Specify a change from per-loss to per-payment basis

Records the exact actuarial change of basis for a collective risk model whose severity is a per-loss payment specification from apply_coverage(). The conversion uses binomial thinning of the loss count and conditioning the payment amount on a strictly positive insurer payment.

Usage

to_payment_basis(x, label = NULL)

Arguments

x: A list-backed collective risk specification containing frequency (a tidyact_frequency_spec) and severity (a tidyact_payment_spec with basis = "per_loss"). The collective class tidyact_collective_risk_spec, as provided by collective_risk_spec(). The input must have count_basis = "loss".
label: Optional nonempty character scalar identifying the transformed model. If NULL, the collective model label is inherited when present.

Value

For q > 0, a tidyact_payment_basis_spec that also inherits from tidyact_collective_risk_spec, with thinned frequency, conditional positive severity, count_basis = "payment", the unchanged source, and metadata. For q = 0, an explicit tidyact_zero_payment_basis_spec with no per-payment severity, since conditioning on zero-probability events is undefined. This degenerate result is not a collective risk specification.

Details

Write N_L for the number of losses and Y_L for the insurer payment per loss, including zero payments. Put $$q = P(Y_L > 0) = 1-F_{Y_L}(0).$$ Assuming independent, identically distributed loss severities independent of N_L, the number of payments has the binomial-thinning law $$N_P \mid N_L=n \sim Binomial(n,q).$$ For q > 0, the per-payment severity is $$Y_P \stackrel{d}{=} (Y_L \mid Y_L>0)$$ and its CDF is 0 for y <= 0 and $$F_{Y_P}(y)=\frac{F_{Y_L}(y)-(1-q)}{q}, \quad y>0.$$ Consequently, the probability generating functions satisfy $$G_{N_P}(z)=G_{N_L}(1-q+qz),$$ and the aggregate payments have the same distribution on either basis.

Poisson counts remain Poisson with mean q * lambda; binomial counts remain binomial with unchanged number of trials and probability multiplied by q. A negative binomial frequency with size = r and mu transforms to the same size and mean q * mu. For mixed Poisson models, the conditional intensity becomes q * Lambda without discarding the mixing variable. Other frequency families must be thinned via the general PGF rule rather than silently assumed to stay in the same family.

When q = 1, both basis representations coincide. When q = 0, payment frequency and aggregate payment are identically zero, while Y_L | Y_L > 0 is undefined. A fabricated point mass at zero must not be used as a per-payment severity.

The conversion is executable for registered Poisson, binomial, negative binomial, geometric and finite empirical frequencies. No unknown/custom frequency family is silently approximated. The probability of a positive payment is resolved from the exact source CDF or finite discrete masses. Continuous per-payment severities inherit only the registered exact operations of their source; unavailable moment orders remain unavailable. For negative binomial and geometric frequencies, thinning uses $p_P=p/(p+q(1-p))$. If a positive payment frequency would round to the degenerate parameter prob = 1, the conversion stops and the per-loss model should be retained; positive risk is never silently removed.

References

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

Bowers, N. L., Gerber, H. U., Hickman, J. C., Jones, D. A., and Nesbitt, C. J. (1997). Actuarial Mathematics, 2nd ed., Chapter 12.

Examples

# Fully executable Poisson collective model with an ordinary deductible.
f <- frequency_spec("poisson", lambda = 4)
x <- severity_spec("discrete", values = c(50, 150, 250),
                   probs = c(0.2, 0.5, 0.3))
paid <- x |> apply_coverage(coverage_spec(deductible = 100))
model <- collective_risk_spec(f, paid)
by_payment <- model |> to_payment_basis()
by_payment$count_basis
#> [1] "payment"
by_payment$thinning$probability
#> [1] 0.8
aggregate_moments(by_payment, order = 1:2)
#> # A tibble: 2 × 4
#>   type  order  value status
#>   <chr> <int>  <dbl> <chr> 
#> 1 raw       1    280 finite
#> 2 raw       2 110400 finite