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Reserving

From a long claims file to a defended IBNR: triangles with inferred columns, four methods, Mack's full standard error, and an ODP bootstrap — the same engine behind the IDE's reserving bridge, which reproduces the published Mack (1993) RAA figures exactly.

Triangles

claims = sc.sample("claims")            # or sc.load("claims.csv")
tri = sc.triangle(claims)               # origin × development, cumulative
tri = sc.triangle(claims, origin="uw_year", value="paid")   # override inference
tri = sc.from_wide(matrix, origins=range(1981, 1991))       # already-wide data
sc.ata(tri)                             # age-to-age factors per origin + averages
sc.latest_diagonal(tri)                 # paid to date per origin
sc.to_incremental(tri); sc.to_cumulative(tri)

claims <- sc_sample("claims")
tri <- sc_triangle(claims)
tri <- sc_triangle(claims, origin = "uw_year", value = "paid")
tri <- sc_from_wide(m, origins = 1981:1990)
sc_ata(tri)
sc_latest_diagonal(tri)
sc_to_incremental(tri); sc_to_cumulative(tri)

dev           0         1         2         3         4         5         6
origin
2018    68919.0   96350.0  178652.0  297389.0  396982.0  537291.0  560463.0
2019    39449.0   74979.0  131950.0  204479.0  320886.0  421532.0       NaN
...
— Cumulative triangle · paid by origin_year × development
  · 7 origin periods × 7 development lags, summed from 79 rows. Input rows treated
    as incremental amounts.

Input rows are incremental amounts by default (incremental_input=False when your file is already cumulative). Development is indexed purely by period — from a lag column, or as payment − origin when you give a calendar payment column — never inferred from dates, so a truncated parallelogram cannot grow phantom origins. Cells inside the observed diagonal with no claims are 0; cells beyond it stay missing.

The methods

sc.chain_ladder(tri)                    # volume-weighted (or simple / regression)
sc.mack(tri)                            # + Mack (1993) SE per origin and in total
sc.bf(tri, premium=prem, elr=0.65)      # Bornhuetter–Ferguson
sc.cape_cod(tri, premium=prem)          # ELR estimated from the triangle itself
sc.bootstrap(tri, n=1000, seed=42)      # England–Verrall ODP, gamma process error
sc.tail(sc.ldf(tri))                    # exponential-decay tail factor
sc.reserve(claims)                      # all four side by side, from the raw file

sc_chain_ladder(tri)
sc_mack(tri)
sc_bf(tri, premium = prem, elr = 0.65)
sc_cape_cod(tri, premium = prem)
sc_bootstrap(tri, n = 1000, seed = 42)
sc_tail(sc_ldf(tri))
sc_reserve(claims)

mack: IBNR 1,532,963 · ultimate 3,407,400 · latest 1,874,437 · SE 346,036 (CV 22.6%)
           latest       cdf  ...             se        cv
origin
2018     560463.0  1.000000  ...       0.000000       NaN
2019     421532.0  1.043127  ...    2082.253100  0.114538
...
total   1874437.0       NaN  ...  346036.429161  0.225730
— Mack chain ladder
  basis: volume-weighted link ratios · Mack (1993) MSE
  · SE is the square root of Mack's MSE: process + estimation error per origin, plus
    the inter-origin covariance in the total. ±1.96·SE is a normal-approximation
    interval, not a tail quantile.

Every method returns a reserving result: the per-origin Table plus ibnr, ultimate, latest, factors, cdf, se, cv and a detail dict of internals (per-origin MSE, sigmas, the bootstrap's simulated totals). What the numbers mean:

  • Chain ladder — volume-weighted link ratios by default (average= for simple or regression; n_periods= to use only recent diagonals; tail_factor= for a tail).
  • Mack — the full 1993 MSE, including the inter-origin covariance term in the total, with Mack's own extrapolation for the last σ².
  • BF — a-priori from, in order: your apriori (a scalar, a per-origin vector, or another result whose ultimates seed it), premium × elr, or the book-average chain-ladder ultimate (a flat ELR would cancel straight back to CL).
  • Cape Cod — the ELR estimated as Σ latest / Σ (premium / CDF), reported in detail.
  • Bootstrap — over-dispersed Poisson (England–Verrall): bias-adjusted Pearson residuals resampled, each pseudo-triangle re-fitted, gamma process error on top. Seeded (seed=42 by default), so a rerun reproduces exactly; detail carries the full simulated distribution and p5 … p99.

reserve runs the four in one line and stacks their summaries — the table shown in the quickstart — with the individual results in its attributes.

Checked against the literature

The test suites (Python and R both) reproduce Mack (1993) on the published RAA triangle: IBNR 52,135, ultimate 213,122, total SE 26,909 (CV 51.6 %), per-origin SEs 0, 206, 623, 747, 1469, 2002, 2209, 5358, 6333, 24566 — the same figures R's ChainLadder gives with est.sigma = "Mack". The scelo[reserving] extra installs the chainladder package purely so you can run that cross-check yourself.

Function list

triangle from_wide is_cumulative to_incremental to_cumulative latest_diagonal ata ldf cdf chain_ladder mack bf cape_cod bootstrap tail reserve — each with the sc_ twin in R.