Interest rate risk and the cross-section of stock returns

Transcription

1 Interest rate risk and the cross-section of stock returns Paulo Maio 1 First draft: November 2009 This draft: December Durham Business School. Corresponding address: Durham Business School, Durham University, Mill Hill Lane, Durham DH1 3LB 06800, UK. paulofmaio@gmail.com. I have benefited from comments by Abraham Lioui, Axel Kind, Yvan Lengwiler, Skander Van den Heuvel, Dayong Huang, Jens Hilscher, and seminar participants at the University of Basel, Warwick Business School, 2010 FMA Meeting, and 2010 CRSP Forum.

2 Abstract In this paper, I derive a consumption-interest capital asset pricing model (CI-CAPM) based on intertemporal Epstein Zin preferences and on an intratemporal Cobb Douglas utility over real consumption and real money balances. When the model is estimated with first-stage generalized method of moments (GMM), the results show that the CI-CAPM explains well the dispersion in the excess returns of 25 size/book-to-market (BM) portfolios, and also 25 portfolios sorted on size and long-term reversal in returns, in addition to the real risk free rate and the excess market return. The change in the log interest ratio (growth in the opportunity cost of money) is a priced factor, and seems to drive most of the explanatory power over the cross-section of quarterly excess returns. The implied estimates for the preference parameters are economically plausible and statistically significant. Under this model, both value stocks and past long-term losers enjoy higher average (excess) returns because they have greater interest rate/monetary risk than growth/past winner stocks. The model significantly outperforms the nested models, i.e. the standard consumption CAPM (C-CAPM), CAPM, and the Epstein and Zin (1991) model. Moreover, the CI-CAPM compares favorably with recent alternative macroeconomic models, that also represent extensions of the baseline C-CAPM. Keywords: Asset pricing models; Consumption CAPM; Interest rates; Opportunity cost of money; Equity premium; Linear multifactor models; Cross-section of stock returns; Size and value anomalies; Long-term reversal in returns JEL classification: G12; E41; E44.

3 1 Introduction The risk-return tradeoff is one of the central topics in finance, and has warranted special attention by both academics in the field and finance practioneers. In rational asset pricing models, the dispersion in expected returns across different assets should be linked to a corresponding dispersion in the sensitivities (factor loadings) to a set of common risk factors. In principle, the C-CAPM (Lucas (1978), Breeden (1979)) provides a coherent theoretical background to explain the cross-section of asset returns. According to the C-CAPM, assets that covary positively with consumption growth (and hence, covary negatively with the marginal utility of consumption) are less attractive since they do not offer a hedge for bad states (i.e., periods of low consumption or high marginal utility of consumption), and therefore investors are willing to hold those assets only if they earn a positive risk premium. Nevertheless, the empirical fit of the standard C-CAPM (based on power utility) over the cross-section of stock returns and also on pricing the (excess) market return, has been rather poor (Breeden, Gibbons and Litzenberger (1989), Mankiw and Shapiro (1986), Mehra and Prescott (1985), Lettau and Ludvigson (2001), Yogo (2006), Cochrane (2007) to name a few). I propose a generalization of the standard C-CAPM model to a monetary economy in which the representative agent features temporal recursive preferences as in Epstein and Zin (1989, 1991) and Weil (1989). This type of preferences allows one to disentangle relative risk aversion from the elasticity of intertemporal substitution, and is by now one of the most extensively used settings in the asset pricing literature. 1 Money is introduced in the economy by employing a money-in-the-utility function (MIUF) framework, in which the representative agent derives intratemporal utility over consumption and real money balances. The MIUF approach is widely used in the monetary economics literature, and it brings a simple and natural transmission mechanism of monetary shocks to the real economy and thus to asset prices, by affecting the marginal utility of consumption (pricing 1 See for example the extensive long-run risks literature Bansal and Yaron (2004), Bansal, Gallant and Tauchen (2007), Hansen, Heaton and Li (2008), Bansal, Dittmar and Kiku (2009), and Malloy, Moskowitz and Vissing-Jørgensen (2009), among others. 1

4 kernel of a representative investor). 23 The purpose in this paper is to build an equilibrium consumption based asset pricing model that allows for the role of the monetary side of the economy, and conduct an empirical test over the cross-section of stock returns. A key innovation of this model is that one makes full use of the optimal conditions to obtain a new stochastic discount factor for the monetary economy. Namely, through the equilibrium first order conditions for consumption and real balances holdings, I obtain a relationship between consumption, real balances and the nominal interest rate, which represents the opportunity cost of holding real balances. As a consequence, I derive a stochastic discount factor (SDF) driven by three factors: consumption growth, real market return (due to the recursiveness of the utility function), and the growth in the nominal interest rate. The latter is new and brought about by the economy being explicitly modeled as a monetary one. This representation is all the more valuable because one does not need to know precisely what real balances are. Alternative equilibrium models of the transmission mechanism suggest the presence of a latent factor in the stock market related to the stance of monetary policy. For example, Marshall (1992), Bakshi and Chen (1996), Chan, Foresi and Lang (1996) and Lioui and Poncet (2004), among others, show that the nominal feature of modern economies brings about an alternative or additional risk factor relative to the traditional consumption growth factor. This factor is related to the growth rate of the real balances (real value of nominal money supply) in the economy. Tests of such models use some proxy for this latent factor, namely one of the monetary aggregates, be it M1, M2 or sometimes even M3. Yet it is generally recognized that real balances are hard to identify, in particular after decades of financial innovations. Chan, Foresi and Lang (1996) and Balvers and Huang (2009) investigate whether real money balance growth is priced in the cross-section of 2 See Walsh (2003) for a detailed analysis of the monetary economics literature. An incomplete list of papers that study the asset pricing implications of money by using this approach includes Danthine and Donaldson (1986), Boyle and Young (1988), Finn, Hoffman and Schlagenhauf (1990), Bufman and Leiderman (1993), Bakshi and Chen (1996), Holman (1998) and Buraschi and Jiltsov (2007). 3 Another class of monetary models are transaction-costs models. Feenstra (1986) shows that MIUF models are equivalent to transaction-costs models. Studies that analyze the asset pricing implications of transaction-costs models include Marshall (1992), Bansal and Coleman (1996), Chan, Foresi and Lang (1996) and Balvers and Huang (2009), among others. 2

