Package for carrying out actuarial reserving
This Reserving package is a small library for carrying out reserving. At the moment it simply consists of the chain ladder function as described by the paper "Distribution-free calculation of the standard error of chain-ladder reserve estimates" by Thomas Mack ASTIN Bulletin, Vol. 23, No. 2, 1993.
It is hoped that at some point this package will be available in the Julia repository so that it can be installed using Pkg.add("Reserving"). Till then the package can be unpackaged into the ".julia/v0.3/" folder holding the packages and then use
Pkg.build("Reserving") # to build the package
using Reserving # to load the package
There are three datasets included in this package:
Two data sets from Thomas Mack's paper mentioned in the Overview
julia> mtr1
10x10 Array{Float64,2}:
357848.0 1.12479e6 1.73533e6 2.21827e6 … 3.60629e6 3.83352e6 3.90146e6
352118.0 1.23614e6 2.17003e6 3.35332e6 4.91404e6 5.33909e6 NaN
290507.0 1.29231e6 2.21853e6 3.0e6 4.90932e6 NaN NaN
310608.0 1.41886e6 2.19505e6 3.75745e6 NaN NaN NaN
443160.0 1.13635e6 2.12833e6 2.89782e6 NaN NaN NaN
396132.0 1.33322e6 2.18072e6 2.98575e6 … NaN NaN NaN
440832.0 1.28846e6 2.41986e6 3.48313e6 NaN NaN NaN
359480.0 1.42113e6 2.8645e6 NaN NaN NaN NaN
376686.0 1.36329e6 NaN NaN NaN NaN NaN
344014.0 NaN NaN NaN NaN NaN NaN
julia> mtr2
9x9 Array{Float64,2}:
58046.0 127970.0 476599.0 1.02769e6 … 1.81918e6 1.90685e6 1.95011e6
24492.0 141767.0 984288.0 2.14266e6 4.0489e6 4.11576e6 NaN
32848.0 274682.0 1.52264e6 3.20343e6 5.34259e6 NaN NaN
21439.0 529828.0 2.9003e6 4.99902e6 NaN NaN NaN
40397.0 763394.0 2.92075e6 4.98957e6 NaN NaN NaN
90748.0 951994.0 4.21064e6 5.86648e6 … NaN NaN NaN
62096.0 868480.0 1.9548e6 NaN NaN NaN NaN
24983.0 284441.0 NaN NaN NaN NaN NaN
13121.0 NaN NaN NaN NaN NaN NaN
and paid auto claims dataset from R's ChainLadder package by Markus Gesmann, Daniel Murphy,Wayne Zhang
julia> auto1
10x10 Array{Float64,2}:
101125.0 209921.0 266618.0 305107.0 327850.0 340669.0 348430.0 351193.0 353353.0 353584.0
102541.0 203213.0 260677.0 303182.0 328932.0 340948.0 347333.0 349813.0 350523.0 NaN
114932.0 227704.0 298120.0 345542.0 367760.0 377999.0 383611.0 385224.0 NaN NaN
114452.0 227761.0 301072.0 340669.0 359979.0 369248.0 373325.0 NaN NaN NaN
115597.0 243611.0 315215.0 354490.0 372376.0 382738.0 NaN NaN NaN NaN
127760.0 259416.0 326975.0 365780.0 386725.0 NaN NaN NaN NaN NaN
135616.0 262294.0 327086.0 367357.0 NaN NaN NaN NaN NaN NaN
127177.0 244249.0 317972.0 NaN NaN NaN NaN NaN NaN NaN
128631.0 246803.0 NaN NaN NaN NaN NaN NaN NaN NaN
126288.0 NaN NaN NaN NaN NaN NaN NaN NaN NaN
The package consists of simply the Chain Ladder algorithm as described in the Overview and analysis revolves around a chain ladder CL object. The CL constructor can accept an Array{Float64,2} containing a claims triangle and Array{Float64,1} containing a claims triangle flattened to 1D. Why flattened? Because the C-code that carries out the analysis crunches 1D flattened triangles.
