Term Structure of Interest Rates with Short-run and Long-run Risks

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Term Structure of Interest Rates with Short-run and Long-run Risks Olesya Grishchenko, Zhaogang Song, and Hao Zhou 2015-095 Please cite this paper as: Grishchenko, Olesya, Zhaogang Song, and Hao Zhou (2015). “Term Structure of Interest Rates with Short-run and Long-run Risks ,” Finance and Economics Discussion Series 2015-095. Washington: Board of Governors of the Federal Reserve System, http://dx.doi.org/10.17016/FEDS.2015.095. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Term Structure of Interest Rates with Short-Run and Long-Run Risks∗ Olesya V. Grishchenko†, Zhaogang Song‡, and Hao Zhou§ October 7, 2015 Abstract Bond returns are time-varying and predictable. What economic forces drive this variation? To answer this long-standing question, we propose a consumption-based model with recursive preferences, long-run risks, and inflation non-neutrality. Our model offers two important insights. First, our model matches well the upward-sloping U.S. Treasury yield curve. Second, consistent with our model’s implication, variance risk premium based on the U.S. interest rate derivatives data emerges as a strong predictor for short-horizon Treasury excess returns, above and beyond the predictive power of other popular factors. In the model equilibrium, the variance risk premium is related to the short-run risks in the economy, while standard forward-rate-based factors are associated with long-run risks in the economy. ∗We are grateful to seminar participants at the financial mathematics seminar at Penn State University, internalMonetaryandFinancialMarketAnalysissectionseminarattheFederalReserveBoard,Luxembourg School of Finance, European Investment Bank, and Banque de France as well as conference participants at the 6th Risk Management Conference in Mont-Tremblant, Canada, the 2014 Midwest Finance Association Meetings, the 2014 Annual Missouri Economics Conference, the 2014 Darla Moore School of Business Fixed Income Conference, the 2014 CEPR Asset Pricing Workshop at Gerzensee, and the 2015 International ConferenceoftheFrenchFinanceAssociationforusefuldiscussionsandcomments. WethankDobriDobrev, Alex Hsu, Don Kim, Canlin Li, Jeremy Stein, Min Wei, and Ram Yamarthi for helpful discussions and comments. Theopinionsexpressedinthispaperarethoseoftheauthorsanddonotnecessarilyreflectthose of others in the Federal Reserve System. †Division of Monetary Affairs, Federal Reserve Board, Washington, DC 20551; Olesya.V.Grishchenko@frb.gov ‡The John Hopkins Carey School of Business, 100 International Drive, Baltimore, MD 21202; jsong8@jhu.edu §PBC School of Finance, Tsinghua University, 43 Chengfu Road, Haidian District, Beijing, 100083, P. R. China; zhouh@pbcsf.tsinghua.edu.cn

JEL Classification: G12, G13, G14 Keywords: Long-run risk, economic uncertainty, term structure of interest rates, bond risk premium, variance risk premium, predictability, interest rate derivatives 2

Term Structure of Interest Rates with Short-Run and Long-Run Risks Abstract Bond returns are time-varying and predictable. What economic forces drive this variation? To answer this long-standing question, we propose a consumptionbasedmodelwithrecursivepreferences,long-runrisks,andinflationnon-neutrality. Our model offers two important insights. First, our model matches well the upward-sloping U.S. Treasury yield curve. Second, consistent with our model’s implication, variance risk premium based on the U.S. interest rate derivatives data emerges as a strong predictor for short-horizon Treasury excess returns, above and beyond the predictive power of other popular factors. In the model equilibrium, the variance risk premium is related to the short-run risks in the economy, while standard forward-rate-based factors are associated with long-run risks in the economy. JEL Classification: G12, G13, G14 Keywords: Long-runrisk,economicuncertainty,termstructureofinterestrates, bond risk premium, variance risk premium, predictability, interest rate derivatives

