EDHEC-Risk Institute October 2016

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

1. Literature and Practice Reviews

We then proceed to performance and risk decomposition. Taking sample averages in both sides of the previous equation, we obtain, for a portfolio (or an individual stock) p :

of their common loadings on the market factor, and eliminating the effect of this factor reduces their correlations. To give a quantitative sense of this effect, the average correlation across the 10 long-only sector returns is 0.72, while it is only 0.37 between the pure sector returns. This reduction in the correlation levels enables us to disentangle the effects of sectors on the portfolio more reliably (i.e. with larger t-statistics). Finally, the method can be straightforwardly extended to include other custom factors such as country benchmarks, in order to capture the effects of geography on performance and risk. To this end, we regress equally-weighted country portfolios returns R c on the market factor returns and we form the pure country effect ν c .

(1.11)

By construction, residuals are orthogonal from factors, so the portfolio variance can be written as:

We decompose the systematic portfolio risk using the Euler decomposition of volatility (see Section 1.2.2.2). denotes the sample covariance matrix of all factors (the market plus the sectors and the additional pricing factors) and we have the following risk contribution of factor k to the portfolio risk: These decompositions hold ex-post, but similar expressions can be written down for ex-ante expected return and variance. They can also be applied by replacing portfolio returns with returns in excess over a benchmark in an attempt to provide a statistical explanation for the outperformance or the underperformance of the portfolio with respect to the benchmark and for the tracking error. This method alleviates the multi-collinearity concern. By eliminating the market from the additional factors (Carhart factors and sectors), it drastically reduces their level of pairwise correlation. Collinearity is less an issue for the Carhart factors, which are already long-short returns with moderate correlations, than for the sectors, which are long-only returns. Much of the correlation between the sectors arises indeed because (1.12)

The more comprehensive model, with all pricing factors, sector and country effects, reads:

Empirical Illustration We illustrate this method by taking as a test portfolio the equally-weighted portfolio of the S&P 500 universe over the period 2002-2015 and the market factor from Ken French’s library. The analysis is conducted with quarterly returns, and the universe’s constituents are classified into ten sectors according to the TRBC classification: Energy, Basic Materials, Industrials, Cyclical Consumer, Non-Cyclical Consumer, Financials, Healthcare, Technology, Telecoms and Utilities. We define sector returns as the returns to equally-weighted portfolios

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