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Venter : Testing the assumption of age-to-age-factors (Key definitions…
Venter
: Testing the assumption of age-to-age-factors
Testable implication of implicit Assumptions of the chain ladder method
1.Statistical significance of the age-to-age factors
Rule of thumb: factor is twice its standard deviation *
(for a normal distribution, this means there is a bout a 4.5% probability of getting a factor of this magnitude value or greater when the true factor is 0.
2.Superiority of factor assumption to alternative emergence patterns
Linear with constant (assumes a 0 constant)
Factor times a parameter
Including a calender year effect
3.Linearity of the model
Analysis of residuals as a function of the cumulative losses:
negative residuals followed by positive residuals is indicative of a non-linear process
4.Stability of factor
Analysis of residuals as a function of time
: sequences of high residuals followed by low residuals is indicative of unstable estimated factors
5.No correlation among columns
(factors)
Analysis of one pair of columns
see formulas sheet for steps
Analysis of whole table of colomns
see formulas sheet for steps
No particularly high of low diagonals
creat a matrix
See example
1.The first column is all incremental losses except for the first development period
2.The following columns correspond to the cumulative losses previous to that incremental amount if that cumulative amount is immediately previous to the incremental amount and 0 otherwise
The first column of this sequence of columns corresponds to the first age of cumulative losses. The following columns move to the right of the triangle up until the before last column
The total number of columns at this point should correspond to the number of columns of the loss triangle
Under the Cape Cod method, the cumulative loss amounts are replaced by 1s to represent that it is the same as the additive chain ladder method and that additive coefficients are fit rather than multiplicative ones
3.Depending on the diagonals we want to test, we set up a “dummy” column for each diagonal to test and assign a value of 1 when the incremental entry is in the diagonal of interest and 0 when it is not. (Two last diagonals by default)
4.Run a regression
Key definitions
(see formulas)
r
is the coefficient of correlation
n
is the number of entries in the shorter colomne
m
number of possible pairs of colomns of factors in the n x n triangle
Alpha
confidence level of the test
N
a set number of standard deviations
(like a Z in normal)
q(w,d)
incremental loss
d
development period
w
accident year
f(d)
proportion of incremental losses from d - 1 to d
h(w)
ulstimate losses a age
w
Itteration starting with chain ladder
BF method : itteration with
q(w,d)= h(w)*f(d)
Cape cod: itteration with
q(w,d)= h*f(d)
.
Only one iteration can be done
Cape cod produces a fit equivalent to the additive CL method -> the predicted incremental losses for any accident year d is equal to the straight average of observed incremental losses at that development age
Testing alternative emergence pattern
Calculate AIC,BIC and SSE
BIC penalizes for over-parametrization , AIC is to permissible