An insurance company sells one-year policies that have uniformly distributed effective dates. The following rate changes have occurred:

Date	Rate Change
June 1, CY1	+10%
August 1, CY2	r

Rates are currently at the level set on August 1, CY2. The earned premium at current rates for CY2 is 1.03 times the CY2 earned premium.

Calculate r.

• 1.7%
• 2.4%
• 3.3%
• 5.3%
• 7.5%

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Key: B

Let P represent the rate level before the rate change on June 1, CY1. The rate level 1.1P takes effect on June 1, CY1. The rate level (1.1P)(1 + r) takes effect on August 1, CY2, so this level is the current rate level.

The parallelogram method is shown in the diagram below.

For CY2, the average rate level for the earned exposure is

[[math]] \begin{aligned} &\frac{1}{2} \left( \frac{5}{12}\right)^2 P + \left [ 1 - \frac{1}{2}\left( \frac{5}{2} \right)^2 - \frac{1}{2} \left( \frac{5}{12}\right)^2\right](1.1P) + \frac{1}{2} \left( \frac{5}{12} \right)^2 (1.1P)(1+r) \\ &= 0.086806 P + 0.909028P + 0.0954861P(1 + r ) = 0.99583 P + 0.0954861 P(1 + r ) \end{aligned} [[/math]]

The ratio of the earned premium at current rates for CY2 to the CY2 earned premium, which is the on-level factor for CY2, is

[[math]] \begin{aligned} &\frac{(1.1P)(1 + r )}{0.99583 P + 0.095486 1 P (1 + r )} = \frac{1.1}{0.99583 /(1 + r ) + 0.095486 1} = 1.03 \\ & 1 + r = 1.02402 \Rightarrow r = 2.402\% \end{aligned} [[/math]]

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.