Chapter 3 Equity Style Drivers: Business Cycle Risk versus
3.2 Econometric framework
In a risk-based multifactor economy, suppose there are N stocks to be priced and M macroeconomic variables containing useful information important for pricing the stocks and are observable by investors. If the market is efficient and in the absence of arbitrage, the N stocks are priced by the pricing kernel, mt, such that:
1 1
[ ] 1
t t t N
E R m (1)
Where 1N is a N1 vector of ones, Rt1 is the N1vector of gross returns of the N stocks in time period t1, and mt1 is the scalar stochastic discount factor (pricing kernel). Assuming that the pricing kernel can be proxied as a linear multivariate function of a set of pricing factors, i.e
~ '
1 1
t t t t
m a b f (2)
Here at and bt are time-varying coefficients that are adapted to
information set of M macroeconomic variables at given time t. Assuming that at and bt are linear functions of the M macroeconomic
variables: ' t t t t a a Z b bZ (3)
Where a and bare M1 andN M , respectively. Ztis a M1vector of
M macroeconomic variables that are observed at time t-1. Hence:
~
' '
1 ( ) 1
t t t t
m a Z bZ f (4)
Thus, at each point of time t, the expected return of an individual stock can be related to the conditional covariance of returns with the
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measure of the pricing kernel. The pricing kernel is proxied by a linear and multivariate structural function based on the M macroeconomic variables, implicitly allowing the time variations in the exposure to macroeconomic variables over the business cycles.
Equation (4) is a dynamic multifactor model and both theoretical and empirical studies support the use of such dynamic multifactor pricing models. The motivation to use a conditional framework is that in a dynamic world the pricing kernel of assets are likely to be time-varying in responding to different information set (Wu, 2002). The multifactor approach is also empirically motivated. Justified for a century of empirical analysis, the fragility of single factor asset pricing model such as CAPM is well recognised. CAPM summarises the expected asset return with a single beta measurement that relates to the comovement with the overall market. Thus higher expected returns should suggest higher betas that act as compensation for higher common risk exposures. The extant literature has however identified asset groups that offer better returns than others but do not necessarily have higher CAPM betas. For example, Fama and French (1996) show that small stocks and value stocks do not have higher market betas, suggesting the major failure of CAPM in explaining the cross-sectional variations in average returns. Hence, as Cochrane (2000) points out, at least since Merton (1971, 1973) asset pricing theory recognises the use of additional factors of the source of priced risks beyond the movement of market portfolio to explain why some assets earn higher returns than others.
Given the foregoing and in the spirit of Chordia and Shivakumar (2002), for each individual stock, the expected returns conditional on the M macroeconomic variable set Zt can be specified as:
, 1 0 , , 1 , , 1
( i t | t) i i M t i M t ( t)
M
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This chapter uses 4 macroeconomic variables as the instruments to proxy the pricing kernel. These variables are:
div - the dividend yield on the overall market;
def - the default spread measured by the yield spread between the lower- to higher- yield bond;
term - the term spread measured by the differential between the yield of long-term government bond and the yield of 3-month T- bills;
yld - the short-term interest rate proxied by the yield on the 3- month T-bills.
Thus Z = (div, yld, term, def) and it is easy to show:
(6)
Therefore, the one-period-ahead predicted stock return is obtained from the following regression:
(7)
Where i,0 i0 i M, M0
M
c
b a and ci j,
Mbi M, aM j, , for j1, 2, ,M(M = 4). The selection of these 4 variables is motivated by the criteria noted in Campbell (1996) that proxies for state variables of time-varying investment opportunities should be chosen based on their ability to forecast market returns and explain the patterns of cross-sectional average asset returns. Prior studies on the relation between stock returns and the business cycles have focused on these 4 variables due to their indicator nature that relate to the business cycle fluctuations. For example, Fama and Schwert (1977), Fama (1981) show that the yield on the 3-month T-bills is negatively related to future market returns. The dividend yield on the overall market is perhaps one of the oldest variables recognised to affect the expected stock returns. Fama (1990) shows that stock prices are low relative to the dividends when73
the discount rate and expected returns are high, and vice versa. Keim and Stambaugh (1986), Cambell and Shiller (1987) and Fama and French (1988) also show that dividend yield is associated with slow mean reversion in stock returns in the business cycles.
