The intertemporal risk-return relation: A bivariate model approach

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Journal of Financial Markets ] (]]]]) ]]]–]]] www.elsevier.com/locate/finmar

The intertemporal risk-return relation: A bivariate model approach$ Xiaoquan Jianga,n, Bong-Soo Leeb,1 a

Department of Finance, College of Business, Florida International University, Miami, FL 33199, USA b Department of Finance, College of Business, Florida State University, Tallahassee, FL 32306, USA Received 12 August 2011; received in revised form 13 February 2013; accepted 13 February 2013

Abstract This paper examines the intertemporal risk-return relation using a more sensible empirical specification that is motivated by two concerns: the theoretical risk-return relation is an ex ante relation and the empirical method used to detect the relation should be reliable. We measure both the expected excess return and conditional variance jointly using the common information set based on a bivariate moving average representation of excess returns and variances. As a result, we can detect a significant positive relation between the expected excess return and the conditional variance. We also find that the positive relation is robust. Published by Elsevier B.V. JEL classifications: G12; C32; E32 Keywords: Intertemporal risk-return relation; Bivariate moving average representation

$ We thank the anonymous referee and Bruce Lehmann (the Editor) for their insightful comments and suggestions which contributed substantially to the quality of this paper. We remain responsible for all errors. n Corresponding author. Tel.: þ1 305 348 7910; fax: þ1 305 348 4245. E-mail addresses: jiangx@fiu.edu (X. Jiang), [email protected] (B.-S. Lee). 1 Tel.: þ1 850 644 4713; fax: þ1 850 644 4225.

1386-4181/$ - see front matter Published by Elsevier B.V. http://dx.doi.org/10.1016/j.finmar.2013.02.002 Please cite this article as: Jiang, X., Lee, B.-S., The intertemporal risk-return relation: A bivariate model approach. Journal of Financial Markets (2013), http://dx.doi.org/10.1016/j.finmar.2013.02.002

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1. Introduction Intertemporal risk-return relation is a fundamental concept in finance. Merton’s (1973) elegant Intertemporal Capital Asset Pricing Model (ICAPM) predicts a positive relation. However, the empirical evidence on the intertemporal risk-return relation is at most mixed. Prior studies tend to employ three different approaches to detect a positive intertemporal risk-return relation: focusing on the measurement of conditional variance and expected return, the model specification, and the small sample issue. First, some studies focus on the measure of the conditional market variance, showing that the relation is highly sensitive to the way the conditional variance is measured. French, Schwert, and Stambaugh (1987) use Autoregressive Integrated Moving Average (ARIMA) and Autoregressive Conditional Heteroskedastic in Mean (ARCH-M) models; Glosten, Jagannathan, and Runkle (1993) employ an asymmetric GARCH model; and Ghysels, Santa-Clara, and Valkanov (2005) develop a proxy for conditional market volatility called ‘‘MIDAS.’’ Recently, researchers tend to focus on the measure of the expected return and show that it is no less important in the relation (e.g., Brav, Lehavy, and Michaely, 2005; Pastor, Sinha, and Swaminathan, 2008; Campello, Chen, and Zhang, 2008; Jiang and Lee, 2009).2 Second, Scruggs (1998) and Guo and Whitelaw (2006) emphasize the importance of the empirical specification that includes a hedge component, which is consistent with Merton’s (1973) original ICAPM specification. Third, Ludvigson and Ng (2007) and Lettau and Ludvigson (2009) argue that the disagreement in the empirical risk-return relation is likely to be attributable to the relatively small amount of conditioning information that is typically used to model the conditional mean and conditional volatility of excess stock market returns. Lundblad (2007) also argues that the primary challenge in estimating the intertemporal risk-return relation is due to small samples.3 We examine the intertemporal risk-return relation motivated by the following two observations. First, in investigating the intertemporal risk-return relation, it is important to make sure that the measures of both the expected excess return and conditional variance of returns are based on the same common information set for consistency of the measurement of the two variables, which is implicitly embedded in Merton’s (1973) theoretical specification. Previous studies in the literature tend to recognize this. However, in our view, previously employed econometric methods might not necessarily be very suitable in this respect. They tend to measure the conditional variance and expected returns separately. For instance, many prior studies focus on the measure of conditional variance but still employ the realized return as measures of the theoretical expected return (For example, French, Schwert, and Stambaugh, 1987; Nelson, 1991; Chan, Karolyi, and Stulz, 1992; Glosten, Jagannathan, and Runkle, 1993; Scruggs, 1998; Ghysels, Santa-Clara, and Valkanov, 2005; Lundblad, 2007). Other studies focus on the measure of expected returns while using the lagged realized variance.4 2

Jiang and Lee (2009) examine the intertemporal risk-return relation empirically. Instead of measuring expected return and conditional variance using common information, they use fundamentals as a proxy for the expected return. 3 Rossi and Timmermann (2009) find a non-monotonic risk-return relation and claim that it helps resolve the disagreement on the empirical risk-return relation. Schwert (1989) argues that a stronger risk aversion in the boom period implies a higher excess stock return and compensation for risk in the expansionary period. 4 See, for example, Pastor, Sinha, and Swaminathan (2008). Harvey (2001) states, ‘‘While economic theory tells us how to link conditional expectations with conditional risk and reward, it does not tell us how the conditional Please cite this article as: Jiang, X., Lee, B.-S., The intertemporal risk-return relation: A bivariate model approach. Journal of Financial Markets (2013), http://dx.doi.org/10.1016/j.finmar.2013.02.002

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Second, prior studies find that the estimate of conditional variance is very persistent, so the statistical inference can be unreliable. Several studies including Nelson and Kim (1993), Stambaugh (1986, 1999), Valkanov (2003), and Ang and Bekaert (2007) show that the statistical inference can be misleading if the regressor is highly persistent. Ferson, Sarkissian, and Simin (2003, 2008) refer to this as a spurious regression, showing that the persistence or high autocorrelation of a regressor tricks the standard test statistics into finding a significant relation where none exists (see also Petersen, 2009). The risk-return relation estimated by GARCH models seems sensitive to model specification and sample period. For example, French, Schwert, and Stambaugh (1987) find positive risk-return relation using ARCH-M. Glosten, Jagannathan, and Runkle (1993) find a negative and significant risk-return relation using a Threshold (asymmetric) GARCH-M (TGARCH-M). Lundblad (2007) argues that using the popular GARCH-M framework, the empirical evidence measured over the past 50 to 75 years of the U.S. market portfolio regarding this relationship is, at best, weak. To address the first concern, we use a bivariate time-series model of excess return and variance to determine information-consistent measures. Due to the law of iterated expectations, the measures of the expected excess return and conditional variance based on the bivariate model provide us with simple but information-consistent measures of the two variables. That is, both the expected return and conditional variance are measured jointly using a common information set.5 There are two representations of the bivariate time-series model: the bivariate autoregressive representation (BARR) and the bivariate moving average representation (BMAR) under the regularity assumption of the BMAR being invertible. To alleviate the second concern, we use the measures based on the BMAR of the two variables while still providing comparisons of its performance with that based on the BARR.6 Recently, Pastor and Stambaugh (2009) develop a framework for estimating expected returns, referred to as a predictive system, which allows predictors to be imperfectly correlated with the conditional expected return. When predictors are imperfect, the estimated expected return depends on past returns in a manner that depends on the correlation between unexpected returns and innovations in expected returns. Our approach of using a BMAR model to measure the expected return and conditional variance is similar in spirit to Pastor and Stambaugh (2009) in that the BMAR model includes past values of innovations (i.e., unexpected changes) in both market returns and market return variances. We also employ a long data sample period and subsample period analysis as a robustness check. With the measures of the expected excess return and conditional variance based on the same common information set, we detect a significant positive intertemporal risk-return (footnote continued) expectations are generated.’’ Pastor, Sinha, and Swaminathan (2008) estimate the intertemporal risk-return tradeoff using the Implied Cost of Capital (ICC) as a proxy for expected market return and find evidence of a positive relation between the conditional mean and variance of market returns in the G-7 countries, at both the country level and the world market level. 5 Campbell (1987), Whitelaw (1994), and Brandt and Kang (2004) use common information in constructing the measures of the conditional mean and variance of stock returns. In measuring the first two conditional moments, Campbell (1987) and Brandt and Kang (2004) use a latent variables approach while Whitelaw (1994) uses a state variables approach. 6 Further evidence on the rationale for choosing BMAR over BARR is provided in Section 2 and 5. Jiang and Lee (2006) investigate the dynamic relation between market returns and idiosyncratic risk using a univariate moving average approach to measure the conditional idiosyncratic volatility. Please cite this article as: Jiang, X., Lee, B.-S., The intertemporal risk-return relation: A bivariate model approach. Journal of Financial Markets (2013), http://dx.doi.org/10.1016/j.finmar.2013.02.002

