per policy-year and losses per $1,000 insured. (Answer: $400 per policy-year and $0.80 per $1,000 insured.)
- Two territories have the same 85% loss ratio, but Territory J has earned premium of $400 per exposure and Territory K has earned premium of $520 per exposure. Using loss ratio equals loss cost divided by premium per exposure, calculate each loss cost. (Answer: J has $340 per exposure; K has $442 per exposure.)