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Standard deviation of the claims development result after one year for the distribution-free chain-ladder model (Mack) and Bootstrap model.

Usage

CDR(x, ...)
# S3 method for class 'MackChainLadder'
CDR(x, dev=1, ...)
# S3 method for class 'BootChainLadder'
CDR(x, probs=c(0.75, 0.95), ...)
# Default S3 method
CDR(x, ...)

Arguments

x

otput of either MackChainLadder or BootChainLadder

dev

vector of development periods or "all". Currently only applicable for MackChainLadder output. Defines the years for which the run off claims development result should be returned.

probs

only applicable for BootChainLadder output. Define quantiles to be returned.

...

other arguments

Details

Merz & Wüthrich (2008) derived analytic formulae for the mean square error of prediction of the claims development result for the Mack chain-ladder model after one year assuming:

  • The opening reserves were set using the pure chain-ladder model (no tail)

  • Claims develop in the year according to the assumptions underlying Mack's model

  • Reserves are set after one year using the pure chain-ladder model (no tail)

Value

A data.frame with various IBNR/reserves and one-year statistics of the claims development result.

References

Michael Merz, Mario V. Wüthrich. Modelling the claims development result for solvency purposes. Casualty Actuarial Society E-Forum, Fall 2008.

Michael Merz, Mario V. Wüthrich. Claims Run-Off Uncertainty: The Full Picture. Swiss Finance Institute Research Paper No. 14-69. https://www.ssrn.com/abstract=2524352. 2014

Author

Mario Wüthrich and Markus Gesmann with contributions from Arthur Charpentier and Arnaud Lacoume for CDR.MackChainLadder and Giuseppe Crupi and Markus Gesmann for CDR.BootChainLadder.

Note

Tail factors are currently not supported.

See also

Examples

# Example from the 2008 Merz, Wuthrich paper mentioned above
MW2008
#>       dev
#> origin       1       2       3       4       5       6       7       8       9
#>      1 2202584 3210449 3468122 3545070 3621627 3644636 3669012 3674511 3678633
#>      2 2350650 3553023 3783846 3840067 3865187 3878744 3898281 3902425      NA
#>      3 2321885 3424190 3700876 3798198 3854755 3878993 3898825      NA      NA
#>      4 2171487 3165274 3395841 3466453 3515703 3548422      NA      NA      NA
#>      5 2140328 3157079 3399262 3500520 3585812      NA      NA      NA      NA
#>      6 2290664 3338197 3550332 3641036      NA      NA      NA      NA      NA
#>      7 2148216 3219775 3428335      NA      NA      NA      NA      NA      NA
#>      8 2143728 3158581      NA      NA      NA      NA      NA      NA      NA
#>      9 2144738      NA      NA      NA      NA      NA      NA      NA      NA
M <- MackChainLadder(MW2008, est.sigma="Mack")
plot(M)