5 asset returns. Chan, Foresi and Lang find some support for this factor although the estimates of the preference parameters are similar to those delivered by the traditional C-CAPM. Moreover, the results are extremely sensitive to the proxy of real balances used. Balvers and Huang have no structural preferences in their model and therefore it is impossible to assess whether the improved explanatory power of a model including real balance growth rate is at the cost of economically implausible values of the preferences parameters. 4 By linearizing the SDF, I obtain a three-factor macroeconomic model, in which there are three (factor) covariances that help to explain the cross-sectional dispersion in stock returns: the covariance with the log consumption-growth; the covariance with the log growth in the opportunity cost of money; and the covariance with the log market return. This model is denoted as the consumption-interest CAPM (CI-CAPM). When the representative agent s parameter that governs relative risk aversion is greater than one (the relative risk aversion of a logarithmic myopic investor), the risk price associated with the log growth in the opportunity cost of money is negative, and this implies that assets that covary positively with the log growth in the opportunity cost of money will earn a lower risk premium, in comparison to assets that covary negatively with this factor. Periods of high interest rates are usually periods of tight market conditions where inflation expectations are high and liquidities are in limited supply. As a consequence, an asset with high returns precisely in such an economic situation (bad times, or periods of high marginal utility of consumption) is worthwhile for the investors who are willing to pay a premium for holding it, thus requiring a lower expected (excess) return. The main empirical results can be summarized as follows. When the model is estimated with first-stage generalized method of moments (GMM), the results show that the CI-CAPM explains well the dispersion in the excess returns of the 25 size/book-to-market (BM) portfolios (SBM25) and 25 size/long term reversal portfolios (SLTR25), in addition to the real risk free rate and the excess market return. Moreover, the change in the log 4 Other researchers focus on the time-series relation between stock returns and monetary variables (Hess and Lee (1999), Flannery and Protopapadakis (2002), Bernanke and Kuttner (2005), and Baele, Bekaert and Inghelbrecht (2010), among others). 3

6 interest ratio is a priced factor, and seems to drive most of the explanatory power over the cross-section of quarterly excess returns, while both consumption and market risks seem to play a secondary role. Note that, as said above, this performance is obtained when constraining the model to help explain the equity premium and the real interest rate in addition to the cross-section of stock returns; thus the empirical test is more demanding than most other comparable studies. Moreover, one obtains similar performance results for a different set of assets (10 size plus 10 BM portfolios (S10+BM10)). The model also performs well in the estimation with second-stage GMM, indicating that a linear combination of the factor-mimicking portfolios is close to being meanvariance efficient. That is, the model is able to explain the Sharpe ratios of the testing assets for both sets of equity portfolios, in addition to the market Sharpe ratio and the real interest rate. In both the first- and second-stage GMM, I obtain implied estimates for the preference parameters risk aversion, elasticity of intertemporal substitution, and share of real balances in the intra-period utility function that are economically plausible, and statistically significant. The implied estimate of the parameter of risk aversion could be as low as 24 compared to 170 for the standard C-CAPM. As to the real balances share in the utility function, most of the estimates vary between 14% and 30%, which shows that money has an important share in total utility. This model is thus able to provide an alternative explanation to the value premium. Value stocks tend to load negatively on the monetary factor, the growth rate of the opportunity cost of money holdings. As a consequence, the additional premium related to the monetary factor is positive, which justifies the higher expected return of value stocks relative to growth stocks. Why are value stocks more sensitive to monetary risk, i.e., unexpected rises in future nominal interest rates? One possible explanation is that many of these value firms are cash cows with stable earnings streams but few growth opportunities, making them act more like long-term bonds, whose current prices are more (negatively) sensitive to rises in future interest rates. On the other hand, the CI-CAPM provides an explanation for the long-term reversal in 4

7 returns anomaly from DeBondt and Thaler (1985). Stocks which have underperformed for a long period in the past (past long-term losers) enjoy higher average excess returns than stocks which outperformed (past long-term winners), because they have higher interest rate/monetary risk. The reason is that stocks with a consistent underperformance for several years, are likely to have a long sequence of negative shocks in their cash flows, and hence have became more financial constrained trough time. Consequently, these firms will be more sensitive to additional negative shocks, namely positive surprises in short-term interest rates. The CI-CAPM model outperforms significantly the nested models standard C- CAPM, CAPM, and the Epstein and Zin (1991) model. Moreover, the CI-CAPM compares favorably with recent alternative macroeconomic models, that also represent extensions of the baseline C-CAPM (Lettau and Ludvigson (2001), Lustig and Van Nieuwerburgh (2005), Yogo (2006), Parker and Julliard (2005), and the long-run risks model from Bansal and Yaron (2004)). Finally, the model has a similar explanatory power over the size/bm and size/long-term reversal portfolios relative to the more ad doc model from Fama and French (1993). Several papers in the empirical asset pricing literature also use models containing risk factors related to interest rates (Chen, Roll and Ross (1986), Campbell (1996), Ferson and Harvey (1999), Brennan, Wang and Xia (2004), Petkova (2006), Hahn and Lee (2006), among others). Nevertheless, a crucial difference in this paper is that I present a macroeconomic model in which the interest rate factor is a nonlinear function of future interest rates, capturing the growth in the opportunity cost of money. Moreover, the risk prices in the model are related to structural preference parameters, and thus, have an economic interpretation per se. Thus, the economic plausibility of those implied preference parameters provides an additional constraint to the model, as advocated by Lewellen, Nagel and Shanken (2010). This paper proceeds as follows. Section 2 presents the theoretical background, Section 3 describes the data and econometric methodology, while Section 4 presents the main empirical results. Section 5 conducts a comparison with alternative macro models, whereas 5

