Mack (1994) - CL assumptions

Question 1

Implicit CL assumptions (3)

Mack

Answer 1

1. linearity - future loss proportional to claims to date&raquo_space; means that development factors are uncorrelated
2. independence - AYs are independent
3. variance - variance of next loss is a function of age and losses to date

Mack’s assumptions apply to cumulative losses

Question 2

Variance of next loss

Answer 2

variance of next cumulative loss = claims to date * alpha-subk^2

Question 3

MSE of ultimate claims

Answer 3

MSE = C_ij^2 * RowSum[(a^2 / f^2) * (1/C_ik + 1/ColSum(C_jk))]

where C_ij^2 is the estimate at ultimate
a^2 is the estimated column parameter
f is the LDF (LS, volume weighted, or simple average; based on Var assumption)
the ColSum consists of elements above the diagonal

Question 4

Alpha-k^2 formula and what it represents

Answer 4

a_k^2 = 1/(I - k - 1) * C_ik * ColSum(C_i,k+1/C_i,k - f_k_hat)^2

C_i,k+1/C_i,k is the empirical LDF in row i col k
f_hat is the LDF estimate for the column (based on Var assumption use simple or weighted average, or LS)
I is the size of the triangle I x I

represents variability in age-to-age factors

Question 5

Estimators for last alpha parameter (3)

Answer 5

1. = 0 - only reasonable if development expected to be finished by end of triangle
2. extrapolate the alpha series using loglinear regression
3. IF a’s are decreasing: Previous * Previous / 2nd Previous
   ex. a3 = a2 * a2 / a1
   IF a’s are increasing: use 2nd Previous a

Question 6

Problems with assuming reserves are normally distributed (2)

Answer 6

1. poor distribution if data is skewed
2. potential for a negative lower bound even if negative reserves are not possible (lognormal distribution corrects this)

Question 7

Lognormal confidence interval for reserves (and lognormal parameters)

Answer 7

sigma^2 = LN [(1 + se(R)/R)^2]

CI = R * exp(-sigma^2 / 2 +/- Z*sigma)

Question 8

Alternate variance assumptions and corresponding variance proportionality implications (3)

Answer 8

1. claims to date^2 weighting - age-to-age factor = sum( current claims * claims in next period ) / sum( current claims^2 )
   &raquo_space; variance proportional to 1
2. normal volume weighted avg
   » variance proportional to claims to date
3. simple average - age-to-age factor = (1 / (I - k)) * sum (claims in next period / current claims)
   » variance proportional to claims-to-date^2

Question 9

Test for linearity assumption

Mack

Answer 9

plot claims in next period against claims to date and look for a straight line through the origin

Question 10

Test for variance assumption

Mack

Answer 10

plot weighted residuals against claims to date and look for a random scatter

weighted residual = (actual claims in next period - fitted claims in next period) / sqrt(variance proportionality assumption)

Question 11

Weaknesses of the CL method (2)

Mack

Answer 11

1. age-to-age factors further out in the tail rely on very few observations
2. known claims in latest AY form an uncertain basis for projecting to ultimate

Question 12

Test for correlation between development factors

Mack

Answer 12

Calculate LDFs for whole triangle
Rank LDFs in each pair of adjacent columns (1 is smallest)
D^2 = squared difference in each pair, sum up D^2s to get S
T_i= 1 - S/(n*(n^2-1)/6) ; n = number of components in comparison
weight the T_i’s and SUMPRODUCT to get T

E(T) = 0
Var(T) = 1 / [(I - 2)(I-3)/2]
where I is the size of the triangle (full loss triangle NOT LDF triangle)

50% CI = +/- 0.67*sigma
Reject H0 if T is outside this interval

Question 13

Correlation coefficient (T) for a pair of columns and global statistic
(Mack)

Answer 13

T-k = 1 - 6 * { sum(squared diff b/w r and s) / [ (I - k)^3 - I + k ] }

T = (I - k - 1) weighted average of T-k’s

Question 14

E[T] and Var(T) formulas for correlation coefficient b/w development factors
(Mack)

Answer 14

E[T] = 0

Var(T) = 1 / [ (I - 2) * (I - 3) / 2]

Question 15

Examples of CY effects and impacted Mack assumption (4)

Mack

Answer 15

1. major claims handling changes
2. case reserving changes
3. substantial court decisions
4. inflation

*CY effects violate AY independence

Question 16

Test for CY effects/AY independence

Mack

Answer 16

Calculate LDFs for whole triangle
RANK() each column with 1 = smallest
Assign S if < Median, L if > Median, * if = Median
N = # elements in diagonal
S = add up S in each diagonal
L = add up L in each diagonal
Zi = MIN(S,L) each row
m = FLOOR((n-1)/2)
c_n = COMBIN(n-1 | m) * n/2^n
E[Zn] = n/2 - c_n
Var[Zn] = n(n-1)/4 - c_n*(n-1) + E[Zn] - E[Zn]^2

95% CI = E[Z] +/- 2*sigma(Z)

Question 17

S, L, Z, n, and m statistics for CY effects/AY independence test

Answer 17

S-j = # smaller than median on diagonal j
L-j = # larger than median on diagonal j
Z-j = min( S-j, L-j )

n = S-j + L-j
m = (n - 1) / 2 (truncated)

Question 18

E[Z] and Var(Z) for CY effects/AY independence test

Mack

Answer 18

E[Z-j] = n / 2 - c

Var(Z-j) = n * (n - 1) / 4 - c*(n - 1) + E[Z-j] - E[Z-j]^2

where c = nCr = COMBIN(n-1 | m) * n/2^n

i.e. “n-1 choose m”