Earned Exposure

Study guide: Exam 5

Earned Exposure is the portion of written exposure for which coverage has actually been provided during the period — the exposure analogue of earned premium, and the correct denominator for a pure premium.

$\text{Earned Exposure} = \text{Written Exposure} - \Delta\text{Unearned Exposure}$

$\text{Pure Premium} = \frac{\text{Losses}}{\text{Earned Exposures}}$

A $12$ -month policy contributes one earned car-year in total, spread pro rata over its term: written the day it is issued, earned $1/12$ per month, in force from effective date to expiry.
Losses in the numerator and exposures in the denominator must cover the same period. Accident-year losses over written exposures inflates the denominator in a growing book and understates the pure premium — the same mismatch that makes written premium wrong for a loss ratio.
With policies written uniformly through the year, a calendar year earns roughly half of that year's written exposure plus half of the prior year's. In a steady-state book the two halves are equal and earned $\approx$ written; in a growing book earned lags written.
Exposures must be earned on a basis consistent with the premium: if premium is earned pro rata, so are exposures. Lines with strongly seasonal risk (crop, some property covers) may earn premium on a non-uniform schedule, and exposures follow the same schedule.
Earned exposures are also the weights in classification analysis and the denominator of claim frequency, so an error here propagates through the entire indication.

Calendar Year Earned Car-Years An insurer writes $1{,}200$ annual auto policies during $2024$ — $100$ on the first of each month — and wrote no business before $2024$ .

How many car-years are earned in calendar year $2024$ ?
Answer The January cohort earns $12/12$ of a year, the February cohort $11/12$ , and so on down to the December cohort's $1/12$ :
 $\begin{align*} \text{Earned} &= 100 \times \frac{12 + 11 + \cdots + 1}{12} \\ &= 100 \times \frac{78}{12} \\ &= 650 \text{ car-years} \end{align*}$ 
Written exposure for $2024$ is $1{,}200$ car-years, so the year earns $54.2\%$ of what it wrote. The remaining $550$ car-years are unearned at $12/31/2024$ and earn during $2025$ .

Using $1{,}200$ as the denominator of the $2024$ pure premium would understate loss cost by nearly half.

Earned Exposure in a Growing Book An insurer writes annual policies uniformly and grows steadily:

Year Written car-years

$2023$ $10{,}000$

$2024$ $12{,}000$

Accident year 2024 losses are $\$4{,}500{,}000$ . Compute the AY 2024 pure premium.
Answer With uniform writings and annual terms, a calendar year earns half of its own writings and half of the prior year's:
 $\begin{align*} \text{Earned}_{2024} &= 0.5 \times 12{,}000 + 0.5 \times 10{,}000 \\ &= 11{,}000 \text{ car-years} \\[6pt] \text{Pure premium} &= \frac{\$4{,}500{,}000}{11{,}000} \\ &= \$409.09 \end{align*}$ 
Had written exposure ( $12{,}000$ ) been used, the pure premium would come out at $\$375.00$ — an $8\%$ understatement, entirely an artefact of growth. The faster the book grows, the larger the error, which is why the earned basis is not optional.