fmVaRDecomp: Decompose VaR into individual factor contributions

In factorAnalytics: Factor Analytics

Description

Compute the factor contributions to Value-at-Risk (VaR) of assets' returns based on Euler's theorem, given the fitted factor model. The partial derivative of VaR w.r.t. factor beta is computed as the expected factor return given fund return is equal to its VaR and approximated by a kernel estimator. Option to choose between non-parametric and Normal.

Usage

fmVaRDecomp(object, ...)

## S3 method for class 'tsfm'
fmVaRDecomp(object, factor.cov, p = 0.95, type = c("np",
  "normal"), use = "pairwise.complete.obs", ...)

## S3 method for class 'sfm'
fmVaRDecomp(object, factor.cov, p = 0.95, type = c("np",
  "normal"), use = "pairwise.complete.obs", ...)

## S3 method for class 'ffm'
fmVaRDecomp(object, factor.cov, p = 0.95, 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	confidence level for calculation. Default is 0.95.
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 VaR of the asset's return is given by:

VaR.fm = sum(cVaR_k) = sum(beta.star_k*mVaR_k)

where, summation is across the K factors and the residual, cVaR and mVaR are the component and marginal contributions to VaR respectively. The marginal contribution to VaR is defined as the expectation of F.star, conditional on the loss being equal to VaR.fm. This is approximated as described in Epperlein & Smillie (2006); a triangular smoothing kernel is used here.

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

Value

A list containing

VaR.fm	length-N vector of factor model VaRs of N-asset returns.
n.exceed	length-N vector of number of observations beyond VaR for each asset.
idx.exceed	list of numeric vector of index values of exceedances.
mVaR	N x (K+1) matrix of marginal contributions to VaR.
cVaR	N x (K+1) matrix of component contributions to VaR.
pcVaR	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 Zivot, Yi-An Chen and Sangeetha Srinivasan

References

Eric Zivot's slides from CFRM 546: Estimating risk measures: Portfolio of Assets, April 28, 2015.

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, fitSfm, fitFfm for the different factor model fitting functions.

fmSdDecomp for factor model SD decomposition. fmEsDecomp for factor model ES decomposition.

Examples

# Time Series Factor Model
data(managers)
fit.macro <- fitTsfm(asset.names=colnames(managers[,(1:6)]),
                     factor.names=colnames(managers[,(7:8)]), data=managers)
                     
VaR.decomp <- fmVaRDecomp(fit.macro)
# get the component contributions
VaR.decomp$cVaR

# Statistical Factor Model
data(StockReturns)
sfm.pca.fit <- fitSfm(r.M, k=2)

VaR.decomp <- fmVaRDecomp(sfm.pca.fit, type="normal")
VaR.decomp$cVaR

# Fundamental Factor Model
data(Stock.df)
exposure.vars <- c("BOOK2MARKET", "LOG.MARKETCAP")
fit <- fitFfm(data=stock, asset.var="TICKER", ret.var="RETURN", 
              date.var="DATE", exposure.vars=exposure.vars)

VaR.decomp <- fmVaRDecomp(fit, type="normal")
VaR.decomp$cVaR

factorAnalytics documentation built on April 15, 2017, 11:18 a.m.