R/Overbeck2.R

In OCA: Optimal Capital Allocations

#' Overbeck type II Allocation Principle
#'
#' @description
#' \loadmathjax{}
#' This function implements the Overbeck type II allocation principle for optimal capital allocation.
#'
#' @param Loss Either a scalar or a vector of size \emph{N} containing the mean losses.
#' @param Capital A scalar representing the capital to be allocated to each loss.
#' @param alpha A numeric value (either a single one or a vector) consisting of the significance
#' level at which the allocation has to be computed, it can either be a single numeric value or a
#' vector of numeric values.
#' @param model A character string indicating which distribution is to be used for computing the
#' VaR underlying the Overbeck type II principle, the default value is the \code{normal} distribution,
#' the other alternative is \code{t-student} distribution with \eqn{\upsilon}{\upsilon} degrees of freedom.
#' When \code{model='both'} \code{'normal'} as well as \code{'t-student'} are used when computing the
#' allocations, see examples.
#' @param df An integer indicating the degrees of freedom for the t-student distribution when setting
#' \code{model='t-student'} and \code{model='both'}. \code{df} must be greater than 2.
#'
#' @details
#' \code{Overbeck2} computes the  capital allocation based on the following formulation:
#'
#' \mjtdeqn{K_{i} = \frac{K}{CTE_{p}[S]} E \left[ X_{i}|S > F_{X_{S}}^{-1}(p) \right], \quad i=1, \ldots, n.}{K_{i} = \frac{K}{CTE_{p}[S]} E \left[ X_{i}|S > F_{X_{S}}^{-1}(p) \right], \quad i=1, \ldots, n.}{}
#'
#' Where \mjteqn{K}{K}{} is the aggregate capital to be allocated, \mjteqn{CTE_{p}[S]}{CTE_p[S] }{} is the
#' Conditional Tail Expectation of the aggregate loss at level \mjteqn{p}{p}{}, \mjteqn{X_{i}}{X_i}{} is the
#' individual loss, \mjteqn{S}{S}{} is the aggregate loss and \mjteqn{F_{X}^{-1}(p)}{ F_X^-1(p)}{} is the quantile
#' function of \mjteqn{X}{X}{} at level  \mjteqn{p.}{p.}{}
#'
#' @return
#' A vector containing the optimal capital allocation,
#' if \code{Capital} is set to 1, then the returned matrix will consist of the proportions of capital
#' each individual loss needs to be optimally faced.
#'
#'@references
#' Dhaene J., Tsanakas A., Valdez E. and Vanduffel S. (2011). \emph{Optimal Capital Allocation Principles}. The Journal of Risk and Insurance. Vol. 00, No. 0, 1-28.
#'
#' Urbina, J. (2013) \emph{Quantifying Optimal Capital Allocation Principles based on Risk Measures.} Master Thesis, Universitat Politècnica de Catalunya.
#'
#' Urbina, J. and Guillén, M. (2014). \emph{An application of capital allocation principles to operational risk and the cost of fraud}. Expert Systems with Applications. 41(16):7023-7031.
#'
#' @seealso
#' \code{\link{hap}}, \code{\link{cap}}
#'
#' @author Jilber Urbina
#' @noMd
#' @export
#' @importFrom mathjaxr preview_rd
#' @importFrom stats cov dnorm dt qnorm qt var
#' @examples
#'data(dat1, dat2)
#'Loss <- cbind(Loss1=dat1[1:400, ], Loss2=unname(dat2))
#' # Proportions of capital to be allocated to each bussines unit
#' Overbeck2(Loss, Capital=1)
#'
#' # Capital allocation,
#' # capital is determined as the empirical VaR of the losses at 99\%
#' K <- quantile(rowSums(Loss),  probs = 0.99)
#' Overbeck2(Loss, Capital=K)





Overbeck2 <- function(Loss, Capital, alpha=0.95, model=c("normal", "t-student", "both"), df=NULL){
  K <- as.numeric(Capital)
  L <- as.matrix(Loss)
  alpha <- as.numeric(alpha)
  S <- rowSums(L)

  # VaR alpha%: F^{-1}_{S}(p)=VaR
  VaRp <- as.numeric(VaR(variance = var(S), alpha=alpha, model=model, df=df))

  # Funcion indicador: S > VaR
  ind <- S > VaRp

  # Capital allocation
  return(K*(colMeans(L[ind, ,drop=FALSE]) / mean(S[ind])))

}