State uncertainty in stock markets: How big is the impact on the cost of equity?

Journal of Banking & Finance 36 (2012) 2575–2592

Contents lists available at SciVerse ScienceDirect

Journal of Banking & Finance journal homepage: www.elsevier.com/locate/jbf

State uncertainty in stock markets: How big is the impact on the cost of equity? Yufeng Han ⇑ University of Colorado Denver, Business School, PO Box 173364, Campus Box 165, Denver, CO 80217-3364, United States

a r t i c l e

i n f o

Article history: Received 2 October 2009 Accepted 29 May 2012 Available online 7 June 2012 JEL classification: G12 G31 C11 Keywords: Cost of capital Time-varying parameters State uncertainty Dirichlet process Bayesian analysis

a b s t r a c t We propose a novel Bayesian framework to incorporate uncertainty about the state of the market. Among others, one advantage of the framework is the ability to model a large collection of time-varying parameters simultaneously. When we apply the framework to estimate the cost of equity we find economically significant effects of state uncertainty. A state-independent pricing model overestimates the cost of equity by about 4% per annum for a utility firm and by as much as 3% for industries. We also observe that the expected return, volatility, risk loading, and pricing error all display state-dependent dynamics that coincide with the business cycle. More interestingly, the forecasted market and Fama–French factor risk premiums can predict the future real GDP growth rate even though the model does not use any macroeconomic variables, which suggests that the proposed Bayesian framework captures the state-dependent dynamics well. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction In a CAPM world, returns are assumed to be normally distributed with constant expected returns and volatilities, and the expected returns are a function of the market betas, the market risk premium, and potential pricing errors. All the parameters are assumed to be constant. However, extensive empirical evidence suggests that the assumption of constant parameters is far from reality. For example, a large body of literature on return predictability suggests that the expected return varies over time.1 There is also a huge amount of research in modeling time-varying volatility.2 Time-varying market risk (beta) is also widely documented.3 In fact, just about every parameter is shown to change over time. One approach to deal with time-varying parameters is to use conditional versions of the pricing model. For example, Ferson and Locke (1998)

⇑ Tel.: +1 303 315 8458; fax: +1 303 315 8084. E-mail address: [email protected] See, e.g., Ferson (1989), Avramov (2003), Guo (2006), and Han (2010) to name a few. 2 For the (G)ARCH model, see the review by Bollerslev et al. (1994) and references therein; for the stochastic volatility model, see Kim et al. (1998), the review by Ghysels et al. (1995) and references therein. 3 See Kothari et al. (1995), Harvey (2001), Campbell and Vuolteenaho (2004), Adrian and Franzoni (2005), ?, Lewellen and Nagel (2006), Andersen, Bollerslev, Diebold and Wu (2006), Basu and Stremme (2007), and Ang and Chen (2007) to name a few. 1

0378-4266/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jbankfin.2012.05.016

use the conditional CAPM to investigate the sources of errors in estimating the cost of equity.4 The difficulty, however, lies in the fact that these time-varying parameters are unobserved, and thus the problem with the conditional approach is that it requires specifying the dynamics of the time-varying parameters in ad hoc manners. Ghysels (1998) finds that the usual specifications of the conditional pricing models often perform worse than the unconditional pricing models. Other studies also find the lack of support for the conditional models by the data.5 More importantly, the conditional approach is difficult to implement when more time-varying parameters need to be modeled.6 In this paper, we propose a novel Bayesian framework that can easily incorporate a large number of unobserved time-varying parameters. Instead of modeling these time-varying parameters separately, we model them collectively and assume that their values depend on a latent state variable. Whenever the state of the market changes, the values of these parameters change as well. With a Bayesian approach, any ad hoc assumptions about the states are unnecessary, and instead we will learn about the states of the market from the data.

4 Other studies include Fama (1996), Brennan (1997), Bakshi and Chen (2001), Ang and Liu (2002), etc. 5 Examples are He et al. (1996), Ferson and Siegel (1999), and Ferson and Harvey (1999). 6 Avramov and Chordia (2006) model time-varying alpha, beta, and factor premiums, but do not model time-varying volatility.

2576

Y. Han / Journal of Banking & Finance 36 (2012) 2575–2592

As the key component of our Bayesian framework, the dynamics of the state and the state-dependent parameters are described by a mixture of Dirichlet process (MDP). Because of the discreteness of the Dirichlet process (due to Dirichlet distribution), all the state-dependent parameters evolve in a discrete fashion and are clustered into distinct values. These distinct clusters represent the distinct states of the market. One major advantage of this framework is that it allows for the arrival of new states in future – the market in the future can fall into one of the existing states or emerge as a completely new state, and thus a financial decision maker faces state uncertainty. For example, let’s assume that there have been three distinct states in the market, say Bear (a bear state), Norm (a normal calm state), and Bull (a bull state). If the current state is Bear, the uncertainty a financial decision maker faces is whether the market next period would be still in Bear, or change to Norm, or change to Bull, or change to a completely new state (say, a even deeper bear state). With our Bayesian framework, a financial decision maker is able to not only learn about the past states from the data but also to forecast future states of the market. Compared to the conditional model, an important advantage of the framework is that no ad hoc specifications and auxiliary instrumental variables are required, hence overcoming the mis-specification problem described by Ghysels (1998). Instead, the parameters evolve to different values according to a latent state indicator. This parsimonious specification can easily accommodate a large collection of state-dependent parameters. Perhaps, a more closely related model is the regime switching model. However, several important differences are worth noting. First, in the most common specification of the regime switching model, only two states are specified and only the expected return and volatility are modeled. On the one hand, this simple specification captures the common classification of the market into Bull and Bear states. On the other hand, it imposes the strong and unrealistic assumption that every bull market is the same and every bear market is the same. Using the MDP model, however, relaxes this assumption so that, for example, the recession in 1970s characterized by ‘‘stagflation’’ because of ‘‘Oil Crisis’’, can be different from the recession in 2001, the burst of the internet bubble. Second, in the regime switching model, the number of states is exogenously specified, whereas in the MDP model the number of states is endogenously determined by the data. Third, the transition probability is thus different. In the MDP model, the probability of a state occurring in future does not explicitly depend on the current state but depends on the likelihood of that state to fit the data and how frequent that state has occurred in the past.7 Finally, the uncertainty in the regime switching model is which of the two states will be realized. The MDP model, however, adds another dimension of uncertainty – not only which of the existing states may be realized, but also how likely a new state may be realized. We apply the Bayesian framework to the estimation of cost of equity focusing on the expected excess rate of return on the firm’s stock. We choose to estimate the cost of equity because it is one of the most important inputs for a firm’s financial decisions, and a common approach to estimating it is to use a factor asset-pricing model such as CAPM or Fama–French three-factor model.8 Furthermore, Pástor and Stambaugh (1999) propose a Bayesian framework

7 Thus in the MDP model, it is impossible to have a transition probability matrix; the persistence of the states is determined by the persistence of the data because similar data will have similar likelihood and will thus fall in the same state. 8 For example, Bower et al. (1984), Bower and Schink (1994), Elton et al. (1994), Fama and French (1997), Ferson and Locke (1998), Koedijk et al. (2001), Barnes and Lopez (2006) have used the CAPM, the Fama–French three-factor model, or the APT model to estimate the cost of equity.

to estimate the cost of equity with the assumption that the expected excess returns are constant. Thus our Bayesian framework provides a natural extension to their framework. Consistent with the existing literature, we observe interesting state-dependent dynamics of the expected excess returns and volatilities, which are evidently related to the business cycles identified by the NBER. While volatilities always change countercyclically, expected excess returns display both cyclic and counter-cyclic dynamics. For example, the market risk premium displays cyclic dynamics consistent with prior findings, whereas some industries display counter-cyclic dynamics in the expected excess return. In addition, we document for the first time the state-dependent dynamics of the SMB and HML portfolios and of their loadings. We find that the expected returns of SMB and HML portfolios increase during volatile periods (counter-cyclic), which is consistent with the conjecture that the SMB and HML are proxies for ‘‘distress risk’’. However, the risk loadings on the SMB and HML portfolios display different dynamics, suggesting that these two portfolios may mimic different risk factors. We observe higher correlations among the factor portfolios and higher market betas for many industries during volatile periods, suggesting that stocks tend to move more closely in volatile periods. The pricing errors often display dynamics similar to those of volatilities, suggesting increased idiosyncratic abnormal returns during volatile periods. Finally, we find that the market risk premium forecasted from the MDP model has a Granger causality relation with the real GDP growth rate. Furthermore, the forecasted Fama–French factor risk premiums predict future GDP growth rate with high adjusted R2s, whereas the realized factor returns fail to predict future GDP growth rate. This is striking given that no GDP growth rate or any other macroeconomic variables are used in the model. This evidence suggests that the MDP model captures the state-dependent dynamics well. Consistent with the findings of Pástor and Stambaugh (1999), we find that the posterior means of the expected excess returns are relatively insensitive to mispricing uncertainty, more so in the presence of the state uncertainty. In contrast, we find that state uncertainty has economically significant effects on the estimation of the cost of equity. Ignoring state uncertainty results in overestimating the posterior means of the expected excess returns – the average overestimation bias is about 4% per annum for an individual utility stock, and about 1.5% per annum for industry portfolios (as high as 3%). Both the pricing error and risk premium, the two components of the expected excess return, contribute to the overestimation, but one may dominate the other in different cases. In addition, the overestimation is insensitive to different prior beliefs about the number of states in the market and persists regardless of the level of prior mispricing uncertainty. State uncertainty also substantially increases the uncertainties about the pricing error, risk loading, and factor risk premium, and thus increases the overall uncertainty about the expected excess return. It should be noted that our results are conservative because we only compare the forecasted expected excess returns for the first month, but in practice, to evaluate a project, the decision maker would have to forecast a term structure of future expected excess returns. As we will discuss later, the expected excess returns over longer periods are functions of the monthly expected excess returns, and hence it is likely that the effect of state uncertainty is much greater due to accumulation. Recently, there emerges a promising literature that uses the discounted Residual Income Valuation (RIV) model to estimate the cost of equity capital. Claus and Thomas (2001) use the residual income model to estimate the market risk premium and conclude that the equity premium is as low as three percent. Gebhardt et al. (2001) use the residual income model to estimate the implied

Y. Han / Journal of Banking & Finance 36 (2012) 2575–2592

costs of equity for individual stocks and relate the estimates to firm characteristics. Pástor et al. (2008) use the implied cost of capital to estimate the intertemporal risk-return tradeoff. Lee et al. (2009) use the implied cost of equity to test the international asset pricing models. However, this approach is also subject to criticism. Easton and Monahan (2005) examine the correlations between the implied cost of equity estimates and realized return and find that none of the correlations are positive. Furthermore, Botosan and Plumlee (2005) find that implied costs of equity estimates often negatively correlated with the market beta, which is clearly inconsistent with the asset pricing theory. We compare the estimated cost of equity from the MDP model with the implied cost of equity estimates and find that the MDP estimates are much higher in magnitude and much less volatile than the implied cost of equity. We also confirm that the implied costs of equity are negatively correlated with the realized returns whereas the MDP estimated costs of equity are positively correlated with the realized returns. Bayesian mixture models including the mixture of Dirichlet process have not been widely used in the Finance literature despite their flexibility. Kacperczyk and Damien (2011) uses the Dirichlet process in asset allocation. Kacperczyk et al. (2005) use Semiparametric Scale Mixture of Betas (SSMBs) models that are based on Dirichlet process for option pricing. Geweke and Keane (2007) use a mixture model where the state probabilities depend on observed covariates and the number of states is fixed a priori. Zarepour et al. (2008) use the Dirichlet process to model returns and calculate the value at risk. In Econometrics literature, Dirichlet process models have been used for various modeling purposes. For example, Jensen (2004) uses the Dirichlet process to model longmemory stochastic volatility models; Hirano (2002) uses the Dirichlet process for autoregressive panel data models. Other examples include Tiwari et al. (1988), Chib and Hamilton (2002), and Conley et al. (2008). The reminder of the paper is organized as follows. The methodology is developed in Section 2, wherein we introduce the MDP model in the context of multifactor pricing models, discuss the term structure of the cost of equity, but leave the details of the model and estimation to the appendix. Section 3 describes the data and explains the empirical Bayes procedure used to obtain the hyperparameters in the prior distributions. Section 4 discusses the estimated market states and state-dependent dynamics of the parameters. Section 5 presents the estimates of the cost of equity capital and compares the estimates with implied costs of equity estimates. Section 6 discusses the predictive power of the model to predict the real GDP growth rate. We conclude in Section 7. 2. Methodology 2.1. Stochastic setting Let rt, t = 1, . . . , n denote the excess returns on the firm’s stock, rpt denote a q  1 vector of excess returns on the benchmark portfolios that are proxies for the market-wide risk factors. Assume that the excess returns rt follow a linear multifactor process:

rt ¼ at þ b0t r pt þ t ; rpt ¼ lpt þ gt ;   t  N 0; r2t ;

