An actuary is establishing reserves for a group of policies as of December 31, CY3. You are given the following table of reserve estimates for AY1 and AY2:

	Reserve estimates as of December 31, CY3
	[math]R_{BF}[/math]	[math]R_{LR}[/math]	[math]R_{CL}[/math]
AY1	400,000	250,000	437,500
AY2	1,120,000	1,200,000	?

where [math]R_{BF}[/math] is the loss reserve under the Bornhuetter-Ferguson method, [math]R_{LR}[/math] is the loss reserve under the Expected Loss Ratio method, and [math]R_{CL}[/math] is the loss reserve under the Chain Ladder method.

If [math]f_2[/math] , the loss development factor from the paid-loss-development triangle at duration 2, equals 1.05, determine the reserve estimate as of December 31, CY3 using the expected loss ratio.

Solution: C

Step 1: Calculate the cumulative development factors for AY1 and AY2

The cumulative factor for AY1 equals [math]F_2[/math] and the cumulative development factor for AY2 equals [math]F_1 = F_2 * f_2 [/math] with [math]f_2 = 1.05 [/math]. To calculate [math]F_2[/math], we use the equation:

[[math]] \begin{aligned} R_{BF} = R_{LR}(1-1/F_2) + 1/F_2 * R_{CL} &\implies 400000 = 250000 * (1-1/F_2) + 1/F_2 * 437500 \\ & \implies F_2 = 1.25, F_1 = 1.05 * 1.25 = 1.3125 \end{aligned} [[/math]]

where [math]R_{BF}[/math] is the reserve estimate for AY1 using the Bornhuetter-Ferguson method.

Step 2: Calculate the reserve for AY2 using the chain ladder method

We use the same formula as above but for AY2:

[[math]] \begin{aligned} R_{BF} = R_{LR}(1-1/F_1) + 1/F_1 * R_{CL} &\implies 1120000 = 1200000 * (1-1/1.3125) + 1/1.3125 * R_{CL} \\ & \implies R_{CL} = 1095000 \end{aligned} [[/math]]