Time-varying Risk-Return Tradeoff Over Two Centuries:

Transcription

1 Time-varying Risk-Return Tradeoff Over Two Centuries: Sungjun Cho 2 Manchester Business School This Version: August 5, Two anonymous referees provided insightful and constructive comments that greatly improve this paper. I would like to thank two anonymous referees, the editor (Eric Ghysels), and seminar participants at the 2013 EFMA-Reading. 2 Correspondence Information: Sungjun Cho, Division of Accounting and Finance at Manchester Business School, M39, The University of Manchester, Booth Street West, Manchester, M15 6PB mailto:sungjun.cho@mbs.ac.uk

2 Abstract Lundblad (2007,JFE) shows that the risk-return tradeoff is unequivocally positive with a two-century history of equity market data. However, a further examination of the relation with UK monthly stock returns from 1836 to 2010 produces a rather weak riskreturn relation. This study shows that the risk-return tradeoff is mostly positive but varies considerably over time based on new nonlinear CAPMs with the time-varying risk aversion. The often observed negative risk-return relation is statistically insignificant with the 90% confidence bounds. These results are robust after controlling for a proxy of hedging components and with data. Keywords: Time-varying Risk-Return Tradeoff and Hedging Coefficient, ICAPM, State- Space models with GARCH JEL Classification: G12, C15, C22

3 1 Introduction Mainstream asset pricing theories such as the CAPM or ICAPM implies a positive risk-return tradeoff or a positive time-series relation between the conditional mean and variance of market returns. While the risk-return tradeoff is fundamental to finance, the empirical evidence has been rather inconclusive. For example, some studies find a positive relation (e.g. Scruggs (1998), Ghysels, Santa-Clara, and Valkanov (2005), Lundblad (2007), Hedegaard and Hodrick (2014b)), but others find a negative relation or mixed evidence (e.g., French, Schwert, and Stambaugh (1987), Glosten, Jagannathan, and Runkle (1993), Scruggs and Glabadanidis (2003), Whitelaw (2000), Brandt and Kang (2004), Ghysels, Plazzi, and Valkanov (2013)). 1 Several approaches have been proposed to explain this puzzling risk-return relation. Some researchers have shown that a positive risk-return relation exists when hedging demands are included in the asset pricing specifications (e.g. Scruggs (1998) and Hedegaard and Hodrick (2014b)). Many other existing papers have argued that the risk-return relation is highly sensitive to the way a proxy of conditional variance is computed (Ghysels, Plazzi, and Valkanov (2013) and Hedegaard and Hodrick (2014a)). However, notably, using the nearly two-century history of the equity market data, Lundblad (2007) argues that the risk-return relationship is positive and significant regardless of conditional variance and without hedging components. His Monte Carlo analysis shows that researchers need more than 100 years of realized return data to estimate the relation between the market risk premium and conditional variance with any precision. He argues that the weak empirical relationship found from the previous research may be viewed as a statistical artefact of small samples and that in the GARCH-M context, one simply requires sufficiently a long span of data in order to detect this relationship. This paper reexamines the risk-return relation over a two-century of stock and bond 1 Lettau and Ludvigson (2010) provide an extensive review of the literature on the risk-return tradeoff. 1

4 market data from 1836:01 to 2010:12. This paper first estimates a univariate GARCH-M model for UK equity data, and presents evidence that a long span of time-series does not seem to guarantee a significantly positive risk-return relation. Following Scruggs (1998) and Scruggs and Glabadanidis (2003), a version of the Merton (1973) ICAPM using Engle (2002) s dynamic conditional correlation model (DCC) is also estimated, but even this model produces a rather weak risk-return relation. 2 Recently, several empirical studies (see e.g., Brandt and Kang (2004) and Bollerslev, Gibson, and Zhou (2011)) have documented unstable or countercyclical variation in the risk-return tradeoff. Moreover, theoretically, the risk-return relation can be time-varying. For example, in Merton (1973) ICAPM, the intertemporal relation between the conditional mean and variance of market returns is interpreted as the relative risk aversion of the representative investor. Many existing applied theory papers have emphasized timevarying risk aversion to examine several puzzles in financial markets: The external habit formation model of Campbell and Cochrane (1999) uses the surplus consumption ratio to proxy for time-varying relative risk aversion, and their model successfully matches the historical equity premium. Further, Wachter (2006) and Verdelhan (2010) extend the Campbell and Cochrane (1999) model to the bond market and the foreign exchange market, respectively, to explain the expectation hypotheses puzzles. These asset pricing models usually presume a positive risk aversion or risk-return tradeoff. However, a negative relation is also easily justified. For example, Whitelaw (2000) shows that the negative relation exists when the market excess returns acts as a proxy for hedging components in a regime switching consumption based model. To incorporate this unstable risk-return relation, this paper develops new nonlinear CAPM and two-factor ICAPM with the time-varying risk-return tradeoff. Because it is 2 Recently Hedegaard and Hodrick (2014b) argues that using cross-section of high-frequency (e.g., daily) stock returns are crucial to reveal the positive risk-return tradeoff. This approach is not taken here because of data availability. 2

5 not easy to determine the sources of time-varying relation in the long run data due to historical data limitation, this paper proposes a novel econometric framework using state space models with conditional variance, the latent time-varying risk-return tradeoff, and hedging component. Based on these new models, this paper addresses the following important questions that have not been addressed in the literature: What degree of time-variation in riskreturn tradeoff is supported by the longest historical data? How important is the issue of the instability for the tradeoff (e.g., relative to the issue of including the hedging components)? By how much do the weak risk-return tradeoff change once the riskreturn instability is taken into account? In summary, this paper finds that the riskreturn relation is indeed time-varying and largely positive across time. Even when the point estimate indicates the negative relation, it is not statistically different from zeros with 90% confidence bounds. These results are robust after controlling for a proxy of hedging components and with data. Monte Carlo simulations are also conducted for univariate nonlinear models to confirm that these models do not report spurious time-varying risk-return tradeoff. This paper concludes that the time-varying risk-return tradeoff is the main culprit for the seemingly weak risk-return relation over two centuries of financial market data. The remainder of this paper is organized as follows. Section 2 provides the asset pricing framework and the empirical models for the risk-return relationship. Section 3 presents the econometric methodology to estimate the proposed nonlinear CAPM and ICAPM. Section 4 describes the data. Section 5 first provides the time-series evidence on the risk-return relationship implied by CAPM and ICAPM with a constant risk-return tradeoff, and then reports the main empirical results for the new nonlinear models with the time-varying risk-return tradeoff. Section 5 also presents the robustness of these results using various specifications and data. Section 6 concludes. 3

