Specify a collective risk model

Creates a lightweight specification of the classical collective risk model

$$S = \sum_{i=1}^{N} X_i,$$

where $N$ is a claim count random variable and $X_1, X_2, \ldots$ are identically distributed individual claim severities. Independence between the frequency variable and the severities, and mutual independence of the severities, are intrinsic assumptions of this specification.

The constructor records the probabilistic model only. It does not calculate aggregate moments, construct the distribution of $S$, discretize a severity distribution, simulate losses, or apply an approximation.

Usage

collective_risk_spec(frequency, severity, count_basis = NULL, label = NULL)

Arguments

frequency: A tidyact_frequency_spec object, typically created with frequency_spec() or empirical_frequency().
severity: A tidyact_severity_spec object, typically created with severity_spec() or empirical_severity().
count_basis: Either NULL, "loss", or "payment". The count basis indicates whether frequency counts underlying losses or positive payments. When NULL, the constructor uses basis information already carried by the component specifications when available; otherwise the default is "loss".
label: Optional nonempty character scalar identifying the model.

Value

An object with classes "tidyact_collective_risk_spec" and "tidyact_risk_spec". It contains the following fields:

schema_version: internal schema version.
frequency: the frequency specification.
severity: the severity specification.
count_basis: either "loss" or "payment".
label: optional user label.
metadata: reserved metadata list.

Details

The collective model is represented by the random sum

$$S = X_1 + \cdots + X_N,$$

with the convention $S = 0$ when $N = 0$.

collective_risk_spec() deliberately keeps the model specification separate from computational methods. In particular, no Panjer recursion, convolution, FFT calculation, Monte Carlo simulation, numerical discretization, or normal approximation is performed by this function.

The count_basis field is part of the model because frequency and severity must refer to the same event basis. A loss-basis specification counts underlying losses, including losses that may later produce a zero payment after coverage is applied. A payment-basis specification counts positive payments. Conversion between these bases is a separate actuarial operation and is not performed silently by this constructor.

The constructor does not expose an independence argument. Independence is part of the definition of this version of the collective risk model rather than an optional computational setting.

References

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

Examples

frequency <- frequency_spec(
  "poisson",
  lambda = 4
)

severity <- severity_spec(
  "gamma",
  shape = 2,
  rate = 0.001
)

model <- collective_risk_spec(
  frequency,
  severity
)

model
#> <tidyact_collective_risk_spec>
#> Model: S = sum(X_i, i = 1, ..., N)
#> Frequency: Poisson
#> Severity: Gamma
#> Count basis: loss

# An explicit payment-basis model.
collective_risk_spec(
  frequency,
  severity,
  count_basis = "payment",
  label = "Illustrative payment model"
)
#> <tidyact_collective_risk_spec>
#> Model: S = sum(X_i, i = 1, ..., N)
#> Frequency: Poisson
#> Severity: Gamma
#> Count basis: payment
#> Label: Illustrative payment model

# Empirical components can be used without fitting a parametric model.
empirical_n <- empirical_frequency(c(0, 0, 1, 1, 2, 3))
empirical_x <- empirical_severity(c(500, 750, 1000, 1500, 3000))

collective_risk_spec(
  empirical_n,
  empirical_x
)
#> <tidyact_collective_risk_spec>
#> Model: S = sum(X_i, i = 1, ..., N)
#> Frequency: Empirical
#> Severity: Empirical
#> Count basis: loss