Package 'Pareto'

Title: The Pareto, Piecewise Pareto and Generalized Pareto Distribution
Description: Utilities for the Pareto, piecewise Pareto and generalized Pareto distribution that are useful for reinsurance pricing. In particular, the package provides a non-trivial algorithm that can be used to match the expected losses of a tower of reinsurance layers with a layer-independent collective risk model. The theoretical background of the matching algorithm and most other methods are described in Ulrich Riegel (2018) <doi:10.1007/s13385-018-0177-3>.
Authors: Ulrich Riegel [aut, cre]
Maintainer: Ulrich Riegel <[email protected]>
License: GPL (>= 2)
Version: 2.4.5
Built: 2024-11-12 03:39:59 UTC
Source: https://github.com/ulrichriegel/pareto

Help Index

Density of the generalized Pareto Distribution

Description

Calculates the density function of the generalized Pareto distribution

Usage

dGenPareto(x, t, alpha_ini, alpha_tail, truncation = NULL)

Arguments

x

Numeric. The function evaluates the density at x.
t

Numeric. Threshold of the Pareto distribution.
alpha_ini

Numeric. Initial Pareto alpha.
alpha_tail

Numeric. Tail Pareto alpha.
truncation

Numeric. If truncation is not NULL and truncation > t, then the generalized Pareto distribution is truncated at truncation.

Value

Density function of the Pareto distribution with parameters t, alpha_ini and alpha_tail evaluated at x

Examples

x <- 0:10 * 1000
dGenPareto(x, 1000, 1, 3)
dGenPareto(x, 1000, 1, 3, truncation = 5000)

Density of the Pareto Distribution

Description

Calculates the density function of the Pareto distribution

Usage

dPareto(x, t, alpha, truncation = NULL)

Arguments

x

Numeric. The function evaluates the density at x.
t

Numeric. Threshold of the Pareto distribution.
alpha

Numeric. Pareto alpha.
truncation

Numeric. If truncation is not NULL and truncation > t, then the Pareto distribution is truncated at truncation.

Value

Density function of the Pareto distribution with parameters t and alpha evaluated at x

Examples

x <- 0:10 * 1000
dPareto(x, 1000, 2)
dPareto(x, 1000, 2, truncation = 5000)

Density of the Piecewise Pareto Distribution

Description

Calculates the density function of the piecewise Pareto distribution

Usage

dPiecewisePareto(x, t, alpha, truncation = NULL, truncation_type = "lp")

Arguments

x

Numeric. The function evaluates the density at x.
t

Numeric vector. Thresholds of the piecewise Pareto distribution.
alpha

Numeric vector. alpha[i] is the Pareto alpha in excess of t[i].
truncation

Numeric. If truncation is not NULL and truncation > t, then the Pareto distribution is truncated at truncation.
truncation_type

Character. If truncation_type = "wd" then the whole distribution is truncated. If truncation_type = "lp" then a truncated Pareto is used for the last piece.

Value

Density function of the piecewise Pareto distribution with parameter vectors t and alpha evaluated at x

Examples

t <- c(1000, 2000, 3000)
alpha <- c(1, 1.5, 2)
x <- 0:10 * 1000
dPiecewisePareto(x, t, alpha)
dPiecewisePareto(x, t, alpha, truncation = 5000, truncation_type = "lp")
dPiecewisePareto(x, t, alpha, truncation = 5000, truncation_type = "wd")

Example data: Attachment Points

Description

Example data: Attachment Points

Usage

Example1_AP

Format

An object of class numeric of length 5.

Example data: Expected Losses

Description

Example data: Expected Losses

Usage

Example1_EL

Format

An object of class numeric of length 5.

Expected Frequency in Excess of a Threshold

Description

Calculates the expected frequency in excess of a threshold for a collective model

Usage

Excess_Frequency(CollectiveModel, x = 0)

Arguments

CollectiveModel

A collective model object. Currently only PPP_Models are handled.
x

Numeric. Threshold.

Value

The expected frequency in excess of x for the given CollectiveModel

Examples

PPPM <- PiecewisePareto_Match_Layer_Losses(Example1_AP, Example1_EL)
PPPM
Excess_Frequency(PPPM, c(-Inf, 0, 1000, 2000, 3000, Inf))

Expected Frequency in Excess of a Threshold

Description

Calculates the expected frequency in excess of a threshold for a PGP_model

Usage

## S3 method for class 'PGP_Model'
Excess_Frequency(CollectiveModel, x = 0)

Arguments

CollectiveModel

PGP_Model object.
x

Numeric. Threshold.

Value

The expected frequency in excess of x for the given CollectiveModel

Examples

PGPM <- PGP_Model(2, 1000, 1, 2, dispersion = 2)
PGPM
Excess_Frequency(PGPM, c(-Inf, 0, 1000, 2000, 3000, Inf))

Expected Frequency in Excess of a Threshold

Description

Calculates the expected frequency in excess of a threshold for a PPP_model

Usage

## S3 method for class 'PPP_Model'
Excess_Frequency(CollectiveModel, x = 0)

Arguments

CollectiveModel

PPP_Model object.
x

Numeric. Threshold.

Value

The expected frequency in excess of x for the given CollectiveModel

Examples

PPPM <- PiecewisePareto_Match_Layer_Losses(Example1_AP, Example1_EL)
PPPM
Excess_Frequency(PPPM, c(-Inf, 0, 1000, 2000, 3000, Inf))

Fits a Collective Model to a PML Curve

Description

Fits a PPP_Model that matches the values of a PML curve

Usage

Fit_PML_Curve(
  return_periods,
  amounts,
  tail_alpha = 2,
  truncation = NULL,
  truncation_type = "lp",
  dispersion = 1
)

Arguments

return_periods

Numeric vector. Vector containing the return periods of the PML curve.
amounts

Numeric vector. Vector containing the loss amounts corresponding to the return periods.
tail_alpha

Numerical. Pareto alpha that is used above the highest amount of the PML curve.
truncation

Numeric. If truncation is not NULL and truncation > max(t), then the distribution is truncated at truncation.
truncation_type

Character. If truncation_type = "wd" then the whole distribution is truncated. If truncation_type = "lp" then a truncated Pareto is used for the last piece.
dispersion

Numerical. Dispersion of the claim count distribution in the resulting PPP_Model.

Value

A PPP_Model object that contains the information about a collective model with a Panjer distributed claim count and a Piecewise Pareto distributed severity. The object contains the following elements:

  • FQ Numerical. Frequency in excess of the lowest threshold of the piecewise Pareto distribution

  • t Numeric vector. Vector containing the thresholds for the piecewise Pareto distribution

  • alpha Numeric vector. Vector containing the Pareto alphas of the piecewise Pareto distribution

  • truncation Numerical. If truncation is not NULL and truncation > max(t), then the distribution is truncated at truncation.

  • truncation_type Character. If truncation_type = "wd" then the whole distribution is truncated. If truncation_type = "lp" then a truncated Pareto is used for the last piece.

  • dispersion Numerical. Dispersion of the Panjer distribution (i.e. variance to mean ratio).

  • Status Numerical indicator: 0 = success, 1 = some information has been ignored, 2 = no solution found

  • Comment Character. Information on whether the fit was successful

Examples

return_periods <- c(1, 5, 10, 20, 50, 100)
amounts <- c(1000, 4000, 7000, 10000, 13000, 14000)

fit <- Fit_PML_Curve(return_periods, amounts)
1 / Excess_Frequency(fit, amounts)

fit <- Fit_PML_Curve(return_periods, amounts, tail_alpha = 1.5,
                     truncation = 20000, truncation_type = "wd")
1 / Excess_Frequency(fit, amounts)

Fit a Collective Model to a Wishlist of References

Description

The function fits a collective model to a wishlist of references (expected layer losses and excess frequencies). The function allows to specify the family of the severity distribution that is used. Depending on this distribution family the function works slightly differently:

  • For the severity distribution PiecewisePareto the function returns a PPP_Model that satisfies all the references

  • For the severity distribution Pareto the function returns a PPP_Model that minimizes the squared relative deviations from the references

  • For the severity distribution GenPareto the function returns a PGP_Model that minimizes the squared relative deviations from the references

Usage

Fit_References(
  Covers = NULL,
  Attachment_Points = NULL,
  Expected_Layer_Losses = NULL,
  Thresholds = NULL,
  Frequencies = NULL,
  model_threshold = min(c(Attachment_Points, Thresholds)),
  default_alpha = 2,
  dispersion = 1,
  alpha_max = 100,
  severity_distribution = "PiecewisePareto",
  ignore_inconsistent_references = FALSE
)

Arguments

Covers

Numeric vector. Vector containing the covers of the layers from the wishlist.
Attachment_Points

Numeric vector. Vector containing the attachment points of the layers from the wishlist.
Expected_Layer_Losses

Numeric vector. Vector containing the expected losses of the layers from the wishlist.
Thresholds

Numeric vector. Contains the thresholds from the wishlist for which excess frequencies are given.
Frequencies

Numeric vector. Expected frequencies excess the Thresholds from the wishlist.
model_threshold

Numerical. Lowest threshold of the fitted piecewise Pareto distribution.
default_alpha

Numerical. Default alpha for situations where an alpha has to be selected.
dispersion

Numerical. Dispersion of the claim count distribution in the resulting PPP_Model.
alpha_max

Numerical. Maximum alpha to be used for the matching.
severity_distribution

Character. Implemented distributions: "PiecewisePareto" (default), "Pareto" and "GenPareto".
ignore_inconsistent_references

Logical. If TRUE then inconsistent references are ignored in case of the piecewise Pareto distribution and the other references are used to fit the model

Value

For severity_distribution = "PiecewisePareto" or "Pareto": A PPP_Model object that contains the information about a collective model with a Panjer distributed claim count and a Piecewise Pareto distributed severity. The object contains the following elements:

  • FQ Numerical. Frequency in excess of the lowest threshold of the piecewise Pareto distribution

  • t Numeric vector. Vector containing the thresholds for the piecewise Pareto distribution

  • alpha Numeric vector. Vector containing the Pareto alphas of the piecewise Pareto distribution

  • truncation Numerical. If truncation is not NULL and truncation > max(t), then the distribution is truncated at truncation.