6 2 Risk and Return in Equilibrium Merton (1973) derives the dynamic risk-return trade-off between the conditional mean of the return on the wealth portfolio, E t [r M,t+1 ], in relation to its conditional variance, σ 2 M,t and the conditional covariance with variation in the investment opportunity set,σ MF,t : [ JW W W E t [r M,t+1 r f,t ] = J W ] [ ] σm,t 2 JW F + σ MF,t (2.1) J W J (W (t), F (t), t) is the indirect utility function in wealth, W (t), and F (t) describing the evolution of the investment opportunity set over time; subscripts denote partial [ ] J derivatives, and W W W J W is the coefficient of relative risk aversion, denoted as λ M, [ ] J which is typically assumed to be positive. The W F W J W in the second component describes the hedging coefficient 3. The sign of the hedging coefficient is indeterminate because it depends on the relationship between the marginal utility of wealth and the state of the world, and the conditional covariance. If the investment opportunity set is time-invariant, Merton (1980) shows that the hedging component is negligible and the conditional excess market return is proportional to its conditional variance. [ JW W W E t [r M,t+1 r f,t ] = J W ] σ 2 M,t (2.2) Since Merton (1980), this conditional CAPM specification has been subject to dozens of empirical investigations. This paper first estimates the following univariate GARCH- M model (Model 1) used in Lundblad (2007). r M,t+1 r f,t = λ 0,M + λ M σ 2 M,t + ε M,t+1 (2.3) where ε M,t+1 N(0, σ 2 M,t ) and σ2 M,t = δ 0,M + δ 1,M ε 2 M,t + δ 2,Mσ 2 M,t 1. r M,t+1 r f,t 3 Cochrane (2014) calls this coefficient as investors state-variable aversion 4

7 is the stock market return in excess of the conditionally risk free rate. Usual parameter restrictions are imposed to guarantee positive definite covariance estimates (δ 0,M > 0, δ 1,M > 0, δ 2,M > 0, and δ 1,M + δ 2,M < 1). λ 0,M is added to account for transaction costs or taxes. 4 Lundblad (2007) finds that in this specification, λ M is positive and statistically significant with a long span of data. In preliminary empirical investigations, I experimented with various asymmetric GARCH specifications but the asymmetric terms are statistically insignificant at any significance level for both UK stock and bond returns data. Therefore, I present empirical results only with the symmetric GARCH specifications for UK data. 5 Scruggs (1998), however, argues that the partial relationship between market risk premia and conditional volatility is masked in the univariate context by failing to account for the additional hedging demands associated with a time-varying investment opportunity set. Scruggs (1998) advocates a two-factor ICAPM by including the return on a long-term Treasury bond as the hedging component as follows. E t [r M,t+1 r f,t ] = λ M σ 2 M,t + λ F σ MF,t where σ MF,t is the conditional covariance estimate computed using the stock market excess return (r M,t+1 r f,t ) and the bond market return in excess of the conditionally risk free rate (r F,t+1 r f,t ). Scruggs (1998) motivates this specification from Merton (1973) s suggestion that the risk-free interest rate is the best candidate as a state variable because investors would desire to hedge against its unexpected changes. However, this one equation model is 4 Scruggs (1998, P.589) further shows that estimating this model without a constant can lead to biased or misleading estimates. 5 For data, the asymmetric term is marginally significant (7%). As a result, several models with asymmetric terms are estimated for data in the robustness analysis. 5

8 only partially correct because Merton (1973) ICAPM implies the following asset pricing form for any financial asset. E t [r i,t+1 r f,t ] = λ M cov [r i,t+1, r M,t+1 ] + λ H cov [r i,t+1, r H,t+1 ] where r H,t+1 is the return to the hedge portfolio. If the bond market return (r F,t+1 ) is used as a proxy for the return to the hedge portfolio, the bond market excess return should be written in the following form. E t [r F,t+1 r f,t ] = λ M cov [r F,t+1, r M,t+1 ] + λ F cov [r F,t+1, r F,t+1 ] Many existing papers (e.g. Scruggs and Glabadanidis (2003) and Hedegaard and Hodrick (2014b)) combine these two equations for the stock and bond excess returns to estimate λ M and λ F jointly after constraining these parameters to be identical, consistent with the ICAPM theory. Following this tradition, this paper uses the long term bond return in excess of the risk free as a proxy for hedging portfolios, and employs the following bivariate GARCH-M (Model 2). r M,t+1 r f,t = λ 0,M + λ M σ 2 M,t + λ F σ MF,t + ε M,t+1 r F,t+1 r f,t = λ 0,F + λ M σ MF,t + λ F σf,t 2 + ε F,t+1 cov [ε M,t+1, ε F,t+1 ψ t ] = Σ t, Σ t = σ2 M,t σ MF,t σ MF,t σ 2 F,t (2.4) where ψ t is the information set up to time t. To describe the time-series evolution of the stock and bond market return conditional covariance matrix, this paper employs the following DCC (1,1) specification. 6 6 I thank an anonymous referee for suggesting this model. The previous version of this paper uses a (more restricted) diagonal BEKK (1,1) model. 6