The importance of default spread and term spread in explaining stock returns is also well documented. Keim and Stambaugh (1986) use the default spread to predict stock and bond returns. Chen et al. (1986) find that the default spread is an indicator to the business cycle. They argue that the default spread is likely to be high when the economy is in good condition, and vice versa. Likewise, Fama and French (1989) and Fama (1990) show that the default spread tracks the long-term business cycle conditions and captures the variations in expected returns within the business cycles. Daniel and Torous (1991) further show that the default spread contains information about future production volatility.
Fama and French (1989) find that the term spread is also closely related to the business cycles. They argue that this variable tends to decrease near peaks of business cycles and increases when the economy troughs. Daniel and Torous (1991) provide evidence that the term spread is primarily informative about future growth prospects. Chen (1991) also use the default and term spread to predict excess returns and contends that the predictability of these variables is due to their business cycle indictors that contain information about the current and future economic conditions.
Overall, the above 4 variables are standard macroeconomic variables containing rich information of the business cycle risks. In particular, the default and term spread variables have long been used as proxies for credit market conditions and the stance of monetary policy, suggesting that innovations in these variables would capture revisions in the market's expectation about future credit market conditions and interest rates. Given that small and large size stocks have different
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accessibility to credit markets (c.f. Chan and Chen (1991); Gertler and Giichrist (1994)), and that value stocks tend to have high financial leverage and cash flow problems than growth stocks (Fama and French (1992, 1995)), it is expected that the default spread would be good state variable capturing the cross-sectional variations in average returns of size and value-growth stocks. Furthermore, this variable may also capture investor’s hedging concerns associated with time- varying risk premia (Jagannathan and Wang (1996)). Similarly, since the term spread is one of the most widely used proxies for market's expectation about future interest rates, it is also expected to capture the hedging concerns to investors associated with changes in interest rates.
While most prior studies use these macroeconomic variables to do empirical tests on the portfolio level, recently there are some studies to implement the same framework but on the individual stock level. Avramov and Chordia (2006a) argue that the use of individual stocks reduces the data-snooping biases raised by Lo and MacKinlay (1990) and can avoid the loss of information in the portfolio sorting process suggested by Litzenberger and Ramaswamy (1979). Recent paper of Chordia and Shivakumar (2002) use these macroeconomic variables to investigate the influence of time variation in risk premia on momentum returns. They first estimate individual stock returns using these variables and subsequently sort stocks based on these predicted returns to investigate if momentum effect still exists. Chordia and Shivakumar (2002) find that momentum profits based on predicted returns are substantially reduced, suggesting that momentum profits could be attributed to higher conditional expected stock returns and hence can be interpreted as compensation for bearing business cycle risks.
This chapter employs a similar methodology to that used in Chordia and Shivakumar (2002). Each month Equation (7) is used to predict
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the one-month-ahead expected returns for individual stocks. The parameters of Equation (7) are estimated using a rolling window based on previous 60-month observed (realised) returns. To obtain the meaningful estimates, only stocks with at least 24-month return observations are included during the parameter estimation procedure. The estimated coefficients of Equation (7) are then used to predict the one-month-ahead expected returns of the underlying stocks.
Under a rational asset pricing framework, any abnormal returns are caused by risk factors. Equation (7) states that expected stock returns are driven by conditional shocks to macroeconomic variables. Given the evidence of significant size and value premiums found in the U.K. stock market, the following two hypotheses can be tested:
Null hypothesis: if business cycle risks are the major driving force to such divergent stock returns, it is expected that return differentials across styles should be substantially decreased once controlling for the exposures to these macroeconomic variables.
Null hypothesis: if firm-specific components are the major sources that underlie the relative returns across different stock groups, controlling for the business cycle effect should not cause material changes for the observed style spreads. Hence simple style investing strategies based on the unexplained parts (i.e. the pricing error) of Equation (7) should generate significant payoffs.