رابطه تجربی ریسک-بازده : یک روش تحلیل عاملی

Existing empirical literature on the risk–return relation uses relatively small amount of conditioning information to model the conditional mean and conditional volatility of excess stock market returns. We use dynamic factor analysis for large data sets, to summarize a large amount of economic information by few estimated factors, and find that three new factors—termed “volatility,” “risk premium,” and “real” factors—contain important information about one-quarter-ahead excess returns and volatility not contained in commonly used predictor variables. Our specifications predict 16–20% of the one-quarter-ahead variation in excess stock market returns, and exhibit stable and statistically significant out-of-sample forecasting power. We also find a positive conditional risk–return correlation.

Financial economists have long been interested in the empirical relation between the conditional mean and conditional volatility of excess stock market returns, often referred to as the risk–return relation. The risk–return relation is an important ingredient in optimal portfolio choice, and is central to the development of theoretical models aimed at explaining observed patterns of stock market predictability and volatility. Among those theoretical models that have become standard-bearers in finance, a positive risk–return relation is the benchmark prediction, so that times of predictably higher risk coincide with times of predictably higher excess returns, and vice versa. Unfortunately, the body of empirical evidence on the risk–return relation is mixed and inconclusive. Some evidence supports the theoretical prediction of a positive risk–return tradeoff, but other evidence suggests a strong negative relation. Yet a third strand of the literature finds that the relation is unstable and varies substantially through time. We summarize the existing evidence below. Several criticisms of the existing empirical literature relate to the relatively small amount of conditioning information used to model the conditional mean and conditional volatility of excess stock market returns. First, the conditional expectations underlying the conditional mean and conditional volatility are typically measured as projections onto predetermined conditioning variables; but, as Harvey (2001) points out, the decision as to which predetermined conditioning variables to use in the econometric analysis can influence the estimated risk–return relation. In practice, researchers are forced to choose among a few conditioning variables because conventional statistical analyses are quickly overwhelmed by degrees-of-freedom problems as the number rises. Such practical constraints introduce an element of arbitrariness into the econometric modeling of expectations and can lead to omitted-information estimation bias, since a small number of conditioning variables is unlikely to span the information sets of financial market participants. If investors have information not reflected in the chosen conditioning variables used to model market expectations, measures of conditional mean and conditional volatility will be misspecified and possibly highly misleading. This point was made forcibly by Hansen and Richard (1987) in the context of estimating and testing dynamic asset pricing models. A second and related criticism of the existing empirical literature is that the estimated relation between the conditional mean and conditional volatility of excess returns often depends on the parametric model of volatility, e.g., GARCH, EGARCH, stochastic volatility, or kernel density estimation (Harvey, 2001). Such procedures can impose potentially restrictive parametric assumptions and they often suffer from a curse-of-dimensionality problem that constrains their ability to accommodate large data sets of conditioning information. Finally, the reliance on a small number of conditioning variables exposes existing analyses to problems of temporal instability in the underlying forecasting relations being modeled. For example, it is commonplace to model market expectations of future stock returns using the fitted values from a forecasting regression of returns on a measure of the market-wide dividend–price ratio. A difficulty with this approach is that the predictive power of the dividend–price ratio for excess stock market returns is unstable and exhibits statistical evidence of a structural break in the mid-1990s (Lettau, Ludvigson, and Wachter, 2005). In this paper, we consider one remedy to these problems using the methodology of dynamic factor analysis for large data sets. Recent research on dynamic factor models finds that the information in a large number of economic time series can be effectively summarized by a relatively small number of estimated factors, affording the opportunity to exploit a much richer information base than what has been possible in prior empirical studies of the risk–return relation. In this methodology, “a large number” can mean hundreds or even more than one thousand economic time series. By summarizing the information from a large number of series in a few estimated factors, we eliminate the arbitrary reliance on a small number of exogenous predictors to estimate the conditional mean and conditional volatility of stock returns, and make feasible the use of a vast set of economic variables that is more likely to span the unobservable information sets of financial market participants. In the words of Stock and Watson (2004), dynamic factor analysis permits us to turn dimensionality from a curse into a blessing. Dynamic factor analysis allows us to escape the limitations of existing empirical analyses on several fronts. First, if a large amount of information can be effectively summarized by a relatively few common factors, then a natural remedy to the omitted information problem is to augment fitted conditional moments with estimated factors. We do so here by including estimated factors in the construction of fitted mean and volatility. Second, by combining dynamic factor analysis with a nonparametric approach to modeling volatility—an approach referred to hereafter as realized volatility—we avoid relying on potentially restrictive parametric structures while at the same time