Here we create a CL object:
julia> ladder1 = CL(mtr1)
Claims triangle of size: 10
10x10 Array{Float64,2}:
357848.0 1.12479e6 1.73533e6 2.21827e6 … 3.60629e6 3.83352e6 3.90146e6
352118.0 1.23614e6 2.17003e6 3.35332e6 4.91404e6 5.33909e6 NaN
290507.0 1.29231e6 2.21853e6 3.0e6 4.90932e6 NaN NaN
310608.0 1.41886e6 2.19505e6 3.75745e6 NaN NaN NaN
443160.0 1.13635e6 2.12833e6 2.89782e6 NaN NaN NaN
396132.0 1.33322e6 2.18072e6 2.98575e6 … NaN NaN NaN
440832.0 1.28846e6 2.41986e6 3.48313e6 NaN NaN NaN
359480.0 1.42113e6 2.8645e6 NaN NaN NaN NaN
376686.0 1.36329e6 NaN NaN NaN NaN NaN
344014.0 NaN NaN NaN NaN NaN NaN
Dev Factors:
1x9 Array{Float64,2}:
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Dev Factor S.E.
1x9 Array{Float64,2}:
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Sigma
1x9 Array{Float64,2}:
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Reserve
1x10 Array{Float64,2}:
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Reserve S.E.
1x10 Array{Float64,2}:
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Total Reserve S.E.
0.0
Notice that the various parts of the CL object are filled out awaiting the analysis. Even though the claims triangle looks like a 10x10 array, it is actually stored as a 100x1 array already flattened before being analysed.
There are four ways to carry out the chainladder analysis
The first is to overwrite an existing CL object using the chainladder!() function:
julia> chainladder!(ladder1);
julia> ladder1
Claims triangle of size: 10
10x10 Array{Float64,2}:
357848.0 1.12479e6 1.73533e6 2.21827e6 2.7456e6 … 3.46634e6 3.60629e6 3.83352e6 3.90146e6
352118.0 1.23614e6 2.17003e6 3.35332e6 3.79907e6 4.64787e6 4.91404e6 5.33909e6 5.43372e6
290507.0 1.29231e6 2.21853e6 3.23518e6 3.986e6 4.62891e6 4.90932e6 5.28515e6 5.37883e6
310608.0 1.41886e6 2.19505e6 3.75745e6 4.02993e6 4.58827e6 4.83546e6 5.20564e6 5.29791e6
443160.0 1.13635e6 2.12833e6 2.89782e6 3.40267e6 4.20746e6 4.43413e6 4.77359e6 4.8582e6
396132.0 1.33322e6 2.18072e6 2.98575e6 3.69171e6 … 4.42655e6 4.66502e6 5.02216e6 5.11117e6
440832.0 1.28846e6 2.41986e6 3.48313e6 4.08868e6 4.90253e6 5.16665e6 5.56218e6 5.66077e6
359480.0 1.42113e6 2.8645e6 4.17476e6 4.90054e6 5.876e6 6.19256e6 6.66663e6 6.7848e6
376686.0 1.36329e6 2.38213e6 3.47174e6 4.07531e6 4.8865e6 5.14976e6 5.544e6 5.64227e6
344014.0 1.20082e6 2.09823e6 3.05798e6 3.58962e6 4.30413e6 4.53601e6 4.88327e6 4.96982e6
Dev Factors:
[3.49061,1.74733,1.45741,1.17385,1.10382,1.08627,1.05387,1.07656,1.01772]
Dev Factor S.E.
[0.219477,0.0606729,0.052809,0.0286883,0.027648,0.0226507,0.00592011,0.0116044,0.0107937]
Sigma
[400.35,194.26,204.854,123.219,117.181,90.4753,21.1333,33.8728,21.1333]
Reserve
[0.0,94633.8,469511.0,709638.0,984889.0,1.41946e6,2.17764e6,3.9203e6,4.27897e6,4.62581e6]
Reserve S.E.
[0.0,75535.0,121699.0,133549.0,261406.0,411010.0,558317.0,875328.0,971258.0,1.36315e6]
Total Reserve S.E.
