Mack - Benktander

Question 1

Bornhuetter/Ferguson (BF) Method

Answer 1

Estimates reserves based on an a priori expectation of ultimate losses
R_BF = q_k * U_0
where q_k = % unpaid after k years of development = 1 - 1/CDF and U_0 is the a priori expectation of ult losses
Assumes current claims amount C_k is NOT predictive of future claims

Question 2

Chain-Ladder (CL) Method

Answer 2

Estimates reserves based on claims to date, assuming C_k is fully predictive of future claims
R_CL = CDF * C_k = q_k * U_CL
where U_CL = C_k / p_k and p_k = % paid

Question 3

Advantage of CL method over the BF method

Answer 3

Different actuaries should obtain similar results when running the CL method

Question 4

Benktander (GB) Method

Answer 4

Credibility weighted average of CL and BF methods
R_GB = p_k * R_CL + q_k * R_BF
U_GB = p_k * U_CL + q_k * U_BF

Question 5

GB method as a credibility-weighted average of CL method and a priori expectation

Answer 5

U_GB = (1 - q_k^2) * U_CL + q_k^2 * U_0

Question 6

Iterative relationships

Answer 6

U(m) = (1 - q_k^m) * U_CL + q_k^m * U_0
R(m) = (1 - q_k^m) * R_CL + q_k^m * R_BF

Iteration 1 is the BF method
Iteration 2 is the GB method
Iteration inf is the CL method

Question 7

Advantages of GB method

Answer 7

• the MSE is almost always smaller than BF or CL methods if the payout pattern is neither extremely volatile nor extremely stable
• better approximation of the exact Bayesian procedure
• superior to CL method since it gives more weight to the a priori expectation of ultimate losses
• superior to BF method since it gives more weight to actual loss experience

Question 8

When does the GB reserve have a smaller MSE than the BF reserve?

Answer 8

When Z > p_k / 2