Long Forward Probabilities, Recovery, and the Term Structure of Bond Risk Premiums

Long Forward Probabilities, Recovery, and the Term Structure of Bond Risk Premiums Abstract This paper examines the assumption of transition independence of the stochastic discount factor (SDF) in the bond market. This assumption underlies the recovery result of Ross 2015. Following the methodology of Alvarez and Jermann 2005 and Hansen and Scheinkman 2009, we estimate the martingale component in the long-term factorization of the SDF using U.S. Treasury data. The empirically estimated martingale component is highly volatile and produces a downward-sloping term structure of bond Sharpe ratios. In contrast, the transition independence assumption implies a degenerate martingale component and an upward-sloping term structure of bond Sharpe ratios. Thus, transition independence is inconsistent with our empirical results. Received April 17, 2016; editorial decision January 17, 2018 by Editor Stijn Van Nieuwerburgh. Ross (2015) shows that under the assumptions that all uncertainty in the economy follows a discrete-time irreducible Markov chain and that the stochastic discount factor (SDF) is transition independent, there exists a unique recovery of subjective transition probabilities of investors from observed Arrow-Debreu prices (Carr and Yu 2012 extend to one-dimensional diffusions on a bounded interval; Walden 2017 extends to more general one-dimensional diffusions; and Qin and Linetsky 2016 extend to general Markov processes). Under the assumption of rational expectations, it leads to the recovery of the data-generating transition probabilities.1 Transition independence of the SDF is the key assumption that allows Ross to appeal to the Perron-Frobenius theory to achieve a unique recovery. The Perron-Frobenius theorem asserts existence and uniqueness of a positive eigenvector of an irreducible nonnegative matrix. It is this unique positive eigenvector that allows Ross to establish a unique recovery under the assumption of transition independence of the SDF. To assess empirical implications of the recovery, it is important to understand economic consequences of the transition independence assumption. Hansen and Scheinkman (2017), Borovička, Hansen, and Scheinkman (2016), Martin and Ross (2013), and Qin and Linetsky (2016) connect Ross’ recovery to the factorization of Hansen and Scheinkman (2009) and show that the transition independence assumption in a Markovian model implies that the martingale component in the long-term factorization of the SDF is degenerate and equal to unity. Hansen and Scheinkman (2017) and Borovička, Hansen, and Scheinkman (2016) point out that such degeneracy is inconsistent with many structural dynamic asset pricing models in the literature, as well as with the empirical evidence in Alvarez and Jermann (2005) and Bakshi and Chabi-Yo (2012) based on bounds on the permanent and transitory components of the SDF. The focus of this paper is on assessing the plausibility of the transition independence assumption in the bond market. This paper explicitly extracts transitory and permanent (martingale) components in the long-term factorization of the stochastic discount factor (SDF) of Alvarez and Jermann (2005) and Hansen and Scheinkman (2009) (see also Qin and Linetsky 2017). We posit an arbitrage-free dynamic term structure model (DTSM), estimate it on the time series of U.S. Treasury yield curves, and explicitly determine the long-term factorization of the SDF via the Perron-Frobenius extraction of the principal eigenfunction following the methodology of Hansen and Scheinkman (2009) (see also Qin and Linetsky 2016). With the estimated long-term factorization in hand, we are able to empirically test the structural assumption of transition independence of the SDF underpinning the recovery result of Ross (2015). Consistent with the calibrated structural example in Borovička, Hansen, and Scheinkman (2016), as well as the empirical literature relying on bounds and finite-maturity proxies for the long-term bond (Alvarez and Jermann 2005; Bakshi and Chabi-Yo 2012; Bakshi, Chabi-Yo, and Gao 2018), we find that the martingale component in the long-term factorization of the SDF is highly volatile, rather than degenerate, as would be implied by the transition independence assumption. The martingale component of the long-term factorization defines the long-term risk-neutral probability measure (Hansen and Scheinkman 2009, 2017; Hansen 2012; Borovička, Hansen, and Scheinkman 2016) that also can be identified with the long forward measure. The long forward measure can be understood as the long-term limit of $$T$$-maturity forward measures associated with $$T$$-maturity pure discount bonds, as the maturity $$T$$ becomes asymptotically large (see Qin and Linetsky 2017 for details). The Ross recovery procedure via the Perron-Frobenius eigenvector yields this long forward measure, rather than the physical measure. The martingale connects the long forward and physical measures. If the martingale is degenerate, then the long forward measure is identified with the physical measure (under the assumption of rational expectations, as discussed above). However, our empirical results show that the martingale is highly volatile, thus driving the wedge between the long forward measure and the physical measure. We now sketch the theory underlying the analysis in the paper. First, we briefly recall the long-term factorization of the SDF (Alvarez and Jermann 2005; Hansen and Scheinkman 2009; Hansen 2012; Borovička, Hansen, and Scheinkman 2016; Qin and Linetsky 2016, 2017): \begin{align} \frac{S_{t+\tau}}{S_t}=\frac{1}{R^\infty_{t,t+\tau}}\frac{M_{t+\tau}}{M_t}, \end{align} (1) where $$S_t$$ is the pricing kernel process, $$R^\infty_{t,t+\tau}$$ is the gross holding period return on the long bond (limit of gross holding period returns $$R_{t,t+\tau}^T=P_{t+\tau,T}/P_{t,T}$$ on pure discount bonds maturing at time $$T$$ as $$T$$ grows asymptotically large), and $$M_t$$ is a martingale. This martingale defines the long forward measure we denote by $${\mathbb L}$$ (in this paper we denote the physical or data-generating measure by $${\mathbb P}$$ and the risk-neutral measure by $${\mathbb Q}$$). Under $${\mathbb L}$$, the long bond serves as the growth optimal numeraire portfolio (see Borovička, Hansen, and Scheinkman 2016; Qin and Linetsky 2017). By Jensen’s inequality, the expected log return on any other asset is dominated by the long bond: \begin{equation} {\mathbb E}_t^{\mathbb L}\left[\log R_{t,t+\tau}\right]\leq {\mathbb E}_t^{\mathbb L}\left[\log R^\infty_{t,t+\tau} \right], \end{equation} (2) where $$R_{t,t+\tau}=V_{t+\tau}/V_t$$ is the gross holding period return on an asset with the value process $$V$$, and the expectation is taken under the long forward measure $${\mathbb L}$$. To put it another way, only the covariance with the long bond is priced under $${\mathbb L}$$, with all other risks neutralized by distorting the probability measure: \begin{equation} {\mathbb E}_t^{\mathbb L}\left[R_{t,t+\tau}\right]-R^f_{t,t+\tau}=-{\rm cov}_t^{\mathbb L}\left(R_{t,t+\tau},\frac{1}{R^\infty_{t,t+\tau}}\right)R^f_{t,t+\tau}, \end{equation} (3) where $$R^f_{t,t+\tau}=1/P_{t,t+\tau}$$ is the gross holding period return on the risk-free discount bond. Dividing both sides by the conditional volatility of the asset return $$\sigma_t^{\mathbb L}(R_{t,t+\tau})$$, the conditional Sharpe ratio under $${\mathbb L}$$ is \begin{equation} {\mathcal S}{\mathcal R}_t^{\mathbb L}(R_{t,t+\tau})=-{\rm corr}_t^{\mathbb L}\left(R_{t,t+\tau},\frac{1}{R^\infty_{t,t+\tau}}\right)R^f_{t,t+\tau}\sigma_t^{\mathbb L}\left(1/R_{t,t+\tau}^\infty\right). \end{equation} (4) The perfect negative correlation gives the Hansen and Jagannathan (1991) bound under $${\mathbb L}$$: \begin{equation} {\mathcal S}{\mathcal R}_t^{\mathbb L}(R_{t,t+\tau})\leq \sigma_t^{\mathbb L}\left(1/R_{t,t+\tau}^\infty\right) R^f_{t,t+\tau}. \end{equation} (5) Assuming that the Markovian SDF is transition independent implies that the martingale component is degenerate, that is, $$S_{t+\tau}/S_t=1/R^\infty_{t,t+\tau}$$. This identifies $${\mathbb P}$$ with $${\mathbb L}$$ and identifies the long bond with the growth optimal numeraire portfolio in the economy (see also result 5 in Martin and Ross 2013 and section 4.3 in Borovička, Hansen, and Scheinkman 2016) and implies that the only priced risk in the economy is the covariance with the long bond. In particular, applying Equation (4) to returns $$R^T_{t,t+\tau}$$ on pure discount bonds, Equation (4) predicts that bond Sharpe ratios are generally increasing in maturity and approach their upper bound (Hansen-Jagannathan bound (5)) at asymptotically long maturities.2 However, this sharply contradicts empirical evidence in the U.S. Treasury bond market. Duffee (2010), Frazzini and Pedersen (2014), and van Binsbergen and Koijen (2017) document that short-maturity bonds have higher Sharpe ratios than do long maturity bonds. Backus, Boyarchenko, and Chernov (forthcoming) and van Binsbergen and Koijen (2017) provide recent bibliographies to the growing literature on the term structure of risk premiums. In this paper we focus on the term structure of bond risk premiums. The empirical term structure of bond Sharpe ratios is generally downward sloping, rather than upward sloping. Frazzini and Pedersen (2014) offer an explanation based on the leverage constraints faced by many bond market participants that result in their preference for longer maturity bonds over leveraged positions in shorter maturity bonds, even if the latter offer higher Sharpe ratios. Furthermore, empirical results in this paper show that leveraged short-maturity bonds achieve substantially higher expected log-returns than long-maturity bonds and, in particular, the (model-implied) long bond. This empirical evidence puts in question the assumption of transition independent and degeneracy of the martingale component in the U.S. Treasury bond market. In Section 1 we estimate an arbitrage-free DTSM on the U.S. Treasury bond data. The zero interest rate policy (ZIRP) in the United States since December of 2008 is an added challenge. Most conventional DTSM do not handle the zero lower bound (ZLB) well. Gaussian models allow unbounded negative rates, whereas CIR-type affine factor models feature vanishing volatility at the ZLB. Shadow rate models are essentially the only class of dynamic term structure models in the literature at present capable of handling the ZLB. The shadow rate idea comes from Black (1995). Gorovoi and Linetsky (2004) provide an analytical solution for single-factor shadow rate models and calibrate them to the term structure of Japanese government bonds (JGB). Kim and Singleton (2012) estimate two-factor shadow rate models on the JGB data. In this paper we estimate the two-factor shadow rate model B-QG2 (Black quadratic Gaussian two factor) shown by Kim and Singleton (2012) to provide the best fit among the model specifications they consider in their investigation of the JGB market. In Section 2 we perform Perron-Frobenius extraction in the estimated model, extract the principal eigenvalue and eigenfunction, construct the long-term factorization of the pricing kernel, and recover the long-term risk-neutral measure (long forward measure) dynamics of the underlying factors. We then directly compare market price of risk processes under the estimated data-generating probability measure and the recovered long-term risk-neutral measure. The difference in these market prices of risk is highly significant. It is identified with the instantaneous volatility of the martingale component. In Section 3 we explore economic implications of our results. We use our model-implied long-term bond dynamics to estimate expected log returns on the long bond and test how far it is from growth optimality implied by the assumption that the martingale component is unity. We find that duration-matched leveraged positions in short and intermediate maturity bonds have significantly higher expected log returns than long maturity bonds and, in particular, the long bond. We also estimate the realized term structure of Sharpe ratios for bonds of different maturities and conclude that it is downward sloping, consistent with the empirical evidence in Duffee (2010), Frazzini and Pedersen (2014), and van Binsbergen and Koijen (2017). We further consider Sharpe ratio forecasts under our estimated probability measures $${\mathbb P}$$ and $${\mathbb L}$$. We find, in particular, that $${\mathbb L}$$ implies forecasts for excess returns on shorter-maturity bonds (up to 3 years) that are essentially zero (risk neutral), while significant excess returns with high Sharpe ratios are observed empirically in this segment of the bond market and are correctly forecast by our estimated $${\mathbb P}$$ measure. Thus, identifying $${\mathbb P}$$ and $${\mathbb L}$$ leads to sharply distorted risk-return trade-offs in the bond market. We also test the null hypothesis that the martingale is equal to unity (and, hence, the data-generating probability measure is identical to the long forward measure) and reject it at the 99.99% level. Finally, in Section 4 we perform robustness check. We note that our econometric approach in this paper is entirely different from the approaches of Alvarez and Jermann (2005), Bakshi and Chabi-Yo (2012), and Bakshi et al. (2018), who rely on bounds on the transitory and martingale components, while we directly estimate a fully specified DTSM, explicitly accomplish the Perron-Frobenius extraction of Hansen and Scheinkman (2009) and obtain the permanent and martingale components in the framework of our DTSM. We also note recent work by Christensen (2017), who develops a nonparametric approach to the Perron-Frobenius extraction and estimates permanent and transitory components under structural (Epstein-Zin and power utility) specifications of the SDF calibrated to real per-capita consumption and real corporate earnings growth. These three lines of inquiry, our parametric modeling and estimation based on asset market data, Christensen’s modeling based on macroeconomic fundamentals, and the approaches of Alvarez and Jermann (2005), Bakshi and Chabi-Yo (2012), and Bakshi, Chabi-Yo, and Gao (2018) based on bounds are complementary, and all result in the conclusion that the martingale component is highly economically significant. 