Returns and GARCH Component Estimated Variance Thierry C.J. Parihala1 Faculty of Economics and Business Administration Tilburg University, NL Final version: 08-12-2010 Abstract This paper tries to contribute to a better understanding of the general risk-return relation by examining the relation between the market variance and market return and the relation between the market variance and the return of a small portfolio. First, the market variance is measured using the GARCH Component model of Engle and Lee (1993). Second, two new asset pricing models are constructed by adding variance factors to the CAPM and to the model of Fama and French (1993), creating the models CAPM+GCOMP and FF+GCOMP respectively. Third, these models are used in regressions on the market and portfolio return. Regressions using CAPM+GCOMP show there is a statistically and economically significant positive relation between the market variance and the returns. Regressions using FF+GCOMP do not give conclusive results. Several motivations are examined that are likely to drive these findings. This paper concludes that the market variance and market returns appear positively related. the market variance and portfolio returns also appear positively related, but this relation seems weaker and does not seem robust. 1 Thierry Parihala, student number 753274, has written this research paper as a student at Tilburg University for the Master Financial Management – enrollment period: 2008-2010. This research paper is written to serve as the Master Thesis for this master and may be freely distributed, copied, and edited. I am grateful to Prof.dr. B.J.M. Werker for mentoring, commenting, Thomas van Steenis for reviewing and commenting, and Esther Vorstenbosch for coaching. The views expressed in this paper are those of the author and do not necessarily represent those of Tilburg University or Prof.dr. B.J.M. Werker. This paper is digitally available via the Library Services of Tilburg University. For comments, questions or a digital copy, please contact the author at [email protected] 1 I Introduction The financial literature on the relation between risk and return is abundant. Theoretical works like that of Sharpe (1964) and Black and Scholes (1974) indicate a relation between stock returns and stock variance or between stock returns and the covariance between stock returns and market returns. The empirical literature remains ambiguous, both regarding the existence of this relation as regarding its nature. For example, French, Schwert and Stambaugh (1987), Adrian and Rosenberg (2008) and Bollerslev and Tauchen (2009) find a positive relation between variance and returns. Glosten, Jagannathan, and Runkle (1993) and Ang, Hodrick, Xing and Zhang (2006) find a negative relation, while Baillie and DeGennaro (1990) find no significant relation between variance and returns. The financial literature shows various approaches have been used to measure the relation between variance and returns. Different methods of measuring variance have been employed such as GARCH, GARCH-M, component models, and “model-free” methods. Despite this, clear consensus has yet to emerge. To complement the various approaches in this strand in literature, and to gain more insight in the risk-return relation, this paper uses a different approach to measure the relation between variance and returns. In the first stage, a version of the GARCH Component model (GCOMP) developed by Engle and Lee (1993) is used to measure the conditional variance. The GCOMP is a twocomponent model which distinguishes between the long-term trend and the short-term trend in variance. Engle and Rosenberg (2000) perform hedging tests on several variance models to compare their performance in explaining realized variance. They rank the ‘GCOMP with leverage’ higher than other specifications of ARCH models, therefore this paper uses this particular version of the GCOMP. In the second stage, the GCOMP variance is used as a pricing factor in several specifications of asset pricing models. Adrian and Rosenberg (2008) find that a pricing model in which short- and long-term variances are added to the model of Fama and French (1993) (FF) performs better in explaining the cross-section of expected returns, than other established pricing models – including the CAPM and FF model. In this context, variance factors are added to the CAPM and to the FF model. The first factor which is added to these models is the conditional variance specified by the GCOMP. The second factor is the permanent component of the GCOMP, q, which is analogous to the long-term trend in variance. Hence, the models CAPM+GCOMP and FF+GCOMP are created by adding these two factors to the CAPM and FF model respectively. In the third stage, these pricing models are used to perform regressions on the market return. Scope limitations of this paper demarcate the area of interest of the empirical part to the effect of market variance on 2 market returns. In this way, this paper tries to gain more insight in the risk-return relation on a broader, market level. The results should answer the question, how is market risk related to market return? The results obtained to answer this question can contribute to a better understanding of the general risk-return relation. In addition, to explore if the contingent effects of market risk on market return are also generalizable to a smaller selection of stocks, the effect of market variance on the return of a small portfolio of randomly selected stocks is also examined. The results of the regressions using CAPM+GCOMP show that market variance is positively related to the market return and the portfolio return. The explanatory power of market variance is stronger for the market return than for the portfolio return. The coefficient on market variance is statistically and economically significant for both the market as well as the portfolio return. One standard deviation increase in market variance results in 1.6 basis points increase in the daily market return and 1.5 basis points increase in the daily portfolio return. The results regarding the market return appear robust using different specifications of ARCH models to measure the variance. The regressions using FF+GCOMP do not give conclusive results regarding the explanatory power of market variance in stock returns. Several motivations for these findings are examined. The possibility that the FF factors do a better job in explaining the effects of risk on return than the conditional market variance is not excluded. Accuracy issues in estimating the variance are also considered as a driver of its low explanatory power regarding asset returns, as financial literature, in the meanwhile, has suggested better performing alternatives to model the conditional variance. For example, Adrian and Rosenberg use a more advanced two-component variance model and do find a statistically and economically significant role for short- and long-term variance when these factors are added to the FF model. The conclusion of this paper is the market variance and market returns appear positively related. The market variance and portfolio returns also appear positively related but this relation seems weaker and does not seem robust, which creates interesting possibilities for further research. The remainder of this paper is organized as follows. Section II provides an overview of literature related to this subject to provide the context for this research. This section describes several theoretical and empirical works regarding the relation between risk and return and reviews various variance models. Section III is the empirical part in which the methodology, the model, and the results are presented. Section IV presents robustness results and gives interpretations to the results of section III. Section V concludes. 3 II Literature Review 2.1 Theories on risk and return Assuming investors are risk averse and rational, the intuitive consequence with respect to the relation between risk and return is that higher risk makes investors require a higher expected return. Obviously, this means that investors who seek higher expected returns must undertake riskier investments. In order to be able to quantify this positive relation between risk and return, both variables must be measurable. For returns this is obvious. For risk we use the widely accepted measure of variance (or standard deviation), which shows the variation from the mean. Assuming investors’ mean-variance utility function, variance in returns is seen as synonymous for risk. Higher variance implies higher fluctuations in asset prices. The more the value of an asset fluctuates, the less certainty there is about its value at the time the investor wishes to sell it. This lack of certainty is, thus, perceived as risk and is, based on the assumptions, unwanted. A paradigm in finance which tries to describe this relation between risk and return more precisely is the Capital Asset Pricing Model (CAPM), as developed by Treynor (1961, 1962), Sharpe (1964), Lintner (1965a,b) and Black (1972). The model assumes all investors are risk averse and rational. Therefore, those who do take risk, expect to be rewarded. Second, it assumes markets are efficient and perfect. And third, investors can borrow and lend unlimitedly against the risk-free rate.2 The CAPM essentially divides risk into systematic risk and idiosyncratic risk. It assumes the latter to be diversifiable through a properly structured portfolio. Hence, investors will only be compensated for an asset’s systematic risk. That is, investors will only be compensated for volatility which cannot be reduced through diversification. The CAPM defines the volatility of the total market portfolio as the unit for systematic risk, because the market portfolio is, by definition, the most diversified portfolio. Inherently, the return on the market in excess of the risk-free rate is the price of one unit of systematic risk. As common in economic theories, this price arises from an equilibrium determined by the supply of and the demand for risk. Here, the demand is a function of investors’ risk aversion. The logic reasoning of the CAPM that follows is that an asset’s return in excess of the risk-free rate is therefore linked to an asset’s exposure to systematic risk and the price of systematic risk. The exposure to systematic risk is determined by 2 To complete the analysis, CAPM also assumes that investors only care about expected returns and volatility, that they have homogeneous beliefs about the market, that there is only one common risk factor which is the systematic risk, and that returns are normally distributed. 