Bornhuetter-Ferguson projection

Fit a Bornhuetter-Ferguson (1972) projection from a "Triangle" object. The BF estimator blends the observed cumulative loss for each cohort with an a priori expected loss ratio (ELR) applied to the cohort's ultimate premium, weighted by the expected unemerged fraction $1 - q_i$:

$$\hat L_{ult, i}^{BF} = L_{obs, i} + (1 - q_i) \cdot \mathrm{ELR}_i \cdot E_i^{ult}$$

where

$L_{obs, i}$: cohort $i$'s observed cumulative loss at its latest observed development period.
$q_i = L_{obs, i} / \hat L_{ult, i}^{CL}$: the expected emerged fraction, equivalent to the inverse of the cumulative loss development factor (LDF) for cohort $i$.
$\mathrm{ELR}_i$: the user-supplied a priori expected loss ratio for cohort $i$ (prior argument).
$E_i^{ult}$: cohort $i$'s ultimate premium, projected via chain ladder on the premium column.

This is a peer worker alongside fit_cl() / fit_ed() / fit_loss(). Standalone for the BF recipe – composition with fit_ratio() is not part of this worker. Point projection is always computed; bootstrap SE / CI is opt-in via bootstrap = TRUE (Phase 3b). Closed-form Mack (2008) MSEP is not yet implemented.

Usage

fit_bf(
  x,
  loss = "loss",
  exposure = "premium",
  prior,
  bootstrap = NULL,
  B = 999L,
  seed = NULL,
  type = c("parametric", "nonparametric", "analytical"),
  residual = c("cell", "link"),
  process = c("gamma", "od_pois", "normal"),
  alpha = 1,
  sigma_method = c("locf", "min_last2", "loglinear", "mack", "none"),
  recent = NULL,
  regime = NULL,
  maturity = NULL,
  credibility = NULL,
  conf_level = 0.95,
  ...
)

Arguments

x

A Triangle object.

loss

A single cumulative loss variable to project. Default "loss".

exposure

A single cumulative premium variable used as the denominator of the prior ELR. Default "premium".

prior

The a priori expected loss ratio. Accepts:

single numeric: Applied uniformly to every cohort.
per-cohort data.frame (cohort + elr): Per-cohort ELR. Must cover every cohort present in x (extras are silently dropped, missing cohorts raise an error).
per-group data.frame (grouping columns + elr): One ELR per group, broadcast to every cohort in that group. Useful when a single a priori ELR is set per line of business. Must cover every group present in x.

A data.frame prior may also carry an optional elr_se column – the standard error of the a priori ELR (a distribution prior). When supplied, the bootstrap path draws a per-replicate ELR from Normal(elr, elr_se) instead of treating the prior as a fixed point. Omit it (or leave NA) for a deterministic prior.

bootstrap

Bootstrap configuration. Five forms accepted:

NULL / FALSE (default): Point estimate only – no bootstrap SE/CI.
TRUE / "auto": Internal bootstrap() calls (one for loss, one for premium) sharing seed so replicate indices align across the two simulations.
Named list list(loss = BootstrapTriangle, premium = BootstrapTriangle): Pre-built objects from bootstrap(). Must have matching meta$B / meta$seed so per-replicate composition is well-defined; meta$target must be "loss" and "premium" respectively.
Function function(tri) -> list(loss = ..., premium = ...): Lazy spec invoked on the input Triangle (leakage-safe for backtest()).

Latest observed cumulative loss is not perturbed in the BF recipe – it is treated as the cohort anchor, mirroring the point-estimate formula.

B

Integer number of bootstrap replicates. Used only when bootstrap resolves to "auto". Default 999.

seed

Optional integer seed for reproducible bootstrap. Default NULL.

type

One of "parametric" (default), "nonparametric", or "analytical". "parametric" / "nonparametric" select the bootstrap residual paradigm; "analytical" skips simulation and uses the closed-form Mack (2008) BF MSEP decomposition for the cohort-level SE / CI. When no bootstrap is requested the analytical path is used regardless of type.

residual

Residual scope for type = "nonparametric". One of "cell" (default) or "link". See bootstrap().

process

One of "gamma" (default), "od_pois", or "normal". See bootstrap().

alpha

Numeric scalar passed through to the inner fit_cl() and fit_premium() calls. Default 1.