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relation for each sample period we consider. We find that the positive relation is robust with respect to different time interval-based measures of realized market variance, to the use of different return data (e.g., the S&P index data and the CRSP index data), and to the inclusion of the hedge component in the relation. Ghysels, Santa-Clara, and Valkanov (2005) also study the intertemporal relation between the conditional mean and the conditional variance of the aggregate stock market returns and find a positive relation between risk and return in the stock market. They obtain the result by using a new estimator that forecasts monthly variance with a weighted average of lagged daily squared returns, the mixed data sampling (or MIDAS) approach, which is different from GARCH models. They use a flexible functional form to parameterize the weight given to each lagged daily squared return. Therefore, their focus is on a new flexible, improved measure of the conditional variance that takes into account its persistence, while our paper focuses on the common information for the joint estimation of the conditional mean and the conditional variance. Both the MIDAS approach in Ghysels, Santa-Clara, and Valkanov (2005) and our simple BARR and BMAR methods incorporate past conditional information in estimating conditional variances. We also measure the conditional mean using the same, common information and address the persistence of the conditional variance based on the relation between bivariate AR and bivariate MA representations.7 This paper is organized as follows. In Section 2, we introduce the intertemporal capital asset pricing model, which is used as a basis for the empirical estimation of the intertemporal risk-return relation, and discuss how to estimate the relation with appropriate measures of the expected excess return and conditional variance. In Section 3, we present the data and descriptive statistics; and in Section 4, we report various estimation results of the intertemporal risk-return relation. Section 5 is the conclusion. 2. Intertemporal risk-return relation Merton (1973, 1980), in his seminal ICAPM, shows that the expected excess market return, E[(RM, tRf, t)9Ot1], is positively related to the conditional market volatility, E(s2M,t9Ot1). His model can be stated, ignoring a hedge component, as follows:8 E½ðRM,t 2Rf ,t ÞjOt1  ¼ m þ gEðs2 M,t jOt1 Þ

ð1Þ

where E denotes the expectations operator, Ot1 denotes the information set available at time t1, and g is a parameter reflecting the relative risk aversion. Both sides of the equation are based on the same information set, Ot1. This suggests that the intertemporal risk-return relation can be misleading if the measures of the expected excess return and conditional variance are not based on the common information. 7 This study is different from Brandt and Kang (2004) in several dimensions. We use a bivariate model of realized returns and volatility as in the standard linear VAR framework to compute the one-step-ahead forecasts of expected excess returns and volatilities. Brandt and Kang (2004) use a bivariate model of the logged conditional mean and volatility in the framework of a latent VAR process. We compute conditional mean and volatility of stock returns as the one-step-ahead forecasts of returns and volatilities as in the standard linear bivariate VAR framework. Brandt and Kang (2004) rely on a sophisticated simulation-based method for nonlinear and nonGaussian state-space models. However, recently Cochrane (2008) shows that the latent VAR model does not necessarily lead to a better prediction using the example of dividend yield model. 8 We discuss the relation with the hedge component in section IV.F.

Please cite this article as: Jiang, X., Lee, B.-S., The intertemporal risk-return relation: A bivariate model approach. Journal of Financial Markets (2013), http://dx.doi.org/10.1016/j.finmar.2013.02.002

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By collecting terms, we can rewrite Eq. (1) as follows: E½fðRM,t 2Rf ,t Þmgs2 M,t gjOt1  ¼ 0:

ð2Þ

It is almost impossible for econometricians to reproduce the information set used by the economic agents, Ot1, to estimate the parameter g and test the intertemporal risk-return relation. However, according to the law of the iterated expectations, econometricians can effectively estimate the parameter g using a smaller information set It1, which is available to the econometricians and is a subset of Ot1 (i.e., It1  Ot1) as long as it is used consistently. Therefore, using a smaller information set It1, Eq. (2) can be rewritten as: E½fðRM,t 2Rf ,t Þmgs2 M,t gjIt1  ¼ 0,

ð3Þ

where It1 is the information set available to econometricians and is a subset of Ot1. Although the theoretical intertemporal risk-return relation is based on the expected excess return and conditional variance of returns, many empirical studies employ the mean and variance of realized returns as proxies for the theoretical variables. This is partly because both the expected market return and the conditional variance are unobservable, and the information surprises tend to cancel out over time so the average of realized market return is an unbiased estimate of expected returns. However, as Elton (1999) in his AFA presidential speech indicates, this intuition can be misleading.9 For instance, during the period from 1927 to 1981 (longer than 50 years), risky long-term bonds, on average, underperformed the risk-free rate. In estimating and testing the intertemporal risk-return relation, previous research takes diverse approaches in measuring the expected return and conditional variance (e.g., French, Schwert, and Stambaugh, 1987; Ghysels, Santa-Clara, and Valkanov, 2005; Campello, Chen, and Zhang, 2008; Pastor, Sinha, and Swaminathan, 2008; Jiang and Lee, 2009). The key issue is how to find a reasonable and information-consistent measure of the expected excess return and conditional variance. We contribute to the literature by proposing a time-series model of jointly measuring the expected excess return and conditional variance using the same common information set. In Eq. (3), It1 is the information set available to econometricians and is a subset of Ot1. A natural candidate for this information set is the past values of excess returns and variances: (RM,tsRf, ts) and s2M,ts, for s ¼ 1, 2,y.. Further, for any stationary stochastic process, it has a Moving Average Representation (MAR) by the Wold representation theorem, and it has an Autoregressive Representation (ARR) as long as it is invertible, which is assumed here. As discussed above, the main motivation for using the past values of excess returns and variances as the information set is the so-called the law of iterated expectations. The law of iterated expectation allows us to use a smaller set, still maintaining the risk-return relation. Given that our main focus is to compute expected returns and conditional variances based on a set of information, a natural candidate for the information set in this case would be the past values of excess returns and variances. This choice seems particularly reasonable for our purpose in that we use a bivariate model to ensure that the same information is 9

Campello, Chen, and Zhang (2008) also provide evidence showing that the average realized return does not converge to the expected return. Mayfield (2004) suggests that the simple historical average of excess market volatility obscures significant variation in the market risk premium and that about half of the measured risk premium is associated with the risk of future changes in the level of market volatility. Please cite this article as: Jiang, X., Lee, B.-S., The intertemporal risk-return relation: A bivariate model approach. Journal of Financial Markets (2013), http://dx.doi.org/10.1016/j.finmar.2013.02.002

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used for the expectation of the two variables and that conditional variances tend to be persistent. Therefore, we consider the following BARR of Zt that includes (RM, tRf, t ) and s2M,t: 1 X As Zts þ ut ¼ AðLÞZt1 þ ut , or Zt ¼ s¼1

"

2X 1

a11s # " # 6 RM,t Rf ,t Z1t 6s¼1 6 ¼ ¼6X 1 s2 M,t Z2t 4 a21s s¼1

3 a12s 7" # " # u1t 7 Z1ts s¼1 7 1 7 Z2ts þ u2t X a22s 5 1 X

ð4Þ

s¼1

0 where Zt is a bivariate vector containing (RM, tRf, t) and s2M,t {i.e., Zt ¼ 1t, Z2t] ¼  [(RM, P[Z 1 2 0 s tRf, t ), sM,t] }; A(L) is a polynomial in the lag operator L [ i:e:,AðLÞ ¼ s ¼ 0 As L ], with L being a lag (or backshift) operator (i.e., Ln Xt ¼ Xtn); A is a 2  2 coefficient matrix (i.e., A ¼ [aij] for i, j¼ 1, 2); and, ut is a bivariate error term {i.e., ut ¼ [u1t, u2t,]0 }. Once we obtain estimate of the above BARR of Zt, we can invert it to a BMAR of Zt: 1 X BS uts , ð5Þ Zt ¼ BðLÞut ¼ s¼0

where B(L)¼ [IA(L)L]1 with I being the identity matrix, and B is a 2  2 coefficient matrix {i.e., B ¼ [bij] for i, j¼ 1, 2}. Two measures of the expected excess return, E[(RM,tRf,t) 9It1], and the conditional variance, E(s2M,t 9 It1), are available from the BARR (Eq. (4)) and BMAR (Eq. (5)), respectively, by taking conditional expectations of the above representations, one based on the BARR (Eq. (4)) and the other based on the BMAR (Eq. (5)): 10 1 X EðZt jIt1 Þ ¼ AðLÞZt1 ¼ As Zts , and ð6Þ s¼1

EðZt jIt1 Þ ¼ BðLÞut1 ¼

1 X

Bs uts :

ð7Þ

s¼1

A simpler proxy for EðZt jIt1 Þwould be to use only one lagged Zt instead of many lagged Zt s in the above BARR (Eq. (6)). This type of measure is often used in prior studies but in a univariate model context. A practical question is which of the above two measures should be used — one based on BARR (Eq. (6)) or the other based on BMAR (Eq. (7)) — for the expected excess return and conditional variance. In general, both are deemed legitimate proxies. However, here we are concerned with the intertemporal relation between risk and return, and the risk measure tends to be quite persistent (i.e., s2M,t is strongly serially correlated). Under this circumstance, using the proxy based on the BMAR is preferred because the innovations in s2M,t are serially uncorrelated by construction and the persistence of conditional variances tends to be lower (see Table 1). As a result, the ICAPM regression becomes more balanced 10 Eqs. (6) and (7) stem from the fact that Eðut jIt1 Þ ¼ 0 by construction and Et ðuts jIt1 Þ ¼ uts for any s Z1. One way to understand the equivalence of the two representations is that the Hilbert space spanned by past Zt s and the one spanned by past innovations uts are equivalent.

