Insurance Premium

Study guides: Exam FAM, Exam ALTAM, Exam 5, Exam CP-321, Exam GI-101, Exam GI-302

Insurance premium is what a policyholder pays an insurer for the coverage an insurance policy gives over its term. In ratemaking it equals the rate per unit of exposure times the number of exposures. Premium is the denominator of the loss ratio, so historical premium has to be restated to today's rates and the future average before it is compared with projected losses.

$\text{Premium} = \text{Rate} \times \text{Exposure}$

$\text{Projected EP} = \text{EP} \times \text{OLF} \times \text{Trend}$

Not the bond premium. The Premium page is Exam FM's bond premium, a price above par. This page is the price of insurance.
Four measures. Written premium is booked at inception. Earned premium ( $\text{EP}$ ) matches the coverage already provided, and unearned premium is what's left. In-force premium is the annual premium of policies in force on a given date. Ratemaking pairs losses with earned premium.
Current rate level. Historical premium was charged at the rates of its time, and on-leveling restates it at current rates. Extension of exposures re-rates every policy at today's rates, which is the most accurate approach but needs policy-level data. The parallelogram method applies an on-level factor to aggregate premium: $\text{OLF}$ is the current rate level divided by the average historical rate level. It assumes policies are written evenly through the year.
Premium development. In some lines the exposure is estimated at inception and audited after expiration (payroll in workers compensation, sales in general liability). There, reported premium keeps changing. A premium audit books the difference, and policy-year premium has to be developed to ultimate.
Premium trend. Average premium drifts even when rates are unchanged, as the mix shifts toward higher deductibles, newer cars or higher insured values. Premium Trend projects that drift to the future policy period. Premium then has to fund losses, expenses and profit in the Overall Rate Level Indication.

On-Level Factor by the Parallelogram Method Annual policies are written evenly through the year, and rates rose $10\%$ on July 1, 2023. Calendar-year 2024 earned premium is $\$5{,}000{,}000$ .

Find the on-level factor and on-level earned premium for CY 2024.
Answer Policies written before July 1, 2023 earn a triangle of CY 2024 with both legs one half, so its area is $\tfrac{1}{2} \times 0.5 \times 0.5 = 0.125$ at the old rate level.
 $\begin{align*} \text{Average level} &= 0.125(1.00) + 0.875(1.10) \\ &= 1.0875 \\[4pt] \text{OLF} &= \frac{1.10}{1.0875} \\ &= 1.0115 \\[4pt] \text{On-level EP} &= \$5{,}000{,}000 \times 1.0115 \\ &= \$5{,}057{,}471 \end{align*}$ 
Most of 2024's premium was already written at the new rate, so its factor is small. For CY 2023, $87.5\%$ was at the old level, which gives $\text{OLF} = 1.10/1.0125 = 1.0864$ .

A Workers Compensation Premium Audit A workers compensation policy is rated at $\$2.00$ per $\$100$ of payroll on an estimated payroll of $\$10{,}000{,}000$ . The audit after expiration finds actual payroll of $\$11{,}500{,}000$ .

Find the deposit premium, the final premium and the audit adjustment.
Answer
 $\begin{align*} \text{Deposit} &= \frac{10{,}000{,}000}{100} \times 2.00 \\ &= \$200{,}000 \\[4pt] \text{Final} &= \frac{11{,}500{,}000}{100} \times 2.00 \\ &= \$230{,}000 \end{align*}$ 
The audit books $\$30{,}000$ of additional premium ( $15\%$ ) in a period after the policy expired. When payrolls are growing, an unaudited policy year understates premium. Pairing that premium with fully developed losses overstates the loss ratio, so premium development has to be recognized.