1 Introduction The failure of the expectations hypothesis first documented by Fama and Bliss (1987) and Campbell and Shiller (1991) has attracted enormous attention in the asset pricing literature over the past decades. Various plausible risk factors that appear to capture bond return predictability (forward spread (Fama and Bliss, 1987), forward rates factor (Cochrane and Piazzesi, 2005), jump risk measure (Wright and Zhou, 2009), hidden term structure factor (Duffee, 2011), and macroeconomic variables’ factor(s) (Ludvigson and Ng, 2009; Huang and Shi, 2012), among many others) have been proposed. Despite this impressive progress the fundamental challenge for uncovering a particular economic mechanism behind bond return variation still stands out. Its resolution is equally important for market participants as well as for monetary policy makers. Our paper focuses on this issue. Tothatextentweproposeastylizedgeneralequilibriummodel. Thismodelcanbeviewed as an extension of long-run risk models in Bansal and Yaron (2004, BY) and Bollerslev, Tauchen, and Zhou (2009, BTZ). Bansal and Yaron (2004) emphasize importance of the long-run risk in consumption growth for explaining the equity premium, while Bollerslev, Tauchen, and Zhou (2009) show that richer volatility dynamics in consumption growth can be successful in capturing future stock return predictability. Our model includes both longrun risk and certain nontrivial volatility dynamics in consumption growth. It generates a two-factor volatility structure for the endogenously determined bond risk premium, in which the factors are explicitly related to the underlying volatility dynamics of consumption growthwheredifferentvolatilityconceptsloaddifferentlyonthefundamentalriskfactorsand capture separately short-run and long-run risks of Treasury excess returns. In particular, the difference between the risk-neutral and objective expectations of variation in interest rates, the factor that we term the interest-rate variance risk premium (IRVRP), effectively isolates the short-run risk factor associated with the volatility-of-volatility of consumption growth. The long-run risk factor associated with volatility of consumption growth appears to be captured by Fama and Bliss (1987) forward spread. 1

Ourmainempiricalfindingsexploittheinformationalcontentoftheinterest-ratevariance risk premium constructed from the U.S. interest rate swaps and swaptions markets. Why do we use this data to measure the short-run risk? First, the interest rate derivatives markets represent the largest segment of the U.S. fixed-income market and are important tools for corporate treasurers, asset managers, and public institutions to hedge interest rate risk.1 AccordingtotheBankofInternationalSettlements,asofJune2012,theoutstandingnotional value of interest rate swaps and swaptions exceeded $379 and $50 trillion, respectively, on a net basis. This outstanding notional value combined together is much larger than the $52 trillion of all exchange-traded interest rate futures and options, such as Treasury futures and futures options traded on Chicago Mercantile Exchange. In addition, a 2009 survey by the International Swaps and Derivatives Association reports that 88.3% of the Fortune Global 500 companies use swaps and swaptions for hedging interest rate risk. Last but not least, Dai and Singleton (2000) pointed to the similarities between the U.S. Treasury yields and swap rates and swaption prices. Thus, it appears likely that the interest rate derivatives markets can be informative for explaining variation in Treasury yields.2 Our first empirical finding connects time variation in the relatively short-horizon (one- to three-month) bond risk premiums to the variation in the interest-rate variance risk premium. The latter is driven entirely by the volatility-of-volatility factor with a positive loading predicted by our theory. Consistent with this prediction, interest-rate variance risk premium always loads positively on Treasury excess returns in the data and high(low) values of the interest-ratevarianceriskpremiumareassociatedwithsubsequenthigh(low)Treasuryexcess returns. Interest-rate variance risk premium alone explains a nontrivial share (40 to 50 percent) of the variation in one- and three-month Treasury excess returns, with the most 1Interest rate swaps play a central role in the whole financial system as swap rates reflect term financing rates of major financial institutions. In fact, the floating leg of a plain vanilla swap is usually tied to the 3-month LIBOR, which serves as a benchmark rate for corporate treasurers, mortgage lenders, and credit card agencies. 2DaiandSingletonnoted, inparticular, that“thoughtheinstitutionalstructuresofdollarswapandU.S. Treasury markets are different, some of the basic distributional characteristics of the associated yields are similar”. 2

explanatory power concentrated at the quarterly horizon. Moreover, combined with other standard predictors such as forward spread or forward rate risk factors, IRVRP generates even higher bond return predictability than on its own and contributes nontrivially (about 20 percent) to standard predictors. Our second empirical result is that the interest-rate variance risk premium has limited forecasting power for longer-maturity excess bond returns, while the forward spread is a more important factor that captures information related to the variation of the long-run risk factor. Thus, our findings are consistent with our model where both long-run and short-run risk factors drive bond risk premium in equilibrium. Finally, our calibration exercise suggests that the model fits the upward-sloping nominal yield curve remarkably well. The key two factors in fitting the nominal yield curve are the presence of the long-run risk in the model and inflation non-neutrality. As such, the longrun risk state variable from the real side of the model affects nominal prices via inflation channel. We reasonably calibrate inflation process while we leave the real side model parameters like the ones in BY and BTZ. The most important feature of the inflation process is the negative correlation with consumption volatility shock, consistent with recent empirical findings (Piazzesi and Shneider, 2007; Campbell, Sunderam, and Viceira, 2013; Bansal and Shaliastovich, 2013). Without this feature, the nominal yield curve is downward-sloping. It stands to reason that aggregate macro series, and consumption growth, in particular, are more volatile when investors are relatively more uncertain about economy growth prospects compared to the periods with relatively low volatility environment. Thus, our two-factor volatility structure of consumption risks can be linked to the studies that model economic uncertainty. The idea of economic uncertainty as a potential risk factor has gained attention recently, both for explaining variation in stock returns (Bollerslev, Tauchen, and Zhou, 2009; Bloom, 2009; Drechsler, 2013) and in bond returns (Wright, 2011; Bansal and Shaliastovich, 2013; Giacoletti, Laursen, and Singleton, 2015). The last two papers are especially relevant to our study. Bansal and Shaliastovich (2013) link bond excess return 3