  • truncation_type Character. If truncation_type = "wd" then the whole distribution is truncated. If truncation_type = "lp" then a truncated Pareto is used for the last piece.

  • dispersion Numerical. Dispersion of the Panjer distribution (i.e. variance to mean ratio).

  • Status Numerical indicator: 0 = success, 1 = some information has been ignored, 2 = no solution found

  • Comment Character. Information on whether the fit was successful

For severity_distribution = "GenPareto": A PGP_Model object that contains the information about a collective model with a Panjer distributed claim count and a Piecewise Pareto distributed severity. The object contains the following elements:

  • FQ Expected claim count of the collective model.

  • t Numeric. Threshold of the Pareto distribution.

  • alpha_ini Numeric. Initial Pareto alpha (at t).

  • alpha_tail Numeric. Tail Pareto alpha.

  • truncation If truncation is not NULL and truncation > t, then the Pareto distribution is truncated at truncation.

  • dispersion Numerical. Dispersion of the Panjer distribution (i.e. variance to mean ratio).

  • Status Numerical indicator: 0 = success, 1 = some information has been ignored, 2 = no solution found

  • Comment Character. Information on whether the fit was successful

Examples

covers <- c(1000, 1000, 1000)
att_points <- c(1000, 2000, 5000)
exp_losses <- c(100, 50, 10)
thresholds <- c(4000, 10000)
fqs <- c(0.04, 0.005)
fit <- Fit_References(covers, att_points, exp_losses, thresholds, fqs)
Layer_Mean(fit, covers, att_points)
Excess_Frequency(fit, thresholds)
fit <- Fit_References(covers, att_points, exp_losses, thresholds, fqs,
                      severity_distribution = "Pareto")
Layer_Mean(fit, covers, att_points)
Excess_Frequency(fit, thresholds)
fit <- Fit_References(covers, att_points, exp_losses,
                      severity_distribution = "GenPareto")
Layer_Mean(fit, covers, att_points)

Layer Mean of the generalized Pareto Distribution

Description

Calculates the expected loss of a generalized Pareto distribution in a reinsurance layer

Usage

GenPareto_Layer_Mean(
  Cover,
  AttachmentPoint,
  t,
  alpha_ini,
  alpha_tail,
  truncation = NULL
)

Arguments

Cover

Numeric. Cover of the reinsurance layer. Use Inf for unlimited layers.
AttachmentPoint

Numeric. Attachment point of the reinsurance layer.
t

Numeric. Threshold of the Pareto distribution. If t is NULL (default) then t <- Attachment Point is used.
alpha_ini

Numeric. Initial Pareto alpha (at t).
alpha_tail

Numeric. Tail Pareto alpha.
truncation

Numeric. If truncation is not NULL and truncation > t, then the Pareto distribution is truncated at truncation.

Value

The expected loss of the (truncated) Pareto distribution with parameters t and alpha in the layer Cover xs AttachmentPoint

Examples

GenPareto_Layer_Mean(4000, 1000, 1000, 1, 3)
GenPareto_Layer_Mean(4000, 1000, t = 1000, alpha_ini = 1, alpha_tail = 3)
GenPareto_Layer_Mean(4000, 1000, t = 5000, alpha_ini = 1, alpha_tail = 3)
GenPareto_Layer_Mean(4000, 1000, t = 1000, alpha_ini = 1, alpha_tail = 3, truncation = 5000)
GenPareto_Layer_Mean(9000, 1000, t = 1000, alpha_ini = 1, alpha_tail = 3, truncation = 5000)

Second Layer Moment of the Generalized Pareto Distribution

Description

Calculates the second moment of a generalized Pareto distribution in a reinsurance layer

Usage

GenPareto_Layer_SM(
  Cover,
  AttachmentPoint,
  t,
  alpha_ini,
  alpha_tail,
  truncation = NULL
)

Arguments

Cover

Numeric. Cover of the reinsurance layer. Use Inf for unlimited layers.
AttachmentPoint

Numeric. Attachment point of the reinsurance layer.
t

Numeric. Threshold of the Pareto distribution. If t is NULL (default) then t <- Attachment Point is used
alpha_ini

Numeric. Initial Pareto alpha (at t).
alpha_tail

Numeric. Tail Pareto alpha.
truncation

Numeric. If truncation is not NULL and truncation > t, then the Pareto distribution is truncated at truncation.

Value

The second moment of the (truncated) generalized Pareto distribution with parameters t, alpha_ini and alpha_tail in the layer Cover xs AttachmentPoint

Examples

GenPareto_Layer_SM(4000, 1000, 1000, 1, 2)
GenPareto_Layer_SM(4000, 1000, t = 1000, alpha_ini = 1, alpha_tail = 3)
GenPareto_Layer_SM(4000, 1000, t = 5000, alpha_ini = 1, alpha_tail = 3)
GenPareto_Layer_SM(4000, 1000, t = 1000, alpha_ini = 1, alpha_tail = 3, truncation = 5000)
GenPareto_Layer_SM(9000, 1000, t = 1000, alpha_ini = 1, alpha_tail = 3, truncation = 5000)

Layer Variance of the Generalized Pareto Distribution

Description

Calculates the variance of a generalized Pareto distribution in a reinsurance layer

Usage

GenPareto_Layer_Var(
  Cover,
  AttachmentPoint,
  t,
  alpha_ini,
  alpha_tail,
  truncation = NULL
)

Arguments

Cover

Numeric. Cover of the reinsurance layer. Use Inf for unlimited layers.
AttachmentPoint

Numeric. Attachment point of the reinsurance layer.
t

Numeric. Threshold of the Pareto distribution. If t is NULL (default) then t <- Attachment Point is used
alpha_ini

Numeric. Initial Pareto alpha (at t).
alpha_tail

Numeric. Tail Pareto alpha.
truncation

Numeric. If truncation is not NULL and truncation > t, then the Pareto distribution is truncated at truncation.

Value

Variance of the (truncated) generalized Pareto distribution with parameters t, alpha_ini and alpha_tail in the layer Cover xs AttachmentPoint

Examples

GenPareto_Layer_Var(4000, 1000, 1000, 1, 2)
GenPareto_Layer_Var(4000, 1000, t = 1000, alpha_ini = 1, alpha_tail = 3)
GenPareto_Layer_Var(4000, 1000, t = 5000, alpha_ini = 1, alpha_tail = 3)
GenPareto_Layer_Var(4000, 1000, t = 1000, alpha_ini = 1, alpha_tail = 3, truncation = 5000)
GenPareto_Layer_Var(9000, 1000, t = 1000, alpha_ini = 1, alpha_tail = 3, truncation = 5000)

Maximum Likelihood Estimation of the Pareto Alphas of a Generalized Pareto Distribution

Description

Calculates the maximum likelihood estimators of the parameters alpha_ini and alpha_tail of a generalized Pareto distribution with known threshold and (if applicable) known truncation

Usage

GenPareto_ML_Estimator_Alpha(
  losses,
  t,
  truncation = NULL,
  reporting_thresholds = NULL,
  is.censored = NULL,
  weights = NULL,
  alpha_min = 0.001,
  alpha_max = 10
)

Arguments

losses

Numeric vector. Losses that are used for the ML estimation.
t

Numeric or numeric vector. Threshold of the generalized Pareto distribution. Alternatively, t can be a vector of same length as losses. In this case t[i] is the reporting threshold of losses[i].
truncation

Numeric. If truncation is not NULL and truncation > t, then the generalized Pareto distribution is truncated at truncation.
reporting_thresholds

Numeric vector. Allows to enter loss specific reporting thresholds. If NULL then all reporting thresholds are assumed to be less than or equal to t.
is.censored

Logical vector. TRUE indicates that a loss has been censored by the policy limit. The assumption is that the uncensored losses are Generalized Pareto distributed with the alphas we are looking for. is.censored = NULL means that no losses are censored.
weights

Numeric vector. Weights for the losses. For instance weights[i] = 2 and weights[j] = 1 for j != i has the same effect as adding another loss of size loss[i].
alpha_min

Numeric. Lower bound for the estimated alphas.
alpha_max

Numeric. Upper bound for the estimated alphas.

Value

Maximum likelihood estimator for the parameters alpha_ini and alpha_tail of a generalized Pareto distribution with threshold t given the observations losses

Examples

losses <- rGenPareto(1000, 1000, 2,3)
GenPareto_ML_Estimator_Alpha(losses, 1000)
losses <- rGenPareto(1000, 1000, 2, 1, truncation = 10000)
GenPareto_ML_Estimator_Alpha(losses, 1000)
GenPareto_ML_Estimator_Alpha(losses, 1000, truncation = 10000)

t <- 1000
alpha_ini <- 1
alpha_tail <- 3
losses <- rGenPareto(5000, t, alpha_ini, alpha_tail)
reporting_thresholds <- rPareto(5000, 1000, 3)
reported <- losses > reporting_thresholds
losses <- losses[reported]
reporting_thresholds <- reporting_thresholds[reported]
GenPareto_ML_Estimator_Alpha(losses, t)
GenPareto_ML_Estimator_Alpha(losses, t, reporting_thresholds = reporting_thresholds)
limit <- 3000
censored <- losses > limit
losses[censored] <- limit
reported <- losses > reporting_thresholds
losses <- losses[reported]
censored <- censored[reported]
reporting_thresholds <- reporting_thresholds[reported]
GenPareto_ML_Estimator_Alpha(losses, t, reporting_thresholds = reporting_thresholds)
GenPareto_ML_Estimator_Alpha(losses, t, reporting_thresholds = reporting_thresholds,
                             is.censored = censored)

losses <- c(190, 600, 120, 270, 180, 120)
w <- rep(1, length(losses))
w[1] <- 3
losses2 <- c(losses, losses[1], losses[1])
GenPareto_ML_Estimator_Alpha(losses, 100, weights = w)
GenPareto_ML_Estimator_Alpha(losses2, 100)

Check if an object is a PGP_Model

Description

Checks if the class of an object is 'PGP_Model'

Usage

is.PGP_Model(x)

Arguments

x

Object to be checked.

