Why Written and Earned Premium Differ
Premium is not always attributable to the same period when it is recorded and when the insurer provides coverage. A policy may be written during one period but provide coverage over several periods. For an indication, the distinction matters because the premium denominator should represent the same experience basis as the losses being measured. Written premium can be appropriate for some policy-based analyses; earned premium is often relevant when the experience period reflects coverage provided during that period.
The central question is not simply which premium figure is available. It is what period and exposure the premium is intended to represent. The experience-period and exposure concepts established in the earlier material on selecting experience periods and selecting an exposure base provide the context for making that choice. This framework is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.
A policy’s written premium is associated with the policy when it is written. Its earned premium accumulates as time passes and coverage is provided. Thus, the same policy can contribute its full written premium to one period’s written-premium total while contributing only part of that amount to a calendar period’s earned-premium total.
For a policy with premium \(P\), a term of \(T\) equal time units, and \(m\) units of coverage overlapping the experience period, pro rata earned premium for that period is \(P(m/T)\). This assumes uniform earning over the term, no relevant midterm changes to premium or coverage, and a consistent time-unit convention. Where policies have different effective dates or terms, calculate the contribution policy by policy, or by groups with the same timing.
Deriving the Pro Rata Earning Calculation
Under uniform pro rata earning, each equal time unit of a policy term earns the same share of the policy premium. If a policy’s total premium is \(P\) and its term contains \(T\) equal units, each unit earns \(P/T\). If \(m\) of those units overlap the period being measured, the period receives \(m\) such shares:
This also explains why premiums for policies written near the end of a calendar year can be substantially written but only partly earned in that calendar year. The policy contributes its full premium to the written total when recorded, but only the fraction corresponding to coverage already provided to the earned total.
The formula is not a universal description of every insurer’s accounting. It is a calculation under the stated pro rata assumption. Premium audits, endorsements, cancellations, short-rate provisions, and other adjustments may change the amount or timing recognized. If such details are given, they must be reflected rather than ignored.
Worked Calendar-Period Calculations
Worked Example: Policies Written at Two Dates
A hypothetical insurer writes 25 one-year policies effective January 1 at an annual premium of $600 each and 20 one-year policies effective October 1 at an annual premium of $900 each. Calculate total written premium for the calendar year and earned premium in that calendar year. Assume all policies remain in force, premiums are earned pro rata by month, and each policy term is 12 equal months.
Method. Add the full premiums of policies written during the year for written premium. For calendar-year earned premium, count only the months of coverage provided within the calendar year.
Written premium. The January cohort contributes \(25 \times \$600 = \$15{,}000\). The October cohort contributes \(20 \times \$900 = \$18{,}000\). Therefore, written premium is \(\$15{,}000 + \$18{,}000 = \$33{,}000\).
Earned premium. Each January policy provides 12 months of coverage during the year, so the cohort earns \(25 \times \$600 \times 12/12 = \$15{,}000\). Each October policy provides 3 months of coverage, October through December, so the cohort earns \(20 \times \$900 \times 3/12 = \$4{,}500\). Total calendar-year earned premium is \(\$15{,}000 + \$4{,}500 = \$19{,}500\).
Check. Earned premium is below written premium because the October policies have nine months of their terms after the calendar year ends. The October cohort’s unearned portion at year-end is \(20 \times \$900 \times 9/12 = \$13{,}500\), and \(\$19{,}500 + \$13{,}500 = \$33{,}000\).
Common mistake avoided. Counting only two months for the October policies would produce \(20 \times \$900 \times 2/12 = \$3{,}000\), and an incorrect total of $18,000. The calendar year includes October, November, and December: three months, not two.
When losses are measured for a calendar period, earned premium can represent the coverage provided during that same period. Written premium for that period may include policies whose coverage mostly falls outside it, and may omit the later portions of policies written earlier. That is why the two totals are not interchangeable merely because they refer to the same calendar year.
Worked Example: Monthly Policy Starts and a Calendar-Year Loss Ratio
A hypothetical book writes 10 one-year policies on the first day of each month from January through December. Every policy has a premium of $600, remains in force, and earns premium uniformly over 12 equal months. Calendar-year losses are $31,200. Calculate the calendar-year written premium, calendar-year earned premium, and loss ratio using each premium measure. Identify the ratio that matches the stated calendar-year loss basis.
Method. Written premium includes all policies written during the year. For earned premium, a cohort beginning on the first day of a month earns for the remaining months of the calendar year, including its effective month. The stated losses are calendar-year losses, so compare them with calendar-year earned premium for a period-aligned ratio.
Written premium. There are 12 monthly cohorts of 10 policies each, for \(12 \times 10 = 120\) policies. Written premium is \(120 \times \$600 = \$72{,}000\).
Earned premium. The January cohort earns for 12 months, the February cohort for 11 months, and so on, through the December cohort for 1 month. Across all policies, the number of policy-months earned is \(10 \times (12+11+10+9+8+7+6+5+4+3+2+1)=10 \times 78=780\). Each policy-month earns \(\$600/12=\$50\). Calendar-year earned premium is \(780 \times \$50=\$39{,}000\).
Loss ratios. Using earned premium gives \(\$31{,}200/\$39{,}000=0.80\), or 80.0%. Using written premium gives \(\$31{,}200/\$72{,}000=0.4333\), or 43.3% to one decimal place. The 80.0% ratio pairs calendar-year losses with calendar-year earned premium.
