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Analyze loss triangle using Clark's LDF (loss development factor) method.
ClarkLDF(Triangle, cumulative = TRUE, maxage = Inf,
adol = TRUE, adol.age = NULL, origin.width = NULL,
G = "loglogistic")A loss triangle in the form of a matrix. The number of columns must be at least four; the number of rows may be as few as 1. The column names of the matrix should be able to be interpreted as the "age" of the losses in that column. The row names of the matrix should uniquely define the year of origin of the losses in that row. Losses may be inception-to-date or incremental.
The "ages" of the triangle can be "phase shifted" --
i.e., the first age need not be as at the end of the origin period.
(See the Examples section.)
Nor need the "ages" be uniformly spaced.
However, when the ages are not uniformly spaced,
it would be prudent to specify the origin.width argument.
If TRUE (the default), values in Triangle are
inception to date.
If FALSE, Triangle holds incremental losses.
The "ultimate" age to which losses should be projected.
If TRUE (the default), the growth function should be applied
to the length of time from the average date of loss ("adol")
of losses in the origin year.
If FALSE, the growth function should be applied
to the length of time since the beginning of the origin year.
Only pertinent if adol is TRUE.
The age of the average date of losses within an origin period
in the same units as the "ages" of the Triangle matrix.
If NULL (the default) it will be assumed to be half the width
of an origin period (which would be the case if losses can be assumed
to occur uniformly over an origin period).
Only pertinent if adol is TRUE.
The width of an origin period
in the same units as the "ages" of the Triangle matrix.
If NULL (the default) it will be assumed to be the mean difference
in the "ages" of the triangle,
with a warning if not all differences are equal.
A character scalar identifying the "growth function."
The two growth functions defined at this time are "loglogistic"
(the default)
and "weibull".
A list of class "ClarkLDF" with the components listed below.
("Key" to naming convention: all caps represent parameters;
mixed case represent origin-level amounts;
all-lower-case represent observation-level (origin, development age) results.)
"LDF"
name of the growth function
names of the rows of the triangle
the most mature value for each row
the most mature "age" for each row
the most mature age used; differs from "CurrentAge" when adol=TRUE
same as 'maxage' argument
the maximum age for development from the average date of loss; differs from MAXAGE when adol=TRUE
the projected loss amounts ("Reserves" in Clark's paper)
the process standard error of the FutureValue
the parameter standard error of the FutureValue
the total standard error (process + parameter) of the FutureValue
a list with amounts that appear on the "Total" row
for components "Origin" (="Total"), "CurrentValue", "FutureValue",
"ProcessSE", "ParameterSE", and "StdError"
the estimated parameters
the estimated parameters for the "ultimate loss" by origin year ("U" in Clark's notation)
the estimated parameters of the growth function
value of the growth function as of the CurrentAge.used
value of the growth function as of the MAXAGE.used
the estimate of the sigma^2 parameter
the "to-ultimate" loss development factor (sometimes called the "cumulative development factor") as defined in Clark's paper for each origin year
the "to-ultimate" loss development factor as of the maximum age used in the model
the "truncated" loss development factor for developing the current diagonal to the maximum age used in the model
the gradient of the FutureValue function
the origin year corresponding to each observed value of incremental loss
the age of each observed value of incremental loss
the expected value of each observed value of incremental loss (the "mu's" of Clark's paper)
the actual minus fitted value for each observed incremental loss
the standardized residuals for each observed incremental loss (= residuals/sqrt(sigma2*fitted), referred to as "normalized residuals" in Clark's paper; see p. 62)
the "Fisher Information" matrix as defined in Clark's paper (i.e., without the sigma^2 value)
the value of the loglikelihood function at the solution point
the number of calls to the loglikelihood function and its gradient function when numerical convergence was achieved
Clark's "LDF method" assumes that the incremental losses across development periods in a loss triangle are independent. He assumes that the expected value of an incremental loss is equal to the theoretical expected ultimate loss (U) (by origin year) times the change in the theoretical underlying growth function over the development period. Clark models the growth function, also called the percent of ultimate, by either the loglogistic function (a.k.a., "the inverse power curve") or the weibull function. Clark completes his incremental loss model by wrapping the expected values within an overdispersed poisson (ODP) process where the "scale factor" sigma^2 is assumed to be a known constant for all development periods.
The parameters of Clark's "LDF method" are therefore: U, and omega and theta (the parameters of the loglogistic and weibull growth functions). Finally, Clark uses maximum likelihood to parameterize his model, uses the ODP process to estimate process risk, and uses the Cramer-Rao theorem and the "delta method" to estimate parameter risk.
Clark recommends inspecting the residuals to help assess the
reasonableness of the model relative to the actual data
(see plot.clark below).
Clark, David R., "LDF Curve-Fitting and Stochastic Reserving: A Maximum Likelihood Approach", Casualty Actuarial Society Forum, Fall, 2003 https://www.casact.org/sites/default/files/database/forum_03fforum_03ff041.pdf
# NOT RUN {
X <- GenIns
ClarkLDF(X, maxage=20)
# Clark's "LDF method" also works with triangles that have
# more development periods than origin periods
ClarkLDF(qincurred, G="loglogistic")
# Method also works for a "triangle" with only one row:
# 1st row of GenIns; need "drop=FALSE" to avoid becoming a vector.
ClarkLDF(GenIns[1, , drop=FALSE], maxage=20)
# The age of the first evaluation may be prior to the end of the origin period.
# Here the ages are in units of "months" and the first evaluation
# is at the end of the third quarter.
X <- GenIns
colnames(X) <- 12 * as.numeric(colnames(X)) - 3
# The indicated liability increases from 1st example above,
# but not significantly.
ClarkLDF(X, maxage=240)
# When maxage is infinite, the phase shift has a more noticeable impact:
# a 4-5% increase of the overall CV.
x <- ClarkLDF(GenIns, maxage=Inf)
y <- ClarkLDF(X, maxage=Inf)
# Percent change in the bottom line CV:
(tail(y$Table65$TotalCV, 1) - tail(x$Table65$TotalCV, 1)) / tail(x$Table65$TotalCV, 1)
# }
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