2.4470948608346656e6
The second and third is to carry out the analysis from an Array{Float64,2} and an Array{Float64,1} object. The outputs will be mutted since you have seen this enough by now. The [:] below is used to flatten the 2D array.
chainladder(mtr1);
chainladder(mtr2[:]);
The last technique is to call use the chainladder() on a CL object (non-overwritten). In the example below, ladder2 remains unwritten and the output is captured in output.
ladder2 = CL(mtr2);
output = chainladder(ladder2);
CL objects can also be copied using the copy constructor:
output2 = copy(output);
The implementation details of the CL object are heald in the chainladder\cl.jl file. The components of the object are:
tri::Array{Float64,1}- the flattened trianglep::Int64- the size of the trianglef::Array{Float64,1}- development factorsf_se::Array{Float64,1}- standard error of development factorssigma::Array{Float64,1}- sigma as described in the Thomas Mack's paperres::Array{Float64,1}- the reserveres_se::Array{Float64,1}- reserve standard errortres_se::Float64- total reserve standard error
There are two ways of accessing any component of the object. The first and native is using the name:
julia> output.f
8-element Array{Float64,1}:
11.1043
4.09227
1.70791
1.27592
1.13891
1.0687
1.02633
1.02268
But you can also access it using the index:
julia> output[8]
3.7288702412577975e6
... use Symbolic names:
julia> output[:f]
8-element Array{Float64,1}:
11.1043
4.09227
1.70791
1.27592
1.13891
1.0687
1.02633
1.02268
... use String names
julia> output["f"]
8-element Array{Float64,1}:
11.1043
4.09227
1.70791
1.27592
1.13891
1.0687
1.02633
1.02268
There is a convenience function provided for when the use simply wants to extract the triangle from the object:
julia> tri(output)
9x9 Array{Float64,2}:
58046.0 127970.0 476599.0 1.02769e6 1.36049e6 1.64731e6 1.81918e6 1.90685e6 1.95011e6
24492.0 141767.0 984288.0 2.14266e6 2.96198e6 3.68394e6 4.0489e6 4.11576e6 4.20912e6
32848.0 274682.0 1.52264e6 3.20343e6 4.44593e6 5.15878e6 5.34259e6 5.48328e6 5.60766e6
21439.0 529828.0 2.9003e6 4.99902e6 6.46011e6 6.8539e6 7.32474e6 7.51764e6 7.68816e6
40397.0 763394.0 2.92075e6 4.98957e6 5.64856e6 6.43322e6 6.87516e6 7.05622e6 7.21627e6
90748.0 951994.0 4.21064e6 5.86648e6 7.48516e6 8.52494e6 9.11058e6 9.35051e6 9.5626e6
62096.0 868480.0 1.9548e6 3.33862e6 4.25982e6 4.85156e6 5.18484e6 5.32139e6 5.44209e6
24983.0 284441.0 1.16401e6 1.98803e6 2.53657e6 2.88893e6 3.08739e6 3.16869e6 3.24057e6
13121.0 145699.0 596240.0 1.01833e6 1.2993e6 1.47979e6 1.58145e6 1.6231e6 1.65991e6
We have gone to much effort to get ensure good performance from this package. The analysis itself is carried out in C, the code is located in deps/cl.c part of the package. Not only is it written in C, but as mentioned before the triangle has been flattened to one dimension as a result the calculation is very fast indeed
Below are the summary statistics for times in microseconds taken to run the chainladder algorithm on the mtr2 data. Bear in mind that the times include the full calculation of the items in the CL object.
ttimes = [@elapsed chainladder(mtr2) for i in 1:1000];
julia> quantile(ttimes, [0, .25, .5, .75, 1])'*1E6
1x5 Array{Float64,2}:
2.854 2.931 3.012 3.52825 12.229
We hope that this is just the beginning. We plan to include more reserving methods and options for the current method as time allows. The next obvious step is to expand the current chain ladder algorithm to that described by Thomas Mack's 1999 paper. For any suggestions or requests contact chibisi@activeanalytics.co.uk.