Please cite this article as: Jiang, X., Lee, B.-S., The intertemporal risk-return relation: A bivariate model approach. Journal of Financial Markets (2013), http://dx.doi.org/10.1016/j.finmar.2013.02.002

Lag of Autocorrelation

Rm ER1 ER2 MV CV1 CV2 CV3 CV4

Mean x100

Variance x100

Skewness

Kurtosis

0.4010 0.4010 0.4040 0.1610 0.1620 0.1610 0.1610 0.1620

0.1710 0.0220 0.0213 0.0011 0.0007 0.0011 0.0010 0.0006

0.593 1.885 2.087 8.299 8.617 8.351 8.240 7.693

17.625 34.516 37.294 80.441 93.633 81.605 80.207 79.951

1

2

3

4

5

6

7

8

9

10

11

12

0.285 0.181 0.194 0.958 0.908 0.956 0.958 0.898

0.006 0.106 0.017 0.914 0.823 0.910 0.911 0.805

0.049 0.135 0.157 0.867 0.709 0.865 0.867 0.685

0.008 0.127 0.049 0.822 0.592 0.815 0.811 0.565

0.067 0.115 0.005 0.771 0.481 0.764 0.762 0.451

0.029 0.038 0.019 0.720 0.375 0.714 0.709 0.345

0.028 0.044 0.015 0.667 0.270 0.662 0.655 0.243

0.040 0.062 0.033 0.616 0.170 0.611 0.606 0.146

0.048 0.173 0.102 0.566 0.065 0.559 0.541 0.039

0.030 0.008 0.004 0.505 0.034 0.499 0.481 0.063

0.019 0.072 0.010 0.445 0.046 0.438 0.422 0.069

0.020 0.181 0.100 0.382 0.083 0.375 0.361 0.106

X. Jiang, B.-S. Lee / Journal of Financial Markets ] (]]]]) ]]]–]]]

Please cite this article as: Jiang, X., Lee, B.-S., The intertemporal risk-return relation: A bivariate model approach. Journal of Financial Markets (2013), http://dx.doi.org/10.1016/j.finmar.2013.02.002

Table 1 Descriptive statistics on returns and risk. This table reports summary statistics including autocorrelation coefficients for three measures of expected returns and four measures of conditional variances. Excess market return is measured by S&P index return in excess of long-term interest rate. Rm denotes S&P index returns. ER1 and ER2 denote the expected excess returns calculated based on bivariate autoregressive representation (BARR) and bivariate moving average representation (BMAR) of the realized market excess return Rm, respectively. MV denotes the realized variance calculated using previous 12-month returns. CV1 and CV2 denote the conditional variances calculated based on a univariate MAR and an ARIMA model of MV, respectively. CV3 and CV4 denote the conditional variances calculated based on BARR and BMAR of the market returns and variances, respectively. The table reports mean, variance, skewness, and kurtosis, and autocorrelations up to 12 lags for the full sample from 1871:02 to 2010:12.

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in terms of the degree of persistence (or stochastic orders) between the first two conditional moments of returns while maintaining the same common information content in the prediction. The conditional expectation in Merton’s ICAPM suggests that the information set used to form the expectation on the market excess return has to be the same as the information set used to form the expectation on the market variance. Using univariate ARR or MAR as a proxy for conditional variance alone may not lead to the use of the common information set embedded in Merton’s ICAPM. With this concern, we employ BARR and BMAR and measure the expected excess return and conditional variance jointly based on the same common information set. However, for comparison we also report estimation results using univariate ARR and MAR. In sum, using mt and s2t for the conditional mean and variance at time t estimated using information available at time t1, we consider the following six cases: Model 1: mt ¼ realized returns, and s2t ¼ realized variance at t1. Model 2: mt ¼ realized returns, and s2t ¼ conditional variance based on univariate MAR. Model 3: mt ¼ realized returns, and s2t ¼ conditional variance based on BARR. Model 4: mt ¼ realized returns, and s2t ¼ conditional variance based on BMAR. Model 5: mt ¼ expected returns based on BARR, and s2t ¼ conditional variance based on BARR. Model 6: mt ¼ expected returns based on BMAR, and s2t ¼ conditional variance based on BMAR.

It follows that a comparison of Models 3 and 5 or 4 and 6 will capture the importance of estimating the expected returns, while keeping the conditional variance model the same. Comparing Models 5 and 6 is a joint test of the conditional mean and conditional variance.

3. Data and descriptive statistics In prior studies, the realized variance is constructed in the two ways: rolling 12-month variance (e.g., Officer, 1973; Merton, 1980) or using daily data adjusted for autocorrelations (e.g., French, Schwert, and Stambaugh, 1987; Ghysels, Santa-Clara, and Valkanov, 2005). There is a tradeoff between the two methods. The former has an advantage of employing a longer sample period that is not restricted by the availability of daily data. The latter can use daily data and is not subject to the concern of an overlapping issue. However, it is restricted to the availability of daily data and is subject to small sample issues (see Lundblad, 2007). In estimating and testing the intertemporal risk-return relation, we employ the historical Standard and Poor’s Composite Stock Price Index monthly return data from 1872:02 to 2010:12 to measure market returns. We measure the excess market return using the S&P index return in excess of the long-term interest rate.11 The monthly realized variance is 11 The S&P index and long-term interest rate are from Shiller’s website. For details, see Shiller’s website: http://www.econ.yale.edu/~shiller/data.htm. We use the long-term interest rates in Shiller’s website to proxy riskfree rates since T-bill rates are not available for such a long sample period.

Please cite this article as: Jiang, X., Lee, B.-S., The intertemporal risk-return relation: A bivariate model approach. Journal of Financial Markets (2013), http://dx.doi.org/10.1016/j.finmar.2013.02.002

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constructed using the previous 12-month market return series, carrying forward by dropping the beginning period return and adding the additional return. For robustness, we also employ monthly CRSP value-weighted index and equalweighted index return data from 1926:01 to 2010:12. In this case, the market excess return is measured by CRSP (both equal-weighted and value-weighted) index returns in excess of one-month T-bill rates. The variance is constructed using previous 12-month excess return series in the same way as in the S&P index data. For another robustness check, we also use the CRSP index daily data to construct the monthly variance as in French, Schwert, and Stambaugh (1987). The construction of the data for hedge components is similar to Guo and Whitelaw (2006) and Pastor, Sinha, and Swaminathan (2008). We use the default spread (Baa-Aaa yield spread obtained from the St. Louis Fed), the relative T-bill rate (the one-month T-bill rate in excess of its 12-month moving average), and the dividend-price ratio (extracted from the value-weighted CRSP market return series with and without dividends). Table 1 provides summary statistics including autocorrelations of excess market return (S&P index return in excess of long-term interest rate) and conditional variances for the full sample period of 1872:02–2010:12. The sample mean of the monthly market excess return is 0.40%, and the variance of the return series is about 0.17. The first-order autocorrelation coefficient of the excess return series is around 0.29, and the market return is not persistent. The sample means of the two expected market returns, ER1 and ER2, which are measured based on BARR and BMAR, respectively, are similar to the mean of the realized excess returns but with lower autocorrelations, which is not surprising. The mean of the variance of the excess market return based on a 12-month window, MV, is 0.16 and its variance is 0.0011. The first-order autocorrelation coefficient of the monthly variance MV is 0.958. The autocorrelation decays slowly as the lag length increases. However, the variance of the conditional variance measured based on a univariate MAR, CV1, is 0.0007 and the first-order autocorrelation is 0.908.12 The autocorrelation decays faster as the lag length increases. When the conditional variance is measured based on an ARIMA model, CV2, it behaves very much like the realized variance, MV, in terms of its variance and autocorrelations. This is not surprising given that the ARIMA model contains a lagged variance as a component. CV3 and CV4 represent the conditional variances measured based on BARR and BMAR, respectively. Both measures have somewhat lower variance and autocorrelations compared to the realized variance MV. In particular, between the two measures, CV4 has a lower variance and lower autocorrelations than CV3. In sum, various measures of the expected return and conditional variance based on time-series models have lower variance and autocorrelations than the realized market return and market variance. In particular, the conditional variance measure based on a BMAR, CV4, has the lowest variance and autocorrelations among various measures of variance of returns.13

12 This pattern of autocorrelation is also observed in the conditional variance measured by GARCH models. For instance, Ghysels, Santa-Clara, and Valkanov (2005) show that the first order autocorrelation of the conditional GARCH variance is 0.97 using the monthly market return data from 1928:01 to 2000:12. 13 In addition, the conditional variance measured based on a BMAR, CV4, has the lowest skewness and kurtosis among the various measures of variance of returns.

