Clark LDF method

Analyze 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.width argument.

cumulative

If TRUE (the default), values in Triangle are inception to date. If FALSE, Triangle holds 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. If FALSE, the growth function should be applied to the length of time since the beginning of the origin year.

adol.age

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).

origin.width

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.

G

A character scalar 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 list with 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

Author

Daniel Murphy

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)