EDHEC-Risk Institute October 2016

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

1. Literature and Practice Reviews

• ε t denotes the error term which is uncorrelated with each of the common factors and vector of error term is serially uncorrelated.

beta coefficients but also on the factors’ variance-covariance matrix. Since the coefficient of determination is a measure of the systematic risk of portfolio, extracting the core, stand alone components of common factors enables us to decompose the systematic risk by disentangling the R-squared, based on factors' volatility and their corresponding betas. In the presence of uncorrelated regressors (factors), the R 2 of a multiple regression is the sum of the R 2 of individual regressions, and the contribution of each factor to the variance of the portfolio is simply measured by the individual R 2 . In Section 1.2.2.2 and Section 1.2.2.3, we present two methods for identifying within the explained component what the contribution of each factor is to the portfolio variance, in the more general case of correlated factors. Tracking Error Portfolio risk is also frequently measured relative to a benchmark: the tracking error is defined as the standard deviation of the difference between the portfolio and benchmark returns. In a return-based factor analysis, the methodology is similar to that employed for absolute risk decomposition, but it is the portfolio returns in excess of a benchmark that are regressed against long/short factor indices:

σ ( R p - R B ) denotes the tracking error of the portfolio against its benchmark.

(1.7)

The coefficient of determination from our model is the ratio of systematic variation to the total return variation:

Thus, any method to decompose absolute risk can also be applied to the decomposition of relative risk, provided one considers the betas of returns in excess of the benchmark portfolio as opposed to the betas of returns in excess of the risk-free rate. 1.2.2.2 First Decomposition: Using Euler Decomposition of Volatility This approach is the one adopted in risk budgeting methodologies for constructing portfolios: the definition of the contributions of various assets to the volatility of a portfolio is based on a mathematical property of volatility, known as Euler decomposition (see Roncalli (2013), and also Qian (2006) for the economic interpretation of such risk decompositions). It is adapted here to a different context, where we are seeking to assess the contribution of different factors to the risk of a portfolio (the two main differences being that the betas do not sum up to one, and that there is a residual term in factor analysis, which is not present when decomposing a portfolio return into the weighted sum of the components' returns).

• R p,t is the return on portfolio minus the return on benchmark index in time period t ( t = 1,..., T ) • is the factor loading or factor beta for the portfolio in excess of benchmark portfolio on the k -th factor, B is the vector of beta. - R B,t

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