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Return decomposition with a two-beta model in the UK stock market
and a comparison with other asset pricing models
Stella Suyan Zhang Supervisor: Rients Galema
Master Thesis – MSc BA Specialization Finance University of Groningen, the Netherlands
21 July 2010
Abstract
In this paper I apply the two-beta model developed in Campbell and Vuolteenaho (2004) to the monthly UK stock returns from October 1980 to December 2008, which decomposes the excess returns into two risk loadings: cash flow beta and discount rate beta. In accordance with Merton’s (1972) ICAPM and the Campbell and Vuolteenaho (2004) study, the ‘bad’ cash-flow beta are found to have a risk premium several times larger than the ‘good’ discount rate beta. UK growth stocks are also found to have larger discount rate betas than value stocks, although the difference is not as large as in the US. Thus the higher CAPM market beta of the growth stocks observed during the sample period is largely of the ‘good’ discount-rate beta. The three-factor model of Fama and French (1993) outperforms the two-beta model as well as the traditional CAPM, although the two-beta model performs relatively well and is robust against an additional state variable and split-sample tests.
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Keywords: Asset pricing models, two-beta model, CAPM, Fama-French three-factor model, ICAPM, Return decomposition, Vector Autoregression
2 1. Introduction
Understanding movements in stock returns has been one of the central lines of research in finance for decades. In one of the most important asset pricing model in the field, the Capital Asset Pricing Model (CAPM; Sharpe, 1964, Lintner, 1965), the stock return is only dependent on the risk of the individual stocks, which is summarized by their beta with the market portfolio containing all traded stocks. Fama and MacBeth (1973) tested the CAPM empirically and confirmed the hypothesis that no other factor in addition to the stock beta should affect average stock returns. The CAPM has since then served as a null hypothesis against which a large number of different empirical implications are tested to explain the observed deviations from the returns predicted by the CAPM. In the process, many empirically determined variables lacking a current theoretical motivation are shown to be reliable in explaining the long-standing anomalies in the cross-section of average returns. For example, Banz (1981) documents a size effect in stock returns, whereby the stocks of firms with low market capitalization outperform those with high market capitalization; Basu (1977) finds a price-earnings ratio (P/E) effect, since the low P/E stocks appeared to earn higher abnormal returns than high P/E stocks. Moreover, Miller and Scholes (1982) find that low priced stocks earn higher expected returns than high priced ones; finally, Brennan (1970) argues that high dividend yield stocks command a differential premium because dividends are taxed at a different rate than capital gains.
In addition, many finance practitioners maintain that relating a stock’s fundamentals such as accounting (book) value of equity to its market value is useful in determining future returns. Stocks with high ratios of book-to-market value of equity are called value stocks, stocks with low book-to-market ratio (BE/ME) are named growth or glamour stocks. Stattman (1980) and Rosenberg, Reid and Lanstein (1985)document that value stocks have higher average returns that are not captured by their betas. Synthesizing the evidence on the empirical failures of the CAPM, Fama and French (1992, 1993 and 1996) convincingly argued that size and book-to-market ratio are additional risk factors that explain cross-sectional stock returns.
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care only about the mean and variance of one-period portfolio return is regarded to be unrealistic, the market beta does not capture an asset’s risk completely, so that the differences in expected return may not be fully explained by differences in beta. As an extension of the CAPM, Merton’s (1973) intertemporal capital asset pricing model (ICAPM) takes the assumption that investors are concerned not only with their end-of-period payoff, but also the investment or consumption opportunities in the future. Additional betas are needed in explaining expected returns to represent the covariances of portfolio returns with state variables related to these future opportunities. In this light, the Fama and French (1993) three factor model can be seen as an indirect approach to implement the ICAPM. They argue that although size and BE/ME are not themselves state variables, the higher average returns on small stocks and high book-to-market stocks reflect unidentified state variables which produce undiversifiable return covariances that are not captured by market betas.
The seminal work by Campbell and Shiller (1988) suggests that unexpected asset returns can be decomposed into two components: news about discount rates and news about cash flows. This is derived from the Rational Valuation Formula’s definition that the price of any asset is written as the sum of its expected future cash flows, discounted to the present using a set of discount rates: the price of the asset changes either when expected cash flows change, or when discount rates change. Implementing a Vector Autoregression (VAR) approach on a dividend-ratio model, Campbell and Shiller (1988) break down the movements in the log dividend-price ratio into components attributable to expected future dividend growth and expected future discount rates.
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‘bad beta’ and may demand a higher premium for assets with a high cash flow beta than for assets with a high discount rate beta.
Using returns of the portfolios sorted by size and BE/ME in the US stock market as test assets, they find indeed that value stocks historically have a considerably higher bad beta than growth stocks, which explains the value stocks’ higher returns. This finding contradicts the classic CAPM, since in the 1963-2001 sample in CV (2004) the value stocks are found to have a lower CAPM beta than growth stocks. A further inspection in their study shows that the higher CAPM beta of growth stocks in this sample is attributed to their disproportionately high discount rate beta, which carries a lower risk premium than the cash flow beta. Also an adjusted version of Merton’s (1973) ICAPM is developed in Campbell (1993, 1996) and CV (2004), in which investors attach more importance to permanent cash-flow driven movements than to temporary discount-rate driven movements in the aggregate stock market.
The question whether the stocks’ market betas are determined by shocks to their cash flows or their discount rate is still a topic of interest for ongoing research, and the expected explanations for the cross-sections of stock returns should hold in markets worldwide. In this paper, I test the feasibility of decomposing stock returns in the UK market into cash flow and discount rate betas and determine which beta has the dominant effect, after applying the VAR system to UK stock returns in extracting both news terms. However, existing studies employing the VAR approach on beta decompositions to cash flow news and discount rate news terms are mostly focused on the US markets. This is in the first place owing to the high availability of a large amount of security price data over a wide time span in the US, and secondly thanks to the ready accessibility of data on the Fama-French portfolios sorted on size and book-to-market ratios (see later the Data section).
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UK market. Applying this recently established methodology to the UK stock market data would be very useful in expanding the scope of the existing studies on this subject, and it should provide valuable out-of-the-sample empirical tests contributing to a further understanding of this return decomposition framework. With the very recent availability of size and BE/ME sorted portfolios in the Fama-French (1992) fashion for the UK market (Gregory et al., 2009), I am well-equipped to study the contribution of the cash flow news and discount rate news on the stocks’ systematic risks for an entirely new market outside the US, and I am able to assess the performance of this two-factor model in comparison with other existing asset pricing models.
