EDHEC-Risk Institute October 2016

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

1. Literature and Practice Reviews

In this section, we start with a reminder on the standard factor models that have been developed in the academic literature. The purpose of these models is to identify the systematic risk factors that explain the differences in expected returns across financial assets. Historically, this has been achieved through theoretical analysis based on economic or statistical arguments, or through empirical studies of the determinants of expected returns. We then show how one of these models, namely the four-factor model of Carhart (1997), can be used to decompose the ex-post performance and volatility of a portfolio. However, industry practices often rely on models that involve a much larger number of factors, such as Barra-type models, to analyse performance and risk. We review these models below, by giving attention to the multi-collinearity issues raised by the simultaneous inclusion of numerous factors, and we provide examples of multi- dimensional decomposition across risk factors and sectors on US equity portfolios. 1.1 Factor Model Theory This section is a reminder on the standard factor models that have been developed in the academic literature. A complete presentation of the theory can be found in Cochrane (2005), who relates the factors to the stochastic discount factor. While our paper is focused on equity portfolios, factor models can be in principle developed for other asset classes, such as bonds and commodities. 1.1.1 The Single-Factor Model The CAPM of Sharpe (1963, 1964) and Treynor (1961) is a model for pricing an individual security or portfolio. In this model, the differences in expected returns across securities are explained by their respective sensitivities to a single factor,

which is the return on the market portfolio. At equilibrium, the returns on assets less the risk-free rate are proportional to their market beta. Mathematically, the CAPM relationship at equilibrium is written as follows:

(1.1)

where R i denotes the excess return on asset i (in excess of the risk-free rate), R m denotes the excess return on the market portfolio. B i , the exposure to the market factor, is defined by where σ i,m denotes the covariance between asset i and the market portfolio and denotes the variance of the market portfolio. The correct measure of risk for an individual asset is therefore the beta, and the reward per unit of risk taken is called the risk premium. The asset betas can be aggregated: the beta of a portfolio is obtained as a linear combination of the betas of the assets that make up the portfolio. In this model, the asset beta is the only driver of the expected return on a stock. Accurate measurement of market exposure is critical for important issues such as performance and risk measurement. However, the model is based on very strong theoretical assumptions which are not satisfied by the market in practice. The model assumes, inter alia, that investors have the same horizon and expectations. Moreover, the prediction of an increasing link between the expected return and the market beta is not validated by examination of the data: the relation tends to be actually decreasing (Frazzini and Pedersen, 2014), and there are a number of empirical regularities in expected returns, such as the size, value and momentum effects, that cannot be explained with the CAPM and thus constitute anomalies for the model. 3

3 - Empirical evidence has linked variations in the cross-section of stock returns to firm characteristics such as market capitalisation and book-to-market values (Fama and French, 1992, 1993) and

short term continuation effect in stock returns (Carhart, 1997).

22

An EDHEC-Risk Institute Publication