Mack 2000 - Benktander Flashcards
State the relationship between Reserve & Ultimate Loss Estimates
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
Calculate reserve using Bornhuetter-Ferguson (BF) method
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.
Calculate Ult Losses using BF method
U_BF = Ck + R_BF
Calculate Ult Losses using Chain-Ladder method
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.
Calculate reserve using Chain-Ladder method
R_CL = qk*U_CL
Briefly explain the main advantage of CL method over BF
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.
Explain how BF method is a mixture of a priori estimate and chain-ladder
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
Explain the Benktander method
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
Complete the sentence:
If we infinitely iterate between reserves and ultimate losses, we will eventually obtain the (…) method.
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
State 4 reasons why the Benktander method is superior to BF and CL methods.
- 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. - Better approx 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.
Briefly explain why it may not be appropriate to use BF method when losses develop downward
Since BF IBNR does not respond to actual loss performance, the downward development will not affect IBNR produced by BF.
If downward development represents real trends, then BF will overstate IBNR.