6 2. Literature review
The body of empirical literature testing various asset pricing models on UK stock markets is growing but still scarce compared to studies based on US stock data. Below I discuss a selection of studies that examines the cross-section of the UK stock returns using the CAPM, the Fama-French three factor model among other models.
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Looking at a different aspect of the CAPM, Fletcher (2001) finds that neither the domestic CAPM or APT are adequate in explaining UK stock returns between 1955 and 1995, but that domestic CAPM captures more of the time-series predictability in stock returns than the international CAPM (based on the World market portfolio such as the MSCI World index or a weighted portfolio of several major currencies). In this study, I use the domestic CAPM with only UK stocks included in the market portfolio.
The CAPM regained some empirical validity when both Fletcher (1997) and Hung et al. (2004) find that separating positive and negative excess market returns gives a higher cross-sectional significance of market beta for UK data, and Hung et al. (2004) shows that this beta effect is robust with respect to the Fama-French size and value factors. They also document that including higher moments of the return distribution (skewness and kurtosis) as factors does not increase the explanatory power of the regression equation. In explaining the time-series return differences between highest beta and lowest beta deciles, with the Fama-French three factors, the market beta and the book-to-market ratio is very significant while the size factor is not significant.
Combining many of the asset pricing models discussed above, Fletcher and Kihanda (2005) evaluate the performance of the standard CAPM, the CAPM with higher moments, and a three-factor model similar to Fama and French (1993) using the stochastic discount factor approach (Cochrane, 2001; Ferson, 2003) on UK stock returns between 1975 and 2001. Using the Hansen and Jagannathan (1997) distance measure, which compares the asset pricing models on how well they price the most mispriced portfolio among the primitive assets, they find that a conditional four-moment CAPM including labor income growth has the best performance of all CAPM models and is even better than the Fama-French model, and that the standard CAPM has the worst performance. At the same time they also highlight the danger that better performance of the conditional models is in part due to overfitting of the data.
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CAPM and the Fama-French model. With regard to the performance evaluation, I expect the Fama-French model to perform better than the traditional CAPM, whilst the performance of two-beta model based on the cash flow and the discount rate beta remains to be examined.
The VAR methodology used in this study to extract the news terms has been applied to the UK stock market by Cuthbertson et al. (1997) at first to test the efficient market hypothesis, where a linear Wald test is used to test the predictability of one-period returns with the null hypothesis of constant expected returns. Their VAR model consists of two variables, the log dividend-price ratio (proxy for change in future discount rates) and the real dividend growth (proxy for change in future cash flow). The VAR approach has the advantage of taking explicit account of the non-stationarity in the data, and allowing tests of the efficient markets hypothesis under a wide variety of assumptions about the determinants of equilibrium returns. A VAR-adapted version of the CAPM is supported by the regression results, as when market return variance is added as additional variable, the other variables in the VAR no longer have predictive power over future returns.
Decomposing unexpected returns to either news influencing expectations of dividends or returns for the UK market, Cuthbertson et al. (1999) use a multivariate VAR framework to predict stock returns. Annual data from Barclays de Zoete Wedd (value-weighted) equity stock price index and related dividends from 1918 to 1993. The state variables in the VAR equations include combinations of returns, dividend-price ratio, return volatility and term yield spread. Highest R2-value (48.89) is obtained with the combination return, dividend-price ratio and return variance as state variables. The results indicate that the dividend-price ratio is a important determinant of one-period returns, whilst return volatility also provides incremental explanatory power. They also found that both news terms are largely independent and the contribution of discount rate news is about four times that of cash flow news, which means that the assumption of constant dividend growth in Poterba and Summers (1986), French et al. (1987) is not unreasonable, since cash flow news has relatively little impact on unexpected changes in stock prices.
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terms by creating two corresponding betas. The VAR model in my study uses a slightly different but more enriched set of economically motivated state variables, which deal with most of the stock characteristics examined in earlier asset pricing literature, but do not require information on short-term dividend variations (see Section 4.1 for more details on the state variables used).
10 3. Data
Following CV (2004) in using a VAR approach to estimate cash-flow and discount-rate flow news, I use close UK proxies of the four state variables chosen in their study. Data for constructing the state variables are collected from Datastream, which contains the rate for 3 month UK Sterling Certificates used as the yield on short-term bonds; the UK Gross Redemption Yield on 20 year Gilts used as the yield on short-term bonds; and the price-earnings ratio of FTSE Total Non-Financial stocks. The market return and returns on small value and small growth stocks are taken from the Gregory et al. (2009) data set (see here directly below).
All data are monthly registered and span the period of October 1980 and December 2008, which is most importantly limited by the availability of the Gregory et al. (2009) dataset, as the size-BE/ME-sorted portfolio returns contained in this dataset form crucial test assets in calculating the cash-flow and discount-rate betas for the Fama-French portfolios (Fama and Fama-French, 1993). This dataset of the UK monthly benchmark returns on Fama-French portfolios is obtained from the website1 containing the working paper by Gregory et al. (2009) from the Xfi Centre for Finance and Investment at the University of Exeter. Until then, there has been no freely downloadable foreign equivalent of the US Fama-French portfolio returns and Fama-French factors such as those available on Kenneth French’s website2
The sources of this dataset come from the following databases (see Gregory et al., 2009): The London Business School Share Price Database (LSPD), which includes data on monthly returns, market capitalization and key dates of first listing and de-listing; Datastream; tailored Hemscott data (from Gregory and Michou (2009) studies of director’s trading); and hand collected data on bankrupt firms.
. This only recently available dataset provides an unique opportunity to conduct out-of-sample tests on the prevailing asset pricing models based on US securities data.
The monthly Fama-French factors from this dataset enable us to evaluate the performance of the Fama-French model among other asset pricing models. The UK proxies of the Fama-French factors are constructed in the following manner. First, the
1
http://xfi.exeter.ac.uk/researchandpublications/portfoliosandfactors/ 2
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UK main-market stocks is divided into two size groups, small (S) and big (B), based on the size breakpoint which is the median firm in the largest 350 firms by market capitalization. The book-to-market (BE/ME) breakpoints are based on the 30th and 70th percentiles of the same universe of 350 firms, with which these stocks are divided into three groups, H, M, and L, denoting high, medium and low BE/ME respectively. This method is in accordance with Fama and French (1992), where the same breakpoints for the size and BE/ME are used.
