# Statistical Methods for the Chain Ladder Technique Revisited

January 15, 2012
By

(This article was first published on PR, and kindly contributed to R-bloggers)

Background

Forecasting outstanding claims and setting up suitable reserves to meet these claims is an important part of the business of a general insurance company. Indeed, the published profits of these companies depend not only on the actual claims paid, but on the forecasts of the claims which will have to he paid. It is essential, therefore, that a reliable estimate is available of the reserve to be set aside to cover claims, in order to ensure the financial stability of the company and its profit and loss account. There are a number of methods which have proved useful in practice, one of which is extensively used and is known as the chain ladder technique.

Chain ladder method is a statistical method of estimating outstanding claims, whereby the weighted average of past claim development is projected into the future. The projection is based on the ratios of cumulative past claims, usually paid or incurred, for successive years of development. It requires the earliest year of origin to be fully run-off or at least that the final outcome for that year can be estimated with confidence. If appropriate, the method can be applied to past claims data that have been explicitly adjusted for past inflation.

In recent years, a statistical framework for analyzing this data has been built up, which encompasses the actuarial method, extending and consolidating it. We hope to bring together there results and to illustrate how the chain ladder technique can be improved and extended, without altering the basic foundations upon which it has been built.

These improvements are designed to overcome two problems with the chain ladder technique. Firstly, that not enough connection is made between the accident years, resulting in an over-parametrized model and unstable forecasts. Secondly, that the development pattern is assumed to be the same for all accident years. No allowance is made by the chain ladder technique for any change in the speed with which Aims are settled, or for any other factors which may change the shape of the run-off pattern.

R Package used

ChainLadder (google code name chainladder) is an R package providing methods which are typically used in insurance claims reserving. The package started out of presentations given at the Stochastic Reserving Seminar at the Institute of Actuaries in 2007 and 2008, followed by talks at CAS meetings in 2008 and 2010. The package started with implementations of the Mack-, Munich- and Bootstrap-chain-ladder methods. Since version 0.1.3-3 it also provides general multivariate chain ladder models by Wayne Zhang. Version 0.1.4-0 introduces new functions on LDF fitting and Cape Cod by Daniel Murphy following a paper by David Clark. Version 0.1.5-0 has added loss reserving models within the generalized linear model framework following a paper by England P. and Verrall R. (1999) implemented by Wayne Zhang.

The package offers also some utility functions to convert quickly tables into triangles, triangles into tables, cumulative into incremental and incremental into cumulative triangles.

The ChainLadder-package comes with an example spreadsheet which demonstrates how to use the ChainLadder functions in Excel. The spreadsheet is located in the Excel folder of the package. The R command system.file(“Excel”, package=”ChainLadder”) will tell you the exact path to the directory. To use the spreadsheet you will need the RExcel-Addin. The package also provides an example SWord file, demonstrating how the the functions of the package can be integrated into a MS Word file via SWord. Again you find the Word file via the command: system.file(“SWord”, package=”ChainLadder”)

Dataset

General Insurance companies will sell insurance policies and receive claims everyday. These claims will be indexed by their business year and the delay.

We will use the following dataset, RAA for illustrative purpose.

RAA is a dataset of Run-off triangle of Automatic Facilitative business in General Liability in a matrix with 10 accident years and 10 development year, taken from Historical Loss Development, Reinsurance Association of America (RAA), 1991, p.96.

Functions

The Mack-chain-ladder model forecasts future claims developments based on a historical cumulative claims development triangle and estimates the standard error around those.

The Mack-chain-ladder model can be regarded as a weighted linear regression through the origin for each development period: lm(y ~ x + 0, weights=w/x^(2-alpha)), where y is the vector of claims at development period k + 1 and x is the vector of claims at development period k.

The BootChainLadder procedure provides a predictive distribution of reserves or IBNRs for accumulative claims development triangle.

