fmSdDecomp {facmodTS}

Decompose standard deviation into individual factor contributions

Description

Compute the factor contributions to standard deviation (SD) of assets' returns based on Euler's theorem, given the fitted factor model.

Usage

fmSdDecomp(object, ...)

## S3 method for class 'tsfm'
fmSdDecomp(object, factor.cov, use = "pairwise.complete.obs", ...)

## S3 method for class 'sfm'
fmSdDecomp(object, factor.cov, use = "pairwise.complete.obs", ...)

## S3 method for class 'ffm'
fmSdDecomp(object, factor.cov, ...)

Arguments

object

fit object of class tsfm or ffm.

...

optional arguments passed to cov.

factor.cov

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

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 standard deviation of the asset's return is given as:

Sd.fm = sum(cSd_k) = sum(beta.star_k*mSd_k)

where, summation is across the K factors and the residual, cSd and mSd are the component and marginal contributions to SD respectively. Computing Sd.fm and mSd is very straight forward. The formulas are given below and details are in the references. The covariance term is approximated by the sample covariance.

Sd.fm = sqrt(beta.star''cov(F.star)beta.star)
mSd = cov(F.star)beta.star / Sd.fm

Value

A list containing

Sd.fm

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

mSd

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

cSd

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

pcSd

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

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

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.

fmCov for factor model covariance. fmVaRDecomp for factor model VaR decomposition. fmEsDecomp for factor model ES decomposition.

Examples

# Time Series Factor Model

 # load data
data(managers, package = 'PerformanceAnalytics')
colnames(managers)
 # Make syntactically valid column names
colnames(managers) <- make.names( colnames(managers))
colnames(managers)

fit.macro <- fitTsfm(asset.names=colnames(managers[,(1:6)]),
                     factor.names=colnames(managers[,(7:9)]),
                     rf.name="US.3m.TR", data=managers)
decomp <- fmSdDecomp(fit.macro)
# get the percentage component contributions
decomp$pcSd