Cape Cod projection (Stanard 1985)

Fit a Cape Cod projection from a "Triangle" object. Cape Cod is the prior-free Bornhuetter-Ferguson variant introduced by Stanard (1985): the a priori expected loss ratio is estimated from the data itself as a portfolio-pooled quantity, then plugged into the BF formula.

$$\widehat{\mathrm{ELR}}^{CC} = \frac{\sum_i L_{obs, i}}{\sum_i E_i^{ult} \cdot q_i}$$

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 (inverse of cumulative LDF).
$E_i^{ult}$: cohort $i$'s ultimate premium (projected via chain ladder on premium).

Given $\widehat{\mathrm{ELR}}^{CC}$, the per-cohort ultimate is obtained from the BF formula with this single pooled ELR:

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

When multiple groups are present, $\widehat{\mathrm{ELR}}^{CC}$ is computed within group (not pooled across groups) so each group retains its own portfolio-level ELR estimate.

This is a peer worker alongside fit_bf() / fit_cl() / fit_ed(). Standalone for the Cape Cod 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). The bootstrap path also produces per-replicate pooled ELR draws (elr_cc_se, elr_cc_cv, elr_cc_ci_lo, elr_cc_ci_hi) since the Cape Cod ELR itself is data-driven and thus uncertain.

Usage

fit_cc(
  x,
  loss = "loss",
  exposure = "premium",
  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,
  credibility = NULL,
  conf_level = 0.95,
  ...
)

Arguments

x: A Triangle object.
loss: A single cumulative loss variable. Default "loss".
exposure: A single cumulative premium variable. Default "premium".
bootstrap: Bootstrap configuration. Same forms as fit_bf()'s bootstrap arg – see there for the full description.
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) MSEP decomposition (with Var(ELR_cc) from the delta method on the pooled ELR). 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".
process: One of "gamma" (default), "od_pois", "normal".
alpha: Numeric scalar passed through to the inner fit_cl() / 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.
credibility: Optional credibility specification. NULL (default) gives the classical CC blend weighted by the emergence fraction q. A list list(method = "bs", K = NULL) switches to a Buehlmann-Straub credibility blend ult = Z * CL + (1 - Z) * prior with the pooled ELR as the prior; Z = K / (K + s^2) shrinks a green / rare-event cohort toward the pooled ELR. See fit_bf() for the full description. A credibility blend uses the analytical SE path.
conf_level: Confidence level for the SE-based CI (bootstrap quantile or analytical normal). Default 0.95.
...: Reserved for future extension (currently unused).

Value

An object of class "CCFit" containing:

call: The matched call.
data: The input Triangle.
method: "cc".
groups, cohort, dev, loss, premium: Metadata.
full, proj, summary: Same shape as BFFit. With bootstrap enabled, $full carries loss_total_se/loss_total_cv/loss_ci_lo/loss_ci_hi on projected cells, and $summary carries the same plus elr_cc_se/elr_cc_cv/elr_cc_ci_lo/elr_cc_ci_hi (uncertainty on the pooled ELR itself).
elr_cc: data.table(group..., elr_cc) – the pooled ELR per group (or scalar if no group).
q: Per-cohort emerged fraction.
credibility: NULL for the classical blend, or a list list(method, weights) with the Buehlmann-Straub Z / K per cohort.
cl_fit, premium_fit: Inner CL / Premium fits.
bootstrap: When bootstrap is enabled, a CCBootstrap helper holding both Triangle-level BootstrapTriangle objects, the per-replicate ultimate replicates, and the per-replicate pooled ELR draws; 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 / loss_ci_hi plus the pooled-ELR columns elr_cc_se / elr_cc_cv / elr_cc_ci_lo / elr_cc_ci_hi.
alpha, sigma_method, recent, regime: Inputs forwarded to the inner fit_cl() / fit_premium() calls.

References

Stanard, J. N. (1985). A simulation test of prediction errors of loss reserve estimation techniques. Proceedings of the Casualty Actuarial Society, 72, 124-148.

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

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"
)
cc <- fit_cc(tri)
summary(cc)
cc$elr_cc   # pooled ELR per group
} # }

Usage

Arguments

Value

References

See also

Examples