PiecewisePareto_Match_Layer_Losses {Pareto}

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)