EDHEC-Risk Institute October 2016

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

1. Literature and Practice Reviews

Figure 7: Absolute and Relative Risk (Volatility) Decomposition Using Euler Decomposition for the Equally-Weighted Portfolio of the S&P 500 Universe on Market, Carhart and Sector Factors Factor returns are from Ken French’s library. Sector returns are equally-weighted portfolios from the S&P 500 universe. We regress quarterly excess returns to each sector and factor portfolio on the market factor returns over the period 2002-2015 in order to extract pure sector and factor effects. We then regress equally-weighted portfolio excess returns on the market factor returns augmented with the pure sectors and factor effects, so as to obtain the factor exposures. For absolute performance, equally-weighted portfolio returns are in excess of the risk-free rate; for relative performance, they are in excess of market returns. We use formula (1.12) to make the risk attribution.

1.3.3 Using the Sector Composition In this approach, a portfolio is viewed as a bundle of sector portfolios, which are themselves projected on risk factors. In this sense, the method is both holding-based (it relies on the expression of the portfolio return as the weighted sum of constituents’ return) and return-based (the sector returns are regressed against the factors). Portfolio weights are usually time-varying, due to rebalancing and price changes. Hence, we now decompose the ex-ante return and volatility conditional on the weights of a given date, as opposed to writing an ex-post decomposition for the realised return and volatility over a period. Thus, the decomposition does not apply to past performance and risk but instead to the expected return and risk, estimated in a forward-looking way.

The starting point is the relation between the return to a portfolio p and those of its N constituents. Stocks can be grouped in the S sectors in this expression:

where w i,t- 1

denotes the weight of asset i

in portfolio p at period t –1, r i,t the return on stock i at period t , and is a dummy variable equal to 1 if stock i belongs to sector j at period t , and 0 otherwise. We then define as the j -th sector weight in portfolio p and as the weight of asset i within sector j :

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