ð1Þ

gt  N ð0; Xt Þ;

where the expected excess return on the stock is determined by the pricing error, at, the sensitivity of the firm’s stock to the risk factors, bt, and the factor risk premiums, lpt, as

lt ¼ at þ b0t lpt :

ð2Þ

2577

The parameters at ; bt ; r2t ; lpt , and Xt, denoted collectively by   wt  at ; bt ; r2t ; lpt ; Xt , are state-dependent; i.e., wt is the state vector. However, the value of wt at each period t is unknown, so is the number of different states that the market has experienced. To deal with this problem, we employ a Bayesian seminonparametric approach. Specifically, we assume

  wt  at ; bt ; r2t ; lt ; Xt  G; G  Dðk; G0 Þ;

ð3Þ

where G is a random distribution and Dðk; G0 Þ is the Dirichlet process with parameter k and G0.9 The state vectors wt, t = 1, . . . , n are sampled from a common distribution, however the common distribution G() is uncertain (random) and sampled from a Dirichlet process with prior G0() and precision parameter k. G0() is the base prior, or prior mean of G(); i.e., E[G(w)] = G0(w) for all w. k determines the concentration of the prior G() around G0(), and therefore measures the ‘‘strength of belief’’ in G0(). One of the important features of the Dirichlet process is that G() is almost surely discrete. Because of the discreteness of G() under the Dirichlet process, W  {wt, t = 1, . . . , n} lies in a set of k < n vectors of distinct values, denoted by H  {h1, . . . , hk}. Therefore, hj,j = 1, . . . , k, correspond to the distinct states of the market, and k is the total number of distinct states having experienced in the market during the sampling period. For example, if k is fixed at 2, h1 and h2 correspond to the Bear and Bull markets. However, it is rather unrealistic to assume only two states in the market because each Bull or Bear market can be very different. Therefore, the number of states k is an unknown parameter in this model, determined by the data and prior beliefs. For simplicity, we assume the base prior G0() contains the usual conjugate priors for the parameters,10 G0 ðw; b0 ; R; m0 ; s0 ; l0 ; s; m; SÞ   r2 R ¼ N b; b0 ; IGðr2 ; m0 =2; s0 =2ÞN ðlp ; l0 ; X=sÞIWðX; m; SÞ; ð4Þ E½r2 

where b = [a; b], a concatenation of vector a and b. Note that for a state-independent model, G0() would be the prior of the parameters.11 2.2. Predicting the term structure of costs of equity The expected excess return on a firm’s stock, lt, is a function of the state-dependent parameters including the pricing error at, risk loading bt, and factor risk premium lpt, and thus is also statedependent. The posterior of lt can be obtained from the posterior distributions of at, bt, and lpt. However, to evaluate the firm’s capital budgeting or investment projects, a decision maker would require the expected excess returns from future periods as input, not from the past. Therefore, a term structure of future expected excess returns, D = {Dt, t = 1, . . . , T}, needs to be estimated, where Dt denotes the accumulative expected excess return over the period from n to n + t. For example, to evaluate a project that has expected cash flows I now, C1 after 1 year, and C2 after 2 years, a decision maker calculates the NPV of the project as follows:

9 We briefly introduce the Dirichlet process in the appendix. For a more thorough discussion of the Dirichlet process, see, for example, Ferguson (1973) and Antoniak (1974). 10 As a robustness test, we estimate the model using non-conjugate priors and obtain similar results. 11 When k is extremely large, i.e., the belief in G0 is extremely strong, the prior is degenerated to G0, and the MDP model degenerates to the state-independent model. In that case, k is the same as n, which does not mean that every period is a distinct state because all the parameters are drawn from the same posterior distribution, making it effectively one state, or state independent. This intuition is made clear in Eq. (A.5) in Appendix A.

2578

NPV ¼ I þ

Y. Han / Journal of Banking & Finance 36 (2012) 2575–2592

C1 C2 þ ; 1 þ D12 þ Rf ;12 1 þ D24 þ Rf ;24

ð5Þ

where Rf,12 and Rf,24 are the 1-year and 2-year spot risk-free rates, respectively, obtained from the term structure of Treasure yields. To determine Dt, note that

1 þ Dt þ Rf ;t ¼

t Y

ð1 þ ljjn þ r f ;j Þ;

ð6Þ

j¼1

where rf,j is the forward risk-free rate at month j (rf,1 is the spot rate of next month), and ljjn is the j-step ahead predicted expected excess return at month n + j, given the information up to month n. Therefore, to estimate the term structure of the future expected excess returns, D = {Dt, t = 1, . . . , T}, it boils down to forecast the expected excess returns ljjn, j = 1, . . . , T. The j-step ahead predicted expected excess return ljjn can be obtained as

ljjn ¼ ajjn þ b0jjn lp;jjn ;

ð7Þ

where the predictive distributions of ajjn, bjjn, and lp,jjn are estimated through iterations. See Appendix A for details. As the estimations of ltjn, t = 1, . . . , T, are similar, we only report the results of l1jn (or D1), in our empirical analysis. 3. Data and prior hyperparameters We estimate the costs of equity for one individual utility firm, Allegheny Energy Inc. (Ticker: AYE) and 40 industry portfolios, following Fama and French (1997).12 We investigate the monthly stock returns from July 1926 to December 2009, a total 1002 observations. One of the most important steps in Bayesian estimation is the construction of prior distributions. Following Pástor and Stambaugh (1999), we take an empirical Bayes approach and determine the hyperparameters using the cross section of 366 utility firms (SIC codes from 4900 to 4999) with at least 48 months of data. Fama and French (1997) employ a similar strategy in computing shrinkage estimates of b for industry portfolios. To determine the hyperparameters for the individual firms from the cross section of utility firms, we first collect returns for all the utility firms in the CRSP monthly NYSE/Amex/Nasdaq stock file that have at least 48 months of data in the period from January 1926 through December 2009. For each firm, we then compute ^ a vector containing a ^ and r ^ and b, ^ 2 using the OLS estimates, b, all the data available. The prior mean of bt is set to be the cross-sec^ whereas the prior mean of at is set to zero. tional average of the b’s, The prior covariance matrix R is constructed as follows. We first compute the matrix



^ r b ¼ VðbÞ ^ 2i X 0i X i R

1

;

ð8Þ

^ is the sample cross-sectional covariance matrix of the where VðbÞ ^ and X is a matrix of covariates containing ones and the excess b’s, i returns of the benchmark portfolios restricted to the period of observations available to stock i. The second term is the average ^ As noted by across stocks of the sampling-error variance of b’s. b is an estimate of the cross-sectional Fama and French (1997), R covariance of the b’s. However, Fama and French (1997) also note b is not always positive definite. Typically, the cross-sectional that R ^ ) is smaller than or very close variance of the estimated intercepts (a ^ . Therefore, to to the average of the sampling-error variance of a

12 We thank Ken French for making the data available on the Internet. Eight industry portfolios are omitted because of missing values.

construct R we set the prior variance of bt equal to the diagonal eleb excluding the first one, and set the prior standard deviments of R ation of at, ra, to 15% per annum, assuming that the decision maker has very high uncertainty about the ability of the pricing models to correctly price individual stocks. (Assuming high prior variance also minimizes the shrinkage toward prior mean (zero) of a.) Of course, we also examine the effect of prior mispricing uncertainty by varying r2a , similar to Pástor and Stambaugh (1999). The covariance terms in R are set to zero. To construct the prior for r2t , note that the inverse gamma density implies

m0 ¼ 4 þ

2ðE½r2 Þ2 ; v arðr2 Þ

ð9Þ

s0 ¼ ðm0  2ÞE½r2 :

ð10Þ

^ 2 ’s for E[r2]. SimWe substitute the cross-sectional average of the r ilarly for var(r2), we substitute the variance computed from the cross-sectional variance and the average sampling-error variance ^ 2, of r

v arðr2 Þ ¼ Vðr^ 2 Þ 

Ti  2 T 2i

r^ 4i ;

ð11Þ

^ 2 Þ is the cross-sectional variance of the r ^ 2 ’s, and Ti is the where Vðr number of observations available for stock i. The value of m0 is rounded up to the next integer. Panel A1 of Table 1 reports the hyperparameters. For the 40 industry portfolios, the hyperparameters of the priors are determined from the cross section of the 40 industry portfolios in a similar fashion. We also set ra to 5.5% per annum. The hyperparameters are reported in Panel A2 of Table 1. To construct the priors for the benchmark portfolios, we set the prior mean for the market portfolio to 6% and the prior mean of the variance of the market portfolio to (19%)2 per annum. The prior means of SMB and HML portfolios are assumed to be zeros, and the prior means of the variances are set to (11%)2. Panel B in Table 1 reports the hyperparameters for the benchmark portfolios. The prior on k reflects our beliefs about the number of states experienced in the market. Typically, the number of states is much smaller than the number of observations. In the empirical analysis that follows, we compare the results of three different prior beliefs. One with very small number of states, one with modest number of states, and the other with relatively large number of states. Table 2 lists the prior specifications of k that conform to these prior beliefs, respectively. Also listed are the implied expected number of states for the sampling periods, which also depends on the number of observations. 4. Market state uncertainty and state-dependent dynamics 4.1. Learning about states in the stock market Over the past 84 years or so, the US market has experienced many business cycles (about 15 cycles). Each cycle is different and unique, and thus the expected returns and volatilities of stock returns are likely different in different cycles. As the decision maker is uncertain about how many states the stock market has gone through, she first forms prior views of the number of states, and then learns about the states from the data and updates her belief about the number of states. We compare three different priors in Table 2: One view is that there are on average two states (E[kjn] = 2.34) similar to the regime switching model, another extreme view is that there are a large number of states in the stock market (E[kjn] = 13.35), and finally a more modest view is that there are a small number of states (E[kjn] = 4.24). Table 2 also lists

2579

Y. Han / Journal of Banking & Finance 36 (2012) 2575–2592 Table 1 Hyperparameters of prior distributions. Hyperparameters for at and bt

Model

b0 Panel A: Hyperparameters for the utility or industries Panel A1: utility stocks CAPM 0 0.747 FF three-factor 0 0.766 0.308 0.340

Hyperparameters for

rb

m0

s0

0.0433 0.466 0.0433 0.419 0.700 0.625

5

0.0515

5

0.0490

10

0.0179

10

0.0165

Panel A2: industry portfolios 0 0.0159 1.052 0.205 0 0.0159 1.005 0.175 0.180 0.269 0.0771 0.229

CAPM FF three-factor

Hyperparameters for

r2t

Hyperparameters for Xt

lp,t l0

s

m

Panel B: Hyperparameters for the benchmark portfolios CAPM 0.005 100 FF three-factor 0.005 100 0 0

5 7

S 0.009 0.009

0 0.003

0 0 0.003

In Panel A1, the hyperparameters are determined using the cross-section of all the utilities with at least 48 months of observations available in the CRSP in the period 7/1926–12/2009. In Panel A2, the hyperparameters are determined using the cross-section of the 40 industry portfolios in the period 7/1926–12/2009. The prior specification is

 G0 ðwt Þ ¼ N bt ; b0 ;



  r2 R IG r2t ; m0 =2; s0 =2 N ðlp;t ; l0 ; Xt =sÞIWðXt ; m; SÞ; 2 E½r 

where rb is the prior standard deviation of bt = [at; bt], and R = diag (rb)Ipdiag(rb), where Ip is an identity matrix with dimension p of 2 or 4, respectively. The prior mean of at is set to zero, and the prior standard deviation ra is set to 15% per annum for the utility stock, 5.5% per annum for industry portfolios.