Please cite this article as: Jiang, X., Lee, B.-S., The intertemporal risk-return relation: A bivariate model approach. Journal of Financial Markets (2013), http://dx.doi.org/10.1016/j.finmar.2013.02.002

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Table 2 Estimates of the intertemporal risk-return relation: GARCH-M approach. This table reports the estimation results of the risk-return relation using S&P index returns from 1872:02 to 2010:12. We use three GARCH-M models to measure the conditional variance. The mean equation is: rM,tþ1 ¼ l0 þ lM s2M,t þ etþ1 : The conditional variance is described by the following GARCH processes: GARCH(1,1): s2M,tþ1 ¼ c þ ae2t þ bs2M,t , EGARCH(1,1): lnðs2M,tþ1 Þ ¼ c þ ajet j=sM,t þ blnðs2M,t Þ þ det =sM,t , and TARCH(1,1): s2M,tþ1 ¼ c þ ae2t þ bs2M,t þ dIeo0 e2t , where Ieo0 ¼ 1 if et1o0; otherwise Ieo0 ¼ 0. Standard errors are provided below the parameter estimates. In GARCH-M and TGARCH-M, the coefficients and standard errors of l0 and c are reported in percentages. n, nn, and nnn denote statistical significance at the 10%, 5% and 1% levels, respectively. Sample

Model

l0

lM

c

1872:02 – 1925:12

GARCH-M

0.0999 0.3400 0.0903 0.0990 0.0205 0.3280

4.2312 3.2202 0.0003nnn 0.0000 3.3873 3.0700

0.0378nnn 0.0127 5.0480nnn 0.0271 0.0362nnn 0.0114

0.2534nnn 0.0726 0.2582nnn 0.0031 0.1698nn 0.0716

0.3997nnn 0.1531 0.2976nnn 0.0023 0.4242nnn 0.1412

0.0034nnn 0.0001 0.1443n 0.0874

1.4010nnn 0.4230 0.0075 0.3300 1.0840n 0.5770

0.2198 1.0246 0.0008nnn 0.0002 0.0758 1.7709

0.0109n 0.0064 3.9760nnn 0.0415 0.0207nn 0.0094

0.2088nnn 0.0496 0.2338nnn 0.0056 0.0311 0.0434

0.8026nnn 0.0324 0.2939nnn 0.0048 0.8004nnn 0.0448

0.0291nnn 0.0009 0.2389nnn 0.0834

0.6620 1.0590 0.3270n 0.1690 0.4840nnn 0.0002

2.6625 11.3196 0.0008nnn 0.0001 3.8034nnn 0.0048

0.0775 0.0654 5.1219nnn 0.0447 0.0899nnn 0.0003

0.1570nn 0.0756 0.2797nnn 0.0056 0.0615nnn 0.0000

0.0020 0.7154 0.2986nnn 0.0036 0.1564nnn 0.0010

0.0458nnn 0.0012 0.3885nnn 0.0012

2.9030 1.9340 0.0294 0.1710 0.5870 0.4670

25.7515 16.1438 0.0003nnn 0.0001 2.3325 3.6244

0.0139nn 0.0057 4.9210nnn 0.0443 0.0064nn 0.0031

0.0582 0.0373 0.3630nnn 0.0081 0.1019nnn 0.0214

0.8278nnn 0.0585 0.2981nnn 0.0039 0.9692nnn 0.0433

0.0321nnn 0.0008 0.1494nnn 0.0450

0.0000 0.0000 0.0000 0.0009 0.0000 0.0000

0.0000 0.2242 0.0000 0.0000 0.0000 0.2197

0.1370nnn 0.0026 6.5918nnn 0.0004 0.1370nnn 0.0028

0.0500 0.0556 0.0500nnn 0.0000 0.0500 0.0556

0.4500nnn 0.0491 0.4500nnn 0.0000 0.4500nnn 0.0491

0.0000 0.0000 0.0000 0.0608

EGARCH-M TGARCH-M

1926:01 – 1946:12

GARCH-M EGARCH-M TGARCH-M

1947:01 – 1966:12

GARCH-M EGARCH-M TGARCH-M

1967:01 – 1986:12

GARCH-M EGARCH-M TGARCH-M

1987:01 – 2010:12

GARCH-M EGARCH-M TGARCH-M

a

b

d

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4. Estimation of the intertemporal risk-return relation 4.1. GARCH-M model approach We first consider the GARCH-M family models to show that earlier studies based on GARCH-M models tend to show some inconsistency in the signs of the intertemporal riskreturn relation. Engle, Lilien, and Robins (1987) and Bollerslev, Engle, and Wooldridge (1988) extend the GARCH framework to allow the mean return to depend on its own conditional variance. This class of model, called GARCH-M, is particularly suited to the study of asset pricing. French, Schwert, and Stambaugh (1987) find a positive and significant risk-return relation in the U.S. stock market using GARCH-M. Previous studies also find that market volatility responds asymmetrically to lagged return innovations (see Black, 1976; Christie, 1982). The ‘‘bad’’ news seems to have a more pronounced effect on volatility than does ‘‘good’’ news. Taking into account the asymmetric effect of good news and bad news, Glosten, Jagannathan, and Runkle (1993) find a negative and significant risk-return relation using a Threshold (asymmetric) GARCH-M (TGARCH-M) (see also Lundblad, 2007). In this section, we use three types of GARCH-M models. The mean equation is given by: rM,tþ1 ¼ l0 þ lM s2M,t þ etþ1 : The conditional variance is described by the following GARCH processes: GARCH(1,1): s2M,tþ1 ¼ c þ ae2t þ bs2M,t , EGARCH(1,1): lnðs2M,tþ1 Þ ¼ c þ ajet j=sM,t þ blnðs2M,t Þ þ det =sM,t , and TGARCH(1,1): s2M,tþ1 ¼ c þ ae2t þ bs2M,t þ d Ieo0 e2t , where Ieo0 ¼ 1 if et1o0; otherwise Ieo0 ¼ 0. In Table 2, we report the estimation results of the intertemporal risk-return relation using S&P index excess market returns.14 We estimate the relation for five sample periods: 1872:02–1925:12, 1926:01–1946:12, 1947:01–1966:12, 1967:01–1986:12, and 1987:01– 2010:12. For the first subsample of 1872:02–1925:12, the CRSP data are not available and it was before the Great Depression. The second subsample of 1926:01–1946:12 includes the Great Depression and World War II. After 1926, we split the sample period into roughly twenty years for each subsample. In Table 2, the estimation results of the three types of GARCH-M models are presented with standard errors below the parameter estimates. In GARCH-M and TGARCH-M, the coefficients and standard errors of l0 and c are reported in percentages. We focus on the risk-return relation represented by the coefficient lM. Using GARCH-M, we find that in each subsample the estimate of the risk-return relation, lM, is not significantly positive at the 10% significance level. Using EGARCH-M and TGARCH-M, we find the estimate of the risk-return relation is not significant, except for the TGARCH-M in the 1947: 01–1966:12 subsample.15 Our results show that the empirical evidence on the risk-return 14

We also conduct the analysis on real market returns and market returns as a robustness check and find qualitatively similar results. 15 In an unreported table, we also use real market return and market return data to implement the GARCH-M family models and the result is similar. Please cite this article as: Jiang, X., Lee, B.-S., The intertemporal risk-return relation: A bivariate model approach. Journal of Financial Markets (2013), http://dx.doi.org/10.1016/j.finmar.2013.02.002

