1. Chain Ladder and Bornhuetter/Ferguson Chain Ladder and Bornhuetter/Ferguson – Some Practical Aspects by Thomas Mack, Munich Re

2. Chain Ladder and Bornhuetter/Ferguson  Consider one single accident year  Paid claims after k years of development  Expected cumulative payout pattern p 1, p 2,..., p k,..., p n = 1 e.g. 10%, 30%, 50%, 70%, 85%, 95%, 100%  U 0 = prior est. of ultimate claims amount R BF = q k U 0 with q k = 1-p k Bornhuetter/F.

3. Chain Ladder and Bornhuetter/Ferguson  C k = claims amount paid up to now (completely ignored by BF)  U BF = C k + R BF posterior estimate (  U 0 )  U = C k + R (axiomatic relationship)  U CL = C k / p k Chain Ladder ult. claims  R CL = U CL – C k = q k U CL CL reserve (ignores U 0 completely)

4. Chain Ladder and Bornhuetter/Ferguson Comparison:  With CL, different actuaries usually come to similar results  With BF, there is no clear way to U 0  U 0 can be manipulated: If you want to have reserve X, simply put U 0 = X / q k

5. Chain Ladder and Bornhuetter/Ferguson  Gunnar Benktander's proposal (1976): R GB = p k R CL + (1-p k )R BF = p k q k C k /p k + q k R BF = q k ( C k + R BF ) = q k U BF  Iterated Bornhuetter/Ferguson  The more the claims develop, the higher the weight p k of R CL.

6. Chain Ladder and Bornhuetter/Ferguson Ultimate U(R) Connection Reserve R(U) U 0 R BF = q k U 0 U 1 = U BF = C k + q k U 0 = (1-q k )U CL + q k U 0 R 1 = q k U 1 = q k U BF = R GB = (1-q k )R CL + q k R BF U 2 = U GB = (1-q k 2 )U CL + q k 2 U 0 *q k C k + *q k

7. Chain Ladder and Bornhuetter/Ferguson Ultimate U(R) Connection Reserve R(U) U n = (1-q k n )U CL + q k n U 0 R n = (1-q k n )R CL + q k n R BF U n+1 = (1-q k n+1 )U CL + q k n+1 U 0.......... U  = U CL R  = R CL *q k C k +

8. Chain Ladder and Bornhuetter/Ferguson R GB is a credibility mixture of R CL and R BF : R c = c R CL + (1-c) R BF with c = p k  [0; 1] It gives R BF for c = 0 and R CL for c = 1. Best mixture if mean squared error is minimized: mse(R c ) = E(R c -R) 2 = min (=> c*)

9. Chain Ladder and Bornhuetter/Ferguson

10.  R c* is always better than R 0 = R BF, R 1 = R CL  R GB is not always better but mostly  How to determine c* ?  How to decide which of R GB, R CL, R BF is best at a given data set ?

11. Chain Ladder and Bornhuetter/Ferguson R c = cR CL + (1-c)R BF = c(R CL -R BF ) + R BF E(R c – R) 2 = E[c(R CL -R BF ) + (R BF -R)] 2 = c 2 E(R CL -R BF ) 2 + 2cE[(R CL -R BF )(R BF -R)] + + E(R BF -R) 2

12. Chain Ladder and Bornhuetter/Ferguson So far, we have not used any assumptions. But for Var(C k ), Cov(C k,R) we need a model. Model A: (with U = C n ) E(C k |U) = p k U, Var(C k |U) = p k q k  2 (U) =>Var(C k ) = p k q k E(  2 (U)) + p k 2 Var(U) Cov(C k,R) = p k q k ( Var(U) – E(  2 (U)) ) But E(  2 (U)) is difficult to estimate.

13. Chain Ladder and Bornhuetter/Ferguson B:Increments S j = C j – C j-1, m j = p j – p j-1 E(S j /m j |  ) = µ(  ), S j |  independent, Var(S j /m j |  ) =  2 (  )/m j, (Bühlmann/S.)  indicates the "quality" of the acc.year =>Var(C k ) = p k q k E(  2 (  )) + p k 2 Var(U) Cov(C k,R) = p k q k ( Var(U) – E(  2 (  )) ) Var(µ(  ))

14. Chain Ladder and Bornhuetter/Ferguson Both models are math. equivalent and lead to E(  2 (  )) = inner variab. random error Var(µ(  )) = level variab. Var(U) Var(U 0 ) = estimation error to be est. by actuary

15. Chain Ladder and Bornhuetter/Ferguson An actuary who presumes to establish a point estimate U 0 should also be able to estimate its uncertainty Var(U 0 ) and the variability Var(U) of the underlying claims process.

