Calculate moments of a collective aggregate loss

Calculates analytical raw moments, central moments, or cumulants of a collective risk model

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

using the moments of the claim count distribution and the individual severity distribution stored in a tidyact_collective_risk_spec object.

No simulation, numerical integration, discretization, or approximation is performed by this function.

Usage

aggregate_moments(x, order = 1:4, type = "raw")

Arguments

x: A tidyact_collective_risk_spec object created by collective_risk_spec().
order: Nonempty numeric vector of nonnegative integer moment orders. Cumulant orders must be positive.
type: One of "raw" (default), "central", or "cumulant".

Value

A tibble::tibble() with one row per requested order and columns: type, order, value, and status.

Details

Let

$$m_j = E[X^j]$$

denote the raw severity moments and

$$\nu_k = E[(N)_k]$$

the factorial moments of the claim count. For positive integer $r$, the raw aggregate moment is evaluated through the random-sum identity

$$ E[S^r] = \sum_{k=1}^{r} \nu_k B_{r,k}(m_1,\ldots,m_{r-k+1}), $$

where $B_{r,k}$ is a partial exponential Bell polynomial.

This formulation includes the familiar identities

$$E[S] = E[N]E[X]$$

and

$$ \operatorname{Var}(S) = E[N]\operatorname{Var}(X) + \operatorname{Var}(N)(E[X])^2. $$

Central moments and cumulants are derived algebraically from aggregate raw moments. For a compound Poisson model, cumulants use the direct identity

$$\kappa_r(S) = \lambda E[X^r]$$

whenever the corresponding severity moment is finite.

The function respects the status information returned by distribution_moments(). Results are classified as:

"finite": a finite numerical result;
"infinite": a mathematically divergent raw or central moment;
"undefined": the requested quantity cannot be defined from the available moments;
"overflow": a theoretically finite calculation exceeded floating-point range.

Because both frequency and severity are nonnegative in this module, a finite first moment equal to zero identifies a degenerate-at-zero component. If either $N=0$ almost surely or $X=0$ almost surely, then the aggregate loss is degenerate at zero and all positive-order aggregate moments and cumulants are zero.

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.

Klugman, S. A., Panjer, H. H., and Willmot, G. E. (2019). Loss Models: From Data to Decisions, 5th ed. Wiley.

Examples

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

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

model <- collective_risk_spec(
  frequency,
  severity
)

aggregate_moments(model)
#> # A tibble: 4 × 4
#>   type  order   value status
#>   <chr> <int>   <dbl> <chr> 
#> 1 raw       1 8   e 3 finite
#> 2 raw       2 8.80e 7 finite
#> 3 raw       3 1.18e12 finite
#> 4 raw       4 1.86e16 finite

aggregate_moments(
  model,
  order = 1:4,
  type = "central"
)
#> # A tibble: 4 × 4
#>   type    order   value status
#>   <chr>   <int>   <dbl> <chr> 
#> 1 central     1 0       finite
#> 2 central     2 2.40e 7 finite
#> 3 central     3 9.60e10 finite
#> 4 central     4 2.21e15 finite

aggregate_moments(
  model,
  order = 1:4,
  type = "cumulant"
)
#> # A tibble: 4 × 4
#>   type     order   value status
#>   <chr>    <int>   <dbl> <chr> 
#> 1 cumulant     1 8   e 3 finite
#> 2 cumulant     2 2.40e 7 finite
#> 3 cumulant     3 9.60e10 finite
#> 4 cumulant     4 4.80e14 finite