9 Σ t = σ M,t 0 0 σ F,t 1 ρ 12,t ρ 12,t 1 σ M,t 0 0 σ F,t where ρ 12,t = q 12,t q11,t q 22,t, σi,t 2 = δ 0,i + δ 1,i ε 2 i,t + δ 2,i σi,t 1 2 and z i,t = ε i,t/ σi,t 1 for i = M, F +c 1 z2 M,t z M,t z F,t +c 2 q 11,t 1 q 11,t q 12,t q 12,t q 22,t = (1 c 1 c 2 ) 1 c 0 c 0 1 z M,t z F,t z 2 F,t q 12,t 1 q 12,t 1 q 22,t 1 In addition to the usual parameter restrictions for conditional variance parameters as in the GARCH-M model presented before, the parameter restrictions for ρ 12,t are imposed ( 1 < c 0 < 1, c 1 > 0, c 2 > 0, and c 1 + c 2 < 1). This specification guarantees the positive definiteness of the conditional covariance matrix Σ t, and yet allows time variation in conditional variances, covariances, and correlations across these markets. Consistent with the univariate GARCH-M analysis of the market portfolio returns and bond returns, I do not include asymmetric terms for UK data. Recently, several empirical studies (see e.g., Brandt and Kang (2004) and Bollerslev, Gibson, and Zhou (2011)) have documented unstable or countercyclical variation in the risk-return tradeoff. Lundblad (2007) also provides some preliminary evidence to show that the fundamental risk-return relationship has changed over time. To incorporate this unstable risk-return relation, this paper first develops an econometric model for the following new nonlinear CAPM with the time-varying risk-return tradeoff (Model 3). r M,t+1 r f,t = λ 0,M + λ M,t σm,t 2 + ε M,t+1 λ M,t = λ M,t 1 + ω M,t where ε M,t+1 N(0, σm,t 2 ), σ2 M,t = δ 0,M + δ 1,M ε 2 M,t 1 + δ 2,MσM,t 2, and ω M,t N(0, σm) 2 While Lundblad (2007) presents evidence only within univariate context, this paper employs a more general two-factor ICAPM model with both time-varying risk-return 7

10 relation and hedging coefficient (Model 4). r M,t+1 r f,t = λ 0,M + λ M,t σ 2 M,t + λ F,tσ MF,t + ε M,t+1 r F,t+1 r f,t = λ 0,F + λ M,t σ MF,t + λ F,t σf,t 2 + ε F,t+1 cov [ε M,t+1, ε F,t+1 ψ t ] = Σ t, Σ t = σ2 M,t σ MF,t σ MF,t σ 2 F,t (2.5) where λ M,t = λ M,t 1 + ω M,t, ω M,t N(0, σm), 2 λ F,t = λ F,t 1 + ω F,t, ω F,t N(0, σf 2), cov [ω M,t ω F,t ] = σ m,f, and Σ t follows the same DCC (1,1) as in the two-factor ICAPM presented before. This paper models time-varying risk-return tradeoff and the hedging coefficient as a driftless random walk. In other words, while these coefficients are exposed to normal random shock, I do not assume systematic movements in these coefficients, and consider changes in them as unpredictable. This random walk specification allows for gradual changes of coefficients, and is well known for capturing a persistent yet slow movement in time-varying coefficient models with stochastic volatility (e.g. Cogley and Sargent (2005)). The random walk specification for these coefficients, however, is theoretically unattractive as a data generating process for the returns on the financial assets. For example, these coefficients might drift to arbitrarily high or low values, and consequently returns could be non-stationary. 7 Nevertheless random walk assumptions are quite common in the recent return predictability literature. For example, Ferreira and Santa-Clara (2011) assume that the dividend-price ratio follows a random walk in their return forecasting model. More importantly, after carefully comparing AR (1) and random walk coefficient models in the return forecasting setup, Dangl and Halling (2012) convincingly demon- 7 I thank an anonymous referee for pointing out this important issue. 8

11 strate that any deviation from the assumption of no predictability (i.e., random walk) in the shocks on coefficients reduces the predictive power of their regression system. They find that the main advantage of the random walk assumption is a substantial reduction in estimation errors as fewer parameters have to be estimated. They also argue that the estimation of return predictability on a monthly basis empirically mitigates any concerns about the random walk assumption while they state that such concerns might become more important when updating at a lower frequency. Interestingly, the empirical models employed in this paper can be regarded as return forecasting models using random coefficients and conditional variance. Indeed, one of Dangl and Halling (2012) s empirical models are directly comparable to the models of this paper while they use realized variance instead of GARCH, and avoid complex econometric issues dealt in this paper. Moreover, the present study uses monthly returns, which could further reduce some concerns on the random walk specification. To further establish the robustness of the time-varying risk-return tradeoff, this paper also presents the empirical results for a univariate nonlinear CAPM with an AR (1) risk-return tradeoff using UK data in the robustness section. Assuringly the implied risk-return tradeoff is time-varying and statistically significant. However, the empirical results using stationary coefficient models seem to request further investigations for the pile-up problem and numerical concerns. 8 8 More details will be provided in the robustness section. In the time-varying coefficient model literature, it is well known that the variance of time-varying coefficient could have a large point mass at 0, implying a constant coefficient model. This phenomenon is called as a pile-up problem. See Stocks and Watson (1998) and references therein for more details. 9

12 3 Estimation Methods for a Two-factor Nonlinear ICAPM I present the estimation framework for the proposed two-factor nonlinar ICAPM (Model 4) as the following state space model with GARCH terms. Measurement equation: r M,t r f,t 1 = r F,t r f,t 1 ε M,t ε F,t ψ t 1 σ2 M,t 1 σ MF,t 1 N 0 2 σ MF,t 1 σf,t 1 2 σ2 M,t 1 σ MF,t 1 λ M,t 1 + λ 0,M σ MF,t 1 σf,t 1 2 λ F,t 1 λ 0,F + ε M,t ε F,t In matrix terms, y t = Σ t 1 β t 1 + A + ε t, ε t N(0, Σ t 1 ) where ψ t 1 is the information set up to time t-1. Σ t = σ2 M,t σ MF,t σ MF,t σ 2 F,t with DCC (1,1), y t is a 2 x 1 vectors of returns observed at time t; β t is a 2 x 1 vector of unobserved state variables; Σ t 1 is a 2 x 2 matrix that links the observed vector y t and the unobserved β t ; A is a 2 x 1 constant vector. Transition equation: λ M,t = 1 0 λ M,t λ F,t λ F,t 1 + ω M,t ω F,t, ω M,t ω F,t N 0 2 σ2 m σ m,f σ m,f σ 2 f In matrix terms, β t = F β t 1 + ω t, ω t N(0, Q) where β t is a 2 x 1 vector of unobserved state variables; F is 2 x 2 ; ω t is 2 x 1. Following Harvey, Ruiz, and Sentana (1992), I augment the heteroskedastic shocks 10