Examples

PGPM <- PGP_Model(2, 1000, 1, 2, dispersion = 2)
PGPM
is.valid.PGP_Model(PGPM)
is.valid.PGP_Model(PGPM, comment = TRUE)

PGPM$alpha_tail <- -2
is.PGP_Model(PGPM)
is.valid.PGP_Model(PGPM)
is.valid.PGP_Model(PGPM, comment = TRUE)

Check if an object is a PPP_Model

Description

Checks if the class of an object is 'PPP_Model'

Usage

is.PPP_Model(x)

Arguments

x

Object to be checked.

Examples

PPPM <- PPP_Model(2, c(1000,2000), c(1,2), dispersion = 2)
PPPM
is.valid.PPP_Model(PPPM)

PPPM$alpha <- 2
is.valid.PPP_Model(PPPM)
is.PPP_Model(PPPM)

Check if an object is a valid PGP_Model

Description

Checks if an object is a PGP_Model object and whether it is valid for the use in functions like Layer_Mean

Usage

is.valid.PGP_Model(x, comment = FALSE)

Arguments

x

Object to be checked.
comment

If FALSE then the function returns a boolean indicating whether x is a valid PGP_Model. If TRUE then the function returns a comment instead.

Examples

PGPM <- PGP_Model(2, 1000, 1, 2, dispersion = 2)
PGPM
is.valid.PGP_Model(PGPM)
is.valid.PGP_Model(PGPM, comment = TRUE)

PGPM$alpha_tail <- -2
is.valid.PGP_Model(PGPM)
is.valid.PGP_Model(PGPM, comment = TRUE)

Check if an object is a valid PPP_Model

Description

Checks if an object is a PPP_Model object and whether it is valid for the use in functions like Layer_Mean

Usage

is.valid.PPP_Model(x, comment = FALSE)

Arguments

x

Object to be checked.
comment

If FALSE then the function returns a boolean indicating whether x is a valid PPP_Model. If TRUE then the function returns a comment instead.

Examples

PPPM <- PPP_Model(2, c(1000,2000), c(1,2), dispersion = 2)
PPPM
is.valid.PPP_Model(PPPM)
is.valid.PPP_Model(PPPM, comment = TRUE)

PPPM$alpha <- 2
is.valid.PPP_Model(PPPM)
is.valid.PPP_Model(PPPM, comment = TRUE)

Expected Loss of a Reinsurance Layer

Description

Calculates the expected loss of a reinsurance layer for a collective model

Usage

Layer_Mean(CollectiveModel, Cover = Inf, AttachmentPoint = 0)

Arguments

CollectiveModel

A collective model object. Currently only PPP_Models are handled.
Cover

Numeric. Cover of the reinsurance layer. Use Inf for unlimited layers.
AttachmentPoint

Numeric. Attachment point of the reinsurance layer.

Value

The expected loss of the layer Cover xs AttachmentPoint for the given CollectiveModel

Examples

PPPM <- PiecewisePareto_Match_Layer_Losses(Example1_AP, Example1_EL)
PPPM
Example1_Cov <- c(diff(Example1_AP), Inf)
Example1_AP
Example1_Cov
Example1_EL
Layer_Mean(PPPM, Example1_Cov, Example1_AP)

Expected Loss of a Reinsurance Layer

Description

Calculates the expected loss of a reinsurance layer for a PGP_Model

Usage

## S3 method for class 'PGP_Model'
Layer_Mean(CollectiveModel, Cover = Inf, AttachmentPoint = 0)

Arguments

CollectiveModel

PGP_Model object.
Cover

Numeric. Cover of the reinsurance layer. Use Inf for unlimited layers.
AttachmentPoint

Numeric. Attachment point of the reinsurance layer.

Value

The expected loss of the layer Cover xs AttachmentPoint for the given CollectiveModel

Examples

PGPM <- PGP_Model(2, 1000, 1, 2, dispersion = 2)
PGPM
Example1_Cov <- c(diff(Example1_AP), Inf)
Example1_AP
Example1_Cov
Example1_EL
Layer_Mean(PGPM, Example1_Cov, Example1_AP)

Expected Loss of a Reinsurance Layer

Description

Calculates the expected loss of a reinsurance layer for a PPP_Model

Usage

## S3 method for class 'PPP_Model'
Layer_Mean(CollectiveModel, Cover = Inf, AttachmentPoint = 0)

Arguments

CollectiveModel

PPP_Model object.
Cover

Numeric. Cover of the reinsurance layer. Use Inf for unlimited layers.
AttachmentPoint

Numeric. Attachment point of the reinsurance layer.

Value

The expected loss of the layer Cover xs AttachmentPoint for the given CollectiveModel

Examples

PPPM <- PiecewisePareto_Match_Layer_Losses(Example1_AP, Example1_EL)
PPPM
Example1_Cov <- c(diff(Example1_AP), Inf)
Example1_AP
Example1_Cov
Example1_EL
Layer_Mean(PPPM, Example1_Cov, Example1_AP)

Standard Deviation of a Reinsurance Layer

Description

Calculates the standard deviation of the loss in a reinsurance layer for a collective model

Usage

Layer_Sd(CollectiveModel, Cover = Inf, AttachmentPoint = 0)

Arguments

CollectiveModel

A collective model object. Currently only PPP_Models are handled.
Cover

Numeric. Cover of the reinsurance layer. Use Inf for unlimited layers.
AttachmentPoint

Numeric. Attachment point of the reinsurance layer.

Value

The standard deviation of the loss in the layer Cover xs AttachmentPoint for the given CollectiveModel

Examples

PPPM <- PiecewisePareto_Match_Layer_Losses(Example1_AP, Example1_EL)
PPPM
Example1_Cov <- c(diff(Example1_AP), Inf)
Layer_Sd(PPPM, Example1_Cov, Example1_AP)

Standard Deviation of a Reinsurance Layer

Description

Calculates the standard deviation of the loss in a reinsurance layer for a PGP_model

Usage

## S3 method for class 'PGP_Model'
Layer_Sd(CollectiveModel, Cover = Inf, AttachmentPoint = 0)

Arguments

CollectiveModel

PGP_Model object.
Cover

Numeric. Cover of the reinsurance layer. Use Inf for unlimited layers.
AttachmentPoint

Numeric. Attachment point of the reinsurance layer.

Value

The standard deviation of the loss in the layer Cover xs AttachmentPoint for the given CollectiveModel

Examples

PGPM <- PGP_Model(2, 1000, 1, 2, dispersion = 2)
PGPM
Example1_Cov <- c(diff(Example1_AP), Inf)
Layer_Sd(PGPM, Example1_Cov, Example1_AP)

Standard Deviation of a Reinsurance Layer

Description

Calculates the standard deviation of the loss in a reinsurance layer for a PPP_model

Usage

## S3 method for class 'PPP_Model'
Layer_Sd(CollectiveModel, Cover = Inf, AttachmentPoint = 0)

Arguments

CollectiveModel

PPP_Model object.
Cover

Numeric. Cover of the reinsurance layer. Use Inf for unlimited layers.
AttachmentPoint

Numeric. Attachment point of the reinsurance layer.

Value

The standard deviation of the loss in the layer Cover xs AttachmentPoint for the given CollectiveModel

Examples

PPPM <- PiecewisePareto_Match_Layer_Losses(Example1_AP, Example1_EL)
PPPM
Example1_Cov <- c(diff(Example1_AP), Inf)
Layer_Sd(PPPM, Example1_Cov, Example1_AP)

Variance of a Reinsurance Layer

Description

Calculates the variance of the loss in a reinsurance layer for a collective model

Usage

Layer_Var(CollectiveModel, Cover = Inf, AttachmentPoint = 0)

Arguments

CollectiveModel

A collective model object. Currently only PPP_Models are handled.
Cover

Numeric. Cover of the reinsurance layer. Use Inf for unlimited layers.
AttachmentPoint

Numeric. Attachment point of the reinsurance layer.

Value

The variance of the loss in the layer Cover xs AttachmentPoint for the given CollectiveModel

Examples

PPPM <- PiecewisePareto_Match_Layer_Losses(Example1_AP, Example1_EL)
PPPM
Example1_Cov <- c(diff(Example1_AP), Inf)
Layer_Var(PPPM, Example1_Cov, Example1_AP)

Variance of a Reinsurance Layer

Description

Calculates the variance of the loss in a reinsurance layer for a PGP_model

Usage

## S3 method for class 'PGP_Model'
Layer_Var(CollectiveModel, Cover = Inf, AttachmentPoint = 0)

Arguments

CollectiveModel

PGP_Model object.
Cover

Numeric. Cover of the reinsurance layer. Use Inf for unlimited layers.
AttachmentPoint

Numeric. Attachment point of the reinsurance layer.