Table 2 Learning about the number of states in the market. Specification

a

c

E[k]

r(k)

n

E[kjn]

Panel A: priors Gð3;4Þ Gð3;2Þ Gð3;0:5Þ

3 3 3

4.0 2.0 0.5

0.75 1.50 6.00

0.188 0.75 12.00

1002 1002 1002

2.34 4.24 13.35

CAPM  k

FF three-factor model s(k)

Panel B: posterior 0.73 0.34 Gð3;4Þ Gð3;2Þ 1.16 0.56 Gð3;0:5Þ 2.05 1.01

 k 5.94 7.80 11.26

estimates of the precision parameter k. For example, under the CAPM, the mean is shrunk from 0.75 to 0.73, from 1.5 to 1.16, and from 6.0 to 2.05, respectively, and the standard deviation is shrunk from 0.188 to 0.34, from 0.75 to 0.56, and from 12.00 to 1.01, respectively, for the three prior views. Under Fama–French three-factor model, similar shrinkage is observed. 4.2. State-dependent dynamics of the benchmark portfolios

s(k)

 k

s(k)

 k

s(k)

2.11 3.09 4.49

1.23 1.86 3.23

0.48 0.74 1.30

10.84 13.13 17.70

2.98 3.83 5.28

Panel A specifies the priors, Gða;cÞ , on the precision parameter k. E[k] and r(k) are the prior mean and standard deviation of k; E[kjn] is the implied prior mean of k, the number of states in the market, conditioned on the number of observations, n. Panel B reports the posteriors of k and k.  k and s(k) are the posterior mean and standard  and s are the posterior mean and standard deviation of k. The deviation of k, and k k sample size of the Gibbs sampling scheme is 10,000. The sample period is from 7/ 1926 to 12/2009.

the posterior means and standard deviations of k and k. The posterior means of k are, respectively, 6, 8, and 11 under the CAPM, and 11, 13 and 18, respectively, under the Fama–French three-factor model. The number of states is increased under the Fama–French model, which is likely due to changes related to SMB and HML factors that are not fully reflected in the market portfolio. If we compare the number of states to the number of business cycles in our sample period, it seems that some of the business cycles are actually similar. There are substantial shrinkages in the

Mounting evidence in the literature suggests that the market risk premium is time-varying and state-dependent. However, the fact that it is unobserved makes it difficult to estimate. The predominant way of estimating it is to assume that the market risk premium is a linear function of lagged economic variables, such as the dividend yield, T-bill yield, term spread, and so on. However, this approach is subject to a number of problems including data snooping and ad hoc assumptions, and many studies have challenged this approach.13 In our framework, we estimate the statedependent market risk premium without using any auxiliary variables and assumptions and thus avoid those problems associated with using observed variables. The upper two panels of Fig. 1 plot the posterior means14 of the market volatility and market risk premium over the period from July 13 See, for example, Nelson and Kim (1993), Goetzmann and Jorion (1993), Kirby (1997), Ferson et al. (2003), Bossaerts and Hillion (1999), Neely and Weller (2000), Goyal and Welch (2003), Cooper and Gulen (2006), Ang and Bekaert (2007), and Welch and Goyal (2008). 14 All quantities of the benchmark portfolios plotted are the averages of the 40 time series of posterior means estimated with the industry portfolios and are converted to quarterly series.

2580

Y. Han / Journal of Banking & Finance 36 (2012) 2575–2592

State-Dependent Market Volatility (%)

State-Dependent Market Expected Excess Returns (%)

50

14

45

12

40

10

35

8

30

6

25

4

20

2

15 10 01/28

0 01/40

01/52

01/64

01/76

01/88

01/00

-2 01/28

State-Dependent SMB Volatility (%)

01/40

01/52

01/64

01/76

01/88

01/00

State-Dependent SMB Expected Returns (%)

40

8

35

7

30

6 5

25

4 20

3

15

2

10

1

5 01/28

01/40

01/52

01/64

01/76

01/88

01/00

0 01/28

01/00

10 9 8 7 6 5 4 3 2 1 01/28

State-Dependent HML Volatility (%) 40 35 30 25 20 15 10 01/40

01/52

01/64

01/76

01/88

01/52

01/64

01/76

01/88

01/00

State-Dependent HML Expected Returns (%)

45

5 01/28

01/40

01/40

01/52

01/64

01/76

01/88

01/00

Fig. 1. State-dependent expected excess returns and volatilities of the benchmark portfolios. Each graph plots the time series of the posterior means of the volatilities (left panels) or expected excess returns (right panels) of the benchmark portfolios during the period of 7/1926–12/2009. The time series plotted are averages of the 40 time series of the posterior means estimated with the 40 industry portfolios, and converted to quarterly series. The top panels are the market portfolio, the middle panels are the SMB portfolio, and the bottom panels are the HML portfolio. Each posterior mean is the average of 10,000 posterior draws, and is annualized and expressed in percentage values. The vertical lines in each graph indicate the business cycles identified by the NBER: solid lines are peaks and dashed lines are troughs.

1926 to December 2009. Also shown in each panel are the business cycles identified by the NBER. The solid vertical lines indicate the peaks, and the dashed vertical lines indicate the troughs. Both the market volatility and market risk premium estimated from the MDP model display business-cycle related dynamics, which suggests that the MDP model captures the state-dependent dynamics quite well. The market volatility displays counter-cyclic dynamics: It increases during recessions and reaches the highest level near the troughs, and then decreases during expansions and bottoms out near the peaks. In the prolonged periods of expansions, however, the market volatility also changes significantly, experiencing drastic ups and downs. There are episodes of extremely high volatility, notably the Great Depression in 1930s, World War II in 1940s, the Oil Crisis in 1970s, the market crash in 1980s, the dot com bubble burst in early 2000s, and the most recent financial crisis (Great Recession). The market risk premium, on the other hand, displays cyclic dynamics: It decreases during recessions and increases during expansions. Similarly, it also changes drastically in the prolonged periods of expansions. Similar observations are reported by Fama and French (1989),

Chauvet and Potter (2001), and Han (2010). In addition, the risk premium and volatility are negatively correlated – high volatilities are associated with low risk premiums.15 It is also worth noting that the market risk premium remains mostly positive throughout the sampling period, even in the Great Recession, and only slightly negative in the Great Depression. Over the sample period from 1926 to 2009, the average market risk premium is about 6.50% per annum, while the lowest risk premium is 0.83% per annum occurring in the fourth quarter of 1929, and the highest risk premium is 13.57% occurring in the third quarter of 1932. In the latest Bear market due to the Great Recession, the market risk premium dropped to as low as 0.58% per annum in the last quarter of 2008, which is about two quarters ahead of the trough in June of 2009, identified by NBER.

15 Han (2010) provides an explanation based on volatility risk premium for this paradoxical negative correlation between the market expected excess return and volatility.

Y. Han / Journal of Banking & Finance 36 (2012) 2575–2592

Fig. 1 also shows the state-dependent dynamics of volatility and expected return for the SMB factor (middle panels) and the HML factor (lower panels). Both portfolios display counter-cyclic dynamics in the expected return and volatility, a result consistent with Zhang’s (2005) model. For the best of our knowledge, we are the first to show that both the expected return and volatility of both Fama–French factors display counter-cyclic dynamics. In the highly volatile periods such as the Great Depression, the World War II and so on, the expected returns and volatilities of both portfolios increase dramatically. These observations are consistent with the academic conjecture that the two portfolios are proxies for ‘‘distress risks’’ (see, e.g. Fama and French, 1993, 1995) because in highly volatile periods, firms are more likely in distress or default, and investors demand higher compensation for bearing the risks. Using GARCH models, Guo et al. (2009) also obtain very similar counter-cyclic dynamics for the HML portfolio, which again confirms that the MDP model works well. Over the sample period from 1926 to 2009, the SMB factor yields an average return of 1.17% per annum. The highest return that occurred in the third quarter of 1932 is 7.09 % per annum; the first quarter of 2000 sees the second highest return as high as 6.32% per annum. Similarly, the HML factor yields an average return of 2.01% per annum. The highest return occurred in the third quarter of 1932 as high as 9.03% per annum, and the second highest return occurred in the first quarter of 2000 as high as 6.88% per annum. Fig. 2 plots the state-dependent correlations among the market, SMB, and HML portfolios. The correlation between the market and SMB portfolios has an average of 0.205 over the entire sampling period, but it varies from 0.055 to 0.361, with substantially higher correlations in the volatile periods. The correlation between the market and HML portfolios also displays prominent counter-cyclic dynamics: It increases during recessions and decreases during expansions. It is interesting to note that even though the average of the correlation is only 0.033, the correlation reaches as high as 0.504 in the Great Depression. The correlation between the SMB and HML portfolios displays similar dynamics and varies in a wide range, even though the average correlation is only 0.004. The fact that all the benchmark portfolios become highly correlated during recessions seems consistent with recent findings that stocks and portfolios tend to move together more closely in bear markets than in bull markets in both domestic and international markets.16

2581

pricing error at increases as the market volatility increases and decreases as the market volatility decreases, which suggests that industry-specific abnormal returns increase during volatile periods. However, some industries display cyclic behavior in the pricing errors instead; examples include Shipping Container industry (Boxes, shown in the middle right), Lab Equipments, Whole Sale, Meals, and Tobacco. The bottom two plots in Panel B draw the dynamics of the expected excess returns of the industry portfolios, Computer (Comps) and Business Services (BusSv), both of which exhibit cyclic dynamics. Most industries display similar cyclic dynamics. The market risk loadings bt of most industries increase during volatile periods. The upper two plots in Panel A of Fig. 4 show two examples: Electric Equipment (ElcEq) and Shipping Containers (Boxes). The case of ElcEq represents the majority of the industry portfolios; that is, the market risk loadings increase in bear markets. On the other hand, the market beta of Boxes decreases during volatile bear markets. This different behavior of the market risk loadings has important implications for portfolio management. As most stocks and portfolios move more closely with the market as volatilities increase, those industries whose risk loadings decrease move less closely with the market, and thus are valuable instruments for diversification when it is needed the most. In addition, the risk loadings on the SMB and HML portfolios also display interesting state-dependent dynamics (Panel B and C of Fig. 4). For most industries, the risk loadings on the SMB portfolio (bs,t) are cyclic (e.g. Insur on the left of Panel B), while a few industries display counter-cyclic patterns (e.g. Toys on the right of Panel B). On the contrary, the risk loadings on the HML portfolio (bh,t) are counter-cyclic for most industries (e.g. ElcEq on the left of Panel C), while a few industries display cyclic patterns (e.g. Mines on the right of Panel C). The behavior of bh,t suggests that industry portfolios load more on the HML during volatile periods, which is consistent with the conjecture that the HML portfolio proxies for ‘‘distress factors’’. However, the behavior of bs,t suggests that the SMB portfolio might mimic a risk factor that is different from the ‘‘distress factor’’ mimicked by the HML portfolio.

5. Costs of equity 5.1. Costs of equity of individual utility firms

4.3. State-dependent dynamics of the utility stock and industry portfolios The expected excess returns of individual stocks and portfolios are state-dependent not only because the expected excess returns on the benchmark portfolios are state-dependent, but also because the pricing errors and risk loadings are state-dependent. However, the dynamics of these state-dependent parameters vary widely across stocks and industries. For the utility stock (AYE) and most industries, the pricing errors at exhibit counter cyclic dynamics similar to those of the market volatility. Panel A of Fig. 3 shows the dynamics of at (left plot), which is counter-cyclic, and the expected excess return (right plot), which is cyclic, for the utility stock AYE. The expected excess return of the utility stock changes dramatically across the sample periods. Under the Fama–French three-factor model, the average of the expected excess return is 6.24% per annum, but the record high and low are 19.61% and 0.826% per annum, respectively. Panel B shows the dynamics of at for four industries. The upper left plot graphs Chemistry (Chems) as a representative example for the majority of the industries. The 16 See, for example, Ang and Chen (2002) for the phenomenon in the domestic stock markets, and Lin et al. (1994), and Karolyi and Stulz (1996), and Longin and Solnik (2001) for the phenomenon in the international stock markets.

