fmEsDecomp {facmodTS}

Decompose ES into individual factor contributions

Description

Compute the factor contributions to Expected Tail Loss or Expected Shortfall (ES) of assets' returns based on Euler's theorem, given the fitted factor model. The partial derivative of ES with respect to factor beta is computed as the expected factor return given fund return is less than or equal to its value-at-risk (VaR). Option to choose between non-parametric and Normal.

Usage

fmEsDecomp(object, ...)

## S3 method for class 'tsfm'
fmEsDecomp(
  object,
  factor.cov,
  p = 0.05,
  type = c("np", "normal"),
  use = "pairwise.complete.obs",
  ...
)

## S3 method for class 'sfm'
fmEsDecomp(
  object,
  factor.cov,
  p = 0.05,
  type = c("np", "normal"),
  use = "pairwise.complete.obs",
  ...
)

## S3 method for class 'ffm'
fmEsDecomp(
  object,
  factor.cov,
  p = 0.05,
  type = c("np", "normal"),
  use = "pairwise.complete.obs",
  ...
)

Arguments

object

fit object of class tsfm, sfm or ffm.

...

other optional arguments passed to quantile.

factor.cov

optional user specified factor covariance matrix with named columns; defaults to the sample covariance matrix.

p

tail probability for calculation. Default is 0.05.

type

one of "np" (non-parametric) or "normal" for calculating VaR. Default is "np".

use

method for computing covariances in the presence of missing values; one of "everything", "all.obs", "complete.obs", "na.or.complete", or "pairwise.complete.obs". Default is "pairwise.complete.obs".

Details

The factor model for an asset's return at time t has the form

R(t) = beta'f(t) + e(t) = beta.star'f.star(t)

where, beta.star=(beta,sig.e) and f.star(t)=[f(t)',z(t)]'. By Euler's theorem, the ES of the asset's return is given by:

ES.fm = sum(cES_k) = sum(beta.star_k*mES_k)

where, summation is across the K factors and the residual, cES and mES are the component and marginal contributions to ES respectively. The marginal contribution to ES is defined as the expected value of F.star, conditional on the loss being less than or equal to VaR.fm. This is estimated as a sample average of the observations in that data window.

Refer to Eric Zivot's slides (referenced) for formulas pertaining to the calculation of Normal ES (adapted from a portfolio context to factor models).

Value

A list containing

ES.fm

length-N vector of factor model ES of N-asset returns.

mES

N x (K+1) matrix of marginal contributions to VaR.

cES

N x (K+1) matrix of component contributions to VaR.

pcES

N x (K+1) matrix of percentage component contributions to VaR.

Where, K is the number of factors and N is the number of assets.

Author(s)

Eric Zviot, Sangeetha Srinivasan and Yi-An Chen

References

Epperlein, E., & Smillie, A. (2006). Portfolio risk analysis Cracking VAR with kernels. RISK-LONDON-RISK MAGAZINE LIMITED-, 19(8), 70.

Hallerback (2003). Decomposing Portfolio Value-at-Risk: A General Analysis. The Journal of Risk, 5(2), 1-18.

Meucci, A. (2007). Risk contributions from generic user-defined factors. RISK-LONDON-RISK MAGAZINE LIMITED-, 20(6), 84.

Yamai, Y., & Yoshiba, T. (2002). Comparative analyses of expected shortfall and value-at-risk: their estimation error, decomposition, and optimization. Monetary and economic studies, 20(1), 87-121.

See Also

fitTsfm for the different factor model fitting functions.

fmSdDecomp for factor model SD decomposition. fmVaRDecomp for factor model VaR decomposition.

Examples

 # Time Series Factor Model
 # load data
data(managers, package = 'PerformanceAnalytics')

fit.macro <- fitTsfm(asset.names=colnames(managers[,(1:6)]),
                     factor.names=colnames(managers[,(7:8)]), 
                     data=managers)
                     
ES.decomp <- fmEsDecomp(fit.macro)

# get the component contributions
ES.decomp$cES