8 Section 6 presents some additional results. Finally, Section 7 concludes. 2 A consumption-capm model with interest rates (CI-CAPM) 2.1 SDF representation The role of interest rates for asset pricing is assessed by deriving the stochastic discount factor (SDF) of a monetary economy. I adopt hereafter the recursive utility setting that allows one to disentangle (relative) risk aversion from the elasticity of intertemporal substitution. This is a fairly standard setting in the asset pricing literature, but it has not been widely applied in the money-in-the-utility function (MIUF) approach from the monetary economics literature. 5 I go on to detail the assumptions for the representative consumer preferences, then the budget constraint and finally the equilibrium. Details of the derivations are in Appendix A. As it is standard in the monetary economics literature, I assume a money-in-utility function (MIUF), in which the representative consumer derives utility from consumption and real balances, that is, { U t = (1 δ) ( Ct 1 ε M ε t ) 1 γ θ + δ [ E t ( U 1 γ t+1 )] 1 } θ 1 γ θ, (1) where C stands for real consumption and M for real money balances. This utility from consumption and real balance holdings is not time separable and has the recursive form introduced by Epstein and Zin (1989, 1991) and Weil (1989). In (1), δ < 1 is the subjective discount rate; γ stands for the coefficient of relative risk aversion (RRA); θ is an auxiliary parameter defined as θ (1 γ)ψ ; ψ is the intertemporal ψ 1 elasticity of substitution; and ε is the real balances share in the utility function. When 5 Bufman and Leiderman (1993) use an intertemporal Epstein-Zin aggregator, and an intratemporal utility over consumption and money. However, their intratemporal utility function is different from the one used in this paper. Eraker (2008) use standard Epstein-Zin preferences over consumption, and define an exogenous process for inflation in order to price nominal asset returns. 6

9 the RRA parameter and the reciprocal of the intertemporal elasticity of substitution are equal (γ = 1 ), we have θ = 1, which corresponds to the standard time separable case ψ with power utility. The intratemporal utility over consumption and real balances has a Cobb Douglas form, as in Finn, Hoffman and Schlagenhauf (1990) and Holman (1998), implying that both quantities are non-separable. The representative consumer s wealth is composed of real balance holdings, risky asset holdings and the risk-free asset, and its dynamics over time is given by ( W t+1 = R w,t+1 W t C t R f,t+1 1 R w,t+1 = N 1 j=1 R f,t+1 M t ), (2) ω j,t (R j,t+1 R r,t+1 ) + R r,t+1 (3) In the formulation above, W t+1 stands for the total wealth at the end of t + 1; R w,t+1 is the return on wealth, which corresponds to the return on the stock market portfolio as in Epstein and Zin (1991) and Yogo (2006) 6 ; ω j is the relative weight associated with risky asset j; R j,t+1 is the real gross return on asset j between t and t + 1; R f,t+1 and R r,t+1 are the nominal and real gross risk-free rates from t to t + 1, which are both known at the beginning of period t + 1; and R f,t+1 1 R f,t+1 cost of money. 7 represents the present value of the opportunity As shown in Appendix B, one can rewrite the intertemporal budget constraint (2) in a more intuitive way, W t+1 = ( N a j,t R j,t+1 + W t C t M t j=1 ) N R f,t+1 a j,t + M t, (4) 1 + π t π t+1 j=1 where a j,t is the real amount invested in the risky asset j, and π t+1 is the inflation rate between t and t + 1. In this specification, the total wealth at the end of the period corresponds to the market value of the risky assets plus the real value of the investment 6 This is subject to the Roll (1977) critique. However, due to significant measurement error in nonfinancial wealth, I use the most usual proxy for return on wealth. See Chen, Favilukis and Ludvigson (2010) for a semiparametric approach that estimates the implied return on wealth. 7 Changes in money supply, as usual, are assumed to be transferred to the agents in the economy by the government through a lump sum transfer. For simplicity, we assume that this happens at the beginning of the period, and thus, the lump sum transfer is part of M t. 7