Following the notion by Agarwal and Taffler (2008) that 22% of the UK firms have March year ends and only 37% of firms have December year ends, the Gregory et al. (2009) dataset uses March year t accounting data and end of September year t market capitalization data. Compare this with Fama and French (1992), who construct BE/ME for year t by dividing book equity of the fiscal year ending in calendar year t-1, by market equity at the end of December of t-1. The portfolios are formed at the beginning of October in year t and financial firms are excluded from portfolios, as are stocks with negative book-to-market ratios.
Six value-weighted portfolios (S/H, S/M, S/L, B/H, B/M, B/L) are then formed from the intersections of the two size and the three BE/ME groups. Following Fama and French (1992), the synthesized portfolio HML represents the average return on the two high-BE/ME portfolios (S/H and B/H) minus the average return on the two low-BE/ME portfolios (S/L and B/L), the portfolio SMB is the average return on the three small portfolios (S/H, S/M, S/L) minus the average return on the three big portfolios (B/H, B/M, B/L). The total return on the FT All Share Index is used for the market return, and the one month return on UK Treasury Bills are used as the risk free rate. The excess market returns used in this paper is then computed as the difference between the market return and the risk free rate as defined above.
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Gregory et al. (2009) reports for this dataset that within size categories, the returns tend to increase as BE/ME increases, although the effect is not completely monotonic in the medium and largest size categories, and a consistent relationship between size and average return cannot be observed from the 25 portfolios. The UK portfolios thus weakly confirm the Fama and French (1992a) finding for the US sample returns that there is a stronger positive relation between average return and BE/ME, compared to the weaker negative relation between size and average return. Later in this study, these 25 sets of returns are used as the dependent variables in the time-series regressions and will be regressed on the cash-flow news and discount-rate news betas, in order to determine whether the two factors from return decomposition capture common factors in stock returns related to size and BE/ME.
13 4. Methodology
The general approach in this paper to study the return decomposition of excess stock returns follows Campbell and Vuolteenaho (2004), while the motivation and derivation of the theoretical framework stem mainly from Campbell and Shiller (1988) and Campbell (1991). I first estimate the two components of excess return caused by flow news and discount-rate news, then I use these components to estimate cash-flow and discount-rate betas for the 25 portfolios sorted on size and BE/ME. Finally, the performance of the two-beta model will be compared with the closely related ICAPM-version of the two-beta model, the Fama-French three factor model, and the traditional CAPM in explaining excess returns across the 25 size-BE/ME-based portfolios.
4.1 Two components of unexpected stock returns
The two components of the aggregate stock market excess return are either cash-flow news, or news on discount rates. Below I introduce the loglinear return decomposition by Campbell and Shiller (1988) and Campbell (1991), and empirically estimate these two components of unexpected return for the UK market portfolio.
4.1.1 Return-Decomposition Framework
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whereht+1is a log stock return, ∆dt+1denotes a one-period change in the log dividend paid
by the stock, E denotes a rational expectation at time t, and t ρ is a discount coefficient. NCF denotes news about future cash flows in the form of dividends or consumption, and
NDR denotes news about future discount rates, or the expected return of the investments.
This equation says that the unexpected stock returns must be associated with changes in expectations of future cash flows or discount rates. The value of the stocks may fall because investors receive bad news about future cash flows, or it may also fall because investors increase the discount rate or cost of capital that they apply to these cash flows. Alternatively, an increase in expected future cash flows is associated with a capital gain today, while an increase in discount rates is associated with a capital loss today, because at a given level of dividend stream, higher future returns can only be achieved by future price appreciation of a lower current price. This explains why the sign of the discount rate news appears to be negative in the equation. The negative of the discount rate news will also be used in subsequent graphs for the ease of interpretation.
Relatively speaking, cash flow news is more related to firm fundamentals and can be interpreted as permanent shocks to wealth, whereas discount rate news can reflect time-varying risk aversion or investor sentiment, which is more transitory in nature. These two components of asset returns have particular significance for a risk-averse, long-term investor, as such an investor is expected to demand a greater reward for bearing the risks brought about by cash flow news than for the discount rate related risks. 4.1.2 Vector Autoregression model
To implement the return decomposition for the UK stock market, I follow Campbell (1991) and estimate the cash flow news and discount rate news series using a vector autoregression (VAR) model. The dynamic structure of the VAR model allows shocks to a variable in the system to be transmitted to all other variables in the system, even though the error variance of this variable will be mostly explained by its own shocks. The VAR approach is different from the regression methods commonly used to test a model of expected stock returns. For example, the standard way to proceed would be to regress the one-period ex post stock return less the discount rate ht - rt on a constant term
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power to detect long-lived deviations of stock prices that are not observable from single-period returns regressions (Shiller 1984, Summers 1986), as it allows us to calculate the effects of today’s shocks over the discounted infinite future, without assuming that these effects die out after two to five years (Campbell et al., 2010).
My task is to rewrite Equation (1) in such a way that it is useful for the VAR model. The detailed description of a similar exercise for a dividend-price ratio model is first presented in Campbell and Schiller (1988), but the derivation for the model used in this study will be given here for the sake of clarity.
I assume that at the start of period t, market participants observe a vector of variables yt, and the information set It is just the history { yt, yt-1...}. I then assume that yt follows a linear stochastic process with constant coefficients that are known to the market participants. It follows from this assumption that any subset of the variables in yt also follows a linear stochastic process with constant coefficients. Let me now define a vector zt that includes the variables in yt that are observable to the financial economists and are likely to have strong forecasting abilities for future stock returns, also referred to as state variables. Including the stock return in the vector of market information set It among other state variables effectively means including all relevant information that are available to the market participants, even if not all of these information variables can be measured statistically (Campbell and Shiller, 1987).
In my model the state variable vector zt contains four state variables, with the first element of the vector being the excess stock return h , and another three state variables t (denoted for simplicity as bt, ct, and dt below) relating stock returns with the general stock
market conditions, which will be dealt with more extensively in the next section.
