EDHEC-Risk Institute October 2016

Multi-Dimensional Risk and Performance Analysis for Equity Portfolios — October 2016

1. Literature and Practice Reviews

We consider a set of assets with a covariance matrix Σ . Let R ( x ) be a risk measure of the portfolio x = ( x 1 ,…, x n ) . If this risk measure is homogenous of degree 1 in the weights, it satisfies the Euler principle, and we have

The problem of assigning correlated components is addressed by attributing half of each correlated component to each one of the two factors. In developed form, the risk contribution of factor k is:

(1.8)

Roncalli and Weisang (2012) define the risk contribution RC i of asset i as the product of the weight by the marginal risk:

In this approach, the factors’ covariance matrix Σ ƒ and the factors’ loading vector B used to compute risk contribution of each factor are not observable. We need to estimate covariances between factors, generating a covariance matrix estimate . The simplest estimate is the sample covariance matrix, but we can also apply an exponentially decreasing weighting scheme to historical observations; it allows placing more weight on recent observations. We estimate the factors loading vector by regressing portfolio returns on factor returns (ordinary least squares). Bhansali et al. (2012) point the benefits of using a factor-based diversification measure with respect to asset-based measures. Using a principal component analysis, they extract two risk factors driven by global growth and global inflation, and argue that these two risk factors dominate asset class risk and return. They extract the risk factors from a sample universe of 9 conventional assets: U.S. equities, International equities, EM equities, REITS, commodities, global bonds, U.S. long Treasury, investment grade corporate bonds and high yield bonds. Then, they decompose the return and the variance of an asset into the following two factors: the growth risk (linked with equity risk), the inflation risk (linked with bond risk) and a residual risk that is not spanned by equity and bond factors. For the variance decomposition, they equally divide the covariance term between the bond and the equity components. The above two-factor variance decomposition

If we use the volatility of the portfolio as the risk measure, it follows that the contribution of the asset i to the portfolio volatility is:

Roncalli and Weisang (2012) define also the risk contribution with respect to the factors. Here, we focus on the contribution of the factor k to the systematic variation of portfolio (the fraction of variance explained by the factors). With the previous notations, we have the following risk contribution of factor k : ,

where the risk measure is the systematic variance of portfolio returns: .

In the presence of uncorrelated factors, there is no correlated component in the factors’ variance-covariance matrix and we have the following risk contribution of factor k :

with

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