Mack 1994

Question 1

Chain Ladder Assumptions

Answer 1

1. expected losses in next development period are proportional to the cumulative losses to date
2. variance of losses in next development period is a fct of the age and the cumulative losses to date
3. losses are independent between AYs

Question 2

when is the 3rd assumption violated?

Answer 2

3 is violated when have CY influences that affect multiple AYs

Examples: reserve strengthening/weakening, changes in payment processes, changes in inflation, changes in claim settlement rates, legislative changes

Question 3

major consequence of #1 assumption

Answer 3

implies subsequent DFs are uncorrelated

Question 4

formula for (s.e.(C_iI))^2 and how its related to R_iI

Answer 4

Question 5

formula for alpha_k^2

Answer 5

Question 6

options for last alpha

Answer 6

Question 7

estimating CI for Reserves

normal and lognormal

Answer 7

lognormal: σ²=ln[1+(se(R)/R)²]

Question 8

if doing CI for cumulative losses

Answer 8

use formula for CI for reserves but shift by paid/rptd loss amount ie std deviation is the same for both

Question 9

Testing assumption 2 and 3 different DFs

Answer 9

fk0: Cik^2 weighted average
fk1: Cik weighted average
fk2: simple average

fk0 and fk2 violate the assumption because they are proportional to 1 and Cik^2, respectively

Question 10

testing variance assumptions: residual plots for 3 DFs

Answer 10

fk0: (C_i,k+1-f*C_ik) vs C_ik
fk1: (C_i,k+1-f*C_ik)/sqrt(C_ik) vs C_ik
fk2: (C_i,k+1-f*C_ik)/C_ik vs C_ik

create 3 residual plots and see if 0&2 are better than 1

Question 11

testing for CY effects (creating triangle)

Answer 11

• calc LDFs for each cell of triangle
• rank factors -> lowest gets rank of 1
• convert table of ranks into S, L, *

* is median, L > *, S < *

-count number of L’s and S’s in each diag Aj (exclude first diag)

Question 12

testing for CY effects (creating table)

Answer 12

-create table of j, Sj, Lj, Zj, n, m, E[Zj], Var(Zj)

**j for all except first diagonal aka only 1 element

Zj=min(S,L)

n=S+J

m=(n-1)/2 truncated

E[Zj]=n/2-(n-1 choose m)*n/(2^n)

Var(Zj)=n(n-1)/4-(n-1 choose m)*n*(n-1)/(2^n)+E[Zj]-E[Zj]^2

E[Z]=sum(E[Zj]), Var(Z)=sum(Var(Zj))

*If sum(Zj) lies outside CI, reject null that there are no CY effects aka AYs are independent

Question 13

Testing for correlations between subsequent DFs

Answer 13

• calc LDFs for each celll
• create table of r_ik and s_ik

calc spearmann’s rank correlation coeff Tk

*value close to 0 indicates DFs between k-1 and k and DFs between k and k+1 are uncorrelated

calc global T

*tests the entire triangle for correlation

calc CI for T

*If T lies outside CI, reject null that you have uncorrelated DFs

Question 14

formula for Tk

Answer 14

Question 15

formula for T, E[T], Var(T), CI for 50%

Answer 15

E[T]=0

Var(T)=2/[(I-3)(I-2)]

+/-0.67*sqrt(Var(T))

Question 16

why is it more important to look at global T?

Answer 16

*more important to know whether correlation globally prevails than finding a small portion of the triangle that is correlated

*At a 10% level of significance, 10% of the pairs of columns could exhibit significant correlation by random chance. Thus, we need to check the whole triangle to determine if correlation is truly present

Question 17

Testing assumption 2 with weighted residuals

Answer 17

weighted residual = (y-yhat)/sqrt(x)

*Scatter plot of x vs weighted residuals -> tests variance assumption (#2)

*If residuals are not randomly scattered 0, assumption is violated

*assumption 2 requires most appropriate selection method for loss development factors = all-year volume-weighted average

Question 18

testing linearity assumption

Answer 18

*Scatter plot of x vs y aka C_ik and C_i,k+1

should be through origin with slope of DF

Question 20

when is lognormal more appropriate for CI for reserves?

Answer 20

lognormal distribution is appropriate if normal distribution results in negative lower bound of CI

when standard error is high % of reserve estimate

Question 21

% std errors aka s.e(R)/R comparison for AYs

Answer 21

should be greatest for older AYs since reserve estimate is smallest there and absolute value of std error is smallest

should increase significantly for most recent AYs because most immature and most uncertain AYs

Question 22

variance of total reserve compared to individual years

Answer 22

MSE of total reserve should be greater than sum of variance of individual years if using CL which Mack uses

this is reasonable and expected because CL uses same LDFs between AYs so reserve estimates for individual AYs are positively correlated causing higher total var than if AY reserve estimates were independent