Value

The variance of the loss in the layer Cover xs AttachmentPoint for the given CollectiveModel

Examples

PGPM <- PGP_Model(2, 1000, 1, 2, dispersion = 2)
PGPM
Example1_Cov <- c(diff(Example1_AP), Inf)
Layer_Var(PGPM, Example1_Cov, Example1_AP)

Variance of a Reinsurance Layer

Description

Calculates the variance of the loss in a reinsurance layer for a PPP_model

Usage

## S3 method for class 'PPP_Model'
Layer_Var(CollectiveModel, Cover = Inf, AttachmentPoint = 0)

Arguments

CollectiveModel

PPP_Model object.
Cover

Numeric. Cover of the reinsurance layer. Use Inf for unlimited layers.
AttachmentPoint

Numeric. Attachment point of the reinsurance layer.

Value

The variance of the loss in the layer Cover xs AttachmentPoint for the given CollectiveModel

Examples

PPPM <- PiecewisePareto_Match_Layer_Losses(Example1_AP, Example1_EL)
PPPM
Example1_Cov <- c(diff(Example1_AP), Inf)
Layer_Var(PPPM, Example1_Cov, Example1_AP)

Local Pareto Alpha

Description

Calculates the local Pareto alpha of the normal, lognormal and gamma distribution

Usage

Local_Pareto_Alpha(x, distribution, ...)

Arguments

x

Numeric. Vector of thresholds at which the local Pareto alpha is calculated.
distribution

Character.

  • 'lnorm' for lognormal distribution (arguments: meanlog, sdlog)

  • 'norm' for normal distribution (arguments: mean, sd)

  • 'gamma' for gamma distribution (arguments: shape, rate, scale)

  • 'weibull' for Weibull distribution (arguments: shape, scale)

  • 'exp' for exponential distribution (arguments: rate)

  • 'Pareto' for Pareto distribution (arguments: t, alpha, truncation = NULL)

  • 'GenPareto' for exp distribution (arguments: t, alpha_ini, alpha_tail, truncation = NULL)

  • 'PiecewisePareto' for exp distribution (arguments: t, alpha, truncation = NULL, truncation_type = 'lp')

...

Arguments for the selected distribution

Value

Local Pareto alpha of the selected distribution at x

References

Riegel, U. (2008) Generalizations of common ILF models. Blaetter der DGVFM 29: 45–71

Examples

x <- 1:10 * 1e6
Local_Pareto_Alpha(x, "norm", mean = 5e6, sd = 2e6)
Local_Pareto_Alpha(x, "lnorm", meanlog = 0, sdlog = 4)
Local_Pareto_Alpha(x, "gamma", shape = 5, rate = 1e-6)
Local_Pareto_Alpha(x, "weibull", shape = 0.5, scale = 1e6)
Local_Pareto_Alpha(x, "exp", rate = 1e-6)
Local_Pareto_Alpha(x, "Pareto", t = 1e6, alpha = 1, truncation = 20e6)
Local_Pareto_Alpha(x, "GenPareto", t = 1e6, alpha_ini = 1, alpha_tail = 2)
Local_Pareto_Alpha(x, "PiecewisePareto", t = c(1e6, 3e6, 5e6), alpha = c(1, 2, 3),
                       truncation = 20e6, truncation_type = "wd")

Distribution Function of the Pareto Distribution

Description

Calculates the cumulative distribution function of a Pareto distribution. This function is deprecated. Use pPareto instead.

Usage

Pareto_CDF(x, t, alpha, truncation = NULL)

Arguments

x

Numeric. The function evaluates the CDF at x.
t

Numeric. Threshold of the Pareto distribution.
alpha

Numeric. Pareto alpha.
truncation

Numeric. If truncation is not NULL and truncation > t, then the Pareto distribution is truncated at truncation.

Value

Distribution function of the Pareto distribution with parameters t and alpha evaluated at x

Examples

x <- 0:10 * 1000
pPareto(x, 1000, 2)
pPareto(x, 1000, 2, truncation = 5000)

Pareto Extrapolation

Description

Uses a Pareto distribution to derive the expected loss of a layer from the expected loss of another layer

Usage

Pareto_Extrapolation(
  Cover_1,
  AttachmentPoint_1,
  Cover_2,
  AttachmentPoint_2,
  alpha,
  ExpLoss_1 = NULL,
  truncation = NULL
)

Arguments

Cover_1

Numeric. Cover of the layer from which we extrapolate. Use Inf for unlimited layers.
AttachmentPoint_1

Numeric. Attachment point of the layer from which we extrapolate.
Cover_2

Numeric. Cover of the layer to which we extrapolate. Use Inf for unlimited layers.
AttachmentPoint_2

Numeric. Attachment point of the layer to which we extrapolate.
alpha

Numeric. Pareto alpha used for the extrapolation.
ExpLoss_1

Numeric. Expected loss of the layer from which we extrapolate. If NULL (default) then the function provides only the ratio between the expected losses of the layers.
truncation

Numeric. If truncation is not NULL and truncation > AttachmentPoint_1, then the Pareto distribution is truncated at truncation.

Value

The expected loss of the layer Cover_2 xs AttachmentPoint_2 given that Cover_1 xs AttachmentPoint_1 has expected loss ExpLoss_1 and assuming a (truncated) Pareto distribution with parameters t and alpha. If missing then ExpLoss_1 == 1 is assumed.

References

Riegel, U. (2018) Matching tower information with piecewise Pareto. European Actuarial Journal 8(2): 437–460

Examples

Pareto_Extrapolation(1000, 1000, 2000, 2000, 2, ExpLoss_1 = 100)
Pareto_Extrapolation(1000, 1000, 2000, 2000, 2) * 100
Pareto_Extrapolation(1000, 1000, 2000, 2000, 2, truncation = 5000, ExpLoss_1 = 100)
Pareto_Extrapolation(1000, 1000, 2000, 2000, 2, truncation = 5000) * 100

Pareto Alpha Between a Frequency and a Layer

Description

Finds the Pareto alpha between an excess frequency and the expected loss of a layer

Usage

Pareto_Find_Alpha_btw_FQ_Layer(
  Threshold,
  Frequency,
  Cover,
  AttachmentPoint,
  ExpLoss,
  max_alpha = 100,
  tolerance = 1e-10,
  truncation = NULL
)

Arguments

Threshold

Numeric. Threshold
Frequency

Numeric. Expected frequency in excess of Thershold
Cover

Numeric. Cover of the second layer.
AttachmentPoint

Numeric. Attachment point of the layer.
ExpLoss

Numeric. Expected loss of the layer.
max_alpha

Numeric. Upper limit for the alpha that is returned.
tolerance

Numeric. Accuracy of the result.
truncation

Numeric. If truncation is not NULL then the Pareto distribution is truncated at truncation.

Value

The Pareto alpha between the expected number of claims Frequency excess Threshold and the layer Cover xs AttachmentPoint with expected loss ExpLoss

References

Riegel, U. (2018) Matching tower information with piecewise Pareto. European Actuarial Journal 8(2): 437–460

Examples

Pareto_Find_Alpha_btw_FQ_Layer(1000, 1, 1000, 1000, 500)
Pareto_Find_Alpha_btw_FQ_Layer(1000, 1, 1000, 1000, 500, truncation = 5000)

Pareto Alpha Between Two Frequencies

Description

Finds the Pareto alpha between two excess frequencies

Usage

Pareto_Find_Alpha_btw_FQs(
  Threshold_1,
  Frequency_1,
  Threshold_2,
  Frequency_2,
  max_alpha = 100,
  tolerance = 1e-10,
  truncation = NULL
)

Arguments

Threshold_1

Numeric. Threshold 1
Frequency_1

Numeric. Expected frequency in excess of Threshold_1
Threshold_2

Numeric. Threshold 2
Frequency_2

Numeric. Expected frequency in excess of Threshold_2
max_alpha

Numeric. Upper limit for the alpha that is returned.
tolerance

Numeric. Accuracy of the result.
truncation

Numeric. If truncation is not NULL then the Pareto distribution is truncated at truncation.

Value

The Pareto alpha between the expected number of claims Frequency_1 excess Threshold_1 and the expected number of claims Frequency_2 excess Threshold_2

References

Riegel, U. (2018) Matching tower information with piecewise Pareto. European Actuarial Journal 8(2): 437–460

Examples

Pareto_Find_Alpha_btw_FQs(1000, 1, 2000, 0.5)
Pareto_Find_Alpha_btw_FQs(1000, 1, 2000, 0.5, truncation = 5000)

Pareto Alpha Between Two Layers

Description

Finds the Pareto alpha between two layers

Usage

Pareto_Find_Alpha_btw_Layers(
  Cover_1,
  AttachmentPoint_1,
  ExpLoss_1,
  Cover_2,
  AttachmentPoint_2,
  ExpLoss_2,
  max_alpha = 100,
  tolerance = 1e-10,
  truncation = NULL
)

Arguments

Cover_1

Numeric. Cover of the first layer.
AttachmentPoint_1

Numeric. Attachment point of the first layer.
ExpLoss_1

Numeric. Expected loss of the first layer.
Cover_2

Numeric. Cover of the second layer.
AttachmentPoint_2

Numeric. Attachment point of the second layer.
ExpLoss_2

Numeric. Expected loss of the second layer.
max_alpha

Numeric. Upper limit for the alpha that is returned.
tolerance

Numeric. Accuracy of the result.
truncation

Numeric. If truncation is not NULL then the Pareto distribution is truncated at truncation.

Value

The Pareto alpha between the layer Cover_1 xs AttachmentPoint_1 with expected loss ExpLoss_1 and the layer Cover_2 xs AttachmentPoint_2 with expected loss ExpLoss_2

References

Riegel, U. (2018) Matching tower information with piecewise Pareto. European Actuarial Journal 8(2): 437–460

Examples

Pareto_Find_Alpha_btw_Layers(100, 100, 100, 200, 200, 50)
Pareto_Find_Alpha_btw_Layers(100, 100, 100, 200, 200, 50, truncation = 500)

Layer Mean of the Pareto Distribution

Description

Calculates the expected loss of a Pareto distribution in a reinsurance layer

Usage

Pareto_Layer_Mean(Cover, AttachmentPoint, alpha, t = NULL, truncation = NULL)

Arguments

Cover

Numeric. Cover of the reinsurance layer. Use Inf for unlimited layers.
AttachmentPoint

Numeric. Attachment point of the reinsurance layer.
alpha

Numeric. Pareto alpha.
t

Numeric. Threshold of the Pareto distribution. If t is NULL (default) then t <- Attachment Point is used.
truncation

Numeric. If truncation is not NULL and truncation > t, then the Pareto distribution is truncated at truncation.