4 measuring the sensitivity of an asset’s returns to the sensitivity of market returns, which is defined by: βi = cov (ri, rm) / var(rm) (1) Where, βi is asset i's sensitivity to the market returns, Cov (ri, rm) is the covariance of the returns of asset i and the returns of the market, and var(rm) is the variance of the returns of the market. The expected return that the CAPM derives is: E(ri) = rf + βi(E(rm) – rf) (2) Where, E(ri) is the expected return of asset i, E(rm) is the return on the market portfolio, and Rf is the risk-free rate.3 According to the CAPM formula, asset returns are related to the volatility of the market to a degree expressed by the beta. A higher beta implies more sensitivity to market returns, a larger part of the volatility that cannot be reduced by diversification, more exposure to systematic risk, and thus a higher expected return. In its attempt to describe the interaction between risk and return, the CAPM proves a positive linear relation with a constant rf and a reward for risk which is a function of the exposure to systematic risk and the price of systematic risk. The CAPM provides a comprehensible analysis of the risk-return relation in a straightforward way. However, the simplicity of the analysis is only allowed for by defining certain assumptions which also form the weakness of the model. Because the derivation of the model is flawless, criticism covers the assumptions. Such as the fact that perfect markets do not exist, one can’t borrow and lend unlimitedly at the risk-free rate, and there is no true risk-free asset. Critique has also reached a conceptual level. Ross (1976b) argues it is difficult to justify the assumption of normality of returns as well as investors’ mean-variance preferences. Furthermore, Roll (1977) explains that the CAPM cannot be tested unless the complete market portfolio, including all assets, is known, which cannot be identified. An alternative to the CAPM is the Arbitrage Pricing Theory (APT), as described by Ross (1976a, 1976b). Like the CAPM, the APT is a one-period model in which stochastic properties of returns are consistent with a factor structure. Both models share the assumptions of perfect capital markets and homogenous expectations. However, the APT does not require assumptions of normally distributed returns and merely one common risk factor. Instead, the model assumes that the returns of asset i can be predicted using the linear relationship 3 The original CAPM is based on arithmetically calculated returns, whereas this study will use log normal returns. 5 between the returns of asset i on the one hand, and k common risk factors and idiosyncratic (firm-specific) risk on the other. Furthermore, the APT assumes the non-existence of arbitrage opportunities. Under these assumptions, Ross argues that the return of a risky asset can be defined as: i= Here, i i + bi1 1 + bi2 2 + … + bik is the expected return on asset i, k + 1, (3) i …, k are the risk factors, the extent to which the return depends on the these factors is given by the factor loadings bi1, …, bik, and i is the idiosyncratic risk. This idiosyncratic risk is assumed to be zero on average, uncorrelated between assets, and uncorrelated with the factors. As such, Ross proves that, in large numbers, the average effect of the idiosyncratic risk can be neglected. The conclusion of the APT is therefore, that the expected risk premium of a risky asset merely depends on the factor loadings: i Where ( – rf = bi1( fk f1 – rf) + bi2( f1 – rf)… + bik( fk – rf) (4) – rf) is the premium on factor k. The intuition behind this is that of Arrow Debreu security pricing (Arrow and Debreu (1954)). If the unexpected part of asset i’s return is linearly related to a set of k factors, then the expected return of that same asset must also be linearly related to that same set of k factors. The model does not specify which factors might have explanatory powers. But after finding these factors and determining the factor loadings through regression, the APT will indicate what an asset’s price should be in accordance to the most recent factor values - assuming the model is well specified. If the indicated price were to deviate from the actual price, an arbitrage opportunity would present itself – which is assumed not to be possible. In addition to having less stringent assumptions, one other advantage of the APT over the CAPM is the absence of the direct need to identify the market portfolio. Numerous studies have tried to identify appropriate common risk factors in an effort to specify a factor model with a good fit. Fama and French (FF) show that the relations between average return and size, and average return and book to market equity are strong and that the explanatory power of the CAPM beta is actually weak (Fama and French (1992)). Fama and French (1993) argue that the expected excess return is largely explained by the sensitivity to three factors: 1) the risk premium on the market; 2) the return on a long position in a portfolio of small stocks, financed with a short position in a portfolio with large stocks (SMB); and 3) the return on a long position in a portfolio of high book-to-market stocks, financed with a 6 short position in a portfolio with low book-to-market stocks (HML). Therefore, they suggest a factor model which accounts for these factors: E(rm) – rf = α + bi(E(rm) – rf) + siE(SMB) + hiE(HML) (5) Here, (E(rm) – rf), E(SMB), and E(HML) are the expected premiums, and bi,, si, and hi are the factor loadings determined by regression. In theory, the constant α is equal to zero as this would otherwise point to the existence of possible other pricing factors. However, empirical evidence does not support this hypothesis. The factor loading of the market risk premium bi is analogous to the CAPM beta, but it is not equal to it due to the addition of the other two factors. The extensions to the CAPM, as the SMB and HML factor can be regarded, are a reaction to observed anomalies with respect to the CAPM. The CAPM can be viewed as a particular one-factor model of the APT. This three-factor model can be seen as Fama and French’s interpretation of the APT and is their way to explain anomalies found in the crosssection of average returns. The fact that the model originated from empirical findings makes it quite applicable for practitioners who require estimates of expected stock returns. However, the theoretical justification for the common risk factors is subject to debate. 4 It can thus be said that the Fama and French model is more descriptive than explanatory. All the same, the discussion is manifested at a conceptual level whereas the value of the SMB and HML risk factors is empirically well proven.5 Likewise, empirical observations, that complement the theories of the CAPM and the APT, have contributed to consensus in literature about the existence of an important relation between asset returns and market risk.6 However, the exact shape of this relation remains subject to the methodology used. 2.2 Modeling Variance There is a conceptual discussion in science about what is the true variance in stock returns. Variance can be estimated using data with different kinds of frequency. Most of the literature stemming from the previous century is based on monthly or daily data. However, advances in technology have made it possible for research to use intra-day data or even transaction data to 4 One economic interpretation of the explanatory power of SMB and HML that is in line with the Effcient Market Hypothesis (EMH), as described by Fama (1969), is that the higher returns on these portfolios are due to the higher risk which is inherent. Fama and French (1996) argue that a high book-to-market ratio signals distress and doubts about future earnings, as do Chan and Chen (1991). A small firm could be less diversified making it more exposed to business risk. Both result in a higher cost of capital. Adversaries of the EMH argue that the higher returns on these portfolios are due to corrections following undervaluation. 5 See Banz (1981), Basu (1983), Rosenber, Reid, and Lanstein (1985), and Lakonishok, Shleifer and Vishny (1994). 6 For empirical evidence see Douglas (1969), Miller and Scholes (1969), Black, Jensen, and Scholes (1972), Fama and French (1993, 1996), Carhart (1997). 