EDHEC-Risk Institute October 2016

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

1. Literature and Practice Reviews

For an equally- or cap-weighted portfolio, the sector portfolios are themselves equally- or cap-weighted. This allows us to express the return on sector portfolio j as:

This method allows us to decompose the conditional expected return of the portfolio p , conditionally on the sector allocation at date t in the portfolio.

We can now rewrite the return on portfolio p as a linear combination of returns on sector portfolios:

The next step is to decompose the returns on sector portfolios according to the Carhart model:

(1.13)

This expression can be immediately extended to the case of time-varying factor premia. Finally we decompose the conditional portfolio risk at date t :

Substituting this expression for each sector in the expression for the portfolio returns:

where

For an easier risk decomposition, we make the assumption that Carhart factors are uncorrelated. This assumption can always be satisfied by applying a suitable rotation to the factors to make them orthogonal (see Section 1.2.2.3). Hence the factors covariance matrix is a diagonal matrix and the risk decomposition is rewritten as follows:

By identifying factor loadings we obtain the following expressions for the portfolio alpha and beta:

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