A recent paper in the journal

*Variance*demonstrated the existence of a broader family of Chain-Ladder Factor Models (CLFM) to which, under certain conditions, many more selected factors conform. If those conditions are satisfied, the selected factor may be considered the "best linear unbiased estimate" (BLUE) of a member of the family, and uncertainty estimates of the chain-ladder central estimates are

__directly applicable__. This post will demonstrate how the ChainLadder package implements CLFM for estimating chain-ladder method uncertainty when link ratios are "selected." (Note: the CLFM's psi function is not yet fully implemented in ChainLadder.)

As with a previous post, our example will consider data from the Institute and Faculty of Actuaries

*Claims Reserving Manual*, Section F, "Case Estimates & the Projection of Incurred Claims", p. 9:
> ReportedLossTriangle <- matrix(c(

2777, 3264, 3452, 3594, 3719, 3717,

3252, 3804, 3973, 4231, 4319, NA,

3725, 4404, 4779, 4946, NA, NA,

4521, 5422, 5676, NA, NA, NA,

5369, 6142, NA, NA, NA, NA,

>
library(ChainLadder)

The observed link ratios, along with the simple and volume-weighted averages, are

> ata(ReportedLossTriangle)

Age

Accident Year 12-24 24-36 36-48 48-60 60-72

2007 1.175 1.058 1.041 1.035 0.999

2008 1.170 1.044 1.065 1.021 NA

2009 1.182 1.085 1.035 NA NA

2010 1.199 1.047 NA NA NA

2011 1.144 NA NA NA NA

smpl 1.174 1.059 1.047 1.028 0.999

vwtd 1.173 1.058 1.046 1.027 0.999

Suppose we made the following selections:

> selected <- c(1.175, 1.059, 1.045, 1.03, .999)

Then we can find each link ratio's "consistent" member of the CLFM family (i.e., the model whose expected value equals the selected factor) with the 'CLFMdelta' function

> CLFMdelta(ReportedLossTriangle, selected)

12 24 36 48 60

3 2 -1 6 1

attr(,"foundSolution")

[1] TRUE TRUE TRUE TRUE TRUE

**Some background**: Each member of the CLFM family is distinguished by the exponent of the beginning value of loss that impacts the variance of the age-to-age projection. The exponent need not be an integer. The fact that the attribute "foundSolution" is all TRUE indicates that a member was successfully found for each element of "selected" relative to the ReportedLossTriangle. A "solution" is not guaranteed. For example, CLFMdelta cannot find a solution for a 24-36 selection of 1.06. Refer to the paper for the necessary conditions.

To continue, save the "solution" (i.e., the index to the CLFM member model) in an R variable:

> delta <- CLFMdelta(ReportedLossTriangle, selected)

Now call the MackChainLadder function with the method's "alpha" parameter equal to 2 - delta. (

**Aside**: The CLFM paper formulates the exponent a-la Murphy and Barnett&Zehnwirth; ChainLadder uses Mack's formulation. "CLFMalpha" would have been a more appropriate name relative to the CLFM paper, but ChainLadder's choice of notation had already been made.)

> MackChainLadder(ReportedLossTriangle, alpha = 2 - delta, est.sigma = "Mack", mse.method = "Independence")

MackChainLadder(Triangle = ReportedLossTriangle, alpha = 2 -

delta, est.sigma = "Mack", mse.method = "Independence")

Latest Dev.To.Date Ultimate IBNR Mack.S.E CV(IBNR)

2007 3,717 1.000 3,717 0.00 0.00e+00 NaN

2008 4,319 1.001 4,317 -2.32 9.06e-21 -3.90e-21

2009 4,946 0.971 5,092 145.61 8.33e+01 5.72e-01

2010 5,676 0.930 6,106 429.87 1.59e+02 3.70e-01

2011 6,142 0.878 6,994 851.72 2.59e+02 3.05e-01

2012 5,818 0.747 7,784 1,966.45 3.64e+02 1.85e-01

Totals

Latest: 30,618.00

Dev: 0.90

Ultimate: 34,009.32

IBNR: 3,391.32

Mack S.E.: 532.88

CV(IBNR): 0.16

In this case, the central estimates for Ultimate Loss and IBNR will be exactly equal to what the actuary would produce under the deterministic Chain-Ladder Method with those selected factors. Furthermore, the total standard error, 532.88, is immediately applicable to the total estimates of Ultimate and IBNR.

**Caveats:**

- Of course, the standard error does not account for model error. Residuals should be analyzed for potential model misspecification.
- The "psi function" of the CLFM approach has not yet been implemented in ChainLadder. This is a future enhancement currently being considered by the package developers. Contributors are welcome!

- dmm

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ChainLadder is a package for the R statistical environment that contains various functions for performing loss reserving for Property/Casualty/General Insurance.

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