Looking more closely at Equation (1), the unexpected excess stock returns are associated with changes in the expected discounted value of dividend growth (denoting cash flow news) and excess returns (which is a natural substitute for the use of short-term interest rates in representing discount rate news, because an increase in discount rates is associated with a capital loss today). In the VAR approach used in this paper, I first estimate the terms Etht+1 and
∑
∞ = ++ + − 1 1 1 ) ( j j t j t t E h
cash-16
flow news. An important advantage of this approach is that one need not to concern with the irregularities in the distribution dates of the dividends for the individual stocks, as only the dynamics of expected returns are needed in obtaining results. The only time-varying variable in the above two terms is the excess return h , as t ρ is assumed to take on a constant value which will be justified at a later stage.
This limits my task to estimating h as part of the state variable vector zt t in the VAR system, thereby obtaining the variance or shocks in the unexpected excess returns as a residual vector. I assume the linear process for zt can be written as a first-order VAR, but in fact this assumption is not restrictive, as zt with p lags greater than 1 are also possible and involves little further loss of generality. As mentioned earlier, zt is a 4-by-1 vector of state variables with h as its first element, and the vector is assumed to be t linearly related to the one-period lag of the four state variables contained, plus a random error term. The first-order VAR system can be written out as:
+ = − − − − t t t t t t t t t t t t u u u u d c b h C C C C C C C C C C C C C C C C d c b h 4 3 2 1 1 1 1 1 44 43 42 41 34 33 32 31 24 23 22 21 14 13 12 11
where the Cjk denotes the coefficient of the j-th variable in zt on the k-th lagged variable, and uj the random error term for each of the state variables. The above matrix can also be
simplified as:
zt+1 =Azt +ut+1 (2)
where A is a 4-by-4 matrix of constant parameters that is also known as the companion matrix of the VAR. The vector zt has the useful property that to forecast it ahead k periods, I can simply multiply zt by the k-th power of the matrix A, which
is t k k t A z z + = t
E . I further define e1 a vector that has one as its first element and zero as all other elements (or [1 0 0 0]'), which means that e1' picks out the element ht from the
vector zt if the vector e1' is applied to zt.
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DR
N , . The discount rate news term can be rewritten as linear functions of the t + 1 shock
vector ut+1 using notations given above substituted in Equation (1) and (2):
1 t u A e1' + ∞ = ∞ = ++ + + ≡ −
∑
=∑
1 1 1 1 1 , ( ) j j j j j t j t t t DR E E h N ρ ρ 1 t 1 t 1 t λu e1' u A I A e1' u A e1' + + − + ∞ = = − = =∑
1 1 ) ( ρ ρ ρ j j jwhere λ ≡ρA(I−ρA)−1, which comes from evaluating the infinite sum
∑
∞ =1 j j j A ρ using a
special form of the Taylor series expansion
x
x
x
m m n n−
=
∑
∞ =1
for x <1 and 0 Ν ∈ m or more explicitly, A A Aρ
ρ
ρ
− =∑
∞ =1 1 j j j .From Equation (1), (2), and the vector notation of the discount rate news, I can now back out the cash-flow news term by
1 t 1 t 1 t u λ e1' e1' u e1' u λ e1' + + + + + + + = + = ⋅ = ) ( 1 , 1 , 1 , t DR t CF t DR N N N (3)
since the shocks e1'ut+1 in unexpected excess returns are mapped to the news terms
1 , 1
,t+ − DRt+
CF N
N . In the news terms notation given in Equation (3), e1'λcaptures the long-run significance of the VAR shock of each individual state variable to discount rate expectations. The greater the absolute value of a state variable’s coefficient in the return prediction equation (the top row of A), the greater the weight the variable receives in the discount-rate-news formula. More persistent variables should also receive more weight as captured by the term (I− A)ρ −1.
The discount factor ρ used in this study follows CV (2004) and takes the value of
1/12
0.95 =
consumption-18
wealth ratio in Campbell (1993). An annualized value of 0.95 for ρ corresponds to a consumption level of approximately 5 percent of total wealth per year, which seems reasonable for a long-term investor (CV, 2004). Regression results are shown to be robust to reasonable variation in ρ in CV (2004) and Campbell et al. (2010).
4.1.3 VAR State Variables and Estimation
The news terms which could explain stock return variance are also sensitive to the other state variables in the VAR system. The variables to be included in the state vector beside the excess stock return itself (or bt, ct, and dt in the above discussion) are direct UK
equivalents of the state variables used in Campbell and Vuolteenaho (2004), in so far data are available. A parsimonious model with four state variables is used, including the excess market return (computed as the difference between FT All Share Index and the one-month Treasury Bills rate, as noted above); the yield spread between long-term and short-term bonds (measured as the difference between the 20-year yield on gilts and the yield on 3-month Sterling Certificates); the market’s price-earnings ratio (measured as the logarithm of the price-earnings ratio of the FTSE Total Non Financial stock index); and the small-stock value spread (measured as the difference between the return on small value and small growth stocks). The last state variable is defined in Campbell and Vuolteenaho as the difference between the log BE/ME ratio of small value and small growth stocks. Although this difference in log BE/ME ratio would be more pronounced than the difference in returns, there is no UK equivalent of data available at this level of detail.
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market-wide discount rates, as episodes of irrational investor optimism are shown to have a particularly powerful effect on small growth stocks (Shiller, 2000).
4.2 Measuring Cash-Flow and Discount-Rate Betas
If the market returns can be decomposed into two components (cash flow news and discount rate news), both of which are highly volatile and not closely correlated with each other, one could expect that different types of stocks may have different betas with the two components of the market returns. The cash-flow beta and the discount-rate beta are defined as follows:
) Var( ) , ( Cov ) Var( ) , ( Cov , 1 , , , , , 1 , , , , e t M t e t M t DR t i DR i e t M t e t M t CF t i CF i r E r N r r E r N r − − − − ≡ − ≡ β β
where ri,t and rM,t denote the returns on stock i and the market portfolio, respectively.
Note that in this case the discount-rate is defined as the covariance of the stock i with good news about the stock market in the form of lower-than-expected discount rates, and that each beta is divided by the total variance of unexpected market returns, rather than the variance of cash-flow news or discount-rate news separately.