13 into the original state vector in the transition equation to get the conditional expectations of the squares of the unobserved shocks. The transformed state space model has modified equations as follows. Measurement equation: r M,t r f,t 1 r F,t r f,t 1 = 0 0 σ2 M,t 1 σ MF,t σ MF,t 1 σf,t λ M,t λ F,t λ M,t 1 λ F,t 1 ε M,t + λ 0,M λ 0,F ε F,t In matrix terms, y t = H t β t + A. Transition equation: λ M,t λ F,t λ M,t 1 λ F,t 1 ε M,t ε F,t = λ M,t 1 λ F,t 1 λ M,t 2 λ F,t 2 ε M,t 1 ε F,t 1 + ω M,t ω F,t λ M,t 1 λ F,t 1 ε M,t ε F,t 11

14 ω M,t ω F,t λ M,t 1 λ F,t 1 ε M,t N 0 6 σm 2 σ m,f σ m,f σf σm,t 1 2 σ MF,t 1 ε F,t σ MF,t 1 σ 2 F,t 1 In matrix terms, the transformed transition equation is stated as βt = F β t 1 + ωt, ωt N(0, Q t ) Given the model s parameters, the linear Kalman filter for the state-space model consists of the following six equations: 9 Prediction: β t t 1 = F β t 1 t 1, p t t 1 = F p t 1 t 1 F + Q t, η t t 1 = y t H t β t t 1 A, f t t 1 = H t p t t 1 H t, where β t t 1 = E[β t ψ t 1 ], β t 1 t 1 = E[β t 1 ψ t 1 ], p t t 1 = E[(β t E[β t ψ t 1 ]) 2 ], p t 1 t 1 = E[(β t 1 E[β t 1 ψ t 1 ]) 2 ], η t t 1 = y t E[y t ψ t 1 ], f t t 1 = E[(η t t 1 ) 2 ]. Updating: β t t = β t t 1 + p t t 1 H t f t t 1 1 η t t 1, p t t = p t t 1 p t 1 t 1 H t f t t 1 1 H t p t t 1, where β t t = E[β t ψ t ], p t t = E[(β t E[β t ψ t ]) 2 ]. 9 This paper closely follows the notations in the chapter 6 of Kim and Nelson (1999). To initialize the Kalman filter for the random walk transition equation, two assumptions are made (β 0 0 = 0 and it takes a huge conditional variance p 0 0 = 0). To reduce the potential problem related to this arbitrary initial value, the real data from 1800:01 to 1835:12 are used to construct the β 1836: :01 and p 1836: :01 while these observations are not used in the maximum likelihood estimation. 12

15 To process the above Kalman filter, following Harvey, Ruiz, and Sentana (1992), I approximate ε 2 M,t 1 (ε2 F,t 1 ) with E[ε2 M,t 1 ψ t 1] (E[ε 2 F,t 1 ψ t 1]) respectively in the Σ t 1 matrix. Given the fact that ε M,t 1 = E[ε M,t 1 ψ t 1 ] + ε M,t 1 E[ε M,t 1 ψ t 1 ], ε F,t 1 = E[ε F,t 1 ψ t 1 ] + ε F,t 1 E[ε F,t 1 ψ t 1 ], we compute E[ε 2 M,t 1 ψ t 1] = E[ε M,t 1 ψ t 1 ] 2 + E[(ε M,t 1 E[ε M,t 1 ψ t 1 ]) 2 ] and E[ε 2 F,t 1 ψ t 1] = E[ε F,t 1 ψ t 1 ] 2 +E[(ε F,t 1 E[ε F,t 1 ψ t 1 ]) 2 ] where E[ε M,t 1 ψ t 1 ] and E[ε F,t 1 ψ t 1 ] are obtained from the last two elements of β t 1 t 1. E[(ε M,t 1 E[ε M,t 1 ψ t 1 ]) 2 ] and E[(ε F,t 1 E[ε F,t 1 ψ t 1 ]) 2 ] are obtained from the last two diagonal elements of p t 1 t 1. Finally, I approximate ε M,t 1 ε F,t 1 as (E[ε 2 M,t 1 ψ t 1]E[ε 2 F,t 1 ψ t 1]) 0.5. Because this last approximation of ε M,t 1 ε F,t 1 implicitly assumes that two shocks should have the same signs, this model is denoted as an approximate DCC model. However, the approximation errors seem small in this paper. The GARCH variance and covariance estimates presented in the Section 5 for this model are almost same as those from a bivariate GARCH-M with DCC specification. As by-products of the above Kalman filter, I obtain the prediction error η t t 1 and its variance f t t 1. This prediction error decomposition induces the approximate log likelihood as follows. ln L = 1 2 T ln((2π) n f t t 1 ) 1 2 t=1 T η t t 1 f 1 t t 1 η t t 1 t=1 which can be maximized with respect to the unknown parameters of the model for an approximate Quasi-MLE. 13