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relation in the literature is weak as Lundblad (2007) points out. In sum, the three types of GARCH-M models provide varying estimates. One possible reason for not detecting a significantly positive relation may be due to lack of consistency in the information used for both the measures of the expected excess return and conditional variance.16 4.2. Univariate and bivariate model approach Now, we consider univariate models. We consider six models. In Models 1 and 2, the expected return is measured by the realized excess return and the conditional variance by one lagged realized variance in Model 1 and by a univariate MAR in Model 2, respectively. In Models 3 and 4, the expected excess return is measured by the realized excess return and the conditional variance by BARR in Model 3 and BMAR in Model 4, respectively. Both the expected excess return and conditional variance are measured by BARR in Model 5 and by BMAR in Model 6.17 For each model, Table 3 reports the coefficient of the intertemporal risk-return relation, b, its t-statistic, which is adjusted for the residual autocorrelation and heteroscedasticity using the Newey-West correction with 12 lags, and adjusted R-square. In Model 1, the point estimate of the risk-return relation b is positive (5.469, 0.516, 0.260, 9.007, and 1.241 for each sample period, respectively). In the five subsamples, the risk-return relation is significant in two subsamples, 1872:02–1925:12 and 1967:01–1986:12. The adjusted R-square of the regression is very low, 0.010, 0.001, 0.004, 0.042, and 0.002, respectively. In Model 2, expected return is measured by the realized return and the conditional variance by a univariate MAR. The estimate of the risk-return relation is very similar to that in Model 1. The finding implies that measuring the conditional variance alone based on a univariate MAR does not help us detect a significant positive risk-return relation, although it helps lower the market return volatility and its autocorrelations. Now, we consider the intertemporal risk-return relation using bivariate models. Columns 5 to 8 in Table 3 report the estimates for Models 3, 4, 5, and 6.18 In each subsample, the estimates in Models 3 and 4, where the expected excess return is measured by the realized excess return and the conditional variance is measured by BARR in Model 3 and BMAR in Model 4, respectively, are similar to the estimates in Models 1 and 2: they are positive but only significant in the subsamples 1872:02–1925:12 and 1967:01–1986:12. The estimate in Model 5, where both the expected excess return and conditional variance are measured by BARR, displays a more significantly positive relation than that in Models 1, 2, 3, and 4. The t-statistic and adjusted R-square for Model 5 are generally higher than 16 Lundblad (2007) points out that ‘‘Using the popular GARCH-in-mean framework, the empirical evidence measured over the past 50 to 75 years of the U.S. market portfolio regarding this relationship is, at best, weak.’’ Harvey (1989) points out that there are two potential disadvantages in GARCH-M family models. 17 We use OLS to estimate BARR. In a BARR, all equations have the same explanatory variables, so OLS applied equation by equation is efficient. In theory, we invert (IA(L)L) to get coefficients B(L). However, in practice, we use innovations (residuals) derived from the BARR as explanatory variables and estimate B(L) coefficients in the BMAR regressions, which are supposed to yield equivalent coefficients. Table 1 shows that the autocorrelation of conditional variance tends to go down to zero around 10 lags. Based on this and considering both the Akaike information criterion and the Schwarz Bayesian criterion, we choose 10 lags for both ARR and MAR for the full sample and the first three subsamples. 18 Note that the estimate of parameter b in Model 3 is the same as the one in Model 5 by construction, although their t-statistics and R squares are different. Similarly, the same estimates are also found for Model 4 and Model 6.

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Table 3 Estimates of the intertemporal risk-return relation. This table reports the estimation results of the intertemporal risk-return relation using S&P index excess market returns for the five subsample periods: 1872:02–1925:12, 1926:01–1946:12, 1947:01–1966:12, 1967:01–1986:12, and 1987:01–2010:12. In Models 1 and 2, the expected return is measured by the realized return and the conditional variance by one lagged realized variance in Model 1 and by a univariate MAR in Model 2, respectively. In Models 3 and 4, the expected return is measured by the realized return and the conditional variance by BARR in Model 3 and BMAR in Model 4, respectively. Both the expected return and conditional variance are measured by BARR in Model 5 and by BMAR in Model 6. ‘b’ denotes the coefficient of the conditional variance. ‘t’ denotes the t-statistic, which is adjusted for the residual autocorrelation and heteroskedasticity using the Newey-West correction with 12 lags. R2 denotes the adjusted R-square. n, nn, and nnn denote statistical significance at the 10%, 5% and 1% levels, respectively. Model 1

Model 2

Model 3

Model 4

Model 5

Model 6

1872:02–1925:12

b t R2

5.469nn 2.402 0.010

5.403nn 2.140 0.007

5.275nn 2.263 0.008

5.948nn 2.238 0.008

5.275nnn 5.559 0.076

5.948nnn 9.285 0.108

1926:01–1946:12

b t R2

0.516 0.709 0.001

0.953 0.973 0.003

0.786 1.001 0.003

1.654 1.382 0.011

0.786n 1.661 0.024

1.654nn 2.239 0.065

1947:01–1966:12

b t R2

0.260 0.079 0.004

1.997 0.466 0.003

0.548 0.157 0.004

2.037 0.435 0.003

0.548 0.434 0.003

2.037nn 2.224 0.011

1967:01–1986:12

b t R2

9.007nnn 2.989 0.042

9.053nnn 2.901 0.034

9.269nnn 3.015 0.041

14.897nnn 3.355 0.032

9.269nnn 9.730 0.274

14.897nnn 10.883 0.345

1987:01–2010:12

b t R2

1.241 0.601 0.002

1.205 0.513 0.002

0.694 0.283 0.003

2.223 0.713 0.001

0.694 0.609 0.000

2.223nnn 2.657 0.019

that in Models 1, 2, 3, and 4. However, the estimate of the risk-return relation is not always significantly positive for each subsample. The estimate is significantly positive in three subsamples of 1872:021925:12, 1926:01–1946:12, and 1967:01–1986:12, respectively, while it is not significant in other two subsamples. We move on to Model 6, where both the expected return and conditional variance are measured by BMAR to alleviate persistence in the conditional variance, as well as to maintain the common information set. The lower variance and less persistence of the BMAR-based conditional variance are shown in Table 1. Table 3 shows that the estimate of the risk-return relation is positive and significant in each subsample. In addition, the adjusted R-square in Model 6 is higher than in any other models in each subsample. The tstatistic (adjusted R2) is 9.285 (0.108), 2.239 (0.065), 2.224 (0.011), 10.883 (0.345), and 2.657 (0.019) in each subsample, respectively. The comparison of Model 6 with Models 1 to 4 suggests that the significantly positive risk-return relation is due to the use of the same common information in a consistent manner in measuring both the expected return and conditional variance based on a bivariate model of market returns and variance. The comparison of Model 6 with Model 5 suggests that the significantly positive risk-return relation is also due to the use of BMAR Please cite this article as: Jiang, X., Lee, B.-S., The intertemporal risk-return relation: A bivariate model approach. Journal of Financial Markets (2013), http://dx.doi.org/10.1016/j.finmar.2013.02.002

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Table 4 Estimates of the intertemporal risk-return relation: Recursive approach. This table reports the estimation results of the intertemporal risk-return relation using S&P index excess market returns and recursive approach. The conditional mean and conditional variance are generated from 1926:01 to 2010:12, using recursive approach, i.e., using all data available to estimate the expected return and conditional variance. Specifically, at each time t, we use all available data to compute one-step ahead forecast of excess return and variance for time tþ1. In Models 1 and 2, the expected return is measured by the realized return and the conditional variance by a univariate MAR. In Models 3 and 4, the expected return is measured by the realized return and the conditional variance by BARR in Model 3 and BMAR in Model 4, respectively. Both the expected return and conditional variance are measured by BARR in Model 5 and by BMAR in Model 6, respectively. ‘b’ denotes the coefficient of the conditional variance. ‘t’ denotes the t-statistic, which is adjusted for the residual autocorrelation and heteroskedasticity using the Newey-West correction with 12 lags. R2 denotes the adjusted R-square. n, nn, and nnn denote statistical significance at the 10%, 5% and 1% levels, respectively. Model 2

Model 3

Model 4

Model 5 nn

Model 6

1926:01–1946:12

b t R2

2.610 1.614 0.006

0.682 0.966 0.002

2.564 1.606 0.005

1.344 2.202 0.038

0.091 0.069 0.004

1947:01–1966:12

b t R2

0.775 0.373 0.004

0.830 0.327 0.004

0.953 0.465 0.003

0.050 0.053 0.004

1.516nnn 3.346 0.020

1967:01–1986:12

b t R2

4.842 1.536 0.006

8.595nn 2.431 0.037

5.255n 1.679 0.007

2.842nnn 2.609 0.037

3.105nnn 4.423 0.039

1987:01–2010:12

b t R2

1.592 0.625 0.001

0.763 0.374 0.003

1.821 0.710 0.000

0.066 0.094 0.003

1.502n 1.826 0.015

rather than BARR in measuring both the expected return and conditional variance in an attempt to mitigate potential persistence in the conditional variance. 4.3. Recursive approach In the above analysis, we construct the conditional mean and conditional variance jointly and examine the risk-return relation using full sample. This may introduce a lookahead bias. To address this concern, we re-estimate the intertemporal risk-return relation using a moving window recursive approach to generate expected returns and conditional variances from 1926:01 to 2010:12 for each model we consider. Specifically, at each time t, we use all available data to compute one-step ahead forecast of excess return and variance for time tþ1. Our estimates of expected returns and conditional variances are not subject to the look-ahead bias in this recursive approach. Then, we examine the risk-return relation in non-overlapping subsample periods of 1926:01–1946:12, 1947:01–1966:12, 1967:01–1986:12, and 1987:01–2010:12. Table 4 reports the results for four non-overlapping subsamples.19 We find that the riskreturn relation based on the recursive estimation is similar to the results we find using the 19

Since Model 1 is based on realized returns and one lagged realized variances, which are observable, there is no need to estimate Model 1 using recursive approaches in the subsequent tables. Please cite this article as: Jiang, X., Lee, B.-S., The intertemporal risk-return relation: A bivariate model approach. Journal of Financial Markets (2013), http://dx.doi.org/10.1016/j.finmar.2013.02.002