7 estimate variance. The discussion can subsequently be shifted to whether to use bid, ask or mid prices. Researchers are given a lot of options in choosing the ‘best’ data frequency and type and there is no convention or consensus about what is appropriate. Data used in financial literature thus lacks certain uniformity. This paper will treat the results of different types of data to be generalizable to results generated by other types of data as to avoid the discussion. In modeling the conditional variance, the only real constant seems to be the continuous evolution of stochastic variance models over time. To motivate the variance model used in section III, the following paragraph will provide an overview of variance models working up to the GCOMP model with leverage of Engle and Lee (1993). Based on the ARCH, as proposed by Engle (1982), Bollerslev (1986) introduced the GARCH(p,q) model which may be written as: rit - rft = µt - 0.5σ ෝ2t + εt εt ~ N(0, σ ෝ2t) (6) σ ෝ2t = α + βσ ෝ2t-1 + Φε2t-1 where rit is ln(pit /pit-1), where pit is the price of stock i at time t, rit - rft is the return in excess of the risk free rate, µ is a constant risk premium, σ ෝ2t is the conditional variance, and Rft is the risk-free rate. This model allows for both autoregressive and moving-average components in the heteroskedastic variance, making it an ARMA process. The parameter β represents the rate of mean reversion, and the parameter Φ determines the relative importance of the lagged squared error. Through the years, the widely used GARCH has uncovered one stylized fact, which is the fact that variance in asset returns is highly persistent, i.e. the sum of the parameters Φ and β is close to one. Despite this however, the mean reversion as modeled in the GARCH also implies an exponential decay of a shock’s persistence. This is contrast with findings in many empirical studies, which have pointed out that shocks to the conditional variance endure in an extreme degree7. In other words, the GARCH assumes markets have a short term memory, whereas empirical findings suggest markets have a long term memory. An extension to the GARCH handling this issue is the IGARCH, as introduced by Bollerslev (1986). When the GARCH model is integrated in variance, the innovations sum up to one: 7 See for example, Bollerslev, Chou, and Kroner (1992). 8 ୀଵ ୀଵ ߚ + ߔ = 1 (7) This way, the IGARCH incorporates infinite persistence of shocks. A side effect of the unit root in this model is that it is non-stationary. That means that there are no mean reverting characteristics. As mean reversion is no longer a property of the variance, predictions using IGARCH will tend to resemble recent variation more than the average historic variation. For example, a large shock in 1929 will be expected to have less effect than an equally sized shock in 1982. Another implication of the infinite persistence is the fact that the variance will increase linearly with the forecast horizon. As Baillie, Bollerslev, and Mikkelsen (1996) argue, such a relation between variance and horizon would cause the pricing of assets to be extremely dependent on the initial conditions, or the current state of the economy, which both contradict with observed pricing behavior. Besides Baillie et al, Bollerslev and Engle (1993) also argue the infinite persistence contradicts to stylized facts. They argue financial markets’ variance depends on long run dependencies, and seem to have long term memory, but no infinite memory. Hence, both the GARCH’s exponential decay and the IGARCH’s infinite persistence seem to contradict with empirical findings. Or to quote Baillie et al., “the knifeedge distinction between I(0) and I(1) processes can be far too restrictive”. In reaction, Baillie et al propose another extension to the GARCH, the FIGARCH, which is a GARCH model, fractionally integrated I(d), with 0 < d < 1. As a first step in separating shortrun and long-run variance, the FIGARCH captures the short run through the GARCH components, and accounts for the long-run through the fractional differencing parameter (d). This model implies a slow hyperbolic rate of decay of the shocks. Hence, it is more in line with empirical findings. To cope with the leverage effect, first mentioned by Black (1976), Glosten et al. (1993) generalize the GARCH(p,q), to allow for an asymmetric effect of negative return shocks to variance. The theoretical explanation for this effect is that the debt-to-equity ratio increases, when the market capitalization of the firm decreases, following a negative return shock. The Glosten et al. model is defined by: rit - rft = µt - 0.5σ ෝ 2t + ε t εt ~ N(0, σ ෝ2t) (8) σ ෝ2t = α + βσ ෝ2t-1 + Φε2t-1 + γMax(0,-εt-1)2 9 Here, γ measures the effect of a negative return shock to the variance. The other coefficients serve the same purpose as in the GARCH without leverage. Engle and Lee (1999) further develop the distinction between variance in the short-run and the long-run, by modeling the long memory behavior of the variance process as the sum of two components. The transitory component, aimed at capturing the short-run, is characterized by an almost complete integration. The permanent component is aimed at capturing the long-run and has much faster time decay. Their model, known as the GCOMP, short for GARCH Component model, is defined by: rit - rft = µt 0.5σ ෝ2t + εt ε t ~ N(0, σ ෝ2t) σ ෝ2t = qt + θ + α(εt-12 – q2t-1)+ β(σ2-1 – q2t-1) + γ[Max(0,-εt-1)2 –qt-1 ] (9) qτ = ω + ρq2t-1 + Ф(εt-12 - σ2t-1) Here, q is the permanent long-run component, α is the shock on the short-run component of variance, β reflects the influence of the lagged variance factor, γ reflects the short-run leverage effect, ρ is the persistence of the long-run component, and Ф reflects the effect of a variance shock on the long-run component. Engle and Rosenberg (2000) rank the performance of stochastic variance models in explaining ‘true’ variance use hedging tests. They derive and implement volatility hedges using different variance models and rank these models based on their performance in hedging shifts in the volatility term structure. According to their findings, this latter model of Engle and Lee outperforms all other stochastic variance models. 2.3 Empirics on the relation between variance and return As the CAPM argues market risk plays an important role in explaining asset returns, a more direct way of examining this relationship is to assess the relation between market variance and returns. Assuming variance varies trough time; changes in variance will change expectations of future market returns and can affect the risk-return trade-off. In this respect, variance can be seen as a systematic risk factor, which, according to the APT, should be priced in the cross-section of stock returns. Merton (1980) was the first to argue that the market risk premium depends on market variance. Based on the exploratory work of Merton (1980), French, Schwert, and Stambaugh (1987) use daily values to predict monthly variance of stock returns. To correct for both autocorrelation caused by non-synchronous trading of securities 10 and heteroskedasticity, French et al. first model the variance using an adjusted GARCH model which allows for both these effects8. Second they estimate the following relation: E( rmt - rft | ଶ ୫୲ , σumt) = µ + b1 ଶ ଶ୳ ୫୲ + b2 ୫୲ (10) where rm,t is the return on the stock market portfolio, rft is the risk-free rate, ଶ ௧ is the predicted variance of the market, the constant µ is can be viewed as an average risk premium, and where ଶ୳ ௧ = ߪଶ௧ - ଶ ௧ is the unpredicted variance, in which ߪଶ௧ is the observed variance. These regressions do support the hypothesis that the unpredicted variance is negatively related to the realized excess holding period return (b2 < 0). This, indirectly, provides evidence for a positive relation between predictable variance and risk premiums9. As French et al. argue, higher than expected observed variance, ߪଶ௧ > ଶ ௧ , implies predictions of variance will be revised upward for all future time periods. A positive relation between standard deviation and risk premiums will cause the discount rate for future cash flows to increase. Ceteris paribus, both the discounted present values as well as the current stock prices will decrease. Therefore, French et al. argue: “a positive relation between the stock market variance and the risk premium induces a negative relation between the unpredicted component of variance and excess holding period returns”.10 However, they argue that the effect of the negative relation between the unpredicted component in observed variance and risk premiums is likely to be larger than the positive relation between the prediction of volatility and risk premiums. Ballie and DeGennaro (1990) use GARCH-M models, like explained in part 2.2, which are adjusted to allow for the implications of delayed delivery. At the time of their 8 To measure the variance French et al use a GARCH type model specified as: ଶ ୫୲ N N ౪ ౪షభ ଶ = ቄ୧ୀଵ r୧୲ଶ + 2 ୧ୀଵ r୧୲ r୲ାଵ,୲ ቅ (11) Here, there are Nt daily returns, rit, in month t. The product of the daily return with an adjacent return serves to capture the autocorrelation effect. French et al. do not subtract the average daily return in the variance calculation because this adjustment so small that it does not impact the results. To prove this, French et al. have experimented with model adjustments that feature the subtractions of within-month mean return from each observation and experienced little effect on the results. Second, omitting this subtraction of the sample mean when values are small is consistent with Merton (1980). 9 Initial regressions on the variance without the unpredicted variance component, provides weak evidence that the standard deviation is positively related to the expected risk premium. 