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 (ultimate claims) = ultimate^2 * sum [ (alpha-k^2 / age-to-age factor^2) * (1 / est. prior cumulative losses + 1 / sum(prior cum claims for all prior AYs x lates diagonal) ) ]

Question 4

Alpha-k^2 formula and what it represents

Answer 4

alpha-k^2 = (1 / (I - k - 1)) * sum down col (current claims * (actual age-to-age - expected age-to-age)^2 )

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. assume the same relative change in subsequent alphas, alpha-I-1^2 = min( alpha-I-2^4 / alpha-I-3^2, min( alpha-I-3^2, alpha-I-2^2 ))

Question 6

95% confidence interval for reserves under a normal distribution

Answer 6

(R-hat - 2 * s.e.(R-hat) , R-hat + 2 * s.e.(R-hat))

Question 7

Problems with assuming reserves are normally distributed (2)

Answer 7

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 8

Lognormal confidence interval for reserves (and lognormal parameters)

Answer 8

exp( mu +/- t * sigma)

mu = ln(R-hat) - sigma^2 / 2

sigma^2 = ln( 1+ (s.e.(R-hat)^2 / R-hat^2) )

Question 9

Alternate variance assumptions and corresponding variance proportionality implications (3)

Answer 9

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 10

Test for linearity assumption

Mack

Answer 10

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

Question 11

Test for variance assumption

Mack

Answer 11

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 12

Weaknesses of the CL method (2)

Mack

Answer 12

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 13

Test for correlation between development factors

Mack

Answer 13

table of r’s and s’s:
use global test statistic to create a 50% confidence interval around T
» reject if T not in CI: (E[T] - .67 * sqrt(var(T)) , E[T] + .67 * sqrt(var(T)) )

Question 14

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

Answer 14

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 15

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

Answer 15

E[T] = 0

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

Question 16

Examples of CY effects and impacted Mack assumption (4)

Mack

Answer 16

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

*CY effects violate AY independence

Question 17

Test for CY effects/AY independence

Mack

Answer 17

triangle of S’s, L’s, and *’s compared to median age-to-age factor, and count number per diagonal:
create a 95% CI around test statistic, Z
» reject if Z not in CI: ( E[Z] - 2 * sqrt(Var(Z)) , E[Z] + 2 * sqrt(Var(Z)) )

Question 18

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

Answer 18

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 19

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

Mack

Answer 19

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

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

E[Z] = sum(E[Z-j]) and Var(Z) = sum(Var(Z-j)) b/c of independence assumption