Compound Poisson Distribution

Computes various risk measures (mean, variance, Value-at-Risk (VaR), and Tail Value-at-Risk (TVaR)) for the compound Poisson distribution.

pCompPois(
  x,
  lambda,
  shape,
  rate = 1/scale,
  scale = 1/rate,
  k0,
  distr_severity = "Gamma"
)

expValCompPois(
  lambda,
  shape,
  rate = 1/scale,
  scale = 1/rate,
  distr_severity = "Gamma"
)

varCompPois(
  lambda,
  shape,
  rate = 1/scale,
  scale = 1/rate,
  distr_severity = "Gamma"
)

VatRCompPois(
  kap,
  lambda,
  shape,
  rate = 1/scale,
  scale = 1/rate,
  k0,
  distr_severity = "Gamma"
)

TVatRCompPois(
  kap,
  lambda,
  shape,
  rate = 1/scale,
  scale = 1/rate,
  vark,
  k0,
  distr_severity = "Gamma"
)

Arguments

x

vector of quantiles

lambda

Rate parameter \(\lambda\).

shape

shape parameter \(\alpha\), must be positive.

rate

rate parameter \(\beta\), must be positive.

scale

alternative parameterization to the rate parameter, scale = 1 / rate.

k0

point up to which to sum the distribution for the approximation.

distr_severity

Choice of severity distribution.

• "gamma" (default)

• "lognormal" only for the expected value and variance.

kap

probability.

vark

Value-at-Risk (VaR) calculated at the given probability kap.

Value

Function :

• pCompPois gives the cumulative density function.

• expValCompPois gives the expected value.

• varCompPois gives the variance.

• TVatRCompPois gives the Tail Value-at-Risk.

• VatRCompPois gives the Value-at-Risk.

Returned values are approximations for the cumulative density function, TVaR, and VaR.

Details

The compound Poisson distribution with parameters ... has density ....

Examples

pCompPois(x = 2, lambda = 2, shape = log(1000) - 0.405,
          rate = 0.9^2, k0 = 1E2, distr_severity = "Gamma")
#> [1] 0.1361652

expValCompPois(lambda = 2, shape = log(1000) - 0.405, rate = 0.9^2,
          distr_severity = "Lognormale")
#> [1] 2000

varCompPois(lambda = 2, shape = log(1000) - 0.405, rate = 0.9^2,
          distr_severity = "Lognormale")
#> [1] 4495816

VatRCompPois(kap = 0.9, lambda = 2, shape = log(1000) - 0.405,
            rate = 0.9^2, k0 = 1E2, distr_severity = "Gamma")
#> [1] 32.63546

vark_calc <- VatRCompPois(kap = 0.9, lambda = 2, shape = 0.59,
            rate = 0.9^2, k0 = 1E2, distr_severity = "Gamma")
TVatRCompPois(kap = 0.9, lambda = 2, shape = 0.59, rate = 0.9^2,
            vark = vark_calc, k0 = 1E2, distr_severity = "Gamma")
#> [1] 5.352432