The definition of the betas here follows Campbell and Mei (1993), which translates to the unconditional covariance of the shocks to excess market return with shocks in a news term, divided by the unconditional variance of the shocks to excess market return. This beta definition is neither a full conditional beta, which would use conditional variances and covariances, nor a straightforward unconditional beta, which would use return itself rather than shocks in returns (take the classic market beta in CAPM: ) var( ) , cov( , M M i M i r r r =
β , wherer andi rM are return on asset i and market return, respectively). Using beta in the above definition is advantageous because it can be broken into components in a relatively simple way. Since the total variance of unexpected market returns are decomposed in only two components, it implies that the cash-flow beta and the discount-rate beta as defined above add up to the total market beta
20 4.3 Pricing Cash-Flow and Discount-rate Betas 4.3.1 The Intertemporal Asset Pricing Model
The intertemporal capital asset pricing model (ICAPM) proposed by Robert Merton (1973) is a generalization of the CAPM to a multi-period world with a time-varying investment opportunity set and possibly multiple dimensions of risk, whereby the expected return is a compensation to investors for bearing not only the market risk (as is included in the traditional CAPM), but also the risk of unfavorable shifts in the investment opportunity set.
Campbell (1993) developed a loglinear discrete-time approximation of the Merton’s (1973) ICAPM. In this approximate solution of the model, the risk premia depend only on the coefficient of relative risk aversion γ and the discount coefficient ρ, and not directly on the elasticity of intertemporal substitution, which is assumed to be close to one. Under an optimal portfolio strategy p, the risk premium on asset i satisfies
) , ( cov ) 1 ( ) , ( cov 2 ] [ , 1 , 1 , 1 , 1 , , 1 2 , 1 , 1 ,+ − + + = t it+ pt+ − t pt+ + − t it+ − pDRt+ t i t f t f h h h Eh h N h E σ γ γ (4)
where p is the optimal portfolio that the investor chooses to hold and
∑
∞ = ++ + + ≡ − 1 1 1 1 , , ( ) j j t j t t t DR p E E hN ρ is the discount rate news on this portfolio. The left-hand side of the Equation (4) is the expected excess log return on asset i over the riskless interest rate, plus one-half of the variance of the excess return to adjust for the effect of Jensen’s inequality. The right-hand side of (4) consists of a weighted average of two covariances, one is the covariance of return on asset i with the return on portfolio p, the other is the covariance of the return on asset i with negative news about future expected returns (or discount rate news) on portfolio p. The second term on the right-hand side weighted by (1-γ) represents the intertemporal hedging component of the asset demand, for if γ = 1, the portfolio choice is myopic and in this case pricing implications is derived for the familiar CAPM.
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Equation (4), based on the equality defined earlier, hp,t+1−Ethp,t+1 =NCF,t+1−NDR,t+1, in
which the excess return shocks are decomposed into two news terms. After multiplying and dividing by the conditional variance σ of portfolio p’s return, I arrive at 2p,t
ft f t it pt iCF t pt iDR t p p h h E 2, , , 2, , , 2 , 1 , 1 , 2 ] [ + − + +σ =γσ β +σ β (5)
where the risk premium for the discount-rate beta should equal the variance of the portfolio return, while the price of risk for the cash-flow beta should be γ times greater, γ being the investor’s coefficient of relative risk aversion. Assuming that portfolio p is fully invested in a value-weighted market index, the above relation then also holds with respect to the market variance. If the investor is risk-averse (γ >1), the cash-flow beta should have a higher price of risk. It is also for this reason that the two betas from the decomposition above are referred to as the ‘good’ and the ‘bad’ beta by Campbell and Vuolteenaho (2004), where the risk of a stock for a long-term investor is found to be primarily determined by its ‘bad’ cash-flow beta with only a secondary influence from its ‘good’ discount-rate beta. Equation (5) implies that the risk premium of the discount rate news should equal the variance of the market return, and this will be implemented in the specifications of one of the asset pricing models later in this study.
4.3.2 Empirical Estimates of Risk Premia
We now compare the performance of the two-beta model with three alternative asset pricing models by testing them on the 25 size- and book-to-market-sorted portfolios. Each model is estimated in two different specifications: one with a restricted zero-beta rate equal to the Treasury-bill rate, and one with an unrestricted zero-beta rate that is allowed to vary freely in the regression equation. The two specifications of each model are related to Fischer Black (1972), where he used a ‘two-factor model’ (Equation 7) instead of a single-factor market model (Equation 6) to restrict the CAPM assumption of unlimited borrowing, which is found to give a better fit for the behavior of well-diversified portfolios at different levels of βi.
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where h is the return on a ‘second factor’ that is independent of the market (it has a zb zero βi). This model thus suggests that when h is positive, the low zb βi portfolios will perform better and the high βi portfolios will perform worse than predicted by Equation
(5). The reverse is true when h is negative: low zb βi portfolios do worse and high βi
portfolios do better than expected from the traditional CAPM. Equation (7) approximate the returns with a weighted average of return on the market portfolio and return on a portfolio of riskless assets. Taking expectations of both sides of Equation (7) and rearranging terms, I obtain
) ( ) 1 ( zb M i zb zb i zb M i zb i M i i Eh Eh Eh Eh Eh Eh Eh Eh Eh − + = − + = − + = β β β β β
where the last equality is very similar to the CAPM model in equation (6), only the risk-free rate h is now replaced by f h , which we will denote from now on as the zero-beta zb rate. In other words, the effect of relaxing the unlimited borrowing assumption in CAMP by Black (1972) is simply replacing the risk-free rate by the zero-beta rate.
When estimating risk premia in this study, a model is first estimated with a zero-beta rate h restricted to be equal to the risk-free rate zb h (which means f h less zb h is set f equal to zero). Next, the same model is estimated again with a zero-beta rate that is allowed to take any value, which is manifested empirically by introducing an intercept term in the regression equation. The first specification of the model includes risk-free assets such as Treasury bills in the set of alternative assets available to the investor, while the in the second specification, risk-free assets are not available and the investor has only access to other equities as alternative assets. As explained in CV (2004), the equity risk premia estimated in the first specification includes both unconditional equity premium and premia to value and small stocks; while in the second specification, the equity premium can be separated from the set of variables to be explained.
two-23
beta model, see Equation 8), the three Fama-French factors (Equation 9), or the excess market return (Equation 10), depending on which model is being investigated. The bar in the equations denotes time-series mean and hie ≡hi −hf denotes the sample average simple excess return on asset i. The 25 test assets are then regressed onto these betas belonging to each model to obtain the risk premia listed in Table 5.
i DR i CF i e i g g g h = 0 + 1βˆ, + 2βˆ, +ε~ (8) i HML SMB M e i g g g g h = 3 + 4βˆ + 5βˆ + 6βˆ +η~ (9) i M e i g g v h = 7 + 8βˆ +~ (10)
Beside the premia estimates, the R statistic is also reported for a cross-sectional ˆ2 regression of average returns on the test assets onto the fitted values from the model. The regression R is computed as in Equation (11) below (the same way as in EViews®), ˆ2 which allows for negative R2 for poorly fitting models estimated under the constraint that the zero-beta rate equals the risk-free rate. The ˆ2
R statistic is used to give an intuitive and transparent measure for the performance of different asset pricing models.
