Minimum Premium

Study guide: Exam 5

Minimum Premium is the floor below which an insurer will not write a policy, whatever the rated exposure. It exists because fixed expenses — issuance, billing, servicing, filing — do not shrink with the size of the risk.

$\text{Premium} = \max\!\left(\text{Rate} \times \text{Exposure},\; \text{Minimum Premium}\right)$

The floor is calibrated to the per-policy fixed expense load plus a margin: below the crossover exposure, an exposure-based rate alone does not recover the cost of putting the policy on the books.
It is one of the implementation levers in Considerations for Implementing Rates, alongside expense fees, rounding rules and rate capping, and it must be encoded in the rating algorithm because where in the sequence it applies changes the answer.
Set too high, it prices the insurer out of desirable small accounts and invites competitors to take the segment; too low, it writes small policies at a loss. Either error is invisible in the aggregate indication, which is why minimum premiums are reviewed separately.
A percentage rate change does not flow through evenly to policies sitting at the minimum: raising the base rate $8\%$ changes nothing for a risk already at the floor, so the achieved rate change on a book with many minimum-premium policies is less than the filed one.
Distinguish from the minimum retrospective premium (the floor in a retro plan, which prices a risk-transfer band rather than expenses) and from a deposit premium (an advance against an auditable exposure — see Premium Audit).

Applying the Minimum Premium A liability rate is $\$4.00$ per $\$1{,}000$ of receipts, with a minimum premium of $\$750$ . A small contractor has $\$150{,}000$ of receipts.

Compute the premium, and find the receipts level at which the minimum stops binding.
Answer
 $\begin{align*} \text{Rated premium} &= \$4.00 \times \frac{\$150{,}000}{1{,}000} \\ &= \$600 \end{align*}$ 
Since $\$600 < \$750$ , the minimum applies and the charge is $\$750$ .

The crossover is where the rated premium reaches the floor:

$\frac{\$750}{\$4.00} \times \$1{,}000 = \$187{,}500 \text{ of receipts}$

Every insured below $\$187{,}500$ of receipts pays the same $\$750$ , so within that band the effective rate per $\$1{,}000$ falls as the risk grows — from $\$7.50$ at $\$100{,}000$ of receipts down to $\$4.00$ at the crossover.

Setting the Floor from the Expense Structure An insurer's fixed expenses are $\$95$ per policy, variable expenses $23\%$ , target profit $5\%$ . The smallest risks it writes have an expected loss cost of about $\$110$ .

What minimum premium is indicated, and what happens if the insurer files $\$200$ instead?

Answer The rate that covers the smallest risk's own costs is

$\frac{\$110 + \$95}{1 - 0.23 - 0.05} = \frac{\$205}{0.72} = \$284.72$

so a minimum around $\$285$ is indicated.

At a $\$200$ minimum, each such policy contributes

$\$200 \times (1 - 0.23) = \$154$

after variable expenses, against $\$110$ of loss and $\$95$ of fixed expense — a loss of $\$51$ per policy, with no profit provision at all.

The insurer would be buying small policies. Worse, a $\$200$ floor against competitors' $\$285$ guarantees it wins that segment: it will write every small risk in the market at a loss, and the volume will look like growth in every management report until the loss ratio arrives.