Mack 2000 - Benktander

Question 1

State the relationship between Reserve & Ultimate Loss Estimates

Answer 1

Uhat = Ck + Rhat
Uhat is Ultimate Loss Estimate
Ck is the Actual Claims amont paid after k years of development
That is the Reserve Estimate

Question 2

Calculate reserve using Bornhuetter-Ferguson (BF) method

Answer 2

R_BF = qk*U0

qk = 1 - 1/CDF is the proportion of ult claims unpaid after k years
U0 is the a priori expectation of Ultimate Losses = prep * ELR

It assumes Ck is NOT predictive of future claims.

Question 3

Calculate Ult Losses using BF method

Answer 3

U_BF = Ck + R_BF

Question 4

Calculate Ult Losses using Chain-Ladder method

Answer 4

U_CL = Ck/pk = Loss * CDF

pk = 1-qk is the proportion of ult claims expected to be paid after k years of development.

Assumes Ck is FULLY predictive of future claims.

Question 5

Calculate reserve using Chain-Ladder method

Answer 5

R_CL = qk*U_CL

Question 6

Briefly explain the main advantage of CL method over BF

Answer 6

Different actuaries should obtain similar results when running the chain-ladder method which is not the case with the BF method due to differences in selection of U0.

Question 7

Explain how BF method is a mixture of a priori estimate and chain-ladder

Answer 7

U_c = cU_CL + (1-c)U0
c is the credibility weight

As Ck develop, credibility should increase.

Setting c = pk:
U_pk = pkU_CL + (1-pk)U0 = Ck + qk*U0 = Ck + R_BF = U_BF

Question 8

Explain the Benktander method

Answer 8

R_GB = qkU_BF = (1-qk)R_CL + qk*R_BF

U_GB = Ck + R_GB = pkU_CL + qkU_BF = (1-qk)U_CL + qkU_BF

Benktander is a credibility-wkeighted average of CL and BF methods.

Thus, it also a credibility-weighted average of the CL and a priori expectation:
U_GB = (1-qk^2)U_CL + qk^2U0

Question 9

Complete the sentence:
If we infinitely iterate between reserves and ultimate losses, we will eventually obtain the (…) method.

Answer 9

Chain-Ladder method

U(m) = (1-qk^m)U_CL + qk^mU0
As m increases, qk^m tends to 0 and U(m) tends to U_CL

Question 10

State 4 reasons why the Benktander method is superior to BF and CL methods.

Answer 10

1. MSE is almost always smaller than BF or CL methods
   Except if pk is small and C* is large at the same time, which occurs if payout is neither extremely volatile nor extremely stable.
2. Better approx of the exact Bayesian procedure
3. Superior to CL method since it gives more weight to the a priori expectation of ultimate losses.
4. Superior to BF method since it gives more weight to actual loss experience.