Overbeck2: Overbeck type II Allocation Principle

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.

Usage

Overbeck2(
  Loss,
  Capital,
  alpha = 0.95,
  model = c("normal", "t-student", "both"),
  df = NULL
)

Arguments

Loss	Either a scalar or a vector of size N containing the mean losses.
Capital	A scalar representing the capital to be allocated to each loss.
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.
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 normal distribution, the other alternative is t-student distribution with υ degrees of freedom. When model='both' 'normal' as well as 't-student' are used when computing the allocations, see examples.
df	An integer indicating the degrees of freedom for the t-student distribution when setting model='t-student' and model='both'. df must be greater than 2.

Details

Overbeck2 computes the capital allocation based on the following formulation:

\mjtdeqn

K_i = \fracKCTE_p[S] E \left[ X_i|S > F_X_S^-1(p) \right], \quad i=1, ..., n.K_i = \fracKCTE_p[S] E \left[ X_i|S > F_X_S^-1(p) \right], \quad i=1, ..., n.

Where \mjteqnKK is the aggregate capital to be allocated, \mjteqnCTE_p[S]CTE_p[S] is the Conditional Tail Expectation of the aggregate loss at level \mjteqnpp, \mjteqnX_iX_i is the individual loss, \mjteqnSS is the aggregate loss and \mjteqnF_X^-1(p) F_X^-1(p) is the quantile function of \mjteqnXX at level \mjteqnp.p.

Value

A vector containing the optimal capital allocation, if Capital is set to 1, then the returned matrix will consist of the proportions of capital each individual loss needs to be optimally faced.

Author(s)

Jilber Urbina

References

Dhaene J., Tsanakas A., Valdez E. and Vanduffel S. (2011). Optimal Capital Allocation Principles. The Journal of Risk and Insurance. Vol. 00, No. 0, 1-28.

Urbina, J. (2013) Quantifying Optimal Capital Allocation Principles based on Risk Measures. Master Thesis, Universitat Politècnica de Catalunya.

Urbina, J. and Guillén, M. (2014). An application of capital allocation principles to operational risk and the cost of fraud. Expert Systems with Applications. 41(16):7023-7031.

See Also

hap, cap

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)

OCA documentation built on Feb. 16, 2023, 8:27 p.m.