Suppose an insurer makes the following rate changes for annual policies during three calendar years:

Effective Date	Rate Change
09/01/CY1	+5%
10/01/CY2	+5%
06/01/CY3	-1%

Calendar year 3 earned premium totalled 5M. Using the parallelogram method, determine the the on-level earned premium for calendar year 3.

• 5,036,500
• 5,142,049
• 5,149,167
• 5,150,159
• 5,197,500

The diagram below describes the rate changes:

Using the parallelogram method, the on-level factor for calendar year 2 equals the cumulative rate index divided by the weighted average index. The cumulative rate index equals 1.05 *0.99 and the weighted average index equals

[[math]] A_1 + 1.05\cdot A_2 + 1.05 \cdot 0.99 \cdot A_3 [[/math]]

with [math]A_1 [/math] the area of region 2 intersected with rectangle CY3, [math]A_2 [/math] the area of region 3 intersected with rectangle CY3 and [math]A_3 [/math] the area of region 4 intersected with rectangle CY3. Elementary geometry implies that [math]A_1 [/math] equals

(9/12)2/2 = 0.28125

, [math]A_3[/math] equals

(7/12)2/2 = 0.3403

, and [math]A_2 [/math] equals

1 - 0.28125 - 0.3403 = 0.37845.

Hence the weighted average index equals

0.28125  + 1.05 * 0.3403 + 1.05 * 0.99 * 0.37845 = 1.032

, the on-level factor equals

1.05*0.99/1.032 = 1.0073

, and finally the on-level earned premium equals 5M multiplied by 1.0073 or 5.0365M.