Value

The expected loss of the (truncated) Pareto distribution with parameters t and alpha in the layer Cover xs AttachmentPoint

Examples

Pareto_Layer_Mean(4000, 1000, 2)
Pareto_Layer_Mean(4000, 1000, alpha = 2, t = 1000)
Pareto_Layer_Mean(4000, 1000, alpha = 2, t = 5000)
Pareto_Layer_Mean(4000, 1000, alpha = 2, t = 1000, truncation = 5000)
Pareto_Layer_Mean(9000, 1000, alpha = 2, t = 1000, truncation = 5000)

Second Layer Moment of the Pareto Distribution

Description

Calculates the second moment of a Pareto distribution in a reinsurance layer

Usage

Pareto_Layer_SM(Cover, AttachmentPoint, alpha, t = NULL, truncation = NULL)

Arguments

Cover

Numeric. Cover of the reinsurance layer. Use Inf for unlimited layers.
AttachmentPoint

Numeric. Attachment point of the reinsurance layer.
alpha

Numeric. Pareto alpha.
t

Numeric. Threshold of the Pareto distribution. If t is NULL (default) then t <- Attachment Point is used
truncation

Numeric. If truncation is not NULL and truncation > t, then the Pareto distribution is truncated at truncation.

Value

The second moment of the (truncated) Pareto distribution with parameters t and alpha in the layer Cover xs AttachmentPoint

Examples

Pareto_Layer_SM(4000, 1000, 2)
Pareto_Layer_SM(4000, 1000, alpha = 2, t = 1000)
Pareto_Layer_SM(4000, 1000, alpha = 2, t = 5000)
Pareto_Layer_SM(4000, 1000, alpha = 2, t = 1000, truncation = 5000)
Pareto_Layer_SM(9000, 1000, alpha = 2, t = 1000, truncation = 5000)

Layer Variance of the Pareto Distribution

Description

Calculates the variance of a Pareto distribution in a reinsurance layer

Usage

Pareto_Layer_Var(Cover, AttachmentPoint, alpha, t = NULL, truncation = NULL)

Arguments

Cover

Numeric. Cover of the reinsurance layer. Use Inf for unlimited layers.
AttachmentPoint

Numeric. Attachment point of the reinsurance layer.
alpha

Numeric. Pareto alpha.
t

Numeric. Threshold of the Pareto distribution. If t is NULL (default) then t <- Attachment Point is used.
truncation

Numeric. If truncation is not NULL and truncation > t, then the Pareto distribution is truncated at truncation.

Value

The variance of the(truncated) Pareto distribution with parameters t and alpha in the layer Cover xs AttachmentPoint

Examples

Pareto_Layer_Var(4000, 1000, 2)
Pareto_Layer_Var(4000, 1000, alpha = 2, t = 1000)
Pareto_Layer_Var(4000, 1000, alpha = 2, t = 5000)
Pareto_Layer_Var(4000, 1000, alpha = 2, t = 1000, truncation = 5000)
Pareto_Layer_Var(9000, 1000, alpha = 2, t = 1000, truncation = 5000)

Maximum Likelihood Estimation of the Alpha of a Pareto distribution

Description

Calculates the maximum likelihood estimator for the parameter alpha of a Pareto distribution with a known threshold and (if applicable) a known truncation

Usage

Pareto_ML_Estimator_Alpha(
  losses,
  t,
  truncation = NULL,
  reporting_thresholds = NULL,
  is.censored = NULL,
  weights = NULL,
  alpha_min = 0.001,
  alpha_max = 10
)

Arguments

losses

Numeric vector. Losses that are used for the ML estimation.
t

Numeric. Threshold of the Pareto distribution.
truncation

Numeric. If truncation is not NULL, then the Pareto distribution is truncated at truncation.
reporting_thresholds

Numeric vector. Allows to enter loss specific reporting thresholds. If NULL then all reporting thresholds are assumed to be less than or equal to t.
is.censored

Logical vector. TRUE indicates that a loss has been censored by the policy limit. The assumption is that the uncensored losses are Pareto distributed with the alpha we are looking for. is.censored = NULL means that no losses are censored.
weights

Numeric vector. Weights for the losses. For instance weights[i] = 2 and weights[j] = 1 for j != i has the same effect as adding another loss of size loss[i].
alpha_min

Numeric. Lower bound for alpha (only used in truncated case).
alpha_max

Numeric. Upper bound for alpha (only used in truncated case).

Value

Maximum likelihood estimator for the parameter alpha of a Pareto distribution with threshold t given the observations losses

Examples

losses <- rPareto(100, 1000, 2)
Pareto_ML_Estimator_Alpha(losses, 1000)
losses <- rPareto(100, 1000, 2, truncation = 2000)
Pareto_ML_Estimator_Alpha(losses, 1000)
Pareto_ML_Estimator_Alpha(losses, 1000, truncation = 2000)

t <- 100
alpha <- 2
losses <- rPareto(10000, t, alpha)
reporting_thresholds <- rPareto(10000, t, 5)
index <- losses > reporting_thresholds
losses <- losses[index]
reporting_thresholds <- reporting_thresholds[index]
Pareto_ML_Estimator_Alpha(losses, t)
Pareto_ML_Estimator_Alpha(losses, t, reporting_thresholds = reporting_thresholds)

losses <- rPareto(10, 1000, 2)
w <- rep(1, 10)
w[1] <- 3
losses2 <- c(losses, losses[1], losses[1])
Pareto_ML_Estimator_Alpha(losses, 1000, weights = w)
Pareto_ML_Estimator_Alpha(losses2, 1000)

Density of the Pareto Distribution

Description

Calculates the density function of the Pareto distribution. This function is deprecated. Use dPareto instead.

Usage

Pareto_PDF(x, t, alpha, truncation = NULL)

Arguments

x

Numeric. The function evaluates the density at x
t

Numeric. Threshold of the Pareto distribution.
alpha

Numeric. Pareto alpha.
truncation

Numeric. If truncation is not NULL and truncation > t, then the Pareto distribution is truncated at truncation.

Value

Density function of the Pareto distribution with parameters t and alpha evaluated at x

Examples

x <- 0:10 * 1000
dPareto(x, 1000, 2)
dPareto(x, 1000, 2, truncation = 5000)

Distribution Function of the generalized Pareto Distribution

Description

Calculates the cumulative distribution function of a generalized Pareto distribution

Usage

pGenPareto(x, t, alpha_ini, alpha_tail, truncation = NULL)

Arguments

x

Numeric. The function evaluates the CDF at x.
t

Numeric. Threshold of the generalized Pareto distribution.
alpha_ini

Numeric. Initial Pareto alpha.
alpha_tail

Numeric. Tail Pareto alpha.
truncation

Numeric. If truncation is not NULL and truncation > t, then the generalized Pareto distribution is truncated at truncation.

Value

Distribution function of the generalized Pareto distribution with parameters t, alpha_ini and alpha_tail evaluated at x

Examples

x <- 0:10 * 1000
pGenPareto(x, 1000, 1, 3)
pGenPareto(x, 1000, 1, 3, truncation = 5000)

PGP_Model (Collective Panjer & Generalized Pareto Model) Object

Description

Constructor function for the PGP_Model object

Usage

PGP_Model(
  FQ = NULL,
  t = NULL,
  alpha_ini = NULL,
  alpha_tail = NULL,
  truncation = NULL,
  dispersion = 1,
  Status = 0,
  Comment = "OK"
)

Arguments

FQ

Numerical. Expected claim count of the collective model.
t

Numeric. Threshold of the Pareto distribution. If t is NULL (default) then t <- Attachment Point is used
alpha_ini

Numeric. Initial Pareto alpha (at t).
alpha_tail

Numeric. Tail Pareto alpha.
truncation

Numeric. If truncation is not NULL and truncation > t, then the Pareto distribution is truncated at truncation.
dispersion

Numerical. Dispersion of the Panjer distribution (i.e. variance to mean ratio).
Status

Numerical indicator if a function returns a PGP_Model object: 0 = success, 1 = some information has been ignored, 2 = no solution found
Comment

Charakter. An optional comment.

Examples

PGPM <- PGP_Model(2, t = 1000, alpha_ini = 1, alpha_tail = 2 , dispersion = 2)
PGPM

Distribution Function of the Piecewise Pareto Distribution

Description

Calculates the cumulative distribution function of a Piecewise Pareto Distribution. This function is deprecated. Use pPiecewisePareto instead.

Usage

PiecewisePareto_CDF(x, t, alpha, truncation = NULL, truncation_type = "lp")

Arguments

x

Numeric. The function evaluates the CDF at x.
t

Numeric vector. Thresholds of the piecewise Pareto distribution.
alpha

Numeric vector. alpha[i] is the Pareto alpha in excess of t[i].
truncation

Numeric. If truncation is not NULL and truncation > t, then the distribution is truncated at truncation.
truncation_type

Character. If truncation_type = "wd" then the whole distribution is truncated. If truncation_type = "lp" then a truncated Pareto is used for the last piece.

