Mack CL

Question 1

Mack CL Assumptions

Answer 1

1. Expected losses in the next period are proportional to losses-to-date
2. Losses are independent between AYs
3. Variance of losses in next period are proportional to losses-to-date with constant alphak^2 that varies by age

Question 2

How to check CL 1

Answer 2

plot Cik+1 against Cik and check for linearity

Question 3

How to check CL 2

Answer 3

test for CY effects -> reserves strengthening or weakening, changes in payment processes, changes in inflation, etc.

Question 4

How to check CL 3

Answer 4

plot weighted residuals against Cik and check for randomness

Question 5

sigma^2 for lognormal distribution

Answer 5

ln( 1 + (stderr(R) / R)^2 )

Question 6

confidence interval for lognormal dist

Answer 6

lower: R * exp(-z * sigma - 0.5 * sigma^2)
upper: R * exp(z * sigma - 0.5 * sigma^2)

Question 7

Assume the variance of the next incremental loss is a constant for each development age

Answer 7

set up cumulative triangle Cik, Cik^2 triangle, and calculate LDFs as sumproduct(Cik, Cik+1)/sum(Cik^2)

then use those LDFs to project cumulative loss to ultimate

Variance constant -> LDF calc weight is Cik^2

Question 8

Assume the variance of the next incremental loss is proportional to the losses reported to date

Answer 8

Mack CL assumption - use volume weighted LDFs

Question 9

Assume the variance of the next incremental loss is proportional to the square of the losses to date

Answer 9

Calculate all the LDFs from the given triangle then take the simple average to make selections

Question 10

CY Effects Calculation

Answer 10

1. From LDF triangle, calc median for each column; rank within each column whether the LDF is large or smaller than the median (or equal to)
2. Switch to diagonal analysis for CY effects. For each diagonal (oldest AY is diag1, then look at diag2+):
   Count the number of S and L
   Zj = min(S, L)
   nj = sum(S, L)
   mj = (nj-1)/2, rounded down

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

3. Sum up Z, E[Zj], and Var(Zj) of each diagonal
4. Set up CI = E[Zj] +/- Z-val * sqrt(Var(Z)) two tailed test for if empirical Z is within the interval (fail to reject)

Question 11

sigma^2 parameter of normal distribution of overall reserve

Answer 11

s.e.(overall reserve)^2

Question 12

confidence interval for reserve using normal distribution

Answer 12

Reserve +/- s.e.(Reserve) * z-value

Question 13

MSE Calculation

Answer 13

1. Calculate age-to-age factors and volume-weighted LDF
2. Calc alpha^2, the variance proportionality constant:
   a) set up intermediate triangle of lossAY,k * (age-to-age factorAY,k - LDFk)^2
   b) alpha for each column = sum of the column / (#AYs - column# - 1)
   c) alphak^2 for the final dev period can be extrapolated from the prev 2 alphak^2: (alphak-1^2)^2 / alphak-2^2 - NOTE that only works if the alphak^2 are decreasing; if that is not the case then just match the final alpha^2 to the min of the two before it
3. Square out the cumulative Reported Loss triangle using the vol-wtd LDFs
4. Calc MSE triangle =UltAY^2 * alpha^2k/LDFk^2 * [1/CAYkhat + 1/SUM(prev CAYs in column) ] starting with curr diagonal & then to future periods (not ult)

Question 14

Overall MSE Calculation

Answer 14

Calc LDFs, CDFs volume-weighted and project Losses to Ult

(1) For each AY after the first row, calc Ulti * SUM(Ult for AYs in rows below i); final row will be zero

(2) For each AY, calc 2 * alphak^2 / LDFk^2 then divide by sum(CAYk of AYs before curr diagonal; (2) is the sum across dev periods

MSE(Ri) = AY MSE + (1) * (2)
Overall MSE = Sum of MSE(Ri)

s.e.(R) = sqrt(MSE(R))

Question 15

Correlation of Adjacent LDFs

Answer 15

1. Within each LDF column, rank the values 1=lowest and counting up to the highest
2. For a development period, compare each AY’s ranking with the dev period (column) before it
   Sk = sum( (rank k - rank prior)^2 )
3. Spearman’s rank correlation coefficient
   Tk = 1 - [Sk / (n * (n^2-1) /6 ) ]
   where n = # rank pairs

table: period | n | S_k | T_k | weight

weightk = #AYs in original triangle - k - 1; you can also remember that the last column has a weight of 1 and it increases going back to the first column

T = SUM(weightk * Tk) / sum(weightk)

Var(T) = 1/ [(#AYs in og triangle - 2) * ((#AYs -3)/2) ]

C.I. = 0 +/- z * sqrt(Var(t))

Question 16

Mack weighted residual

Answer 16

(Ci,k+1 - Ci,k * fhat) / sqrt(Var of Loss)

Loss k+1 - Lossk * LDF / sqrt(Var assumption of Loss)

Question 17

E[Zj] in CY Effects Calc

Answer 17

n/2 - COMBIN(n-1, m) * n / (2^n)

Question 18

Var(Zj) in CY Effects Calc

Answer 18

n * (n-1) / 4 - COMBIN(n-1, m) * n * (n-1) / (2^n) + E[Zj] - E[Zj]^2

Question 19

m in CY Effects Calc

Answer 19

(n-1)/2 rounded down to the nearest whole number

Question 20

Tk in LDF Correlation Calc

Answer 20

1 - Sk/ (n * (n^2-1)/6)

Question 21

Var(T) in LDF Correlation Calc

Answer 21

1 / ( (#OG AYs - 2) * (#OG AYs - 3)/2)