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non-recursive estimation in Table 3. The risk-return relation based on Model 6 is always positive. The t-statistics of Model 6 are 0.069, 3.346, 4.423, and 1.826 for the subsample periods of 1926:01–1946:12, 1947:01–1966:12, 1967:01–1986:12, and 1987:01–2010:12, respectively. They are significant at the 10% level for all subsample periods except for the first subsample period from 1926:01 to 1946:12. The adjusted R-squares in Model 6 are the largest among the models considered for all subsample periods except for the first subsample period from 1926:01 to 1946:12. The insignificant risk-return relation in the period from 1926:01 to 1946:12 may not be surprising given the unexpected historical events (e.g., the Great Depression, World War II, etc.) during this period. In an unreported table, we also examine the risk-return relation starting from 1940:01 using a moving window recursive approach for the subsample periods from 1940:01 to 1959:12, 1960:01 to 1979:12, and 1980:01 to 2010:12. We find qualitatively similar results. Therefore, the significantly positive risk-return relation is not likely attributable to the look-ahead bias. In sum, the estimation results of the recursive approach are generally consistent with those in Table 3 in that the intertemporal risk-return relation is significantly positive. 4.4. Generalized method of moments (GMM) with recursive approach Eq. (3) can be fitted to a typical generalized method of moments (GMM) framework as follows: E½f ðXt ,gÞjIt1  ¼ 0

ð8Þ

where Xt ¼ [1, (RM, t–Rf, t ), s2M,t] and f(Xt, g)¼ (RM, t–Rf, t)mg s2M,t. When we understand this equation in the context of a GMM framework, both expected excess returns, E½ðRM,t Rf ,t ÞjIt1 , and conditional variance of returns, E½ðs2M,t jIt1 , are based on the same common information set It1. Therefore, we can maintain consistency in the information set when measuring both the expected excess return and conditional variance of returns.20 The GMM framework is very flexible in the functional form of f(Xt, g). It not only allows for a non-linear functional form but also can easily accommodate an additional term (e.g., a hedge component) by reformulating f(Xt, g) as follows: f ðXt ,gÞ ¼ ðRM,t Rf ,t Þmg1 s2 M,t g2 covðRM,t ,St Þ,

ð9Þ

where St is a set of state variables to be specified. Given that the intertemporal risk-return relation can be nicely fitted to a GMM framework with a common information set for both the expected return and conditional variance and that both potential heteroscedasticity and autocorrelation can be taken into 20

In addition, the GMM framework provides the following useful features. First, the GMM procedure does not require a complete explicit representation of the economic environment, and Eq. (8) provides an adequate basis for estimation and testing. Second, the GMM procedure is distribution-free while maximum likelihood estimation requires an additional distributional assumption and ARCH models require some distributional assumption. Third, the GMM procedure allows for a general form of conditional heteroscedasticity of disturbance terms. It does not require the dependence of the conditional mean or variance on the information set to be explicitly characterized, whereas the ARCH model requires it (e.g., in an AR form) to be specified a priori. Fourth, the GMM procedure is very flexible in selecting instrumental variables for estimation. The instrumental variables do not have to be explicitly exogenous. They only need to be predetermined so any variable in information set It1 that is observable by an econometrician can be a candidate for instrumental variables. Please cite this article as: Jiang, X., Lee, B.-S., The intertemporal risk-return relation: A bivariate model approach. Journal of Financial Markets (2013), http://dx.doi.org/10.1016/j.finmar.2013.02.002

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Table 5 Estimates of the intertemporal risk-return relation: GMM with recursive approach. This table reports the GMM estimation results of the intertemporal risk-return relation using S&P index excess market returns and recursive approach. The conditional mean and conditional variance are generated from 1926:01 to 2010:12, using recursive approach, i.e., using all data available to estimate the expected return and conditional variance. Specifically, at each time t, we use all available data to compute one-step ahead forecast of excess return and variance for time tþ1. In Models 1 and 2, the expected return is measured by the realized return and the conditional variance by a univariate MAR. In Models 3 and 4, the expected return is measured by the realized return and the conditional variance by BARR in Model 3 and BMAR in Model 4, respectively. Both the expected return and conditional variance are measured by BARR in Model 5 and by BMAR in Model 6, respectively. ‘g’ denotes the coefficient of the conditional variance in Eq. (8). ‘t’ denotes the t-statistic, which is adjusted for the residual autocorrelation and heteroskedasticity using the GMM correction with 12 lags. R2 denotes the adjusted Rsquare. n, nn, and nnn denote statistical significance at the 10%, 5% and 1% levels, respectively. Model 2

Model 3

Model 4

Model 5

Model 6

Panel A: The instrument variables are a constant, lagged market returns, and lagged market variances. 1926:01–1946:12

g t R2

5.237nnn 3.309 0.003

0.232 0.434 0.005

5.388nnn 3.294 0.005

0.512 1.225 0.022

0.100 0.091 0.005

1947:01–1966:12

g t R2

0.480 0.234 0.008

4.501n 1.648 0.014

0.420 0.202 0.008

1.345 1.567 0.013

1.030nn 2.077 0.017

1967:01–1986:12

g t R2

2.276 0.604 0.000

7.874nnn 3.219 0.036

2.762 0.762 0.002

2.897nnn 3.641 0.036

1.712n 1.959 0.030

1987:01–2010:12

g t R2

1.550 0.733 0.007

0.496 0.276 0.007

1.660 0.782 0.007

0.551 0.970 0.009

1.873nnn 3.161 0.014

Panel B: The instrument variables are a constant, lagged market returns, and lagged market variances, and lagged dividend price ratios. 1926:01–1946:12

g t R2

5.304nnn 3.352 0.005

0.345 0.640 0.004

5.389nnn 3.355 0.007

0.470 1.111 0.019

0.036 0.033 0.004

1947:01–1966:12

g t R2

0.781 0.386 0.008

3.349 1.268 0.009

0.681 0.331 0.008

1.342 1.563 0.013

1.029nn 2.075 0.017

1967:01–1986:12

g t R2

2.421 0.712 0.000

8.193nnn 3.359 0.036

2.837 0.870 0.002

2.750nnn 3.506 0.037

1.842nn 2.160 0.031

1987:01–2010:12

g t R2

1.657 0.785 0.010

0.117 0.068 0.008

1.826 0.862 0.009

0.590 1.054 0.009

1.914nnn 3.240 0.013

Panel C: The instrument variables are a constant, lagged market returns, and lagged market variances, and lagged term spreads. 1926:01–1946:12

g t

4.922nnn 3.236

0.299 0.559

5.012nnn 3.264

0.429 1.012

0.037 0.034

Please cite this article as: Jiang, X., Lee, B.-S., The intertemporal risk-return relation: A bivariate model approach. Journal of Financial Markets (2013), http://dx.doi.org/10.1016/j.finmar.2013.02.002

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Table 5 (continued ) Model 2

Model 3

Model 4

Model 5

Model 6

R2

0.004

0.004

0.005

0.017

0.005

1947:01–1966:12

g t R2

0.563 0.274 0.009

4.478 1.638 0.013

0.513 0.247 0.009

1.289 1.508 0.013

1.049nn 2.119 0.017

1967:01–1986:12

g t R2

3.114 0.847 0.002

8.198nnn 3.356 0.036

3.580 1.022 0.003

3.004nnn 3.915 0.036

1.923nn 2.230 0.032

1987:01–2010:12

g t R2

1.543 0.730 0.005

1.120 0.645 0.008

1.668 0.787 0.005

0.635 1.263 0.012

1.800nnn 3.077 0.013

Table 6 Estimates of the intertemporal risk-return relation with hedge components and recursive approach. This table reports the estimation results of the intertemporal risk-return relation using monthly CRSP index returns from 1925:01 to 2010:12 including hedge components and recursive approach. Realized variance is measured using previous 12 month return series. The excess market return is measured by the difference between CRSP index return and one-month T-bill rates. The conditional mean and conditional variance are generated from 1947:01 to 2010:12, using recursive approach, i.e., using all data available to estimate the expected return and conditional variance. Specifically, at each time t, we use all available data to compute one-step ahead forecast of excess return and variance for time tþ1. In Model 2, the expected return is measured by the realized return and the conditional variance by a univariate MAR. In Models 3 and 4, the expected return is measured by the realized return and the conditional variance by BARR in Model 3 and BMAR in Model 4, respectively. Both the expected return and conditional variance are measured by BARR in Model 5 and by BMAR in Model 6. ‘g1’ denotes the coefficient of the conditional variance (or standard deviation) in Eq. (9). ‘t’ denotes the t-statistic, which is adjusted for the residual autocorrelation and heteroskedasticity using the Newey-West correction with 12 lags. R2 denotes the adjusted R-square. n, nn, and nnn denote statistical significance at the 10%, 5% and 1% levels, respectively. Model 2