The fact that the expected returns are state-dependent and have experienced dramatic changes over the years highlights the potential problem of using a state-independent pricing model to estimate the costs of equity. As discussed earlier, for a decision maker, it is necessary to take a predictive perspective when estimating the cost of equity, and a key quantity in estimating the future cost of equity is the predicted expected excess return, which is a function of the pricing error, the risk loadings, and the factor risk premiums in future periods. Because of state uncertainty, the future states may fall into any of the existing states or emerge from a new distinct state. Therefore, the decision maker must take it into account when estimating the future expected excess returns as discussed in Section 2.2. Table 3 reports the estimated one-period ahead expected excess return (l1jn) on the utility stock (AYE) under the three different prior views of k. Panel A reports the expected excess return (l1jn) under the CAPM, along with the predicted pricing error (a1jn), risk loading (bm,1jn), and market risk premium (lm,1jn). In all cases, the expected excess returns are overestimated by the state-independent (normal) CAPM. For example, the expected excess return is 9.30% per annum under the state-independent CAPM, but is reduced to 6.11% ðk  Gð3; 4ÞÞ, 6.11% ðlambda  Gð3; 2ÞÞ, and 6.22% ðk  Gð3:0:5ÞÞ, respectively, under the state-dependent CAPM. The

2582

Y. Han / Journal of Banking & Finance 36 (2012) 2575–2592

State-Dependent Correlations of Mkt and SMB 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 01/28

01/40

01/52

01/64

01/76

01/88

01/00

01/88

01/00

01/88

01/00

State-Dependent Correlations of Mkt and HML 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 01/28

01/40

01/52

01/64

01/76

State-Dependent Correlations of SMB and HML 0.3 0.25 0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 01/28

01/40

01/52

01/64

01/76

Fig. 2. State-dependent correlations among the market portfolio and Fama–French SMB and HML portfolios. Each graph plots the time series of the average posterior means of the correlations averaged over 40 time series of posterior means estimated with the 40 industry portfolios during the period of 7/1926–12/2009. The plotted are quarterly series averaged from the monthly series. The correlations are the market and SMB portfolios (top), the market and HML portfolios (middle), and the SMB and HML portfolios (bottom). Each posterior mean is the average of 10000 posterior draws. The vertical lines in each graph indicate the business cycles identified by the NBER: solid lines are peaks and dashed lines are troughs.

overestimation in the expected excess return is due to the overestimation of the pricing error (a1jn) and the market beta (bm,1jn) by the state-independent CAPM. For example, bm,1jn of AYE is 0.77 under the MDP CAPM versus 0.94 under the normal CAPM. The standard deviations indicate that the state uncertainty also substantially increases the uncertainties about the pricing errors a1jn and the risk premiums b0m;1jn lm;1jn , and thus the overall uncertainty about the expected excess returns. For example, without state uncertainty, the standard deviation of l1jn is about 0.91% per annum, whereas with state uncertainty, the standard deviation increases to about 1.90% per annum. Another striking result is that the overestimation is robust to the different priors of k or of the number of states in the markets. For example, the difference in the expected excess return between the two extreme prior views is merely 0.11% per annum. The robustness suggests that the data have enough information such that two decision makers having completely different prior beliefs about the number of states in the market will nevertheless update their prior beliefs to similar beliefs about the state and obtain similar estimates of the expected excess returns. Similar observations apply to the Fama–French three-factor model as well. The pricing error and the market risk loading are both overestimated by the state-independent Fama–French model.

However, unlike the market risk premium estimated under the CAPM, which is slightly underestimated, the expected (excess) returns on the factor portfolios are significantly overestimated. The loading on the HML factor is also overestimated. The overestimation bias is thus more pronounced under the Fama–French model; on average the expected excess return is overestimated by 3.8% per annum.

5.2. Costs of equity of industries The estimated one-period ahead expected excess returns required for estimating the industry costs of equity are listed in Table 4 for the Fama–French three-factor model.17 Because the pricing error a1jn and risk premium b01jn lp;1jn may change in different directions when state uncertainty is incorporated, the change of the expected excess return is a net outcome of changes in these two components. For almost every industry portfolio, the risk premiums b01jn lp;1jn are overestimated by the state-independent Fama–French three-factor model. The pricing error a1jn are overestimated for 17 The results for the CAPM are similar but slightly weaker. We drop the results to save space, but the results are available upon request.

2583

Y. Han / Journal of Banking & Finance 36 (2012) 2575–2592

State-Dependent Pricing Errors -- AYE 2 1.5

μt (%)

αt (%)

1 0.5 0 -0.5 -1 01/28

01/40

01/52

01/64

01/76

01/88

01/00

State-Dependent Stock Expected Excess Returns -- AYE 20 18 16 14 12 10 8 6 4 2 0 01/28 01/40 01/52 01/64 01/76 01/88 01/00

State-Dependent Pricing Errors -- Chems

State-Dependent Pricing Errors -- Boxes

0.1

0.2

0.08

0.15

αt (%)

0.02 0

0.05 0

-0.02

-0.05

-0.04

-0.1

-0.06 01/28

μt (%)

0.1

0.04

01/40

01/52

01/64

01/76

01/88

-0.15 01/28

01/00

State-Dependent Stock Expected Excess Returns -- Comps 18 16 14 12 10 8 6 4 2 0 01/28 01/40 01/52 01/64 01/76 01/88 01/00

01/40

01/52

01/64

01/76

01/88

01/00

State-Dependent Stock Expected Excess Returns -- BusSv 12 10 8 μt (%)

αt (%)

0.06

6 4 2 0 01/28

01/40

01/52

01/64

01/76

01/88

01/00

Fig. 3. State-dependent pricing errors and expected excess returns of the utility stock and industry portfolios under the Fama–French three-factor model. In each graph, the plotted is the quarterly time series of the posterior means converted from the monthly series during the period of 7/1926–12/2009. Each posterior mean is the average of 10,000 posterior draws. The vertical lines in each graph indicate the business cycles identified by the NBER: solid lines are peaks and dashed lines are troughs.

many industries, while for a few industries the pricing errors are underestimated. For example, for Toys, both the pricing error and the risk premium are overestimated, 0.02% versus 0.67% for a1jn, and 8.33% versus 7.44% for b01jn lp;1jn . As a result, the expected excess return is overestimated by 1.58% (8.35% versus 6.77%) per annum. For Telcom (Telcm) industry, the pricing error is underestimated (0.32% versus 0.89%), whereas the market beta is overestimated (7.67% versus 4.18%). The net effect is that the expected excess return is overestimated by 2.3% per annum (7.36% versus 5.06%). Similar to the case of the individual firms, the overall uncertainty about the expected excess returns increases substantially when state uncertainty is incorporated. Increased uncertainties about both a1jn and b01jn lp;1jn contribute to the overall uncertainty in various degrees. Fig. 5 plots the estimation biases in the expected excess returns for the 40 industry portfolios under the Fama–French model. Panel A plots the bias in magnitude, and Panel B plots the bias in percent-

age. As we discussed previously, most industries experience overestimation in the expected excess returns when estimated using the state-independent model. The biggest bias is for Real Estate (Industry #38), with a magnitude of 2.86% per annum, which accounts about 58% of the estimates under the state-dependent model. Out of the 40 industries, only three industries experience underestimation; they are Aero, Coal, and Comps. In all three cases, the bias is small and is caused by the underestimation of the pricing error. 5.3. Mispricing uncertainty and state uncertainty Pástor and Stambaugh (1999) find that the expected excess returns are relatively insensitive to mispricing uncertainty due to the shrinkage effect. For instance, the expected excess return changes by less than 1% when ra increases to 5%. In this subsection, we examine the effect of mispricing uncertainty in the presence of

2584

Y. Han / Journal of Banking & Finance 36 (2012) 2575–2592

State-Dependent Risk Loadings on Mkt -- ElcEq

State-Dependent Risk Loadings on Mkt -- Boxes

1.01

1.005

1.0095

1.0045

1.009 1.004 βm,t

βm,t

1.0085 1.008 1.0075

1.0035 1.003

1.007 1.0025

1.0065 1.006 01/28 01/40 01/52 01/64 01/76 01/88 01/00

1.002 01/28 01/40 01/52 01/64 01/76 01/88 01/00

State-Dependent Risk Loadings on SMB -- Insur

State-Dependent Risk Loadings on SMB -- Toys

0.179

0.205

0.178

0.2 0.195

0.176

βs,t

βs,t

0.177

0.175

0.19

0.174 0.185

0.173 0.172 01/28

01/40

01/52

01/64

01/76

01/88

0.18 01/28

01/00

State-Dependent Risk Loadings on HML -- ElcEq 0.078

0.081

0.0775

0.08

01/64

01/76

01/88

01/00

0.077

0.079

βh,t

βh,t

01/52

State-Dependent Risk Loadings on HML -- Mines

0.082

0.078

0.0765 0.076

0.077

0.0755

0.076 0.075 01/28

01/40

01/40

01/52

01/64

01/76

01/88

01/00

0.075 01/28 01/40 01/52 01/64 01/76 01/88 01/00

Fig. 4. State-dependent risk loadings on the market, SMB and HML portfolios. The quarterly time series of the posterior means of the risk loadings converted from the monthly series during the period of 7/1926–12/2009 are plotted for the market (Panel A), SMB (Panel B), and HML (Panel C). Each posterior mean is the average of 10,000 posterior draws. The vertical lines in each graph indicate the business cycles identified by the NBER: solid lines are peaks and dashed lines are troughs.

state uncertainty, and the sensitivity of the expected excess returns to state uncertainty at different levels of mispricing uncertainty. Table 5 compare results of no state uncertainty and two different degrees of state uncertainty at three different levels of prior uncertainty about the pricing error, 3%, 5% and 50% per annum. Consistent with the findings of Pástor and Stambaugh (1999), under the normal CAPM, the posterior means of the expected excess returns do not change much when the level of mispricing uncertainty increases from 3% to 5%. For example, the posterior mean of AYE is 7.04% at ra = 3% and 7.44% at ra = 5%. When the prior mispricing uncertainty is extremely large (e.g., 50%), the pricing error is close to the sample mean. More interestingly, the changes in the posterior means are even smaller in the presence of state uncertainty. For example, the posterior mean of the expected excess return of AYE, l1jn, changes only slightly, from 5.79% to

5.67% under the prior Gð3; 0:5Þ of k, and from 5.83% to 5.86% under the prior Gð10; 0:5Þ of k, as the prior mispricing uncertainty increases from 3% to 5%. This result suggests that the findings of Pástor and Stambaugh (1999) can be generalized to the case with state uncertainty. On the other hand, the effects of state uncertainty are significant under all levels of mispricing uncertainty as the overestimation biases of the expected excess returns remain substantial. For example, the overestimation remains at 1–2% for AYE regardless of the prior mispricing uncertainty. Panel B in Table 5 reports the results under the Fama–French three-factor model. In general, the observations made above for the CAPM apply to the Fama–French model as well. In particular, the expected excess return is robust to mispricing uncertainty even in the presence of state uncertainty, and the overestimation bias

2585

Y. Han / Journal of Banking & Finance 36 (2012) 2575–2592 Table 3 Cost of equity for utility aye. Gð3;4Þ

N

Gð3;2Þ

Gð3;0:5Þ

Mean

Std. dev.

Mean

Std. dev.

Mean

Std. dev.

Mean

Std. dev.

lm,1jn bm,1jnlm,1jn

Panel A: CAPM 2.48 0.94 7.25 6.82

0.74 0.03 0.56 0.53

0.41 0.77 7.61 5.70

1.37 0.07 1.87 1.53

0.38 0.77 7.65 5.74

1.32 0.07 1.92 1.57

0.46 0.77 7.68 5.75

1.32 0.07 1.89 1.54

l1jn

9.30

0.91

6.11

1.94

6.11

1.93

6.22

1.90

b0p;1jn lp;1jn

Panel B: Fama–French three-factor model 0.71 0.72 0.05 0.89 0.03 0.77 0.25 0.05 0.29 0.49 0.04 0.35 7.27 0.57 6.50 2.59 0.35 1.17 4.41 0.38 2.11 9.26 0.61 6.03

1.21 0.04 0.06 0.05 2.14 1.10 1.16 1.98

0.09 0.78 0.29 0.35 6.59 1.23 2.24 6.16

1.18 0.04 0.06 0.05 2.19 1.08 1.18 2.03

0.17 0.78 0.29 0.36 6.49 1.28 2.32 6. 14

1.21 0.05 0.06 0.06 2.24 1.07 1.18 2.10

l1jn

9.97

2.28

6.24

2.28

6.31

2.35

a1jn bm,1jn

a1jn bm,1jn bs,1jn bh,1jn

lm,1jn ls,1jn lh,1jn

0.94

6.09

The means and standard deviations of the predictive distributions of the pricing error a1jn, risk loading on the market bm,1jn, risk loading on the SMB bs,1jn, risk loading on the   HML bh,1jn, the market risk premium lm,1jn, risk premium on the SMB ls,1jn, risk premium on the HML lh,1jn, stock risk premium bm;1jn lm;1jn b0p;1jn lp;1jn , and expected excess return l1jn are reported. Except for the moments of the risk loadings, all moments are reported as annualized percentage values. N denotes the state-independent pricing model. The sample size of the Gibbs sampling scheme is 10,000. The sample period is from 7/1926 to 12/2009.

