Chain ladder reserving

fit_cl() is the dedicated chain ladder fit for a single value column. Unlike fit_lr() (which projects loss / exposure jointly to get loss ratio), fit_cl() projects one cumulative metric forward and computes Mack-style standard errors per cohort.

Basic usage

For brevity this vignette uses the SUR group only — every step generalises to multi-group input.

library(lossratio)
data(experience)
exp <- as_experience(experience)
tri <- build_triangle(exp[cv_nm == "SUR"], group_var = cv_nm)

cl <- fit_cl(tri, value_var = "closs", method = "mack")
print(cl)
#> <CLFit>
#> method      : mack 
#> value_var   : closs 
#> weight_var  : none 
#> alpha       : 1 
#> sigma_method: min_last2 
#> recent      : all 
#> use_maturity: FALSE 
#> tail_factor : 1 
#> groups      : cv_nm 
#> periods     : 30

value_var selects the cumulative column to project — typically "closs" (cumulative loss) for reserving, or "crp" (cumulative risk premium) for exposure projection.

Method: basic vs Mack

Two estimation methods are available:

`method`	What it computes
`"basic"`	Point projection only (selected age-to-age factors)
`"mack"`	Point projection + factor / process / parameter SE

cl_basic <- fit_cl(tri, value_var = "closs", method = "basic")
cl_mack  <- fit_cl(tri, value_var = "closs", method = "mack")

names(cl_basic)
#>  [1] "call"          "data"          "method"        "group_var"    
#>  [5] "cohort_var"    "dev_var"       "value_var"     "full"         
#>  [9] "pred"          "ata"           "summary"       "factor"       
#> [13] "selected"      "maturity"      "alpha"         "sigma_method" 
#> [17] "weight_var"    "recent"        "use_maturity"  "maturity_args"
#> [21] "tail"          "tail_factor"

# Mack adds variance estimates to $full and $summary
head(cl_mack$summary)
#>     cv_nm     cohort     latest   ultimate   reserve     proc_se    param_se
#>    <char>     <Date>      <num>      <num>     <num>       <num>       <num>
#> 1:    SUR 2023-04-01 2442597048 2442597048         0         0.0         0.0
#> 2:    SUR 2023-05-01 2423543638 2600462324 176918686    270023.8    278555.7
#> 3:    SUR 2023-06-01 3211045460 3634951626 423906166    461673.6    481436.4
#> 4:    SUR 2023-07-01 2552396709 3106052713 553656004 217960839.6 130390124.6
#> 5:    SUR 2023-08-01 2472997706 3159902325 686904619 235800277.9 139803599.9
#> 6:    SUR 2023-09-01 2014222417 2712676349 698453932 230925350.1 124174149.3
#>             se           cv
#>          <num>        <num>
#> 1:         0.0 0.0000000000
#> 2:    387951.2 0.0001491855
#> 3:    667025.8 0.0001835034
#> 4: 253985259.7 0.0817710719
#> 5: 274129198.7 0.0867524279
#> 6: 262194082.1 0.0966551289

method = "mack" enables the projection plot’s confidence bands (show_interval = TRUE):

plot(cl_mack, type = "projection", show_interval = TRUE)

Tail factor

For triangles where the latest observed development period is still developing, an extrapolated tail factor estimates ultimate:

# Log-linear extrapolation from the selected ATA factors
cl_tail <- fit_cl(tri, value_var = "closs", method = "mack", tail = TRUE)

# Or supply a literal tail factor
cl_tail <- fit_cl(tri, value_var = "closs", method = "mack", tail = 1.025)

The extrapolation fits $\log(f_k - 1) \sim k$ to projected factors and extends the projection by the cumulative product of extrapolated $f_k$ values. Disabled by default (tail = FALSE).