24 5. Empirical Results
5.1 Computing the effect of cash-flow and discount-rate news
Estimating the first-order VAR model with the defined state vector enables us to observe the effect of the state variables on the excess returns and provides means to construct the cash flow news and discount rate news terms by linear mapping as described earlier. Table 2 reports parameter estimates for the VAR model over the full sample period November 1980 to December 2008. Each row of the table corresponds to a different equation of the VAR model. The first five columns report coefficients on the four state variables and the constant. Ordinary least squares (OLS) t-statistics for the estimated coefficients are reported in parentheses. In the last two columns of the table, R2 and F statistics for each regression are reported. The bottom of the table contains correlation matrix of the equation residuals, with standard deviations of each residual presented on the diagonal.
The first row of Table 2 shows although all signs of the parameters in the excess market return estimation equation matches with the CV (2004) values, none of the four VAR state variables is statistically significant in predicting excess returns on the aggregate stocks market, while all four of the state variables in CV (2004) have some ability to predict the excess returns. The 1.1-percent R2 is also lower than the 2.6 percent reported for the similar return forecasting equation in CV (2004). Even though the absolute values of the R2 appear to be very low, this is still acceptable for a monthly model which has more volatility in data points. In the Campbell et al. (2010) study, the VAR parameters are estimated using the same set of state variables but based on annual data, where three out of the four state variables (with the exception of the lagged excess market return) are statistically significant when regressed on excess market return, with an slightly improved R2 value of 10.8 percent.
25
short-term bonds. In both CV (2004) and Campbell et al. (2010) however, the term yield spread are mainly explained by the lagged term spread and the lagged small-stock value spread. The price-earnings ratio is also highly persistent with an autoregressive coefficient as large as 0.98, but the PE ratio is not also predicted by the lagged market return, as it is the case in CV (2004). Finally, the small-stock value spread could only be predicted by its own lagged value, although the coefficient is not as highly persistent as the VAR model in CV (2004), because there the small-stock value-spread is defined as the difference in log book-to-market ratios of small value and small growth stocks; whereas in this study, the small-stock value-spread is the return difference between the small value and small growth stocks.
Following the estimation results from Table 2, the behavior of the implied cash-flow news and discount-rate news components of the market return are reported in Table 3. The top right panel shows that the discount-rate news has a standard deviation of about 3.3 percent per month, slightly larger than the 2.7-percent standard deviation of cash-flow news. This result is inconclusive to the hypothesis that discount-rate news is the dominant component of the market return variation, which is found by Campbell (1991) and CV (2004), where the discount rate news has a much larger standard deviation than the cash flow news (5 percent versus 2.5-percent in CV (2004)). More importantly, this table also shows that the two components of return are almost uncorrelated with each other, which justifies the decomposition framework and is consistent with the CV (2004) forecasting model using the same amount of state variables.
26
between either the market return shocks or the P/E shocks and the cash flow news as in my study, even though the signs of both correlations with the cash flow news agree between CV (2004) and my results. Innovation in term yield spread are significantly positively correlated with both news terms, while it is only significantly correlated with discount rate news in CV (2004). Finally, the significant and negative relationship between shocks in small-stock value spread and cash flow news found in CV (2004) is not observed here, while the significantly positive correlation between value spread innovation and discount rate news in my study does not corresponded with the weakly negative correlation in CV (2004).
The coefficients that map innovations in state variables to cash flow and discount rate news are shown in the lower right part of Table 3 under ‘Functions’, which are the numerical values of the vectors (e1'+e1'λ) and e1'λderived in the methodology section. All coefficients here have the same sign as in CV (2004) and Campbell et al. (2010), although the values are almost always smaller in Table 3.
An illustration of the news terms implied by the VAR model and recessions in UK during the sample period is given in Figure 1b. The recession periods are from the Office for National Statistics in the United Kingdom, the first vertical line from the left denotes the end of the 1979-1983 recession (monthly average drop in GDP of 3%), the second and third line from the left denote the 1990-1992 recession (monthly average drop in GDP of 2%), and the vertical line on the most right signals the start of the most recent recession starting in 2008 caused by the financial crisis. Because the sample period contain only one full period of recession in the UK history, the effect of the cash flow and discount rate news on recessions is not very clear.
27
A first view of the long-term trends in cash flow news and discount rate news in both graphs (Figure 1a) gives the impression that there has not been many strong upward or downward trends in both news terms during the examined time span of almost thirty years. However, when I extract and zoom in on the long-term trends in cash flow news and the negative of the discount rate news in a separate graph (Figure 1b), the trends in both news terms become visible. Toward the end of the 1979-1983 recession, the cash flow news and the negative of discount rate news are both moving upward, apparently back from a crisis-low to a long-term average zero-level. This is also when interest rates fell and the inflation in the UK dropped from an earlier high of 18% to 8.6%3
Due to the short sample period and the fact that both news terms may behave differently in each recession, no clear-cut behavior of the movements of the news terms can be concluded from Figure 1b, although one might conjecture that at the beginning of a crisis, cash flow news generally trends downwards due to deteriorating business confidence and financial position of the private sector, and discount rate perceived by investors increases sharply due to the high inflation prevailing in many recessions; conversely towards the end of a crisis, investors receive positive news about firms’ future cash flows and inflation falls back to the pre-crisis level (both cash flow news and the negative of discount rate news move upwards). In the periods that are not registered as recessions, cash flow news and the negative of discount rate news fluctuate in opposite . On the other hand, at the start of the current financial crisis in 2007-2008, caused by a shock to the availability of credit and a massive build up of debt, both cash flow and the negative of discount rate news are strongly decreasing starting around the year 2005. The only full period recession included in the sample period is the 1990-1992 recession, during which a sharp devaluation of the pound took place led by the savings and loan crisis in the US. Here the news terms move in opposite directions, with the discount rate steadily decreasing (the negative of the discount rate news moves upwards), while the cash flow news reaches a trough in the middle of the crisis period as large numbers of business failed and the housing market crashed caused by the high interest rates.
