Pure Premium

Study guides: Exam 5, Exam GI-101

Pure Premium (also loss cost or burning cost) is expected loss and LAE per unit of exposure — the portion of the rate that pays claims, before any provision for expenses or profit.

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

$\text{Pure Premium} = \text{Frequency} \times \text{Severity}$

The identity holds because $\dfrac{\text{Losses}}{\text{Exposures}} = \dfrac{\text{Claims}}{\text{Exposures}} \times \dfrac{\text{Losses}}{\text{Claims}}$ — and it is what allows the two components to be trended, developed and diagnosed separately (Frequency, Severity).
Pure premium is the building block of the Pure Premium Method indication, where the indicated rate is the projected pure premium plus fixed expenses, grossed up for variable expenses and profit.
It is also how the loss ratio connects to the rate: $\text{Loss Ratio} = \text{Pure Premium} / \text{Average Premium}$ , so a class whose pure premium is twice the book's needs twice the average premium to run the same loss ratio.
Advisory organizations (ISO, NCCI) file loss costs — pure premiums — rather than rates, leaving each insurer to add its own expense and profit provisions through a loss cost multiplier.
The pure premium used in a rate is always projected: historical losses developed to ultimate, trended to the forecast period, adjusted for large losses and catastrophes, and divided by exposures that are themselves trended if the base inflates (Exposure Trend).

Pure Premium from Frequency and Severity A homeowners book has $12{,}000$ earned house-years, $960$ ultimate claims and $\$5{,}280{,}000$ of ultimate losses and LAE.

Compute the pure premium two ways.
Answer Directly:

$\text{Pure premium} = \frac{\$5{,}280{,}000}{12{,}000} = \$440 \text{ per house-year}$

Through the components:
 $\begin{align*} \text{Frequency} &= \frac{960}{12{,}000} = 0.080 \\[4pt] \text{Severity} &= \frac{\$5{,}280{,}000}{960} = \$5{,}500 \\[4pt] \text{Pure premium} &= 0.080 \times \$5{,}500 = \$440 \;\checkmark \end{align*}$ 
If the average premium is $\$680$ , the loss ratio is $\$440/\$680 = 64.7\%$ — the third view of the same fact.

Trending Pure Premium Through Its Components The same book has a frequency trend of $-1.5\%$ a year and a severity trend of $+6.0\%$ a year. The trend period is $2.5$ years.

Project the pure premium, and compare with trending the pure premium directly at a selected $4.4\%$ .
Answer Component trending:
 $\begin{align*} \text{Frequency factor} &= 0.985^{2.5} = 0.9629 \\ \text{Severity factor} &= 1.060^{2.5} = 1.1568 \\[4pt] \text{Combined} &= 0.9629 \times 1.1568 = 1.1139 \end{align*}$ 
$\text{Projected PP} = \$440 \times 1.1139 = \$490.12$

Direct pure premium trending at $4.4\%$ :

$1.044^{2.5} = 1.1137 \Rightarrow \$440 \times 1.1137 = \$490.03$

The two are within $0.02\%$ , because $1.06 \times 0.985 = 1.0441$ — the implied pure premium trend. The component approach is preferred not because it gives a different answer but because it is diagnosable: an actuary can defend $-1.5\%$ frequency and $+6.0\%$ severity against external evidence (repair costs, safety technology, litigation rates) in a way that a single blended $4.4\%$ cannot be defended.