EDHEC-Risk Institute October 2016

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

1. Literature and Practice Reviews

1.2.1 Expected Return Decomposition Factor allocation is generally defined as the exposures of an investor's portfolio to a number of risk factors. These risk factors are rewarded over long period (for pricing factors) and the estimation of portfolio exposures to these factors augmented with the risk-adjusted performance enable us to decompose the portfolio expected return over the period. In a factor-based decomposition, a multi-factor model allows to measure the impact on performance of the portfolio manager’s decision to implement biases towards certain factors such as value, size and momentum. Using expression (1.3), we can decompose stock expected return through its factor exposures. We obtain, with the Carhart model, the following performance decomposition for the stock or portfolio i : The risk premia are given by: λ k = E ( F k ) for long-short factors or λ k = E ( F k ) - r ƒ for long-only factors. The decomposition can also be applied to ex-post average return which is an estimator for the expected return. In this case, we first estimate the four factor loadings and the alpha using OLS regression for each stock/portfolio over the period. Then we measure the realised risk premia over the period:

and finally we decompose the ex-post average return:

(1.5)

The contribution of each factor to the performance of the stock or portfolio is given by the following formula:

To provide an empirical illustration, we consider four US equity mutual funds from large providers, with different investment styles or themes: these funds are labelled by their respective providers as “long-short”, “defensive”, “dynamic” and “green”. We regress monthly fund returns against the Carhart factors from Ken French’s library, over the period 2001-2015. Figure 1 shows the decomposition of performance. Over the period considered, all ex-post risk premia are positive, with annual premia of 5.9% for the market factor, 3.8% for the size factor, 1.6% for the value factor and 2.3% for the momentum factor. Because the market premium is the largest and because long-only funds have market betas around one, the market factor explains most of the realised performance of these funds. It explains less for the long-short fund, whose market exposure is closer to zero. The abnormal performance, measured by the alpha in the regression (1.5), is actually negative. After analysing the factors contribution to portfolio performance, we need to consider the risk that funds managers took to achieve those returns by considering factor contribution to overall risk portfolio.

for long-short factors or

for long-only factors.

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