3
BBC: On This Day ‘26 January 1982’
28
directions. More data from a longer time span is required to confirm the above hypothesis, which may become valuable in recognizing and predicting other period of recessions.
5.2 Measuring cash-flow and discount-rate betas
We have seen above that the two components of the market return indeed display substantial volatility and are not highly correlated with each other. In order to measure the possibly different betas of different types of stocks with both news terms, I use the cash flow beta and discount rate beta defined in section 4.2, which are obtained by regressing the returns of each of the 25 size- and book-to-market portfolio on the scaled news series ) var( ) var( , , DR M e M DR M N r N × and ) var( ) var( , , CF M e M CF M N r
N × , following the approach by CV (2004). The scaling normalizes the regression coefficients to correspond to the definitions of βi,DR and βi,CF, which ensures them adding up to the CAPM beta.
Table 4 shows the estimated betas for the 25 portfolios over the 1980:11-2008:12 period. The portfolios are organized in a square matrix with horizontally increasing BE/ME categories (from growth stocks to value stocks) and vertically increasing size of market capitalization (from large to small). The differences between the smallest and largest portfolios in each BE/ME category and between most extreme growth and value portfolios of each size are reported along the right margin and the bottom of the matrix. The top matrix displays cash-flow betas while the bottom matrix displays discount-rate betas. After each beta estimate the t-statistic is reported in parenthesis.
29
the growth-value categories of the stock portfolios. Large stocks have both significant and higher cash-flow and higher discount-rate betas than small stocks (by 0.05 and 0.04, respectively, for an equal-weighted average across the five growth/value portfolios), which is contrasted by the higher cash-flow and discount-rate betas for small stocks than for large stocks in CV (2004).
Consistent with the definition that the cash-flow and discount-rate betas sum up to the CAPM beta (‘market beta’), when I regressed the 25 portfolio returns on market returns in an usual way to get market betas, I obtained values that are very much the same as the sum value of the related cash flow and discount rate betas. These two components of the market beta are plotted in Figure 2 for each BE/ME portfolio category as the average value of the betas across the five size portfolios. From this plot, one can see that the difference between the CAPM beta of value and growth portfolio is relatively smaller than shown in CV (2004), but a subtle trend is still visible: growth stocks (BE/ME category 1) tend to have higher market betas than value stocks, and this difference is mainly caused by the discount-rate beta which is about 11 percentage points higher for growth stocks than for value stocks. The absolute value of the discount-rate betas are also larger than the cash-flow betas. Following the CV (2004) interpretation, the higher market beta of growth stocks are disproportionately of the ‘good’ discount-rate variety rather than the ‘bad’ cash-flow variety. This finding agrees with the results obtained in CV (2004).
5.3 Pricing cash-flow and discount-rate betas
30
Table 5 shows the risk prices estimated for the four asset pricing models studied in this text. Each model has two columns corresponding to the two specifications of the model. The table also has 6 sets of 3 rows. The first set of 3 rows corresponds to the zero-beta rate in excess of the risk-free rate, the second and the third set to the premium on cash-flow beta and discount-rate beta, respectively. The last three sets of 3 rows are specific premiums for the three Fama-French factor betas. Within each set of rows, the first row gives the point estimate of the risk premium per month, and the second row gives the annualized premium by multiplying the number in the first row by 1,200. The third row reports the t-statistic for each estimated parameter. The row in the middle reports the R2 statistic of each model specification.
In the period of examination, the explanatory power of the two-beta model with zero-beta rate set equal to the risk-free rate is shown to be only slightly higher than the two-beta model with unrestricted risk prices, where the zero-beta rates are allowed to obtain a non-zero value. The cash flow betas in both model specifications have explanatory power for the portfolio excess returns, which is consistent with CV (2004), where no other coefficient, except for the cash flow betas, is statistically significant. Therefore restricting the zero-beta rate to the risk-free rate does not have much effect for the model, neither are discount rate betas useful in predicting excess stock returns. The R2 statistics in CV (2004) for the same two-beta model specifications are generally 20 percentage points higher than results obtained here, which may be caused by the less pronounced difference in cash flow betas and discount rate betas across different size- and book-to-market portfolio categories in my study.
Looking at the estimated risk prices for the cash-flow beta in both restricted and unrestricted two-factor model, it is clear that the premium on cash-flow beta is high at between 35% to 50% per year and statistically significant. In contrast, the risk prices for the discount-rate beta is both low and statistically insignificant for these two models, similar as in CV (2004). This situation is possible even if the model is correct, as the estimated discount-rate betas are noisy and could cause the model to behave badly.
31
although the restricted zero-beta form of the model has a lower R2 statistic, while the two-beta ICAPM model in CV (2004) has approximately the same performance when either the restricted or free zero-beta rates are used. Again in accordance with CV (2004), the coefficients of all factors that are allowed to vary are highly significant in the model, and all signs here agrees with the two-beta ICAPM in the modern sample (1963-2001) in CV (2004).
The three-factor model of Fama and French (1993) greatly outperforms both restricted and unrestricted two-beta models studied in this paper, as well as the traditional CAPM. The CAPM has very low explanatory power for the test assets, and when the zero-beta rate is set equal to the risk-free rate, the R2 statistic of the CAPM even becomes negative, which means that the model has a larger pricing error than the null hypothesis that all portfolios have equal expected returns. Contrary to this result, the cross-sectional R2 statistic as high as 70 percent for the Fama-French three-factor model is more than 40 percentage points higher than that of the two-beta model with both unconstrained zero-beta rate and the zero-zero-beta rate constrained to the risk-free rate. This finding agrees with CV (2004), where the outperformance of the Fama-French model with regard to the two-beta model is about 30 percentage points for the U.S. sample during the period 1963-2001.
