Clark LDF method
ClarkLDF.RdAnalyze loss triangle using Clark's LDF (loss development factor) method.
Usage
ClarkLDF(Triangle, cumulative = TRUE, maxage = Inf,
adol = TRUE, adol.age = NULL, origin.width = NULL,
G = "loglogistic")Arguments
- Triangle
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.widthargument.- cumulative
If
TRUE(the default), values inTriangleare inception to date. IfFALSE,Triangleholds incremental losses.- maxage
The "ultimate" age to which losses should be projected.
- adol
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. IfFALSE, the growth function should be applied to the length of time since the beginning of the origin year.- adol.age
Only pertinent if
adolisTRUE. The age of the average date of losses within an origin period in the same units as the "ages" of theTrianglematrix. IfNULL(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).- origin.width
Only pertinent if
adolisTRUE. The width of an origin period in the same units as the "ages" of theTrianglematrix. IfNULL(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.- G
A
characterscalar identifying the "growth function." The two growth functions defined at this time are "loglogistic" (the default) and "weibull".
Details
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).
Value
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.)
- method
"LDF"
- growthFunction
name of the growth function
- Origin
names of the rows of the triangle
- CurrentValue
the most mature value for each row
- CurrentAge
the most mature "age" for each row
- CurrentAge.used
the most mature age used; differs from "CurrentAge" when adol=TRUE
- MAXAGE
same as 'maxage' argument
- MAXAGE.USED
the maximum age for development from the average date of loss; differs from MAXAGE when adol=TRUE
- FutureValue
the projected loss amounts ("Reserves" in Clark's paper)
- ProcessSE
the process standard error of the FutureValue
- ParameterSE
the parameter standard error of the FutureValue
- StdError
the total standard error (process + parameter) of the FutureValue
- Total
a
listwith amounts that appear on the "Total" row for components "Origin" (="Total"), "CurrentValue", "FutureValue", "ProcessSE", "ParameterSE", and "StdError"- PAR
the estimated parameters
- THETAU
the estimated parameters for the "ultimate loss" by origin year ("U" in Clark's notation)
- THETAG
the estimated parameters of the growth function
- GrowthFunction
value of the growth function as of the CurrentAge.used
- GrowthFunctionMAXAGE
value of the growth function as of the MAXAGE.used
- SIGMA2
the estimate of the sigma^2 parameter
- Ldf
the "to-ultimate" loss development factor (sometimes called the "cumulative development factor") as defined in Clark's paper for each origin year
- LdfMAXAGE
the "to-ultimate" loss development factor as of the maximum age used in the model
- TruncatedLdf
the "truncated" loss development factor for developing the current diagonal to the maximum age used in the model
- FutureValueGradient
the gradient of the FutureValue function
- origin
the origin year corresponding to each observed value of incremental loss
- age
the age of each observed value of incremental loss
- fitted
the expected value of each observed value of incremental loss (the "mu's" of Clark's paper)
- residuals
the actual minus fitted value for each observed incremental loss
- stdresid
the standardized residuals for each observed incremental loss (= residuals/sqrt(sigma2*fitted), referred to as "normalized residuals" in Clark's paper; see p. 62)
- FI
the "Fisher Information" matrix as defined in Clark's paper (i.e., without the sigma^2 value)
- value
the value of the loglikelihood function at the solution point
- counts
the number of calls to the loglikelihood function and its gradient function when numerical convergence was achieved
References
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
Examples
X <- GenIns
ClarkLDF(X, maxage=20)
#> Origin CurrentValue Ldf UltimateValue FutureValue StdError CV%
#> 1 3,901,463 1.171 4,567,994 666,531 261,622 39.3
#> 2 5,339,085 1.217 6,496,508 1,157,423 375,333 32.4
#> 3 4,909,315 1.278 6,274,401 1,365,086 420,492 30.8
#> 4 4,588,268 1.363 6,253,962 1,665,694 483,350 29.0
#> 5 3,873,311 1.487 5,759,792 1,886,481 530,086 28.1
#> 6 3,691,712 1.681 6,206,592 2,514,880 653,577 26.0
#> 7 3,483,130 2.018 7,029,915 3,546,785 847,276 23.9
#> 8 2,864,498 2.709 7,760,999 4,896,501 1,113,865 22.7
#> 9 1,363,294 4.661 6,354,929 4,991,635 1,344,652 26.9
#> 10 344,014 19.091 6,567,720 6,223,706 2,892,103 46.5
#> Total 34,358,090 63,272,813 28,914,723 4,848,938 16.8
# Clark's "LDF method" also works with triangles that have
# more development periods than origin periods
ClarkLDF(qincurred, G="loglogistic")
#> Origin CurrentValue Ldf UltimateValue FutureValue StdError CV%
#> 1995 1,100 1.006 1,107 7 17 242.9
#> 1996 1,300 1.008 1,310 10 21 200.1
#> 1997 1,200 1.010 1,212 12 23 184.0
#> 1998 1,298 1.014 1,316 18 27 154.5
#> 1999 1,583 1.019 1,613 30 36 120.5
#> 2000 1,066 1.027 1,095 29 35 122.3
#> 2001 1,411 1.042 1,470 59 51 87.4
#> 2002 1,820 1.070 1,948 128 78 61.2
#> 2003 1,221 1.138 1,389 168 92 54.7
#> 2004 1,212 1.352 1,638 426 162 38.0
#> 2005 422 2.643 1,115 693 280 40.4
#> 2006 13 339.514 4,414 4,401 7,891 179.3
#> Total 13,646 19,627 5,981 7,891 131.9
# 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)
#> Origin CurrentValue Ldf UltimateValue FutureValue StdError CV%
#> 1 3,901,463 1.176 4,589,676 688,213 334,290 48.6
#> Total 3,901,463 4,589,676 688,213 334,290 48.6
# 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)
#> Origin CurrentValue Ldf UltimateValue FutureValue StdError CV%
#> 1 3,901,463 1.189 4,638,605 737,142 277,588 37.7
#> 2 5,339,085 1.237 6,605,692 1,266,607 397,311 31.4
#> 3 4,909,315 1.301 6,386,651 1,477,336 442,507 30.0
#> 4 4,588,268 1.388 6,369,609 1,781,341 505,579 28.4
#> 5 3,873,311 1.514 5,865,118 1,991,807 550,324 27.6
#> 6 3,691,712 1.710 6,311,201 2,619,489 673,463 25.7
#> 7 3,483,130 2.047 7,128,319 3,645,189 866,256 23.8
#> 8 2,864,498 2.742 7,853,050 4,988,552 1,131,966 22.7
#> 9 1,363,294 4.810 6,556,934 5,193,640 1,389,410 26.8
#> 10 344,014 19.039 6,549,543 6,205,529 2,862,366 46.1
#> Total 34,358,090 64,264,722 29,906,632 4,983,456 16.7
# 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)
#> numeric(0)