10 To strengthen their argument, French et al. use the following GARCH-in-mean model, E( rmt - rft) = α + b3 m,t + εt – b4εt-1, (12) as proposed by Engle, Lilien and Robins (1987), with which they show a significant positive relation (b3 > 0 ) between predicted volatility and risk premium in a more direct way. 11 research, delivery of stocks usually took place six days after the purchase. Baillie and Degenarro argue that the opportunity costs of six days of interest could play a role in the riskreturn relation and that unexpected changes in this rate could influence variance. Therefore, they use the change in Federal Funds Rate as a regressor in both the mean and variance equation. Using daily CRSP data from 1970 to 1984 they investigate eight specifications of GARCH-M models, in which the mean equation and variance equation are estimated at the same time, but find only one specification to have a significant coefficient on variance at the five percent level. The lack of evidence of a relationship between the mean returns and the variance leads them to conclude that: “simple mean-variance models are inappropriate”. They do not reject the hypothesis that a relation exists between risk and return, but they suspect investors to use other types of risk measures which are more important than the variance of portfolio returns. Glosten, Jagannathan, and Runkle (1993) also investigate the tradeoff between risk and return. They recognize “the general agreement that investors, within a given time period, require a larger expected return from a security that is riskier”. But they stress that such agreement does not exist concerning the intertemporal relation between risk and return. They argue that, in theory, riskier periods could coincide with periods in which investors are better able to bear risk. In such a case, a higher risk premium would not be a necessity. They also suppose that in a world without true risk free assets, the risk premium on assets would be reduced as the price of risky assets would bid up. They note that most of the empirical works that find a positive risk-return relation use GARCH and GARCH-M models, while empirical works that find a negative relation often use other techniques. Hence, they suggest that the standard GARCH and GARCH-M models may not be rich enough to capture the time series properties of monthly excess returns. Glosten et al. use an adapted GARCH-M model in which they allow for seasonal effects, by adding a monthly dummy to the GARCH-M, and by allowing for asymmetries to control for the leverage effect. Indeed their model suggests a significant negative relation. They also find that the unexpected part of the returns and the following period’s variance are negatively related. As market return is defined as a weighted average return on all assets, the relation between market variance and market returns could be induced by a relation between market variance and individual asset returns. Ang, Hodrick, Xing and Zhang (2006) study this conjecture by examining the effect of market risk on the cross-section of stock returns, investigating relations of the form: 12 E(rit – rft | (rmt – γmt),(σmt – γvt)) = α + ܾଵ (rmt – γmt) + ܾଶ (σmt – γvt) (13) where (rmt – γmt) is the excess return on the market, where γmt is the conditional mean of the market, and (σmt – γvt) is the innovation in market volatility, where γvt is the conditional mean of the market volatility.11 Ang et al. find lower average returns for stocks that have higher sensitivities, ܾଶ , to innovations of market volatility.12 Ang et al. explain this effect using several economic theories. Investment opportunities deteriorate with a rise in volatility, leaving investors with a demand for a hedge (Chen, 2002). French et al. (1987) show that large volatility shocks usually coincide with a downward movement of the market. Hence, stocks with higher sensitivities to innovations in market volatility risk form a hedge (Bakshi and Kapadia (2003)) which results in a higher demand and thus a lower return. Ang et al. thus find a negative relation between risk and return. Though their model outperforms the CAPM, it is outperformed by the Fama and French model. Adrian and Rosenberg (2008) observe that recent component models for variance, in which the variance is divided into a short-run part and a long-run part, demonstrate unique performance in option pricing. They use the following model to estimate volatility: Market return: rit - rft = µt-1 + σ ෝtηt Market volatility: σ ෝt = exp(St + Lt) ηt ~ N(0, 1) (14) Short-run volatility: St = Ф1St-1 + Ф2Max(0, -εt) + Ф3(|ηt |-√(2/π) Long-run volatility: Lt = Ф4 + Ф5Lt-1 + Ф6Max(0, -εt) + Ф7(|ηt |-√(2/π) where, (|ηt |-√(2/π) are the shocks to the volatility components, with expected value zero, Ф4 is the mean of the long-run component, and Ф2 and Ф6 are the sensitivities to the leverage effect. The main difference between the short and long run component is that each component has a different rate of mean reversion, thus capturing shocks to systematic risk at different horizons. The asset pricing model they construct uses both the variance’s short-run and longrun component: To measure the innovations in aggregate volatility, (σm,t – γv,t), Ang et al. use changes in the VIX index, ∆VIX, from the Chicago Board Options Exchange as a proxy. The VIX index is a derivative of traded options, whose prices directly reflect volatility risk. Therefore, ∆VIX makes a good proxy for innovations in volatility risk. However, use of options also implies implied volatility from the Black-Scholes (1973) model which is a derived volatility. As a robustness test, Ang et al. have also used sample volatility like French, Schwert and Stambaugh (1987), as well as other proxies for innovations in volatility risk but found little difference in their results. 12 These results are adjusted for several known cross-sectional effects, such as size, value, momentum and liquidity. 11 13 E( rit - rft |(E(rm) – rf), St, Lt) = b1(E(rm) – rf) + b2St + b3Lt (15) where (E(rm) – rf) is the market risk premium, St is the short term volatility and Lt is the long term volatility, and b1 to b3 are the factor loadings. St and Lt are both derived using (14). To motivate the use of St and Lt in their model, they relate them to tightness of financial constraints and to the business cycle respectively, both for which they present empirical evidence. Adrian and Rosenberg find evidence for a positive relation between risk and return. More remarkably, they construct an asset pricing model which performs better than the FF model. Their model reports lower pricing errors than the Fama and French model suggesting a better specified model. A second model, in which the St and Lt factors are added to the FF model, outperforms the first model of Adrian and Rosenberg. This does not contradict the conjecture that the completion of the FF model with variance factors can lead to a better specified model to explain stock returns. III Empirical Research 3.1 Context Section 2.1 shows the basis of asset pricing models that try to identify the factors which drive common stock returns. This subject still is within the limelight of financial literature as thorough understanding of this relation remains unattained. Section 2.3 shows some of the work that approaches the risk-return relationship empirically. French et al. find a positive relation between variance and returns using a GARCH model. Glosten finds this relation to be negative, using a GARCH-M model. Meanwhile Baillie finds no convincing evidence suggesting that this relation should be examined using other more precise measures of risk. More recently, Ang et al. find a negative relation using another type of risk measure, changes in the VIX, and Adrian and Rosenberg find a positive relation using a complicated short- and long run variance model. The latter find that a model with the Fama and French factors supplemented with the short- and long-run variance as pricing factors outperforms all other model specifications. The discussion continues. And though one might think of the prevailing thought to relate to the positive relation between risk and return, empirics finding a negative relation are abundant13. 13 For empirical work finding a negative relation between risk and return Glosten et al. refer to Fama and Schwert (1977), Campbell (1987), Pagan and Hong (1991), Breen, Glosten, and Jagannathan (1989), Turner Startz, and Nelson (1989), and Nelson (1991). 14 To complement this strand of financial literature and improve the understanding of the general risk-return relation, this paper will examine the relation between market risk and market return using an approach different to those in the aforementioned studies. In this way, this paper tries to answer the question; how is market risk related to market return? The results obtained to answer this question can contribute to a better understanding of the general risk-return relation. To explore the generalizability of the contingent effects of market variance to smaller selections of stock returns, this paper also investigates the effects of market variance on a small portfolio. 3.2 Methodology Adrian and Rosenberg have found that asset pricing models in which variance factors are priced factors exhibit good performance in explaining realized stock returns. To investigate the relation between market risk and market return, an asset pricing model is constructed in which market variance is a complementary pricing factor to both the CAPM and the Fama and French Model. Engle and Rosenberg have evaluated the different variance models using volatility term structure (VTS) hedging tests. These test asses the performance of different variance models in predicting variance (or its square root) over the complete VTS. Furthermore, Engle and Rosenberg provide several arguments to emphasize why this method is preferred over the comparison of forecasted VTS to realized variance VTS.14 They rank the GCOMP model with leverage (9) higher than other models, which is the reason that a particular version of this model is chosen to measure variance in this research. Particularly, both the conditional variance (σ2t) as well as the permanent component (qt) will be used in the regressions on returns, because Adrian and Rosenberg also distinguish between trends in variance at different horizons. Hence, the two asset pricing models used in the regressions are: CAPM+GCOMP: E( ri,t - rf,t |(E(rm) – rf), σ2, q) = bi1(E(rm) – rf) + bi2σ2t + bi3qt FF+GCOMP: E( ri,t - rf,t |(E(rm) – rf), HML, SMB, σ2, q) = bi1(E(rm) – rf) + bi2HMLt + bi3SMBt + bi4σ2t + bi5qt Here, E(rm) – rf is the market excess return, HML and SMB are the Fama and French factors, σ2 is the conditional variance of the market, qt is the permanent component of the conditional 14 First, hedging test may be able to better distinguish between models with different term structure shapes but similar levels of unconditional variance. Second, “hedging tests may be superior at identifying omitted variables or interrelationships in the volatility model, because hedging performance depends on eliminating sensitivity to all sources of volatility” – Engle and Rosenberg (2000). 