10 on the risk-free asset, in addition to real money holdings. The term, the fact that real balances at time t are equal to M t Π t M t 1+π t+1, stems from assuming M is the nominal money holding and Π represents the general price level. Therefore, real balances at time t + 1 are given by M t Π t+1, leading to Mt Π t Π t+1 Π t = Mt 1+π t+1. I show in Appendix A that at the equilibrium, the Euler equation for risky asset returns is given by E t ( ) 1 γ θ δ θ Ct+1 C t R f,t+2 1 R f,t+2 R f,t+1 1 R f,t+1 ε(γ 1) Rw,t+1R θ 1 j,t+1 = 1, (5) which implies that the SDF is as follows, Q t+1 = δ θ ( Ct+1 C t ) 1 γ θ R f,t+2 1 R f,t+2 R f,t+1 1 R f,t+1 ε(γ 1) R θ 1 w,t+1. (6) The SDF is driven by the traditional real consumption growth, the lead growth or lead ratio in the nominal interest rate (which corresponds to the growth in the opportunity cost of money), and finally the market return, which is also present in the Epstein Zin pricing kernel. Thus, the SDF is a function of a nominal variable, the nominal interest rate. The particular case of a time separable utility function (power utility) is obtained by setting θ = 1 in the previous expression. Relative to the standard Epstein Zin SDF, the new factor is the growth in the opportunity cost of money, IRR t+1 = F t+2 F t+1 R f,t+2 1 R f,t+2 R f,t+1 1 R f,t+1, (7) where F t+1 R f,t+1 1 R f,t+1 denotes the opportunity cost of money between t and t + 1, which is known at the beginning of the period. In order to derive this model, I use the fact that, at the equilibrium, real consumption, real balances and the nominal interest rate are related by a simple relationship, 1 ε Ct 1 M t = R f,t+1 ε R f,t+1 1, (8) 8

11 which states that the intratemporal marginal rate of substitution between real money balances and real consumption (left hand side) is equal to the relative price of consumption over money (right hand side). 8 Given the usual normalization that the price of consumption is 1, the relative price of real money balances is the (present value of the) opportunity cost of holding one unit of money, and it is equal to R f,t+1 1 R f,t+1, that is, the nominal interest rate discounted to the beginning of the period. An advantage of this representation is that it does not require measuring real balances, which are difficult to measure in practice, while interest rates can be fairly precisely measured Linearizing the model Since the focus in this paper is to price the cross-section of stock returns, as is common in the asset pricing literature I derive and test a linear version of the model presented above. From an econometric perspective, the linear specification is more appropriate to price a large number of (excess) returns, and, in addition, it provides greater intuition of the model s explanatory power over the dispersion of average (excess) returns. One can linearize the model represented in SDF form (5), by applying a methodology similar to the one employed in Yogo (2006) that leads to an expected return-covariance representation in unconditional form as follows, E (R j,t+1 R r,t+1 ) = Cov (q t+1, R j,t+1 ), (9) 8 This equation is obtained by combining the focs for consumption and money, as shown in Appendix A. 9 There is no consensus in the monetary economics literature on which monetary aggregate to use (M1, M2 or M3). Moreover, it is also not clear which price deflator should be used to compute real balances. 9

12 where q t+1 ln (Q t+1 ) denotes the log SDF. By applying the general pricing equation (9) to (5), and using the following definitions, c t ln (C t ), c t+1 = c t+1 c t, f t+1 ln (F t+1 ), f t+1 = f t+2 f t+1, r w,t+1 ln (R w,t+1 ), one obtains the expected return covariance representation of the CI-CAPM model, 10 E (R j,t+1 R r,t+1 ) = γ 0 + γ c Cov (R j,t+1, c t+1 ) + γ f Cov (R j,t+1, f t+1 ) + γ w Cov (R j,t+1, r w,t+1 ), (10) where the covariance risk prices (γ c, γ f, γ w ) are linked to the preference parameters in the following way, γ c θ ψ = 1 γ ψ 1, γ f ε (1 γ), γ w (θ 1) = γψ 1 ψ 1. (11) Given these definitions, the implied structural parameters can be backed up by the following relations, ψ = 1 γw γ c, γ = γ c + γ w, ε = γ f 1 γ c γ w. (12) In the above representation, there are three covariances that help to explain the crosssectional dispersion in expected stock returns: The covariance with the log consumption- 10 Lioui and Rangvid (2009) derive a related linear model, in which the factors are consumption growth and the nominal interest rate. However, their SDF is based on temporal CRRA preferences, and hence the factor risk prices are different from those in this model. 10

13 growth ( c t+1 ); covariance with the log interest ratio growth ( f t+1 ); and covariance with the log wealth return (r w,t+1 ). The signs of the covariance risk prices are restricted by economic theory. By assuming that the representative investor is more risk averse than an investor with log utility, that is γ > 1, then the risk price for the log growth in the opportunity cost of money is negative, γ f ε(1 γ) < 0, given that ε is positive, by definition. On the other hand, the risk price for the log market return can be greater or less than zero, depending on whether γψ > 1 or γψ < 1, respectively. These conditions mean that with γ > 1, if ψ is low enough, we can have a negative estimate for the market risk price. Regarding consumption growth, with γ > 1 the consumption risk price will be positive if ψ < 1, and negative otherwise. Given that ψ < 1 is the relevant empirical case in this paper (as we will see in the empirical section), then the consumption risk price will always be positive. In Equation (10), an asset that covaries positively with the log growth in the opportunity cost of money earns a lower risk premium than an asset that covaries negatively with this factor. The intuition is as follows. High interest rates are associated with low real balance holdings and thus higher marginal utility of consumption. As a consequence, consumers require a higher risk premium to hold assets that are negatively correlated with interest rate growth, since they don t provide an hedge (that is, they pay badly in bad times). The previous reasoning is correct as long as the representative investor has an RRA parameter γ, greater than one. On the other hand, given two assets that are negatively correlated with the log growth in the opportunity cost of money, the asset with a more negative covariance will earn a higher risk premium, since it offers a worse hedge against bad times (that is, times with high marginal utility of consumption). 2.3 Nested models The model stated in Equation (10) represents a rich specification, since it nests other well known models in the asset pricing literature, as special cases. The first special case is the Epstein and Zin (1991) model with one consumption good, which results from (10) 11