Maturity filtering

If selected ATA factors are volatile, restrict projection to the mature region only:

cl_mat <- fit_cl(
  tri,
  value_var     = "closs",
  method        = "mack",
  maturity_args = list(cv_threshold = 0.10, rse_threshold = 0.05)
)

cl_mat$maturity
#> Key: <cv_nm>
#>     cv_nm ata_from ata_to ata_link  mean median    wt    cv     f  f_se   rse
#>    <char>    <num>  <num>   <char> <num>  <num> <num> <num> <num> <num> <num>
#> 1:    SUR        9     10     9-10 1.188  1.172 1.165 0.097 1.165 0.022 0.019
#>       sigma n_obs n_valid n_inf n_nan valid_ratio
#>       <num> <num>   <num> <num> <num>       <num>
#> 1: 1774.278    21      21     0     0           1

maturity_args is forwarded to find_ata_maturity().

Variance components (Mack)

fit_cl(method = "mack") decomposes the projection variance into:

proc_se — process variance, from $\sigma^2_k$ (residual link variance per development period).
param_se — parameter variance, from the uncertainty of the selected age-to-age factors $\hat{f}_k$ .
se — total standard error, $\sqrt{\mathrm{proc\_se}^2 + \mathrm{param\_se}^2}$ .
cv — coefficient of variation, se / value_proj.

summary(cl_mack)
#>      cv_nm     cohort     latest   ultimate    reserve      proc_se    param_se
#>     <char>     <Date>      <num>      <num>      <num>        <num>       <num>
#>  1:    SUR 2023-04-01 2442597048 2442597048          0          0.0         0.0
#>  2:    SUR 2023-05-01 2423543638 2600462324  176918686     270023.8    278555.7
#>  3:    SUR 2023-06-01 3211045460 3634951626  423906166     461673.6    481436.4
#>  4:    SUR 2023-07-01 2552396709 3106052713  553656004  217960839.6 130390124.6
#>  5:    SUR 2023-08-01 2472997706 3159902325  686904619  235800277.9 139803599.9
#>  6:    SUR 2023-09-01 2014222417 2712676349  698453932  230925350.1 124174149.3
#>  7:    SUR 2023-10-01 2422172254 3464336723 1042164469  276909538.1 163800531.1
#>  8:    SUR 2023-11-01 2157147612 3350616805 1193469193  347646646.1 180286627.2
#>  9:    SUR 2023-12-01 2062030017 3510350121 1448320104  379204803.4 195291838.6
#> 10:    SUR 2024-01-01 1803809914 3316423447 1512613533  371903391.7 185291186.8
#> 11:    SUR 2024-02-01 1627213157 3293904272 1666691115  406210629.2 191768888.6
#> 12:    SUR 2024-03-01 1006624213 2212909862 1206285649  348109439.7 131371151.5
#> 13:    SUR 2024-04-01  707083237 1712964993 1005881756  316686848.8 103164315.5
#> 14:    SUR 2024-05-01  398857325 1069653556  670796231  262671178.8  65778222.8
#> 15:    SUR 2024-06-01  558855276 1654603718 1095748442  342800640.3 103939643.9
#> 16:    SUR 2024-07-01  423131371 1378042306  954910935  336548946.6  89486750.7
#> 17:    SUR 2024-08-01  457705980 1642689597 1184983617  387322897.0 109347246.8
#> 18:    SUR 2024-09-01  278007651 1166380310  888372659  360265104.1  81491440.0
#> 19:    SUR 2024-10-01  214811381 1027414214  812602833  358796880.9  74015042.1
#> 20:    SUR 2024-11-01  251273971 1400108561 1148834590  451728990.6 105050619.0
#> 21:    SUR 2024-12-01  322678179 2168358632 1845680453  619598522.3 171876903.1
#> 22:    SUR 2025-01-01  179253475 1403314539 1224061064  523388251.8 114399770.9
#> 23:    SUR 2025-02-01  100816665  954214626  853397961  497168458.4  84734261.7
#> 24:    SUR 2025-03-01  111279087 1488227953 1376948866  843027195.9 163376024.6
#> 25:    SUR 2025-04-01   55914454  958667529  902753075  751897091.4 113958278.9
#> 26:    SUR 2025-05-01   41578391 1041506115  999927724  978637599.4 147793339.1
#> 27:    SUR 2025-06-01   14997314  484991215  469993901  813441277.3  81273429.5
#> 28:    SUR 2025-07-01    6232031  436725855  430493824 5630776358.8 495928426.0
#> 29:    SUR 2025-08-01          0          0          0          0.0         0.0
#> 30:    SUR 2025-09-01          0          0          0          0.0         0.0
#>      cv_nm     cohort     latest   ultimate    reserve      proc_se    param_se
#>     <char>     <Date>      <num>      <num>      <num>        <num>       <num>
#>               se           cv
#>            <num>        <num>
#>  1:          0.0 0.000000e+00
#>  2:     387951.2 1.491855e-04
#>  3:     667025.8 1.835034e-04
#>  4:  253985259.7 8.177107e-02
#>  5:  274129198.7 8.675243e-02
#>  6:  262194082.1 9.665513e-02
#>  7:  321728932.9 9.286884e-02
#>  8:  391613915.1 1.168782e-01
#>  9:  426538609.2 1.215089e-01
#> 10:  415505663.8 1.252873e-01
#> 11:  449201938.9 1.363737e-01
#> 12:  372073328.0 1.681376e-01
#> 13:  333066714.3 1.944387e-01
#> 14:  270782057.7 2.531493e-01
#> 15:  358211848.8 2.164940e-01
#> 16:  348242834.9 2.527084e-01
#> 17:  402462230.4 2.450020e-01
#> 18:  369366755.4 3.166778e-01
#> 19:  366351509.1 3.565763e-01
#> 20:  463783045.7 3.312479e-01
#> 21:  642996110.9 2.965359e-01
#> 22:  535744873.7 3.817711e-01
#> 23:  504337556.8 5.285368e-01
#> 24:  858712162.8 5.770031e-01
#> 25:  760483875.8 7.932718e-01
#> 26:  989734521.0 9.502916e-01
#> 27:  817491334.5 1.685580e+00
#> 28: 5652573520.7 1.294307e+01
#> 29:          0.0           NA
#> 30:          0.0           NA
#>               se           cv
#>            <num>        <num>

