MACRAME Based Development Profile Reserve
mcReserve.RdThe function takes a cumulative (or incremental) run-off triangle (partially or completely observed) and returns the reserve prediction obtained by the MACRAME algorithm (see Maciak, Mizera, and Pešta (2022) for further details).
Arguments
- chainLadder
a cumulative or incremental run-off triangle (the triangle must be of the class
triangleormatrix) in terms of a square matrix with a fully observed upper-left triangular part. If the lower-right part is also provided the function will also return standard residuals but only the upper-left (run-off) triangle is be used for the reserve prediction purposes- cum
logical to indicate the type of the input triangle that is provided (DEFAULT value is
TRUEfor the cumulative triangle,FALSEifchainLadderis of the incremental type)- residuals
logical to indicate whether (incremental) residuals should be provided in output or not. If the run-off triangle is completely observed then the residuals are obtained in terms of the true increments minus the predicted ones. If the bottom-right triangle is not available (which is a typical situation in practice) then the residuals are obtained in terms of a back-fitting approach (see Maciak, Mizera, and Pešta (2022) for further details). However, the back-fitted residuals are only calculated when no user specification of the states (in
states) and breaks (inbreaks) is provided (as it is usually not appropriate to use the same states/breaks for the flipped run-off triangle)- states
numeric value to provide either the number of the Markov states to be used or it can specify an explicit set of the states instead. The default setting (
states = NULL) provides the set of states in a fully data-driven manner as proposed in Maciak, Mizera, and Pešta (2022) while any choice ofbreaksis ignored. If the number of states is specified bystates, the states are obtained analogously as in Maciak, Mizera, and Pešta (2022), however, the number of actual states is adjusted and the parameterbreaksis again ignoredIf parameter
statesprovides an explicit vector of Markov chain states (the smallest state should be larger than the smallest observed increment in the run-off triangle and, similarly, the largest state should be smaller than the largest observed increment) then the corresponding bins (breaks) for the run-off triangle increments are defined automatically by the midpoints between the provided states (withbreaksbeing set toNULLDEFAULT)- breaks
vector parameter which provides explicit (unique and monotonly increasing) break points (disjoint bins) for the run-off triangle incremenets. Each bin should be represented by the corresponding Markov chain state—either the values given in
statesor provided automatically ifstatesis not a valid vector of the Markov states. If the breaks are provided asbreaks = c(-Inf, ... , Inf)definingkbins all together thenstatesshould be a vector of the same lengthk. Alternatively, the breaks can be also specified by a set of finite numbers defining againkbins—in such cases, the parameterstatesshould be of the lengthlength(states) = k + 1. Each value instatesshould represent one bin defined bybreaks
Value
An object of the type list with with the following elements:
- reserve
numeric vector with four values: Total paid amount (i.e., the sum of the last observed diagonal in a cumulative run-off triangle); Estimated ultimate (i.e., the sum of the last column in the completed cumulative run-off triangle); Estimated reserve (i.e., the sum of the last column in the completed cumulative triangle minus the sum of the last observed diagonal in
chainLadder); True reserve if a completedchainLadderis provided for the input (i.e., the sum of the last column inchainLadderminus the sum of the last diagonal inchainLadder)- method
algorithm used for the reserve estimation
- Triangle
the input run-off triangle provided in
chainLadder- FullTriangle
completed run-off triangle (the upper-left triangular part is identical with the input triangle in
chainLadderand the lower-right trianglular part is completed by the MACRAME algorithm- trueCompleted
true completed triangle (if available) where the upper-left part is used by the MACRAME algorithm to estimate the reserve and the lower-right part is provided for some evaluation purposes. If the full triangle is not available
NAis returned instead- residuals
a triangle with the corresponding residuals (for
residuals = TRUE). The residuals are either provided in the upper-left triangle (so-called back-fitted incremental residuals if true completed triangle is not available) or the residuals are given in the lower-right triangle (i,e., standard incremental residuals—if the true completed triangle is given)
References
Maciak, M., Mizera, I., and Pešta, M. (2022). Functional Profile Techniques for Claims Reserving. ASTIN Bulletin, 52(2), 449-482. DOI:10.1017/asb.2022.4
Examples
## run-off (upper-left) triangle with NA values
data(MW2014, package = "ChainLadder")
print(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