14 by setting ε = 0 (that is, there is no role for real balances in asset pricing), E (R j,t+1 R r,t+1 ) = γ 0 + γ c Cov (R j,t+1, c t+1 ) + γ w Cov (R j,t+1, r w,t+1 ), (13) γ c θ ψ = 1 γ ψ 1, γ w (θ 1) = γψ 1 ψ 1, with both γ and ψ having the same expressions as in (12). The second special case of CI-CAPM is the Power utility C-CAPM from Lucas (1978) and Breeden (1979), which arises as a special case of (10) by imposing θ = 1 (which means that γ = 1/ψ) and ε = 0, E (R j,t+1 R r,t+1 ) = γ 0 + γ c Cov (R j,t+1, c t+1 ), (14) γ c γ. The third nested case is the CAPM from Sharpe (1964) and Lintner (1965), which is obtained by imposing that ψ + and ε = 0 in (10), E (R j,t+1 R r,t+1 ) = γ 0 + γ w Cov (R j,t+1, r w,t+1 ), (15) γ w γ. To obtain the CAPM pricing equation, notice that ψ + implies that γ c 0 and also that θ 1 γ, which in turn leads to γ w γ. In all above expected return-covariance equations, an intercept is included, although strictly speaking, this intercept should be equal to zero if there exists a real risk-free rate, as is assumed in the derivation of the Euler equation for real returns. 12

15 3 Data and econometric methodology 3.1 Data and variables I use quarterly data spanning the period from 1963:III to 2008:III. All the macroeconomic and interest rate data are obtained from the FRED R database (St. Louis FED). Consumption is equal to non-durables plus services, and in order to compute per capita real consumption, I deflate it by the consumer price index (CPI) and total population (also available from FRED R ). Given that the key factor in the model is the future growth in the opportunity cost of money, two interest rate proxies are employed to construct the opportunity cost of money, which are the three-month Treasury bill rate (TB) and the effective federal funds rate (FED). These interest rates are also used in the calculation of the real risk-free rate and the excess market return. In order to compute the real interest rate and real ex post market return, I use the CPI inflation rate. The equity market return corresponds to the value-weighted average available from the Center for Research in Security Prices (CRSP), while the returns on the 25 size/bm portfolio returns (SBM25) and 25 portfolios sorted on both size and long-term return reversal (SLTR25) are obtained from the webpage of Kenneth French. To compute the portfolio quarterly returns I compound the monthly returns. To calculate the portfolio excess returns I subtract the compounded one-month TB rate, which is available from Kenneth French s website. 11 The standard descriptive statistics for the log consumption growth, log interest growth, and log market return are presented in Table 1. We can see that the log market return is significantly more volatile than log consumption growth, and on the other hand, the log interest growth is more volatile than the market return. Figure 1 shows the time-series for the two versions of the log interest growth. It seems that the growth in the opportunity cost of money exhibits a clear business cycle behavior, presenting higher values at the beginning of a recession and lower values at the bottom of a recession. The high volatility of the interest rate factor around recessions suggests that it represents a valid risk factor 11 To compute real excess returns one can use nominal returns, since the CPI inflation cancels out. 13

16 to explain equity risk premia, which are strongly countercyclical. All the variables that represent risk factors seem to be stationary, as indicated by the first-order autocorrelation coefficients, which are significantly below one in all cases. The cross-correlations reported in panel B show that the two versions of the log interest ratio are highly correlated, as expected, with a correlation coefficient of 88%. On the other hand, consumption growth is moderately correlated with the interest factors (correlation coefficients between 37% and 39%), while the market return is not correlated with any of the log interest growth proxies. 3.2 Econometric framework The CI-CAPM is estimated by a two-stage generalized method of moments (GMM, Hansen (1982)) procedure, which uses equally weighted moments in the first-stage (equivalent to an ordinary least squares (OLS) cross-sectional regression), and in the second stage the moments are weighted according to the inverse of the Spectral density matrix (equivalent to a generalized least squares (GLS) cross-sectional regression). The firststage estimation enables to assess whether the model can explain the returns of a set of interesting portfolios (e.g., size/bm portfolios), while, in the second stage, one seeks to obtain the most efficient estimates of the parameters, i.e., lowest standard errors. The GMM system contains N +3 moment conditions, with the first N sample moments corresponding to the pricing errors for each of the N testing returns, g T (b) (R j,t+1 R r,t+1 ) γ 0 γ c R j,t+1 ( c t+1 µ c ) γ f R j,t+1 ( f t+1 µ f ) 1 T T t=0 γ w R j,t+1 (r w,t+1 µ w ) c t+1 µ c = 0, f t+1 µ f r w,t+1 µ w i = 1,..., N, (16) 14

17 where (µ c, µ f, µ w ) denote the means of ( c t+1, f t+1, r w,t+1 ). The last three moment conditions in system (16) allow one to estimate the factor means. Thus, the estimated covariance risk prices from the first N moments do account for the estimation error in the factors means, as in Cochrane (2005) (Chapter 13) and Yogo (2006). 12 The standard errors for the parameter estimates and pricing errors, and the remaining GMM formulas are presented in Appendix C. I use one-sided p-values for the tests of individual significance of the risk prices, since the respective signs are constrained by theory, as stressed in the theoretical section. The asymptotic test that the pricing errors are jointly equal to zero (test of overidentifying conditions) with ˆα g T,N (ˆb) representing the first N moments, i.e., the pricing errors associated with the N testing assets is given by ˆα Var( ˆα) ˆα χ 2 (N K), (17) with N K denoting the number of overidentifying conditions (N moments and K parameters to estimate, with K being the number of risk factors), and Var( ˆα) being a pseudo-inverse of Var( ˆα). 13 In addition to the formal test statistic (17), I compute an alternative and more robust goodness-of-fit measure to evaluate the overall pricing ability of the model, the crosssectional OLS coefficient of determination, R 2 OLS = 1 N i=1 ˆα2 i, N i=1 R2 i R OLS 2 = 1 ( 1 R 2) ( ) N 1, N K with R i = 1 T T 1 t=0 (R i,t+1 R f,t+1 ) 1 N N i=1 [ ] 1 T 1 T t=0 (R i,t+1 R f,t+1 ) denoting the 12 In the case of the benchmark specification that does not include a constant term, the GMM system is obtained by making γ 0 = Following Cochrane (1996, 2005), and given the fact that Var( ˆα) is singular in many applications, I perform an eigenvalue decomposition of the moments variance-covariance matrix, Var( ˆα) = QΛQ, where Q is a matrix containing the eigenvectors of Var( ˆα) on its columns, and Λ is a diagonal matrix of eigenvalues, and then only the non-zero eigenvalues of Λ are inverted producing a pseudo inverse. I use the GAUSS command pinv. 15