Value

Distribution function of the piecewise Pareto distribution with parameter vectors t and alpha evaluated at x

References

Riegel, U. (2018) Matching tower information with piecewise Pareto. European Actuarial Journal 8(2): 437–460

Examples

t <- c(1000, 2000, 3000)
alpha <- c(1, 1.5, 2)
x <- 0:10 * 1000
pPiecewisePareto(x, t, alpha)
pPiecewisePareto(x, t, alpha, truncation = 5000, truncation_type = "lp")
pPiecewisePareto(x, t, alpha, truncation = 5000, truncation_type = "wd")

Layer Mean of the Piecewise Pareto Distribution

Description

Calculates the expected loss of a piecewise Pareto distribution in a reinsurance layer

Usage

PiecewisePareto_Layer_Mean(
  Cover,
  AttachmentPoint,
  t,
  alpha,
  truncation = NULL,
  truncation_type = "lp"
)

Arguments

Cover

Numeric. Cover of the reinsurance layer.
AttachmentPoint

Numeric. Attachment point of the reinsurance layer.
t

Numeric vector. Thresholds of the piecewise Pareto distribution.
alpha

Numeric vector. alpha[i] is the Pareto alpha in excess of t[i].
truncation

Numeric. If truncation is not NULL and truncation > t, then the Pareto distribution is truncated at truncation.
truncation_type

Character. If truncation_type = "wd" then the whole distribution is truncated. If truncation_type = "lp" then a truncated Pareto is used for the last piece.

Value

The expected loss of the (truncated) piecewise Pareto distribution with parameter vectors t and alpha in the layer Cover xs AttachmentPoint

References

Riegel, U. (2018) Matching tower information with piecewise Pareto. European Actuarial Journal 8(2): 437–460

Examples

t <- c(1000, 2000, 3000)
alpha <- c(1, 1.5, 2)
PiecewisePareto_Layer_Mean(4000, 1000, t, alpha)
PiecewisePareto_Layer_Mean(4000, 1000, t, alpha, truncation = 5000)
PiecewisePareto_Layer_Mean(4000, 1000, t, alpha, truncation = 5000, truncation_type = "lp")
PiecewisePareto_Layer_Mean(4000, 1000, t, alpha, truncation = 5000, truncation_type = "wd")

Second Layer Moment of the Piecewise Pareto Distribution

Description

Calculates the second moment of a piecewise Pareto distribution in a reinsurance layer

Usage

PiecewisePareto_Layer_SM(
  Cover,
  AttachmentPoint,
  t,
  alpha,
  truncation = NULL,
  truncation_type = "lp"
)

Arguments

Cover

Numeric. Cover of the reinsurance layer.
AttachmentPoint

Numeric. Attachment point of the reinsurance layer.
t

Numeric vector. Thresholds of the piecewise Pareto distribution.
alpha

Numeric vector. alpha[i] is the Pareto alpha in excess of t[i].
truncation

Numeric. If truncation is not NULL and truncation > t, then the Pareto distribution is truncated at truncation.
truncation_type

Character. If truncation_type = "wd" then the whole distribution is truncated. If truncation_type = "lp" then a truncated Pareto is used for the last piece.

Value

The second moment of the (truncated) piecewise Pareto distribution with parameter vectors t and alpha in the layer Cover xs AttachmentPoint

Examples

t <- c(1000, 2000, 3000)
alpha <- c(1, 1.5, 2)
PiecewisePareto_Layer_SM(4000, 1000, t, alpha)
PiecewisePareto_Layer_SM(4000, 1000, t, alpha, truncation = 5000)
PiecewisePareto_Layer_SM(4000, 1000, t, alpha, truncation = 5000, truncation_type = "lp")
PiecewisePareto_Layer_SM(4000, 1000, t, alpha, truncation = 5000, truncation_type = "wd")

Layer Variance of the Piecewise Pareto Distribution

Description

Calculate the variance of a piecewise Pareto distribution in a reinsurance layer

Usage

PiecewisePareto_Layer_Var(
  Cover,
  AttachmentPoint,
  t,
  alpha,
  truncation = NULL,
  truncation_type = "lp"
)

Arguments

Cover

Numeric. Cover of the reinsurance layer.
AttachmentPoint

Numeric. Attachment point of the reinsurance layer.
t

Numeric vector. Thresholds of the piecewise Pareto distribution.
alpha

Numeric vector. alpha[i] is the Pareto alpha in excess of t[i].
truncation

Numeric. If truncation is not NULL and truncation > t, then the Pareto distribution is truncated at truncation.
truncation_type

Character. If truncation_type = "wd" then the whole distribution is truncated. If truncation_type = "lp" then a truncated Pareto is used for the last piece.

Value

The variance of the (truncated) piecewise Pareto distribution with parameter vectors t and alpha in the layer Cover xs AttachmentPoint

Examples

t <- c(1000, 2000, 3000)
alpha <- c(1, 1.5, 2)
PiecewisePareto_Layer_Var(4000, 1000, t, alpha)
PiecewisePareto_Layer_SM(4000, 1000, t, alpha) - PiecewisePareto_Layer_Mean(4000, 1000, t, alpha)^2
PiecewisePareto_Layer_Var(4000, 1000, t, alpha, truncation = 5000)
PiecewisePareto_Layer_Var(4000, 1000, t, alpha, truncation = 5000, truncation_type = "lp")
PiecewisePareto_Layer_Var(4000, 1000, t, alpha, truncation = 5000, truncation_type = "wd")

Match a Tower of Expected Layers Losses

Description

Matches the expected losses of a tower of reinsurance layers using a piecewise Pareto severity

Usage

PiecewisePareto_Match_Layer_Losses(
  Attachment_Points,
  Expected_Layer_Losses,
  Unlimited_Layers = FALSE,
  Frequencies = NULL,
  FQ_at_lowest_AttPt = NULL,
  FQ_at_highest_AttPt = NULL,
  TotalLoss_Frequencies = NULL,
  minimize_ratios = TRUE,
  Use_unlimited_Layer_for_FQ = TRUE,
  truncation = NULL,
  truncation_type = "lp",
  dispersion = 1,
  tolerance = 1e-10,
  alpha_max = 100,
  merge_tolerance = 1e-06,
  RoL_tolerance = 1e-06
)

Arguments

Attachment_Points

Numeric vector. Vector containing the attachment points of consecutive layers in increasing order
Expected_Layer_Losses

Numeric vector. Vector containing the expected losses of layers xs the attachment points.
Unlimited_Layers

Logical. If TRUE, then Expected_Layer_Losses[i] contains the expected loss of Inf xs Attachment_Points[i]. If FALSE then Expected_Layer_Losses[i] contains the expected loss of the layer Attachment_Points[i+1] xs Attachment_Points[i]
Frequencies

Numeric vector. Expected frequencies excess the attachment points. The vector may contain NAs. If NULL then the function calculates frequencies.
FQ_at_lowest_AttPt

Numerical. Expected frequency excess Attachment_Points[1]. Overrules first entry in Frequencies.
FQ_at_highest_AttPt

Numerical. Expected frequency excess Attachment_Points[k]. Overrules last entry in Frequencies.
TotalLoss_Frequencies

Numeric vector. TotalLoss_Frequencies[i] is the frequency of total losses to layer i (i.e. Attachment_Points[i+1] - Attachment_Points[i] xs Attachment_Points[i]). TotalLoss_Frequencies[i] is the frequency for losses larger than or equal to Attachment_Points[i+1], whereas Frequencies[i] is the frequency of losses larger than Attachment_Points[i]. TotalLoss_Frequencies[i] > Frequencies[i+1] means that there is a point mass of the severity at Attachment_Points[i+1].
minimize_ratios

Logical. If TRUE then ratios between alphas are minimized.
Use_unlimited_Layer_for_FQ

Logical. Only relevant if no frequency is provided for the highest attachment point by the user. If TRUE then the frequency is calculated using the Pareto alpha between the last two layers.
truncation

Numeric. If truncation is not NULL, then the distribution is truncated at truncation.
truncation_type

Character. If truncation_type = "wd" then the whole distribution is truncated. If truncation_type = "lp" then a truncated Pareto is used for the last piece.
dispersion

Numerical. Dispersion of the claim count distribution in the resulting PPP_Model.
tolerance

Numeric. Numerical tolerance.
alpha_max

Numerical. Maximum alpha to be used for the matching.
merge_tolerance

Numerical. Consecutive Pareto pieces are merged if the alphas deviate by less than merge_tolerance.
RoL_tolerance

Numerical. Consecutive layers are merged if RoL decreases less than factor 1 - RoL_tolerance.

Value

A PPP_Model object that contains the information about a collective model with a Panjer distributed claim count and a Piecewise Pareto distributed severity. The object contains the following elements:

  • FQ Numerical. Frequency in excess of the lowest threshold of the piecewise Pareto distribution

  • t Numeric vector. Vector containing the thresholds for the piecewise Pareto distribution

  • alpha Numeric vector. Vector containing the Pareto alphas of the piecewise Pareto distribution

  • truncation Numerical. If truncation is not NULL and truncation > max(t), then the distribution is truncated at truncation.

  • truncation_type Character. If truncation_type = "wd" then the whole distribution is truncated. If truncation_type = "lp" then a truncated Pareto is used for the last piece.

  • dispersion Numerical. Dispersion of the Panjer distribution (i.e. variance to mean ratio).

