You are given:

• Accident Year	Cumulative Paid Losses through Development Year						Earned premium
  	0	1	2	3	4	5
  AY4	1,400	5,200	7,300	8,800	9,800	9,800	18,000
  AY5	2,200	6,400	8,800	10,200	11,500		20,000
  AY6	2,500	7,500	10,700	12,600			25,000
  AY7	2,800	8,700	12,900				26,000
  AY8	2,500	7,900					27,000
  AY9	2,600						28,000

• The expected loss ratio for each Accident Year is 0.550.

Calculate the total loss reserve using the Bornhuetter-Ferguson method and three-year arithmetic average paid loss development factors.

• 21,800
• 22,500
• 23,600
• 24,700
• 25,400

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Key: D

The development factors are:

[[math]] 0-1: \frac{1}{3} \left( \frac{7,900}{2,500} + \frac{8,700}{2,800} + \frac{7,500}{2,500}\right) = 3.089 \quad 1-2: \frac{1}{3} \left( \frac{12,900}{8,700} + \frac{10,700}{7,500} + \frac{8,800}{6,400}\right) = 1.428 [[/math]]

[[math]] 2-3: \frac{1}{3} \left (\frac{12,600}{10,700} + \frac{10,200}{8,800} + \frac{8,800}{7,300} \right) = 1.181 \quad 3-4: \frac{1}{2} \left( \frac{11,500}{10,200} + \frac{9,800}{8,800} \right) = 1.1205 \quad 4-5: 1 [[/math]]

The cumulative factors are 0-5: 5.8372, 1-5: 1.8897, 2-5: 1.3233, 3-5: 1.1205; 4-5: 1.

The B-F reserve is [25,000(1 – 1/1.1205) + 26,000(1 – 1/1.3233) + 27,000(1 – 1/1.8897) + 28,000(1 – 1/5.8372)](0.55) = 24,726.

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.