16. Chain Ladder and Bornhuetter/Ferguson For E(  2 (  )), we have an unbiased estimate based on the data observed: Note that and

17. Chain Ladder and Bornhuetter/Ferguson Having estimated we can compare the precisions: mse(R BF ) = E(  2 (  )) (q k + q k 2 / t) mse(R CL ) = E(  2 (  )) q k / p k mse(R c ) = c 2 mse(R CL ) + (1-c) 2 mse(R BF ) + + 2c(1-c)q k E(  2 (  ))

18. Chain Ladder and Bornhuetter/Ferguson and obtain the following results: mse(R BF ) p k < t i.e.use BF for green years use CL for rather mature years mse(R GB ) t < 2-p k mse(R GB ) t > p k q k /(1-p k )

19. Chain Ladder and Bornhuetter/Ferguson

20. Example:U 0 = 90%, k = 3 {p j } = 10%, 30%, 50%, (70, 85, 95, 100 %) {C j } = 15%, 27%, 55% (of the premium) => R BF = 45%,U CL = 110%, R CL = 55% {m j } = 10%, 20%, 20%, (20%, 15%, 10%, 5%) {S j } = 15%, 12%, 28%

21. Chain Ladder and Bornhuetter/Ferguson Inner variability S 1 /m 1 = 1.5, S 2 /m 2 = 0.6, S 3 /m 3 = 1.4 E(  2 (  )) = = 0.042 = (20.5%)²

22. Chain Ladder and Bornhuetter/Ferguson Actuary's estimates: Var(U) = (35%) 2 (e.g. lognormal with 5% above 150%) Var(U 0 ) = (15%) 2 => Var(µ(  )) = (35%) 2 – (20.5%) 2 = (28.4) 2 t = (20.5%) 2 / ( (28.4%) 2 + (15%) 2 ) = 0.408

23. Chain Ladder and Bornhuetter/Ferguson Results: R BF = 45.0%  21.6% R CL = 55.0%  20.5% R GB = 50.0%  18.1% R c* = 50.5%  18.0%with c* = 0.55 Note the high standard errors!

24. Chain Ladder and Bornhuetter/Ferguson Check by distributional assumptions: U ~ Lognormal(µ,  2 ) with E(U) = 90%, Var(U) = (35%) 2 => µ = -0.176,  2 = 0.141 C k |U ~ Lognormal(,  2 ) with E(C k |U) = p k U, Var(C k |U) = p k q k  2 U 2 where  2 is such that Var(C k ) is as before =>  2 = 0.045,  2 = 0.044

25. Chain Ladder and Bornhuetter/Ferguson Then, according to Bayes' theorem: U|C k ~ Lognormal(µ 1,  1 2 ) with µ 1 = z(  2 + ln(C k /p k )) + (1-z)µ = 0.0643  1 2 = z  2 = 0.0335 z =  2 / (  2 +  2 ) = 0.762 => E(U|C k ) = exp(µ 1 +  1 2 /2) = 108.4% => E(R|C k ) = 53.4% Var(U|C k ) = (20.0%) 2 = Var(R|C k )

26. Chain Ladder and Bornhuetter/Ferguson No estimation error standard error still very high Results: R BF = 45.0%  21.6% R CL = 55.0%  20.5% E(R|C k ) = 53.4%  20.0% R GB = 50.0%  18.1% R c* = 50.5%  18.0%with c* = 0.55

27. Chain Ladder and Bornhuetter/Ferguson Conclusions: Use of a priori knowledge (U 0 ) better than distributional assumptions A way is shown how to assess the variability of the Bornhuetter/Ferguson reserve, too.

28. Chain Ladder and Bornhuetter/Ferguson Conclusions (ctd.): Benktander's credibility mixture of BF and CL is simple to apply and gives almost always a more precise estimate. The volatility measure t is not too difficult to estimate and improves the precision even more or helps to decide on BF, CL, GB.