Please enable JavaScript.
Coggle requires JavaScript to display documents.
Mack 1994: mesuring variability of chain ladder reserve estimates (key…
Mack 1994
: mesuring variability of chain ladder reserve estimates
Chain ladder implicit assumptions
Independence between accident years
Expected Incremental Losses are proportional to the losses reported to date
Variance of incremental losses is proportional to losses reported to date (by age)
key definitions
( see formulas sheet)
c
is the accumulated claims amount od accident year i in developpement year k
( can be paid or incurred)
f
unknown true factor of increase which is the same for all AY
(is estimated with available data)
Alpha(k)^2
is a proportionality constant
n
represent the ultimate period
i
represent the accident years
(rows)
j
represent the calendar years
(diagonal)
k
represent developement period
(columns)
When there is a
^
, this means it is estimated
R
is the reserve
Z
is the percentile of the normal distribution
f
age developpement factors
(are estimated)
Facts
Chain ladder method produces a point estimate for a cumulative loss amount in the future, which will normally turn out to be more or less wrong
(very unlikely to be exact)
Since the ultimat = last diagonal of
c
+ Reserve and the last diagonal of
c
is known , the probability distribution entirely rely on the reserve.
We use lognormal distribution to approximate the distribution of the reserve because of it skewed shape
If the standard error of the reserve to the serserve estimate is generally les then 50%, th normal distribution can be use to approximate the distribution of ultimate losses or reserves. Otherwise, stick to lognormal
Empirical LDF method
Selects the largest age to age factor to estimate the upper bound of the Ultimate losses and the smallest factors for the lower bound
This methode tend to understate the size of the confidence of interval for older ages and overstate it for younger ages
(due to the number of factors available)
Residial estimation
We can plot against
c
the residual to test the variance assumption. If the residuals seem to vary randomly about 0 then the assumption is met
the last factors of
f
rely on very few observations and the fact that the known claim amount
c
of the last accident year forms a very uncertain basis for projecting ultimates
Testing for correlations between subsequent development factors
(procedure)
1.Set up a triangle of observed development factors
2.In a separate table, for the first set of age-to-age factors (i.e. from 12 to 24 months), create an r column with the rank of each age factor. For each subsequent set of factors until the set with only two factors in it, create an s and r column. The s column reranks the previous age’s factors, but excludes the bottom one. The r column ranks with the same logic as the first development age, but for its age’s factors.
To be clear, although the s column is based on the previous set of age-to-age factors, it is still associated to the age of the following r column
3.Determine the test statistic T
4.Compare the test statitic to 50% confidence interval
Briefly describe two reasons why it is more appropriate to test the triangle as a whole rather than
correlation between pairs of columns.
Avoid an accumulation of error probabilities
Want to know if correlations globally prevail, rather than just a single pair of columns
Testing for calender year effects
Procedure
1.Set up a triangle of observed development factors
2.For each age group of development factors, rank each observation as being smaller or larger than the median (label the median when an age group has an odd number of observations). Create a new triangle with these ranks
3.For each diagonal (except the observation in the top left corner, (i.e. j > 2), count the number of small and large labels. For each diagonal, calculate the diagonal statistic
Z
, which is the minimum between the number of small and large labels. The total test statistic is the sum of the
Z
Compare with the 95% confidence interval
A calender yesr influence affects one of the diagonals of the loss triangle and therefore also influences the adjacent development factors
(development factors from either sides)
If due to a calender year influence the elements of the diagonal are larger then (maller) than usual, then the observed development factors on the left of the diagonal will be larger (smaller than usual and the observed development factors on the right will be smaller (larger) than usual
Potential causes
Changing inflation
Changes in claim settlement
Change in legal environment