This higher explanatory power was attributed to the two more additional degrees of freedom of the Fama-French model over the two-beta ICAPM model. Since for the latter model, the only free parameter in the regression equation (Equation 5) is the coefficient of relative risk aversion γ. With regard to the earlier established implication when discussing the ICAPM, where risk premium of the cash flow beta should be γ times larger than the discount rate beta in the case of a risk-averse investor, I can extract from the coefficients of the two-beta ICAPM that the measure for relative risk aversion γ are implied to be 15 and 3, respectively, for the free zero-beta and restricted zero-beta rate versions of the model. The corresponding values for γ are 29 and 24 in CV (2004), confirming the assumption that a risk-averse investor considers cash flow news beta as the ‘bad beta’, resulting in a higher risk premium demanded for the cash flow beta.
32
thus an extra degree of freedom, is allowed. The relaxed two-beta model might be a better approximation to reality as it allows the rational investor’s portfolio to include Treasury bills beside equities (CV, 2004). Given these observations and the poor performance of the traditional CAPM from Table 6, the relative performance of the two-beta ICAPM is considered to be quite impressive.
To provide a visual aid to the comparisons done above, I plot the 25 test assets for each model in Table 6 with either the free or the constrained zero-beta rate. The results are presented in Figure 2 to Figure 5. Each figure plots a portfolio’s predicted annual average excess return from a particular model on the horizontal axis and the actual sample annual average excess return realized by each portfolio on the vertical axis. The strong degree of ‘correlation’ between the predicted and the realized average excess returns are easily visible for the Fama-French three-factor model (Figure 5), which corresponds to a high R2 value of more than 70 percent. The same effect is visible to a lesser extent for the two-beta model and the two-beta ICAPM with free zero-beta rate (Figure 2 and 3). The worst performing pricing models in the test, the CAPM with both specifications and the two-beta ICAPM with zero-beta rate constrained to the risk-free rate, predict average excess returns of around 5% annually for all test assets, although the realized average excess returns range from slightly above 10% and slightly below 0%.
5.4 Additional Robustness Checks
5.4.1 Sensitivity to additional state variables
33
The main result after including SMB in the VAR system, which I do not report here, is largely similar to the result obtained earlier. This is possibly the consequence of very little and insignificant correlation between the shocks in SMB and both news terms. The SMB is also not at all persistent (autocorrelation of 0.15) as the other state variables such as the term yield and price earnings ratio (with autocorrelations larger than 0.95), which receive more weight in the VAR system by the term (I− A)ρ −1.
This reminds us that in the future search of new, economically meaningful state variables or proxies to improve or simulate the current VAR system, potential explanatory variables with a large autoregressive coefficient should be considered first, or otherwise the effect of additional or replaced state variable on the regression result will be very small. However, future studies on alternative state variables are needed to strike a balance between economic motivation and strong effect in regression result, as dominant state variables with an autocorrelation of more than 0.99 (the 10-year smoothed P/E ratio in CV(2004)) raises serious stationarity concerns (Chen and Zhao, 2009).
5.4.2 Split-sample tests
In order to look out for different behaviour in the beta estimates in the two subsamples, I split the stock returns data and the news terms into two periods of equal length (November 1981 to November 1994 for the early sample, December 1994 to December 2008 for the late sample). Conducting split-sample tests also helps in detecting potentially spurious explanatory power of the betas induced by the regression setup based initial sample size.
34
1963-2001 sample, value stocks have much lower discount rate betas, and only slightly higher cash flow betas than growth stocks.
From Figure 6 and 8 in the appendix, one first notices that the average excess returns for the 25 size and BE/ME portfolios generally are smaller in the late sample period than in the early sample period. The two-beta model with both constrained and unconstrained zero-beta rate have unchanged explanatory power for the average excess returns of the test assets for both sub-sample periods, while the performance of the two-beta ICAPM with free zero-two-beta rate of both sub-sample periods are worse than for the whole sample period. The Fama-French three-factor model performs well in both sample periods and similarly for both constrained and free zero-beta rates as shown in Figure 7 and 9. However, although the CAPM performs badly for both model specifications in the early sample, a sudden increase in explanatory power is witnessed for the unconstrained zero-beta rate version of the model, with a R2 value of almost 30 percent. In the CV (2004) study, the CAPM performs well in the 1929-1963 sample period (R2 of at least 40 percent) due to the fact that the cash flow news remains roughly a constant fraction of the CAPM beta, but the CAPM is also found to fail dramatically for the 1963-2001 period (R2 values largely negative).
35 6. Conclusion
In this paper I apply the two-beta model developed in Campbell and Vuolteenaho (2004) to a recent set of data on UK stock returns, which decomposes the excess returns into two risk loadings: one reflecting news about the market’s future cash flows, and the other reflecting news about the market’s discount rates. I estimate the cash flow and discount rate betas of value and growth stocks from the behavior of their returns, without the need of dissecting how these betas are linked to the underlying cash flows of individual companies. In accordance with the ICAPM and the CV (2004) study, I found that the ‘bad’ cash-flow beta indeed have a risk premium several times larger than the ‘good’ discount rate beta. Also in agreement with the finding for the post-1963 US data in CV (2004), UK growth stocks in my sample have larger discount rate betas than value stocks, although the difference is not as large as in the US. Thus the higher CAPM market beta of the growth stocks observed during the sample period is largely of the ‘good’ discount-rate beta.
Following the approach in CV (2004), I compared the performance of the two-beta model with a two-two-beta version of the ICAPM, the traditional CAPM, and the Fama-French three-factor model in explaining average excess returns of sorted size-BE/ME portfolios. I find that although the Fama-French model is superior in explaining stock returns, the two-beta model performs relatively well and is robust against an additional state variable and split-sample tests.
36
A possible extension to the study might follow Campbell and Mei (1993) and Campbell et al. (2010), where each of the stock’s cash flow and discount rate beta can be further decomposed into two parts relating to either cash flow and discount rate news, which may yield additional insights on the beta decomposition framework.
Recognizing the data limitations encountered in the current study, firstly due to the availability of UK-area Fama-French factors and portfolios covering only a period less than 30 years, secondly due to the difficulty in obtaining monthly compiled BE/ME ratio for individual stocks in the UK, I expect further empirical tests of the two-beta model on data covering a larger time span or a set of different countries outside the US are required to facilitate a better understanding of this increasingly influential asset pricing model.
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