Model 3

Model 4

Model 5

Model 6

1.772 1.545 0.024 0.133 0.981 0.021

0.153 0.509 0.002 0.017 0.524 0.003

1.377nnn 6.678 0.047 0.146nnn 5.429 0.026

1.613nn 2.234 0.035 0.127 1.279 0.030

1.343nnn 6.109 0.054 0.110nnn 4.115 0.017

1.566nnn 8.425 0.109 0.206nnn 7.096 0.054

Panel A: Monthly value weighted CRSP index excess return s2

s

g1 t R2 g1 t R2

0.845 0.804 0.020 0.068 0.620 0.020

1.828 1.320 0.023 0.146 1.004 0.021

Panel B: Monthly equal weighted CRSP index excess return s2

s

g1 t R2 g1 t R2

0.778 1.012 0.029 0.078 0.821 0.029

1.863 1.573 0.034 0.192 1.329 0.032

Please cite this article as: Jiang, X., Lee, B.-S., The intertemporal risk-return relation: A bivariate model approach. Journal of Financial Markets (2013), http://dx.doi.org/10.1016/j.finmar.2013.02.002

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account without explicitly specifying the serial correlations, we employ a GMM procedure with a recursive approach to estimate the intertemporal risk-return coefficients. Table 5 reports the estimate of the intertemporal risk-return relation b, t-statistics, and adjusted R-squares for the five models using the GMM procedure with a recursive approach. We consider different sets of instrument variables.21 When instrument variables are a constant, lagged market returns, and lagged market variances in Panel A, the t-statistics (adjusted R-squares) of Model 6 are 0.091 (0.005), 2.077 (0.017), 1.959 (0.030), and 3.161 (0.014), respectively, for each subsample of 1926:01–1946:12, 1947:01–1966:12, 1967:01–1986:12, and 1987:01–2010:12. Consistent with Table 4, only the estimate of the risk-return relation in Model 6 is positive and significant in all subsamples except for the subsample from 1926:01 to 1946:12. In Model 5, the estimate is positive and significant only in the subsample of 1967:01–1986:12. The results are similar when instrument variables are a constant, lagged market returns, lagged market variances, and lagged dividend price ratios in Panel B, and when instrument variables are a constant, lagged market returns, lagged market variances, and lagged term spreads in Panel C. Again, the findings in Table 5 suggest that when investigating the intertemporal risk-return relation, it is important to take into account both the same information set and the persistence issue in the conditional variance. 4.5. Robustness with a hedge component and recursive approach In the original ICAPM, Merton (1973) shows that expected market excess return depends not only on the conditional variance of market return (risk component) but also on the covariance with the time-varying future investment opportunities (hedge component). Scruggs (1998) and Guo and Whitelaw (2006) argue that a hedging component is important in detecting a positive risk-return relation. Although we find a positive relation even without a hedging component in our estimation, it seems prudent to examine whether the relation is robust with the hedge component. Following Guo and Whitelaw (2006), we include four state variables in the relation: the default spread (Baa-Aaa yield spread, obtained from the St. Louis Fed), the term spread (30-year minus one-month Treasury yield spread, obtained from CRSP), the detrended risk-free rate (the one-month T-bill rate in excess of its 12-month moving average), and the dividend-price ratio (extracted from the value-weighted or equal-weighted CRSP market return series with and without dividends). Since most of these state variables are not available for the long sample with the S&P index, Table 6 reports the estimation results of the intertemporal risk-return relation using monthly CRSP index returns from 1925:01 to 2010:12 including hedge components. We consider both CRSP value-weighted and equal-weighted index excess return and use both variance and standard deviation of market return as measure of realized volatility. We re-estimate the risk-return relation with the recursive approach in all five models by adding the state variables. Table 6 reports the estimation results using the CRSP index data. Panel A reports the results with value-weighted CRSP index excess return. The riskreturn relation is significantly positive in only Model 6 for both variance and standard deviation. Panel B reports the result with equal-weighted CRSP index excess returns. The risk-return relation is significantly positive in Models 5 and 6 for both variance and 21

We thank a referee for this suggestion.

Please cite this article as: Jiang, X., Lee, B.-S., The intertemporal risk-return relation: A bivariate model approach. Journal of Financial Markets (2013), http://dx.doi.org/10.1016/j.finmar.2013.02.002

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standard deviation. Even in the presence of the hedge component, the estimate of the riskreturn relation in Model 6 is still positive and significant using either value-weighted or equal-weighted CRSP index excess returns and using both variance and standard deviation. Again, the adjusted R-square is the highest in Model 6 regardless of whether we use the value-weighted index or the equal-weighted CRSP index, and whether we use conditional variance or conditional standard deviation. The findings in Table 6 suggest that the significantly positive risk-return relation in Model 6 is robust with respect to the inclusion of the hedge component. Pastor, Sinha, and Swaminathan (2008) also find that the hedge component does not affect the intertemporal risk-return relation.

Table 7 Estimates of the intertemporal risk-return relation: ARIMA model approach. This table reports the estimation results of the intertemporal risk-return relation using S&P index excess market returns for the five subsample periods: 1872:02–1925:12, 1926:01–1946:12, 1947:01–1966:12, 1967:01–1986:12, and 1987:01-2010:12. We consider Models 2–6. In Model 2, the expected return is measured by the realized return and the conditional variance by a univariate ARIMA model. In Models 3 and 4, the expected return is measured by the realized return and the conditional variance by BARR in Model 3 and BMAR in Model 4, respectively. Both expected return and conditional variance are measured by BARR in Model 5 and by BMAR in Model 6. To incorporate ARIMA process for the conditional variance, we use actual return and first differenced variance in the bivariate ARR and MAR. ‘b’ denotes the coefficient of the conditional variance. ‘t’ denotes the t-statistic, which is adjusted for the residual autocorrelation and heteroskedasticity using the Newey-West correction with 12 lags. R2 denotes the adjusted R-square. n, nn, and nnn denote statistical significance at the 10%, 5% and 1% levels, respectively. Model 2

Model 3

Model 4

Model 5

Model 6

1872:02–1925:12 b 5.278nn t 2.441 R2 0.009

5.244nn 2.380 0.009

5.297nn 2.428 0.009

1.379nn 2.239 0.007

1.772nnn 3.195 0.013

1926:01–1946:12 b 0.511 t 0.705 0.001 R2

0.616 0.823 0.000

0.473 0.636 0.001

0.314 1.620 0.004

0.361n 1.652 0.009

1947:01–1966:12 b 0.317 t 0.097 0.004 R2

0.350 0.106 0.004

1.669 0.529 0.003

0.717 0.823 0.000

0.142 0.245 0.004

1967:01–1986:12 b 9.023nnn t 3.023 R2 0.044

9.266nnn 2.966 0.045

9.115nnn 3.032 0.044

4.205nnn 3.324 0.117

3.447nnn 2.885 0.083

1987:01-2010:12 b 2.603 t 1.030 0.001 R2

3.201 1.256 0.003

3.025 1.194 0.003

1.720 1.403 0.020

1.754 1.460 0.021

Please cite this article as: Jiang, X., Lee, B.-S., The intertemporal risk-return relation: A bivariate model approach. Journal of Financial Markets (2013), http://dx.doi.org/10.1016/j.finmar.2013.02.002

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4.6. ARIMA model approach French, Schwert, and Stambaugh (1987) use an autoregressive integrated moving average (ARIMA) model to measure conditional volatility and the mean of realized market returns to measure the expected market return. They find that the estimate of the intertemporal risk-return relation is either positive or negative depending on the sample periods, but none of them is significant. In this section, following French, Schwert, and Stambaugh (1987), we also implement the estimation using an ARIMA model to measure the conditional variance. The estimation results are presented in Table 7. In Model 2, the expected return is measured by the realized return and the conditional variance by a univariate ARIMA model. In Models 3–6, where the conditional variance is measured by bivariate models, we use the realized return and the first differenced variance in the bivariate ARR and MAR to incorporate the ARIMA process for the conditional variance. We do not include Model 1 in Table 7 because the expected return is measured by the realized return and the conditional variance by one lagged realized variance. We find similar results as in French, Schwert, and Stambaugh (1987) in that the riskreturn relations are not significant in the majority subsample periods. Again Model 6 seems to detect most positive and significant risk-return relations. One of the potential reasons that the ARIMA model fails to detect positive and significant risk-return relations is that the measure of the conditional variance in the ARIMA model contains the lagged realized variance that does not help smooth the volatility nor reduce its persistence. Again the findings in Table 7 suggests the importance of taking care of the persistence in the conditional variance when estimating the intertemporal risk-return relation.