16 4 Data As Lundblad (2007) carefully demonstrates, we need a long span of data to investigate the risk-return trade-off to enhance the power of the time-series analysis. To maximize the power of the time-series analysis, this paper employs the longest monthly UK equity and bond market data from 1836:01 to 2010:12. Earlier UK data extending back to 1800 are also available, but I exclude this sample in the maximum likelihood estimation because the stock market data only represent a simple equal weighted average of three shares: the Bank of England, the East India Company, and the South Sea Company. For a more detailed explanation on the data sources, see the documentation from The UK historical stock data (ticker symbol: TFTASD) are taken from the Global Financial Data provider, and represent the FTSE All Shares historical index. I also collect total return data for the UK short-term bill (ticker symbol: TRGBRBIM) and long-term consol bond (ticker symbol: TRGBRGCM) from the same provider. Shortterm bill data will serve as the conditionally risk-free rate in my analysis. As suggested by many existing studies (e.g., Merton (1973) and Scruggs (1998)), I collect long-term U.K. bond returns to capture variation in the investment opportunity set over time. Table 1 reports summary statistics on the total returns for the UK equity market, r M,t, the bond market, r F,t, and the short bill return (the conditionally risk-free rate), r f,t, for the full sample. All variables are expressed as continuously compounded returns. The return data for each series are also displayed in Figure 1. In the whole sample, the mean return on the U.K. stock market portfolio is about 0.57% per month. As expected, the stock market return is highly volatile (3.6% per month). Long term bond and short term bill returns have similar lower mean return around 0.36% per month and also lower volatility as expected. 14

17 5 Empirical Analysis 5.1 Constant Expected Return - Variance Tradeoff A Conditional CAPM Many previous studies on the risk-return tradeoff employ a univariate GARCH-M framework (Model 1). In summary, these previous papers typically find a statistically insignificant or a negative relationship between the market risk premium and its expected variance. A notable exception is Lundblad (2007). Using simulations, he demonstrates that even 100 years of data constitute a small sample that may easily lead to this puzzling insignificant or negative risk-return relation even though the true risk-return tradeoff is positive. Using the nearly two century history of equity market returns, Lundblad estimates a positive and statistically significant risk-return tradeoff across every specification considered. Table 2 presents evidence on the risk-return tradeoff in the univariate context with a long span of UK data from 1836:01 to 2010:12. I use continuously compounded returns as in Scruggs (1998) and report the estimates with the usual symmetric GARCH-M because preliminary investigations (untabulated) reveal that asymmetric terms in the GARCH specifications are not statistically significant. 10 The point estimate, presented in panel A, is 1.66 with a t statistics of 1.81, which is quite different from Lundblad s estimate using simple returns (2.469 with a standard error of in his table). 11 To reconcile the difference, I also estimate the same model with simple returns, and the estimates are provided in panel B. The mean variance tradeoff becomes positive (2.2522) and 10 Results are available upon request. Table 4 in Lundblad (2007) also presents evidence that asymmetric GARCH models are unnecessary for UK data. 11 The smaller coefficient for the GARCH-M model using continuously compounded returns can be understood based on the Jensen s inequality. I thank an anonymous referee for pointing out this interpretation. 15

18 statistically significant. Because many existing papers in this literature such as Scruggs (1998) employ continuously compounded returns, these conflicting results from different ways of computing returns are unconvincing. 12 Figure 2 displays the estimates of conditional market variance implied by the GARCH- M model. Elevated volatility is most pronounced during the Great Depression, stock market crash, the recent global financial crises of the late This volatility pattern seems to capture historical episodes well. For example, during the stock market crash, London Stock Exchange s FT 30 lost 73% of its value. The UK went into recession in 1974, with GDP falling by 1.1 %. At the time, the UK s property market was going through a major crisis, and a secondary banking crisis forced the Bank of England to bail out a number of lenders. After the definitive market low for the FT30 Index on January 6th 1975 when the index closed at 146, the market almost doubled over next 3 months A Two-factor ICAPM Scruggs (1998) argues that the partial relationship between risk premia and conditional volatility can be masked in the univariate context by failing to account for the additional hedging demands associated with a time varying investment opportunity set (essentially generating an omitted variable bias). This issue will be explored in this section based on the intuition of Merton (1973) and the empirical model of Scruggs and Glabadanidis (2003). Specifically, a two-factor ICAPM with DCC (1,1) is estimated and presented in panel A, using the market portfolio excess return and the bond market excess return, after imposing the restrictions consistent with the ICAPM (Model 2). To conduct an explicit test on the ICAPM, the following unrestricted model is also estimated and 12 As one of robustness analysis, a similar model using continuously compounded return data is estimated in the robustness section. Contrary to the findings of Lundblad (2007), the risk-return tradeoff is also not statistically significant. 16

19 presented in panel B. r M,t+1 r f,t = λ 0,M + λ 1,M σm,t 2 + λ 1,F σ MF,t + ε M,t+1 r F,t+1 r f,t = λ 0,F + λ 2,M σ MF,t + λ 2,F σf,t 2 + ε F,t+1 Following the univariate GARCH-M analysis of the market portfolio returns and bond returns, 13 asymmetric terms are not included in the DCC specification. Panel A in Table 3 presents estimation results for the two-factor ICAPM with DCC (1,1) specification (Model 2). First, the conditional variances for the bond and stock markets are persistent in the full historical record. Figure 3 presents time-series plots of the conditional stock and bond return heteroskedasticities and the conditional covariance. The equity variance displays similar patterns presented in the univariate model (see Figure 2). The bond variance is quite low compared with the stock variance but it increases dramatically after 1980 s and especially during the recent crisis. Panel A in Table 3 also presents evidence on the intertemporal relationship between risk and expected return. The partial relationships between the expected market excess return and market conditional variance is positive and statistically significant with t statistics of However, the partial relationship between expected market excess returns and covariance with variation in the investment opportunity set is not statistically significant. Furthermore, the ICAPM restrictions (λ 1,M = λ 2,M and λ 1,F = λ 2,F ) are rejected at 1% level. 14 As a result, a two factor ICAPM does not seem to be supported by the data. 13 The empirical results from a univariate GARCH-M model for the bond excess returns are available upon request. 14 This test is included to see if the ICAPM restrictions are indeed supported. I thank an anonymous referee for the motivating comment. 17

