Mack (2000)

Question 1

Key Notation Mack(2000)

Answer 1

p_k = proportion of the ultimate claims amount which is expected to be paid after k years of development

q_k = the proportion of ultimate claims which is expected to remain unpaid after k years of development

U_o=U⁽⁰⁾= the a priori expection of ultimate losses

U_BF=U⁽¹⁾=the BF ultimate claims estimate

U_GB=U⁽²⁾= the Gunner Benktander ultimate claims estimate

U_CL=U^(infinity)=the CL ultimate claims estimate

U^(hat) is any ultimate claims estimate

R^(hat) is any reserve estimate

C_k is the actual claims amount paid after k years of development

Question 2

General relationship between reserve estimate and ultimate claims

Answer 2

U^(hat)=C_k+R^(hat)

Question 3

Bornhuetter/Ferguson Method

Answer 3

Reserve based on a priori exp: R_BF=q_kU_o

U_BF=C_k+R_BF

Bornhuetter reserve assumes that current claim amounts is not predictive of the future

Question 4

Chain Ladder Method

Answer 4

U_CL=C_k/p_k

R_CL=q_kU_CL=U_CL-C_k

Reserve assumes the current claims amount is fully predictive of future claims.

Advantage over BF: Using CL, different actuaries obtain similar results.

Question 5

Gunner Benktander Method (Iterated BF) Ultimate

Answer 5

Credibility weighted average of extreme positions of BF and CL methods

U_GB=C_k + R_GB = (1-q_k)U_CL + q_kU_BF

Question 6

Gunner Benktander Reserve

Answer 6

R_GB=q_kU_BF

Question 7

Multiple iterations of GB method

Answer 7

U^(m)=(1-q_k^(m))U_CL + q_k^(m)U_o

R^(m) =(1-q_k^(m))R_CL + q_k^(m)R_BF

If we iterate between reserves and ultimates indefinitely we will eventually end up with CL result

Question 8

Why is Benktander method superior to BF and CL?

Answer 8

1. Lower mean squared error (MSE)
2. Better approximation of the exact Bayesian procedure
3. Superior to CL since it gives more weight to the a priori expectation of the ultimate losses
4. Superior to BF since it gives more weiht to actual loss experience

Question 9

Mean Squared Error between GB and BF

Answer 9

MSE(GB)<mse>
</mse>