Apply an insurance contract to a loss severity specification

Creates a per-loss payment severity specification by combining an existing ground-up severity model with a contract created by coverage_spec(). The original model is retained and not discretized, fitted, or simulated.

Usage

apply_coverage(x, coverage, label = NULL)

Arguments

x: A severity specification inheriting from tidyact_severity_spec. It must represent nonnegative ground-up losses, not an existing payment specification.
coverage: A tidyact_coverage_spec object from coverage_spec().
label: Optional nonempty character scalar. If NULL, the source severity label is inherited when present; otherwise the coverage label is used when present.

Value

An S3 object with classes tidyact_payment_spec, tidyact_severity_spec, and tidyact_distribution_spec. The list stores source, coverage, basis = "per_loss", label, and a normalized severity specification (dist = "discrete" for finite discrete losses or dist = "custom" for continuous and mixed losses). The support bound is not asserted to be the smallest possible bound under the original distribution.

Details

The payment variable is defined on a per-loss basis, retaining every loss event even when the associated insurer payment is zero. This is not the conditional payment-per-payment severity distribution.

Let X >= 0 be the ground-up severity, f > 0 the inflation factor, d_f the indexed (or fixed) deductible, L_f the indexed (or fixed) layer limit, and c the insurer coinsurance fraction. The exact transformation is $$Y_L = c \min\{(fX-d_f)_+, L_f\}$$ for an ordinary deductible and $$Y_L = c \min\{fX,L_f\} 1(fX>d_f)$$ for a franchise deductible. The limit caps the covered layer before coinsurance; it is not a final insurer-payment limit.

A continuous ground-up distribution can become mixed: probability mass arises at zero and, when the limit is finite and binding, at the maximum payment c * L_f. Pre-existing atoms are preserved under the pushforward and may merge. When coinsurance = 0 or limit = 0, the payment is identically zero. No smoothing or implicit discretization is performed.

The result is an executable per-loss payment distribution specification; its source and coverage fields contain everything necessary to recover the payment law from the underlying severity provider. The internal pure functions .coverage_payment_amount(), .payment_cdf_from_loss(), .payment_boundary_atoms(), and .payment_discrete_pushforward() implement the transformation exactly given loss values or exact source probabilities. Finite discrete sources are transformed exactly into a finite discrete distribution. Continuous or mixed sources use the existing custom-provider contract for CDF, quantiles, random generation and available exact moments. Moment orders without an exact registered formula are reported unavailable; no numerical integration or discretization is introduced silently.

References

Klugman, S. A., Panjer, H. H., and Willmot, G. E. (2012). Loss Models: From Data to Decisions, 4th ed., Chapter 8.

Frees, E. W., Lee, G., and Yang, L., Loss Data Analytics, chapters on claim severity and coverage modifications.

Examples

# Full, registered severity model.
ground_up <- severity_spec("gamma", shape = 2, rate = 0.001,
                           label = "Ground-up severity")
contract <- coverage_spec(deductible = 500, limit = 10000,
                          coinsurance = 0.8)
payments <- ground_up |> apply_coverage(contract)
inherits(payments, "tidyact_payment_spec")
#> [1] TRUE
payments$basis
#> [1] "per_loss"
payments$coverage$deductible
#> [1] 500