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

Effective Date	Rate Change
09/01/CY1	+3%
05/01/CY2	+7%

Calendar year 2 earned premium totaled 10M. Using the parallelogram method, determine the the on-level earned premium for calendar year 2.

• 10,250,000
• 10,509,200
• 10,556,176
• 10,591,769
• 11,021,000

The diagram below describes the rate changes:

Using the parallelogram method, the on-level factor for calendar year 2 equals 1.03*1.07, the cumulative rate index, divided by the weighted average index. The weighted average index equals

[[math]] A_1 + 1.03\cdot A_2 + 1.03 \cdot 1.07 \cdot A_3 [[/math]]

with [math]A_1 [/math] the area of region 1 intersected with rectangle CY2, [math]A_2 [/math] the area of region 2 intersected with rectangle CY2 and [math]A_3 [/math] the area of region 3 intersected with rectangle CY3. Elementary geometry implies that [math]A_1 [/math] and [math]A_3[/math] equals

(8/12)2/2 = 0.4444

and [math]A_2 [/math] equals

1-2*0.4444 = 0.1122.

Hence the weighted average index equals

0.4444  + 1.03 * 0.1122 + 1.03 * 1.07 * 0.4444 = 1.0487

, the on-level factor equals

1.03*1.07/1.0487 = 1.05092

, and finally the on-level earned premium equals 10M multiplied by 1.05092 or 10,509,200.