1. Dynamic Term Structure Model Estimation We use the data set of daily constant maturity (CMT) U.S. Treasury bond yields from October 1, 1993 to August 19, 2015. The data include daily yields for Treasury constant maturities of 1, 3, and 6 months and 1, 2, 3, 5, 7, 10, 20, and 30 years. Since our focus is on the Perron-Frobenius extraction of the principal eigenvalue and eigenfunction governing the long-term factorization, we include the long end of the yield curve with 20- and 30-year maturities. Thirty year yield data are missing from February 19, 2002 to February 8, 2006. One-month yield data are missing from October 1, 1993 to July 30, 2001, where the data start with 3-month yields. These missing data do not pose any challenges to our estimation procedure. We obtain zero-coupon yield curves from CMT yield curves via cubic splines bootstrap. Figure 1 shows our time series of bootstrapped zero-coupon yield curves. Figure 1 View largeDownload slide U.S. Treasury zero-coupon yield curves bootstrapped from CMT yield curves Figure 1 View largeDownload slide U.S. Treasury zero-coupon yield curves bootstrapped from CMT yield curves We assume that the state of the economy is governed by a two-factor continuous-time Gaussian diffusion under the data-generating probability measure $$\mathbb{P}$$: \begin{equation} dX_t=K^\mathbb{P}(\theta^\mathbb{P}-X_t) dt+\Sigma dB_t^\mathbb{P}, \end{equation} (6) where $$X_t$$ is a two-dimensional (column) vector, $$B_t^\mathbb{P}$$ is a two-dimensional standard Brownian motion, $$\theta^\mathbb{P}$$ is a two-dimensional vector, and $$K^\mathbb{P}$$ and $$\Sigma$$ are $$2\times2$$ matrices. We assume an affine market price of risk specification $$\lambda^\mathbb{P}(X_t)=\lambda^\mathbb{P}+\Lambda^\mathbb{P} X_t,$$ where $$\lambda^\mathbb{P}$$ is a two-dimensional vector and $$\Lambda^\mathbb{P}$$ is a 2x2 matrix, so that $$X_t$$ remains Gaussian under the risk-neutral probability measure $$\mathbb{Q}$$: \begin{equation} dX_t=K^\mathbb{Q}(\theta^\mathbb{Q}-X_t) dt+\Sigma dB_t^\mathbb{Q}, \end{equation} (7) where $$K^\mathbb{Q}=K^\mathbb{P}+\Sigma\Lambda^\mathbb{P}$$ and $$K^\mathbb{Q}\theta^\mathbb{Q}=K^\mathbb{P}\theta^\mathbb{P}-\Sigma\lambda^\mathbb{P}$$. To handle the ZIRP since December of 2008, we follow Kim and Singleton (2012) and specify Black (1995) shadow rate as the shifted quadratic form of the Gaussian state vector, and the nominal short rate as its positive part (here $$'$$ denotes matrix transposition and $$(x)^+=\max(x,0)$$: \begin{equation} r(X_t)=(\rho+\delta' X_t+ X_t' \Phi X_t)^+. \end{equation} (8) This is the B-QG2 (Black-Quadratic Gaussian two-factor) specification of Kim and Singleton (2012). Following Kim and Singleton (2012), we impose the following conditions to achieve identification: $$K^\mathbb{P}_{12}=0, \delta=0, \Sigma=0.1 I_2$$, where $$I_2$$ is the $$2\times2$$ identity matrix. To ensure existence of the long-term limit (see Qin and Linetsky 2016), we impose two additional restrictions. We require that the eigenvalues of $$K^\mathbb{P}$$ have positive real parts, and $$\Phi$$ is positive semidefinite. The first restriction ensures that $$X$$ is mean-reverting under the data-generating measure $$\mathbb{P}$$ and possesses a stationary distribution. The second restriction ensures that the short rate does not vanish in the long run. The mode of the short rate under the stationary distribution is $$(\rho+(\theta^\mathbb{P})'\Phi\theta^\mathbb{P})^+$$. If $$\Phi$$ is not positive semidefinite, the mode of the short rate under the stationary distribution can be zero. We decompose \begin{equation} \Phi=\begin{bmatrix} 1&0\\ A & 1\end{bmatrix} \begin{bmatrix} D_{1}&0\\0 & D_{2}\end{bmatrix}\begin{bmatrix} 1& A \\0 & 1\end{bmatrix}, \end{equation} (9) and require that $$D_{1},D_{2}\geq0$$ and $$D_1 D_2>0$$. Because of the positive part in the short rate specification, in contrast to one-factor shadow rate models that admit analytical solutions (Gorovoi and Linetsky 2004), the two-factor model does not possess an analytic solution for bond prices. Consider the time-$$t$$ price of the zero-coupon bond with maturity at time $$t+\tau$$ and unit face value: \begin{equation} P_{t,t+\tau}=P(\tau,X_t)=\mathbb{E}^\mathbb{Q}_t[e^{-\int_t^{t+\tau} r(X_s) ds}]. \end{equation} (10) Since the state process is time-homogeneous Markov, the bond pricing function $$P(\tau,x)$$ satisfies the pricing PDE \begin{equation} \frac{\partial P}{\partial \tau}-\frac{1}{2}\text{tr}(\Sigma\Sigma'\frac{\partial^2 P}{\partial x\partial x'} )-\frac{\partial P'}{\partial x} K^\mathbb{Q}(\theta^\mathbb{Q}-x)+r(x)P=0 \end{equation} (11) with the initial condition $$P(0,x)=1$$. We compute bond prices by solving the PDE numerically via an operator splitting finite-difference scheme following the approach in appendix A of Kim and Singleton (2012). Our estimation strategy follows Kim and Singleton (2012). Observed bond yields $$Y^O_{t,\tau_i}$$ are assumed to equal their model-implied counterparts $$Y_{t,\tau_i}=Y(\tau_i,X_t)=-(1/\tau_i)\log P(\tau_i,X_t)$$ plus mutually and serially independent Gaussian measurement errors $$e_{t,\tau_i}\sim N(0, \sigma_{\tau_i}^2)$$. The model is estimated using the extended Kalman-filter based quasi-maximum likelihood function. We follow Kim and Priebsch (2013) in estimating standard errors using the approach of Bollerslev and Wooldridge (1992). Table 1 gives the parameter estimates and standard errors. Table 2 gives the average pricing errors. Our pricing errors are slightly higher than those reported by Kim and Singleton (2012), where the model is estimated on weekly JGB data. It is not surprising, since we use daily data for all maturities from 1 month to 30 years, whereas Kim and Singleton (2012) use weekly data with JGB maturities up to 10 years. Table 1 Model parameter estimates and standard errors (in parentheses) $$K^\mathbb{Q}$$ 0.3253 (0.0034) 0.0420 (0.0005) 0.6388 (0.0081) 0.0813 (0.0018) $$\theta^\mathbb{Q}$$ 0.9316 (0.0175) $$-$$5.9180 (0.1157) $$\rho$$ $$-$$0.0046 (0.0002) $$D_1$$ 0.2839 (0.0085) $$D_2$$ 0.0227 (0.0009) $$A$$ 0.3347 (0.0068) $$\lambda^\mathbb{P}$$ $$-$$0.9085 (0.0785) $$-$$1.0503 (0.0468) $$\Lambda^\mathbb{P}$$ $$-$$3.0535 (1.4167) 0.4203 (0.0051) 4.4189 (1.8992) 0.3964 (0.0173) $$K^\mathbb{Q}$$ 0.3253 (0.0034) 0.0420 (0.0005) 0.6388 (0.0081) 0.0813 (0.0018) $$\theta^\mathbb{Q}$$ 0.9316 (0.0175) $$-$$5.9180 (0.1157) $$\rho$$ $$-$$0.0046 (0.0002) $$D_1$$ 0.2839 (0.0085) $$D_2$$ 0.0227 (0.0009) $$A$$ 0.3347 (0.0068) $$\lambda^\mathbb{P}$$ $$-$$0.9085 (0.0785) $$-$$1.0503 (0.0468) $$\Lambda^\mathbb{P}$$ $$-$$3.0535 (1.4167) 0.4203 (0.0051) 4.4189 (1.8992) 0.3964 (0.0173) Table 1 Model parameter estimates and standard errors (in parentheses) $$K^\mathbb{Q}$$ 0.3253 (0.0034) 0.0420 (0.0005) 0.6388 (0.0081) 0.0813 (0.0018) $$\theta^\mathbb{Q}$$ 0.9316 (0.0175) $$-$$5.9180 (0.1157) $$\rho$$ $$-$$0.0046 (0.0002) $$D_1$$ 0.2839 (0.0085) $$D_2$$ 0.0227 (0.0009) $$A$$ 0.3347 (0.0068) $$\lambda^\mathbb{P}$$ $$-$$0.9085 (0.0785) $$-$$1.0503 (0.0468) $$\Lambda^\mathbb{P}$$ $$-$$3.0535 (1.4167) 0.4203 (0.0051) 4.4189 (1.8992) 0.3964 (0.0173) $$K^\mathbb{Q}$$ 0.3253 (0.0034) 0.0420 (0.0005) 0.6388 (0.0081) 0.0813 (0.0018) $$\theta^\mathbb{Q}$$ 0.9316 (0.0175) $$-$$5.9180 (0.1157) $$\rho$$ $$-$$0.0046 (0.0002) $$D_1$$ 0.2839 (0.0085) $$D_2$$ 0.0227 (0.0009) $$A$$ 0.3347 (0.0068) $$\lambda^\mathbb{P}$$ $$-$$0.9085 (0.0785) $$-$$1.0503 (0.0468) $$\Lambda^\mathbb{P}$$ $$-$$3.0535 (1.4167) 0.4203 (0.0051) 4.4189 (1.8992) 0.3964 (0.0173) Table 2 Average pricing errors (in basis points) 1 m 3 m 6 m 1 yr 2 yr 3 yr 5 yr 7 yr 10 yr 20 yr 30 yr 20 16 8 2 11 11 6 2 14 26 25 1 m 3 m 6 m 1 yr 2 yr 3 yr 5 yr 7 yr 10 yr 20 yr 30 yr 20 16 8 2 11 11 6 2 14 26 25 Table 2 Average pricing errors (in basis points) 1 m 3 m 6 m 1 yr 2 yr 3 yr 5 yr 7 yr 10 yr 20 yr 30 yr 20 16 8 2 11 11 6 2 14 26 25 1 m 3 m 6 m 1 yr 2 yr 3 yr 5 yr 7 yr 10 yr 20 yr 30 yr 20 16 8 2 11 11 6 2 14 26 25 In Table 3 we also report the goodness of fit for bond holding period returns. Specifically, suppose $$R_{t, t+1/4}^\tau$$ is the realized quarterly holding period return of a zero-coupon bond with initial maturity $$\tau$$. Let $$\hat{R}_{t,t+1/4}^\tau$$ denote the corresponding model-implied return $$\hat{R}_{t,t+1/4}^\tau:=\frac{P(\tau-1/4, X_{t+1/4})}{P(\tau, X_t)}-1,$$ where $$P(\tau,X_t)$$ is the price of the bond with time to maturity $$\tau$$ when the state is $$X_t$$ at time $$t$$. Here $$X_t$$ is the filtered state. Table 3 displays average realized returns, model-implied returns, as well as the slope and $$R^2$$ of regressions of realized returns on model-implied returns $$R_{t, t+1/4}^\tau=a+b\hat{R}_{t,t+1/4}^\tau+{\rm noise}$$. Table 3 Goodness of fit for bond returns Maturity 1 yr 2 yr 3 yr 5 yr 7 yr 10 yr 20 yr 30 yr Realized (%) $$0.90$$ $$1.08$$ $$1.24$$ $$1.49$$ $$1.69$$ $$1.95$$ $$2.58$$ $$3.09$$ Model-implied (%) $$0.88$$ $$1.06$$ $$1.22$$ $$1.47$$ $$1.63$$ $$1.78$$ $$2.02$$ $$2.14$$ Regression slope 0.9897 1.0841 1.1188 1.0980 1.0029 0.8693 0.6803 0.6806 $$R^2$$ (%) $$98.54$$ $$95.16$$ $$95.57$$ $$97.96$$ $$99.28$$ $$95.84$$ $$86.03$$ $$69.78$$ Maturity 1 yr 2 yr 3 yr 5 yr 7 yr 10 yr 20 yr 30 yr Realized (%) $$0.90$$ $$1.08$$ $$1.24$$ $$1.49$$ $$1.69$$ $$1.95$$ $$2.58$$ $$3.09$$ Model-implied (%) $$0.88$$ $$1.06$$ $$1.22$$ $$1.47$$ $$1.63$$ $$1.78$$ $$2.02$$ $$2.14$$ Regression slope 0.9897 1.0841 1.1188 1.0980 1.0029 0.8693 0.6803 0.6806 $$R^2$$ (%) $$98.54$$ $$95.16$$ $$95.57$$ $$97.96$$ $$99.28$$ $$95.84$$ $$86.03$$ $$69.78$$ Realized returns are computed as discussed in Section 4 and averaged over the time series. Model-implied returns are computed using the estimated model parameters and filtered states. The regressions are of the realized returns on the model-implied returns. Table 3 Goodness of fit for bond returns Maturity 1 yr 2 yr 3 yr 5 yr 7 yr 10 yr 20 yr 30 yr Realized (%) $$0.90$$ $$1.08$$ $$1.24$$ $$1.49$$ $$1.69$$ $$1.95$$ $$2.58$$ $$3.09$$ Model-implied (%) $$0.88$$ $$1.06$$ $$1.22$$ $$1.47$$ $$1.63$$ $$1.78$$ $$2.02$$ $$2.14$$ Regression slope 0.9897 1.0841 1.1188 1.0980 1.0029 0.8693 0.6803 0.6806 $$R^2$$ (%) $$98.54$$ $$95.16$$ $$95.57$$ $$97.96$$ $$99.28$$ $$95.84$$ $$86.03$$ $$69.78$$ Maturity 1 yr 2 yr 3 yr 5 yr 7 yr 10 yr 20 yr 30 yr Realized (%) $$0.90$$ $$1.08$$ $$1.24$$ $$1.49$$ $$1.69$$ $$1.95$$ $$2.58$$ $$3.09$$ Model-implied (%) $$0.88$$ $$1.06$$ $$1.22$$ $$1.47$$ $$1.63$$ $$1.78$$ $$2.02$$ $$2.14$$ Regression slope 0.9897 1.0841 1.1188 1.0980 1.0029 0.8693 0.6803 0.6806 $$R^2$$ (%) $$98.54$$ $$95.16$$ $$95.57$$ $$97.96$$ $$99.28$$ $$95.84$$ $$86.03$$ $$69.78$$ Realized returns are computed as discussed in Section 4 and averaged over the time series. Model-implied returns are computed using the estimated model parameters and filtered states. The regressions are of the realized returns on the model-implied returns. 2. Long-term Factorization We now turn to constructing the long-term factorization of the SDF process \begin{equation} S_t=e^{-\int_0^t r(X_s)ds}e^{-\int_0^t \lambda^{\mathbb P}(X_s)dB_s^{\mathbb P}-\frac{1}{2}\int_0^t {\vert\vert}\lambda^{\mathbb P}(X_s){\vert\vert}^2ds} \end{equation} (12) in the estimated dynamic term structure model. Consider the gross holding period return on the zero-coupon bond with maturity at time $$T$$ over the period from $$s$$ to $$s+t$$, $$R^T_{s,s+t}=P(T-s-t,X_{s+t})/P(T-s,X_s)$$. We are interested in the limit as $$T$$ goes to infinity (holding period return on the zero-coupon bond of asymptotically long maturity). In Markovian models, if the long-term limit exists (see Qin and Linetsky 2017 for sufficient conditions and mathematical details), then \begin{equation} \lim_{T\rightarrow \infty}R^T_{s,s+t}=e^{\lambda t}\frac{\pi(X_{s+t})}{\pi(X_s)} \end{equation} (13) for some $$\lambda$$ and a positive function $$\pi(x)$$, with $$\pi(x)$$ serving as the positive (principal) eigenfunction of the (time-homogeneous Markovian) pricing operator with the eigenvalue $$e^{-\lambda t}$$: \begin{equation} {\mathbb E}^{\mathbb P}_0[S_t \pi(X_t)]= e^{-\lambda t}\pi(X_0), \end{equation} (14) where $$S_t$$ is the SDF. For the sake of brevity, here we do not repeat the theory of long-term factorization and its connection to the Perron-Frobenius theory and refer the reader to Hansen and Scheinkman (2009), Hansen (2012), Borovička, Hansen, and Scheinkman (2016), and Qin and Linetsky (2016, 2017). In the framework of our model the bond pricing function $$P(t,x)$$ is numerically determined by solving the bond pricing PDE by finite differences. We also determine the principal eigenfunction $$\pi(x)$$ numerically as follows. Choosing some error tolerance $$\epsilon$$, we solve the bond pricing PDE for an increasing sequence of times to maturity indexed by integers $$n$$, consider the ratios $$P(n+1,x)/P(n,x)$$ as $$n$$ increases, and stop at $$n=N$$ such that $$M_N-m_N \leq \epsilon$$ for the first time, where $$M_n=\max_{x\in \Omega} P(n+1,x)/P(n,x)$$ and $$m_n=\min_{x\in \Omega} P(n+1,x)/P(n,x)$$ and the max and min are computed over the grid in the domain $$\Omega$$ where we approximate the bond pricing function by the computed numerical solution of the PDE. The eigenvalue and the principal eigenfunction are then approximately given by $$e^{-\lambda} = (m_N+M_N)/2$$ and $$\pi(x)=e^{\lambda N}P(N,x)$$ in the domain $$x\in\Omega$$ (with the error tolerance $$\epsilon$$). Figure 2 plots the computed eigenfunction $$\pi(x)$$. The corresponding principal eigenvalue is $$\lambda=0.0283$$.3 Figure 2 View largeDownload slide Principal eigenfunction $$\pi(x_1,x_2)$$ with eigenvalue $$\lambda=0.0283$$ Figure 2 View largeDownload slide Principal eigenfunction $$\pi(x_1,x_2)$$ with eigenvalue $$\lambda=0.0283$$ With the principal eigenfunction $$\pi(x)$$ and eigenvalue $$\lambda$$ in hand, we explicitly obtain the long-term factorization: \begin{equation} S_t = \frac{1}{L_t}M_t, \quad L_t = e^{\lambda t}\frac{\pi(X_t)}{\pi(X_0)}, \quad M_t = S_t e^{\lambda t}\frac{\pi(X_t)}{\pi(X_0)}, \end{equation} (15) where $$L_t = R^\infty_{0,t}$$ is the long bond process (gross return from time zero to time $$t$$ on the zero-coupon bond of asymptotically long maturity) determining the transitory component $$1/L_t$$, and $$M_t$$ is the martingale (permanent) component of the long-term factorization. In particular, we can now recover the $${\mathbb L}$$ measure by applying Girsanov’s theorem. First, applying Itô’s formula to $$\log\pi(x)$$ and using the SDE for $$X$$ under $${\mathbb P}$$ we can write \begin{align} \log \frac{\pi(X_t)}{\pi(X_0)}& = \int_0^t \frac{\partial \log \pi}{\partial x'}(X_s) \Sigma dB_s^{\mathbb P}\notag \\ &\quad + \int_0^t \left(\frac{1}{2}\text{tr}(\Sigma\Sigma'\frac{\partial^2 \log\pi}{\partial x\partial x'})(X_s)+ \frac{\partial \log \pi}{\partial x'}(X_s) b^\mathbb{P}(X_s) \right) ds, \end{align} (16) where $$b^{\mathbb P}(x)=K^{\mathbb P}(\theta^{\mathbb P}-x)$$ is the drift under the data-generating measure. Next, we recall that the eigenfunction satisfies the (elliptic) PDE (without the time derivative): \begin{equation} \frac{1}{2}\text{tr}(\Sigma\Sigma'\frac{\partial^2 \pi}{\partial x\partial x'})(x)+\frac{\partial \pi}{\partial x'}(x) b^\mathbb{Q}(x)+(\lambda-r(x))\pi(x)=0, \end{equation} (17) where $$b^{\mathbb Q}(x)=K^{\mathbb Q}(\theta^{\mathbb Q}-x)$$ is the drift under the risk-neutral measure. Using the identity $$\frac{\partial^2 \log\pi}{\partial x\partial x'}=\frac{1}{\pi}\frac{\partial^2 \pi}{\partial x\partial x'}-\frac{1}{\pi^2}\frac{\partial \pi}{\partial x} \frac{\partial \pi}{\partial x'}$$ and the PDE, we can write \begin{align} \log \frac{\pi(X_t)}{\pi(X_0)} & =\int_0^t (\frac{1}{\pi}\frac{\partial \pi}{\partial x'})(X_s) \Sigma dB_s^{\mathbb P}\notag\\ &\quad + \int_0^t \left(r(X_s)-\lambda-(\frac{1}{2\pi^2}\frac{\partial \pi}{\partial x'}\Sigma\Sigma' \frac{\partial \pi}{\partial x})(X_s)+(\frac{1}{\pi}\frac{\partial\pi}{\partial x'}\Sigma\lambda^\mathbb{P})(X_s)\right)ds. \end{align} (18) Substituting this into the expression in Equation (15) for the martingale $$M_t$$, we obtain \begin{equation} M_t=e^{\int_0^t v_s dB^{\mathbb P}_s-\frac{1}{2}\int_0^t {\vert\vert}v_s{\vert\vert}^2ds} \end{equation} (19) with the instantaneous volatility process: \begin{equation} v_t=-\lambda^{\mathbb P}(X_t)+\lambda^{\mathbb L}(X_t), \end{equation} (20) where $$\lambda^{\mathbb P}(x)$$ is the drift of the state vector under the data-generating measure $${\mathbb P}$$, and we introduced the following notation \begin{equation} \lambda^{\mathbb L}(x):=\frac{1}{\pi(x)}\Sigma'\frac{\partial \pi}{\partial x}(x). \end{equation} (21) The martingale defines the long-term risk-neutral measure $${\mathbb L}$$. Applying Girsanov’s theorem, we obtain the drift of the state vector $$X$$ under $${\mathbb L}$$: \begin{equation} b^{\mathbb L}(x)=b^{\mathbb Q}(x)+\Sigma\lambda^{\mathbb L}(x), \end{equation} (22) where $$\lambda^{\mathbb L}(X_t)$$ is thus identified with the market price of risk process under the long-term risk-neutral measure $${\mathbb L}$$. The instantaneous volatility $$v_t=v(X_t)$$ of the martingale component is equal to the difference between the market prices of risk under the long-term risk-neutral measure $${\mathbb L}$$ and the data-generating measure $${\mathbb P}$$ and is explicitly expressed in terms of the principal eigenfunction: \begin{equation} v_t=-\lambda^{\mathbb P}(X_t)+\frac{1}{\pi(X_t)}\Sigma'\frac{\partial \pi}{\partial x}(X_t). \end{equation} (23) To compute the $$\mathbb{L}$$-market price of risk $$\lambda^\mathbb{L}(x)$$, we use Equation (21) with numerically computed eigenfunction. $$\lambda^\mathbb{L}(x)$$ can be well approximated by a linear function on the domain $$[-0.3, 0.2]\times[-0.1, 1.2]$$ of values containing the filtered paths of the state variables as depicted in Figure 3. In Table 4, we compute the linear coefficients of $$\lambda^\mathbb{L}(x)$$ using linear regression on the domain $$[-0.3, 0.2]\times[-0.1, 1.2]$$. Figure 3 View largeDownload slide Numerically computed $$\lambda^L(x)$$ Figure 3 View largeDownload slide Numerically computed $$\lambda^L(x)$$ Table 4 Least-squares fit to $$\lambda^\mathbb{L}(x)$$ Intercept $$x_1$$ $$x_2$$ $$R^2 (\%)$$ 0.158 $$-$$0.385 0.168 $$100$$ $$-$$0.095 0.168 $$-$$0.123 $$100$$ Intercept $$x_1$$ $$x_2$$ $$R^2 (\%)$$ 0.158 $$-$$0.385 0.168 $$100$$ $$-$$0.095 0.168 $$-$$0.123 $$100$$ Table 4 Least-squares fit to $$\lambda^\mathbb{L}(x)$$ Intercept $$x_1$$ $$x_2$$ $$R^2 (\%)$$ 0.158 $$-$$0.385 0.168 $$100$$ $$-$$0.095 0.168 $$-$$0.123 $$100$$ Intercept $$x_1$$ $$x_2$$ $$R^2 (\%)$$ 0.158 $$-$$0.385 0.168 $$100$$ $$-$$0.095 0.168 $$-$$0.123 $$100$$ As a result, the eigenfunction in this shadow rate model is well approximated by an exponential-quadratic function of the form \begin{equation} \pi(x)\approx Ce^{-1.93x_1^2-0.61x_2^2+1.68x_1 x_2 +1.58x_1-0.95x_2}, \end{equation} (24) similar to quadratic term structure models (QTSM) (see Qin and Linetsky 2016 for details on positive eigenfunctions in ATSM and QTSM). Substituting $$\lambda^{\mathbb{L}}(x)$$ into the expression for the drift of the state variables under $${\mathbb L}$$ (Equation (22)), we obtain a Gaussian approximation for the dynamics of the state variables under $$\mathbb{L}$$. We can now explicitly compare the data-generating and long-term risk-neutral dynamics. By inspection we see that all the parameters entering the market prices of risk under $$\mathbb{L}$$ in Table 4 are significantly smaller in magnitude than the parameters in the market prices of risk under the data-generating measure $$\mathbb{P}$$: \begin{equation} \lambda^{\mathbb{P}}(x)=\begin{bmatrix} -0.908\\ -1.050\\ \end{bmatrix}+\begin{bmatrix} -3.054 & 0.420 \\ 4.419 & 0.396\\ \end{bmatrix}\begin{bmatrix} x_1\\ x_2\\ \end{bmatrix}. \end{equation} (25) Thus, we obtain the instantaneous volatility of the martingale component as a function of the state variable via Equation (20): \begin{equation} v(x)\approx\begin{bmatrix} 1.066\\ 0.955\\ \end{bmatrix}+\begin{bmatrix} 2.670 & -0.253 \\ -4.251 & -0.519\\ \end{bmatrix}\begin{bmatrix} x_1\\ x_2\\ \end{bmatrix}. \end{equation} (26) Using our estimated model we can now compute an implied path of long bond returns as follows. Recall that the long bond gross return process is given by $$L_t=R_{0,t}^\infty=e^{\lambda t}\pi(X_t)/\pi(X_0)$$ (see Equation (15)). Figure 4 displays the model-implied path of the long bond in our estimated DTSM obtained by evaluating the expression $$e^{\lambda t}\pi(X_t)/\pi(X_0)$$ on the filtered path of the state vector $$X_t$$, where the principal eigenfunction and eigenvalue are given in Figure 2. The figure also displays the wealth (gross return) processes of investing in 20- and 30-year constant maturity zero-coupon bonds for comparison. The plot is separated into two subperiods since the 30-year bond was discontinued in 2002 and resumed in 2006. Specifically, the 30-year time series shows the value over time of the initial investment of one dollar in the 30-year zero-coupon bond rolled over at 3-month intervals back into the 30-year bond. Figure 4 View largeDownload slide Wealth processes investing in 20- and 30-year zero-coupon constant maturity bonds and the long bond Figure 4 View largeDownload slide Wealth processes investing in 20- and 30-year zero-coupon constant maturity bonds and the long bond In the previous literature researchers use 20- or 30-year bonds as proxies for the long bond. In our framework of the fully specified DTSM, we have access to the model-implied long bond dynamics and can use it as a model-based proxy for the unobservable long bond. Figure 4 shows a still noticeable difference between 30-year bonds and the long bond. 3. The Term Structure of Bond Risk Premiums and the Magnitude of the Martingale Component We now turn to the empirical examination of the term structure of bond risk premiums. Table 5 displays realized average quarterly excess returns, standard deviations and Sharpe ratios for zero-coupon bonds of maturities from one to thirty years over the period where all of them are available. Excess holding period returns are computed over the 3-month zero-coupon bond yields known at the beginning of each quarter. We observe that the term structure of Sharpe ratios is downward sloping, with the 1-year bond earning the quarterly Sharpe ratio of 0.46. This is more than two times the Sharpe ratio of the zero-coupon 30-year bond. This shape of the term structure of Sharpe ratios is in broad agreement with the findings of Duffee (2010), Frazzini and Pedersen (2014), and van Binsbergen and Koijen (2017) and is incompatible with the increasing term structure of Sharpe ratios arising under the assumption of transition independence and degeneracy of the martingale component in the long-term factorization. Using model-implied long bond series, we can also compute the long bond’s excess return, standard deviation and Sharpe ratio, which are $$2.70\%$$, $$20.29\%$$ and $$0.13$$, respectively. In particular, the Sharpe ratio for the long bond is lower than for the 30-year bond. Table 5 Realized average quarterly excess returns, standard deviations, and Sharpe ratios for zero-coupon bonds Maturity (yr) 1 2 3 5 7 10 20 30 Exc. ret. (%) $$0.16$$ $$0.35$$ $$0.50$$ $$0.76$$ $$0.95$$ $$1.22$$ $$1.84$$ $$2.36$$ SD (%) $$0.35$$ $$0.92$$ $$1.47$$ $$2.54$$ $$3.48$$ $$4.74$$ $$8.15$$ $$12.07$$ Sharpe 0.46 0.38 0.34 0.30 0.27 0.26 0.23 0.20 Maturity (yr) 1 2 3 5 7 10 20 30 Exc. ret. (%) $$0.16$$ $$0.35$$ $$0.50$$ $$0.76$$ $$0.95$$ $$1.22$$ $$1.84$$ $$2.36$$ SD (%) $$0.35$$ $$0.92$$ $$1.47$$ $$2.54$$ $$3.48$$ $$4.74$$ $$8.15$$ $$12.07$$ Sharpe 0.46 0.38 0.34 0.30 0.27 0.26 0.23 0.20 Maturities are from 1 to 30 years over the period from October 1, 1993 to February 15, 2002 and from February 9, 2006 to May 26, 2016. Excess returns are computed over the 3-month zero-coupon bond yield known at the beginning of each quarter. Table 5 Realized average quarterly excess returns, standard deviations, and Sharpe ratios for zero-coupon bonds Maturity (yr) 1 2 3 5 7 10 20 30 Exc. ret. (%) $$0.16$$ $$0.35$$ $$0.50$$ $$0.76$$ $$0.95$$ $$1.22$$ $$1.84$$ $$2.36$$ SD (%) $$0.35$$ $$0.92$$ $$1.47$$ $$2.54$$ $$3.48$$ $$4.74$$ $$8.15$$ $$12.07$$ Sharpe 0.46 0.38 0.34 0.30 0.27 0.26 0.23 0.20 Maturity (yr) 1 2 3 5 7 10 20 30 Exc. ret. (%) $$0.16$$ $$0.35$$ $$0.50$$ $$0.76$$ $$0.95$$ $$1.22$$ $$1.84$$ $$2.36$$ SD (%) $$0.35$$ $$0.92$$ $$1.47$$ $$2.54$$ $$3.48$$ $$4.74$$ $$8.15$$ $$12.07$$ Sharpe 0.46 0.38 0.34 0.30 0.27 0.26 0.23 0.20 Maturities are from 1 to 30 years over the period from October 1, 1993 to February 15, 2002 and from February 9, 2006 to May 26, 2016. Excess returns are computed over the 3-month zero-coupon bond yield known at the beginning of each quarter. Table 6 displays average realized quarterly log-returns for duration-matched leveraged or deleveraged investments in zero-coupon bonds of different maturities that match the duration of the 10- and 20-year bond over the period when all of them are available. We observe that leveraged investments in shorter-maturity bonds produce significantly higher average log-returns than duration-matched deleveraged investments in longer maturity bonds. Using our model-implied long bond time series displayed in Figure 4, we estimate the average log-return on the long bond to equal 1.50% over this period. Comparing this with the data in Table 6, we see that most of these duration-matched bond portfolios deliver higher log-returns than the long bond. In particular, all of the investments in bonds of maturities from 1 to 30 years leveraged or deleveraged to match the 20-year duration produce significantly higher average log-returns. These results strongly reject growth optimality of the long bond, consistent with the high volatility of the martingale component in the long-term factorization established in Section 2. Table 6 Realized average quarterly log-returns for leveraged or deleveraged investment in zero-coupon bonds Maturity (yr) 1 2 3 5 7 10 20 30 log ret. (10-yr dur.) (%) $$2.28$$ $$2.33$$ $$2.27$$ $$2.10$$ $$1.96$$ $$1.83$$ $$1.56$$ $$1.43$$ log ret. (20-yr dur.) (%) $$3.72$$ $$3.73$$ $$3.59$$ $$3.22$$ $$2.95$$ $$2.71$$ $$2.23$$ $$1.97$$ Maturity (yr) 1 2 3 5 7 10 20 30 log ret. (10-yr dur.) (%) $$2.28$$ $$2.33$$ $$2.27$$ $$2.10$$ $$1.96$$ $$1.83$$ $$1.56$$ $$1.43$$ log ret. (20-yr dur.) (%) $$3.72$$ $$3.73$$ $$3.59$$ $$3.22$$ $$2.95$$ $$2.71$$ $$2.23$$ $$1.97$$ Different maturities are matched to 10- and 20-year durations. The period is from October 1, 1993 to February 15, 2002, and from February 9, 2006 to May 26, 2016. Table 6 Realized average quarterly log-returns for leveraged or deleveraged investment in zero-coupon bonds Maturity (yr) 1 2 3 5 7 10 20 30 log ret. (10-yr dur.) (%) $$2.28$$ $$2.33$$ $$2.27$$ $$2.10$$ $$1.96$$ $$1.83$$ $$1.56$$ $$1.43$$ log ret. (20-yr dur.) (%) $$3.72$$ $$3.73$$ $$3.59$$ $$3.22$$ $$2.95$$ $$2.71$$ $$2.23$$ $$1.97$$ Maturity (yr) 1 2 3 5 7 10 20 30 log ret. (10-yr dur.) (%) $$2.28$$ $$2.33$$ $$2.27$$ $$2.10$$ $$1.96$$ $$1.83$$ $$1.56$$ $$1.43$$ log ret. (20-yr dur.) (%) $$3.72$$ $$3.73$$ $$3.59$$ $$3.22$$ $$2.95$$ $$2.71$$ $$2.23$$ $$1.97$$ Different maturities are matched to 10- and 20-year durations. The period is from October 1, 1993 to February 15, 2002, and from February 9, 2006 to May 26, 2016. Next, we compare model-based conditional forecasts of excess returns, volatility and Sharpe ratios of zero-coupon bonds of different maturities under the data-generating measure $${\mathbb P}$$ estimated in Section 1 and the long forward measure $${\mathbb L}$$ obtained via Perron-Frobenius extracton in Section 2. Table 7 displays average conditional excess return, volatility and Sharpe ratio forecasts under $${\mathbb P}$$ and $${\mathbb L}$$. Reported values are obtained by calculating conditional forecasts along the filtered sample path of the state vector $$X_t$$ and taking the averages over the time period. Excess return forecasts are over the 3-month zero-coupon bond yield known at the beginning of each quarter. Sharpe ratio forecasts are computed as the ratios of excess return forecast to the volatility forecast. Comparing Sharpe ratio forecasts in Table 7 with Table 5, we observe that $$\mathbb{P}$$-measure Sharpe ratio forecasts exhibit the downward-sloping term structure broadly comparable with the downward-sloping term structure of realized Sharpe ratios in Table 5. In contrast, the $${\mathbb L}$$-measure forecasts exhibit a generally upward-sloping term structure that starts near zero for 1- to 3-year maturities ($${\mathbb L}$$-measure forecasts are essentially risk-neutral for these shorter maturities) and increases toward the Hansen-Jagannathan bound in Equation (5) discussed in the Introduction. The bound is approximately attained by the long bond. Although the long bond is growth optimal, it does not generally maximize the Sharpe ratio since $${\rm corr}_t^{\mathbb L}\left(R^\infty_{t,t+1},1/R^\infty_{t,t+1}\right)$$ is not generally equal to $$-1$$. However, for sufficiently small holding periods this correlation is close to $$-1$$. As is clear in Table 7, the empirically estimated average quarterly $${\mathbb L}$$-Sharpe ratio of the long bond of 0.17 is comparable to its average quarterly volatility estimated to be 0.187. Table 7 Average conditional 3-month excess return, volatility, and Sharpe ratio $${\mathbb P}$$- and $${\mathbb L}$$-forecasts for zero-coupon bonds Maturity (yr) 1 3 5 10 20 30 LB $$\mathbb{P}$$ Ex. ret. (%) $$0.17$$ $$0.49$$ $$0.65$$ $$0.74$$ $$0.67$$ $$0.61$$ $$0.72$$ SD (%) $$0.42$$ $$1.07$$ $$1.64$$ $$4.05$$ $$9.39$$ $$12.86$$ $$17.61$$ Sharpe $$0.40$$ $$0.46$$ $$0.40$$ $$0.18$$ $$0.07$$ $$0.05$$ $$0.04$$ $$\mathbb{L}$$ Ex. ret. (%) $$-0.02$$ $$0.02$$ $$0.16$$ $$0.68$$ $$1.68$$ $$2.31$$ $$3.24$$ SD (%) $$0.44$$ $$1.10$$ $$1.65$$ $$4.16$$ $$9.79$$ $$13.51$$ $$18.70$$ Sharpe $$-0.05$$ $$0.02$$ $$0.10$$ $$0.16$$ $$0.17$$ $$0.17$$ $$0.17$$ Maturity (yr) 1 3 5 10 20 30 LB $$\mathbb{P}$$ Ex. ret. (%) $$0.17$$ $$0.49$$ $$0.65$$ $$0.74$$ $$0.67$$ $$0.61$$ $$0.72$$ SD (%) $$0.42$$ $$1.07$$ $$1.64$$ $$4.05$$ $$9.39$$ $$12.86$$ $$17.61$$ Sharpe $$0.40$$ $$0.46$$ $$0.40$$ $$0.18$$ $$0.07$$ $$0.05$$ $$0.04$$ $$\mathbb{L}$$ Ex. ret. (%) $$-0.02$$ $$0.02$$ $$0.16$$ $$0.68$$ $$1.68$$ $$2.31$$ $$3.24$$ SD (%) $$0.44$$ $$1.10$$ $$1.65$$ $$4.16$$ $$9.79$$ $$13.51$$ $$18.70$$ Sharpe $$-0.05$$ $$0.02$$ $$0.10$$ $$0.16$$ $$0.17$$ $$0.17$$ $$0.17$$ Maturities are from 1 to 30 years, and the model-implied long bond (LB) is over the period from October 1, 1993 to February 15, 2002, and from February 9, 2006 to May 26, 2016. Excess return forecasts are in excess of the 3-month zero-coupon bond yield known at the beginning of each quarter. Table 7 Average conditional 3-month excess return, volatility, and Sharpe ratio $${\mathbb P}$$- and $${\mathbb L}$$-forecasts for zero-coupon bonds Maturity (yr) 1 3 5 10 20 30 LB $$\mathbb{P}$$ Ex. ret. (%) $$0.17$$ $$0.49$$ $$0.65$$ $$0.74$$ $$0.67$$ $$0.61$$ $$0.72$$ SD (%) $$0.42$$ $$1.07$$ $$1.64$$ $$4.05$$ $$9.39$$ $$12.86$$ $$17.61$$ Sharpe $$0.40$$ $$0.46$$ $$0.40$$ $$0.18$$ $$0.07$$ $$0.05$$ $$0.04$$ $$\mathbb{L}$$ Ex. ret. (%) $$-0.02$$ $$0.02$$ $$0.16$$ $$0.68$$ $$1.68$$ $$2.31$$ $$3.24$$ SD (%) $$0.44$$ $$1.10$$ $$1.65$$ $$4.16$$ $$9.79$$ $$13.51$$ $$18.70$$ Sharpe $$-0.05$$ $$0.02$$ $$0.10$$ $$0.16$$ $$0.17$$ $$0.17$$ $$0.17$$ Maturity (yr) 1 3 5 10 20 30 LB $$\mathbb{P}$$ Ex. ret. (%) $$0.17$$ $$0.49$$ $$0.65$$ $$0.74$$ $$0.67$$ $$0.61$$ $$0.72$$ SD (%) $$0.42$$ $$1.07$$ $$1.64$$ $$4.05$$ $$9.39$$ $$12.86$$ $$17.61$$ Sharpe $$0.40$$ $$0.46$$ $$0.40$$ $$0.18$$ $$0.07$$ $$0.05$$ $$0.04$$ $$\mathbb{L}$$ Ex. ret. (%) $$-0.02$$ $$0.02$$ $$0.16$$ $$0.68$$ $$1.68$$ $$2.31$$ $$3.24$$ SD (%) $$0.44$$ $$1.10$$ $$1.65$$ $$4.16$$ $$9.79$$ $$13.51$$ $$18.70$$ Sharpe $$-0.05$$ $$0.02$$ $$0.10$$ $$0.16$$ $$0.17$$ $$0.17$$ $$0.17$$ Maturities are from 1 to 30 years, and the model-implied long bond (LB) is over the period from October 1, 1993 to February 15, 2002, and from February 9, 2006 to May 26, 2016. Excess return forecasts are in excess of the 3-month zero-coupon bond yield known at the beginning of each quarter. We remark that, by comparing Table 5 and 7, we observe that the model-based conditional forecasts of excess returns for bonds with maturities longer than 10 years are lower than their realized counterparts. This suggests that the filtered sample path of the state variables estimated from the given time period may not be representative of the ergodic path. This is anticipated, given that in our sample period bond yields were declining overall. However, this does not affect our conclusion about the volatility of the martingale component because the model-based conditional forecasts of the 1-year bond’s $$\mathbb{P}$$ excess return, standard deviation and Sharpe ratio match their realized counterparts sufficiently well. As we will show below (see Equation (30)), these quantities are already sufficient to assert the magnitude of the volatility of the martingale component. We now discuss the relationship between the term structure of bond Sharpe ratios and the magnitude of the volatility of the martingale component. We first develop a general connection between Sharpe ratios and the magnitude of the volatility of the martingale component in the spirit of Hansen-Jagannathan bounds. Consider any risky asset with the value process (with dividends reinvested) $$V_t$$ in an economy driven by $$d$$ independent standard Brownian motions and with the volatility vector $$\sigma_t$$ loading on these Brownian motions, $$dV_t=(r_t + \sigma_t\cdot \lambda_t^\mathbb{P})V_tdt+V_t\sigma_t\cdot dB_t^{\mathbb P}$$. The instantaneous Sharpe ratio (ISR) of the asset is then \begin{equation} \text{ISR}_t^\mathbb{P}(V)=\frac{\sigma_t\cdot\lambda^\mathbb{P}_t}{{\vert\vert}\sigma_t{\vert\vert}_2}={\vert\vert}\lambda^\mathbb{P}_t{\vert\vert}_2 \cos \theta_{\sigma_t,\lambda^\mathbb{P}_t}. \end{equation} (27) Thus, the norm of the market price of risk vector $${\vert\vert}\lambda^\mathbb{P}_t{\vert\vert}_2$$ determines the upper bound on the ISR in the economy. Similarly, we can write for the ISR under the long forward measure: \begin{equation} \text{ISR}_t^\mathbb{L}(V)=\frac{\sigma_t\cdot\lambda^\mathbb{L}_t}{{\vert\vert}\sigma_t{\vert\vert}_2}={\vert\vert}\lambda^\mathbb{L}_t{\vert\vert}_2 \cos \theta_{\sigma_t,\lambda^\mathbb{L}_t}. \end{equation} (28) We can then decompose the ISR under the data-generating measure in terms of the ISR under the long forward measure and an additional term: \begin{equation} \text{ISR}_t^\mathbb{P}(V)=\text{ISR}_t^\mathbb{L}(V)+\frac{\sigma_t\cdot(\lambda^\mathbb{P}_t-\lambda^\mathbb{L}_t)}{{\vert\vert}\sigma_t{\vert\vert}_2}=\text{ISR}_t^\mathbb{L}(V)-{\vert\vert}v_t{\vert\vert}_2\cos\theta_{v_t,\sigma_t}, \end{equation} (29) where $$v_t=-\lambda_t^\mathbb{P}+\lambda_t^\mathbb{L}$$ is the volatility of the martingale component in the long-term factorization Equation (20). Thus, the volatility of the martingale component in the long-term factorization furnishes the upper bound for the difference in instantaneous Sharpe ratios of an asset under $$\mathbb{P}$$ and $$\mathbb{L}$$: \begin{equation} {\vert\vert}v_t{\vert\vert}_2\geq |\text{ISR}_t^\mathbb{P}(V)-\text{ISR}_t^\mathbb{L}(V)|. \end{equation} (30) This is closely related to proposition 2 in Alvarez and Jermann (2005) and proposition 1 in Bakshi and Chabi-Yo (2012), who also derive bounds on the volatility of the permanent component. The main difference here is that we use the Sharpe ratio instead of return or log-return to establish the bound. Now coming back to our empirical results on the Sharpe ratios of bond returns, recall that the term structure of Sharpe ratios in Table 7 is downward sloping under $${\mathbb P}$$, while it slopes upward under $${\mathbb L}$$. Thus, the 1-year bond has the largest gap between its $${\mathbb P}$$- and $${\mathbb L}$$-Sharpe ratios. Using the estimated values in Table 7 yields a lower bound for the average over the path of the volatility of the martingale component equal to $$2*(0.40-(-0.05))=0.90$$. Thus, we expect the martingale component to have volatility of at least $$90\%$$ on average. With our estimated model in hand, we can also directly estimate the magnitude of the volatility of the martingale component to compare with this lower bound. Recall that the instantaneous volatility as a function of the state is approximately given by Equation (27). Consider the quadratic variation of the log of the martingale component $$ \langle \log M\rangle_t = \int_0^t {\vert\vert}v(X_s){\vert\vert}^2ds. $$ Its expectation under the stationary distribution, which we denote by $$\nu$$, is equal to $$ \mathbb{E}^\mathbb{P}_\nu[{\vert\vert}v(X){\vert\vert}^2]t. $$ Thus, under the stationary distribution the constant $$ \sqrt{\mathbb{E}^\mathbb{P}_\nu[{\vert\vert}v(X){\vert\vert}^2]} $$ summarizes the magnitude of the annualized volatility of the martingale component. Using our parameter estimates, we compute this quantity to equal $$1.32$$. Thus, the estimated magnitude of the volatility of the martingale component is highly significant at $$132\%$$, is larger than the lower bound of $$90\%$$ we previously obtained, and is also much larger than the volatility of the transitory component in the long-term factorization (the reciprocal of the long bond). We next formally test the null hypothesis $$\mathbb{P}=\mathbb{L}$$ (equivalently, degeneracy of the martingale component, $$v_t=0$$) that corresponds to the transition independence assumption on the pricing kernel. The market price of risk under $${\mathbb P}$$ contains six independent parameters that are estimated with standard errors given in Table 1. The market price of risk parameters under the long-term risk-neutral measure are uniquely determined (recovered) from the risk-neutral parameters. We use the delta method to compute the standard errors of our estimated parameters of the volatility of the martingale component $$v_i(x)=v_i+\sum_j v_{ij}x_j$$, $$v_i=-\lambda^\mathbb{P}_{i}+\lambda^\mathbb{L}_{i}$$ and $$v_{ij}=-\Lambda^\mathbb{P}_{ij}+\Lambda^\mathbb{L}_{ij}$$. We stack $$v_i$$ and $$v_{ij}$$ into a column vector $$v$$ and denote the derivative of $$v$$ with respect to the parameters as $$D$$. Then the variance-covariance matrix of $$v$$ is given by $$D\Omega D'$$, where $$\Omega$$ is the variance-covariance matrix of the model parameters. Taking the square root of the diagonal elements, we obtain the standard errors of the parameters $$v_i$$ and $$v_{ij}$$. The first six columns of Table 8 summarize the results. Note that the standard errors of $$v_i$$ and $$v_{ij}$$ from the delta method are very close to the standard errors of $$\mathbb{P}$$-market price of risk from Table 1, as the $$\mathbb{P}$$-market price of risk has much larger estimation errors than the errors for risk-neutral parameters, and dominates the estimation errors of $$\mathbb{Q}$$ parameters. We then compute the $$p$$-values for the joint hypothesis $$v_i=v_{ij}=0$$ for all $$i,j=1,2$$ using the test statistics $$v'(D\Omega D')^{-1}v$$ (which follows $$\chi^2(6)$$ under the null). The last column of Table 8 summarizes the results. The null hypothesis that the long-term risk-neutral measure is identified with the data-generating measure is rejected at least at the $$99.99\%$$ confidence level.4 Table 8 Estimated parameters of the volatility $$v(x)$$ of the martingale component $$v_1$$ $$v_2$$ $$v_{11}$$ $$v_{12}$$ $$v_{21}$$ $$v_{22}$$ Joint test $$v=0$$ Parameters 1.066 0.955 2.669 $$-$$0.253 $$-$$4.251 $$-$$0.519 $$t$$-stat 4,226.07 SE 0.078 0.047 1.416 0.005 1.900 0.173 $$p$$-value $$.00$$ $$v_1$$ $$v_2$$ $$v_{11}$$ $$v_{12}$$ $$v_{21}$$ $$v_{22}$$ Joint test $$v=0$$ Parameters 1.066 0.955 2.669 $$-$$0.253 $$-$$4.251 $$-$$0.519 $$t$$-stat 4,226.07 SE 0.078 0.047 1.416 0.005 1.900 0.173 $$p$$-value $$.00$$ The table presents standard errors, the test statistic, and the $$p$$-value for the joint hypothesis test that all parameters vanish. Table 8 Estimated parameters of the volatility $$v(x)$$ of the martingale component $$v_1$$ $$v_2$$ $$v_{11}$$ $$v_{12}$$ $$v_{21}$$ $$v_{22}$$ Joint test $$v=0$$ Parameters 1.066 0.955 2.669 $$-$$0.253 $$-$$4.251 $$-$$0.519 $$t$$-stat 4,226.07 SE 0.078 0.047 1.416 0.005 1.900 0.173 $$p$$-value $$.00$$ $$v_1$$ $$v_2$$ $$v_{11}$$ $$v_{12}$$ $$v_{21}$$ $$v_{22}$$ Joint test $$v=0$$ Parameters 1.066 0.955 2.669 $$-$$0.253 $$-$$4.251 $$-$$0.519 $$t$$-stat 4,226.07 SE 0.078 0.047 1.416 0.005 1.900 0.173 $$p$$-value $$.00$$ The table presents standard errors, the test statistic, and the $$p$$-value for the joint hypothesis test that all parameters vanish. 4. Robustness Checks In this section we perform several robustness checks. Our data sample includes the financial crisis of 2008 and the subsequent zero interest rate policy regime (ZIRP). To check that our results are not an artifact of special features of the ZIRP regime, we consider the pre-crisis period separately. We also extend our data set to 1987 and consider the pre-crisis period from 1987 to 2008, along with the longer period 1987–2016, which includes the crisis and the subsequent ZIRP. We limit ourselves to data starting in 1987 because of concerns of potential structural breaks in U.S. interest rates due to changes in U.S. monetary policy in the eighties. A substantial literature points out structural changes in the 1980s (e.g., Stock and Watson 2003; Rudebusch and Wu 2007; Smith and Taylor 2009; Joslin, Priebsch, and Singleton 2014). In particular, Joslin, Priebsch, and Singleton (2014) point out that after mid-1980s the long-run mean of the short rate is lower under the risk-neutral measure, yields and macroeconomic variables are more persistent and have lower volatility. Specifically, here we choose to follow Rudebusch and Wu (2007) and take 1987 as the starting year of our data to estimate our DTSM. We nevertheless point out that Frazzini and Pedersen (2014) and van Binsbergen and Koijen (2017) consider a much broader scope of data going back to 1950s and report that the qualitative pattern of the downward-sloping term structure of bond Sharpe ratios has held in this longer time series. Table 9 summarizes the $$p$$-values of our hypothesis test, as well as the magnitude of the annualized volatility of the martingale component. While the precise numerical values of model parameter estimates change with the time period considered, the hypothesis test that $$\mathbb{P}=\mathbb{L}$$ is rejected at least at the $$99.99\%$$ level and the annualized volatility of the martingale component is greater than $$100\%$$ for all periods considered. Table 9 Hypothesis test analysis 87-16 87-16$$^{a}$$ 87-08 87-08$$^{a}$$ 93-15 93-15$$^{a}$$ p-value $$.00$$ $$.00$$ $$.00$$ $$.00$$ $$.00$$ $$.00$$ $$\sqrt{\mathbb{E}_\nu[{\vert\vert}v(X){\vert\vert}^2]} (\%)$$ $$129.27$$ $$209.68$$ $$173.85$$ $$250.86$$ $$132.39$$ $$153.68$$ 87-16 87-16$$^{a}$$ 87-08 87-08$$^{a}$$ 93-15 93-15$$^{a}$$ p-value $$.00$$ $$.00$$ $$.00$$ $$.00$$ $$.00$$ $$.00$$ $$\sqrt{\mathbb{E}_\nu[{\vert\vert}v(X){\vert\vert}^2]} (\%)$$ $$129.27$$ $$209.68$$ $$173.85$$ $$250.86$$ $$132.39$$ $$153.68$$ $$p$$-values of the hypothesis test $${\mathbb P}={\mathbb L}$$ ($$v_i=0$$ and $$V_{ij}=0$$) and the annualized volatility of the martingale component. $$^{a}$$Excludes 20- and 30-year maturities. Table 9 Hypothesis test analysis 87-16 87-16$$^{a}$$ 87-08 87-08$$^{a}$$ 93-15 93-15$$^{a}$$ p-value $$.00$$ $$.00$$ $$.00$$ $$.00$$ $$.00$$ $$.00$$ $$\sqrt{\mathbb{E}_\nu[{\vert\vert}v(X){\vert\vert}^2]} (\%)$$ $$129.27$$ $$209.68$$ $$173.85$$ $$250.86$$ $$132.39$$ $$153.68$$ 87-16 87-16$$^{a}$$ 87-08 87-08$$^{a}$$ 93-15 93-15$$^{a}$$ p-value $$.00$$ $$.00$$ $$.00$$ $$.00$$ $$.00$$ $$.00$$ $$\sqrt{\mathbb{E}_\nu[{\vert\vert}v(X){\vert\vert}^2]} (\%)$$ $$129.27$$ $$209.68$$ $$173.85$$ $$250.86$$ $$132.39$$ $$153.68$$ $$p$$-values of the hypothesis test $${\mathbb P}={\mathbb L}$$ ($$v_i=0$$ and $$V_{ij}=0$$) and the annualized volatility of the martingale component. $$^{a}$$Excludes 20- and 30-year maturities. Because of liquidity concerns with bonds of long maturities, econometric studies of DTSM often consider maturities up to 10 years (e.g., Kim and Singleton 2012). To be consistent with this literature, in the next set of robustness checks, we exclude 20- and 30-year bonds and reestimate the model and recompute all our results on the data set including maturities up to 10 years. We remark that our estimation procedure assumes that bond yield errors are mutually and serially independent, as is standard in the dynamic term structure modeling literature. In contrast, our estimated fitting errors for bond yields of long maturities exhibit some positive serial correlations. This issue is well known in the literature on estimating affine models (see Hamilton and Wu 2014; Duffee 2011). This is not surprising, since our model, though nonlinear, is nevertheless close to the affine class. However, this does not pose a challenge to our result on the magnitude of the martingale component. On one hand, the estimated volatility of the martingale component is very large and in excess of 100% annually. On the other hand, as we demonstrated in Section 3, the downward-sloping term structure of bond Sharpe ratios implies large volatility of the martingale component. It is not subjected to some possible misspecification in measurement errors. In particular, the model-free lower bound on the volatility of the martingale component from Equation (30) is already around 90%, while our estimated volatility is over 100%. More generally, our results in this paper are based on estimating a particular DTSM. While choosing a different model specification would result in some quantitative differences, our qualitative conclusions that the martingale component is highly volatile and produces the generally downward-sloping term structure of bond Sharpe ratios, as opposed to the long forward probability forecast of generally upward-sloping term structure of bond Sharpe ratios, are robust to choosing a particular model specification and an estimation strategy. In particular, the results of Bakshi, Chabi-Yo, and Gao (2018) based on an entirely different procedure of estimating the martingale component from the data including Treasury bond futures and options offer further confirmation that the martingale component is highly volatile. 5. Concluding Remarks This paper has demonstrated that the martingale component in the long-term factorization of the stochastic discount factor (SDF) studied in Alvarez and Jermann (2005) and Hansen and Scheinkman (2009) is highly volatile, produces a downward-sloping term structure of bond Sharpe ratios as a function of bond maturity, and implies that the long bond is far from growth optimality. In contrast, the long forward probabilities forecast a generally upward-sloping term structure of bond Sharpe ratios that starts from near zero for short-term bonds and increases toward the Sharpe ratio of the long bond. This forecast implies that the long bond is growth optimal. Our empirical findings in the U.S. Treasury bond market are inconsistent with the assumption of transition independence of the SDF and degeneracy of the martingale component in its long-term factorization. This paper is based on research supported by grants from the National Science Foundation [CMMI-1536503 and DMS-1514698]. Footnotes 1 We stress that the assumption of rational expectations is critical to the discussion in this paper. Without assuming rational expectations, one would also need to model subjective beliefs $${\mathbb P}^*$$. 2 Although the long bond maximizes the expected log return, it does not generally maximize the Sharpe ratio since $${\rm corr}_t^{\mathbb L}\left(R^\infty_{t,t+\tau},1/R^\infty_{t,t+\tau}\right)$$ is not generally equal to $$-1$$. However, for sufficiently small holding periods, this correlation is close to $$-1$$ in diffusion models. In the empirical results in this paper, for 3-month holding periods, the empirically estimated $${\mathbb L}$$-Sharpe ratio of the long bond is close to its upper bound as discussed in Section 4. 3 We note that the value of $$2.83\%$$ for the asymptotic yield is in broad agreement with the results of Giglio, Maggiori, and Stroebel (2014) on very long-run discount rates. Based on the U.K. data, they argue that real long-run risk-free discount rates are under $$1\%$$ per annum, and perhaps as low as $$0.40\%$$ (see section V.A in their paper). The realized inflation rate in the United States over our estimation period from 1993 to 2015 was $$2.26\%$$. Subtracting this inflation rate from our estimate for the long-run nominal rate of $$2.83\%$$, we arrive at the real rate of $$0.57\%$$, which is consistent with that of Giglio, Maggiori, and Stroebel (2014). 4 We note that our hypothesis test relies on the linear approximation of the $${\mathbb L}$$-market price of risk that itself follows from the exponential-quadratic approximation for the principal eigenfunction. As discussed in Section 2, this approximation is highly accurate in the domain of relevant values of the state variables (in particular, containing all filtered values of the state variables), and, hence, the hypothesis test is robust to this approximation. References Alvarez, F., and Jermann. U. J. 2005 . Using asset prices to measure the persistence of the marginal utility of wealth. Econometrica 73 : 1977 – 2016 . Google Scholar CrossRef Search ADS Backus, D., Boyarchenko, N. and Chernov. M. Forthcoming . Term structures of asset prices and returns. 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Risk, return and Ross recovery. Journal of Derivatives 12 : 38 – 59 . Google Scholar CrossRef Search ADS Christensen, T. M. 2017 . Nonparametric stochastic discount factor decomposition. Econometrica 85 : 1501 – 36 . Google Scholar CrossRef Search ADS Duffee, G. R. 2010 . Sharpe ratios in term structure models. Working papers. Duffee, G. R. 2011 . Forecasting with the term structure: The role of no-arbitrage restrictions. Working papers , Johns Hopkins University. Frazzini, A., and Pedersen. L. H. 2014 . Betting against beta. Journal of Financial Economics 111 : 1 – 25 . Google Scholar CrossRef Search ADS Giglio, S., Maggiori, M. and Stroebel. J. 2014 . Very long-run discount rates. Quarterly Journal of Economics 130 : 1 – 53 . Google Scholar CrossRef Search ADS Gorovoi, V., and Linetsky. V. 2004 . Black’s model of interest rates as options, eigenfunction expansions and Japanese interest rates. Mathematical Finance 14 : 49 – 78 . Google Scholar CrossRef Search ADS Hamilton, J. D., and Wu. J. C. 2014 . Testable implications of affine term structure models. Journal of Econometrics 178 : 231 – 42 . Google Scholar CrossRef Search ADS Hansen, L. P. 2012 . Dynamic valuation decomposition within stochastic economies. Econometrica 80 : 911 – 67 . Google Scholar CrossRef Search ADS Hansen, L. P., and Jagannathan. R. 1991 . Implications of security market data for models of dynamic economies. Journal of Politicial Economy 99 : 225 – 62 . Google Scholar CrossRef Search ADS Hansen, L. P. and Scheinkman. J. 2017 . Stochastic compounding and uncertain valuation. In After the flood , pp. 21 – 50 . Chicago : University of Chicago Press . Google Scholar CrossRef Search ADS Hansen, L. P. and Scheinkman. J. 2009 . Long-term risk: An operator approach. Econometrica 77 : 177 – 234 . Google Scholar CrossRef Search ADS Joslin, S., Priebsch, M. and Singleton. K. J. 2014 . Risk premiums in dynamic term structure models with unspanned macro risks. Journal of Finance 69 : 1197 – 233 . Google Scholar CrossRef Search ADS Kim, D. H., and Priebsch. M. 2013 . Estimation of multi-factor shadow-rate term structure models. Board of Governors of the Federal Reserve System, Washington, DC, October , 9 , 2013 . Kim, D. H., and Singleton. K. J. 2012 . Term structure models and the zero bound: an empirical investigation of Japanese yields. Journal of Econometrics 170 : 32 – 49 . Google Scholar CrossRef Search ADS Martin, I., and Ross. S. 2013 . The long bond. Working Paper . Qin, L., and Linetsky. V. 2016 . Positive eigenfunctions of Markovian pricing operators: Hansen-Scheinkman factorization, Ross recovery and long-term pricing. Operations Research 64 : 99 – 117 . Google Scholar CrossRef Search ADS Qin, L., and Linetsky. V. 2017 . Long term risk: A martingale approach. Econometrica 85 : 299 – 312 . Google Scholar CrossRef Search ADS Ross, S. A. 2015 . The recovery theorem. Journal of Finance 70 : 615 – 48 . 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Long Forward Probabilities, Recovery, and the Term Structure of Bond Risk Premiums