15 variance of the market, and bi…bj are the factor loadings on the dependent variables. Hypotheses with respect to both models are: H 0: b1 = b2 = … = bj = 0 H 1: bj ≠0 for at least one j Though theory to justify the role of the factors in both my models is existent, the choice to include them into these models is mainly driven by empirics dealt with in section II. This pragmatic approach brings the use of these models into the realm of the APT. Hence, the framework and the assumptions of the APT are applicable to this research. The methodology uses the following sequence. First, the GCOMP model will be estimated using Maximum Likelihood Estimation (MLE), so that the conditional variance series and permanent component series for the market can be calculated. Second, using Ordinary Least Square regression (OLS), both models of interest will be estimated and the significance of their factor loadings will be determined. As depicted in the specification of both models, the variance of stock returns, not the standard deviation, will be used in the regressions. French et al. have shown that using variance or standard deviation in regressions does not influence the results and that conclusions on effects of variance as dependant variable in regressions can be generalized to effects of volatility – and vice versa. These regressions can be performed on lots of types of stocks, portfolios, and the market. As said, scope limitations demarcate the area of interest on the market return, for which the S&P 500 Composite is a proxy. In addition, a portfolio of six randomly selected stocks will be used to explore if the results obtained using market returns are somewhat generalizable to a smaller selection of stock returns. 3.3 Data The S&P 500 Composite is the proxy for the market. Hence, it will be used for calculating the market variance and it will be used as a dependent variable in the regressions. The stocks used to form a portfolio are randomly selected. For the return data, DataStream ‘Total Return’ indexes are used for both the S&P 500 Composite and the US stock returns as to incorporate the effects of dividends. To reduce estimation errors resulting from a wrongly specified volatility model, Nelson (1992) suggest using higher frequency data. Therefore this study uses daily values. Higher frequency data, such as intraday data, is not available over the sample 16 period. The Fama and French Factors are obtained from Kenneth French’s online database15. To save time, this study uses the market return and risk free rate from French’s database for the regressions as these figures are ready available16. Table 1 shows yearly expected returns of the selection of US stocks vary between 6% and 17 % and standard deviations vary between 25% and 37%. The stock returns are negatively skewed and have a high Kurtosis17. One implication is that extreme negative values are more likely than in a normal distribution, but this is a normal empirical finding in stock return data. The size factor (SMB) shows an average return of -0,04%. If this is the true value of the SMB factor then this would go against Fama and French’s arguments for its relevance. However, the 95% confidence interval ranges from approximately -17% to +17%, which does not reject Fama and French’s observation that small stocks outperform large stocks. Meanwhile, empirics also suggest a changing strength of the SMB factor through the years, underpinning the fact that the SMB factor is not found significantly positive in this sample18. The skewness of the portfolio which consists of six randomly selected stocks, each with an equal weight, is positive. A large part of literature argues the shape of stock return distribution to be negatively skewed, these studies are mainly focused on the aggregate of stock market returns. 19 Singleton and Wingender (1986) find empirical evidence for portfolios of stock returns to be positively skewed for certain periods in time, but not persistently. Without persistence of skewness and in efficient markets this skewness should not explain much about the pricing of stocks. This does not explain the positives skewness of 15 http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html Though it is not likely that the market return of French’s database, which is based on the total CRSP database, is exactly equal to that of the S&P, I assume that the difference is of such a magnitude that it will not influence the results. 17 Kurtosis is high relative to that of a normal distribution which is 3. Also, another measure of Tabachnik and Fidell (1996), who argue that data with a Kurtosis of more than two times the standard error of Kurtosis can be considered not normally distributed, considers these Kurtosis values high. 18 As Fama and French (1993) argue, their observation that size is related to profitability stems mostly from the 1980s part of their data. Fama and French (1993): 16 “Until 1981, controlling for BE/ME, small firms are only slightly less profitable than big firms. But for small firms, the 1980-1982 recession turns into a prolonged earnings depression. For some reason, small firms do not participate in the economic boom of the middle and late 1980s”. The difference in earnings between small and big firms creates a common risk factor, size, which is negatively related to returns. However, this strong negative relation also seems not as strong after the 1980s. Zhang (2008) observes that, during the 1992-2007 period, the Fama and French data shows that the SMB factor is much less strong than during the 1963-1991 period. For lower book-to-market portfolios, the size effect even changes its sign. As the majority of my sample period covers the period after 1990, Zhangs’ observations might underpin why the SMB factor is not found significantly positive over the 1981-2005 sample period. 19 I refer to Fama (1965), Black (1976), Christie (1982), Blanchard and Watson (1982), Pindyck (1984), French et al. (1987) and Hong and Stein (2003). 17 Table I Descriptive Statistics S&P 500 Portfolio SMB HML MKT -/RF RF 0,1307 0,1309 -0,0004 0,0605 0,0702 0,0554 Std. Dev. 0,1665 0,1883 0,0887 0,0823 0,1538 0,0019 Mean 0,0005 0,0003 0,0000 0,0002 0,0003 0,0002 Std. Dev. Median 0,0105 0,0006 0,0119 0,0004 0,0056 0,0002 0,0052 0,0001 0,0097 0,0005 0,0001 0,0002 Maximum Minimum 0,0831 -0,2121 0,1138 -0,2419 0,0643 -0,1124 0,0388 -0,0489 0,0863 -0,1716 0,0006 0,0000 Skewness Kurtosis -1,45 33,99 1,35 34,29 -1,43 34,46 0,04 9,44 -1,02 23,83 0,88 4,12 6311,00 6311,00 6311,00 6311,00 6311,00 6311,00 Coca Cola General Dynamics General Electric Goodyear Merck Walt Disney 0,1606 0,1353 0,1603 0,0631 0,1323 0,1340 Std. Dev. 0,2617 0,2822 0,2582 0,3617 0,2684 0,3218 Mean 0,0006 0,0005 0,0006 0,0003 0,0005 0,0005 Std. Dev. Median 0,0165 0,0000 0,0178 0,0000 0,0162 0,0000 0,0228 0,0000 0,0169 0,0000 0,0203 0,0000 Maximum Minimum 0,1796 -0,2836 0,1823 -0,1542 0,1174 -0,1922 0,1654 -0,3365 0,1225 -0,3117 0,1748 -0,3439 Skewness Kurtosis -0,61 22,21 0,10 10,38 -0,17 9,69 -0,47 15,58 -1,13 24,85 -1,10 25,05 6311,00 6311,00 6311,00 6311,00 6311,00 6311,00 Yearly Mean Daily Observations Yearly Mean Daily Observations This table shows the descriptive statistics for the sample used in the empirical part. I use daily return data over the sample period 02-01-1981 to 30-12-2005 which gives 6311 observations. The Fama and French factors, small minus big (SMB), high minus low (HML), and market return minus risk free rate (mkt-/-rf), are obtained from Kenneth French's online database. The remaining data is obtained from the DataStream database. I have used 'Total Return' indexes for both the S&P 500 Composite as the individual stocks, to incorporate the effects of dividends. The portfolio consists of the six individual stocks, in which all are equally weighted. General Dynamics (GD). GD is the only stock for which the maximum return is higher than its minimum return. Though it would be easy to impute the positive skewness to this positive outlier, the histogram (see Figure 2 in the appendix) shows a combination of more positive outliers as well as more positive observations in the proximity of the mean are accountable for this positive skewness20. 20 The positive outliers are market responses to the sales of GD units or the announcements of restructuring. This was the part of the strategy of downsizing, restructuring, and liquidation put in place by the new management which was installed early in 1991. The highest outlier of 18 percent is the market response to the sale of the Fort Worth division to Lockheed in March 1991. Subsequently, this management and its strategy 18 Table II GARCH Component Model Estimates return equasion Variable µ Coefficient 0,0006 Std. Error 0,0001 z-Statistic 6,3216 Prob. 0,0000 variance equasion ω 0,0001 0,0000 11,1112 0,0000 ρ φ 0,9943 0,0281 0,0011 0,0035 879,3999 7,9501 0,0000 0,0000 θ α -0,0223 0,1204 0,0073 0,0072 -3,0486 16,7727 0,0023 0,0000 β 0,8484 0,0157 54,1002 0,0000 R-squared -0,0001 Adjusted R-squared S.E. of regression -0,0001 0,0105 Sum squared resid Log likelihood 0,6934 20713,55 Durbin-Watson stat 1950657 To estimate the variance of the market which will be used in the regressions, the GARCH Component Model with leverage is used as shown in (16): rit - rft = µ t + εt εt ~ N(0, σ2) σ2 = qt + θ + α[ Max(0,-εt-1)2 – qt-1]+ β(σ2t-1 – q2t-1) qt = ω + ρ(q2t-1 – ω) + Ф(ε2t-1 - σ2t-1) Using Maximum Likelihood Estimation, I estimate the GCOMP model in Eviews on daily return data of the S&P 500 between 02-01-1981 and 30-12-2005. The estimates of the coefficients are shown in Table II. The alpha and beta sum up to 0.97 and are thus close to one like is often seen in a GARCH type variance model of stock return data. All variables are statistically significant at the 1 percent level. 