Check and common mistake avoided. The 120 policies generate $72,000 written, while only 780 of the possible \(120 \times 12=1{,}440\) policy-months fall in the calendar year. Thus, only \(780/1{,}440\) of the written premium is earned in that year: \(\$72{,}000 \times 780/1{,}440=\$39{,}000\). Dividing calendar-year losses by written premium without explaining the basis produces a different ratio, not a like-for-like calendar-period measure.
Policy Timing Can Change the Premium Mix
Written and earned premium can differ not only because of the number of months earned, but also because policies written at different times can carry different premiums. For example, a hypothetical rate change could make later-written policies more expensive than earlier-written policies. In a calendar year, the later policies may contribute their full premium to written premium but only a portion to earned premium. This tutorial focuses on the timing distinction; the treatment of rate levels in an indication is a separate calculation.
Worked Example: Timing Across a Hypothetical Rate Change
A hypothetical insurer writes 100 one-year policies effective January 1 at $800 each and 100 one-year policies effective July 1 at $880 each. Assume the July premium reflects a hypothetical 10% increase, all policies remain in force, and premium is earned pro rata by month. Calendar-year losses are $99,200. Calculate written premium, earned premium, and the calendar-period loss ratio using earned premium.
Method. Calculate each cohort separately because the cohorts have different premiums and different coverage durations in the calendar year.
Written premium. The January cohort contributes \(100 \times \$800=\$80{,}000\). The July cohort contributes \(100 \times \$880=\$88{,}000\). Total written premium is \(\$80{,}000+\$88{,}000=\$168{,}000\).
Earned premium. The January cohort earns 12 months, or \(100 \times \$800 \times 12/12=\$80{,}000\). The July cohort earns 6 months, or \(100 \times \$880 \times 6/12=\$44{,}000\). Total calendar-year earned premium is \(\$80{,}000+\$44{,}000=\$124{,}000\).
Loss ratio and check. The calendar-period loss ratio on earned premium is \(\$99{,}200/\$124{,}000=0.80\), or 80.0%. The July cohort’s $44,000 earned amount is half its $88,000 written amount, consistent with six months of a 12-month term. The written total is higher than the earned total because it includes all of that cohort’s premium.
Common mistake avoided. Applying the January cohort’s $800 premium to the July cohort would ignore the stated premium difference. Applying the full $88,000 July written premium to calendar-year earned premium would instead ignore that only six months of its coverage fall in the calendar year.
Choosing a Premium Measure for an Indication
The proper measure depends on how the experience is defined and what the premium is intended to represent. Written premium may be a reasonable base when the selected experience is organized around policies written during a period and the corresponding losses are assigned on a compatible policy basis. For a fully completed policy cohort with no premium adjustments, the full premium for each policy has been earned over the entire policy term, so total earned premium over those completed terms equals the total written premium.
Earned premium is often the natural measure when losses and exposure are associated with coverage provided during a calendar period. But the label alone does not guarantee a consistent comparison: the loss period, premium period, and exposure basis still need to be checked. If written premium is used with losses measured on a different basis, the resulting ratio can reflect timing differences as well as underlying experience.
Common Mistakes and Exam Traps
- Treating written premium as earned immediately. A policy written late in the period contributes its full written premium, but only the portion for coverage already provided is earned under pro rata earning.
- Miscounting the overlap months. State the effective date, the end of the experience period, and which months are included before applying the fraction of term.
- Using a convenient denominator without checking the loss basis. A ratio of losses to written premium can be calculated, but its interpretation depends on whether the written premium represents the same experience basis as those losses.
- Assuming every policy earns uniformly in all circumstances. Pro rata earning is an assumption. If the problem supplies audits, endorsements, cancellations, or another earning convention, account for it.
- Combining cohorts too early. Different effective dates, terms, or premiums can produce different earned fractions. Calculate each cohort’s contribution first, then sum.
In an exam calculation, a brief statement of the period and earning assumption can prevent a substantial denominator error: identify the policies included, determine each policy’s coverage overlap, calculate the applicable premium measure, and check that it matches the stated loss basis.
Check Your Understanding
For each problem, state the relevant premium measure and show the timing calculation before checking the answer.
- A hypothetical insurer writes 15 one-year policies effective January 1 at $720 each and 10 one-year policies effective October 1 at $960 each. Assume pro rata earning by month and no cancellations. Find calendar-year written and earned premium. (Answer: Written premium $20,400; earned premium $13,200.)
- A hypothetical insurer writes 12 one-year policies effective April 1 and 12 effective September 1. Each policy has an $840 premium, remains in force, and earns pro rata by month. Find calendar-year written premium and earned premium, counting the effective month as covered. (Answer: Written premium $20,160; earned premium $10,920.)
- Calendar-year losses are $45,000. Calendar-year earned premium is $60,000, while written premium is $90,000. Calculate both loss ratios and identify the ratio using earned premium. (Answer: Loss ratio on earned premium 75.0%; on written premium 50.0%.)
- A hypothetical insurer writes 40 one-year policies effective January 1 at $1,000 each and 40 effective July 1 at $1,200 each. Assume pro rata earning by month, no cancellations, and calendar-year losses of $51,200. Calculate written premium, earned premium, and the loss ratio on earned premium. (Answer: Written premium $88,000; earned premium $64,000; loss ratio 80.0%.)
- A fully completed hypothetical policy-year cohort contains 75 one-year policies, each written for $1,400. Assume every policy completed its term, no premium adjustments occurred, and ultimate losses for the cohort are $84,000. Find the cohort’s total written premium, total earned premium over its completed terms, and loss ratio. (Answer: Written premium $105,000; earned premium $105,000; loss ratio 80.0%.)