  • Status Numerical indicator: 0 = success, 1 = some information has been ignored, 2 = no solution found

  • Comment Character. Information on whether the fit was successful

References

Riegel, U. (2018) Matching tower information with piecewise Pareto. European Actuarial Journal 8(2): 437–460

Examples

AP <- Example1_AP
EL <- Example1_EL
PiecewisePareto_Match_Layer_Losses(AP, EL)
EL_unlimited <- rev(cumsum(rev(Example1_EL)))
PiecewisePareto_Match_Layer_Losses(AP, EL_unlimited, Unlimited_Layers = TRUE)
PiecewisePareto_Match_Layer_Losses(AP, EL, FQ_at_lowest_AttPt = 0.5)
Example1_FQ <- c(0.3, 0.15, 0.08, 0.02, 0.005)
PiecewisePareto_Match_Layer_Losses(AP, EL, Frequencies = Example1_FQ)

Maximum Likelihood Estimation of the Alphas of the Piecewise Pareto Distribution

Description

Calculates the maximum likelihood estimator of the parameter vector alpha for a a piecewise Pareto distribution with given vector t and (if applicable) a known truncation

Usage

PiecewisePareto_ML_Estimator_Alpha(
  losses,
  t,
  truncation = NULL,
  truncation_type = "lp",
  reporting_thresholds = NULL,
  is.censored = NULL,
  weights = NULL,
  alpha_min = 0.001,
  alpha_max = 10
)

Arguments

losses

Numeric vector. Losses that are used for the ML estimation.
t

Numeric vector. Thresholds of the piecewise Pareto distribution.
truncation

Numeric. If truncation is not NULL and truncation > max(t), then the distribution is truncated at truncation.
truncation_type

Character. If truncation_type = "wd" then the whole distribution is truncated. If truncation_type = "lp" then a truncated Pareto is used for the last piece.
reporting_thresholds

Numeric vector. Allows to enter loss specific reporting thresholds. If NULL then all reporting thresholds are assumed to be less than or equal to t[1].
is.censored

Logical vector. TRUE indicates that a loss has been censored by the policy limit. The assumption is that the uncensored losses are piecewise Pareto distributed with the alphas we are looking for. is.censored = NULL means that no losses are censored.
weights

Numeric vector. Weights for the losses. For instance weights[i] = 2 and weights[j] = 1 for j != i has the same effect as adding another loss of size loss[i].
alpha_min

Numeric. Lower bound for the estimated alphas (only used in truncated case).
alpha_max

Numeric. Upper bound for the estimated alphas (only used in truncated case).

Value

Maximum likelihood estimator for the parameter alpha of a Pareto distribution with threshold t given the observations losses

Examples

losses <- rPiecewisePareto(10000, t = c(100,200,300), alpha = c(1,2,3))
PiecewisePareto_ML_Estimator_Alpha(losses, c(100,200,300))

losses <- rPiecewisePareto(10000, t = c(100,200,300), alpha = c(1,2,3),
                           truncation = 500, truncation_type = "wd")
PiecewisePareto_ML_Estimator_Alpha(losses, c(100,200,300))
PiecewisePareto_ML_Estimator_Alpha(losses, c(100,200,300),
                                 truncation = 500, truncation_type = "wd")
reporting_thresholds <- rPareto(10000, 100, 3)
index <- losses > reporting_thresholds
losses <- losses[index]
reporting_thresholds <- reporting_thresholds[index]
PiecewisePareto_ML_Estimator_Alpha(losses, c(100,200,300),
                                 truncation = 500, truncation_type = "wd")
PiecewisePareto_ML_Estimator_Alpha(losses, c(100,200,300),
                                 truncation = 500, truncation_type = "wd",
                                 reporting_thresholds = reporting_thresholds)

losses <- c(140, 240, 490, 200, 110, 710, 120, 190, 210, 310)
w <- rep(1, length(losses))
w[1] <- 2
losses2 <- c(losses, losses[1])
PiecewisePareto_ML_Estimator_Alpha(losses, c(100,200,300), weights = w)
PiecewisePareto_ML_Estimator_Alpha(losses2, c(100,200,300))

Density of the Piecewise Pareto Distribution

Description

Calculates the density function of the piecewise Pareto distribution. This function is deprecated. Use dPiecewisePareto instead.

Usage

PiecewisePareto_PDF(x, t, alpha, truncation = NULL, truncation_type = "lp")

Arguments

x

Numeric. The function evaluates the density at x.
t

Numeric vector. Thresholds of the piecewise Pareto distribution.
alpha

Numeric vector. alpha[i] is the Pareto alpha in excess of t[i].
truncation

Numeric. If truncation is not NULL and truncation > t, then the distribution is truncated at truncation.
truncation_type

Character. If truncation_type = "wd" then the whole distribution is truncated. If truncation_type = "lp" then a truncated Pareto is used for the last piece.

Value

Density function of the piecewise Pareto distribution with parameter vectors t and alpha evaluated at x

Examples

t <- c(1000, 2000, 3000)
alpha <- c(1, 1.5, 2)
x <- 0:10 * 1000
dPiecewisePareto(x, t, alpha)
dPiecewisePareto(x, t, alpha, truncation = 5000, truncation_type = "lp")
dPiecewisePareto(x, t, alpha, truncation = 5000, truncation_type = "wd")

Distribution Function of the Pareto Distribution

Description

Calculates the cumulative distribution function of a Pareto distribution

Usage

pPareto(x, t, alpha, truncation = NULL)

Arguments

x

Numeric. The function evaluates the CDF at x.
t

Numeric. Threshold of the Pareto distribution.
alpha

Numeric. Pareto alpha.
truncation

Numeric. If truncation is not NULL and truncation > t, then the Pareto distribution is truncated at truncation.

Value

Distribution function of the Pareto distribution with parameters t and alpha evaluated at x

Examples

x <- 0:10 * 1000
pPareto(x, 1000, 2)
pPareto(x, 1000, 2, truncation = 5000)

Distribution Function of the Piecewise Pareto Distribution

Description

Calculates the cumulative distribution function of a Piecewise Pareto Distribution

Usage

pPiecewisePareto(x, t, alpha, truncation = NULL, truncation_type = "lp")

Arguments

x

Numeric. The function evaluates the CDF at x.
t

Numeric vector. Thresholds of the piecewise Pareto distribution.
alpha

Numeric vector. alpha[i] is the Pareto alpha in excess of t[i].
truncation

Numeric. If truncation is not NULL and truncation > t, then the distribution is truncated at truncation.
truncation_type

Character. If truncation_type = "wd" then the whole distribution is truncated. If truncation_type = "lp" then a truncated Pareto is used for the last piece.

Value

Distribution function of the piecewise Pareto distribution with parameter vectors t and alpha evaluated at x

References

Riegel, U. (2018) Matching tower information with piecewise Pareto. European Actuarial Journal 8(2): 437–460

Examples

t <- c(1000, 2000, 3000)
alpha <- c(1, 1.5, 2)
x <- 0:10 * 1000
pPiecewisePareto(x, t, alpha)
pPiecewisePareto(x, t, alpha, truncation = 5000, truncation_type = "lp")
pPiecewisePareto(x, t, alpha, truncation = 5000, truncation_type = "wd")

PPP_Model (Collective Panjer & Piecewise Pareto Model) Object

Description

Constructor function for the PPP_Model object

Usage

PPP_Model(
  FQ = NULL,
  t = NULL,
  alpha = NULL,
  truncation = NULL,
  truncation_type = "lp",
  dispersion = 1,
  Status = 0,
  Comment = "OK"
)

Arguments

FQ

Numerical. Expected claim count of the collective model.
t

Numeric vector. Vector containing the thresholds of the Piecewise Pareto distribution.
alpha

Numeric vector. Vector containing the alphas of the Piecewise Pareto distribution.
truncation

Numerical. If truncation is not NULL and truncation > max(t), then the distribution is truncated at truncation.
truncation_type

Character. If truncation_type = "wd" then the whole distribution is truncated. If truncation_type = "lp" then a truncated Pareto is used for the last piece.
dispersion

Numerical. Dispersion of the Panjer distribution (i.e. variance to mean ratio).
Status

Numerical indicator if a function returns a PPP_Model object: 0 = success, 1 = some information has been ignored, 2 = no solution found
Comment

Charakter. An optional comment.

Examples

PPPM <- PPP_Model(2, c(1000,2000), c(1,2), dispersion = 2)
PPPM

Expected Frequency in Excess of a Threshold

Description

Calculates the expected frequency in excess of a threshold for a PPP_Model

Usage

PPP_Model_Excess_Frequency(x, PPP_Model)

Arguments

x

Numeric. Threshold.
PPP_Model

PPP_Model object.

Value

The expected frequency in excess of x for the given PPP_Model

Examples

PPPM <- PiecewisePareto_Match_Layer_Losses(Example1_AP, Example1_EL)
PPPM
Excess_Frequency(PPPM, c(-Inf, 0, 1000, 2000, 3000, Inf))

Expected Loss of a Reinsurance Layer

Description

Calculates the expected loss of a reinsurance layer for a PPP_Model. This function is deprecated. Use Layer_Mean instead.

Usage

PPP_Model_Exp_Layer_Loss(Cover, AttachmentPoint, PPP_Model)

Arguments

Cover

Numeric. Cover of the reinsurance layer. Use Inf for unlimited layers.
AttachmentPoint

Numeric. Attachment point of the reinsurance layer.
PPP_Model

PPP_Model object.

Value

The expected loss of the layer Cover xs AttachmentPoint for the given PPP_Model

Examples

PPPM <- PiecewisePareto_Match_Layer_Losses(Example1_AP, Example1_EL)
PPPM
Example1_Cov <- c(diff(Example1_AP), Inf)
Example1_AP
Example1_Cov
Example1_EL
Layer_Mean(PPPM, Example1_Cov, Example1_AP)

Standard Deviation of a Reinsurance Layer

Description

Calculates the standard deviation of the loss in a reinsurance layer for a PPP_Model

Usage

PPP_Model_Layer_Sd(Cover, AttachmentPoint, PPP_Model)

Arguments

Cover

Numeric. Cover of the reinsurance layer. Use Inf for unlimited layers.
AttachmentPoint

Numeric. Attachment point of the reinsurance layer.
PPP_Model

PPP_Model object.