3.4 GARCH Component Model The daily variance series for the S&P 500 which will be used in the regressions are estimated using Maximum Likelihood Estimation in Eviews. The variance of the S&P is estimated in using the following version of the GARCH Component model of Engle and Lee (1993): rit - rft = µt + εt ε t ~ N(0, σ2) (16) σ2 = qt + θ + α[ Max(0,-εt-1)2 –qt-1 ]+ β(σ2t-1 – q2t-1) qτ = ω + ρ(q2t-1 – ω) + Ф(ε2t-1 - σ2t-1) The coefficients are estimated on daily log normal returns on DataStreams’ Total Return index of the S&P 500 for the period 1981-2005. Estimates of the coefficients of the GCOMP realized a dividend-reinvested return of 533% by 1993 – though the influence of US participation in the Gulf War on GD’s stock returns should not be undervalued. 19 Table III Comparison of S&P 500 sample variance estimated by different models Var (daily) GCOMP GARCH(1,1) GARCH(1,1) with leverage Descriptives 0,0001 0,0001 0,0001 0,0001 Var (yearly) 0,0270 0,0280 0,0286 0,0277 Std.Dev. (yearly) 16,44% 16,74% 16,90% 16,65% Table III compares the sample variance of the S&P 500 estimated by different models. The first model is the model of interest and is the GCOMP model as shown in (16), which will be used in the regressions. The second model is a GARCH(1,1) model as shown in (6): rit - rft = µ t + εt εt ~ N(0, σ2t) σ2t = α + βσ2t-1 + Φε2t-1 The third model is a GARCH(1,1) model adjusted to cope with the asymmetric effects of leverage as shown in (8): rit - rft = µt - 0.5σ2t + εt εt ~ N(0, σ2t) 2 2 2 σ t = α + βσ t-1 + Φε t-1 + γMax(0,-εt-1)2 The descriptives are derived from the sample descriptives as generated by Eviews. All measures of the sample variance are reasonably in line with each other showing yearly standard deviations of approximately sixteen percent. Table IV Descriptive Statistics of the S&P 500 daily variance estimated by the GCOMP Model σ2 q Mean 0,0270 0,0253 Std. Dev. 0,29% 0,14% Mean Median 0,0001 0,0001 0,0001 0,0001 Maximum Minimum 0,0062 0,0000 0,0015 0,0000 Std. Dev. Skewness 0,0002 18,55 0,0001 5,80 Kurtosis Sum 469,41 0,6743 63,78 0,6313 Sum Sq. Dev. Observations 0,0002 6311 0,0000 6311 Horizon Descriptive yearly daily Table 4 depicts the descriptive statistics of the daily variance series of the S&P 500 from 1981 to 2005 as measured with the GCOMP model. These series are created using the model estimates from Table 2. The σ2 column depicts the descriptives for the conditional variance series and the q column depicts the descriptives of the series of the permanent component of the GCOMP model. The yearly mean of the conditional variance of 0,0270 is analogous to the yearly var of the S&P 500 of Table 3. 20 model are shown in Table 2. All coefficients are significant at a one percent level. The alpha and beta sum up to 0.97 and are thus close to one like we would expect from a GARCH type variance model. Table 3 shows the mean variance and standard deviation of the S&P 500 estimated with the GCOMP model. The values are in line with the GARCH(1,1) variance and the GARCH(1,1) with leverage variance which I also estimated in Eviews using MLE. The values also compare to the descriptives of table 1. Table 4 shows the descriptive statistics of the daily variance series estimated with the GCOMP. Obviously, a large part of the conditional variance is explained by the long run component. Inherently to the way the GCOMP is constructed, q’s standard deviation is lower than that of the conditional variance. Though the GCOMP model contains a long run component, there is no assignable short run component. In this respect the model is sometimes referred to as a GARCH(2,2) model. Extreme values are controlled for by the conditional variance, hence its higher skewness and Kurtosis. 3.5 Regression Results Using the GCOMP variance time series, CAPM+GCOMP and FF+GCOMP are estimated on the returns of the S&P 500 and the returns of the portfolio of randomly selected stocks using OLS regression technique. Table 5 shows the results with t-statistics indented. All coefficients of the benchmark models CAPM and Fama and French are found to be statistically significant. Though the Fama and French factors increase the R-square of the model for both the S&P and the portfolio, the increase is not as large as reported in Fama and French (1996). For CAPM+GCOMP the coefficients on the GCOMP factors are significant for both the S&P 500 and the portfolio. The coefficient on the conditional market variance is positive, suggesting a positive relation between risk and return. The permanent component of the GCOMP model, q, shows a significant negative coefficient. At first sight, this might seem contradicting to a positive relation between risk and return. However, the variable q is also a constant in the variance model (see (9)). In this respect, the variable q is thus priced twice in CAPM+GCOMP; once as part of the conditional variance, and once individually. Another implication of the fact that q is part of the conditional variance is a high correlation between both variables of .85 percent (see table 6). Therefore, the negative coefficient on q is likely to offset part of the correlation. The 95% confidence ellipses of the coefficients of CAPM+GCOMP for both the S&P 500 as the portfolio confirm this, see Figure 1. It shows that the coefficient on q (C3 in Figure1) is likely to increase with a decrease in the market 21 Table V OLS Results Rm - Rf HML SMB σ2 q S&P 500 CAPM+ CAPM+σ2 GCOMP CAPM σ2 1,07* … 1,07* 1,06* 1,01* 1,01* 1,01* … (455,05) … (439,21) … (451,67) -0,02* (449,27) -0,02* (451,64) -0,02* … (-4,1) -0,27* (-3,99) -0,27* (-4,1) -0,27* (-89,22) … (-86,76) 0,15 (-88,51) 0,01 … (1,02) -0,25 (0,14) … (454,3) … … … … … 2,71* 2,92* 0,92* … (4,38) … (13,77) -3,78* (8,33) … FF (-11,01) FF+GCOMP FF+σ2 (-1,11) R-squared 0,9704 0,0022 0,9690 0,9684 0,9867 0,9867 0,9867 Sum squared residuals 0,0198 0,6919 0,0215 0,0219 0,0092 0,0092 0,0092 FF FF+GCOMP FF+σ2 Rm - Rf HML SMB σ2 q Portfolio CAPM+ CAPM+σ2 GCOMP CAPM σ2 0,98* … 0,98* 0,98* 0,98* 0,98* 0,98* … (106,99) … (106,85) … (79,79) 0,20* (79,39) 0,20* (79,81) 0,20* … (9,00) -0,37* (9,00) -0,37* (9,01) -0,37* (-21,65) -0,51 (-22,08) -0,67 (-0,66) -0,30 (1,7) … (106,97) … … … … … 2,46* 3,36* 0,81* (-22,04) … … (3,52) … (4,14) -4,81* (1,94) … … (-3,67) R-squared Sum squared residuals (-0,24) 0,6443 0,0013 0,6453 0,6446 0,6849 0,6850 0,6850 0,3151 0,8849 0,3143 0,0071 0,2792 0,2791 0,2791 Table 5 shows the results of various Ordinary Least Squares regressions on the daily excess returns of the S&P 500 (upper half) and the daily excess returns of the small portfolio (lower half) between 1981 and 2005. The columns show the composition of the model used in the regressions and the coefficients estimated for the independent variables with t-statistics in parentheses. Rm-Rf is the market return minus the risk free rate, HML and SMB are the high-minus-low and small-minus-big factors of Fama and French (1993), σ2 is the conditional market variance measured by the GCOMP model as depicted in (16) of which the model estimates are shown in Table 2, and q is the permanent component of the GCOMP model also explained in (16). Coefficients tagged with a (*) are significant at the one percent level. 22 Figure 1 95% Confidence elipses for coefficients of the CAPM+GCOMP model S&P 500 Portfolio 3.4 5 3.2 C(2) C(2) 4 3.0 2.8 2.6 3 2 -2 -3.2 -3.6 C(3) C(3) -4 -4.0 -6 -4.4 0 2 4 3. 3. 8 C(1) 3. 6 1.070 2. 1.065 2. -8 1.060 0.97 0.98 0.99 1.00 C(1) C(2) 2 3 4 5 C(2) Figure 1 shows the 95% confidence ellipses for the coefficients of the CAPM+GCOMP model, which are estimated using OLS on the daily excess returns of the S&P 500 and the portfolio. The regressions cover the sample period 1981 to 2005. C(1) is the coefficient on mkt-/-rf, C(2) is the coefficient on the conditional market variance measured by the GCOMP model, and c(3) is the coefficient on q which is the long term component of the market variance as measured by GCOMP model. The ellipse in the lowerright quadrant shows the interaction between the conditional market variance and the long term component of the market variance q. It shows that the coefficient on q is likely to be less negative when the conditional market variance is less positive. Table VI Correlation Matrix of Independent Variables Rm - Rf hml smb σ2 q Rm - Rf 1 -0,6226 -0,3007 0,0262 0,0235 hml smb σ2 -0,6226 -0,3007 1 -0,0804 -0,0804 1 -0,0019 -0,1619 -0,0154 -0,0638 0,0262 0,0235 -0,0019 -0,0154 -0,1619 -0,0638 1 0,8477 0,8477 1 q Table VI shows the correlation between the various independent variables used to construct the asset pricing models. The correlation is determined based on daily data between 1981 and 2005. Rm - Rf, hml, and smb are the Fama and French factors for this period of which the daily data is obtained from Kenneth French's database. The GCMP variance factors, σ2 and q, are the conditional variance of the S&P 500 and the permanent component of that variance, respectively. These daily series are calculated using (16) and its estimates in Table 2. 