4.7. Bootstrap approach As discussed above, we try to provide a measure of conditional variance that alleviates persistence, which plagues the empirical tests. In theory, BARR and BMAR provide equivalent representation of variables (in this case, excess return and variance of returns). However, in practice, with a finite sample, they may provide somewhat different measures of expected excess returns and conditional variance of returns. In the absence of knowledge of the true data-generating process, the purpose of a parametric model is to provide a flexible framework to approximate the dynamics of the predictor variable in finite samples. If we have a choice, we may want to use one with a less persistence. This is because with a persistent regressor, the standard inference is unreliable due to a skewed distribution and tests tend to over reject the null hypothesis (Stambaugh, 1999). Further, in the ICAPM regression, while conditional variance is very persistent, expected excess return is not persistent at all (see Table 1). So there is a substantial discrepancy in the degree of persistence between the two variables, which leads to an unreliable inference. Using expected excess return and conditional variance of returns based on either BARR or BMAR, we can obtain a positive intertemporal relation (see Tables 3 and 4). However, while BARR is based on realized lagged data, BMAR is based on fitted lagged residuals (or innovations). As a result, BMAR usually generates conditional variance, which is closer to a stationary process with a less persistence (i.e., lower autocorrelations), as we report in Table 1. Please cite this article as: Jiang, X., Lee, B.-S., The intertemporal risk-return relation: A bivariate model approach. Journal of Financial Markets (2013), http://dx.doi.org/10.1016/j.finmar.2013.02.002

Panel A: Autocorrelations Lag of autocorrelation

Bootstrap Moving block bootstrap

Variable

1

2

3

4

5

6

7

8

9

10

11

12

CV4 CV5 CV4 CV5

0.959 0.894 0.776 0.816

0.914 0.783 0.726 0.715

0.869 0.692 0.700 0.618

0.818 0.590 0.637 0.521

0.768 0.492 0.598 0.438

0.715 0.393 0.556 0.357

0.662 0.295 0.515 0.282

0.609 0.207 0.483 0.207

0.547 0.099 0.413 0.130

0.485 0.002 0.363 0.056

0.426 0.002 0.304 0.028

0.374 0.002 0.198 0.065

Panel B: Risk-return relation Model 5

Model 6 b

t

R2

0.012

1.302nn

2.026

0.010

0.012

1.297nnn

3.004

0.010

b

t

R

Bootstrap

1.778nnn

2.632

Moving block bootstrap

1.766nnn

2.779

2

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Please cite this article as: Jiang, X., Lee, B.-S., The intertemporal risk-return relation: A bivariate model approach. Journal of Financial Markets (2013), http://dx.doi.org/10.1016/j.finmar.2013.02.002

Table 8 Bootstrapping results. This table provides the autocorrelations and risk-return relation for the first two conditional moments based on BARR and BMAR using bootstrap and moving block bootstrap approaches with S&P index excess market returns from 1872:02–2010:12. CV4 and CV5 are the conditional variances based on BARR and BMAR, respectively. Both the expected return and conditional variance are measured by BARR in model 5 and by BMAR in model 6. Autocorrelations are the mean of the bootstrap autocorrelations. b, t, and R2 are the risk-return coefficient, t-statistics, and adjusted R-square based on bootstrapping estimates.

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With a less persistent measure of (or proxy for) the conditional variance, which is based on a BMAR representation, the ICAPM regression becomes more balanced in terms of the degree of persistence (or stochastic orders) between the first two conditional moments of returns while maintaining the same common information content in the prediction. As a result, we tend to obtain higher R-squares in the intertemporal relation regressions with more significant positive coefficients. To provide further evidence that conditional variance based on BMAR is less persistent than the one based on BARR so that the risk-return relation can be less subject to the small sample bias, we implement the bootstrap inference. Table 8 reports the results. We find that the autocorrelations of the bootstrapped conditional variances based on BARR and BMAR are very close to the ones using the realized data reported in Table 1. The conditional variance based on BMAR not only has lower autocorrelations, but also is less persistent. The first autocorrelations are 0.96 and 0.89 for BARR- and BMAR-based conditional variances, respectively. The autocorrelation of the BMAR-based conditional variance is close to zero after nine months while that of the BARR-based conditional variance is still 0.37 after 12 months, consistent with the results in Table 1. The bootstrap estimate of the risk-return relation is positive for both Model 5 and Model 6 when full sample from 1872:01 to 2010:12 is used. The intertemporal relation is significant for Model 5 and Model 6. The t-statistics and R-squares for Model 5 (Model 6) are 2.632 and 0.012 (2.026 and 0.010), respectively. However, Table 1 and the above bootstrap show that conditional variance in Models 5 and 6 are quite persistent. To address potential serial dependence and heteroscedasticity, we also implement a practically useful moving block bootstrap method developed by Politis and White (2004).22 This method is characterized by the fastest possible rate of convergence, which is adaptive on the strength of the correlation of the time series as measured by the correlogram.23 With the moving block bootstrap procedure, the autocorrelation decays faster than the one without using the moving block bootstrap procedure. The moving block bootstrap procedure also shows a significantly positive intertemporal risk-return relation for Models 5 and 6. Particularly in Model 6, the moving block bootstrapped standard error is smaller. Overall, we confirm that the conditional variance based on BMAR is a better proxy for the conditional variance in terms of persistence. It also confirms that the risk-return relation tends to be more positive and significant with BMAR-based conditional mean and conditional variance. For robustness, we also examine the intertemporal risk-return relation by using CRSP daily and monthly data.24 The results suggest that the significantly positive risk-return relation in Model 6 is robust with respect to the use of different market index return, different data frequency, and a different econometric approach. It is worth noting that the bootstrap analysis provides further evidence that conditional variance based on BMAR is less persistent than the one based on BARR so that the risk-return relation can be less subject to the small sample bias. With a less persistent measure of (or proxy for) the conditional variance, which is based on a BMAR, the ICAPM regression becomes more balanced in terms of the degree of persistence (or stochastic orders) between the first two conditional moments of returns while maintaining the same, common information content 22

We appreciate the suggestion of the editor. Following Politis and White (2004), we choose block size as 29.40. 24 We do not report the tables to save space. The tables are available from the authors. 23

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in the prediction. As a result, we tend to obtain higher R-squares in the intertemporal relation regressions with more significant positive coefficients. 5. Conclusion The intertemporal risk-return relation is an important topic in finance. While the theory in general predicts a positive relation, empirical evidence is at most mixed, which is quite puzzling. Using a more sensible empirical specification, we find a strong positive relation. Our motivation is simple: the theoretical risk-return relation is an ex ante relation and empirical method to detect whether or not the relation should be reliable. This suggests that the measures of both the expected excess return and conditional variance should be based on the same information set used by investors, and the measure of the conditional variance should be less persistent to yield a more reliable inference on the relation. To address the first concern, in measuring both the expected excess return and conditional variance we use the common information set based on a bivariate model of time series of excess returns and variances in a consistent manner. To address the second concern, we measure the conditional variance based on a bivariate moving average representation of excess returns and variances among various available measures. As a result, we can detect a significant positive relation between expected excess return and the conditional variance, which is not feasible with other alternative measures. We also find that the positive relation is robust with respect to different time interval-based measures of realized market variance, to the use of different return data (e.g., S&P or CRSP index return data), and to the inclusion of the hedge component in the relation. References Ang, A., Bekaert, G., 2007. Stock return predictability: is it there? Review of Financial Studies 20 (3), 651–707. Black, F., 1976. Studies of stock price volatility changes. Proceedings of the Business and Economic Statistics, American Statistical Association, 177–181. Bollerslev, T., Engle, R., Wooldridge, J., 1988. A capital asset pricing model with time varying covariances. Journal of Political Economy 96, 116–131. Brandt, M., Kang, Q., 2004. On the relationship between the conditional mean and volatility of stock returns: a latent VAR approach. Journal of Financial Economics 72, 217–257. Brav, A., Lehavy, R., Michaely, R., 2005. Using expectations to test asset pricing models. Financial Management 34, 31–64. Campbell, J., 1987. Stock returns and the term structure. Journal of Financial Economics 18, 373–399. Campello, M., Chen, L., Zhang, L., 2008. Expected returns, yield spreads, and asset pricing tests. Review of Financial Studies 21, 1297–1338. Chan, K., Karolyi, A., Stulz, R., 1992. Global financial markets and the risk premium. Journal of Financial Economics 32, 137–167. Christie, A., 1982. The stochastic behavior of common stock variances: value, leverage and interest rate effects. Journal of Financial Economics 10, 407–432. Cochrane, J., 2008. State-space vs. VAR models of stock returns. Working Paper. University of Chicago. Elton, E., 1999. Expected return, realized return, and asset pricing tests. Journal of Finance 51, 1199–1220. Engle, R., Lilien, D., Robins, R., 1987. Estimation of time varying risk premia in the term structure: the ARCHM model. Econometrica 55, 391–407. Ferson, W., Sarkissian, S., Simin, T., 2003. Spurious regressions in financial economics? Journal of Finance 8, 1393–1413. Ferson, W., Sarkissian, S., Simin, T., 2008. Asset pricing models with conditional alphas and betas: the effects of data snooping and spurious regression. Journal of Financial and Quantitative Analysis 43, 331–354. French, K., Schwert, G., Stambaugh, R., 1987. Expected stock returns and volatility. Journal of Financial Economics 19, 3–30. Please cite this article as: Jiang, X., Lee, B.-S., The intertemporal risk-return relation: A bivariate model approach. Journal of Financial Markets (2013), http://dx.doi.org/10.1016/j.finmar.2013.02.002

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