Why Premium and Exposure Must Match
A premium amount is meaningful for comparison only when it represents the coverage and risks described by its exposure denominator. If a premium total includes several coverages but the exposure represents just one, dividing one by the other does not produce a measure for either coverage alone. The same problem arises when premium and exposure refer to different time periods or different insured units.
The goal is to define the experience segment before doing the division. A segment specifies the coverage being studied, the time period, and the insured risks or units included. The exposure unit must be meaningful for that coverage: examples include a policy-year, vehicle-year, or acre-season. A policy count by itself is not a time-adjusted exposure measure when policies cover different lengths of time.
For a consistent segment, a descriptive premium-per-exposure measure is:
This ratio describes the premium attached to each unit of exposure in the selected data. It is not automatically a rate adequacy conclusion: that requires further analysis. Nor does the ratio make unlike coverage comparable. First establish what is included in both its numerator and denominator.
Define the Segment Before Calculating
A practical matching check asks four questions. First, what coverage is included? Second, what dates define the experience period? Third, which insured risks or units are represented? Fourth, what exposure unit is being counted? Answering them prevents a superficially plausible division from hiding a mismatch.
Identify the particular protection and terms represented, such as liability coverage at a stated limit. Do not include premium for other coverages unless their exposures are also part of the segment.
Choose the experience period and include premium earned for coverage provided during that period, together with exposure from that same period.
Decide whether exposure means policy-years, vehicle-years, or another unit. Count the covered units and the time for which they were covered.
Check that transactions and exposure records belong to the selected segment. A transaction record is not, by itself, proof that the corresponding amount was earned in the period.
The earlier material on assigning premium transactions to experience periods distinguishes when a premium transaction is assigned from when coverage is provided. For a measure based on earned premium, the relevant amount is the portion earned for coverage in the selected period, not simply an amount billed, collected, or written then. When coverage varies during a policy term, the exposure measure should reflect the actual covered quantity and time.
How Earned Exposure Connects to Premium
Suppose a risk is covered continuously for a fraction \(t\) of a year, and the exposure unit is one policy-year. Its earned exposure is \(t\) policy-years. If the coverage and exposure quantity are unchanged, and premium is earned uniformly over the covered time, the corresponding earned premium is the same fraction \(t\) of the annual premium.
This relation follows from the meaning of a policy-year. One full year contributes one unit; covering a fraction \(t\) of that year contributes \(t\) units. Under uniform earning, the premium for that fraction is also \(t\) times the full-year premium. Thus the proportionate amounts refer to the same coverage time. The calculation does not apply unchanged when earning is nonuniform or when the covered quantity changes; in those cases, use the applicable earning pattern and measure the exposure actually provided.
Worked Example: Partial-Year Policies and Policy-Year Exposure
A hypothetical insurer reviews one liability coverage with unchanged terms. Forty policies provide coverage for a full 12 months, and another 40 provide coverage for six months before ending. Assume a 12-month policy-year, uniform earning over covered time, and an annual premium of $1,200 per policy-year. Calculate the earned exposure and premium per policy-year for the group.
Method: measure covered time in policy-years. Each of the 40 full-year policies contributes \(12/12=1\) policy-year, so that group contributes \(40 \times 1=40\) policy-years. Each of the 40 six-month policies contributes \(6/12=0.5\) policy-year, so that group contributes \(40 \times 0.5=20\) policy-years.
Total earned exposure is \(40+20=60\) policy-years. The full-year policies earn \(40 \times \$1{,}200=\$48{,}000\). Each six-month policy earns \(0.5 \times \$1{,}200=\$600\), so those policies earn \(40 \times \$600=\$24{,}000\). Total earned premium is \(\$48{,}000+\$24{,}000=\$72{,}000\).
The matched measure is \(\$72{,}000/60=\$1{,}200\) per policy-year. Check: the full-year and six-month policies each produce the stated annual premium per unit of covered time. Dividing by the 80 policies instead would give $900 per policy, but that denominator counts a six-month policy as much exposure as a full-year policy. It is not a policy-year measure.
Match Coverage Components, Not Just Policy Counts
A policy can provide more than one coverage, and its premium may be recorded as a combined amount. If the experience question concerns only one coverage, the premium for that coverage must be paired with exposure for that coverage. A common set of insured units does not make a combined premium a single-coverage premium.
Worked Example: Separating a Coverage Component
A hypothetical commercial auto portfolio has 600 vehicle-years of liability exposure at a specified limit. For the same vehicles and period, earned premium is $210,000 for liability and $150,000 for physical damage. Calculate the liability premium per vehicle-year. Also find the result of incorrectly using the combined premium.
Method: select the premium component for the target coverage. The target is liability, so its earned premium is $210,000. Divide by the matching 600 liability vehicle-years: \(\$210{,}000/600=\$350\) per liability vehicle-year.