persists at different levels of prior mispricing uncertainty. Furthermore, as we have seen before, the overestimation biases are larger under the Fama–French model than under the CAPM. 5.4. Compared to implied costs of equity We follow Pástor et al. (2008) and Gebhardt et al. (2001) to estimate the implied cost of equity capital. The sample period is limited by the data availability of Compustat and IBES and starts from 1980. We base the first 3 years’ earnings growth rates on analyst forecasts, and estimate the earnings growth rates from the fourth year and later by assuming a mean-reverting process for the growth rate between the third year and the steady-state growth rate, which is assumed to be the long-term growth rate. The long-term growth rate for individual firms is estimated using the 24-month moving average of the median growth rate of the industry portfolios. In the previous analysis, we use the full sample to estimate the MDP model and to predict 1-month ahead expected excess return. To compare to the implied costs of equity, we make several changes to the estimation procedure. First, we estimate the MDP model recursively, starting from January 1981. In other words, starting from December 1980, we estimate the MDP model each year with additional 12 month observations added to the sample. Second, For each estimation, we predict up to 12-month ahead expected excess returns, i.e., from l1jn up to l12jn, and convert the predicted monthly expected excess returns to annual expected excess returns. Table 6 reports the comparison of the expected annual excess returns predicted with the MDP model (lMdp) with the implied expected annual excess returns inferred from the implied costs of equity estimated from the RIV model (lIcc). The implied expected excess returns are estimated by subtracting the 1-year T-bill rate from the implied costs of equity. We report the comparison results for the utility stock (AYE), and randomly selected four industries, Chips, Trans, Rtail, and Fin.18 The mean of MDP expected excess return, lMdp, is always much higher than the mean of the implied ex18

Because recursive estimation takes much longer to run, we only estimate the utility stock and the four industries.

pected excess return, lIcc. For example, the expected excess return of AYE estimated from the MDP model is 6.21% per annum, compared to 5.97% per annum estimated from the RIV model. The difference is much larger for industry portfolios. On average, the MDP estimated expected excess returns are about 7%, whereas the RIV estimated expected excess returns are about two percent. More notably, the RIV estimated expected excess returns are much more volatile than are the MDP estimated expected excess returns. For example, the standard deviation of the MDP estimated expected excess return on AYE is about 0.20%, compared to 3.64% for the RIV estimated expected excess return. The MDP estimated expected excess returns are also less skewed, with less kurtosis, and with much smaller extreme values. Finally MDP estimated expected excess returns are more serially correlated with a first-order autocorrelation (q1) above 0.90, which is remarkable given that the model itself does not impose any serial correlation. Consistent with Easton and Monahan (2005), we find the correlations (qret) between impled costs of equity and realized returns are negative, whereas the correlations between MDP estimated expected returns and realized returns are positive. 6. The relation between the latent states and the real GDP growth rate In Section 4, we have shown that the market portfolio, Fama– French factor portfolios, individual stocks, and industry portfolios display state-dependent dynamics that seem to coincide with the business cycles identified by the NBER. However, unlike the twostate regime switching model, there is no clear-cut one-to-one correspondence between the states identified in the MDP model and the business cycles (recession and expansion). Therefore, we further examine the relation of the market with the real GDP growth rate.19 Fig. 6 compares the dynamics of the posterior means of the market risk premium to the dynamics of the real GDP growth rate. The market risk premium is positively correlated with the GDP growth rate. That is, consistent with the empirical evidence, low 19 The real quarterly GDP data are obtained from Federal Reserve Economic Data (FRED) hosted by St. Louis Federal Reserve Bank. We estimate the MDP model recursively every quarter and forecast the quarterly market returns.

2586

Y. Han / Journal of Banking & Finance 36 (2012) 2575–2592

Table 4 Cost of equity for industry portfolios in the Fama–French three-factor model. Fama–French three-factor model Industry

Agric Food Beer Smoke Toys Fun Books Hshld Clths MedEq Drugs Chems Txtls BldMt Cnstr Steel Mach ElcEq Autos Aero Ships Mines Coal Oil Util Telcm BusSv Comps Chips LabEq Boxes Trans Whlsl Rtail Meals Banks Insur RlEst Fin Other

Gð3;0:5Þ

N

a1jn

ra

blp

rblp

l1jn

rl

a1jn

ra

blp

rblp

l1jn

rl

0.01 0.04 0.58 0.46 0.02 0.42 0.03 0.11 0.25 0.32 0.31 0.14 0.17 0.12 0.00 0.13 0.08 0.46 0.19 0.95 0.18 0.16 0.77 0.24 0.24 0.32 0.17 0.61 0.26 0.29 0.21 0.27 0.48 0.01 0.12 0.48 0.10 0.63 0.24 0.60

0.28 0.16 0.26 0.26 0.37 0.27 0.25 0.18 0.21 0.22 0.19 0.15 0.23 0.15 0.30 0.22 0.14 0.18 0.22 0.31 0.24 0.23 0.39 0.20 0.19 0.18 0.27 0.22 0.25 0.22 0.18 0.18 0.21 0.16 0.21 0.22 0.23 0.34 0.18 0.23

7.93 7.79 8.03 7.72 8.33 8.43 8.11 7.83 7.88 7.84 7.82 7.99 8.32 8.14 8.47 8.41 8.24 8.18 8.24 8.35 8.25 8.03 7.94 7.88 7.82 7.67 7.90 7.98 8.33 7.87 7.92 8.23 8.16 7.91 7.99 8.06 8.12 8.40 8.31 8.04

0.59 0.58 0.60 0.57 0.62 0.63 0.61 0.59 0.59 0.59 0.59 0.59 0.62 0.61 0.62 0.62 0.61 0.60 0.61 0.62 0.61 0.60 0.59 0.59 0.58 0.57 0.60 0.59 0.62 0.60 0.59 0.61 0.61 0.59 0.60 0.61 0.60 0.62 0.61 0.60

7.94 7.83 8.61 8.18 8.35 8.84 8.08 7.73 7.63 8.17 8.13 8.13 8.15 8.02 8.47 8.28 8.33 8.65 8.43 9.30 8.07 8.19 8.71 8.12 7.58 7.36 8.06 8.60 8.58 8.15 8.13 7.96 7.69 7.92 8.10 8.54 8.22 7.77 8.55 7.44

0.65 0.60 0.65 0.63 0.72 0.68 0.66 0.61 0.63 0.63 0.62 0.61 0.66 0.62 0.69 0.65 0.63 0.63 0.65 0.69 0.65 0.64 0.71 0.62 0.61 0.60 0.66 0.63 0.67 0.64 0.62 0.64 0.65 0.61 0.63 0.65 0.64 0.71 0.63 0.64

0.12 1.77 1.13 2.03 0.67 1.03 0.46 0.89 0.72 1.43 2.09 0.16 1.32 0.74 1.89 1.65 0.53 1.28 0.34 1.55 1.56 0.15 1.11 0.58 0.77 0.89 0.71 2.31 0.04 1.92 1.37 2.37 1.15 0.98 0.73 1.34 0.34 3.65 0.48 0.04

1.66 0.77 1.51 1.44 1.94 1.57 1.48 1.00 1.20 1.34 1.04 0.89 1.26 0.87 1.97 1.29 0.83 1.05 1.35 2.54 1.60 1.50 2.55 1.19 1.00 0.81 1.52 1.31 1.40 1.25 1.24 1.03 1.28 0.91 1.24 1.72 1.47 2.11 1.01 0.26

6.18 5.01 6.48 5.75 7.44 7.12 7.04 6.27 6.51 6.58 5.43 7.61 8.14 7.85 8.60 8.71 7.79 7.01 7.85 7.87 7.50 7.33 7.81 6.87 5.71 4.18 6.93 6.89 8.08 6.15 6.58 7.99 7.32 6.27 7.04 6.79 6.96 8.56 8.33 7.12

1.92 1.86 1.96 1.66 2.77 3.06 2.22 1.84 1.83 2.12 2.23 2.19 2.40 2.32 3.04 2.54 2.83 2.98 2.57 2.51 2.35 2.03 2.12 1.85 1.75 1.87 2.12 2.30 3.18 2.48 2.14 2.48 2.25 2.08 2.12 2.21 2.19 2.95 2.76 2.23

6.29 6.78 7.61 7.78 6.77 8.14 6.58 7.15 5.78 8.01 7.52 7.77 6.82 7.11 6.70 7.07 7.26 8.29 7.51 9.42 5.94 7.18 8.92 7.44 6.48 5.06 7.64 9.20 8.11 8.07 7.95 5.62 6.17 7.25 7.78 8.14 6.62 4.91 7.85 7.07

2.48 2.04 2.49 1.97 3.30 3.34 2.58 2.07 1.97 2.60 2.47 2.33 2.58 2.28 3.69 2.80 2.81 3.00 2.74 3.55 2.56 2.49 3.27 2.25 1.98 1.94 2.50 2.60 3.62 2.50 2.70 2.44 2.49 2.22 2.56 2.31 2.58 3.94 2.92 2.25

For each industry portfolio, the table reports the means and standard deviations of the predictive distributions of the pricing error a1jn, the stock risk premium blp, and the expected excess return l1jn. All moments are reported as annualized percentage values. N denotes the state-independent model. The sample size of the Gibbs sampling scheme is 10,000. The sample period is from 7/1926 to 12/2009.

market risk premium associates with low GDP growth rate, and vice versa. In addition, the market risk premium seems to lead the GDP growth rate. In other words, changes of the market lead changes of the GDP growth rate. To confirm this observation, we next conduct Granger causality test on the market risk premium and GDP growth rate. We conduct Granger test with four lags given that we use the quarterly GDP growth rate and market returns. The results are reported in Table 7. Panel A of Table 7 shows that the realized market returns Granger cause the GDP growth rate as the market returns are significant with one, two, and three lags in the regression of the GDP growth rate. However, the GDP growth rate fails to Granger cause the market returns as the GDP growth rate is insignificant for any lags in the regression of the market returns. Panel B of Table 7 shows that the forecasted market risk premiums Granger cause the GDP growth rate as the forecasted market risk premiums are significant in the regression of the GDP growth rate.20 In contrast, the GDP growth rate fails to Granger cause the forecasted market risk premiums. 20 By definition, the forecasted market risk premiums are based on the returns up to the previous quarter (one lag).

We further compare the predictive power on the GDP growth rate of the forecasted expected returns on the Fama–French factors with the lagged GDP growth rate and realized factor returns. Table 8 shows that the adjusted R2 is 21.2% with one-quarter lagged GDP growth rate; adding the two-quarter lagged GDP grwoth rate increases the adjusted R2 to 31.5%. On the other hand, the onequarter lagged realized Fama–French factor returns have very low predictive power with an adjusted R2 about 6.6%. The forecasted factor expected returns, however, yield an adjusted R2 of 31.0%, similar to that with two lagged GDP growth rate, which is remarkable. In addition, even the one-quarter lagged forecasted factor expected returns produce an adjusted R2 of 21.5%. These results provide strong external evidence for the validity of the model because GDP growth rate is not used in the model, and also show the potential of the model in capturing the state of the economy and the market. 7. Conclusion Modeling time-varying parameters has become a crucial component of a well-specified empirical asset pricing model as more and more evidence shows that expected return, volatility, market beta, etc. change over time. We propose a novel Bayesian frame-

2587

Y. Han / Journal of Banking & Finance 36 (2012) 2575–2592

Fig. 5. Biases in the expected excess returns for industry portfolios under the Fama–French three-factor model. The biases of the posterior means of the expected excess returns between the state-independent pricing model and the MDP pricing model for the 40 industries are plotted in Panel A, and Panel B plots the biases in percentage of the true expected excess returns. The pricing model used is the Fama–French three-factor model. The posterior means of the expected excess returns are the averages of 10,000 predictive draws. The biases in Panel A are annualized and expressed in percentage values.

