EDHEC-Risk Institute October 2016

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

2. From Historical Betas (and Alphas) to Fundamental Betas (and Alphas)

to run a time-series regression of the stock (excess) returns on a market factor (excess return to a proxy for the market portfolio). The exact choice of the factor depends on the application of the measure. In equity portfolio management, a fund manager is concerned with the exposure to the stock market, and a broad cap-weighted stock index is a good representation of market movements. For asset pricing purposes, more care is needed in the definition of the market factor. Indeed, in the CAPM of Sharpe (1964), the market portfolio represents the aggregate wealth of agents, which includes not only stock holdings but also non-tradable assets such as human capital. This point is related to Roll’s (1977) critique of empirical tests of the CAPM, which are joint tests of the model itself and of the quality of the proxy used for the unobservable true market portfolio. Once a suitable proxy has been specified, one has to estimate the conditional beta. This is done in general by running a regression of stock returns on the market over a rolling window. If the joint distribution of stock and market returns were constant over time, the sample beta at date t – 1 would be a consistent estimator of the conditional beta on this date, and the variation in rolling-window estimates would be due to sampling errors only. But returns are not identically distributed: there are clusters in volatility (Harvey, 1989; Bollerslev, Engle and Nelson, 1994) and stock returns exhibit some predictability (Fama, 1981; Keim and Stambaugh, 1986; Fama and French, 1989; Cochrane, 2008), which is another way of saying that expected returns are not constant. Moreover, the beta itself is not constant (Rosenberg and Marathe, 1976). This has led researchers to look for alternative techniques to measure the

Asset managers need to estimate the sensitivity of each stock to the market in order to implement an allocation consistent with their views about factor returns or lack thereof. Specifically, they must estimate the beta of each stock conditional on the information available to date. Mathematically, this quantity is defined as: (2.1) denotes the return on stock i in period [ t – 1 , t ] in excess of risk-free rate, R m,t is the excess return on the market portfolio and Φ t -1 is the information set available at date t – 1. The traditional measure of conditional market exposure is the beta estimated over a sample period, but if the distributions of stock and market returns change over time, the sample estimates are not good estimators of the true conditional moments. By shifting the sample period (rolling window estimation), one generates time dependency in the beta, but due to the overlap between estimation windows, the historical beta changes relatively slowly. This section is precisely devoted to the construction of an alternative estimator for the conditional beta. We name this estimator a “fundamental beta” because it is defined as a function of the stock’s fundamental characteristics. In Section 2.1, we start by reviewing the competing approaches to estimate market betas that have been proposed in the literature. Next, we define and estimate two versions of the fundamental beta in Section 2.2 and Section 2.3. where R i,t

2.1 Measuring Market Beta The traditional approach to measuring the market exposure of a stock or a portfolio is

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