insuring that our measure of conditional volatility effectively summarizes a large amount of information that could be important for predicting the variance of the stock market. Third, there is some evidence (discussed below) that dynamic factor analysis provides robustness against the temporal instability that often plagues low-dimensional forecasting regressions. Indeed, our application appears supportive of this evidence, since the factor-augmented predictive relations we employ are remarkably stable over time, despite the observed temporal instability of many commonly used predictor variables over the sample period we study. An important question of our study is the degree to which estimated common factors add information about the conditional mean and conditional volatility of stock returns that is not already contained in commonly used predictor variables. If, on the one hand, we find that the factors provide new information, then we have evidence that previous estimates of conditional moments are misspecified and the estimated risk–return relation is potentially contaminated. On the other hand, if we find that the information provided by the factors is largely contained in commonly used predictor variables, then we have evidence that previous estimates are likely to be well specified. Either way, our study contributes to the empirical literature on the risk–return relation by evaluating both the potential role of omitted information in the estimated risk–return relation as well as the robustness of previous results to conditioning on richer information sets. We estimate common factors from two quarterly post-war data sets of economic activity using the method of principal components. The first data set consists of 209 primarily macroeconomic indicators; the second data set consists of 172 financial indicators. As a result of investigating these data, we find a number of results particularly interesting. First, in modeling the conditional mean of excess stock market returns, we introduce two new financial factors that are particularly important for forecasting quarterly excess returns on the aggregate stock market. In doing so, we contribute to the continuing debate over the predictability of stock market returns. See, e.g., Campbell and Yogo (2002), Campbell and Thompson (2005), Goyal and Welch (2004), and Lewellen (2004). The first financial factor is the square of the first common factor of the data set comprised of financial indicators. This factor explains almost 80 percent of the contemporaneous variation in squared stock market returns, so we label it a “volatility factor.” The second financial factor is the third common factor from the data set comprised of financial indicators and is highly correlated with a linear combination of three state variables widely used in the empirical asset pricing literature to explain cross-sectional variation in risk premia. These state variables are market return and the Fama-French factors SMBtSMBt, and HMLtHMLt (Fama and French, 1993). Thus, our second factor connects the time series with the cross-section of expected excess stock market returns. For this reason, we call this second factor a “risk premium factor.” When the volatility and risk premium factors are included with the consumption-wealth variable caytcayt, found elsewhere to predict quarterly stock returns (Lettau and Ludvigson, 2001a), the statistical model predicts an unusually high 16% of the variation in one-quarter-ahead excess returns. Moreover, the two factors on their own exhibit remarkably stable, strongly statistically significant out-of-sample forecasting power for quarterly excess returns that is found to be strongest in data after 1995, a period in which the predictive power of many traditional forecasting variables is exceptionally poor. Second, in modeling the conditional volatility of excess stock market returns, we find one macroeconomic factor that, when combined with other predictor variables, is especially useful for forecasting stock market volatility. This factor is the first common factor from the macroeconomic data set, known to be a “real factor,” since it is highly correlated with measures of real output and employment but not highly correlated with prices (Stock and Watson, 2002b). Third, we find that distinguishing between the conditional correlation (conditional on lagged mean and lagged volatility) and unconditional correlation between the conditional mean stock return and its conditional volatility is crucial for understanding the empirical risk–return relation. This finding is consistent with that of Brandt and Kang (2004) who argue that the distinction could explain the disagreement in the literature about the contemporaneous correlation between risk and return. In contrast to some previous studies, however (e.g., Brandt and Kang, 2004 and Lettau and Ludvigson, 2003) we find a positive conditional correlation that is strongly statistically significant, whereas the unconditional correlation is weakly negative and statistically insignificant. We show here that the findings in Lettau and Ludvigson (2003) can be attributed to the omission of the volatility and risk premium factors, which contain important information about one-quarter-ahead returns. Finally, our results imply that the conditional Sharpe ratio has an unmistakable countercyclical pattern, increasing sharply in recessions and declining at the onset of expansions. These findings are consistent with those in Brandt and Kang (2004) and Lettau and Ludvigson (2003). The rest of this paper is organized as follows. In the next section we briefly review related literature. Section 3 lays out the econometric framework, discusses the use of principal components analysis to estimate common factors, and explains how factors are chosen for modeling the conditional mean and conditional volatility of stock returns. Section 4 explains the empirical implementation and describes the data. We move on in Section 5 to present our empirical findings, including the results of one quarter-ahead predictive relations and our results for the estimated risk–return relation. Two additional analyses are performed as robustness checks: out-of-sample investigations and small-sample inference. Section 6 concludes.