Reserve plot

type = "reserve" shows reserve per cohort with optional error bars (Mack only):

plot(cl_mack, type = "reserve", conf_level = 0.95)

Triangle visualisation

plot_triangle() displays the cohort × dev cells as a heatmap, distinguishing observed cells from projected:

plot_triangle(cl_mack, what = "full")    # observed + projected

plot_triangle(cl_mack, what = "pred")    # projected only

plot_triangle(cl_mack, what = "data")    # observed only

The label_style = "cv" mode shows coefficient of variation per cell, useful for spotting unreliable cells:

plot_triangle(cl_mack, label_style = "cv")

plot_triangle(cl_mack, label_style = "se")

plot_triangle(cl_mack, label_style = "ci")

Sigma extrapolation methods

Mack variance requires $\sigma_k$ at all development links, including the last where it cannot be estimated directly. sigma_method controls the extrapolation:

`sigma_method`	Behaviour
`"min_last2"`	(default) min of the last two estimable $\sigma$ values — conservative
`"locf"`	Last observation carried forward
`"loglinear"`	Log-linear extrapolation from the observed $\sigma_k$ sequence

fit_cl(tri, value_var = "closs", method = "mack", sigma_method = "loglinear")
#> <CLFit>
#> method      : mack 
#> value_var   : closs 
#> weight_var  : none 
#> alpha       : 1 
#> sigma_method: loglinear 
#> recent      : all 
#> use_maturity: FALSE 
#> tail_factor : 1 
#> groups      : cv_nm 
#> periods     : 30