An insurer is considering a rate change that will be in effect during calendar year 4. The insurer uses the loss ratio method for ratemaking and has access to the following historical loss data for accident year 1:

Date	Incremental Payments	Case Outstanding
12/31/CY1	$600,000	$2,000,000
12/31/CY2	$800,000	$410,000
12/31/CY3	$375,000	$0

Historical rate changes are given below:

Effective Date	Rate Change
07/01/CY1	+10%
07/01/CY2	+5%

The following is assumed:

• Earned premium for accident year 1 equals $3,000,000.
• Policies are annual and written evenly throughout the year.
• Loss cost inflation equals 3% per annum.
• Accident year losses are fully developed three years after the end of the accident year.
• There are no fixed or variable underwriting expenses.
• The target profit percentage is 15%.

Determine the rate change.

• -34.33%
• -33.75%
• -32.33%
• -31.5%
• -29.75%

Since policies are assumed to be written at a uniform rate through time, we can use the parallelogram method to calculate the earned premium at current rates for accident year 1. The following diagram gives a graphical depiction of the historical rate changes:

The on-level factor for calendar year 1 equals the cumulative rate index, 1.1*1.05, divided by the weighted average index. The weighted average index equals

[[math]] A_1 + 1.1\cdot A_2 [[/math]]

with [math]A_1 [/math] the area of region 1 intersected with rectangle CY1 and [math]A_2 [/math] the area of region 2 intersected with rectangle CY1. Elementary geometry implies that [math]A_2 [/math] equals 1/8 and [math]A_1[/math] equals 7/8. Hence the weighted average index equals

7/8  + 1.1 /8  = 1.0125

, the on-level factor equals

 1.1*1.05/1.0125 = 1.1407

, and the on-level earned premium for accident year 1 equals $3,000,000 multiplied by 1.1407 or $3,422,100. Since it is assumed that accident year 1 losses are fully developed by the start of calendar year 4, the projected ultimate losses for accident year 1 equals the case outstanding, $0, plus the sum of the payments, $1,775,000. The midpoint of the experience period is 07/01/CY1 and the midpoint of the forecasting period is the end of calendar year 4; hence the trend factor equals 1.03 ^3.5 = 1.109 and the inflation adjusted projected ultimate losses for accident year 1 equals $1,968,475. According to the loss ratio method, the indicated change factor equals

[[math]] ICF = \frac{(L + E_L)/P_C + E_F/P_C}{1 - V - Q_T} = \frac{L/P_C}{0.85} [[/math]]

with [math]L/P_C [/math] equal to $1,968,475 divided by $3,422,100. Hence the indicated change factor equals 0.6767 and the rate change is -32.33%.