EDHEC-Risk Institute October 2016

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

1. Literature and Practice Reviews

1.1.2 Multi-Factor Models and the Arbitrage Pricing Theory The current consensus tends towards the idea that a single factor is not sufficient for explaining the cross-section of expected returns. In 1976, Ross introduced the Arbitrage Pricing Theory for the purpose of valuing assets under the assumptions that there is no arbitrage and that asset returns can be decomposed into a systematic part and an idiosyncratic part. Unlike in the CAPM, no assumption is made regarding the investment decisions made by individual agents. This model is also linear and employs K factors and it nests the CAPM as a special case. The APT model postulates that a linear relationship exists between the realised returns of the assets and the K common factors: ) denotes the expected return for asset i • B i,k denotes the sensitivity (or exposure) of asset i to factor k • F k,t denotes the return on factor k at period t with E ( F k )=0 • ε i,t denotes the residual (or specific) return of asset i , i.e. the fraction of return that is not explained by the factors, with E ( ε i )=0. The residuals returns of the different assets are assumed to be uncorrelated from each other and uncorrelated from the factors. (1.2) • R i,t denotes the excess return for asset i in period t • E ( R i

Arbitrage reasoning then allows us to end up with the following equilibrium relationship:

(1.3)

where r ƒ denotes the risk-free rate. This relationship explains the average asset return as a function of the exposures to different risk factors and the market’s remuneration for those factors. The B i,k are the sensitivities to the factors (factor loadings) and can be obtained by regressing realised excess returns on factors. λ k is interpreted as the factor k risk premium. Multi-factor models do not explicitly indicate the number or nature of the factors. We discuss this question in Section 1.1.3. The APT suggests two ways for the search for meaningful factors. First, the requirement of having zero, or at least low, correlation between idiosyncratic risks calls for identifying the common sources of risk in returns. From a statistical standpoint, this is equivalent to having high R 2 and significant betas in the regression Equation (1.2). Second, the linear relationship between expected returns and the betas implies that factor exposures should have explanatory power for the cross-section of average returns. This links the model to the literature on empirical patterns arising in stock returns, the most notorious of which being the size, value, momentum, low volatility and profitability and investment effects. The two sets of factors do not necessarily coincide. For instance, Chan, Karceski and Lakonishok (1998) evaluate the performance of various proposed factors in capturing return comovements which provide sources of portfolio risk. They 1.1.3 Common Factors versus Pricing Factors

We therefore have: •  cov( ε i , ε j

) = 0, for i ≠ j

•  cov( ε i

, F k ) = 0, for all i and k .

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