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Abstract

Abstract This paper examines the assumption of transition independence of the stochastic discount factor (SDF) in the bond market. This assumption underlies the recovery result of Ross 2015. Following the methodology of Alvarez and Jermann 2005 and Hansen and Scheinkman 2009, we estimate the martingale component in the long-term factorization of the SDF using U.S. Treasury data. The empirically estimated martingale component is highly volatile and produces a downward-sloping term structure of bond Sharpe ratios. In contrast, the transition independence assumption implies a degenerate martingale component and an upward-sloping term structure of bond Sharpe ratios. Thus, transition independence is inconsistent with our empirical results. Received April 17, 2016; editorial decision January 17, 2018 by Editor Stijn Van Nieuwerburgh. Ross (2015) shows that under the assumptions that all uncertainty in the economy follows a discrete-time irreducible Markov chain and that the stochastic discount factor (SDF) is transition independent, there exists a unique recovery of subjective transition probabilities of investors from observed Arrow-Debreu prices (Carr and Yu 2012 extend to one-dimensional diffusions on a bounded interval; Walden 2017 extends to more general one-dimensional diffusions; and Qin and Linetsky 2016 extend to general Markov processes). Under the assumption of rational expectations, it leads to the recovery of the data-generating transition probabilities.1 Transition independence of the SDF is the key assumption that allows Ross to appeal to the Perron-Frobenius theory to achieve a unique recovery. The Perron-Frobenius theorem asserts existence and uniqueness of a positive eigenvector of an irreducible nonnegative matrix. It is this unique positive eigenvector that allows Ross to establish a unique recovery under the assumption of transition independence of the SDF. To assess empirical implications of the recovery, it is important to understand economic consequences of the transition independence assumption. Hansen and Scheinkman (2017), Borovička, Hansen, and Scheinkman (2016), Martin and Ross (2013), and Qin and Linetsky (2016) connect Ross’ recovery to the factorization of Hansen and Scheinkman (2009) and show that the transition independence assumption in a Markovian model implies that the martingale component in the long-term factorization of the SDF is degenerate and equal to unity. Hansen and Scheinkman (2017) and Borovička, Hansen, and Scheinkman (2016) point out that such degeneracy is inconsistent with many structural dynamic asset pricing models in the literature, as well as with the empirical evidence in Alvarez and Jermann (2005) and Bakshi and Chabi-Yo (2012) based on bounds on the permanent and transitory components of the SDF. The focus of this paper is on assessing the plausibility of the transition independence assumption in the bond market. This paper explicitly extracts transitory and permanent (martingale) components in the long-term factorization of the stochastic discount factor (SDF) of Alvarez and Jermann (2005) and Hansen and Scheinkman (2009) (see also Qin and Linetsky 2017). We posit an arbitrage-free dynamic term structure model (DTSM), estimate it on the time series of U.S. Treasury yield curves, and explicitly determine the long-term factorization of the SDF via the Perron-Frobenius extraction of the principal eigenfunction following the methodology of Hansen and Scheinkman (2009) (see also Qin and Linetsky 2016). With the estimated long-term factorization in hand, we are able to empirically test the structural assumption of transition independence of the SDF underpinning the recovery result of Ross (2015). Consistent with the calibrated structural example in Borovička, Hansen, and Scheinkman (2016), as well as the empirical literature relying on bounds and finite-maturity proxies for the long-term bond (Alvarez and Jermann 2005; Bakshi and Chabi-Yo 2012; Bakshi, Chabi-Yo, and Gao 2018), we find that the martingale component in the long-term factorization of the SDF is highly volatile, rather than degenerate, as would be implied by the transition independence assumption. The martingale component of the long-term factorization defines the long-term risk-neutral probability measure (Hansen and Scheinkman 2009, 2017; Hansen 2012; Borovička, Hansen, and Scheinkman 2016) that also can be identified with the long forward measure. The long forward measure can be understood as the long-term limit of $$T$$-maturity forward measures associated with $$T$$-maturity pure discount bonds, as the maturity $$T$$ becomes asymptotically large (see Qin and Linetsky 2017 for details). The Ross recovery procedure via the Perron-Frobenius eigenvector yields this long forward measure, rather than the physical measure. The martingale connects the long forward and physical measures. If the martingale is degenerate, then the long forward measure is identified with the physical measure (under the assumption of rational expectations, as discussed above). However, our empirical results show that the martingale is highly volatile, thus driving the wedge between the long forward measure and the physical measure. We now sketch the theory underlying the analysis in the paper. First, we briefly recall the long-term factorization of the SDF (Alvarez and Jermann 2005; Hansen and Scheinkman 2009; Hansen 2012; Borovička, Hansen, and Scheinkman 2016; Qin and Linetsky 2016, 2017): \begin{align} \frac{S_{t+\tau}}{S_t}=\frac{1}{R^\infty_{t,t+\tau}}\frac{M_{t+\tau}}{M_t}, \end{align} (1) where $$S_t$$ is the pricing kernel process, $$R^\infty_{t,t+\tau}$$ is the gross holding period return on the long bond (limit of gross holding period returns $$R_{t,t+\tau}^T=P_{t+\tau,T}/P_{t,T}$$ on pure discount bonds maturing at time $$T$$ as $$T$$ grows asymptotically large), and $$M_t$$ is a martingale. This martingale defines the long forward measure we denote by $${\mathbb L}$$ (in this paper we denote the physical or data-generating measure by $${\mathbb P}$$ and the risk-neutral measure by $${\mathbb Q}$$). Under $${\mathbb L}$$, the long bond serves as the growth optimal numeraire portfolio (see Borovička, Hansen, and Scheinkman 2016; Qin and Linetsky 2017). By Jensen’s inequality, the expected log return on any other asset is dominated by the long bond: \begin{equation} {\mathbb E}_t^{\mathbb L}\left[\log R_{t,t+\tau}\right]\leq {\mathbb E}_t^{\mathbb L}\left[\log R^\infty_{t,t+\tau} \right], \end{equation} (2) where $$R_{t,t+\tau}=V_{t+\tau}/V_t$$ is the gross holding period return on an asset with the value process $$V$$, and the expectation is taken under the long forward measure $${\mathbb L}$$. To put it another way, only the covariance with the long bond is priced under $${\mathbb L}$$, with all other risks neutralized by distorting the probability measure: \begin{equation} {\mathbb E}_t^{\mathbb L}\left[R_{t,t+\tau}\right]-R^f_{t,t+\tau}=-{\rm cov}_t^{\mathbb L}\left(R_{t,t+\tau},\frac{1}{R^\infty_{t,t+\tau}}\right)R^f_{t,t+\tau}, \end{equation} (3) where $$R^f_{t,t+\tau}=1/P_{t,t+\tau}$$ is the gross holding period return on the risk-free discount bond. Dividing both sides by the conditional volatility of the asset return $$\sigma_t^{\mathbb L}(R_{t,t+\tau})$$, the conditional Sharpe ratio under $${\mathbb L}$$ is \begin{equation} {\mathcal S}{\mathcal R}_t^{\mathbb L}(R_{t,t+\tau})=-{\rm corr}_t^{\mathbb L}\left(R_{t,t+\tau},\frac{1}{R^\infty_{t,t+\tau}}\right)R^f_{t,t+\tau}\sigma_t^{\mathbb L}\left(1/R_{t,t+\tau}^\infty\right). \end{equation} (4) The perfect negative correlation gives the Hansen and Jagannathan (1991) bound under $${\mathbb L}$$: \begin{equation} {\mathcal S}{\mathcal R}_t^{\mathbb L}(R_{t,t+\tau})\leq \sigma_t^{\mathbb L}\left(1/R_{t,t+\tau}^\infty\right) R^f_{t,t+\tau}. \end{equation} (5) Assuming that the Markovian SDF is transition independent implies that the martingale component is degenerate, that is, $$S_{t+\tau}/S_t=1/R^\infty_{t,t+\tau}$$. This identifies $${\mathbb P}$$ with $${\mathbb L}$$ and identifies the long bond with the growth optimal numeraire portfolio in the economy (see also result 5 in Martin and Ross 2013 and section 4.3 in Borovička, Hansen, and Scheinkman 2016) and implies that the only priced risk in the economy is the covariance with the long bond. In particular, applying Equation (4) to returns $$R^T_{t,t+\tau}$$ on pure discount bonds, Equation (4) predicts that bond Sharpe ratios are generally increasing in maturity and approach their upper bound (Hansen-Jagannathan bound (5)) at asymptotically long maturities.2 However, this sharply contradicts empirical evidence in the U.S. Treasury bond market. Duffee (2010), Frazzini and Pedersen (2014), and van Binsbergen and Koijen (2017) document that short-maturity bonds have higher Sharpe ratios than do long maturity bonds. Backus, Boyarchenko, and Chernov (forthcoming) and van Binsbergen and Koijen (2017) provide recent bibliographies to the growing literature on the term structure of risk premiums. In this paper we focus on the term structure of bond risk premiums. The empirical term structure of bond Sharpe ratios is generally downward sloping, rather than upward sloping. Frazzini and Pedersen (2014) offer an explanation based on the leverage constraints faced by many bond market participants that result in their preference for longer maturity bonds over leveraged positions in shorter maturity bonds, even if the latter offer higher Sharpe ratios. Furthermore, empirical results in this paper show that leveraged short-maturity bonds achieve substantially higher expected log-returns than long-maturity bonds and, in particular, the (model-implied) long bond. This empirical evidence puts in question the assumption of transition independent and degeneracy of the martingale component in the U.S. Treasury bond market. In Section 1 we estimate an arbitrage-free DTSM on the U.S. Treasury bond data. The zero interest rate policy (ZIRP) in the United States since December of 2008 is an added challenge. Most conventional DTSM do not handle the zero lower bound (ZLB) well. Gaussian models allow unbounded negative rates, whereas CIR-type affine factor models feature vanishing volatility at the ZLB. Shadow rate models are essentially the only class of dynamic term structure models in the literature at present capable of handling the ZLB. The shadow rate idea comes from Black (1995). Gorovoi and Linetsky (2004) provide an analytical solution for single-factor shadow rate models and calibrate them to the term structure of Japanese government bonds (JGB). Kim and Singleton (2012) estimate two-factor shadow rate models on the JGB data. In this paper we estimate the two-factor shadow rate model B-QG2 (Black quadratic Gaussian two factor) shown by Kim and Singleton (2012) to provide the best fit among the model specifications they consider in their investigation of the JGB market. In Section 2 we perform Perron-Frobenius extraction in the estimated model, extract the principal eigenvalue and eigenfunction, construct the long-term factorization of the pricing kernel, and recover the long-term risk-neutral measure (long forward measure) dynamics of the underlying factors. We then directly compare market price of risk processes under the estimated data-generating probability measure and the recovered long-term risk-neutral measure. The difference in these market prices of risk is highly significant. It is identified with the instantaneous volatility of the martingale component. In Section 3 we explore economic implications of our results. We use our model-implied long-term bond dynamics to estimate expected log returns on the long bond and test how far it is from growth optimality implied by the assumption that the martingale component is unity. We find that duration-matched leveraged positions in short and intermediate maturity bonds have significantly higher expected log returns than long maturity bonds and, in particular, the long bond. We also estimate the realized term structure of Sharpe ratios for bonds of different maturities and conclude that it is downward sloping, consistent with the empirical evidence in Duffee (2010), Frazzini and Pedersen (2014), and van Binsbergen and Koijen (2017). We further consider Sharpe ratio forecasts under our estimated probability measures $${\mathbb P}$$ and $${\mathbb L}$$. We find, in particular, that $${\mathbb L}$$ implies forecasts for excess returns on shorter-maturity bonds (up to 3 years) that are essentially zero (risk neutral), while significant excess returns with high Sharpe ratios are observed empirically in this segment of the bond market and are correctly forecast by our estimated $${\mathbb P}$$ measure. Thus, identifying $${\mathbb P}$$ and $${\mathbb L}$$ leads to sharply distorted risk-return trade-offs in the bond market. We also test the null hypothesis that the martingale is equal to unity (and, hence, the data-generating probability measure is identical to the long forward measure) and reject it at the 99.99% level. Finally, in Section 4 we perform robustness check. We note that our econometric approach in this paper is entirely different from the approaches of Alvarez and Jermann (2005), Bakshi and Chabi-Yo (2012), and Bakshi et al. (2018), who rely on bounds on the transitory and martingale components, while we directly estimate a fully specified DTSM, explicitly accomplish the Perron-Frobenius extraction of Hansen and Scheinkman (2009) and obtain the permanent and martingale components in the framework of our DTSM. We also note recent work by Christensen (2017), who develops a nonparametric approach to the Perron-Frobenius extraction and estimates permanent and transitory components under structural (Epstein-Zin and power utility) specifications of the SDF calibrated to real per-capita consumption and real corporate earnings growth. These three lines of inquiry, our parametric modeling and estimation based on asset market data, Christensen’s modeling based on macroeconomic fundamentals, and the approaches of Alvarez and Jermann (2005), Bakshi and Chabi-Yo (2012), and Bakshi, Chabi-Yo, and Gao (2018) based on bounds are complementary, and all result in the conclusion that the martingale component is highly economically significant. 