Table 5 Mispricing uncertainty versus state uncertainty. 3%

5% Gð3;0:5Þ

50%

Gð10;0:5Þ

N

Gð3;0:5Þ

Gð10;0:5Þ

lm,1jn bm,1jnlm,1jn

Panel A: CAPM 0.24 0.01 0.94 0.77 7.22 7.71 6.80 5.78

0.00 0.76 7.80 5.83

0.64 0.94 7.23 6.81

0.01 0.77 7.59 5.68

0.01 0.76 7.85 5.87

l1jn

7.04

5.83

7.44

5.67

b0p;1jn lp;1jn

Panel B: Fama–French three-factor model 0.08 0.00 0.01 0.89 0.78 0.77 0.25 0.29 0.29 0.49 0.35 0.35 7.24 6.57 6.70 2.61 1.19 1.17 4.41 2.14 1.94 9.24 6.11 6.14

0.20 0.89 0.25 0.49 7.20 2.58 4.41 9.20

l1jn

9.32

9.40

N

a1jn bm,1jn

a1jn bm,1jn bs,1jn bh,1jn

lm,1jn ls,1jn lh,1jn

5.79

6.11

6.15

Gð3;0:5Þ

Gð10;0:5Þ

3.66 0.94 7.24 6.80

2.66 0.77 7.90 5.94

2.24 0.76 8.00 6. 00

5.86

10.46

8.60

8.24

0.00 0.78 0.29 0.35 6.62 1.23 2.19 6.19

0.01 0.77 0.29 0.35 6.84 1.18 1.95 6.27

1.08 0.89 0.25 0.49 7.23 2.59 4.40 9.21

0.60 0.77 0.29 0.35 6.67 1.28 2.08 6.20

0.28 0.77 0.30 0.35 6.77 1.21 1.89 6. 20

6.19

6.26

10.29

6.80

6.48

N

This table reports results for three levels of prior uncertainty about the pricing error and two different specifications of prior state uncertainty. For each exercise, the means of the predictive distributions of the pricing error a1jn, risk loading on the market bm,1jn, risk loading on the SMB bs,1jn, risk loading on the HML bh,1jn, the market risk premium   lm,1jn, risk premium on the SMB ls,1jn, risk premium on the HML lh,1jn, stock risk premium bm,1jnlm,1jn b0p;1jn lp;1jn , and expected excess return l1jn are reported. All quantities except for the risk loadings are reported as annualized percentage values. N denotes the state-independent model. Panel A reports results for AYE under the CAPM, and Panel B reports the results under the Fama–French three-factor model. The sample size of the Gibbs sampling scheme is 10,000. The sample period is from 7/1926 to 12/2009.

work that models the time-varying parameters collectively and assumes the dynamics of these parameters depend on a latent state variable. The dynamics of the state and the state-dependent parameters are described by a mixture of Dirichlet process (MDP). A prominent feature of this framework is that the number

of states is determined endogenously by the data. Another important feature is that with a positive probability a new state can always emerge in future. We extend Pástor and Stambaugh’s (1999) Bayesian framework to incorporate state uncertainty into the CAPM and the Fama–

2588

Y. Han / Journal of Banking & Finance 36 (2012) 2575–2592

Table 6 Comparison between the state uncertainty Fama–French model and residual income valuation model. Variable

Mean

Std. dev

Median

P25

P75

Skewness

Kurtosis

Min

Max

q1

qret

Industry

lMdp lIcc lMdp lIcc lMdp lIcc lMdp lIcc lMdp lIcc

6.21 5.97 7.06 0.77 7.11 2.79 7.12 2.06 7.53 2.68

0.20 3.64 0.17 6.95 0.20 3.72 0.19 4.16 0.21 3.85

6.16 4.94 7.04 1.39 7.09 2.74 7.09 1.92 7.50 2.01

6.08 3.35 6.96 0.73 6.98 1.13 6.98 0.57 7.41 0.19

6.38 8.22 7.14 5.13 7.20 5.56 7.24 5.18 7.62 5.68

0.47 1.23 0.05 2.45 0.57 0.87 0.51 0.39 0.50 0.11

0.70 1.81 1.03 8.75 0.42 1.70 0.26 0.55 0.24 0.55

5.79 0.66 6.34 35.16 6.49 10.40 6.71 14.24 6.97 13.87

6.62 21.27 7.48 10.16 7.66 12.61 7.62 10.25 8.09 15.02

0.95 0.92 0.91 0.45 0.93 0.75 0.95 0.88 0.95 0.95

0.01 0.38 0.07 0.08 0.07 0.06 0.01 0.12 0.12 0.10

AYE AYE Chips Chips Trans Trans Rtail Rtail Fin Fin

This table reports the comparison results of the expected annual excess returns either forecasted from the Fama–French MDP model (lMdp) or inferred from the implied costs of equity estimated from the Residual Income Valuation (RIV) model (lIcc). The implied expected excess returns (lIcc) are estimated by subtracting the 1-year T-bill rate from the implied costs of equity. We report the summary statistics of the two estimates of the expected annual excess returns for the utility stock (AYE) and randomly selected four industries, Chips, Trans, Rtail, and Fin. The summary statistics reported are Mean, Standard Deviation (Std. dev.), Median, 25th percentile (P25), 75th percentile (P75), Skewness, Kurtosis, Minimum (Min), Maximum (Max), first order autocorrelation coefficient (q1), and the correlation coefficient with the realized returns (qret) over the sample period from January 1981 to December 2009. For each year, lMdp is the mean of the predictive distribution (draws) obtained after 12-step (12-month ahead) predictions in the recursive estimation of the MDP model. (See the text for the detailed description.) The sample size of the Gibbs sampling scheme is 10,000. The RIV model used to estimate the implied cost of equity is described in the text.

Expected Market Returns and Real GDP Growth Rate (%)

6

3.5

5

3

4 2.5 2

2

1

1.5

μm

Gdp

3

0 1 -1 0.5

-2

Return GDP

-3

0

01/52 01/58 01/64 01/70 01/76 01/82 01/88 01/94 01/00 01/06 Fig. 6. Relation of the forecasted market risk premium with the real GDP growth rate. The predictive distributions (formed from 10,000 draws) of the market quarterly risk premium are obtained each quarter from the recursive estimation of the MDP Fama–French three-factor model. The time-series of the means of the predictive distributions and the quarterly real GDP growth rate from 1947:q1 to 2009:q4 are plotted together. The means of the market risk premium are annualized and expressed in percentage values. The vertical lines in each graph indicate the business cycles identified by the NBER: solid lines are peaks and dashed lines are troughs.

French three-factor model. We find that expected excess returns, volatilities, pricing errors and risk loadings, all display businesscycle related dynamics. Some of the findings are consistent with the existing literature, and some are new findings. For example, we observe counter-cyclic behavior of the expected returns on both the SMB and HML portfolios, which is not documented before. Further investigation of the dynamics of the expected returns of the SMB and HML portfolios and their risk loadings may shed new light on the economic underpinning of the Fama–French three-factor model. We also find that there is a Granger causality relation between the forecasted market risk premium and the GDP growth rate, and the forecasted expected returns on the Fama–French three factors predict the GDP growth rate. This provides external validation to the latent states in the framework. When estimating the costs of equity for individual utility firms and industries, we find that the state-independent pricing models substantially overestimate the posterior means of the expected excess returns. In addition, the overestimation is insensitive to different prior beliefs about the number of states in the market and robust to different levels of prior mispricing uncertainty. State uncertainty also substantially increases the overall uncertainty of the expected excess returns. Both components, the pricing error a and stock risk premium b0 lp, contribute to the overestimation of the posterior means and increased uncertainties of the expected excess returns.

The framework introduced in this paper provides a convenient and efficient way of modeling a collection of time-varying parameters and has some unique features for incorporating state uncertainty. It can be easily applied to other interesting problems. For example, an investor certainly cares about state uncertainty, and her asset-allocation decisions likely depend on her beliefs about the future states of the market.

Acknowledgements The author thanks Siddhartha Chib, Alexander David, Heber Farnsworth, Jung-Wook Kim, Lˇuboš Pástor, Jonathan Taylor, Jian Yang, Guofu Zhou and seminar participants at Washington University in St. Louis and 2009 FMA Conference for their helpful comments. The author also would like to thank the anonymous referee and the editor (Ike Mathur) for their insightful comments that have significantly improved the paper.

Appendix A. Dirichlet process In a seminal paper, Ferguson (1973) defines a random process whose sample functions are almost surely probability measures.

2589

Y. Han / Journal of Banking & Finance 36 (2012) 2575–2592 Table 7 Granger causality test.

Table 8 Predicting real GDP growth rate using forecasted factor premiums.

Panel A: realized market return Rgdp,q

Rm,q

Rm,q1

0.559⁄⁄⁄ (3.64) 0.336⁄⁄⁄ (3.37) 0.106 (1.06) 0.081 (0.86) 0.003 (0.03) 1.715⁄⁄

0.028 (1.17) 0.021 (1.41) 0.022 (1.42) 0.015 (1.03) 0.008 (0.62) 0.030

Rm,q2

(2.59) 1.486⁄⁄

(0.30) 0.006

Rm,q3

(2.19) 1.324⁄

(0.06) 0.037

Rm,q4

(1.94) 0.226

(0.35) 0.023

(0.33)

(0.22)

0.379

0.024

Intercept Rgdp,q1 Rgdp,q2 Rgdp,q3 Rgdp,q4

Adj. R2

Panel B: forecasted market return

Intercept (1.33) Rgdp,q1 Rgdp,q2 Rgdp,q3 Rgdp,q4 Rfm;q Rfm;q1 Rfm;q2 Rfm;q3 adj. R2

Rgdp,q

Rgdp,q

Rfm;q

Intercept

2.304 (1.49) 0.359⁄⁄⁄ (3.76) 0.181⁄ (1.86) 0.071 (0.78) 0.046 (0.55) 533.300⁄⁄

0.001

Rgdp,q1

0.000 (1.25) 0.00 (0.47) 0.000 (0.09) 0.000 (0.33)

Rgdp,q2

Rgdp,q ⁄⁄⁄

0.747 (3.03) 0.434⁄⁄⁄ (2.58)

Rgdp,q ⁄⁄⁄

0.469 (3.05) 0.390⁄⁄⁄ (2.98) 0.255⁄⁄⁄ (3.00)

1. 271 (10.74)

0.636⁄⁄⁄

(0.80) 353.500

(6.31) 0.311⁄⁄⁄

(1.38) 487.800⁄⁄

(2.75) 0.017

(2.14)

(0.16)

0.392

0.802

Panel A reports the result of the Granger test between the realized market return (Rm,q) and the real GDP growth rate (Rgdp,q). We use four lagged variables, Rm,q1, Rm,q2, Rm,q3, Rm,q4 and Rgdp,q1, Rgdp,q2, Rgdp,q3, Rgdp,q4, respectively. Panel B reports the result of the Granger test between the forecasted market risk premium   Rfm;q and the real GDP growth rate. Because the forecasted market risk premium in quarter q is based on information available in quarter q  1, we include Rfm;q and   three lags Rfm;q1 ; Rfm;q2 ; Rfm;q3 in the regression. For each quarter, we recursively estimate the Fama–French MDP model, make 3-step ahead (3-month ahead) predictions, and obtain the predictive distribution (draws) for the quarterly market risk premium. Rfm;q is the mean of the quarterly predictive distributions after 10,000 draws. The t-statistics are included in the parentheses beneath the coefficients and significance at the 1% level, 5%, and 10% level is given by an ⁄⁄⁄, an ⁄⁄, and an ⁄, respectively. The sample period is from the first quarter of 1947 to the last quarter of 2009.