sigma_method

Sigma extrapolation method forwarded to fit_cl() / fit_premium(). Default "locf".

recent

Optional positive integer; calendar-diagonal filter forwarded to the inner fits. Default NULL.

regime

Optional regime specification forwarded to the inner loss and premium fits. See fit_cl() for the four-type dispatch.

maturity

Optional maturity specification forwarded to the inner loss fit. See fit_cl() for the four-type dispatch.

credibility

Optional credibility specification. NULL (default) gives the classical BF blend with weight equal to the emergence fraction q. A list list(method = "bs", K = NULL) switches to a Buehlmann-Straub credibility blend ult = Z * CL + (1 - Z) * prior, where Z = K / (K + s^2), s^2 is the variance of the cohort's own CL loss-ratio estimate, and K is the variance of the hypothetical means (the genuine between-cohort spread). K is estimated per group when NULL, or supplied as a non-negative numeric scalar. The credibility weight protects rare-event cohorts: a green cohort with a CL estimate built on almost no data has a large s^2, so Z shrinks toward 0 and the cohort is pulled to the prior even when its q is high. A credibility blend always uses the analytical SE path (the SE is approximate – the credibility factor is treated as a fixed plug-in).

conf_level

Confidence level for the bootstrap quantile CI on loss_ult. Default 0.95.

...

Reserved for future extension (currently unused).

Value

An object of class "BFFit" containing:

call: The matched call.
data: The input Triangle.
method: "bf".
groups: Grouping variable names.
cohort: Raw cohort variable name.
dev: Raw development variable name.
loss, premium: Loss / premium variable names.
full: data.table [group, cohort, dev, loss_obs, loss_proj, premium_obs, premium_proj, is_observed, incr_loss_proj, incr_premium_proj]. When bootstrap is enabled, additional columns loss_total_se, loss_total_cv, loss_ci_lo, loss_ci_hi carry per-cell bootstrap SE / CI on projected cells (observed cells stay NA).
proj: Same shape as full, with observed-cell projection columns NA'd out.
summary: Cohort-level reserve summary: [group, cohort, latest, loss_ult, reserve, elr, q]. When bootstrap is enabled, additional columns loss_total_se, loss_total_cv, loss_ci_lo, loss_ci_hi carry bootstrap SE / CI on loss_ult.
prior: Resolved data.table(group..., cohort, elr).
q: data.table(group..., cohort, q) of expected emerged fractions.
credibility: NULL for the classical blend, or a list list(method, weights) where weights is a data.table [group..., cohort, Z, K] of the Buehlmann-Straub credibility factors used in place of q.
cl_fit: The inner CLFit used to derive $q_i$.
premium_fit: The inner PremiumFit used to derive $E_i^{ult}$.
bootstrap: When bootstrap is enabled, a BFBootstrap helper holding both Triangle-level BootstrapTriangle objects and the per-replicate ultimate replicates; NULL otherwise.
ci_type: "bootstrap" when a bootstrap was run, "analytical" when the closed-form Mack (2008) MSEP was used. In the analytical case $summary carries loss_total_se, loss_total_cv, loss_ci_lo, and loss_ci_hi.
alpha, sigma_method, recent, regime, maturity: Inputs forwarded to the inner fit_cl() / fit_premium() calls.

References

Bornhuetter, R. L. and Ferguson, R. E. (1972). The actuary and IBNR. Proceedings of the Casualty Actuarial Society, 59, 181-195.

Mack, T. (2008). The prediction error of Bornhuetter/Ferguson. ASTIN Bulletin, 38(1), 87-103. (MSEP – not yet implemented.)

Examples

if (FALSE) { # \dontrun{
data(experience)
tri <- as_triangle(
  experience[coverage == "surgery"],
  groups   = "coverage",
  cohort   = "uy_m",
  calendar = "cy_m",
  loss     = "incr_loss",
  premium = "incr_premium"
)

# Scalar prior: 0.7 ELR for every cohort
bf1 <- fit_bf(tri, prior = 0.7)
summary(bf1)

# Per-cohort prior table
prior_tbl <- data.frame(
  cohort = unique(tri$cohort),
  elr    = c(0.6, 0.65, 0.7, 0.72, 0.75)
)
bf2 <- fit_bf(tri, prior = prior_tbl)
} # }

Usage

Arguments

Value

References

See also

Examples