1. Dynamic Term Structure Model Estimation We use the data set of daily constant maturity (CMT) U.S. Treasury bond yields from October 1, 1993 to August 19, 2015. The data include daily yields for Treasury constant maturities of 1, 3, and 6 months and 1, 2, 3, 5, 7, 10, 20, and 30 years. Since our focus is on the Perron-Frobenius extraction of the principal eigenvalue and eigenfunction governing the long-term factorization, we include the long end of the yield curve with 20- and 30-year maturities. Thirty year yield data are missing from February 19, 2002 to February 8, 2006. One-month yield data are missing from October 1, 1993 to July 30, 2001, where the data start with 3-month yields. These missing data do not pose any challenges to our estimation procedure. We obtain zero-coupon yield curves from CMT yield curves via cubic splines bootstrap. Figure 1 shows our time series of bootstrapped zero-coupon yield curves. Figure 1 View largeDownload slide U.S. Treasury zero-coupon yield curves bootstrapped from CMT yield curves Figure 1 View largeDownload slide U.S. Treasury zero-coupon yield curves bootstrapped from CMT yield curves We assume that the state of the economy is governed by a two-factor continuous-time Gaussian diffusion under the data-generating probability measure $$\mathbb{P}$$: \begin{equation} dX_t=K^\mathbb{P}(\theta^\mathbb{P}-X_t) dt+\Sigma dB_t^\mathbb{P}, \end{equation} (6) where $$X_t$$ is a two-dimensional (column) vector, $$B_t^\mathbb{P}$$ is a two-dimensional standard Brownian motion, $$\theta^\mathbb{P}$$ is a two-dimensional vector, and $$K^\mathbb{P}$$ and $$\Sigma$$ are $$2\times2$$ matrices. We assume an affine market price of risk specification $$\lambda^\mathbb{P}(X_t)=\lambda^\mathbb{P}+\Lambda^\mathbb{P} X_t,$$ where $$\lambda^\mathbb{P}$$ is a two-dimensional vector and $$\Lambda^\mathbb{P}$$ is a 2x2 matrix, so that $$X_t$$ remains Gaussian under the risk-neutral probability measure $$\mathbb{Q}$$: \begin{equation} dX_t=K^\mathbb{Q}(\theta^\mathbb{Q}-X_t) dt+\Sigma dB_t^\mathbb{Q}, \end{equation} (7) where $$K^\mathbb{Q}=K^\mathbb{P}+\Sigma\Lambda^\mathbb{P}$$ and $$K^\mathbb{Q}\theta^\mathbb{Q}=K^\mathbb{P}\theta^\mathbb{P}-\Sigma\lambda^\mathbb{P}$$. To handle the ZIRP since December of 2008, we follow Kim and Singleton (2012) and specify Black (1995) shadow rate as the shifted quadratic form of the Gaussian state vector, and the nominal short rate as its positive part (here $$'$$ denotes matrix transposition and $$(x)^+=\max(x,0)$$: \begin{equation} r(X_t)=(\rho+\delta' X_t+ X_t' \Phi X_t)^+. \end{equation} (8) This is the B-QG2 (Black-Quadratic Gaussian two-factor) specification of Kim and Singleton (2012). Following Kim and Singleton (2012), we impose the following conditions to achieve identification: $$K^\mathbb{P}_{12}=0, \delta=0, \Sigma=0.1 I_2$$, where $$I_2$$ is the $$2\times2$$ identity matrix. To ensure existence of the long-term limit (see Qin and Linetsky 2016), we impose two additional restrictions. We require that the eigenvalues of $$K^\mathbb{P}$$ have positive real parts, and $$\Phi$$ is positive semidefinite. The first restriction ensures that $$X$$ is mean-reverting under the data-generating measure $$\mathbb{P}$$ and possesses a stationary distribution. The second restriction ensures that the short rate does not vanish in the long run. The mode of the short rate under the stationary distribution is $$(\rho+(\theta^\mathbb{P})'\Phi\theta^\mathbb{P})^+$$. If $$\Phi$$ is not positive semidefinite, the mode of the short rate under the stationary distribution can be zero. We decompose \begin{equation} \Phi=\begin{bmatrix} 1&0\\ A & 1\end{bmatrix} \begin{bmatrix} D_{1}&0\\0 & D_{2}\end{bmatrix}\begin{bmatrix} 1& A \\0 & 1\end{bmatrix}, \end{equation} (9) and require that $$D_{1},D_{2}\geq0$$ and $$D_1 D_2>0$$. Because of the positive part in the short rate specification, in contrast to one-factor shadow rate models that admit analytical solutions (Gorovoi and Linetsky 2004), the two-factor model does not possess an analytic solution for bond prices. Consider the time-$$t$$ price of the zero-coupon bond with maturity at time $$t+\tau$$ and unit face value: \begin{equation} P_{t,t+\tau}=P(\tau,X_t)=\mathbb{E}^\mathbb{Q}_t[e^{-\int_t^{t+\tau} r(X_s) ds}]. \end{equation} (10) Since the state process is time-homogeneous Markov, the bond pricing function $$P(\tau,x)$$ satisfies the pricing PDE \begin{equation} \frac{\partial P}{\partial \tau}-\frac{1}{2}\text{tr}(\Sigma\Sigma'\frac{\partial^2 P}{\partial x\partial x'} )-\frac{\partial P'}{\partial x} K^\mathbb{Q}(\theta^\mathbb{Q}-x)+r(x)P=0 \end{equation} (11) with the initial condition $$P(0,x)=1$$. We compute bond prices by solving the PDE numerically via an operator splitting finite-difference scheme following the approach in appendix A of Kim and Singleton (2012). Our estimation strategy follows Kim and Singleton (2012). Observed bond yields $$Y^O_{t,\tau_i}$$ are assumed to equal their model-implied counterparts $$Y_{t,\tau_i}=Y(\tau_i,X_t)=-(1/\tau_i)\log P(\tau_i,X_t)$$ plus mutually and serially independent Gaussian measurement errors $$e_{t,\tau_i}\sim N(0, \sigma_{\tau_i}^2)$$. The model is estimated using the extended Kalman-filter based quasi-maximum likelihood function. We follow Kim and Priebsch (2013) in estimating standard errors using the approach of Bollerslev and Wooldridge (1992). Table 1 gives the parameter estimates and standard errors. Table 2 gives the average pricing errors. Our pricing errors are slightly higher than those reported by Kim and Singleton (2012), where the model is estimated on weekly JGB data. It is not surprising, since we use daily data for all maturities from 1 month to 30 years, whereas Kim and Singleton (2012) use weekly data with JGB maturities up to 10 years. Table 1 Model parameter estimates and standard errors (in parentheses) $$K^\mathbb{Q}$$ 0.3253 (0.0034) 0.0420 (0.0005) 0.6388 (0.0081) 0.0813 (0.0018) $$\theta^\mathbb{Q}$$ 0.9316 (0.0175) $$-$$5.9180 (0.1157) $$\rho$$ $$-$$0.0046 (0.0002) $$D_1$$ 0.2839 (0.0085) $$D_2$$ 0.0227 (0.0009) $$A$$ 0.3347 (0.0068) $$\lambda^\mathbb{P}$$ $$-$$0.9085 (0.0785) $$-$$1.0503 (0.0468) $$\Lambda^\mathbb{P}$$ $$-$$3.0535 (1.4167) 0.4203 (0.0051) 4.4189 (1.8992) 0.3964 (0.0173) $$K^\mathbb{Q}$$ 0.3253 (0.0034) 0.0420 (0.0005) 0.6388 (0.0081) 0.0813 (0.0018) $$\theta^\mathbb{Q}$$ 0.9316 (0.0175) $$-$$5.9180 (0.1157) $$\rho$$ $$-$$0.0046 (0.0002) $$D_1$$ 0.2839 (0.0085) $$D_2$$ 0.0227 (0.0009) $$A$$ 0.3347 (0.0068) $$\lambda^\mathbb{P}$$ $$-$$0.9085 (0.0785) $$-$$1.0503 (0.0468) $$\Lambda^\mathbb{P}$$ $$-$$3.0535 (1.4167) 0.4203 (0.0051) 4.4189 (1.8992) 0.3964 (0.0173) Table 1 Model parameter estimates and standard errors (in parentheses) $$K^\mathbb{Q}$$ 0.3253 (0.0034) 0.0420 (0.0005) 0.6388 (0.0081) 0.0813 (0.0018) $$\theta^\mathbb{Q}$$ 0.9316 (0.0175) $$-$$5.9180 (0.1157) $$\rho$$ $$-$$0.0046 (0.0002) $$D_1$$ 0.2839 (0.0085) $$D_2$$ 0.0227 (0.0009) $$A$$ 0.3347 (0.0068) $$\lambda^\mathbb{P}$$ $$-$$0.9085 (0.0785) $$-$$1.0503 (0.0468) $$\Lambda^\mathbb{P}$$ $$-$$3.0535 (1.4167) 0.4203 (0.0051) 4.4189 (1.8992) 0.3964 (0.0173) $$K^\mathbb{Q}$$ 0.3253 (0.0034) 0.0420 (0.0005) 0.6388 (0.0081) 0.0813 (0.0018) $$\theta^\mathbb{Q}$$ 0.9316 (0.0175) $$-$$5.9180 (0.1157) $$\rho$$ $$-$$0.0046 (0.0002) $$D_1$$ 0.2839 (0.0085) $$D_2$$ 0.0227 (0.0009) $$A$$ 0.3347 (0.0068) $$\lambda^\mathbb{P}$$ $$-$$0.9085 (0.0785) $$-$$1.0503 (0.0468) $$\Lambda^\mathbb{P}$$ $$-$$3.0535 (1.4167) 0.4203 (0.0051) 4.4189 (1.8992) 0.3964 (0.0173) Table 2 Average pricing errors (in basis points) 1 m 3 m 6 m 1 yr 2 yr 3 yr 5 yr 7 yr 10 yr 20 yr 30 yr 20 16 8 2 11 11 6 2 14 26 25 1 m 3 m 6 m 1 yr 2 yr 3 yr 5 yr 7 yr 10 yr 20 yr 30 yr 20 16 8 2 11 11 6 2 14 26 25 Table 2 Average pricing errors (in basis points) 1 m 3 m 6 m 1 yr 2 yr 3 yr 5 yr 7 yr 10 yr 20 yr 30 yr 20 16 8 2 11 11 6 2 14 26 25 1 m 3 m 6 m 1 yr 2 yr 3 yr 5 yr 7 yr 10 yr 20 yr 30 yr 20 16 8 2 11 11 6 2 14 26 25 In Table 3 we also report the goodness of fit for bond holding period returns. Specifically, suppose $$R_{t, t+1/4}^\tau$$ is the realized quarterly holding period return of a zero-coupon bond with initial maturity $$\tau$$. Let $$\hat{R}_{t,t+1/4}^\tau$$ denote the corresponding model-implied return $$\hat{R}_{t,t+1/4}^\tau:=\frac{P(\tau-1/4, X_{t+1/4})}{P(\tau, X_t)}-1,$$ where $$P(\tau,X_t)$$ is the price of the bond with time to maturity $$\tau$$ when the state is $$X_t$$ at time $$t$$. Here $$X_t$$ is the filtered state. Table 3 displays average realized returns, model-implied returns, as well as the slope and $$R^2$$ of regressions of realized returns on model-implied returns $$R_{t, t+1/4}^\tau=a+b\hat{R}_{t,t+1/4}^\tau+{\rm noise}$$. Table 3 Goodness of fit for bond returns Maturity 1 yr 2 yr 3 yr 5 yr 7 yr 10 yr 20 yr 30 yr Realized (%) $$0.90$$ $$1.08$$ $$1.24$$ $$1.49$$ $$1.69$$ $$1.95$$ $$2.58$$ $$3.09$$ Model-implied (%) $$0.88$$ $$1.06$$ $$1.22$$ $$1.47$$ $$1.63$$ $$1.78$$ $$2.02$$ $$2.14$$ Regression slope 0.9897 1.0841 1.1188 1.0980 1.0029 0.8693 0.6803 0.6806 $$R^2$$ (%) $$98.54$$ $$95.16$$ $$95.57$$ $$97.96$$ $$99.28$$ $$95.84$$ $$86.03$$ $$69.78$$ Maturity 1 yr 2 yr 3 yr 5 yr 7 yr 10 yr 20 yr 30 yr Realized (%) $$0.90$$ $$1.08$$ $$1.24$$ $$1.49$$ $$1.69$$ $$1.95$$ $$2.58$$ $$3.09$$ Model-implied (%) $$0.88$$ $$1.06$$ $$1.22$$ $$1.47$$ $$1.63$$ $$1.78$$ $$2.02$$ $$2.14$$ Regression slope 0.9897 1.0841 1.1188 1.0980 1.0029 0.8693 0.6803 0.6806 $$R^2$$ (%) $$98.54$$ $$95.16$$ $$95.57$$ $$97.96$$ $$99.28$$ $$95.84$$ $$86.03$$ $$69.78$$ Realized returns are computed as discussed in Section 4 and averaged over the time series. Model-implied returns are computed using the estimated model parameters and filtered states. The regressions are of the realized returns on the model-implied returns. Table 3 Goodness of fit for bond returns Maturity 1 yr 2 yr 3 yr 5 yr 7 yr 10 yr 20 yr 30 yr Realized (%) $$0.90$$ $$1.08$$ $$1.24$$ $$1.49$$ $$1.69$$ $$1.95$$ $$2.58$$ $$3.09$$ Model-implied (%) $$0.88$$ $$1.06$$ $$1.22$$ $$1.47$$ $$1.63$$ $$1.78$$ $$2.02$$ $$2.14$$ Regression slope 0.9897 1.0841 1.1188 1.0980 1.0029 0.8693 0.6803 0.6806 $$R^2$$ (%) $$98.54$$ $$95.16$$ $$95.57$$ $$97.96$$ $$99.28$$ $$95.84$$ $$86.03$$ $$69.78$$ Maturity 1 yr 2 yr 3 yr 5 yr 7 yr 10 yr 20 yr 30 yr Realized (%) $$0.90$$ $$1.08$$ $$1.24$$ $$1.49$$ $$1.69$$ $$1.95$$ $$2.58$$ $$3.09$$ Model-implied (%) $$0.88$$ $$1.06$$ $$1.22$$ $$1.47$$ $$1.63$$ $$1.78$$ $$2.02$$ $$2.14$$ Regression slope 0.9897 1.0841 1.1188 1.0980 1.0029 0.8693 0.6803 0.6806 $$R^2$$ (%) $$98.54$$ $$95.16$$ $$95.57$$ $$97.96$$ $$99.28$$ $$95.84$$ $$86.03$$ $$69.78$$ Realized returns are computed as discussed in Section 4 and averaged over the time series. Model-implied returns are computed using the estimated model parameters and filtered states. The regressions are of the realized returns on the model-implied returns. 2. Long-term Factorization We now turn to constructing the long-term factorization of the SDF process \begin{equation} S_t=e^{-\int_0^t r(X_s)ds}e^{-\int_0^t \lambda^{\mathbb P}(X_s)dB_s^{\mathbb P}-\frac{1}{2}\int_0^t {\vert\vert}\lambda^{\mathbb P}(X_s){\vert\vert}^2ds} \end{equation} (12) in the estimated dynamic term structure model. Consider the gross holding period return on the zero-coupon bond with maturity at time $$T$$ over the period from $$s$$ to $$s+t$$, $$R^T_{s,s+t}=P(T-s-t,X_{s+t})/P(T-s,X_s)$$. We are interested in the limit as $$T$$ goes to infinity (holding period return on the zero-coupon bond of asymptotically long maturity). In Markovian models, if the long-term limit exists (see Qin and Linetsky 2017 for sufficient conditions and mathematical details), then \begin{equation} \lim_{T\rightarrow \infty}R^T_{s,s+t}=e^{\lambda t}\frac{\pi(X_{s+t})}{\pi(X_s)} \end{equation} (13) for some $$\lambda$$ and a positive function $$\pi(x)$$, with $$\pi(x)$$ serving as the positive (principal) eigenfunction of the (time-homogeneous Markovian) pricing operator with the eigenvalue $$e^{-\lambda t}$$: \begin{equation} {\mathbb E}^{\mathbb P}_0[S_t \pi(X_t)]= e^{-\lambda t}\pi(X_0), \end{equation} (14) where $$S_t$$ is the SDF. For the sake of brevity, here we do not repeat the theory of long-term factorization and its connection to the Perron-Frobenius theory and refer the reader to Hansen and Scheinkman (2009), Hansen (2012), Borovička, Hansen, and Scheinkman (2016), and Qin and Linetsky (2016, 2017). In the framework of our model the bond pricing function $$P(t,x)$$ is numerically determined by solving the bond pricing PDE by finite differences. We also determine the principal eigenfunction $$\pi(x)$$ numerically as follows. Choosing some error tolerance $$\epsilon$$, we solve the bond pricing PDE for an increasing sequence of times to maturity indexed by integers $$n$$, consider the ratios $$P(n+1,x)/P(n,x)$$ as $$n$$ increases, and stop at $$n=N$$ such that $$M_N-m_N \leq \epsilon$$ for the first time, where $$M_n=\max_{x\in \Omega} P(n+1,x)/P(n,x)$$ and $$m_n=\min_{x\in \Omega} P(n+1,x)/P(n,x)$$ and the max and min are computed over the grid in the domain $$\Omega$$ where we approximate the bond pricing function by the computed numerical solution of the PDE. The eigenvalue and the principal eigenfunction are then approximately given by $$e^{-\lambda} = (m_N+M_N)/2$$ and $$\pi(x)=e^{\lambda N}P(N,x)$$ in the domain $$x\in\Omega$$ (with the error tolerance $$\epsilon$$). Figure 2 plots the computed eigenfunction $$\pi(x)$$. The corresponding principal eigenvalue is $$\lambda=0.0283$$.3 Figure 2 View largeDownload slide Principal eigenfunction $$\pi(x_1,x_2)$$ with eigenvalue $$\lambda=0.0283$$ Figure 2 View largeDownload slide Principal eigenfunction $$\pi(x_1,x_2)$$ with eigenvalue $$\lambda=0.0283$$ With the principal eigenfunction $$\pi(x)$$ and eigenvalue $$\lambda$$ in hand, we explicitly obtain the long-term factorization: \begin{equation} S_t = \frac{1}{L_t}M_t, \quad L_t = e^{\lambda t}\frac{\pi(X_t)}{\pi(X_0)}, \quad M_t = S_t e^{\lambda t}\frac{\pi(X_t)}{\pi(X_0)}, \end{equation} (15) where $$L_t = R^\infty_{0,t}$$ is the long bond process (gross return from time zero to time $$t$$ on the zero-coupon bond of asymptotically long maturity) determining the transitory component $$1/L_t$$, and $$M_t$$ is the martingale (permanent) component of the long-term factorization. In particular, we can now recover the $${\mathbb L}$$ measure by applying Girsanov’s theorem. First, applying Itô’s formula to $$\log\pi(x)$$ and using the SDE for $$X$$ under $${\mathbb P}$$ we can write \begin{align} \log \frac{\pi(X_t)}{\pi(X_0)}& = \int_0^t \frac{\partial \log \pi}{\partial x'}(X_s) \Sigma dB_s^{\mathbb P}\notag \\ &\quad + \int_0^t \left(\frac{1}{2}\text{tr}(\Sigma\Sigma'\frac{\partial^2 \log\pi}{\partial x\partial x'})(X_s)+ \frac{\partial \log \pi}{\partial x'}(X_s) b^\mathbb{P}(X_s) \right) ds, \end{align} (16) where $$b^{\mathbb P}(x)=K^{\mathbb P}(\theta^{\mathbb P}-x)$$ is the drift under the data-generating measure. Next, we recall that the eigenfunction satisfies the (elliptic) PDE (without the time derivative): \begin{equation} \frac{1}{2}\text{tr}(\Sigma\Sigma'\frac{\partial^2 \pi}{\partial x\partial x'})(x)+\frac{\partial \pi}{\partial x'}(x) b^\mathbb{Q}(x)+(\lambda-r(x))\pi(x)=0, \end{equation} (17) where $$b^{\mathbb Q}(x)=K^{\mathbb Q}(\theta^{\mathbb Q}-x)$$ is the drift under the risk-neutral measure. Using the identity $$\frac{\partial^2 \log\pi}{\partial x\partial x'}=\frac{1}{\pi}\frac{\partial^2 \pi}{\partial x\partial x'}-\frac{1}{\pi^2}\frac{\partial \pi}{\partial x} \frac{\partial \pi}{\partial x'}$$ and the PDE, we can write \begin{align} \log \frac{\pi(X_t)}{\pi(X_0)} & =\int_0^t (\frac{1}{\pi}\frac{\partial \pi}{\partial x'})(X_s) \Sigma dB_s^{\mathbb P}\notag\\ &\quad + \int_0^t \left(r(X_s)-\lambda-(\frac{1}{2\pi^2}\frac{\partial \pi}{\partial x'}\Sigma\Sigma' \frac{\partial \pi}{\partial x})(X_s)+(\frac{1}{\pi}\frac{\partial\pi}{\partial x'}\Sigma\lambda^\mathbb{P})(X_s)\right)ds. \end{align} (18) Substituting this into the expression in Equation (15) for the martingale $$M_t$$, we obtain \begin{equation} M_t=e^{\int_0^t v_s dB^{\mathbb P}_s-\frac{1}{2}\int_0^t {\vert\vert}v_s{\vert\vert}^2ds} \end{equation} (19) with the instantaneous volatility process: \begin{equation} v_t=-\lambda^{\mathbb P}(X_t)+\lambda^{\mathbb L}(X_t), \end{equation} (20) where $$\lambda^{\mathbb P}(x)$$ is the drift of the state vector under the data-generating measure $${\mathbb P}$$, and we introduced the following notation \begin{equation} \lambda^{\mathbb L}(x):=\frac{1}{\pi(x)}\Sigma'\frac{\partial \pi}{\partial x}(x). \end{equation} (21) The martingale defines the long-term risk-neutral measure $${\mathbb L}$$. Applying Girsanov’s theorem, we obtain the drift of the state vector $$X$$ under $${\mathbb L}$$: \begin{equation} b^{\mathbb L}(x)=b^{\mathbb Q}(x)+\Sigma\lambda^{\mathbb L}(x), \end{equation} (22) where $$\lambda^{\mathbb L}(X_t)$$ is thus identified with the market price of risk process under the long-term risk-neutral measure $${\mathbb L}$$. The instantaneous volatility $$v_t=v(X_t)$$ of the martingale component is equal to the difference between the market prices of risk under the long-term risk-neutral measure $${\mathbb L}$$ and the data-generating measure $${\mathbb P}$$ and is explicitly expressed in terms of the principal eigenfunction: \begin{equation} v_t=-\lambda^{\mathbb P}(X_t)+\frac{1}{\pi(X_t)}\Sigma'\frac{\partial \pi}{\partial x}(X_t). \end{equation} (23) To compute the $$\mathbb{L}$$-market price of risk $$\lambda^\mathbb{L}(x)$$, we use Equation (21) with numerically computed eigenfunction. $$\lambda^\mathbb{L}(x)$$ can be well approximated by a linear function on the domain $$[-0.3, 0.2]\times[-0.1, 1.2]$$ of values containing the filtered paths of the state variables as depicted in Figure 3. In Table 4, we compute the linear coefficients of $$\lambda^\mathbb{L}(x)$$ using linear regression on the domain $$[-0.3, 0.2]\times[-0.1, 1.2]$$. Figure 3 View largeDownload slide Numerically computed $$\lambda^L(x)$$ Figure 3 View largeDownload slide Numerically computed $$\lambda^L(x)$$ Table 4 Least-squares fit to $$\lambda^\mathbb{L}(x)$$ Intercept $$x_1$$ $$x_2$$ $$R^2 (\%)$$ 0.158 $$-$$0.385 0.168 $$100$$ $$-$$0.095 0.168 $$-$$0.123 $$100$$ Intercept $$x_1$$ $$x_2$$ $$R^2 (\%)$$ 0.158 $$-$$0.385 0.168 $$100$$ $$-$$0.095 0.168 $$-$$0.123 $$100$$ Table 4 Least-squares fit to $$\lambda^\mathbb{L}(x)$$ Intercept $$x_1$$ $$x_2$$ $$R^2 (\%)$$ 0.158 $$-$$0.385 0.168 $$100$$ $$-$$0.095 0.168 $$-$$0.123 $$100$$ Intercept $$x_1$$ $$x_2$$ $$R^2 (\%)$$ 0.158 $$-$$0.385 0.168 $$100$$ $$-$$0.095 0.168 $$-$$0.123 $$100$$ As a result, the eigenfunction in this shadow rate model is well approximated by an exponential-quadratic function of the form \begin{equation} \pi(x)\approx Ce^{-1.93x_1^2-0.61x_2^2+1.68x_1 x_2 +1.58x_1-0.95x_2}, \end{equation} (24) similar to quadratic term structure models (QTSM) (see Qin and Linetsky 2016 for details on positive eigenfunctions in ATSM and QTSM). Substituting $$\lambda^{\mathbb{L}}(x)$$ into the expression for the drift of the state variables under $${\mathbb L}$$ (Equation (22)), we obtain a Gaussian approximation for the dynamics of the state variables under $$\mathbb{L}$$. We can now explicitly compare the data-generating and long-term risk-neutral dynamics. By inspection we see that all the parameters entering the market prices of risk under $$\mathbb{L}$$ in Table 4 are significantly smaller in magnitude than the parameters in the market prices of risk under the data-generating measure $$\mathbb{P}$$: \begin{equation} \lambda^{\mathbb{P}}(x)=\begin{bmatrix} -0.908\\ -1.050\\ \end{bmatrix}+\begin{bmatrix} -3.054 & 0.420 \\ 4.419 & 0.396\\ \end{bmatrix}\begin{bmatrix} x_1\\ x_2\\ \end{bmatrix}. \end{equation} (25) Thus, we obtain the instantaneous volatility of the martingale component as a function of the state variable via Equation (20): \begin{equation} v(x)\approx\begin{bmatrix} 1.066\\ 0.955\\ \end{bmatrix}+\begin{bmatrix} 2.670 & -0.253 \\ -4.251 & -0.519\\ \end{bmatrix}\begin{bmatrix} x_1\\ x_2\\ \end{bmatrix}. \end{equation} (26) Using our estimated model we can now compute an implied path of long bond returns as follows. Recall that the long bond gross return process is given by $$L_t=R_{0,t}^\infty=e^{\lambda t}\pi(X_t)/\pi(X_0)$$ (see Equation (15)). Figure 4 displays the model-implied path of the long bond in our estimated DTSM obtained by evaluating the expression $$e^{\lambda t}\pi(X_t)/\pi(X_0)$$ on the filtered path of the state vector $$X_t$$, where the principal eigenfunction and eigenvalue are given in Figure 2. The figure also displays the wealth (gross return) processes of investing in 20- and 30-year constant maturity zero-coupon bonds for comparison. The plot is separated into two subperiods since the 30-year bond was discontinued in 2002 and resumed in 2006. Specifically, the 30-year time series shows the value over time of the initial investment of one dollar in the 30-year zero-coupon bond rolled over at 3-month intervals back into the 30-year bond. Figure 4 View largeDownload slide Wealth processes investing in 20- and 30-year zero-coupon constant maturity bonds and the long bond Figure 4 View largeDownload slide Wealth processes investing in 20- and 30-year zero-coupon constant maturity bonds and the long bond In the previous literature researchers use 20- or 30-year bonds as proxies for the long bond. In our framework of the fully specified DTSM, we have access to the model-implied long bond dynamics and can use it as a model-based proxy for the unobservable long bond. Figure 4 shows a still noticeable difference between 30-year bonds and the long bond. 3. The Term Structure of Bond Risk Premiums and the Magnitude of the Martingale Component We now turn to the empirical examination of the term structure of bond risk premiums. Table 5 displays realized average quarterly excess returns, standard deviations and Sharpe ratios for zero-coupon bonds of maturities from one to thirty years over the period where all of them are available. Excess holding period returns are computed over the 3-month zero-coupon bond yields known at the beginning of each quarter. We observe that the term structure of Sharpe ratios is downward sloping, with the 1-year bond earning the quarterly Sharpe ratio of 0.46. This is more than two times the Sharpe ratio of the zero-coupon 30-year bond. This shape of the term structure of Sharpe ratios is in broad agreement with the findings of Duffee (2010), Frazzini and Pedersen (2014), and van Binsbergen and Koijen (2017) and is incompatible with the increasing term structure of Sharpe ratios arising under the assumption of transition independence and degeneracy of the martingale component in the long-term factorization. Using model-implied long bond series, we can also compute the long bond’s excess return, standard deviation and Sharpe ratio, which are $$2.70\%$$, $$20.29\%$$ and $$0.13$$, respectively. In particular, the Sharpe ratio for the long bond is lower than for the 30-year bond. Table 5 Realized average quarterly excess returns, standard deviations, and Sharpe ratios for zero-coupon bonds Maturity (yr) 1 2 3 5 7 10 20 30 Exc. ret. (%) $$0.16$$ $$0.35$$ $$0.50$$ $$0.76$$ $$0.95$$ $$1.22$$ $$1.84$$ $$2.36$$ SD (%) $$0.35$$ $$0.92$$ $$1.47$$ $$2.54$$ $$3.48$$ $$4.74$$ $$8.15$$ $$12.07$$ Sharpe 0.46 0.38 0.34 0.30 0.27 0.26 0.23 0.20 Maturity (yr) 1 2 3 5 7 10 20 30 Exc. ret. (%) $$0.16$$ $$0.35$$ $$0.50$$ $$0.76$$ $$0.95$$ $$1.22$$ $$1.84$$ $$2.36$$ SD (%) $$0.35$$ $$0.92$$ $$1.47$$ $$2.54$$ $$3.48$$ $$4.74$$ $$8.15$$ $$12.07$$ Sharpe 0.46 0.38 0.34 0.30 0.27 0.26 0.23 0.20 Maturities are from 1 to 30 years over the period from October 1, 1993 to February 15, 2002 and from February 9, 2006 to May 26, 2016. Excess returns are computed over the 3-month zero-coupon bond yield known at the beginning of each quarter. Table 5 Realized average quarterly excess returns, standard deviations, and Sharpe ratios for zero-coupon bonds Maturity (yr) 1 2 3 5 7 10 20 30 Exc. ret. (%) $$0.16$$ $$0.35$$ $$0.50$$ $$0.76$$ $$0.95$$ $$1.22$$ $$1.84$$ $$2.36$$ SD (%) $$0.35$$ $$0.92$$ $$1.47$$ $$2.54$$ $$3.48$$ $$4.74$$ $$8.15$$ $$12.07$$ Sharpe 0.46 0.38 0.34 0.30 0.27 0.26 0.23 0.20 Maturity (yr) 1 2 3 5 7 10 20 30 Exc. ret. (%) $$0.16$$ $$0.35$$ $$0.50$$ $$0.76$$ $$0.95$$ $$1.22$$ $$1.84$$ $$2.36$$ SD (%) $$0.35$$ $$0.92$$ $$1.47$$ $$2.54$$ $$3.48$$ $$4.74$$ $$8.15$$ $$12.07$$ Sharpe 0.46 0.38 0.34 0.30 0.27 0.26 0.23 0.20 Maturities are from 1 to 30 years over the period from October 1, 1993 to February 15, 2002 and from February 9, 2006 to May 26, 2016. Excess returns are computed over the 3-month zero-coupon bond yield known at the beginning of each quarter. Table 6 displays average realized quarterly log-returns for duration-matched leveraged or deleveraged investments in zero-coupon bonds of different maturities that match the duration of the 10- and 20-year bond over the period when all of them are available. We observe that leveraged investments in shorter-maturity bonds produce significantly higher average log-returns than duration-matched deleveraged investments in longer maturity bonds. Using our model-implied long bond time series displayed in Figure 4, we estimate the average log-return on the long bond to equal 1.50% over this period. Comparing this with the data in Table 6, we see that most of these duration-matched bond portfolios deliver higher log-returns than the long bond. In particular, all of the investments in bonds of maturities from 1 to 30 years leveraged or deleveraged to match the 20-year duration produce significantly higher average log-returns. These results strongly reject growth optimality of the long bond, consistent with the high volatility of the martingale component in the long-term factorization established in Section 2. Table 6 Realized average quarterly log-returns for leveraged or deleveraged investment in zero-coupon bonds Maturity (yr) 1 2 3 5 7 10 20 30 log ret. (10-yr dur.) (%) $$2.28$$ $$2.33$$ $$2.27$$ $$2.10$$ $$1.96$$ $$1.83$$ $$1.56$$ $$1.43$$ log ret. (20-yr dur.) (%) $$3.72$$ $$3.73$$ $$3.59$$ $$3.22$$ $$2.95$$ $$2.71$$ $$2.23$$ $$1.97$$ Maturity (yr) 1 2 3 5 7 10 20 30 log ret. (10-yr dur.) (%) $$2.28$$ $$2.33$$ $$2.27$$ $$2.10$$ $$1.96$$ $$1.83$$ $$1.56$$ $$1.43$$ log ret. (20-yr dur.) (%) $$3.72$$ $$3.73$$ $$3.59$$ $$3.22$$ $$2.95$$ $$2.71$$ $$2.23$$ $$1.97$$ Different maturities are matched to 10- and 20-year durations. The period is from October 1, 1993 to February 15, 2002, and from February 9, 2006 to May 26, 2016. Table 6 Realized average quarterly log-returns for leveraged or deleveraged investment in zero-coupon bonds Maturity (yr) 1 2 3 5 7 10 20 30 log ret. (10-yr dur.) (%) $$2.28$$ $$2.33$$ $$2.27$$ $$2.10$$ $$1.96$$ $$1.83$$ $$1.56$$ $$1.43$$ log ret. (20-yr dur.) (%) $$3.72$$ $$3.73$$ $$3.59$$ $$3.22$$ $$2.95$$ $$2.71$$ $$2.23$$ $$1.97$$ Maturity (yr) 1 2 3 5 7 10 20 30 log ret. (10-yr dur.) (%) $$2.28$$ $$2.33$$ $$2.27$$ $$2.10$$ $$1.96$$ $$1.83$$ $$1.56$$ $$1.43$$ log ret. (20-yr dur.) (%) $$3.72$$ $$3.73$$ $$3.59$$ $$3.22$$ $$2.95$$ $$2.71$$ $$2.23$$ $$1.97$$ Different maturities are matched to 10- and 20-year durations. The period is from October 1, 1993 to February 15, 2002, and from February 9, 2006 to May 26, 2016. Next, we compare model-based conditional forecasts of excess returns, volatility and Sharpe ratios of zero-coupon bonds of different maturities under the data-generating measure $${\mathbb P}$$ estimated in Section 1 and the long forward measure $${\mathbb L}$$ obtained via Perron-Frobenius extracton in Section 2. Table 7 displays average conditional excess return, volatility and Sharpe ratio forecasts under $${\mathbb P}$$ and $${\mathbb L}$$. Reported values are obtained by calculating conditional forecasts along the filtered sample path of the state vector $$X_t$$ and taking the averages over the time period. Excess return forecasts are over the 3-month zero-coupon bond yield known at the beginning of each quarter. Sharpe ratio forecasts are computed as the ratios of excess return forecast to the volatility forecast. Comparing Sharpe ratio forecasts in Table 7 with Table 5, we observe that $$\mathbb{P}$$-measure Sharpe ratio forecasts exhibit the downward-sloping term structure broadly comparable with the downward-sloping term structure of realized Sharpe ratios in Table 5. In contrast, the $${\mathbb L}$$-measure forecasts exhibit a generally upward-sloping term structure that starts near zero for 1- to 3-year maturities ($${\mathbb L}$$-measure forecasts are essentially risk-neutral for these shorter maturities) and increases toward the Hansen-Jagannathan bound in Equation (5) discussed in the Introduction. The bound is approximately attained by the long bond. Although the long bond is growth optimal, it does not generally maximize the Sharpe ratio since $${\rm corr}_t^{\mathbb L}\left(R^\infty_{t,t+1},1/R^\infty_{t,t+1}\right)$$ is not generally equal to $$-1$$. However, for sufficiently small holding periods this correlation is close to $$-1$$. As is clear in Table 7, the empirically estimated average quarterly $${\mathbb L}$$-Sharpe ratio of the long bond of 0.17 is comparable to its average quarterly volatility estimated to be 0.187. Table 7 Average conditional 3-month excess return, volatility, and Sharpe ratio $${\mathbb P}$$- and $${\mathbb L}$$-forecasts for zero-coupon bonds Maturity (yr) 1 3 5 10 20 30 LB $$\mathbb{P}$$ Ex. ret. (%) $$0.17$$ $$0.49$$ $$0.65$$ $$0.74$$ $$0.67$$ $$0.61$$ $$0.72$$ SD (%) $$0.42$$ $$1.07$$ $$1.64$$ $$4.05$$ $$9.39$$ $$12.86$$ $$17.61$$ Sharpe $$0.40$$ $$0.46$$ $$0.40$$ $$0.18$$ $$0.07$$ $$0.05$$ $$0.04$$ $$\mathbb{L}$$ Ex. ret. (%) $$-0.02$$ $$0.02$$ $$0.16$$ $$0.68$$ $$1.68$$ $$2.31$$ $$3.24$$ SD (%) $$0.44$$ $$1.10$$ $$1.65$$ $$4.16$$ $$9.79$$ $$13.51$$ $$18.70$$ Sharpe $$-0.05$$ $$0.02$$ $$0.10$$ $$0.16$$ $$0.17$$ $$0.17$$ $$0.17$$ Maturity (yr) 1 3 5 10 20 30 LB $$\mathbb{P}$$ Ex. ret. (%) $$0.17$$ $$0.49$$ $$0.65$$ $$0.74$$ $$0.67$$ $$0.61$$ $$0.72$$ SD (%) $$0.42$$ $$1.07$$ $$1.64$$ $$4.05$$ $$9.39$$ $$12.86$$ $$17.61$$ Sharpe $$0.40$$ $$0.46$$ $$0.40$$ $$0.18$$ $$0.07$$ $$0.05$$ $$0.04$$ $$\mathbb{L}$$ Ex. ret. (%) $$-0.02$$ $$0.02$$ $$0.16$$ $$0.68$$ $$1.68$$ $$2.31$$ $$3.24$$ SD (%) $$0.44$$ $$1.10$$ $$1.65$$ $$4.16$$ $$9.79$$ $$13.51$$ $$18.70$$ Sharpe $$-0.05$$ $$0.02$$ $$0.10$$ $$0.16$$ $$0.17$$ $$0.17$$ $$0.17$$ Maturities are from 1 to 30 years, and the model-implied long bond (LB) is over the period from October 1, 1993 to February 15, 2002, and from February 9, 2006 to May 26, 2016. Excess return forecasts are in excess of the 3-month zero-coupon bond yield known at the beginning of each quarter. Table 7 Average conditional 3-month excess return, volatility, and Sharpe ratio $${\mathbb P}$$- and $${\mathbb L}$$-forecasts for zero-coupon bonds Maturity (yr) 1 3 5 10 20 30 LB $$\mathbb{P}$$ Ex. ret. (%) $$0.17$$ $$0.49$$ $$0.65$$ $$0.74$$ $$0.67$$ $$0.61$$ $$0.72$$ SD (%) $$0.42$$ $$1.07$$ $$1.64$$ $$4.05$$ $$9.39$$ $$12.86$$ $$17.61$$ Sharpe $$0.40$$ $$0.46$$ $$0.40$$ $$0.18$$ $$0.07$$ $$0.05$$ $$0.04$$ $$\mathbb{L}$$ Ex. ret. (%) $$-0.02$$ $$0.02$$ $$0.16$$ $$0.68$$ $$1.68$$ $$2.31$$ $$3.24$$ SD (%) $$0.44$$ $$1.10$$ $$1.65$$ $$4.16$$ $$9.79$$ $$13.51$$ $$18.70$$ Sharpe $$-0.05$$ $$0.02$$ $$0.10$$ $$0.16$$ $$0.17$$ $$0.17$$ $$0.17$$ Maturity (yr) 1 3 5 10 20 30 LB $$\mathbb{P}$$ Ex. ret. (%) $$0.17$$ $$0.49$$ $$0.65$$ $$0.74$$ $$0.67$$ $$0.61$$ $$0.72$$ SD (%) $$0.42$$ $$1.07$$ $$1.64$$ $$4.05$$ $$9.39$$ $$12.86$$ $$17.61$$ Sharpe $$0.40$$ $$0.46$$ $$0.40$$ $$0.18$$ $$0.07$$ $$0.05$$ $$0.04$$ $$\mathbb{L}$$ Ex. ret. (%) $$-0.02$$ $$0.02$$ $$0.16$$ $$0.68$$ $$1.68$$ $$2.31$$ $$3.24$$ SD (%) $$0.44$$ $$1.10$$ $$1.65$$ $$4.16$$ $$9.79$$ $$13.51$$ $$18.70$$ Sharpe $$-0.05$$ $$0.02$$ $$0.10$$ $$0.16$$ $$0.17$$ $$0.17$$ $$0.17$$ Maturities are from 1 to 30 years, and the model-implied long bond (LB) is over the period from October 1, 1993 to February 15, 2002, and from February 9, 2006 to May 26, 2016. Excess return forecasts are in excess of the 3-month zero-coupon bond yield known at the beginning of each quarter. We remark that, by comparing Table 5 and 7, we observe that the model-based conditional forecasts of excess returns for bonds with maturities longer than 10 years are lower than their realized counterparts. This suggests that the filtered sample path of the state variables estimated from the given time period may not be representative of the ergodic path. This is anticipated, given that in our sample period bond yields were declining overall. However, this does not affect our conclusion about the volatility of the martingale component because the model-based conditional forecasts of the 1-year bond’s $$\mathbb{P}$$ excess return, standard deviation and Sharpe ratio match their realized counterparts sufficiently well. As we will show below (see Equation (30)), these quantities are already sufficient to assert the magnitude of the volatility of the martingale component. We now discuss the relationship between the term structure of bond Sharpe ratios and the magnitude of the volatility of the martingale component. We first develop a general connection between Sharpe ratios and the magnitude of the volatility of the martingale component in the spirit of Hansen-Jagannathan bounds. Consider any risky asset with the value process (with dividends reinvested) $$V_t$$ in an economy driven by $$d$$ independent standard Brownian motions