Mack (Chain Ladder)

Question 1

Chain Ladder Method Assumptions

Answer 1

1. Expected losses in the next development periods are proportional to losses to date
2. Losses in one accident year are independent of losses in another year
3. The variance of the next incremental loss is proportional to the losses reported to date

Question 2

Variance Assumptions

How to calculate LDF?

Answer 2

The variance of the next incremental loss is proportional to the losses reported to date
* Weighted average w/ weights = losses

The variance of the next incremental loss is proportional to 1 aka constant for all years
* Weighted average but weights = loss^2

The variance of the next incremental loss is proportional to the square of the losses reported to date
* Straight average

Question 3

Calendar Year Effects Test

Sj, Lj, Zj, Z, nj, mj

Formula

Answer 3

Sj = # of < medium in diagonal
Lj = # of > median in diagonal
Zj = min(Lj, Sj)
Z = sum(Zj)
nj = Lj + Sj
mj = ROUNDDOWN((n-1)/2,0)

Remember to skip diagonal 1 (only 1 value)

Question 4

Calendar Year Effects Test

E(Zj), Var(Zj) and H0 Test

Formula

Answer 4

E(Zj) = n/2 - combin(n-1, m) * n/(2^n)
Var(Zj) = n(n-1)/4 - combin(n-1, m) * n(n-1)/2^n + E(Zj) - E(Zj)^2

If Z is within (95%) confidence interval
E(Z) - 1.96 * sd(Z) ≤ Z ≤ E(Z) + 1.96 * sd(Z)
then fail to reject H0, there are no calendar year effects

E(Z) and Var(Z) is sum of the Zj’s

Question 5

Reserve Confidence Intervals

Lognormal Confidence Interval

Formula

Answer 5

reserves * exp(-Var/2 ± Z * SD)

Question 6

Reserve Confidence Intervals

Variance, σ^2

Formula

Answer 6

ln(1+ SE(reserves)^2 / reserves^2)

Question 7

Reserve Confidence Intervals

Alpha^2 for each k, column

All steps and formulas, what about last column?

Answer 7

1. Calculate LDF triangle
2. Calculate wtd avg LDFs for each k
3. Alpha^2 triangle = loss * (LDF - Avg LDF)^2
4. Alpha^2 for each k = sum[loss * (LDF - Avg LDF)^2] / (I-k-1)
5. Alpha^2 for K-1 = (alpha^2 for previous)^2 / (alpha^2 for previous previous). Make sure alpha^2 is descending, set = to alpha^2 for previous

where I is # of AYs