CDR(M)
#>              IBNR CDR(1)S.E.   Mack.S.E.
#> 1           0.000     0.0000      0.0000
#> 2        4377.670   566.1744    566.1744
#> 3        9347.477  1486.5603   1563.8075
#> 4       28392.406  3923.0986   4157.2733
#> 5       51444.021  9722.8598  10536.4380
#> 6      111811.123 28442.6216  30319.4638
#> 7      187084.178 20954.2870  35967.0384
#> 8      411864.225 28119.3180  45090.1821
#> 9     1433505.008 53320.8210  69552.3397
#> Total 2237826.107 81080.5468 108401.3875
# Return all run-off result developments
CDR(M, dev="all")
#>              IBNR CDR(1)S.E. CDR(2)S.E. CDR(3)S.E. CDR(4)S.E. CDR(5)S.E.
#> 1           0.000     0.0000     0.0000     0.0000     0.0000     0.0000
#> 2        4377.670   566.1744     0.0000     0.0000     0.0000     0.0000
#> 3        9347.477  1486.5603   485.4195     0.0000     0.0000     0.0000
#> 4       28392.406  3923.0986  1305.9869   431.9915     0.0000     0.0000
#> 5       51444.021  9722.8598  3830.3960  1277.0783   423.8641     0.0000
#> 6      111811.123 28442.6216  9689.5440  3820.5641  1274.3261   423.4215
#> 7      187084.178 20954.2870 27423.5036  9340.4725  3684.4291  1229.1060
#> 8      411864.225 28119.3180 20421.8007 26951.8084  9178.4040  3621.3651
#> 9     1433505.008 53320.8210 27782.3071 20193.6339 26778.3919  9118.5624
#> Total 2237826.107 81080.5468 52222.0516 38517.4943 29104.1066 10109.0020
#>       CDR(6)S.E. CDR(7)S.E. CDR(8)S.E. CDR(9)S.E.   Mack.S.E.
#> 1         0.0000     0.0000     0.0000          0      0.0000
#> 2         0.0000     0.0000     0.0000          0    566.1744
#> 3         0.0000     0.0000     0.0000          0   1563.8075
#> 4         0.0000     0.0000     0.0000          0   4157.2733
#> 5         0.0000     0.0000     0.0000          0  10536.4380
#> 6         0.0000     0.0000     0.0000          0  30319.4638
#> 7       408.6786     0.0000     0.0000          0  35967.0384
#> 8      1208.1604   401.9014     0.0000          0  45090.1821
#> 9      3598.2399  1200.5081   399.4584          0  69552.3397
#> Total  3876.0093  1281.3024   399.4584          0 108401.3875

# Example from the 2014 Merz, Wuthrich paper mentioned above
MW2014
#>       dev
#> origin     0     1     2     3     4     5     6     7     8     9    10    11
#>     1  13109 20355 21337 22043 22401 22658 22997 23158 23492 23664 23699 23904
#>     2  14457 22038 22627 23114 23238 23312 23440 23490 23964 23976 24048 24111
#>     3  16075 22672 23753 24052 24206 24757 24786 24807 24823 24888 24986 25401
#>     4  15682 23464 24465 25052 25529 25708 25752 25770 25835 26075 26082 26146
#>     5  16551 23706 24627 25573 26046 26115 26283 26481 26701 26718 26724 26728
#>     6  15439 23796 24866 25317 26139 26154 26175 26205 26764 26818 26836 26959
#>     7  14629 21645 22826 23599 24992 25434 25476 25549 25604 25709 25723    NA
#>     8  17585 26288 27623 27939 28335 28638 28715 28759 29525 30302    NA    NA
#>     9  17419 25941 27066 27761 28043 28477 28721 28878 28948    NA    NA    NA
#>     10 16665 25370 26909 27611 27729 27861 29830 29844    NA    NA    NA    NA
#>     11 15471 23745 25117 26378 26971 27396 27480    NA    NA    NA    NA    NA
#>     12 15103 23393 26809 27691 28061 29183    NA    NA    NA    NA    NA    NA
#>     13 14540 22642 23571 24127 24210    NA    NA    NA    NA    NA    NA    NA
#>     14 14590 22336 23440 24029    NA    NA    NA    NA    NA    NA    NA    NA
#>     15 13967 21515 22603    NA    NA    NA    NA    NA    NA    NA    NA    NA
#>     16 12930 20111    NA    NA    NA    NA    NA    NA    NA    NA    NA    NA
#>     17 12539    NA    NA    NA    NA    NA    NA    NA    NA    NA    NA    NA
#>       dev
#> origin    12    13    14    15    16
#>     1  23960 23992 23994 24001 24002
#>     2  24252 24538 24540 24550    NA
#>     3  25681 25705 25732    NA    NA
#>     4  26150 26167    NA    NA    NA
#>     5  26735    NA    NA    NA    NA
#>     6     NA    NA    NA    NA    NA
#>     7     NA    NA    NA    NA    NA
#>     8     NA    NA    NA    NA    NA
#>     9     NA    NA    NA    NA    NA
#>     10    NA    NA    NA    NA    NA
#>     11    NA    NA    NA    NA    NA
#>     12    NA    NA    NA    NA    NA
#>     13    NA    NA    NA    NA    NA
#>     14    NA    NA    NA    NA    NA
#>     15    NA    NA    NA    NA    NA
#>     16    NA    NA    NA    NA    NA
#>     17    NA    NA    NA    NA    NA
W <- MackChainLadder(MW2014, est.sigma="Mack")
plot(W)