18 (cross-sectionally) demeaned (average) excess returns, and ˆα i standing for the (crosssectionally) demeaned pricing errors. R 2 OLS measures the fraction of the cross-sectional variance in average excess returns explained by the model, and R 2 OLS stands for the adjusted cross-sectional R 2, which corrects for the number of factors in the model. by In the efficient GMM estimation, the test of overidentifying conditions (J-test) is given T ˆα Ŝ 1 N ˆα χ2 (N K), (18) where ŜN denotes the block of the spectral density matrix associated with the N pricing errors (the first N moments in the system). The GLS coefficient of determination is equal to RGLS 2 = 1 ˆα Ŝ 1 N ˆα R (19) Ŝ 1 N R, where ˆα is the vector of demeaned pricing errors, and R is the vector of demeaned (average) excess returns. The GLS R 2 assigns less weight to more noisy pricing errors, i.e., pricing errors with larger variance. The asymptotic theory embedded on the GMM robust standard errors might suffer from several problems in the current application. More specifically, the small sample size in the time-series (181 observations) can imply that the asymptotic approximation is not valid, and this might be especially relevant in the second stage GMM, since the inverse of the spectral density matrix is poorly behaved when the number of moments is high relative to the number of observations. On the other hand, the moment functions might not be distributed as a martingale difference sequence (MDS) as it is assumed in the standard GMM theory used in this paper. In both cases, the asymptotic p-vales associated with the t-statistics and J-test statistics will not be close to the true p-values. To account for this problem, I conduct a bootstrap simulation to produce more robust (empirical) p-values for the tests of individual significance of the parameters and also for the J-test. The bootstrap simulation consists of 10,000 replications, and in each replication, the portfolio return data and the factors are simulated independently, without imposing the 16

19 asset pricing model s restrictions, that is, the data is simulated under the hypothesis that the model is not true. Details of the bootstrap simulation algorithm are provided in Appendix D. 4 Cross-sectional test of the CI-CAPM 4.1 Empirical tests main results The testing assets used to test the consumption-interest CAPM are the Fama French 25 size/bm portfolios (SBM25), and in alternative, 25 portfolios sorted on both size and long-term return reversal (SLTR25). SLTR25 are produced from the intersection of 5 portfolios formed on size (market equity) and 5 portfolios formed on past returns (13 to 60 months before the portfolio formation date). These portfolios are used by DeBondt and Thaler (1985) and Fama and French (1996) to advocate that past losers (portfolios with lower returns in the long-term past) tend to have subsequent higher returns, while past long-term winners have lower future returns. This pattern corresponds to a longterm mean reversion in stock returns, which is not explained by the CAPM. 14 The use of the SLTR25 portfolios represents an important empirical check on the model, since there is some recent criticism on the validity of cross-sectional asset pricing tests that rely exclusively on the SBM25 portfolios (see Lewellen, Nagel and Shanken (2010)). In addition to the equity portfolio returns, I force the model to price both the real risk-free rate and the real stock market return. It seems reasonable to require any macroeconomic asset pricing model to explain not only the cross-section of excess equity returns, but also the equity market excess return and the risk-free rate, that is, one wants to assess how successful the model is in explaining simultaneously the cross-section of excess returns in addition to the joint equity premium-risk-free rate puzzle. 15 This represents one innovation in the empirical test design conducted in this paper, since most of the litera- 14 An incomplete list of studies that conduct asset pricing tests over portfolios sorted on prior long-term returns include Fama and French (1996), Da (2009), and Da and Warachka (2009). 15 There is also a theoretical argument, since the expected return-covariance equations of the C-CAPM, and other models that represent extensions of the C-CAPM, are derived in most cases under the assumption that there is a risk-free asset. 17

