Pure Premium Method

Study guide: Exam 5

Pure Premium Method derives the indicated rate directly: project loss and LAE per exposure, add the fixed expense per exposure, and gross up for variable expenses and profit. It never divides by premium, so it requires no on-levelling.

$\text{Indicated Rate} = \frac{\text{Pure Premium} + F}{1 - V - Q_T}$

$\text{Indicated Change} = \frac{\text{Indicated Rate}}{\text{Current Rate}} - 1$

The method produces an absolute rate, which is why it is the only choice for a new line or programme where there is no current rate to change. The Loss Ratio Method can only produce a change to something that already exists.
Inputs: losses developed to ultimate and trended, earned exposures (trended if the base inflates — see Exposure Trend), the fixed expense per exposure $F$ , the variable expense ratio $V$ , and the target profit provision $Q_T$ .
The method's requirement is a well-defined, stable exposure base. Where exposures are not consistently recorded, or where the base changed mid-history, the pure premium is unreliable and the loss ratio method is preferred.
Both methods are algebraically equivalent given consistent inputs, since $\text{Loss Ratio} = \text{Pure Premium} / \text{Average Premium}$ . They differ in the data each demands: pure premium needs exposures, loss ratio needs premium on level.
Advisory organizations file loss costs — pure premiums — leaving each insurer to apply its own expense and profit provisions through a loss cost multiplier. That division of labour only works in the pure premium framework.

Pure Premium Indication Accident year $2023$ : reported losses $\$1{,}200{,}000$ at $24$ months on $5{,}000$ earned exposures. The CDF to ultimate is $1.15$ ; loss trend is $4\%$ a year; exposure trend is $1\%$ a year; the trend period is $2.5$ years. Fixed expenses are $\$25$ per exposure, variable expenses $22\%$ , target profit $5\%$ . The current rate is $\$425$ .

Compute the indicated rate change.
Answer
 $\begin{align*} \text{Ultimate losses} &= \$1{,}200{,}000 \times 1.15 \\ &= \$1{,}380{,}000 \\[4pt] \text{Trended losses} &= \$1{,}380{,}000 \times 1.04^{2.5} \\ &= \$1{,}380{,}000 \times 1.1030 \\ &= \$1{,}522{,}167 \\[4pt] \text{Trended exposures} &= 5{,}000 \times 1.01^{2.5} \\ &= 5{,}126 \\[6pt] \text{Pure premium} &= \frac{\$1{,}522{,}167}{5{,}126} \\ &= \$296.95 \end{align*}$ 
 $\begin{align*} \text{Indicated rate} &= \frac{\$296.95 + \$25.00}{1 - 0.22 - 0.05} \\ &= \frac{\$321.95}{0.73} \\ &= \$441.03 \\[6pt] \text{Indicated change} &= \frac{\$441.03}{\$425.00} - 1 \\ &= +3.8\% \end{align*}$ 
Note that exposure trend appears in the denominator of the pure premium, partially offsetting the loss trend. Omitting it (dividing by $5{,}000$ instead of $5{,}126$ ) would give a pure premium of $\$304.43$ and an indication of $+6.2\%$ — an increase the insurer would collect automatically as its inflating exposure base grows.

Pure Premium and Loss Ratio Methods Agree Same book: trended on-level earned premium is $\$2{,}178{,}550$ ( $5{,}126$ trended exposures at the current $\$425$ rate), fixed expense ratio $5.7\%$ , variable expenses $22\%$ , target profit $5\%$ .

Show the two methods give the same answer.
Answer Loss ratio method:
 $\begin{align*} \text{Projected LR} &= \frac{\$1{,}522{,}167}{\$2{,}178{,}550} = 69.87\% \\[6pt] \text{Indicated factor} &= \frac{0.6987 + 0.057}{1 - 0.22 - 0.05} \\ &= \frac{0.7557}{0.73} \\ &= 1.0352 \end{align*}$ 
an indication of $+3.5\%$ , against $+3.8\%$ from the pure premium method.

The small residual is the fixed expense treatment: the pure premium method carries $F$ as $\$25$ per exposure, while the loss ratio method carries it as $5.7\%$ of the current premium ( $\$24.23$ ). Force them to agree — set $F\% = \$25/\$425 = 5.88\%$ — and both give $+3.8\%$ .

The lesson is that the methods are equivalent only if the inputs are consistent. When an exam question produces different answers from the two methods, the discrepancy is almost always in the fixed expense provision or in whether exposures were trended.