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Clark : LDF curve fitting and Stochastic Reserving (Definitions (see…
Clark
: LDF curve fitting and Stochastic Reserving
Definitions
(see formulas sheet)
G(x)
is the cumulative percentage reported or paid :
x
is the average date of claims for an accident period in months
(depends on AY or CY)
G(x | w, teta)
is use when reporting/payement pattern follows a distribution with a 2 parameter random variable and
values are increasing with no reversal
u
is the
expected
incremental loss emergece in accident year AY
P
premium earned for period i
c
represent the
actual
incremental loss emergence in accident year AY
p
number. of parameter of the methd
N
is the number of observations
(# cells in loss triangle)
y
truncation period
2 Methodes of expected loss emergence
Cape cod
Assumes that ther is a known relationship between the amount of ultimate loss expected in each of the years in the historical period. This relationship is defined by an exposure basis
It uses 3 parameter to calculate. expected loss emergence
(ELR, teta, w)
May have higher process variance estimated but significintly lower estimation error. This is do to the fact that this method uses more information and a more stable LR for immature years instead of relying on highly levreged LDFs
LDF
LDF method
(chain ladder)
assumes that ultimate loss amount in each accident year is independaent of losses of other years
It uses
n+2
parameters
(w, teta , one parameter per accident year which are the LDFs)
There is a problem with over-parametrixation
Advantages of using a growth function versus implied age-to-age and LDF factors
IEIE
Handles data better when it comes to irregular valuation periods
Works better when handling data at the end of the triangle
Naturally extrapolates beyond the triangle (to determine terminal LDFs)
Naturally interpolates between points that don’t exist in the triangle
Distribution of actual loss emergence and maximum of likelihood
we assume that the process variance to mean ratio
(scale factor)
is constant at any time , which is equivalent to a chi-square error term
If calculating the variance of discounter reserves, the present value factors must be squared in the calcullation of the expected discounted reserves
Note
: the reserves are bunched up at the average accident date when discounting
We assume
c
follows an over-dispersed poisson (ODP) with scaling factor discribed above
The use of a discrete distribution allows for a mass point at 0, representing the cases in which no change in loss is seen in a given development increment
Using ODP allows flexibility for moment-matching
(vs using the poisson where mean and variance must be identical)
MLE estimate of
ELR
(for cape cod)
and
ULT
( for LDF)
is equivalent to the estimates under each methodologie
Key assomptions of the model
The variance mean scale parameter is fixed and known
Can be tested by graphing the expected incremental losses against their normalized residuals. If the residuals are closer to x axis in one side of the graph, this assumption can be rejected
Variance estimates are based on approximation to the Rao-Cramer lower bound
The estimate based on the information matrix is only exact when we are using linear functions. But in this case it is not linear, the variance estimate is a Rao-Cramer lower bound
Incremental losses are iid
Practical applications of the model
We can overcome irregular evaluation periods via use of a table
For an incomplete accident period, the growth function must be annualized by dividing the fraction of the accident period that has been evaluated
When evaluating a model fit, the desired outcome is always that the residuals appear to be randomly scattered around the zero line
A refinement on cape cod method would include an adjustment for loss trend net of exposure trend so that all years are the same cost level as well as rate level
Extrapolation
(from the CDF to estimate loss emergence )
should always be used cautiously. For practical purposes, we should only rely on an extrapolation to a finite point (ex:10 years from now)
Steps for calculation
R
(do not use when estimation the scale factor)
Calculate the value of
G
corresponding to twice the length of the claim triangle, Call it
G*
2.If
G*
is too small
( there is a lot of development still at that point)
, truncate
Under LDF method use
G'(x) = G(x)/G*
rather than
G
Under Cape Cod, let
G(infinit) = G
. For all other non-ultimate or if
G
is closer to 1 then use
G