The random process is based on the discrete Dirichlet distribution (probability measure), and thus called the Dirichlet process. A random probability measure G is a Dirichlet process with parameter k and G0, denoted Dðk; G0 Þ, if for any measurable partition B1, . . . , Bm on the space of support of G0, the joint distribution of the random probabilities (G(B1), . . . , G(Bm)) is a Dirichlet distribution with a parameter vector (kG0(B1), . . . , kG0(Bm)). Antoniak (1974) extends the basic Dirichlet process to a mixture of Dirichlet processes, which is more suitable for Bayesian analysis. In the mixture of Dirichlet processes, the parameter k and/or the parameters in G0 are random as well, and follow a mixing distribution H() (e.g., priors). The two processes share many properties that are important for Bayesian analysis, and thus we will not differentiate the mixture of Dirichlet processes from the basic Dirichlet process, unless otherwise noted. A.1. Return generating process As assumed in the text, the excess returns rt follow a linear multifactor process:

10.680 (2.70)

Rgdp,q ⁄⁄⁄

8.049⁄⁄⁄ (3.14)

2.736⁄ (1.95) 0.167 (0.08) 2.074⁄ (1.74)

Rm,q1 Rsmb,q1 Rhml,q1

641.000⁄⁄⁄

Rfm;q

(2.48) 208.400

Rgdp,q ⁄⁄⁄

(3.17) 860.700⁄⁄⁄

Rfsmb;q

(4.20) 290.900⁄

Rfhml;q

(1.86) 486.500⁄⁄⁄

Rfm;q1

(3.67) 708.800⁄⁄⁄

Rfsmb;q1

(3.94) 204.400

Rfhml;q1

(0.91) Adj. R2

0.212

0.315

0.066

0.310

0.215

This table reports the results of forecasting the real GDP growth rate (Rgdp,q) using the lagged real GDP growth rates (Rgdp,q1 and Rgdp,q2), or using the one-quarter lagged Fama–French factor realized returns (Rm,q1, Rsmb,q1, and Rhml,q1), or using the forecasted Fama–French factor quarterly risk premiums from the MDP model   Rfm;q ; Rfsmb;q ; and Rfhml;q , or using the forecasted Fama–French factor quarterly risk   premiums at quarter q  1 Rfm;q1 ; Rfsmb;q1 ; and Rfhml;q1 . Newey and West (1987) autocorrelation and heteroscedasticity robust t-statistics with seven lags are reported in parentheses and significance at the 1% level, 5%, and 10% level is given by an ⁄⁄⁄, an ⁄⁄, and an ⁄, respectively. The forecasted factor quarterly risk premiums are the means of the quarterly predictive distributions obtained from the recursive estimation of the Fama–French MDP model with 3-month ahead predictions. The sample period is from the first quarter of 1947 to the last quarter of 2009.

where Dðk; G0 Þ is the Dirichlet process with parameter k and G0. Because of the discreteness of G() under the Dirichlet process, W  {wt, t = 1, . . . , n} lies in a set of k < n vectors of distinct values, denoted by H  {h1, . . . , hk}. The prior of k is implicitly determined and depends only on the precision parameter k and the number of periods n. For moderately large n, E[kjk,n] = k ln (1 + n/k) (Antoniak, 1974). Given n, when k is large, the prior mean of k is large and vice versa. Conversely, our prior belief about the number of states in the market can be used to form the prior for k. The prior of k is assumed as

k  Gða; cÞ;

ðA:3Þ

where a and c are the shape and inverse scale of the gamma distribution GðÞ. The base prior G0() is specified in the text as, G0 ðw; b0 ; R; m0 ; s0 ; l0 ; s; m; SÞ   r2 R IGðr2 ; m0 =2; s0 =2ÞN ðlp ; l0 ; X=sÞIWðX; m; SÞ: ¼ N b; b0 ; E½r2 

ðA:4Þ

rt ¼ at þ b0t r pt þ t ; rpt ¼ lpt þ gt ;   t  N 0; r2t ;

ðA:1Þ

gt  N ð0; Xt Þ;

  The state vector wt  at ; bt ; r2t ; lpt ; Xt has a random prior, G(),

  wt  at ; bt ; r2t ; lt ; Xt  G; G  Dðk; G0 Þ;

ðA:2Þ

A.2. State indicators and state-dependent parameters Conditioning on k, we introduce the state indicator st, such that st = j, if and only if wt = hj. Denote S = {st, t = 1, . . . , n}, which determines the assignment of the parameters and the excess returns to the k distinct states. One important feature of the Dirichlet process is the Polya Urn scheme representation introduced by Black-

2590

Y. Han / Journal of Banking & Finance 36 (2012) 2575–2592

well and MacQueen (1973). For clarity, let wt denote W without wt, and st denote S without st. Then the conditional prior is

wt jwt ; st ; k; k 

n X k 1 G0 ðwt Þ þ dw ðw Þ; kþn1 k þ n  1 i¼1;i–t i t

ðA:5Þ

where dwi ðwÞ denotes a unit point mass at w = wi. Because of the clustering structure, W is partitioned into k states, and each state has nj observations; i.e. nj = #{st = j}. The conditional prior reduces to a mixture of fewer components, 

wt jwt ; st ; k; k 

k X k 1 G0 ðwt Þ þ n dh ðw Þ; kþn1 k þ n  1 j¼1 j j t

ðA:6Þ

 where n j denotes the size of j-th state with wt removed; nst ¼ nst  1  and for other j; n ¼ n , and k denotes the number of states when wt j j  is removed. If n st ¼ 0; k ¼ k  1. The conditional prior suggests that at any period, wt is either an existing state drawn from the k existing states, or a new state drawn from the base prior G0(w), in which case the number of states k is updated accordingly, k = k + 1. To simplify notation, let yt denote a vector of excess returns on both the stock and the benchmark portfolios, yt = [rt;rpt], and Y denote the data available in the sample period, Y = {yt, t = 1, . . . , n}. It is easy to verify that the posterior of wt and st given the data Y, k, k, wt and st is

hj jS; k; k; Y  hj jY j 

Y

! f ðyt jhj Þ G0 ðhj Þ;

j ¼ 1; . . . ; k

where Yj = {yt:st = j}, the collection of excess returns in state j. Given S and H, W are determined by wt ¼ hst . A.3. Learning about the precision parameter The precision parameter k implicitly determines the number of distinct states in the market; larger k implies more distinct states and vice versa.22 Because the decision maker is uncertain about the number of states in the market, she expresses her uncertainty through the prior for k, and subsequently learns from the data about k and k and updates her beliefs accordingly. Given the number of states k, the posterior of k is independent of the data and all the other parameters. Under the assumption of a gamma prior, Gða; cÞ, West (1992) shows that the posterior of k can be sampled by data augmentation. Let n be a latent variable with conditional posterior

njk; k  Bðk þ 1; nÞ;

ðA:7Þ

j¼1

kjn; k  pGða þ k; c  lnðnÞÞ þ ð1  pÞGða þ k  1; c  lnðnÞÞ; ðA:13Þ

(

qt;j /

nj f ðyt jhj Þ if j > 0; kht ðyt Þ

if j ¼ 0;

where p is determined by

p

where the probabilities qt,j are given by

1p ðA:8Þ

ht ðyt Þ ¼

f ðyt jwt ÞdG0 ðwt Þ;

¼

aþk1 : nðc  lnðnÞÞ

A.4. Predicting the t-step ahead expected excess return

where ht(yt) is the marginal density of yt,

Z

ðA:12Þ

where BðÞ is a beta distribution. The conditional posterior of k is given by



k X wt jwt ; st ; k; k; Y  qt;0 Gt;0 ðwt Þ þ qt;j dhj ðwt Þ;

ðA:11Þ

t2fst ¼jg

The 1-month-ahead expected excess return l1jn can be obtained as

ðA:9Þ

l1jn ¼ a1jn þ b01jn lp;1jn :

ðA:14Þ

and Gt,0(wt) is the posterior obtained by updating the base prior G0(wt) via the likelihood function f(ytjwt), Gt,0(wt) / f(ytjwt)G0(wt). The posterior of the state indicator st is

To obtain the predictive distributions of a1jn, b1jn, and lp,1jn, note that the conditional prior of wn+1,

Pðst ¼ jjwt ; st ; k; k; YÞ ¼ qt;j :

wnþ1 jW; S; k; k 

ðA:10Þ

Therefore, the posterior state probability is proportional to the frequency of occurrence of the state and the likelihood of the state. Unlike in the regime-switching model, the state probability in the MDP model does not explicitly depend on the current state. For example, the prior probability of changing from state 1 to state 2 is the same as the prior probability of changing from state 2 to state 2 (see Eq. (A.6)). It may seem an undesirable feature, but what it means is that the model does not impose any a priori restrictions on the state probability, e.g., persistence of the states, and instead lets the data tell whether the states are persistent or not. If states are persistent, two highly correlated adjacent periods will have similar likelihood ordinates and will fall into the same state. Our empirical analysis confirms that the dynamics of the latent state are quite persistent. The precision parameter k plays an important role in both the prior and posterior of the state vector wt and state indicator st. It determines the probability that a new distinct state would appear. The larger k is, the more likely a new distinct state will appear. Another important feature of the Dirichlet process is that the distinct values of H are independent random samples from G0().21 Therefore, hj are conditionally independent with posterior densities 21 The state vectors wt, t = 1, . . . , n are nevertheless correlated. As we argue in the preceding text, two adjacent observations likely fall into the same state if they are highly correlated. We find that the posterior dynamics of the states are quite persistent, which can be seen from Table 6, for example.

k k nj X G0 ðwnþ1 Þ þ dh ðw Þ; kþn k þ n j¼1 j nþ1

ðA:15Þ

and

pðw1jn Þ  pðwnþ1 jYÞ ¼

Z

pðwnþ1 jW; S; k; kÞpðW; S; k; kjYÞdW:

ðA:16Þ

Thus we can obtain sample draws from the predictive density p(wn+1jY) by sampling from Eq. (A.15), given the sample draws from the posterior distribution p(W, S, k, kjY) via the method of composition. This suggests that the future state wn+1 is either chosen from the k existing states, or is a new state drawn from the base prior G0. The probability of picking from the existing state hj is proportional to the size of the state, nj, which suggests that the more frequently a state has happened before, the more likely it will happen in future. However, a positive probability that the future state is a new distinct state always exists, and the probability is proportional to the precision parameter k. In general, to forecast the expected excess return, ltjn, t > 1, note similarly,

22 The model degenerates to a state-independent model when k is infinitely large. See footNote 11.

2591

Y. Han / Journal of Banking & Finance 36 (2012) 2575–2592

Xj jY j  Wðm~; eSÞ;

þ

wnþt jWnþt1 ; S; k; k 

k X nþj k G0 ðwnþt Þ þ dh ðw Þ; kþnþt1 k þ n þ t  1 j¼1 j nþt

ðA:17Þ

and

pðwnþt jYÞ ¼

Z

pðwnþt jWnþt1 ; S; k; kÞpðWnþt1 ; S; k; kjYÞdWnþt1 ;

ðA:18Þ

where Wn+t1 denotes the collection of the parameters up to n + t  1, k+ and nþ j denote, respectively, the number of states and the size of state j up to n + t  1. Thus, given the posterior draws of W, S, k, k and predictive draws of wn+1, . . . , wn+t1, the predictive sample draws of wn+t can be obtained similarly by sampling from Eq. (A.17), and the sampling is done recursively.

~ ¼ m þ nj ; where m X ns e ðrpt  r p Þ0 ðr pt  r p Þ þ ðrp  l0 Þðr p  l0 Þ0 ; S ¼Sþ n þs t2fst ¼jg X r p ¼ rpt : t2fst ¼jg

ðB:4Þ  Sampling lpj from the conjugate Normal distribution.

lpj jXj ; Y j  N ðl~ ; X=s~Þ; ~ ¼ s þ nj ; where s

1 X s l~ ¼ ~ r þ l : s t2fst ¼jg pt s~ 0

ðB:5Þ

3. Sampling the precision parameter k with augmentation.  Sampling n given k.

Appendix B. Gibbs sampling scheme The Gibbs sampling scheme to estimate the mixture of Dirichlet process (MDP) model is described below; detailed discussion of the algorithm can be found in West et al. (1994), MacEachern and Müller (1994), Escobar and West (1998), and MacEachern and Müller (2000). For simplicity, we rewrite the excess returns on the stock 0 as r t ¼ bt xt þ t , where bt = (at; bt), and xt = (1; rpt).

njk; k  Bðk þ 1; nÞ:

ðB:6Þ

 Sampling k given n.

kjn; k  pGða þ k; c  lnðnÞÞ þ ð1  pÞGða þ k  1; c  lnðnÞÞ; where

p 1p

¼

aþk1 : nðc  lnðnÞÞ ðB:7Þ

1. Sampling st, for all t = 1, . . . , n given the data and all other parameters. The state indicator st is sampled marginalized over wt from

Pðst ¼ jjwt ; st ; k; YÞ ¼ qt;j ;

ðB:1Þ

where qt,j is given by Eq. (A.8). In particular, under the conjugate prior specifications, the marginal density of yt, ht(yt), is a multivariate student-t distribution. If n j equals zero, we relabel the last state k as the state j and let k = k  1. If st is sampled from a new state, we let k = k + 1 and draw a new wt from Gt,0(wt) and set hk = wt, otherwise, let k = k. 2. Sampling hj, j = 1, . . . , k given the data and all other parameters. The distinct values of hj are sampled from Eq. (A.11). As the posterior of hj contains the base prior G0() as the prior and likelihood function of Yj, it is conjugate with the prior G0. Specifically, aj ; bj ; r2j ; lpj , and Xj are sampled from the conjugate forms of normal, inverse gamma, multivariate normal, and inverse Wishart,respectively. The sample for each hj, j = 1, . . . , k is restricted to the observations whose state belongs to the distinct state j. To simplify notation, denote E[r2]R as R1.  Sampling r2j from the conjugate Inverse Gamma distribution.

r2j jY j  IGðm^=2; ^s=2Þ; ^ ¼ m0 þ nj ; where m ^s ¼ s0 þ

X

^0 xt Þ2 þ ðb ^  b0 Þ0 X 0 XðIqþ1  R ^  b0 Þ; e X 0 XÞðb ðr t  b

t2fst ¼jg

X0X ¼

X

xt x0t ;

e 1 ¼ R1 þ X 0 X; R 1

^ ¼ ðX 0 XÞ1 b

t2fst ¼jg

X

xt r t :

t2fst ¼jg

ðB:2Þ  Sampling bj from the conjugate Normal distribution.