20 ture on empirical testing of consumption-based models over the cross-section of returns focuses only on the equity portfolios (in most cases, the SBM25 portfolios) and ignores the market and risk-free asset returns. 16 On the other hand, the literature on econometric tests of the equity premium largely ignores the cross-section of stock returns. Therefore, there is a total of 27 testing returns when the equity portfolios are either SBM25 or SLTR25. The results for the CI-CAPM asset pricing equation (10) estimated by first-stage GMM over SBM25 are displayed in Table 2. There are two versions of CI-CAPM, each one corresponding to a different measure of the log interest ratio growth. The interest rate proxies are the three-month TB rate (TB, Panel A), and the FED Funds rate (FED, Panel B). By using two proxies for f one can assess the sensitivity of the model s fit to the nominal interest rate proxy. In each panel, Row 1 presents the results for the benchmark model, while Row 2 shows the estimation results when an intercept is added to the expected return-covariance equation. As discussed in the theoretical section, under the assumption that there exists a risk-free asset that earns the real interest rate, the intercept should be undistinguishable from zero, that is, it should not be statistically different from zero. For the two versions of the benchmark model (Row 1), the covariance risk price for the log interest ratio growth is negative as expected, with values that range between (FED) and (TB). Moreover, both the asymptotic t-statistic and the bootstrapped p-value indicate that the estimate for γ f is strongly statistically significant (at the 1% level), when the proxy for the log interest ratio growth is FED. When one uses f tb, the t-ratio indicates significance at the 5% level, while the empirical p-value points to strong statistical significance (1% level). The estimates for the covariance risk price associated with log consumption growth, γ c, are positive, whereas the estimates for the covariance risk price associated with the log market return, γ w, are positive and negative in the versions with TB and FED, respectively. On the other hand, both the asymptotic t- 16 The inclusion of the market return and risk-free rate on testing assets also address the recommendation by Lewellen, Nagel and Shanken (2010) of forcing a given asset pricing model to price its factors when the factors are returns. 18

21 statistics and empirical p-values show that both γ c and γ w are strongly insignificant, which means that both the log consumption growth and the log market return are not priced in the cross-section of returns, when in the presence of the interest factor. The CI-CAPM is not rejected by the χ 2 test-statistic, with asymptotic and empirical p-values well above 5%. In terms of global fit, the model explains nearly 80% of the cross-section variation in average excess returns as indicated by the R 2 OLS estimate of 78%, when one uses f tb. In the version with f f, the overall fit is only slightly lower, with a R 2 OLS estimate of 66%. The average pricing error estimates are 0.29% and 0.37% per quarter, in the versions with TB and FED, respectively. The results for the model with a constant (Row 2) show that the intercept estimates in the two versions of the model have small magnitudes, with values that range between 0.6% per quarter (TB) and 0.8% per quarter (FED), which translate into 2.4% and 3.2% per year, respectively. To get a sense of the economic significance of these intercept estimates, the cross-sectional mean of the average excess returns of the 25 portfolios is 2.18% per quarter. These intercept estimates are highly non-significant in the case of TB, whereas in the case of FED, the intercept is significant at the 5% level according to the asymptotic t-statistic, but based on the more robust empirical p-value (1.00), this point estimate is clearly non-significant. The ROLS 2 estimate is only marginally higher in comparison to the corresponding value in the benchmark model without intercept, in the version with TB. In the version with FED, the coefficient of determination estimate increases by around 5% relative to the benchmark model. The fact that the ROLS 2 estimates in the model with intercept are sensibly the same as in the baseline model represents clear evidence that, by adding the constant factor, the overall model s fit does not change in a meaningful way. Therefore, this result suggests that there is no misspecification in the benchmark model (that is, missing factors). Moreover, the point estimates for γ f, as well as the associated asymptotic t-statistics and empirical p-values, are very close to the corresponding values from the benchmark model, signaling that the inclusion of the intercept does not cause a material change in the contribution of the interest growth risk factor to the overall model s explanatory power. 19

22 The estimation results for the CI-CAPM over SLTR25 (from first-stage GMM) are displayed in Table 3, which is similar to Table 2. The results show that the global fit of the restricted CI-CAPM is high with R 2 OLS estimates close to 70%, which are not significantly different from the corresponding values in the test over SBM25. The estimates for the average pricing error are 0.30% (TB) and 0.34% (FED) per quarter, which are also similar to the corresponding values in the test with SBM25. Moreover, the model is not rejected by the χ 2 -test as suggested by the empirical p-values, which are significantly above 5% in the two versions of the model. The risk price estimates for the interest growth factor have smaller magnitudes than in the test with SBM25, but they are statistically significant at the 5% (TB) or 1% levels (FED), according to the asymptotic t-stats. Based on the empirical p-values, the interest risk price is significant at the 10% level. On the other hand, the point estimates for γ c are positive but not statistically significant in the two cases, while the estimates for γ w are also positive, being significant (5% level) in the version with TB. In the case of the unrestricted model, the corresponding intercept estimates have very small magnitudes, varying between 0.2% per quarter (TB) and 0.3% per quarter (FED). To help interpret these estimates, the cross-sectional mean of the average excess returns of the SLTR25 portfolios is 2.29% per quarter. Moreover, these point estimates are strongly non-significant, as suggested by the empirical p-values of Another sign that the intercept does not play a significant role in the model is the fact that the coefficient of determination estimates are very similar to those in the benchmark restricted model. Hence, these results show that the good fit of the CI-CAPM is not confined to the SBM25 portfolios and generalizes to the SLTR25 portfolios as well. We can summarize the results from Tables 2 and 3 in the following way. First, the CI-CAPM seems to explain well the dispersion in the excess returns of the 25 size/bm portfolios and also the 25 portfolios sorted on size and long-term return reversal, in addition to the real risk free rate and the excess market return. Second, the change in the log interest ratio is a priced factor, and seems to drive most of the explanatory power over the cross-section of quarterly excess returns, while consumption and market risks seem to 20