20 5.2 Time-varying Expected Return - Variance Tradeoff Time-varying Risk-return Tradeoff? As discussed before, there are several concerns associated with the potentially strong assumption of a time-invariant risk-return tradeoff. First, in equilibrium, the mean variance tradeoff can be interpreted as risk aversion and may exhibit countercyclical variation through time as implied by habit models such as Campbell and Cochrane (1999), Wachter (2006), and Verdelhan (2010). Moreover, risk aversion may also vary because of the evolution of financial markets and improved risk sharing. Recently, several empirical studies (see e.g., Brandt and Kang (2004) and Bollerslev, Gibson, and Zhou (2011)) have documented unstable or countercyclical variation in the risk-return tradeoff. Before conducting a formal econometric analysis in the next section, this paper first presents preliminary evidence of risk-return instability by recursively (with expanding window) estimating the univariate GARCH-M model starting from 1876:01 at least with 40 years of data. Figure 4 presents these recursive sample estimates of the risk-return tradeoff. While it is difficult to argue for the time-varying risk-return relation with any formal statistics at this stage, it looks as if the risk-return tradeoff varies a lot Time-varying risk-return tradeoff with a nonlinear CAPM To incorporate this unstable risk-return relation, this paper first develops a new nonlinear CAPM with the time-varying risk-return tradeoff (Model 3), and investigates whether the proposed new model sheds light on the puzzling weak risk-return relation. Table 4 presents the parameter estimates, and Figure 5 plots the monthly time-series of the time-varying risk-return tradeoff along with 90 % confidence bounds and conditional variance estimates of the nonlinear CAPM. First, the GARCH parameters in Table 4 are similar in magnitudes to those presented in Table 2 for the simpler CAPM with a 18

21 constant risk-return relation. The second figure in Figure 5 also confirms that the stock return variance follows the same patterns presented (Figure 2) in the conditional CAPM. Second, Table 4 shows that the standard deviation estimates in the time-varying risk-return (σ m ) is and statistically significant with t-statistics If this estimate equals zero statistically, the risk-return tradeoff is constant over time. Thus, the nonlinear CAPM with GARCH-M (Model 3) nests the specification of the conditional CAPM (Model 1). The statistically significant σ m parameter directly indicates that the Model 3 is superior to the Model Figure 5 shows the time-varying riskreturn relation and its 90% confidence bands estimated from the nonlinear CAPM. The estimated expected market return variance tradeoff is largely positive and statistically significant for the full historical record. Further, this figure shows that the seemingly negative risk-return relation could be entirely spurious because the estimated relation is not statistically different from zeros with the 90% confidence bounds. This evidence indicates that incorporating time-varying risk-return relation is crucial to understand the puzzling risk-return tradeoff. The next section investigates robustness of the results to minimize potential misspecification problems. First, as robustness tests for Guo, Wang, and Yang (2013) s argument, a two-factor nonlinear ICAPM (Model 4) is developed and estimated by combining Models 2 and 3. Second, a univariate nonlinear CAPM with the stationary AR (1) risk-return tradeoff is estimated. The time-varying risk-return tradeoff is confirmed with these specifications. Third, Monte Carlo simulations are conducted to confirm the validity of univariate nonlinear CAPMs. Finally, the time-varying risk-return tradeoff is also confirmed (albeit weakly) using data. 15 I thank an anonymous referee for suggesting this clarification. 19

22 5.3 Robustness checks A Time-varying risk-return Tradeoff with a Nonlinear ICAPM Recently Guo, Wang, and Yang (2013) use a version of the conditional ICAPM and argue that the relative risk aversion (the partial relation between expected return and variance) is constant if a hedging component is included in the ICAPM. In other words, they argue that it is unclear if the time-varying relation exist or it just indicates misspecifications of the model without hedging components. To reduce this possibility, this paper estimates a two-factor nonlinear ICAPM with DCC (1,1) (Model 4) by combining Models 2 and 3. Preliminary investigations reveal that the variance estimate in the hedging coefficient is not statistically significant. 16 Therefore, estimation results for the nonlinear ICAPM with the time-varying risk-return tradeoff and constant hedging coefficient are presented in this section. Table 5 presents the parameter estimates of this nonlinear ICAPM. First, the variance estimate in the time-varying risk-return (σ v ) is statistically significant at 6% level. If this estimate equals zero statistically, the risk-return tradeoff is constant over time. Thus, the nonlinear ICAPM (Model 4) nests the specification of the conditional CAPM (Model 2). The statistically significant σ m parameter directly indicates that the Model 4 is superior to the Model 2. However, the λ F parameter is not statistically significant, indicating that this two-factor ICAPM is not empirically supported. At the minimum, the empirical results of this section seem to support the time-varying risk-return tradeoff with a reasonable hedging component, contrary to the empirical findings of Guo, Wang, and Yang (2013). Figure 6 presents time-series plots of the conditional stock and bond return variances and the conditional stock-bond return covariance estimated from this model. The stock return variance follows the same patterns presented in the conditional 16 Results are available upon request. 20

23 CAPM and ICAPM (see Figure 2 and 3). The bond variance displays clustering consistent with the estimates from the conditional ICAPM (see Figure 3) as well, increasing dramatically during the early 1980 s. Over the full historical record, the conditional covariance between the stock and bond market is a small positive number. Figure 7 shows the time-varying risk-return relation and its 90% confidence bands estimated from this two-factor nonlinear ICAPM. The estimated partial expected market return variance tradeoff varies across time and is largely positive for the full historical record. While the magnitude of the time-varying risk-return relation is smaller than that estimated from the univariate nonlinear CAPM, the overall pattern looks quite similar. This figure also shows that the seemingly negative relation could be entirely spurious because the estimated relation is not statistically different from zeros with the 90% confidence bounds A Stationary Time-varying Risk-return Tradeoff with a Nonlinear CAPM This subsection reports estimation results for the following conditional CAPM with the AR(1) risk-return tradeoff. r M,t+1 r f,t = λ 0,M + λ M,t σm,t 2 + ε M,t+1 λ M,t = µ λ + ϕ λ λ M,t 1 + ω M,t Table 6 presents the parameter estimates of this model. First, the GARCH parameters are similar in magnitudes to those presented in Table 4 for the nonlinear CAPM with the random walk risk-return tradeoff. Interestingly, the ϕ λ estimate is close to one, i.e., the random walk assumption, and the standard deviation estimates in the timevarying risk-return (σ m ) is statistically significant. As in the nonlinear CAPM with the random walk risk-return tradeoff (Model 3), this would indicate statistically significant 21

