**Edhec-Risk Institute**. The article also considers multi-factor investing. ‘Factor investing’ recommends that allocation decisions be expressed in terms of risk factors, as opposed to standard asset class decompositions. While intuitively appealing, this approach poses a major challenge, namely the choice of the meaningful factors and the corresponding investable proxies. In simple terms, factor investing regards each constituent in an investor’s portfolio, and therefore the whole portfolio as a bundle of factor exposures. There are as many factors as individual securities and the factors are themselves portfolios of such securities, so thinking in terms of factors is strictly equivalent to thinking in terms of asset classes, and therefore would not add any value. More relevant is the situation where a parsimonious factor model is used, with a number of factors smaller than the number of constituents.

**MEANINGFUL FACTORS**

The first challenge posed for investors is the identification of meaningful factors. In this perspective, the theoretical section of recent research that we have conducted as part of the Lyxor Risk Allocation Solutions research chair at Edhec-Risk Institute reviews the academic literature on asset pricing and makes a list of conditions that such factors should satisfy. We then survey a vast empirical literature in order to identify the most consensual factors in three major asset classes, namely stocks, bonds and commodities. The second challenge in factor investing is the implementation of decisions in a cost-efficient way with investable proxies. Asset pricing theory makes a distinction between ‘pricing factors’, which explain differences in expected returns across assets, and ‘priced factors’, which earn a premium over the long run. The theory expresses the risk premium on an asset, that is, the expected return on this asset in excess of the risk-free rate, as a function of the covariance between the payoff and an abstract quantity known as the ‘stochastic discount factor’. Pricing factors arise when one attempts to find observable proxies for the aggregate investor’s marginal utility. In a factor model, the risk premium on an asset is a linear combination of the factor risk premia, weighted by the betas of the asset with respect to the factor. As a consequence, all alphas are zero and the cross-sectional differences between expected returns are entirely explained by the differences in factor exposures. As for an asset, the premium of a factor is determined by its covariance with the stochastic discount factor, so that a factor deserves a positive premium if, and only if, it is high in bad times and low in good times. A factor is said to be ‘priced’ if it has a non-zero premium. It can be shown that there is no loss of generality from searching for pricing factors among returns, but further assumptions are needed to identify their economic nature. Two main classes of theoretical models have been developed to this end. A first category uses economic equilibrium arguments. In the static Capital Asset Pricing Model (CAPM), the only factor, or ‘market factor’, is the return on aggregate wealth. The inter-temporal version (ICAPM) adds as new factors the variables that predict changes in expected returns and volatilities. A second class of models refers to the Arbitrage Pricing Theory (APT) and characterises factors as variables that explain returns from a statistical standpoint. One of the questions studied in the recent asset pricing literature is whether the factors proposed in empirical asset pricing models do meet these theoretical criteria. The property that assets have zero alphas with respect to the factors has an interesting implication. The most meaningful way for grouping individual securities may not be by forming arbitrary asset class indices, but instead by forming factor indices, that is replicating portfolios for a set of indices that can collectively be regarded as linear proxies for the unobservable stochastic discount factor, thus providing a theoretical justification for factor investing.

**MULTI-CLASS FACTORS**

Empirically, the search for pricing factors in asset classes such as stocks, bonds and commodities begins with the identification of persistent and economically interpretable patterns in average returns. Recent research has subsequently started to look for multi-class factors. Multi-factor models derived from the ICAPM or the APT do not provide an explicit definition of their factors. Thus, the traditional approach in empirical asset pricing has been to examine the determinants of cross-sectional differences in expected returns and to find sound economic interpretations for regular patterns, such as behavioral biases. Most of the empirical asset pricing literature has focused on factors explaining equity returns. This literature starts in the early 1970s with empirical verifications of the CAPM and concludes that the model’s central prediction, namely the positive and linear relationship between expected excess return and the covariance with aggregate wealth, is not well validated by the data for two reasons. First, there exist patterns that are not explained by the market exposure, and second, the relation between expected returns and the market betas is at best flat, or even negative. The most consensual patterns are those that have shown to be robust to various statistical tests, to exist in almost all international equity markets, to persist over time, in particular after their discovery, and to admit plausible economic explanations. They include the size and value effects, which are historically among the first reported anomalies: small-cap stocks tend to outperform their large-cap counterparts, and there is a positive relationship between the book-to-market ratio and future average returns. The size and value factors are used together with the market factor in the model of Fama and French (1993). Another remarkably robust pattern is the momentum effect: the winners (resp., losers) of the past three to 12 months tend to outperform (resp., underperform) over the next three to 12 months. The number of reported empirical regularities has grown fast in the recent literature. Among them is the controversial ‘low-volatility puzzle’, namely the documented outperformance of low-volatility stocks over high-volatility stocks, the existence and persistence of which remains somewhat debated in the academic literature. Among the other noticeable patterns are the investment and profitability effects. A class-by-class study reveals that some patterns exist repeatedly in various classes. This is the case for short-term momentum and long-term reversal in equities, bonds, commodities and currencies. Furthermore, the single-class momentum factors are positively correlated, and the same goes for value factors. Taken together, these findings justify a new approach, which is the construction of multi-class value and momentum factors, obtained by aggregating the corresponding single-class components. Empirical tests show that investable proxies for factors add value in single-class or multi-class portfolios when they are used as complements or substitutes for broad asset class indices. Moreover, in the equity class, a portfolio of factor indices dominates a portfolio of sector indices. Our empirical study focuses on the following factors, which have been selected because they have well-documented historical performance, are theoretically grounded and are widely accepted by practitioners: size, value, momentum and volatility for equities; term and credit for bonds; term structure and momentum for commodities. In addition, we test multi-class value and momentum factors. A first analysis of the descriptive statistics for these factors highlights a few simple but important facts. Each long-only factor outperforms its opposite tilt, in line with the theoretical and empirical literature, and outperforms the corresponding broad asset class index. Correlations within a class are high (above 75%), although they are lower across classes, and they are much lower for long-short versions of the factors. The analysis extended to a multi-class context by comparing ‘policy-neutral’ portfolios of equity, bond and commodity factors to a fixed-mix policy portfolio of 60% equities, 30% bonds and 10% commodities. Again, both the average return and the Sharpe ratio are improved. To conclude, there are theoretical arguments in favour of factor investing. Extensive empirical literature has documented a number of recurring patterns in the returns of equities, bonds and commodity futures, and provides investors with a rich list of insights regarding the choice of meaningful factors in each of these classes. On the practical side, a challenge is to develop factor indices that aim to capture factor risk premia at reasonable implementation costs. It is being addressed in the equity class with a new generation of ‘smart beta’ indices, but similar products are not as widely developed in other classes and no multiple-class products are available to date. This is an edited version of an article by Lionel Martellini, professor of finance, Edhec Business School, and Vincent Milhau, deputy scientific director, Edhec-Risk Institute

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