  ~ r2 R e ; bj jr2j ; Y j  N b; j

~¼R e R1 b0 þ where b 1

X

!

ðB:3Þ

xt r t :

t2fst ¼jg

 Sampling Xj distribution.

from

the

conjugate

Inverse

Wishart

4. Predicting the t-step ahead expected excess return. Given the posterior draws of the parameters, we generate onestep ahead predictive draws of the parameters from Eq. (A.15) and obtain the one-step ahead predictive draws of the expected excess returns using Eq. (A.14). For t-step ahead predictive draws of the expected excess returns, the t-step ahead predictive draws of the parameters are obtained from the (t  1)-step ahead predictive draws iteratively using Eq. (A.17). We repeat each of the steps 10,500 times, and the posterior draws and predictive draws are collected after the first 500 burnin periods. References Adrian, T., Franzoni, F., 2005. Learning about Beta: Time-varying Factor Loadings, Expected Returns, and the Conditional CAPM. Working Paper, Federal Reserve Bank of New York. Andersen, T.G., Bollerslev, T., Diebold, F.X., Wu, J., 2006. Realized beta: Persistence and predictability. Advances in Econometrics 20, 1–39. Ang, A., Bekaert, G., 2007. Stock return predictability: is it there? Review of Financial Studies 20, 651–707. Ang, A., Chen, J., 2002. Asymmetric correlations of equity portfolios. Journal of Financial Economics 63, 443–494. Ang, A., Chen, J., 2007. CAPM over the long run: 1926–2001. Journal of Empirical Finance 14, 1–40. Ang, A., Liu, J., 2002. How to discount cashflows with time-varying expected returns. Journal of Finance 59, 2745–2783. Antoniak, C.E., 1974. Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems. Annals of Statistics 2, 1152–1174. Avramov, D., 2003. Stock Returns are Predictable: A Firm Level Analysis. Working Paper, University of Maryland. Avramov, D., Chordia, T., 2006. Predicting stock returns. Journal of Financial Economics 82, 387–415. Bakshi, G., Chen, Z., 2001. Stock Valuation in Dynamic Economies. Working Paper, Yale University. Barnes, M.L., Lopez, J.A., 2006. Alternative measures of the federal reserve banks’ cost of equity capital. Journal of Banking and Finance 30, 1687–1711. Basu, D., Stremme, A., 2007. CAPM and Time-Varying Beta: The Cross-section of Expected Returns. Working Paper, City University, London, University of Warwick. Blackwell, D., MacQueen, J.B., 1973. Ferguson distribution via Polya Urn schemes. Annals of Statistics 1, 353–355. Bollerslev, T., Engle, R.F., Nelson, D.B., 1994. ARCH models. In: Engle, R.F., McFadden, D. (Eds.), Handbook of Econometrics, vol. IV. Elsevier, Amsterdam, NorthHolland, pp. 2959–3038. Bossaerts, P., Hillion, P., 1999. Implementing statistical criteria to select return forecasting models: What do we learn? Review of Financial Studies 12, 405–428. Botosan, C.A., Plumlee, M.A., 2005. Assessing alternative proxies for the expected risk premium. Accounting Review 80, 21–53.

2592

Y. Han / Journal of Banking & Finance 36 (2012) 2575–2592

Bower, D.H., Bower, R.S., Logue, D.E., 1984. Arbitrage pricing theory and utility stock returns. Journal of Finance 39, 1041–1051. Bower, R.S., Schink, G.R., 1994. Application of the Fama–French model to utility stocks. Financial Markets, Institutions and Instruments 3, 74–96. Brennan, M.J., 1997. The term structure of discount rates. Financial Management 26, 81–90. Campbell, J.Y., Vuolteenaho, T., 2004. Bad beta, good beta. American Economic Review 94, 1249–1275. Chauvet, M., Potter, S., 2001. Nonlinear risk. Macroeconomic Dynamics 5, 621–646. Chib, S., Hamilton, B.H., 2002. Semiparametric Bayes analysis of longitudinal data treatment models. Journal of Econometrics 110, 67–89. Claus, J., Thomas, J., 2001. Equity premia as low as three percent? Evidence from analysts’ earnings forecasts for domestic and international stock markets. Journal of Finance 56, 1629–1666. Conley, T.G., Hansen, C.B., Mcculloch, R.E., Rossi, P.E., 2008. A semi-parametric Bayesian approach to the instrumental variable problem. Journal of Econometrics 144, 276–305. Cooper, M.J., Gulen, H., 2006. Is time-series based predictability evident in realtime? Journal of Business 79, 1263–1292. Easton, P.D., Monahan, S.J., 2005. An evaluation of accounting-based measures of expected returns. Accounting Review 80, 501–538. Elton, E.J., Gruber, M.J., Mei, J., 1994. Cost of capital using arbitrage pricing theory: a case study of nine New York utilities. Financial Markets, Institutions and Instruments 3, 46–73. Escobar, M.D., West, M., 1998. Computing nonparametric hierarchical models. In: Dey, D., Muller, P., Sinha, D. (Eds.), Practical Nonparametric and Semiparametric Bayesian Statistics. Springer, New York, pp. 1–22. Fama, E.F., 1996. Discounting under uncertainty. Journal of Business 69, 415–428. Fama, E.F., French, K.R., 1989. Business conditions and expected returns on stocks and bonds. Journal of Financial Economics 25, 23–49. Fama, E.F., French, K.R., 1993. Common risk factors in the returns on stocks and bonds. Journal of Financial Economics 33, 3–56. Fama, E.F., French, K.R., 1995. Size and book-to-market factors in earnings and returns. Journal of Finance 50, 131–155. Fama, E.F., French, K.R., 1997. Industry costs of equity. Journal of Financial Economics 43, 153–193. Ferguson, T.S., 1973. A Bayesian analysis of some nonparametric problems. Annals of Statistics 1, 209–230. Ferson, W.E., 1989. Changes in expected security returns, risk, and the level of interest rates. Journal of Finance 44, 1191–1217. Ferson, W.E., Harvey, C.R., 1999. Conditioning variables and the cross section of stock returns. Journal of Finance 54, 1325–1360. Ferson, W.E., Locke, D.H., 1998. Estimating the cost of capital through time: an analysis of the sources of error. Management Science 44, 485–800. Ferson, W.E., Sarkissian, S., Simin, T., 2003. Spurious regressions in financial economics. Journal of Investment Management 1, 1–10. Ferson, W.E., Siegel, A.F., 1999. Stochastic Discount Factor Bounds with Conditioning Information. Working Paper, University of Washington. Gebhardt, W.R., Lee, C.M.C., Swaminathan, B., 2001. Toward an implied cost of capital. Journal of Accounting Research 39, 135–176. Geweke, J., Keane, M., 2007. Smoothly mixing regressions. Journal of Econometrics 138, 252–290. Ghysels, E., 1998. On stable factor structures in the pricing of risk: do time-varying betas help or hurt? Journal of Finance 53, 549–573. Ghysels, E., Harvey, A.C., Renault, E., 1995. Stochastic volatility. In: Maddala, G.S., Rao, C.R. (Eds.), Handbook of Statistics, vol. 14. Elsevier, Amsterdam, NorthHolland, pp. 119–191. Goetzmann, W.N., Jorion, P., 1993. Testing the predictive power of dividend yields. Journal of Finance 48, 663–679. Goyal, A., Welch, I., 2003. Predicting the equity premium with dividend ratios. Management Science 49, 639–654. Guo, H., 2006. The risk-return relation in international stock markets. Financial Review 41, 565–587. Guo, H., Savickas, R., Wang, Z., Yang, J., 2009. Is the value premium a proxy for timevarying investment opportunities? Some time-series evidence. Journal of Financial and Quantitative Analysis 44, 133–154.

Han, Y., 2010. On the economic value of return predictability. Annals of Economics and Finance 11, 1–33. Harvey, C.R., 2001. The specification of conditional expectations. Journal of Empirical Finance 8, 573–638. He, J., Kan, R., Ng, L., Zhang, C., 1996. Tests of the relations among market-wide factors, firm-specific variables, and stock returns using a conditional asset pricing model. Journal of Finance 60, 1891–1908. Hirano, K., 2002. Semiparametric Bayesian inference in autoregressive panel data models. Econometrica 70, 781–799. Jensen, M.J., 2004. Semiparametric Bayesian inference of long-memory stochastic volatility models. Journal of Time Series Analysis 25, 895–922. Kacperczyk, M., Damien, P., Walker, S.G., 2005. A New Class of Bayesian Semiparametric Models with Applications to Option Pricing. Working paper, University of British Columbia. Karolyi, A.G., Stulz, R.M., 1996. Why do markets move together? An investigation of US–Japan stock return comovements. Journal of Finance 51, 951–986. Kacperczyk, M.T., Damien, P., 2011. Asset Allocation Under Distribution Uncertainty. Working paper, University of Texas at Austin. Kim, S., Shephard, N., Chib, S., 1998. Stochastic volatility: optimal likelihood inference and comparison with ARCH models. Review of Economic Studies 65, 361–393. Kirby, C., 1997. Measuring the predictable variation in stock and bond returns. Review of Financial Studies 10, 579–630. Koedijk, K.G., Kool, C.J.M., Schotman, P.C., van Dijk, M.A., 2001. The Cost of Capital in International Financial Markets: Local or Global? Working Paper. Kothari, S.P., Shanken, J.A., Sloan, R.G., 1995. Another look at the cross-section of expected stock returns. Journal of Finance 50, 185–224. Lee, C.M.C., Ng, D., Swaminathan, B., 2009. Testing international asset pricing models using implied costs of capital. Journal of Financial and Quantitative Analysis 44, 307. Lewellen, J., Nagel, S., 2006. The conditional CAPM does not explain asset-pricing anomalies. Journal of Financial Economics 82, 289–314. Lin, W.L., Engle, R.F., Ito, T., 1994. Do bulls and bears move across borders? International transmission of stock returns and volatility. Review of Financial Studies 7, 507–538. Longin, F., Solnik, B., 2001. Extreme correlation of international equity markets. Journal of Finance 56, 649–676. MacEachern, S.N., Müller, P., 1994. Efficient Estimation of Mixture of Dirichlet Process Models. Discussion Paper 94-38, ISDS, Duke University. MacEachern, S.N., Müller, P., 2000. Efficient MCMC schemes for robust model extensions using encompassing Dirichlet process mixture models. In: Insua, D.R., Ruggeri, F. (Eds.), Robust Bayesian Analysis. Springer-Verlag, New York, pp. 295–315. Neely, C.J., Weller, P., 2000. Predictability in international asset returns: a reexamination. Journal of Financial and Quantitative Analysis 35, 601–620. Nelson, C.R., Kim, M.J., 1993. Predictable stock returns: the role of small sample bias. Journal of Finance 48, 641–661. Pástor, L., Sinha, M., Swaminathan, B., 2008. Estimating the intertemporal riskreturn tradeoff using the implied cost of capital. Journal of Finance 63, 2859– 2897. Pástor, L., Stambaugh, R.F., 1999. Costs of equity capital and model mispricing. Journal of Finance 54, 67–121. Tiwari, R.C., Jammalamadaka, S.R., Chib, S., 1988. Bayes prediction density and regression estimation – a semiparametric approach. Empirical Economics 13, 209–222. Welch, I., Goyal, A., 2008. A comprehensive look at the empirical performance of equity premium prediction. Review of Financial Studies 21, 1455–1508. West, M., 1992. Hyperparameter Estimation in Dirichlet Process Mixture Models. Discussion Paper #92-A03, ISDS, Duke University. West, M., Müller, P., Escobar, M.D., 1994. Hierarchical priors and mixture models, with application in regress and density estimation. In: Smith, A.F.M., Freeman, P.R. (Eds.), Aspects of Uncertainty: A Tribute to D.V. Lindley. John Wiley and Sons, New York, New York, pp. 363–386. Zarepour, M., Bédard, T., Dabrowski, A.R., 2008. Return and value at risk using the Dirichlet process. Applied Mathematical Finance 15, 205–218. Zhang, L., 2005. The value premium. Journal of Finance 60, 67–103.