R: Decompose standard deviation into individual factor...

fmSdDecomp {facmodCS}

R Documentation

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, factor.cov, ...)

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

Arguments

`object`	fit object of class `tsfm` or `ffm`.
`factor.cov`	optional user specified factor covariance matrix with named columns; defaults to the sample covariance matrix.
`...`	optional arguments passed to `cov`.

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.