Time-variant CAPM: Learning about Factor Loading Jiho Han April 2006 ABSTRACTAdrian and Franzoni (2005) suggest that beta of a stock is determined by the factor loading of its unobservable long-run beta. The fundamental idea behind this model is that investors engage in a learning process of estimating this long-run beta. In this paper, a variation of this model is tested. Instead of estimating long-run beta, investors try to learn about factor loading. Given the long-run beta and the upcoming level of risk, investors must form the optimal factor loading that minimizes the pricing error. The evolution of the factor loading assumes to follow an AR(1) process. The mispricing results are as equally successful as Adrian’s model. The momentum in the optimal factor loading is confirmed. During the transitional business-cycle periods, the factor loading seems to be more volatile.1.Introduction Sharpe (1964) and Linter (1965) introduce the Capital Asset Pricing Model (CAPM), which provide a first comprehensive mechanism of rational investors’ behaviors. Despite its theoretical elegance and simplicity, CAPM makes many assumptions that are quite unrealistic in the derivation of the model. For example, CAPM assumes an ideal, market-efficient situation where all the investors share same information and same utility function. From structural point of view, the most serious problem of CAPM is to use OLS procedure to estimate betas. Each observation of historical return of a specific stock/portfolio is a time-series data, which inevitably involves non-zero autocorrelation. However, in spite of all these problems, the reason that CAPM is scorned by many investors is that CAPM have failed to cope with the empirical observations, especially for the periods after 1960s. Fama and French (1992) find pervasive evidence that CAPM is unable to explain return on size and book-to-market (BE/ME) sorted portfolios. Naturally, many researchers and scholars begin to look for alternative theories. Two mainstream branches of researchers arise in order to improve CAPM: (1) Multi-factor models and (2) Conditional CAPM. The former group renounces a univariate structure and extends to a multi-dimensional model. Fama and French (1996) suggest an influential three-factor model, which includes market risk factor (β), size factor (SMB), value factor (HML) to account for the problem they address previously. While this model shows highly improved predictive power, it lacks an underlying explanation of why market behaves in this particular way (Campbell 2004). Another branch of researches preserves a one-factor model and proposes that CAPM holds only conditionally. In other words, although the cross-sectional CAPM holds true at any given time t, the unconditional CAPM may fail unconditionally. Jagannathan and Wang (1996) model the evolution of the conditional distribution returns at time t as a function of lagged state variables. From this structure, theyderive that the covariance between the market and portfolio returns, i.e., betas, as affine functions of these variables. Lettau and Ludvigson (2001) find that the betas of value stocks are highly correlated with consumption growth rate. Hence, they suggest the use of a conditional variable CAY1 as a controlling variable when estimating betas. Upon these research results, Adrian and Franzoni (2005) take learning into consideration. They argue that the unobservability of true betas causes investors to engage in learning process. In their Learning-CAPM, a beta evolves by finding a weighted average of the previous period long-run and current betas. Investors continuously “update” their long-run and current betas since these values are not observable. It is important to note that factor loading, or weight, remains fixed in Learning-CAPM. Inspired by Adrian and Franzoni, this paper provides a variation of Learning-CAPM under totally different assumptions. Instead of assuming the unobservability of betas, investors as a whole are assumed to have the estimates for long-run and forecasted upcoming period risk. From these two established risk measures, investors must form the optimal weight, or the factor loading Ft, that minimizes the pricing error. The evolution of factor loading is designed to have a momentum in a sense that it follows an AR(1) process. While the assumption of having upcoming period risk seems rather bold, many researches show that investors have some ability to forecast near-future risk/return. Moreover, this assumption does not imply that individual investors have same upcoming level of risk; it is an aggregated market estimate of risk for the next period. The point of this paper is to examine a market’s response when forecasted upcoming risk is available. In that sense, the evolution of factor loading describes how an efficient market works because it represents stock’s sensitivity to current market news. The learning factor loading model is differentiated from the presented literature in two ways. First, it endorses an active investment situation. Unlike Learning-CAPM where investors merely updates their 1 Lettau and Ludvigson (2001). CAY is an acronym for log consumption, C, log asset, A, and log labor income, Y.risk estimates, the factor loading model hypothesizes that investors develop new index of risk each time period by properly mixing up their long-run and forecasted risk. Secondly, in extension of the first point, the learning factor loading model admits investors’ forecasting ability and its momentum effect on returns. This is a departure from Jagannathan and Wang (1996) in a sense that forecasted risk is used to model betas instead of lagged state variables. From structural point, the learning factor loading model also has two limitations that pertain to the points made in the previous paragraph. First, the learning factor model does not provide answers on how a market evaluates long-run and upcoming level of risk. In this paper, the OLS betas estimated from past 60-months returns are assumed to describe the investors’ estimates of long-run risk whereas the forecasted risks are calculated under Bayesian setting given the next-period returns. In future research, these values can be replaced by macro factors or some investors’ consensus data such as, IBES forecasts. Secondly, the importance of factor loading depends on the relationship between long-run and upcoming betas. Suppose the situation