and with the volatility vector $$\sigma_t$$ loading on these Brownian motions, $$dV_t=(r_t + \sigma_t\cdot \lambda_t^\mathbb{P})V_tdt+V_t\sigma_t\cdot dB_t^{\mathbb P}$$. The instantaneous Sharpe ratio (ISR) of the asset is then \begin{equation} \text{ISR}_t^\mathbb{P}(V)=\frac{\sigma_t\cdot\lambda^\mathbb{P}_t}{{\vert\vert}\sigma_t{\vert\vert}_2}={\vert\vert}\lambda^\mathbb{P}_t{\vert\vert}_2 \cos \theta_{\sigma_t,\lambda^\mathbb{P}_t}. \end{equation} (27) Thus, the norm of the market price of risk vector $${\vert\vert}\lambda^\mathbb{P}_t{\vert\vert}_2$$ determines the upper bound on the ISR in the economy. Similarly, we can write for the ISR under the long forward measure: \begin{equation} \text{ISR}_t^\mathbb{L}(V)=\frac{\sigma_t\cdot\lambda^\mathbb{L}_t}{{\vert\vert}\sigma_t{\vert\vert}_2}={\vert\vert}\lambda^\mathbb{L}_t{\vert\vert}_2 \cos \theta_{\sigma_t,\lambda^\mathbb{L}_t}. \end{equation} (28) We can then decompose the ISR under the data-generating measure in terms of the ISR under the long forward measure and an additional term: \begin{equation} \text{ISR}_t^\mathbb{P}(V)=\text{ISR}_t^\mathbb{L}(V)+\frac{\sigma_t\cdot(\lambda^\mathbb{P}_t-\lambda^\mathbb{L}_t)}{{\vert\vert}\sigma_t{\vert\vert}_2}=\text{ISR}_t^\mathbb{L}(V)-{\vert\vert}v_t{\vert\vert}_2\cos\theta_{v_t,\sigma_t}, \end{equation} (29) where $$v_t=-\lambda_t^\mathbb{P}+\lambda_t^\mathbb{L}$$ is the volatility of the martingale component in the long-term factorization Equation (20). Thus, the volatility of the martingale component in the long-term factorization furnishes the upper bound for the difference in instantaneous Sharpe ratios of an asset under $$\mathbb{P}$$ and $$\mathbb{L}$$: \begin{equation} {\vert\vert}v_t{\vert\vert}_2\geq |\text{ISR}_t^\mathbb{P}(V)-\text{ISR}_t^\mathbb{L}(V)|. \end{equation} (30) This is closely related to proposition 2 in Alvarez and Jermann (2005) and proposition 1 in Bakshi and Chabi-Yo (2012), who also derive bounds on the volatility of the permanent component. The main difference here is that we use the Sharpe ratio instead of return or log-return to establish the bound. Now coming back to our empirical results on the Sharpe ratios of bond returns, recall that the term structure of Sharpe ratios in Table 7 is downward sloping under $${\mathbb P}$$, while it slopes upward under $${\mathbb L}$$. Thus, the 1-year bond has the largest gap between its $${\mathbb P}$$- and $${\mathbb L}$$-Sharpe ratios. Using the estimated values in Table 7 yields a lower bound for the average over the path of the volatility of the martingale component equal to $$2*(0.40-(-0.05))=0.90$$. Thus, we expect the martingale component to have volatility of at least $$90\%$$ on average. With our estimated model in hand, we can also directly estimate the magnitude of the volatility of the martingale component to compare with this lower bound. Recall that the instantaneous volatility as a function of the state is approximately given by Equation (27). Consider the quadratic variation of the log of the martingale component $$ \langle \log M\rangle_t = \int_0^t {\vert\vert}v(X_s){\vert\vert}^2ds. $$ Its expectation under the stationary distribution, which we denote by $$\nu$$, is equal to $$ \mathbb{E}^\mathbb{P}_\nu[{\vert\vert}v(X){\vert\vert}^2]t. $$ Thus, under the stationary distribution the constant $$ \sqrt{\mathbb{E}^\mathbb{P}_\nu[{\vert\vert}v(X){\vert\vert}^2]} $$ summarizes the magnitude of the annualized volatility of the martingale component. Using our parameter estimates, we compute this quantity to equal $$1.32$$. Thus, the estimated magnitude of the volatility of the martingale component is highly significant at $$132\%$$, is larger than the lower bound of $$90\%$$ we previously obtained, and is also much larger than the volatility of the transitory component in the long-term factorization (the reciprocal of the long bond). We next formally test the null hypothesis $$\mathbb{P}=\mathbb{L}$$ (equivalently, degeneracy of the martingale component, $$v_t=0$$) that corresponds to the transition independence assumption on the pricing kernel. The market price of risk under $${\mathbb P}$$ contains six independent parameters that are estimated with standard errors given in Table 1. The market price of risk parameters under the long-term risk-neutral measure are uniquely determined (recovered) from the risk-neutral parameters. We use the delta method to compute the standard errors of our estimated parameters of the volatility of the martingale component $$v_i(x)=v_i+\sum_j v_{ij}x_j$$, $$v_i=-\lambda^\mathbb{P}_{i}+\lambda^\mathbb{L}_{i}$$ and $$v_{ij}=-\Lambda^\mathbb{P}_{ij}+\Lambda^\mathbb{L}_{ij}$$. We stack $$v_i$$ and $$v_{ij}$$ into a column vector $$v$$ and denote the derivative of $$v$$ with respect to the parameters as $$D$$. Then the variance-covariance matrix of $$v$$ is given by $$D\Omega D'$$, where $$\Omega$$ is the variance-covariance matrix of the model parameters. Taking the square root of the diagonal elements, we obtain the standard errors of the parameters $$v_i$$ and $$v_{ij}$$. The first six columns of Table 8 summarize the results. Note that the standard errors of $$v_i$$ and $$v_{ij}$$ from the delta method are very close to the standard errors of $$\mathbb{P}$$-market price of risk from Table 1, as the $$\mathbb{P}$$-market price of risk has much larger estimation errors than the errors for risk-neutral parameters, and dominates the estimation errors of $$\mathbb{Q}$$ parameters. We then compute the $$p$$-values for the joint hypothesis $$v_i=v_{ij}=0$$ for all $$i,j=1,2$$ using the test statistics $$v'(D\Omega D')^{-1}v$$ (which follows $$\chi^2(6)$$ under the null). The last column of Table 8 summarizes the results. The null hypothesis that the long-term risk-neutral measure is identified with the data-generating measure is rejected at least at the $$99.99\%$$ confidence level.4 Table 8 Estimated parameters of the volatility $$v(x)$$ of the martingale component $$v_1$$ $$v_2$$ $$v_{11}$$ $$v_{12}$$ $$v_{21}$$ $$v_{22}$$ Joint test $$v=0$$ Parameters 1.066 0.955 2.669 $$-$$0.253 $$-$$4.251 $$-$$0.519 $$t$$-stat 4,226.07 SE 0.078 0.047 1.416 0.005 1.900 0.173 $$p$$-value $$.00$$ $$v_1$$ $$v_2$$ $$v_{11}$$ $$v_{12}$$ $$v_{21}$$ $$v_{22}$$ Joint test $$v=0$$ Parameters 1.066 0.955 2.669 $$-$$0.253 $$-$$4.251 $$-$$0.519 $$t$$-stat 4,226.07 SE 0.078 0.047 1.416 0.005 1.900 0.173 $$p$$-value $$.00$$ The table presents standard errors, the test statistic, and the $$p$$-value for the joint hypothesis test that all parameters vanish. Table 8 Estimated parameters of the volatility $$v(x)$$ of the martingale component $$v_1$$ $$v_2$$ $$v_{11}$$ $$v_{12}$$ $$v_{21}$$ $$v_{22}$$ Joint test $$v=0$$ Parameters 1.066 0.955 2.669 $$-$$0.253 $$-$$4.251 $$-$$0.519 $$t$$-stat 4,226.07 SE 0.078 0.047 1.416 0.005 1.900 0.173 $$p$$-value $$.00$$ $$v_1$$ $$v_2$$ $$v_{11}$$ $$v_{12}$$ $$v_{21}$$ $$v_{22}$$ Joint test $$v=0$$ Parameters 1.066 0.955 2.669 $$-$$0.253 $$-$$4.251 $$-$$0.519 $$t$$-stat 4,226.07 SE 0.078 0.047 1.416 0.005 1.900 0.173 $$p$$-value $$.00$$ The table presents standard errors, the test statistic, and the $$p$$-value for the joint hypothesis test that all parameters vanish. 4. Robustness Checks In this section we perform several robustness checks. Our data sample includes the financial crisis of 2008 and the subsequent zero interest rate policy regime (ZIRP). To check that our results are not an artifact of special features of the ZIRP regime, we consider the pre-crisis period separately. We also extend our data set to 1987 and consider the pre-crisis period from 1987 to 2008, along with the longer period 1987–2016, which includes the crisis and the subsequent ZIRP. We limit ourselves to data starting in 1987 because of concerns of potential structural breaks in U.S. interest rates due to changes in U.S. monetary policy in the eighties. A substantial literature points out structural changes in the 1980s (e.g., Stock and Watson 2003; Rudebusch and Wu 2007; Smith and Taylor 2009; Joslin, Priebsch, and Singleton 2014). In particular, Joslin, Priebsch, and Singleton (2014) point out that after mid-1980s the long-run mean of the short rate is lower under the risk-neutral measure, yields and macroeconomic variables are more persistent and have lower volatility. Specifically, here we choose to follow Rudebusch and Wu (2007) and take 1987 as the starting year of our data to estimate our DTSM. We nevertheless point out that Frazzini and Pedersen (2014) and van Binsbergen and Koijen (2017) consider a much broader scope of data going back to 1950s and report that the qualitative pattern of the downward-sloping term structure of bond Sharpe ratios has held in this longer time series. Table 9 summarizes the $$p$$-values of our hypothesis test, as well as the magnitude of the annualized volatility of the martingale component. While the precise numerical values of model parameter estimates change with the time period considered, the hypothesis test that $$\mathbb{P}=\mathbb{L}$$ is rejected at least at the $$99.99\%$$ level and the annualized volatility of the martingale component is greater than $$100\%$$ for all periods considered. Table 9 Hypothesis test analysis 87-16 87-16$$^{a}$$ 87-08 87-08$$^{a}$$ 93-15 93-15$$^{a}$$ p-value $$.00$$ $$.00$$ $$.00$$ $$.00$$ $$.00$$ $$.00$$ $$\sqrt{\mathbb{E}_\nu[{\vert\vert}v(X){\vert\vert}^2]} (\%)$$ $$129.27$$ $$209.68$$ $$173.85$$ $$250.86$$ $$132.39$$ $$153.68$$ 87-16 87-16$$^{a}$$ 87-08 87-08$$^{a}$$ 93-15 93-15$$^{a}$$ p-value $$.00$$ $$.00$$ $$.00$$ $$.00$$ $$.00$$ $$.00$$ $$\sqrt{\mathbb{E}_\nu[{\vert\vert}v(X){\vert\vert}^2]} (\%)$$ $$129.27$$ $$209.68$$ $$173.85$$ $$250.86$$ $$132.39$$ $$153.68$$ $$p$$-values of the hypothesis test $${\mathbb P}={\mathbb L}$$ ($$v_i=0$$ and $$V_{ij}=0$$) and the annualized volatility of the martingale component. $$^{a}$$Excludes 20- and 30-year maturities. Table 9 Hypothesis test analysis 87-16 87-16$$^{a}$$ 87-08 87-08$$^{a}$$ 93-15 93-15$$^{a}$$ p-value $$.00$$ $$.00$$ $$.00$$ $$.00$$ $$.00$$ $$.00$$ $$\sqrt{\mathbb{E}_\nu[{\vert\vert}v(X){\vert\vert}^2]} (\%)$$ $$129.27$$ $$209.68$$ $$173.85$$ $$250.86$$ $$132.39$$ $$153.68$$ 87-16 87-16$$^{a}$$ 87-08 87-08$$^{a}$$ 93-15 93-15$$^{a}$$ p-value $$.00$$ $$.00$$ $$.00$$ $$.00$$ $$.00$$ $$.00$$ $$\sqrt{\mathbb{E}_\nu[{\vert\vert}v(X){\vert\vert}^2]} (\%)$$ $$129.27$$ $$209.68$$ $$173.85$$ $$250.86$$ $$132.39$$ $$153.68$$ $$p$$-values of the hypothesis test $${\mathbb P}={\mathbb L}$$ ($$v_i=0$$ and $$V_{ij}=0$$) and the annualized volatility of the martingale component. $$^{a}$$Excludes 20- and 30-year maturities. Because of liquidity concerns with bonds of long maturities, econometric studies of DTSM often consider maturities up to 10 years (e.g., Kim and Singleton 2012). To be consistent with this literature, in the next set of robustness checks, we exclude 20- and 30-year bonds and reestimate the model and recompute all our results on the data set including maturities up to 10 years. We remark that our estimation procedure assumes that bond yield errors are mutually and serially independent, as is standard in the dynamic term structure modeling literature. In contrast, our estimated fitting errors for bond yields of long maturities exhibit some positive serial correlations. This issue is well known in the literature on estimating affine models (see Hamilton and Wu 2014; Duffee 2011). This is not surprising, since our model, though nonlinear, is nevertheless close to the affine class. However, this does not pose a challenge to our result on the magnitude of the martingale component. On one hand, the estimated volatility of the martingale component is very large and in excess of 100% annually. On the other hand, as we demonstrated in Section 3, the downward-sloping term structure of bond Sharpe ratios implies large volatility of the martingale component. It is not subjected to some possible misspecification in measurement errors. In particular, the model-free lower bound on the volatility of the martingale component from Equation (30) is already around 90%, while our estimated volatility is over 100%. More generally, our results in this paper are based on estimating a particular DTSM. While choosing a different model specification would result in some quantitative differences, our qualitative conclusions that the martingale component is highly volatile and produces the generally downward-sloping term structure of bond Sharpe ratios, as opposed to the long forward probability forecast of generally upward-sloping term structure of bond Sharpe ratios, are robust to choosing a particular model specification and an estimation strategy. In particular, the results of Bakshi, Chabi-Yo, and Gao (2018) based on an entirely different procedure of estimating the martingale component from the data including Treasury bond futures and options offer further confirmation that the martingale component is highly volatile. 5. Concluding Remarks This paper has demonstrated that the martingale component in the long-term factorization of the stochastic discount factor (SDF) studied in Alvarez and Jermann (2005) and Hansen and Scheinkman (2009) is highly volatile, produces a downward-sloping term structure of bond Sharpe ratios as a function of bond maturity, and implies that the long bond is far from growth optimality. In contrast, the long forward probabilities forecast a generally upward-sloping term structure of bond Sharpe ratios that starts from near zero for short-term bonds and increases toward the Sharpe ratio of the long bond. This forecast implies that the long bond is growth optimal. Our empirical findings in the U.S. Treasury bond market are inconsistent with the assumption of transition independence of the SDF and degeneracy of the martingale component in its long-term factorization. This paper is based on research supported by grants from the National Science Foundation [CMMI-1536503 and DMS-1514698]. Footnotes 1 We stress that the assumption of rational expectations is critical to the discussion in this paper. Without assuming rational expectations, one would also need to model subjective beliefs $${\mathbb P}^*$$. 2 Although the long bond maximizes the expected log return, it does not generally maximize the Sharpe ratio since $${\rm corr}_t^{\mathbb L}\left(R^\infty_{t,t+\tau},1/R^\infty_{t,t+\tau}\right)$$ is not generally equal to $$-1$$. However, for sufficiently small holding periods, this correlation is close to $$-1$$ in diffusion models. In the empirical results in this paper, for 3-month holding periods, the empirically estimated $${\mathbb L}$$-Sharpe ratio of the long bond is close to its upper bound as discussed in Section 4. 3 We note that the value of $$2.83\%$$ for the asymptotic yield is in broad agreement with the results of Giglio, Maggiori, and Stroebel (2014) on very long-run discount rates. Based on the U.K. data, they argue that real long-run risk-free discount rates are under $$1\%$$ per annum, and perhaps as low as $$0.40\%$$ (see section V.A in their paper). The realized inflation rate in the United States over our estimation period from 1993 to 2015 was $$2.26\%$$. Subtracting this inflation rate from our estimate for the long-run nominal rate of $$2.83\%$$, we arrive at the real rate of $$0.57\%$$, which is consistent with that of Giglio, Maggiori, and Stroebel (2014). 4 We note that our hypothesis test relies on the linear approximation of the $${\mathbb L}$$-market price of risk that itself follows from the exponential-quadratic approximation for the principal eigenfunction. As discussed in Section 2, this approximation is highly accurate in the domain of relevant values of the state variables (in particular, containing all filtered values of the state variables), and, hence, the hypothesis test is robust to this approximation. References Alvarez, F., and Jermann. U. J. 2005 . Using asset prices to measure the persistence of the marginal utility of wealth. Econometrica 73 : 1977 – 2016 . Google Scholar CrossRef Search ADS Backus, D., Boyarchenko, N. and Chernov. M. Forthcoming . Term structures of asset prices and returns. 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For Permissions, please e-mail: journals.permissions@oup.com. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices)

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Published: Apr 6, 2018

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