24 time-varying relation. However, the empirical results using stationary coefficient models seem to request further investigations. First, the magnitude of estimated variance of the time-varying coefficient is extremely small (0.0016). This would reflect the pile-up problem in the time-varying coefficient model literature. In other words, the variance of time-varying coefficient could have a large point mass at 0. Comparing the magnitudes of the σ m estimates from the two nonlinear models seem to suggest that this problem is more severe for stationary coefficient models. Furthermore, my investigations (untabulated) reveal that AR(1) models for these coefficients induce many numerical problems, including identification issues for a two-factor nonlinear ICAPM Monte Carlo Analysis This paper argues that the time-varying risk-return relation is the main culprit of the seemingly weak risk-return relation. However, as documented in Lundblad (2007), it is difficult to argue for the time-varying risk-return tradeoff without simulation studies given the noisy realized market returns. Specifically, one could argue that it is hard to see if the time-variation found when estimating the non-linear model is truly generated by the data, or if it is simply noise induced by a more complicated model. 17 To explore this issue further, I conduct a Monte Carlo analysis on the distribution of the risk-return tradeoff coefficient. Specifically, I simulate 2,100 (i.e. corresponding to the sample size in this paper) monthly excess returns from the GARCH-M model with the parameters given in Table 2 for 1,000 times. Data generating process: a conditional CAPM with GARCH-M r M,t+1 r f,t = σ 2 M,t + ε M,t+1, where ε M,t+1 N(0, σ 2 M,t ) σ 2 M,t = ε2 M,t σ2 M,t 1 17 I thank an anonymous referee for suggesting this investigation. 22

25 For each simulation, two non-linear conditional CAPMs are estimated, one which assumes the random walk risk-return tradeoff and one which takes the stationary AR (1) assumption for the tradeoff. In other words, I estimate these models with the time-varying risk-return tradeoff on simulated data from the GARCH-M model to see how much timevariation these models would predict on simulated data with no time-variation in the risk-return relation. Table 7 presents several statistics computed from this simulation study. First, average parameter estimates and t-statistics are reported. Second, the number of rejections (Count) of the constant risk-return tradeoff from two models are presented, i.e., Count takes one if t-statistics for σ m is larger than 2. For the random walk risk-return tradeoff model (panel A), usual averages of the estimated parameters and t-statistics are computed and reported. However, for the stationary risk-return tradeoff model in panel B, numerical difficulties are sometimes observed: the standard errors of estimated parameters are not computed for 13 out of 1000 simulations, and extremely low or high standard errors are sometimes observed. Therefore, for the stationary tradeoff model in panel B, the averages of parameters and t-statistics for the estimated parameters are computed after removing 13 cases first, and trimming 1% of highest and lowest values (t-statistics) of the remaining 987 cases. First, the λ 0,M and GARCH parameters from two models are remarkably similar in magnitudes to the true GARCH parameters used in the data simulation. This evidence suggests that the volatility specification in the GARCH-M context is a second order effect for inference about the risk return tradeoff. Second, both models produce insignificant average t-statistics for the risk-return tradeoff, (0.5599) for the random walk (stationary) model, respectively. More importantly, the Count takes only 10 out of 1000 simulations for the random walk model. While there is still a 1% chance of spuriously identifying the time-varying risk-return tradeoff, this Monte Carlo analysis indicates 23

26 that the spurious relation is extremely rarely observed. However, for the stationary riskreturn tradeoff model, almost a 6% rejection rate is documented. While a 6% rejection rate should not be considered as a failure of the model, the stationary risk-return tradeoff model seems to generate at least more severe problems, warranting the caution for using the model. In sum, this Monte Carlo study shows that the time-variation of risk-return tradeoff found when estimating the non-linear model in the previous section is truly generated by the data, and it is hard to interpret it as a statistical artifact of simple noise induced by a more complicated model Risk-return Tradeoff in To further investigate the robustness of the time-varying risk-return tradeoff, this section estimates several empirical asset pricing models using data from 1836:01 to 2010:12. The historical stock data (ticker symbol: SPXTRD) are taken from the Global Financial Data provider, and represent the S&P Total Return index. I also collect total return data for the short-term bill (ticker symbol: TRABIM) and long-term consol bond (ticker symbol: TRG10M) from the same provider. Short-term bill data will serve as the conditionally risk-free rate in my analysis. 18 First, the conditional CAPM with the following asymmetric GARCH term is estimated ( Model 1). Model 1: rm,t+1 r f,t = λ 0,M + λ M,t σ2, M,t + ε M,t+1 σ 2, M,t = δ0,m + (δ 1,M + δ 3,M dum t)ε,2 M,t + δ2,m σ2, M,t 1 18 The overall pattern for data is similar to that for the equivalent UK data (untabulated). In the whole sample, the mean return on the stock market portfolio is higher than the other two returns, and is highly volatile. Long term bond and short term bill returns have much lower volatility as expected. 24