23 play a secondary role. Third, when one estimates the CI-CAPM with an intercept, the excess zero-beta rate (relative to the risk-free rate) is strongly non-significant, suggesting that the model is correctly specified. Figure 2 offers an overview of the CI-CAPM s fit over the SBM25 portfolios by plotting the relation between realized and estimated average excess returns for the 25 portfolios (indicated by squares) plus the excess market return and the real risk-free rate. The better the model s fit, the closer are the pairs of realized and predicted average excess returns to the 45 line. We can see that the fitted points are slightly closer to the diagonal line in the version with TB relative to the version with FED, consistent to the OLS coefficients of determination estimates presented in Table 2. Among the 25 portfolios the biggest negative outlier in the graphs is the extreme small-growth portfolio (11, southern point) with pricing errors of -0.82% and -1.06% in the versions with TB and FED, respectively. Therefore, the CI-CAPM is not very successful in pricing the small-growth portfolio, similarly to most models in the empirical asset pricing literature. 17 The biggest positive outliers are the small-value portfolio (15, northeastern point) with pricing errors of 0.93% and 1.13% for TB and FED, respectively, and the large-intermediate BM portfolio (53, northwestern point) with pricing errors of 0.81% and 1.05%, respectively. The pricing errors for the real interest rate are negative but relatively small in magnitude, with values between -0.11% (TB) and -0.22% (FED). On the other hand, the residuals for the excess market return are positive, but also of small size, in the versions with TB (alpha=0.24%) and FED (alpha=0.21%). Figure 3, which is similar to Figure 2, provides a visual inspection of the model s fit in the test with the SLTR25 portfolios. We can see that most pairs of realized and predicted excess returns are very close to the 45 line, as in the test with the SBM25 portfolios. The main outlier in the two versions of the model is the small-past winner portfolio (15), which has negative pricing errors that vary between -1.34% (FED) and -1.48% per quarter (TB). As in the test with SBM25, the pricing errors for the real interest rate are negative but 17 For example, Fama and French (1993) stress that their model is formally rejected when tested over the SBM25 portfolios because it fails to price the small-growth portfolio, while Campbell and Vuolteenaho (2004) report that their two-factor ICAPM also fails to price the small-growth portfolio. 21

24 of even smaller magnitude, ranging between -0.04% (TB) and -0.11% (FED). In contrast to the test with SBM25, the pricing errors associated with the excess market return are negative (-0.22% and -0.25% for TB and FED, respectively), although the magnitudes are relatively small in both cases. Therefore, the CI-CAPM does a relatively good job in pricing both the real interest rate and the excess market return, in the tests with both sets of equity portfolios. 4.2 Empirical tests second-stage GMM The next step is to estimate the CI-CAPM by second-stage or efficient GMM, and put in perspective the results from first-stage GMM with equally weighted testing assets. Some authors (e.g. Lewellen, Nagel and Shanken (2010)) had justified the use of efficient GMM (or equivalently, GLS cross-sectional regressions of average returns on factor betas/covariances) over first-stage GMM (or equivalently, OLS cross-sectional regressions) with the argument that it might represent a greater hurdle to obtain a large cross-sectional GLS R 2 than a large OLS R 2, and they present some simulations that confirm those predictions. Furthermore, the GLS R 2 represents a proxy for how near the factor-mimicking portfolios are to the mean-variance frontier constructed from the testing returns. In addition, if one of the factors is a return, then its risk price should be equal to the corresponding average excess return. However, as argued by Cochrane (1996, 2005), in the efficient GMM estimation one is pricing repackaging portfolios, which often represent extreme linear combinations of the original portfolios. Thus, not only does one lose the economic content of the original portfolios, but also these efficient portfolios might be uninteresting for the average investor. With the previous caveat in mind, I estimate CI-CAPM by efficient GMM, and report the results in Tables 4 (SBM25) and 5 (SLTR25). The results for the 25 size/bm portfolios show that the point estimates for γ f have significantly smaller magnitudes in comparison with the corresponding first-stage estimates. Nevertheless, γ f is significant at the 10% (TB) or 1% level (FED) according to the asymptotic t-stats, while based on the empirical p-values it is significant at the 5% level. The point estimates for γ c also decrease in 22

25 magnitude relative to the first-stage test, but the respective p-values are quite large. On the other hand, the estimates for γ w are positive, being significant around the 5% level, the point estimate in the version with TB (based on the asymptotic inference). Both the asymptotic and empirical p-values associated with the J-statistic (18) are numerically equal to the corresponding values from the first-stage statistic (17). The estimates for the GLS R 2 are quite large for the two versions of the model, with values very close to one. This represents evidence that the CI-CAPM is able to price a meanvariance efficient combination of the original testing assets. The estimates for γ 0 presented in Row 2 are slightly negative, although they are highly non-significant in the two model s versions. Moreover, both the point estimates for γ f and R 2 GLS are relatively close to the corresponding values associated with the benchmark or restricted model. In the second-stage estimation with the SLTR25 portfolios, we have a similar pattern to the test with SBM25. The estimates for R 2 GLS are above 90% in the version with TB, and equal to 51% in the case of FED. While the R 2 GLS estimate in the version with FED is lower than the corresponding R 2 OLS estimate (68%), it still represents a significant explanatory power. The point estimates for γ f have lower magnitudes than the corresponding first stage estimates, but they are significant at the 10% or 1% levels, based on the asymptotic t-stats. However, the empirical p-values suggest that the monetary risk price is not significant, in this case. Thus, there are some signs of multicollinearity in the efficient test over SLTR25, which implies high p-values for γ f, despite the large explanatory ratio of the model, and this is more relevant in the version with TB. The results for the unrestricted model show that the point estimates for γ 0 have very small magnitudes and are statistically insignificant. On the other hand, both the interest risk price estimates and the R 2 GLS estimates are sensibly the same as the corresponding values for the restricted model. Therefore, as in the case of SBM25, the efficient estimation results associated with the SLTR25 portfolios seem to indicate that there is no misspecification in the CI-CAPM. Overall, the results in Tables 4 and 5 indicate that a linear combination of the factor mimicking portfolios is close to being mean-variance efficient, that is, the model is able 23