23 variance. Estimations of CAPM+GCOMP without the permanent component q also do not reject this theory. These regressions show the values of the coefficients on market variance remain significantly positive but become much smaller, 0.92 and 0.81 for the S&P 500 and the portfolio respectively, with almost equal R-square values for the model. Hence, a positive relation between risk and return is not rejected by the negative coefficient on q. The R-square value of CAPM+GCOMP is almost equal to that of the CAPM for both the S&P 500 and the portfolio, suggesting that adding variance to the CAPM as a regressor does not drastically increase the performance of the asset pricing model. Statistical significance in finance is of course of less relevance without economic significance. To isolate any effects of multicollinearity between the variance and q, we consider the coefficient on variance of the model CAPM+σ2 on the S&P 500, which is 0,91. Table 4 shows the standard deviation of the variance is 0,000185. One standard deviation increase in the variance results in 1,6 basis points increase in the return of the S&P 500 on a daily basis21. That is equivalent to a yearly increase in risk premium of 420 basis points. This is a considerable economic value. Assuming transaction costs of 25 basis points, this leaves 395 basis points per year as a trading opportunity, emphasizing the role of market variance in returns. In contrast to CAPM+GCOMP, the coefficients on the GCOMP variables are not found significant in FF+GCOMP. When observed singly, this result could be an argument for the insignificance of variance in explaining asset returns. But when observed in conjunction with the result on CAPM+GCOMP, one interpretation could be that most of the variation explained by the GCOMP factors in CAPM+GCOMP is accounted for by the Fama and French factors in FF+GCOMP, thus making the GCOMP factors insignificant. The strongly declined coefficient on variance and q in FF+GCOMP, in comparison to CAPM+GCOMP, does not falsify this conjecture. Furthermore, the coefficients on HML and SMB show virtually no decrease in FF+GCOMP, when compared with the Fama and French Model. This suggests that their explanatory power is not affected by the addition of variance to the model. However, Table 6 shows the GCOMP factors and the Fama and French factors are almost uncorrelated. Besides, the very small correlation coefficients show negative values which are not supportive to this conjecture. One other explanation for the loss of significance of the GCOMP factors in FF+GCOMP could be the lack of economic significance. However, CAPM+GCOMP shows that that the variance does play an significant role economically. 21 In the sample data, a one percent return is denoted as 0,01. Hence, the impact of one standard deviation increase in variance is 0,91*0,000185 = 0,000168 on the daily return, which, multiplied with 252 trading days, is 0,042 percent or 420 basis points annually. 24 With respect to the hypotheses, I reject H0 for both models. Consistent with a large part of literature, these regression results suggest a positive relation between risk and return. IV Analysis 4.1 CAPM+GCOMP The results for CAPM+GCOMP are consistent with French et al. and Adrian and Rosenberg, as variance is found positively related to returns. In addition to the 95% confidence ellipse of the coefficients of CAPM+GCOMP, a Wald test also rejects the hypothesis that the coefficient on variance is zero both in CAPM+GCOMP as in the CAPM+σ2 model at the one percent level. The coefficient on variance is economically significant, but adding market variance to the CAPM does not increase the model’s performance drastically. The effect of adding market variance to the CAPM on the R-square of the model is small, which is in line with the regression on variance only. In contrary, the R-square of the traditional CAPM factor confirms its reputation for being a model which can explain a large part of variation in stock returns. For CAPM+GCOMP, one standard deviation increase in the CAPM factor results in 103 basis points increase in the daily return for the S&P 500 and 116 basis points increase in the daily return for the portfolio. Correspondingly, the R-square of the CAPM model is already at 0.97 and 0.64 for the S&P 500 and the portfolio respectively. To verify the robustness of these results, these regressions are tested using other, less complicated variance models. The models used in the tests for robustness are a standard GARCH(1,1) model (see (6) for the specification) and a GARCH(1,1) ‘with leverage’ model (see (8) for the specification). The means of the time series estimated with these models are reported in Table 3. Table 7 shows the regression results. For the regressions on the S&P 500, CAPM+σ2 shows significant positive coefficients on the variance for both variance specifications. Both coefficients are smaller than those reported for CAPM+GCOMP and the CAPM+σ2 model in Table 5. However, Engle and Rosenberg have already emphasized the weaker performance of these variance models in comparison to the GCOMP. The R-squares are also smaller, though with a minimal amount. The results using the portfolio do not pass the robustness test. The coefficients on variance are also smaller than those reported for CAPM+GCOMP and the CAPM+σ2 model in Table 5, and remain positive, however they are not significant. One possible explanation for this result could be the weaker variance models used. One other explanation could be the weaker explanatory power of market variance in explaining small portfolio returns compared to their explanatory power with respect to market 25 Table VII Testing for robustness using different variance specifications in the regressions CAPM+σ2 Rm - Rf HML SMB σ2garch(1,1) σ2garch(1,1) with leverage S&P 500 CAPM+σ2 FF+σ2 FF+σ2 1,07* 1,07* 1,01* 1,01* (437,63) … (438,62) … (451,66) -0,02* (451,66) -0,02* … … (-4,08) -0,73* (-4,10) -0,27* 0,49* … (-89,08) -0,10 (-88,72) … (4,28) … 0,65* (-1,36) … -0,06 (6,91) (-0,90) R-squared 0,9681 0,9683 0,9867 0,9867 Sum squared residuals 0,0229 0,0019 0,0092 0,0092 FF+σ2 FF+σ2 Portfolio CAPM+σ CAPM+σ2 2 Rm - Rf HML SMB 2 σ garch(1,1) σ2garch(1,1) with leverage R-squared Sum squared residuals 0,98* (106,88) 0,98* (106,88) 0,98* (79,82) 0,98* (79,80) … … 0,20* (9,03) 0,20* (9,00) … … -0,37* (-22,11) -0,37* (-22,09) 0,29 (0,67) … -0,72 (-1,76) … … 0,55 … -0,59 (1,53) (-1,73) 0,9681 0,9683 0,9867 0,9867 0,0229 0,0019 0,0092 0,0092 Table 7 shows results of various Ordinary Least Squares regressions on the daily excess returns of the S&P 500 (upper half) and the daily excess returns of a portfolio of six randomly selected stocks (lower half) over the period 1981 to 2005. These regressions serve as tests for robustness for the results obtained in Table 5. The columns show the composition of the model used in the regressions and the coefficients estimated for the independent variables with t-statistics in parentheses. Rm-Rf is the market return minus the risk free rate, HML and SMB are the highminus-low and small-minus-big factors of Fama and French (1993), σ2GARCH(1,1) is the conditional variance of the S&P 500 as measured by (6), and σ2GARCH(1,1) with leverage is the conditional variance of the S&P 500 as measured by (8). Coefficients tagged with a (*) are significant at the one percent level. returns. The results in Table 5 do not reject this conjecture, showing variance values for the portfolio are smaller than for the S&P 500 return. It is possible that a combination of these effects explain the results on the robustness test of the portfolio. The robustness tests reject 26 the hypothesis that the coefficient on market variance is significantly different from zero for the FF+σ2 model, which is in line with the results from Section 3. In addition, Granger causality tests show that the hypothesis that the conditional variance does not drive the returns can be rejected for both the S&P 500 and the portfolio. Breusch-Godfrey serial correlation LM tests on both CAPM+GCOMP and FF+GCOMP show no signs of serial correlation or autocorrelation, strengthening the reliability of the results. More tests for robustness is also considered. For example, the variance measured by both models shows extreme values during the crash of 1987. Another test for robustness could therefore exclude these extreme values out of the sample to examine the degree in which the results are driven by these extreme values. However, these tests are not performed due to limitations in time. Based on all tests on CAPM+GCOMP, variance appears to be positively related to returns. Specifically, a statistically significant positive relation between the market variance and market return is found as well as between market variance and portfolio return. The latter of these appears not as strong as the first and does not pass tests for robustness. 4.2 FF+GCOMP Section 3 shows that H0 can be rejected for FF+GCOMP due to the statistically significant Fama and French factors. The GCOMP factors do not seem to play a significant role in explaining returns - judging from their t-statistic and judging from the R-square of FF+GCOMP in comparison to that of the Fama and French model. Based on the results of CAPM+GCOMP, it is unlikely that the economic significance of the market variance is a determinant of that fact. The reason why the GCOMP factors are insignificant in FF+GCOMP could originate in the interaction between the GCOMP factors and the Fama and French factors. A theoretical approach on how this interaction manifests might emphasize Fama and French’s motivation for their relation with returns: “Stocks with low long-term past returns (losers) tend to have positive SMB and HML slopes (they are smaller and relatively distressed) and higher future average returns. Conversely, long-term winners tend to be strong stocks that have negative slopes on HML and low future returns.” - Fama and French (1993) – According to Fama and French the HML and SMB factors do a good job in pricing risk which mainly originates from higher probability of distress due to lower earnings. Combining the good performance of these factors with the possibility of a relative weak model specification of the GCOMP creates a basis for the insignificance of the GCOMP factors in FF+GCOMP. 