The combined premium is \(\$210{,}000+\$150{,}000=\$360{,}000\). Dividing that by the liability exposure gives \(\$360{,}000/600=\$600\) per vehicle-year, but this amount includes premium for physical damage while the stated measure is for liability. It is not a liability-only measure.
Check: the incorrect result exceeds the liability-only result by \(\$150{,}000/600=\$250\) per vehicle-year, exactly the physical-damage premium per vehicle-year included in error. The mistake is a coverage-scope mismatch, not an arithmetic error.
If the available records provide only a combined premium, they do not by themselves establish the premium for one component. The appropriate response is to obtain a supported coverage allocation or to define a segment that includes the combined coverages and corresponding exposure basis. Assigning the entire combined amount to one coverage merely because it is available does not resolve the mismatch.
Choose an Exposure Unit That Describes the Risk
An exposure unit is a measurement of the amount of risk covered. It is not a premium amount and is not necessarily the number of policies. A fleet policy may cover several vehicles; an annual policy may cover several locations or machines. In those cases, a count of policy-years and a count of vehicle-years or machine-years measure different things.
Worked Example: Fleet Policies and Vehicle-Year Exposure
A hypothetical liability portfolio contains 25 fleet policies, each covering four vehicles for a full year, and 10 fleet policies, each covering four vehicles for six months. Assume unchanged liability coverage, uniform earning, and annual earned premium of $600 per vehicle-year. Calculate total earned vehicle-year exposure and premium per vehicle-year.
Method: count covered vehicles over time. The first group contributes \(25 \times 4 \times 12/12=100\) vehicle-years. The second contributes \(10 \times 4 \times 6/12=20\) vehicle-years. Total earned exposure is \(100+20=120\) vehicle-years.
The first group earns \(100 \times \$600=\$60{,}000\). The second earns \(20 \times \$600=\$12{,}000\). Total earned premium is \(\$72{,}000\). The matched measure is \(\$72{,}000/120=\$600\) per vehicle-year.
There are \(25+10=35\) policies, but that count does not represent 120 vehicle-years. Dividing the premium by the policy count gives about $2,057.14 per policy, a different measure with a different denominator. Check: the premium is $600 for each of 120 vehicle-years, totaling $72,000.
Common Mismatches and Their Consequences
A division can be numerically correct and still answer the wrong question. Before interpreting a premium-per-exposure result, check the dimensions of both quantities.
| Mismatch | What goes wrong | What to check |
|---|---|---|
| Written or billed premium with earned exposure | Premium transactions may fall in a different time period from the coverage represented by exposure. | Use the premium measure appropriate to the period and earning basis. |
| Combined premium with single-coverage exposure | The numerator includes amounts for coverage absent from the stated denominator. | Separate the relevant coverage component or redefine the segment and exposure consistently. |
| Policy count with policy-year exposure | Policies of different durations receive equal weight in the count. | Measure each policy’s covered time as a fraction of a policy-year. |
| Policy-years with vehicle-years | One policy can represent several vehicles, so the units are not interchangeable. | Use the exposure unit specified for the intended measure. |
| Different experience periods | Premium and exposure may describe different slices of coverage time. | Reconcile both records to the same period. |
Another useful check is dimensional: if the requested result is dollars per vehicle-year, the numerator must be dollars for the selected coverage and the denominator must be vehicle-years for that coverage. The units in the answer do not repair an inconsistent numerator or denominator; they only make the intended measure explicit.
Check Your Understanding
For each problem, identify the matching exposure basis and check that the premium scope agrees with it.
- A hypothetical portfolio has 30 policies covered for a full year and 60 covered for six months. The same coverage earns $48,000 in total. What is the earned exposure in policy-years and the premium per policy-year? (Answer: 60 policy-years; $800 per policy-year.)
- A hypothetical package has $96,000 of earned premium for the target liability coverage and $48,000 for another coverage. The target coverage has 240 vehicle-years of exposure. What is the target-coverage premium per vehicle-year, and what incorrect result would come from using package premium? (Answer: $400 per vehicle-year; the mismatched package calculation is $600 per vehicle-year.)
- A hypothetical fleet has 50 policies, each covering three devices for nine months. For one unchanged coverage, earned premium is $67,500. What is the exposure in device-years and premium per device-year? (Answer: 112.5 device-years; $600 per device-year.)
- A hypothetical report gives earned premium for a full calendar year but exposure for only the first six months. Can dividing these figures produce a matched full-year premium-per-exposure measure? State the required correction. (Answer: No; align both premium and exposure to the same period, either the full year or the six-month period.)
- A hypothetical portfolio includes 120 insured risks for a full year and 30 additional risks covered for eight months. All have the same target coverage, and matching earned premium is $84,000. What is the earned exposure and premium per risk-year? (Answer: 140 risk-years; $600 per risk-year.)