Ratemaking

Study guides: Exam FAM, Exam 5, Exam 9, Exam GI-101, Exam GI-201, Exam GI-301

Ratemaking is the process of determining the appropriate price (the rate) to charge for insurance coverage. It is a prospective process that uses historical data and actuarial projections to ensure the premium collected is sufficient to cover future losses, expenses, and a margin for profit, while remaining competitive and regulatory-compliant. Contrast with loss reserving, which estimates the unpaid cost of claims that have already occurred.

The fundamental insurance equation states that, over the future policy period:

$\text{Premium} = \text{Losses} + \text{LAE} + \text{UW Expenses} + \text{UW Profit}$

The overall indication restates that equation in one of two ways. The Pure Premium Method gives an indicated average rate per exposure, compared with the current average rate to get a change; the Loss Ratio Method gives the indicated change directly:

$\text{Indicated Average Rate} = \frac{\bar{L} + \bar{E}_F}{1 - V - Q_T}, \qquad \text{Indicated Change} = \frac{\text{Indicated Average Rate}}{\text{Current Average Rate}} - 1$

$\text{Indicated Change} = \frac{\text{LR} + F}{1 - V - Q_T} - 1$

$\bar{L} = \text{pure premium incl. LAE},\ \bar{E}_F = \text{fixed expense per exposure},\ \text{LR} = \text{proj. loss \& LAE ratio},\ F = \text{fixed expense ratio}$

$V = \text{variable expense \%},\ Q_T = \text{target UW profit \%}$

Two equivalent constructions produce this indication: the Pure Premium Method (works from exposures and losses, needs no premium) and the Loss Ratio Method (works from on-level premium and a Permissible Loss Ratio). They agree when given consistent inputs.
Historical data must be adjusted on three axes before it can be used: developed to ultimate, trended to the future period, and brought to current rate level. Skipping any one biases the indication; applying both development and trend is not a double count (Werner & Modlin's overlap fallacy).
Ratemaking data is organized by calendar, accident, policy, or report period, each meeting Werner & Modlin's three objectives — match losses and premium, use the most recent data, minimize the cost of collection — to a different degree.
The overall indication is then allocated to individual risks through classification, territory, limit, and deductible relativities, and finalized through the rating algorithm.
The four CAS principles frame the whole exercise: a rate estimates expected future costs, provides for all costs of the risk transfer, provides for the costs of an individual risk transfer, and is actuarially sound — and therefore not excessive, inadequate, or unfairly discriminatory — when it does all three.
The calculated indication need not be the filed change: regulatory, operational, and marketing constraints and customer lifetime value shape the selected rate change.

Calculating a Pure Premium

An actuary expects a fleet of $1{,}000$ delivery vans to incur $ $250{,}000$ in total losses over the next year. If the administrative expenses are estimated at $ $50$ per van, what is the Pure Premium?

Answer

Projected Losses ( $L$ ): $ $250{,}000$

Exposure Units ( $N$ ): $1{,}000$ vans

The Pure Premium ( $PP$ ) represents the portion of the rate required to pay for losses and loss adjustment expenses:

$PP = \frac{L}{N} = \frac{250{,}000}{1{,}000} = \$250$

The "Gross" Difference The Pure Premium is not the final price the customer pays. To get the Gross Premium, you must "load" the Pure Premium for taxes, commissions, and profit — see Expense Provisions.

Indicated Rate Change Projected pure premium is $\$250$ per exposure, fixed expenses are $\$40$ per exposure, variable expenses are $22\%$ of premium, and the target underwriting profit provision is $5\%$ . The current average rate is $\$385$ .

What is the indicated rate change?

Answer Build the indicated average rate:

$\text{Indicated Rate} = \frac{\$250 + \$40}{1 - 0.22 - 0.05} = \frac{\$290}{0.73} = \$397.26$

Compare to the current average rate:

$\text{Indicated Change} = \frac{\$397.26}{\$385.00} - 1 = +3.2\%$

Note that fixed expenses are added in the numerator while variable expenses gross up the denominator. Loading both as percentages — or adding both as dollars — is the most common error on this calculation.