## MACRAME reserve prediction with the DEFAULT Markov chain setting
mcReserve(MW2014, residuals = TRUE)
#> MACRAME Reserving
#> Estimated Reserve Estimated Ultimate Paid Amount
#> 27861.23 456978.23 429117.00
#> True Reserve
#> NA
#>
#> MACRAME method (functional profile completion)
#> 13109 20355 21337 22043 22401 22658 22997 23158 23492 23664 23699 23904 23960 23992 23994 24001 24002
#> 14457 22038 22627 23114 23238 23312 23440 23490 23964 23976 24048 24111 24252 24538 24540 24550 24597
#> 16075 22672 23753 24052 24206 24757 24786 24807 24823 24888 24986 25401 25681 25705 25732 25774 25899
#> 15682 23464 24465 25052 25529 25708 25752 25770 25835 26075 26082 26146 26150 26167 26209 26334 26472
#> 16551 23706 24627 25573 26046 26115 26283 26481 26701 26718 26724 26728 26735 26782 26856 26993 27140
#> 15439 23796 24866 25317 26139 26154 26175 26205 26764 26818 26836 26959 27310 27573 27804 28007 28194
#> 14629 21645 22826 23599 24992 25434 25476 25549 25604 25709 25723 25770 25844 25981 26128 26288 26452
#> 17585 26288 27623 27939 28335 28638 28715 28759 29525 30302 30935 31310 31570 31788 31981 32165 32341
#> 17419 25941 27066 27761 28043 28477 28721 28878 28948 29050 29209 29382 29548 29720 29890 30061 30231
#> 16665 25370 26909 27611 27729 27861 29830 29844 29891 29965 30102 30249 30409 30573 30741 30909 31079
#> 15471 23745 25117 26378 26971 27396 27480 27596 27799 27972 28159 28331 28507 28678 28850 29021 29192
#> 15103 23393 26809 27691 28061 29183 29704 29971 30220 30427 30625 30808 30986 31161 31334 31506 31677
#> 14540 22642 23571 24127 24210 24326 24529 24702 24889 25061 25237 25408 25580 25751 25922 26093 26263
#> 14590 22336 23440 24029 24340 24602 24817 25021 25206 25387 25562 25736 25908 26079 26250 26421 26592
#> 13967 21515 22603 23124 23391 23640 23847 24045 24228 24406 24581 24754 24926 25097 25268 25438 25609
#> 12930 20111 21134 21657 21953 22202 22411 22607 22789 22967 23141 23314 23486 23657 23828 23998 24169
#> 12539 13743 14333 14663 14924 15138 15335 15518 15696 15870 16043 16215 16386 16557 16727 16898 17069
## complete run-off triangle with 'unknown' truth (lower-bottom run-off triangle)
## with incremental residuals (true increments minus predicted ones)
data(CameronMutual)
mcReserve(CameronMutual, residuals = TRUE)
#> MACRAME Reserving
#> Estimated Reserve Estimated Ultimate Paid Amount
#> 8081.963 113240.963 105159.000
#> True Reserve
#> 7963.000
#>
#> MACRAME method (functional profile completion)
#> 5244 9228 10823 11352 11791 12082 12120 12199 12215 12215
#> 5984 9939 11725 12346 12746 12909 13034 13109 13113 13160
#> 7452 12421 14171 14752 15066 15354 15637 15720 15756 15799
#> 7115 11117 12488 13274 13662 13859 13872 13919 13960 14003
#> 5753 8969 9917 10697 11135 11282 11340 11380 11422 11464
#> 3937 6524 7989 8543 8757 8815 8855 8897 8939 8982
#> 5127 8212 8976 9325 9496 9588 9646 9693 9737 9780
#> 5046 8006 8984 9677 10157 10458 10641 10759 10840 10902
#> 5129 8202 9244 9808 10165 10376 10502 10586 10649 10701
#> 3689 4731 5295 5652 5863 5989 6073 6136 6188 6236
## the same output in terms of the reserve prediction but back-fitted residuals
## are provided instead (as the run-off triangle only is provided)
data(observed(CameronMutual))
#> Warning: data set ‘observed(CameronMutual)’ not found
mcReserve(observed(CameronMutual), residuals = TRUE)
#> MACRAME Reserving
#> Estimated Reserve Estimated Ultimate Paid Amount
#> 8081.963 113240.963 105159.000
#> True Reserve
#> NA
#>
#> MACRAME method (functional profile completion)
#> 5244 9228 10823 11352 11791 12082 12120 12199 12215 12215
#> 5984 9939 11725 12346 12746 12909 13034 13109 13113 13160
#> 7452 12421 14171 14752 15066 15354 15637 15720 15756 15799
#> 7115 11117 12488 13274 13662 13859 13872 13919 13960 14003
#> 5753 8969 9917 10697 11135 11282 11340 11380 11422 11464
#> 3937 6524 7989 8543 8757 8815 8855 8897 8939 8982
#> 5127 8212 8976 9325 9496 9588 9646 9693 9737 9780
#> 5046 8006 8984 9677 10157 10458 10641 10759 10840 10902
#> 5129 8202 9244 9808 10165 10376 10502 10586 10649 10701
#> 3689 4731 5295 5652 5863 5989 6073 6136 6188 6236
## MACRAME reserve prediction with the underlying Markov chain with five
## explicit Markov chain states
mcReserve(CameronMutual, residuals = TRUE, states = c(200, 600, 1000))
#> MACRAME Reserving
#> Estimated Reserve Estimated Ultimate Paid Amount
#> 13089.81 118248.81 105159.00
#> True Reserve
#> 7963.00
#>
#> MACRAME method (functional profile completion)
#> 5244 9228 10823 11352 11791 12082 12120 12199 12215 12215
#> 5984 9939 11725 12346 12746 12909 13034 13109 13113 13313
#> 7452 12421 14171 14752 15066 15354 15637 15720 15920 16120
#> 7115 11117 12488 13274 13662 13859 13872 14072 14272 14472
#> 5753 8969 9917 10697 11135 11282 11482 11682 11882 12082
#> 3937 6524 7989 8543 8757 8957 9157 9357 9557 9757
#> 5127 8212 8976 9325 9525 9725 9925 10125 10325 10525
#> 5046 8006 8984 9784 10344 10742 11046 11300 11528 11741
#> 5129 8202 9002 9562 9960 10264 10518 10746 10959 11166
#> 3689 4489 5049 5447 5751 6005 6233 6446 6653 6857