Revision as of 22:45, 15 May 2023 by Admin

Exercise


ABy Admin
May 15'23

Answer

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.

% of policyearned100%50%0%CY1CY2CY36/1/CY1+10%18/1/CY2r23

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.

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