The BootChainLadder function uses a two-stage bootstrapping/simulation approach. In the first stage an ordinary chain-ladder methods is applied to the cumulative claims triangle. From this we calculate the scaled Pearson residuals which we bootstrap R times to forecast future incremental claims payments via the standard chain-ladder method. In the second stage we simulate the process error with the bootstrap value as the mean and using the process distribution assumed. The set of reserves obtained in this way forms the predictive distribution, from which summary statistics such as mean, prediction error or quartiles can be derived.

### Code

```library(ChainLadder)
RAA
plot(RAA)
plot(RAA, lattice=TRUE)

M = MackChainLadder(Triangle = RAA, est.sigma = "Mack")
M
plot(M)
plot(M, lattice=TRUE)

set.seed(1)
B = BootChainLadder(Triangle = RAA, R = 999, process.distr = "od.pois")
B
plot(B)

predict(RAA2)				```

Result:

### Output

```					> library(ChainLadder)
Hmisc library by Frank E Harrell Jr
Type library(help="Hmisc"), ?Overview, or ?Hmisc.Overview")
to see overall documentation.
NOTE:Hmisc no longer redefines [.factor to drop unused levels when
subsetting.  To get the old behavior of Hmisc type dropUnusedLevels().
Attaching package: "Hmisc"
The following object(s) are masked from "package:survival":
untangle.specials
The following object(s) are masked from "package:base":
format.pval, round.POSIXt, trunc.POSIXt, units
Attaching package: "Matrix"
The following object(s) are masked from "package:base":
det
Attaching package: "car"
The following object(s) are masked from "package:Hmisc":
recode
Attaching package: "zoo"
The following object(s) are masked from "package:base":
as.Date, as.Date.numeric
Attaching package: "actuar"
The following object(s) are masked from "package:grDevices":
cm
Markus Gesmann
Wayne Zhang
Daniel Murphy
to see overall documentation.
Type demo(ChainLadder) to get an idea of the functionality of this package.
See demo(package="ChainLadder") for a list of more demos.
Feel free to send us an email if you would like to keep informed of
new versions or if you have any feedback, ideas, suggestions or would
like to collaborate.
To suppress this message use the statement:
> RAA
dev
origin    1     2     3     4     5     6     7     8     9    10
1981 5012  8269 10907 11805 13539 16181 18009 18608 18662 18834
1982  106  4285  5396 10666 13782 15599 15496 16169 16704    NA
1983 3410  8992 13873 16141 18735 22214 22863 23466    NA    NA
1984 5655 11555 15766 21266 23425 26083 27067    NA    NA    NA
1985 1092  9565 15836 22169 25955 26180    NA    NA    NA    NA
1986 1513  6445 11702 12935 15852    NA    NA    NA    NA    NA
1987  557  4020 10946 12314    NA    NA    NA    NA    NA    NA
1988 1351  6947 13112    NA    NA    NA    NA    NA    NA    NA
1989 3133  5395    NA    NA    NA    NA    NA    NA    NA    NA
1990 2063    NA    NA    NA    NA    NA    NA    NA    NA    NA
> filename <- paste(tempfile(tmpdir="C:/R/tmp"), ".png", sep="")
> png(file=filename)
[1] "C:/R/tmp\\file5ba83bc5.png"
> filename <- paste(tempfile(tmpdir="C:/R/tmp"), ".png", sep="")
> png(file=filename)
[1] "C:/R/tmp\\filefe13c35.png"
>
> M = MackChainLadder(Triangle = RAA, est.sigma = "Mack")
> M
MackChainLadder(Triangle = RAA, est.sigma = "Mack")
Latest Dev.To.Date Ultimate   IBNR Mack.S.E CV(IBNR)