CDR(W)
#>               IBNR   CDR(1)S.E.    Mack.S.E.
#> 1         0.000000    0.0000000    0.0000000
#> 2         1.022874    0.4083149    0.4083149
#> 3        10.085643    2.5393857    2.5652899
#> 4        21.187574   16.7232632   16.8984949
#> 5       117.662565  156.4022713  157.2756452
#> 6       223.279748  137.6522771  207.1650862
#> 7       361.808180  171.1812092  261.9266093
#> 8       469.408830   70.3161155  292.2622285
#> 9       653.504225  271.6352221  390.5874717
#> 10     1008.763182  310.1268449  502.0606072
#> 11     1011.859648  103.3834357  486.0911099
#> 12     1406.702133  632.6388191  806.9028971
#> 13     1492.903495  315.0489135  793.9381916
#> 14     1917.636398  406.1424672  891.6613403
#> 15     2458.152208  285.2076540  916.4940218
#> 16     3384.341045  668.2337878 1106.1262716
#> 17     9596.552341  733.2222786 1295.6909824
#> Total 24134.870088 1842.8507073 3233.6807352

# Example with the BootChainLadder function, assuming overdispered Poisson model
B <- BootChainLadder(MW2008, process.distr=c("od.pois"))
B
#> BootChainLadder(Triangle = MW2008, process.distr = c("od.pois"))
#> 
#>      Latest Mean Ultimate Mean IBNR IBNR.S.E  IBNR 75%  IBNR 95%
#> 1 3,678,633     3,678,633         0        0         0         0
#> 2 3,902,425     3,906,478     4,053    5,483     6,479    14,431
#> 3 3,898,825     3,908,158     9,333    7,691    13,576    22,970
#> 4 3,548,422     3,576,411    27,989   11,954    35,437    49,473
#> 5 3,585,812     3,637,022    51,210   15,773    60,950    79,241
#> 6 3,641,036     3,752,372   111,336   22,264   127,314   148,710
#> 7 3,428,335     3,614,528   186,193   28,383   202,102   232,830
#> 8 3,158,581     3,570,202   411,621   43,594   441,683   487,770
#> 9 2,144,738     3,574,294 1,429,556  101,382 1,491,662 1,602,032
#> 
#>                     Totals
#> Latest:         30,986,807
#> Mean Ultimate:  33,218,098
#> Mean IBNR:       2,231,291
#> IBNR.S.E           132,952
#> Total IBNR 75%:  2,319,624
#> Total IBNR 95%:  2,456,423
CDR(B)
#>              IBNR   IBNR.S.E  CDR(1)S.E  CDR(1)75%  CDR(1)95%
#> 1           0.000      0.000      0.000       0.00       0.00
#> 2        4053.352   5483.375   5483.375    6479.00   14431.30
#> 3        9332.589   7690.852   6101.008   12265.95   20594.63
#> 4       27989.047  11954.354  10215.147   34481.45   46854.14
#> 5       51210.211  15772.669  10686.849   57025.21   70172.51
#> 6      111336.207  22263.812  16677.877  122893.82  140810.58
#> 7      186193.260  28383.341  18957.276  199120.62  217810.52
#> 8      411620.635  43594.083  33865.996  432812.75  471242.46
#> 9     1429556.088 101382.435  90832.645 1487533.82 1581666.14
#> Total 2231291.389 132952.187 110453.804 2303423.49 2423741.73