excans:82aee4d3ba: Difference between revisions
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Latest revision as of 01:29, 8 August 2024
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]]