1981 18,834       1.000   18,834      0        0      NaN
1982 16,704       0.991   16,858    154      206    1.339
1983 23,466       0.974   24,083    617      623    1.010
1984 27,067       0.943   28,703  1,636      747    0.457
1985 26,180       0.905   28,927  2,747    1,469    0.535
1986 15,852       0.813   19,501  3,649    2,002    0.549
1987 12,314       0.694   17,749  5,435    2,209    0.406
1988 13,112       0.546   24,019 10,907    5,358    0.491
1989  5,395       0.336   16,045 10,650    6,333    0.595
1990  2,063       0.112   18,402 16,339   24,566    1.503
Totals
Dev:                    0.76
Ultimate:         213,122.23
IBNR:              52,135.23
Mack S.E.:         26,909.01
CV(IBNR):  0.516138742518263
> filename <- paste(tempfile(tmpdir="C:/R/tmp"), ".png", sep="")
> png(file=filename)
[1] "C:/R/tmp\\file34325318.png"
> filename <- paste(tempfile(tmpdir="C:/R/tmp"), ".png", sep="")
> png(file=filename)
[1] "C:/R/tmp\\file3e8f2054.png"
>
> set.seed(1)
> B = BootChainLadder(Triangle = RAA, R = 999, process.distr = "od.pois")
> B
BootChainLadder(Triangle = RAA, R = 999, process.distr = "od.pois")
Latest Mean Ultimate Mean IBNR SD IBNR IBNR 75% IBNR 95%
1981 18,834        18,834         0       0        0        0
1982 16,704        16,921       217     710      253    1,597
1983 23,466        24,108       642   1,340    1,074    3,205
1984 27,067        28,739     1,672   1,949    2,679    4,980
1985 26,180        29,077     2,897   2,467    4,149    7,298
1986 15,852        19,611     3,759   2,447    4,976    8,645
1987 12,314        17,724     5,410   3,157    7,214   11,232
1988 13,112        24,219    11,107   5,072   14,140   20,651
1989  5,395        16,119    10,724   6,052   14,094   21,817
1990  2,063        18,714    16,651  13,426   24,459   42,339
Totals
Latest:         160,987
Mean Ultimate:  214,066
Mean IBNR:       53,079
SD IBNR:         18,884
Total IBNR 75%:  64,788
Total IBNR 95%:  88,037
> filename <- paste(tempfile(tmpdir="C:/R/tmp"), ".png", sep="")
> png(file=filename)
[1] "C:/R/tmp\\file11e7492a.png"
>
> RAA2 = chainladder(RAA, weights=RAA, delta=1)
> predict(RAA2)
dev
origin    1         2         3         4        5        6        7        8
1981 5012  8269.000 10907.000 11805.000 13539.00 16181.00 18009.00 18608.00
1982  106  4285.000  5396.000 10666.000 13782.00 15599.00 15496.00 16169.00
1983 3410  8992.000 13873.000 16141.000 18735.00 22214.00 22863.00 23466.00
1984 5655 11555.000 15766.000 21266.000 23425.00 26083.00 27067.00 27938.45
1985 1092  9565.000 15836.000 22169.000 25955.00 26180.00 27241.19 28118.25
1986 1513  6445.000 11702.000 12935.000 15852.00 17432.56 18139.18 18723.19
1987  557  4020.000 10946.000 12314.000 14308.52 15735.19 16373.00 16900.15
1988 1351  6947.000 13112.000 16532.776 19210.62 21126.06 21982.39 22690.14
1989 3133  5395.000  8464.494 10672.786 12401.48 13638.00 14190.80 14647.69
1990 2063  4574.169  7176.649  9048.957 10514.63 11563.02 12031.72 12419.09
dev
origin        9       10
1981 18662.00 18834.00
1982 16704.00 16857.95
1983 23838.84 24058.55
1984 28382.35 28643.94
1985 28565.00 28828.28
1986 19020.67 19195.98
1987 17168.66 17326.90
1988 23050.65 23263.10
1989 14880.42 15017.57
1990 12616.41 12732.69
>
```

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