27 Though Engle and Lee (1993) have found the GCOMP better performing in modeling variance than other GARCH types, more recent option literature favors ‘real’ two component models, like the one used by Adrian and Rosenberg (2008), as well as other methods of risk measuring22. One such other measure, implied volatility from options, is compared to the performance in forecasting variance of ARCH models by Blair, Poon, and Taylor (2001). They find that implied variance forecasts are a better predictor of realized variance than ARCH forecasts. There are more possible explanations for the results in section 3 that demand consideration. The varying results between CAPM+GCOMP and FF+GCOMP could be generated by insufficient accuracy of the estimations due to the frequency of the data. Nelson (1992) argues higher frequency of data results in less estimation errors. Andersen and Bollerslev (1998) prove that the use of intraday returns significantly improves the forecasting performance of ARCH models. French et al., Glosten et al., and Ang et al also use daily data, but their methods use a monthly return horizon for which the monthly variance is calculated as an aggregate of the daily variances. And the selected return horizon can contribute to the varying results between CAPM+GCOMP and FF+GCOMP. This research uses daily data and also asses the explanatory contribution of variance in the daily return. Bollerslev and Tachen (2009) observe that the explanatory power of their measure of variance is strongest at the quarterly horizon, and that the explanatory value of variance across other horizons varies strongly. Adrian and Rosenberg (2008) find “wide dispersion in sensitivity to the volatility components, which generates cross sectional variation in the risk premia attributed to these facts”. So the explanatory power of variance has shown to vary in the cross section of returns, suggesting the sample period characteristics might play a role. The fact that the SMB’s mean of the sample used for this research deviates from other empirical work can be recalled in this context. Hence, issues concerning the, return horizon, accuracy stemming from data frequency, as well as the characteristics of the sample period could influence the results. Moreover it is likely that the difference in results between CAPM+GCOMP and FF+GCOMP originates from the SMB and HML factors capturing risk better than the variance specified by an model for which better alternatives have been found in the mean time23. In summary, based 22 For option literature favoring two-component variance models, Adrian and Rosenberg refer to Xu and Taylor (2004), Bates (2000), and Christofferssen, Jacobs, and Wang (2006). 23 A performance test of the GCOMP model as a complement to this analysis is not implanted due to limitations in resources and scope of this paper. 28 on the results obtained with FF+GCOMP, conclusions cannot be drawn with certainty regarding the relationship between market variance and stock returns. 4.3 Model adjustments In hindsight, the initial specifications of CAPM+GCOMP and FF+GCOMP could have been constructed different from their current structure. Like Adrian and Rosenberg (2008), this research tried adding two volatility components to the return regressions to better differentiate among trends in variance at different horizons. However, unlike Adrian and Rosenberg’s model which truly distinguishes between short- and long-run variance, the GCOMP model generates a long term trend which is added to the conditional variance. The correlation and multicollinearity effects that are likely to drive some of the results could therefore have been anticipated by omitting the variable q from both models. This would leave only the conditional variance as an addition to the CAPM (CAPM+σ2) and the FF model (FF+ σ2). The risk-return relation could also have been examined in several other ways. French et al. use a prediction of market variance, predicted at t-1, as a regressor on stock returns at time t. The model used to predict the variance inherently exerts a lot of influence on the results. This could be one reason why French et al. do not find a direct positive relation between their predicted variance and returns. With his modified GARCH-M model, Baillie also uses a direct way to measure this relation and finds no significant relation. However, the indirect measure used by French et al., the difference between their prediction of variance and realized variance, does give some indications about the relation between risk and return. The difference between what variance derived from ex ante data and variance derived from ex post data also plays a role in Bollerslev and Tauchen. They use the difference between the “model-free” implied variance, derived from a large collection of option prices, and the “model-free” realized variance, computed by summating high-frequency intraday squared returns, which they call the variance risk premium. This variance risk premium is significantly positively related to the returns. In this respect, another way of examining the risk-return relationship would be to examine the difference between variance derived from ex ante and ex post data. The focus of this paper is on the influence of market variance at time t on the market return at time t, in the context of the CAPM and FF model. The regressions on the portfolio support the results obtained with regressions on the S&P 500, but show signs of a weaker relationship between market variance and portfolio returns than between market variance and market returns. Besides the other possible modifications described in this part, it would be 29 interesting to see if portfolio returns perhaps exhibit a stronger relationship to portfolio variance. It would also be interesting to examine how the individual stock returns are related to their idiosyncratic risk. V Conclusions To gain more insight into the general risk-return relation, this paper examines the relation between market risk and market return and market risk and portfolio return. In the first stage, the conditional variance of the market using the GCOMP with leverage model is estimated. In the second stage, two factors are added to well known asset pricing models. The first factor is the conditional market variance as modeled by the GCOMP. The second factor is the long term trend of the market variance, q, which is the ‘permanent’ variable within this GCOMP model. These factors are added to the CAPM and the FF model, thus creating model CAPM+GCOMP and model FF+GCOMP respectively. In the third stage, regressions using these models are performed on the market return and the return of a portfolio of six randomly selected US stocks. Consistent with a large part of the literature, the regressions using CAPM+GCOMP show that the market variance is positively related to market returns and the portfolio return. The explanatory power of the market variance is larger for market returns than for the portfolio return, but both regressions indicate that market variance plays an economic significant role in explaining returns. To be precise, one standard deviation increase in the variance of the market results in 1.6 basis points increase in the daily market return and 1.5 basis points increase in the daily portfolio return. In contrary to the results obtained with CAPM+GCOMP, the regressions using FF+GCOMP do not exhibit statistically significant coefficients for the GCOMP factors. Several motivations for these findings are examined. I theorize these results are possible when the FF factors do a better job in explaining the effects of risk on return than the conditional variance. However, Adrian and Rosenberg do find the coefficients on short- and long-term market variance to be significantly positive when added to the FF model. The results could therefore also be influenced by the use of a variance model for which literature, in the meanwhile, has suggested better performing alternatives. This paper concludes that variance appears to be positively related to returns. 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Zhang Chu, 2008, Decomposed Fama-French factors for the size and book-to-market effects, Department of Finance, Hong Kong University of Science and Technology. 33 Appendix Figure 2 Histogram of General Dynamic's daily stock return 1981-2005 This histogram shows the distribution of the General Dynamics daily stock return between 1981 and 2005. Over this period, the skewness of the stock returns for GD is positive which contrasts to what is normally observed in the empirics of individual stock returns. This figure shows that the positive skewness is not only due to more positive outliers, but also due to more positive observations in the proximity of the mean. The highest outlier is the market response to the sale of the Fort Worth division to Lockheed in March 1991. This was the part of the strategy of downsizing, restructuring, and liquidation put in place by the new management which was installed early in 1991. Subsequently, this management and its strategy realized a dividend-reinvested return of 533% by 1993 – though the influence of US participation in the Gulf War on GD’s stock returns should not be undervalued. 34

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