27 where ε M,t+1 N(0, σ2, M,t ), σ2, M,t = δ0,m + (δ 1,M + δ 3,M dum t)ε,2 M,t + δ2,m σ2, M,t 1, ω M,t N(0, σ 2, m ), and dum t = 1 if ε M,t < 0. To guarantee positive definite covariance estimate, the following restrictions are imposed (δ 0,M δ 1,M + 0.5δ 3,M + δ 2,M < 1). > 0, δ 1,M > 0, δ 2,M > 0, and Panel A of Table 8 presents evidence on the risk-return tradeoff in the univariate context with a long span of data from 1836:01 to 2010:12. As before, I use continuously compounded returns. Contrary to the findings of Lundblad (2007), the risk-return tradeoff (λ M ) is not statistically significant. One potential explanation for the different results is the different data source. While the present study collects the data from the Global Financial Data provider, Lundblad (2007) combines various historical data. At the least, the positive risk-return tradeoff doesn t seem to be robust. Notably, for data, the asymmetric term (δ3,m ) is statistically significant at 7% level, implying the so-called leverage effect. Second, the following nonlinear conditional CAPM with the time-varying risk-return tradeoff is estimated using data from 1846:01 to 2010:12 ( Model 2). 19 Model 2: rm,t+1 r f,t = λ 0,M + λ M,t σ2, M,t + ε M,t+1 λ M,t = λ M,t 1 + ω M,t As in Model 1, the asymmetric GARCH term is included in the model. However, this new term complicates the econometric framework. Specifically, Section 2 demonstrates that ε 2 M,t is not observed, and should be approximated. For the same reason, we need to approximate ε M,t to estimate the above model. One natural candidate seems to be (E[ε M,t ψ t]. While it is unclear how much approximation errors are induced, the 19 The first 10 years of data are used only to initialize the Kalman filter for this and the next models and discarded for the maximum likelihood estimation. 25

28 approximation errors seem small in this particular application. The GARCH parameter estimates presented in panel B of Table 8 are almost same as those in panel A of the same table. Panel B also shows that the standard deviation estimates in the time-varying risk-return (σ m ) is highly significant, implying the nonlinear CAPM with GARCH-M ( Model 2) is the better statistical model over the conventional GARCH-M model. However, σ m is close to zero (0.0105), implying that the proposed nonlinear model is largely indistinguishable from a constant risk-return tradeoff model. Figure 8 shows the time-varying risk-return relation and its 90% confidence bands. This figure also indicates that while the risk-return tradeoff varies across time, the magnitude of the variations is much smaller compared to that for the equivalent model for UK data. While certainly the time-varying risk-return tradeoff exists in a statistical sense, further investigations are warranted for resolving unsatisfactory and inconclusive results. Finally, the following two-factor nonlinear ICAPM with DCC (1,1) is estimated ( Model 3). Model 3: rm,t+1 r f,t = λ 0,M + λ M,t σ2, M,t + λ F,t σ MF,t + ε M,t+1 rf,t+1 r f,t = λ 0,F + λ M,t σ MF,t + λ F,t σ2, F,t + ε F,t+1 λ M,t = λ M,t 1 + ω M,t and λ F,t = λ F,t 1 + ω F,t [ cov t ε M,t+1, ] ε F,t+1 where ω M,t σ 2, i,t σ2, M,t σ MF,t N(0, σ2, m ), ωf,t σ MF,t σ 2, F,t = = δ0,i + (δ1,i + δ3,i dum i,t )ε 2, i,t + δ if ε i,t < 0 for i = M, F q 11,t q12,t = (1 c 1 c 2 ) 1 c 0 q12,t q22,t c 0 1 σ M,t 0 0 σ F,t N(0, σ 2, f ), cov [ ωm,t ω F,t 2,i σ 2, i,t 1 +c 1, and z i,t z2, M,t zm,t z F,t 1 ρ 12,t ρ 12,t 1 ] = σ m,f, ρ / = ε i,t zm,t z F,t z 2, F,t σ 12,t = σ M,t 0 0 σ F,t q 12,t q 11,t q 22,t i,t 1 and dum i,t = 1 +c 2 q 11,t 1 q 12,t 1 q 12,t 1 q 22,t 1, Table 9 presents parameter estimates of this nonlinear ICAPM. First, the variance 26

29 estimates in the time-varying risk-return (σm ) and hedging coefficients (σf U S ) are statistically significant at 1% level. Statistically this evidence is reassuring because it supports the time-varying risk-return trade off along with the changing hedging coefficient. However, the magnitudes for these estimates are again close to zero, implying that the estimated time-varying coefficients are almost indistinguishable (at least economically) from the conventional CAPM or ICAPM. Figure 10 shows the time-varying risk-return relation and hedging coefficient with their 90% confidence bands estimated from the nonlinear ICAPM. As in Model 2, this figure also indicates that while the risk-return tradeoff and hedging coefficient vary across time, the magnitude of the variations is too much small with wider confidence bands. I defer this topic to the future studies. 6 Conclusion While the risk-return tradeoff is fundamental to finance, the empirical evidence on the relationship between the risk premium on aggregate stock market and the variance of its return is ambiguous at best. Lundblad (2007) argues the main culprit of this puzzling relationships is the small sample problem. He finds a statistically significant positive risk-return tradeoff using information from two century history of stock market returns in all of the econometric specifications used in his paper. However, this paper presents new evidence that the risk-return tradeoff is rather weak even with the two century history of UK continuously compounded return data, when this paper employs a time-invariant conditional CAPM or two-factor ICAPM. In the conditional CAPM, the risk-return tradeoff parameter is positive yet statistically insignificant at 5% level. When the time-varying investment opportunity set is explicitly accounted as the hedging component, the risk-return tradeoff becomes positive and statistically significant at 5% level. But the hedging coefficient is insignificant even at 27

30 10% level, and the ICAPM restrictions are strongly rejected. Motivated by existing theoretical arguments, empirical evidence, and preliminary recursive estimates, this paper develops and estimates new nonlinear CAPM and ICAPM with the time-varying risk-return tradeoff. Based on these new models, this paper addresses the following important questions that have not been addressed in the literature: What degree of time-variation in risk-return tradeoff is supported by the longest historical data? How important is the issue of the instability for the tradeoff (e.g., relative to the issue of including the hedging components)? By how much do the weak risk-return tradeoff change once the risk-return instability is taken into account? In summary, this paper finds that the risk-return relation is indeed time-varying and largely positive across time. Even when the point estimate indicates the negative relation, it is not statistically different from zeros with 90% confidence bounds. These results are robust after controlling for a proxy of hedging components and with data (albeit weakly). Monte Carlo simulations are also conducted for univariate nonlinear models to confirm that these models do not report spurious results. This paper concludes that the time-varying risk-return tradeoff is the main culprit for the seemingly weak riskreturn relation over two centuries of financial market data. 28