Find the development period at which the loss ratio estimate stabilises

Identify the first dev $k^{**}$ from which the projected portfolio loss ratio is observed to be stable up to the maximum available development period $V$. Three complementary stability criteria are computed on the LR backtest path; the user selects which one defines $k^{**}$ via method =.

Notation mapping (code <-> math):

Standard chain-ladder convention: $i$ indexes cohort (origin period), $k$ indexes development period. The maturity point $k^*$ and convergence point $k^{**}$ live on the $k$ axis. Earlier paper drafts used $v$ (valuation) for the same index in Section 11; we unify on $k$ for consistency.

`dev_max`	$K_{\max}$	Maximum observable dev (a scalar)
`dev_cand`	$k \in [k^*, K_{\max}-2]$	Integer vector of candidate dev points
`ratio[i]`	$LR_k$	Portfolio LR projection at dev = `dev_cand[i]`
`revision[i]`	$R_k = \|LR_k - LR_{k-1}\|$	Adjacent-step revision (diagnostic)
`drift_window[i]`	$\max - \min$ of $LR$ over $[k, k+W-1]$	Local window range
`drift_tail[i]`	$\max - \min$ of $LR$ over $[k, K_{\max}]$	Tail range
`slope[i]`	$\hat\beta_k$, OLS slope of $LR \sim k$ on $[k, K_{\max}]$	Trend test
`dispersion[i]`	$\hat{D}_k$	Robust cross-cohort spread of incremental LR

Stability methods (which sequence drives pass):

"window": Local stability: drift_window[i] < max_drift. Fast, but misses a slow monotone drift that fits under max_drift per step.
"tail": (default, reserving-safe) Global stability: drift_tail[i] < max_drift. Catches monotone drift. The first passing dev is later (more conservative) than "window".
"slope": Trend test: |slope[i]| < max_slope. Explicit no-trend check; sensitive to non-linear trajectories.
"all": Strictest: all three pass simultaneously.

All four pass vectors (pass_window, pass_tail, pass_slope, pass) and the underlying diagnostic series are returned regardless of the chosen method, so the analyst can inspect every criterion and re-decide.

Across all methods, a cross-cohort agreement clause dispersion[i] < max_dispersion is required in addition.

This replaces an earlier formulation $R_k < c \cdot \hat{SE}^{param}_k$ (paper Section 11). The paper's SE-normalised form is asymptotically broken on large portfolios: $\hat{SE}^{param}$ shrinks as $1/\sqrt{n}$ while $R_k$ has a structural noise floor, so the ratio diverges and the criterion never fires.

Caveat (reserving): detected conv_k reflects stability up to dev_max ($K_{\max}$) only – it is not an asymptotic guarantee that the projection will not drift past $K_{\max}$. Treat conv_k as a diagnostic for "from here on, what we observe is stable", not as a guarantee of future stability. For reserving applications, prefer method = "tail" or "all" over "window", attach an IBNR margin via fit_ratio$summary SE/CI columns, and weigh the evidence span (dev_max - conv_k): a conv_k near dev_max has thin evidence.

Usage

detect_convergence(
  triangle,
  method = c("tail", "window", "slope", "all"),
  max_drift = 0.01,
  max_slope = 0.001,
  max_dispersion = 0.15,
  window = 5L,
  mat_k = NULL,
  holdout_max = NULL,
  min_n_cohorts = 5L,
  ...
)

Arguments

triangle: A Triangle object (typically from as_triangle()).
method: Which stability criterion defines conv_k. One of "tail" (default), "window", "slope", or "all". See the description for semantics and the reserving caveat.
max_drift: Upper bound on the drift metric (window or tail), in LR units. Default 0.01 (1pp). Raise for noisier or longer-tail books.
max_slope: Upper bound on |slope[i]|, the OLS slope of $LR \sim k$ on $[k, K_{\max}]$, in LR-per-dev units. Default 1e-3 (0.1pp per dev). Used by method = "slope" / "all".
max_dispersion: Upper bound on the cross-cohort dispersion $\hat{D}_k$. Default 0.15.
window: Drift window length $W$ (in dev steps): the number of consecutive valuations used by the "window" method to compute drift_window. Default 5L. Note: other functions in the package also expose a window argument (e.g. detect_regime() for e-divisive segment width); here it controls only the drift metric, not the e-divisive algorithm.
mat_k: Pre-computed maturity point. When NULL, computed via detect_maturity() applied to a ratio-based ATA.
holdout_max: Maximum holdout depth used for the rolling backtest. When NULL, set to max(window, floor((dev_max - mat_k) / 2)).
min_n_cohorts: Minimum number of cohorts required to compute $\hat{D}_v$. Default 5L.
...: Additional arguments forwarded to backtest() (and thence to fit_ratio()), e.g. loss_method, recent, loss_regime.

Value

An object of class Convergence (named list). Includes the slots tabulated in the notation mapping above (dev_max, dev_cand, ratio, revision, drift_window, drift_tail, slope, dispersion), per-method pass vectors (pass_window, pass_tail, pass_slope, pass), the threshold parameters, and metadata attributes (groups, loss, dispatcher).

`dev_max`	\(K_{\max}\)	Maximum observable dev (a scalar)
`dev_cand`	\(k \in [k^*, K_{\max}-2]\)	Integer vector of candidate dev points
`ratio[i]`	\(LR_k\)	Portfolio LR projection at dev = `dev_cand[i]`
`revision[i]`	\(R_k = \|LR_k - LR_{k-1}\|\)	Adjacent-step revision (diagnostic)
`drift_window[i]`	\(\max - \min\) of \(LR\) over \([k, k+W-1]\)	Local window range
`drift_tail[i]`	\(\max - \min\) of \(LR\) over \([k, K_{\max}]\)	Tail range
`slope[i]`	\(\hat\beta_k\), OLS slope of \(LR \sim k\) on \([k, K_{\max}]\)	Trend test
`dispersion[i]`	\(\hat{D}_k\)	Robust cross-cohort spread of incremental LR

Find the development period at which the loss ratio estimate stabilises

Usage

Arguments

Value

See also