Variable Expenses

Study guide: Exam 5

Variable Expenses are underwriting expenses that vary directly with premium — premium taxes and commissions are the standard examples — and are therefore expressed as a percentage $V$ of premium and carried in the denominator of the rate formula.

$V = \frac{\text{Variable Expenses}}{\text{Premium}}$

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

The denominator placement is not a convention but arithmetic: if commission is $15\%$ of the rate being solved for, the rate must be grossed up by dividing, not by adding $15\%$ of something else. Adding instead of dividing under-recovers the expense.
The size and make-up of $V$ vary by line and by company. Commissions tend to be much higher in lines that need a comprehensive inspection at the onset (large commercial property) than in lines that do not (personal auto); and within a line, an agency-based company pays higher commissions than a national direct writer, which spends more on other acquisition such as advertising — the distribution channel shifts cost between the two.
Because commissions and taxes attach when a policy is written, these ratios are computed against written premium; expenses incurred as coverage is provided use earned premium. Mixing the denominators misstates the provision in a growing or shrinking book.
A structural change in distribution — moving business direct, renegotiating commission scales, adding a contingent commission — must be reflected prospectively rather than from historical ratios, since the ratemaking question is what the expenses will be.
Commission rates may differ between new and renewal business, and contingent commissions vary the commission with the quality of the business written (e.g. a loss ratio) or its amount (e.g. predetermined volume goals).

Building the Variable Expense Ratio and the Rate A personal auto insurer's expense study shows commissions $15\%$ of written premium, premium taxes $2\%$ , other variable acquisition $3\%$ . The projected pure premium is $\$300$ , fixed expenses are $\$40$ per exposure, and the target underwriting profit provision is $5\%$ .

Compute the indicated rate.
Answer
 $\begin{align*} V &= 15\% + 2\% + 3\% = 20\% \\[6pt] \text{Rate} &= \frac{\$300 + \$40}{1 - 0.20 - 0.05} \\ &= \frac{\$340}{0.75} \\ &= \$453.33 \end{align*}$ 
Verify that the rate actually funds everything:
 $\begin{align*} \text{Variable expenses} &= 0.20 \times \$453.33 = \$90.67 \\ \text{Profit} &= 0.05 \times \$453.33 = \$22.67 \\ \text{Losses} + \text{fixed} &= \$340.00 \\[4pt] \text{Total} &= \$453.34 \;\checkmark \end{align*}$ 
Had the $25\%$ been added to $\$340$ instead ( $\$340 \times 1.25 = \$425$ ), commissions and profit would consume $\$106.25$ and only $\$318.75$ would be left for a $\$340$ obligation — a $\$21$ shortfall on every policy.

A Distribution Channel Shift An insurer currently writes entirely through independent agents at $17\%$ commission. It will move $40\%$ of its business to a direct channel with no commission but $\$18$ per policy of additional fixed marketing cost. Premium taxes are $2.5\%$ , other variable expenses $2\%$ , fixed expenses $\$45$ per exposure, target profit $5\%$ , pure premium $\$420$ .

Compute the indicated rate before and after the shift.
Answer Before:
 $\begin{align*} V &= 17\% + 2.5\% + 2\% = 21.5\% \\[4pt] \text{Rate} &= \frac{\$420 + \$45}{1 - 0.215 - 0.05} = \frac{\$465}{0.735} = \$632.65 \end{align*}$ 
After — the blended commission is $0.60 \times 17\% = 10.2\%$ , and fixed expenses rise by $0.40 \times \$18 = \$7.20$ :
 $\begin{align*} V &= 10.2\% + 2.5\% + 2\% = 14.7\% \\[4pt] \text{Rate} &= \frac{\$420 + \$52.20}{1 - 0.147 - 0.05} = \frac{\$472.20}{0.803} = \$588.04 \end{align*}$ 
The indicated rate falls $7.1\%$ — the commission saving net of the added acquisition cost.

Two cautions. First, the change must be reflected prospectively: historical expense ratios still contain the old channel mix and would overstate $V$ . Second, the direct and agency books may not be the same risks; if the direct channel attracts a different mix, the loss side changes too and the expense saving cannot be taken in isolation.