Value

The standard deviation of the loss in the layer Cover xs AttachmentPoint for the given PPP_Model

Examples

PPPM <- PiecewisePareto_Match_Layer_Losses(Example1_AP, Example1_EL)
PPPM
Example1_Cov <- c(diff(Example1_AP), Inf)
Layer_Sd(PPPM, Example1_Cov, Example1_AP)

Variance of a Reinsurance Layer

Description

Calculates the variance of the loss in a reinsurance layer for a PPP_Model. This function is deprecated. Use Layer_Var instead.

Usage

PPP_Model_Layer_Var(Cover, AttachmentPoint, PPP_Model)

Arguments

Cover

Numeric. Cover of the reinsurance layer. Use Inf for unlimited layers.
AttachmentPoint

Numeric. Attachment point of the reinsurance layer.
PPP_Model

PPP_Model object.

Value

The variance of the loss in the layer Cover xs AttachmentPoint for the given PPP_Model

Examples

PPPM <- PiecewisePareto_Match_Layer_Losses(Example1_AP, Example1_EL)
PPPM
Example1_Cov <- c(diff(Example1_AP), Inf)
Layer_Var(PPPM, Example1_Cov, Example1_AP)

Simulate Losses with a PPP_Model

Description

Simulates losses of a PPP_Model

Usage

PPP_Model_Simulate(n, PPP_Model)

Arguments

n

Integer. Number of Simulations.
PPP_Model

PPP_Model object.

Value

A matrix where row k contains the simulated losses of the kth simulation.

Examples

PPPM <- PiecewisePareto_Match_Layer_Losses(c(1000, 2000, 3000), c(2000, 1000, 500),
                                           truncation = 10000, truncation_type = "wd")
PPPM
Simulate_Losses(PPPM, 100)

Print a PGP_Model Object (Collective Panjer & Generalized Pareto Model) Object

Description

Print method for PGP_Model objects

Usage

## S3 method for class 'PGP_Model'
print(x, ...)

Arguments

x

PGP_Model object.
...

Other arguments, all currently ignored.

Print a PPP_Model Object (Collective Panjer & Piecewise Pareto Model) Object

Description

Print method for PPP_Model objects

Usage

## S3 method for class 'PPP_Model'
print(x, ...)

Arguments

x

PPP_Model object.
...

Other arguments, all currently ignored.

Quantile Function of the generalized Pareto Distribution

Description

Calculates the quantile function of a generalized Pareto distribution

Usage

qGenPareto(p, t, alpha_ini, alpha_tail, truncation = NULL)

Arguments

p

Numeric. The function evaluates the inverse CDF at p.
t

Numeric. Threshold of the piecewise Pareto distribution.
alpha_ini

Numeric. Initial Pareto alpha.
alpha_tail

Numeric. Tail Pareto alpha.
truncation

Numeric. If truncation is not NULL and truncation > t, then the generalized Pareto distribution is truncated at truncation.

Value

Quantile function of the Pareto distribution with parameters t, alpha_ini and alpha_tail, evaluated at p

Examples

p <- 0:10 * 0.1
qGenPareto(p, 1000, 2, 3)
qGenPareto(p, 1000, 2, 3, truncation = 5000)

Quantile Function of the Pareto Distribution

Description

Calculates the quantile function of a Pareto distribution

Usage

qPareto(p, t, alpha, truncation = NULL)

Arguments

p

Numeric. The function evaluates the inverse CDF at p.
t

Numeric. Threshold of the piecewise Pareto distribution.
alpha

Numeric. Pareto alpha.
truncation

Numeric. If truncation is not NULL and truncation > t, then the Pareto distribution is truncated at truncation.

Value

Quantile function of the Pareto distribution with parameters t and alpha, evaluated at p

Examples

p <- 0:10 * 0.1
qPareto(p, 1000, 2)
qPareto(p, 1000, 2, truncation = 5000)

Quantile Function of the Piecewise Pareto Distribution

Description

Calculates the quantile function of a piecewise Pareto distribution

Usage

qPiecewisePareto(p, t, alpha, truncation = NULL, truncation_type = "lp")

Arguments

p

Numeric. The function evaluates the quantile function at p.
t

Numeric vector. Thresholds of the piecewise Pareto distribution.
alpha

Numeric vector. alpha[i] is the Pareto alpha in excess of t[i].
truncation

Numeric. If truncation is not NULL and truncation > t, then the distribution is truncated at truncation.
truncation_type

Character. If truncation_type = "wd" then the whole distribution is truncated. If truncation_type = "lp" then a truncated Pareto is used for the last piece.

Value

Quantile function of the piecewise Pareto distribution with parameter vectors t and alpha evaluated at p

Examples

t <- c(1000, 2000, 3000)
alpha <- c(1, 1.5, 2)
p <- 0:10 * 0.1
qPiecewisePareto(p, t, alpha)
qPiecewisePareto(p, t, alpha, truncation = 5000, truncation_type = "lp")
qPiecewisePareto(p, t, alpha, truncation = 5000, truncation_type = "wd")

Simulation of the generalized Pareto Distribution

Description

Generates random deviates of a generalized Pareto distribution

Usage

rGenPareto(n, t, alpha_ini, alpha_tail, truncation = NULL)

Arguments

n

Numeric. Number of observations.
t

Numeric vector. Thresholds of the generalized Pareto distributions
alpha_ini

Numeric vector. Initial Pareto alphas of the generalized Pareto distributions.
alpha_tail

Numeric vector. Tail Pareto alphas of the generalized Pareto distributions.
truncation

NULL or Numeric vector. If truncation is not NULL and truncation > t, then the generalized Pareto distributions are truncated at truncation (resampled generalized Pareto)

Value

A vector of n samples from the (truncated) generalized Pareto distribution with parameters t, alpha_ini and alpha_tail

Examples

rGenPareto(100, 1000, 2, 3)
rGenPareto(100, 1000, 2, 3, truncation = 2000)
rGenPareto(100, t = c(1, 10, 100, 1000), alpha_ini = 1, alpha_tail = c(2, 5))

Simulation of the Pareto Distribution

Description

Generates random deviates of a Pareto distribution

Usage

rPareto(n, t, alpha, truncation = NULL)

Arguments

n

Numeric. Number of observations.
t

Numeric vector. Thresholds of the Pareto distributions
alpha

Numeric vector. Pareto alphas of the Pareto distributions.
truncation

NULL or Numeric vector. If truncation is not NULL and truncation > t, then the Pareto distribution is truncated at truncation (resampled Pareto)

Value

A vector of n samples from the (truncated) Pareto distribution with parameters t and alpha

Examples

rPareto(100, 1000, 2)
rPareto(100, 1000, 2, truncation = 2000)
rPareto(100, t = c(1, 10, 100, 1000, 10000), alpha = c(1, 2, 4, 8, 16))

Simulation of the Piecewise Pareto Distribution

Description

Generates random deviates of a piecewise Pareto distribution

Usage

rPiecewisePareto(
  n,
  t,
  alpha,
  truncation = NULL,
  truncation_type = "lp",
  scale_pieces = NULL
)

Arguments

n

Numeric. Number of simulations
t

Numeric vector. Thresholds of the piecewise Pareto distribution.
alpha

Numeric vector. alpha[i] is the Pareto alpha in excess of t[i].
truncation

Numeric. If truncation is not NULL and truncation > t, then the distribution is truncated at truncation.
truncation_type

Character. If truncation_type = "wd" then the whole distribution is truncated. If truncation_type = "lp" then a truncated Pareto is used for the last piece.
scale_pieces

Numeric vector. If not NULL then the density of the i-th Pareto piece (on the interval (t[i], t[i+1])) is scaled with the factor const * scale_pieces[i] (where const is a normalization constant)

Value

A vector of n samples from the (truncated) piecewise Pareto distribution with parameter vectors t and alpha

Examples

t <- c(1000, 2000, 3000)
alpha <- c(1, 1.5, 2)
rPiecewisePareto(100, t, alpha)
rPiecewisePareto(100, t, alpha, truncation = 5000)
rPiecewisePareto(100, t, alpha, truncation = 5000, truncation_type = "lp")
rPiecewisePareto(100, t, alpha, truncation = 5000, truncation_type = "wd")

Simulate Losses with a Collective Model

Description

Simulates losses with a collective model

Usage

Simulate_Losses(CollectiveModel, nyears = 1)

Arguments

CollectiveModel

A collective model object. Currently only PPP_Models are handled.
nyears

Integer. Number of simulated years.

Value

A matrix where row k contains the simulated losses of the kth simulated year.

Examples

PPPM <- PiecewisePareto_Match_Layer_Losses(c(1000, 2000, 3000), c(2000, 1000, 500),
                                           truncation = 10000, truncation_type = "wd")
PPPM
Simulate_Losses(PPPM, 100)

Simulate Losses with a PGP_Model

Description

Simulates losses with a PGP_Model

Usage

## S3 method for class 'PGP_Model'
Simulate_Losses(CollectiveModel, nyears = 1)

Arguments

CollectiveModel

PGP_Model object.
nyears

Integer. Number of simulated years.

Value

A matrix where row k contains the simulated losses of the kth simulated year.

Examples

PGPM <- PGP_Model(2, 1000, 1, 2, dispersion = 2)
PGPM
Simulate_Losses(PGPM, 100)

Simulate Losses with a PPP_Model

Description

Simulates losses with a PPP_Model

Usage

## S3 method for class 'PPP_Model'
Simulate_Losses(CollectiveModel, nyears = 1)

Arguments

CollectiveModel

PPP_Model object.
nyears

Integer. Number of simulated years.

Value

A matrix where row k contains the simulated losses of the kth simulated year.

Examples

PPPM <- PiecewisePareto_Match_Layer_Losses(c(1000, 2000, 3000), c(2000, 1000, 500),
                                           truncation = 10000, truncation_type = "wd")
PPPM
Simulate_Losses(PPPM, 100)