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 2:

Date	Incremental Payments	Case Outstanding
12/31/CY2	$400,000	$1,000,000
12/31/CY3	$500,000	$350,000

Historical rate changes are given below:

Effective Date	Rate Change
06/01/CY1	+4%
06/01/CY2	+5%

The following is assumed:

• Ultimate losses for accident year 2 losses are projected to be 25% higher than accident year 2 reported losses.
• Earned premium for accident year 2 equals $1,500,000.
• Policies are annual and written evenly throughout the year.
• Loss cost inflation equals 3% per annum.
• There are no fixed or variable underwriting expenses.
• The target profit percentage is 15%.

Determine the rate change.

• +25.17%
• +26.31%
• +27.11%
• +28.25%
• +29.15%

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, 1.04*1.05, divided by the weighted average index. The weighted average index equals

[[math]] A_1 + 1.04 \cdot A_2 + 1.04 \cdot 1.05 \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 of 3 intersected with rectangle CY3. Elementary geometry implies that [math]A_1 [/math] equals (5/12)²/2 = 0.0868, [math]A_3 [/math] equals (7/12)²/2 = 0.1701, and [math]A_2 [/math] equals

1-0.0868-0.1701 = 0.7431. 

Hence the weighted average index equals

0.0868 + 1.04* 0.7431 + 1.04*1.05*0.1701 = 1.0454

, the on-level factor equals

1.04*1.05/1.0454 = 1.0446

, and the on-level earned premium for accident year 2 equals $1,500,000 multiplied by 1.0446 or $1,566,900. The projected ultimate losses for accident year 2 equals the reported losses for accident year 2 multiplied by the loss development factor 1.25. The reported losses for accident year 2 equals the case outstanding ($350,000) plus the sum of the payments ($900,000). Hence the projected ultimate losses for accident year 2 equals $1,562,500. The midpoint of the experience period is 07/01/CY2 and the midpoint of the forecasting period is the end of calendar year 4; hence, the trend factor equals 1.03^2.5 = 1.0767 and the projection for the ultimate inflated adjusted losses for calendar year 2 equals $1,682,344. 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 $1,682,344. divided by $1,566,900. Hence the indicated change factor equals 1.2631 and the rate change is +26.31%.