variation to a variation in volatility in real activity and inflation – variables they interpret as uncertainty - but they do not model uncertainty process explicitly. Giacoletti, Laursen, and Singleton (2015) find that dispersion of beliefs about future interest rates that can be loosely linked to investors’ uncertainty about interest rates is distinct from information about the macroeconomy and can be useful in explaining variation in bond returns. While the bond pricing empirical literature (Fama and Bliss, 1987; Campbell and Shiller, 1991) has documented predictability of long-horizon bond returns, bond predictability in the short run did not receive much attention until recently. A growing literature argues for the existence of the short-run and long-run risk components of the aggregate volatility to study the variation of stock returns (Adrian and Rosenberg, 2008; Christoffersen, Jacobs, Ornthanalai, and Wang, 2008; Branger, Rodrigues, and Schlag, 2011; Zhou and Zhu, 2012, 2013). Zhou (2009), in particular, emphasized importance of the short-run risk factor for short-horizon (such as one-month) stock returns. A recent related paper by Ghysels, Le, Park, and Zhu (2014) emphasises a short-run volatility component of bond yields as a useful predictor for future excess returns, as opposed to a long-run volatility component that does not predict Treasury excess returns. In the frame of this literature and to the best of our knowledge, our paper is the first that explores short-horizon bond return predictability and explains empirical findings within a structural two-factor volatility model framework. It appears that while the volatility-of-volatility of consumption growth (short-run risk factor) drivesthevariationintheshort-horizonTreasuryexcessreturns,thevariationinlong-horizon returns appears to be related to a different kind, possibly more to a longer-run growth factor. The long-run risk factor is important for matching the term structure of nominal interest rates. The rest of the paper is organized as follows. Section 2 presents our long-run risk model with two different volatility types and inflation non-neutrality, and derives asset pricing implications of the model; Section 3 discusses the calibration of the U.S. Treasury yield curve implied by our model; Section 4 provides overview of the interest rate swaptions market and 4

introduces the interest-rate variance risk premium; Section 5 describes all relevant data to our empirical exercise; Section 6 discusses empirical results. Finally, Section 7 concludes. 2 Model and Asset Pricing 2.1 Preferences We consider a discrete-time endowment economy with recursive preferences for early resolution of uncertainty introduced by Kreps and Porteus (1978), Epstein and Zin (1989), and Weil (1989): U = (cid:104) (1−δ)C 1− θ γ +δ (cid:0)E U1−γ(cid:1) θ 1(cid:105) 1− θ γ , (1) t t t t+1 where δ is the time discount factor, γ ≥ 0 is the risk aversion parameter, ψ ≥ 0 is the intertemporal elasticity of substitution (IES), and θ = 1−γ . Preference for early resolution 1−1 ψ of uncertainty implies γ > 1, which, in general, implies θ < 1. We will assume throughout ψ the paper that γ > 1 and ψ > 1, which implies θ < 0 and refer to preference for early resolution of uncertainty as consistent with θ < 0. A special case of recursive preferences expected utility - corresponds to the case of γ = 1 (θ = 1). ψ Epstein and Zin (1989) show that the log-linearized form of the associated real stochastic discount factor m is given by: t θ m = θlnδ − g +(θ−1)r , (2) t+1 t+1 c,t+1 ψ (cid:16) (cid:17) where g = log Ct+1 is the log growth of the aggregate consumption, r is the log t+1 Ct c,t+1 return on an aggregate wealth portfolio that delivers aggregate consumption as its dividend each time period. Note that the return on wealth is different from the observed return on the market portfolio because aggregate consumption is not equal to aggregate dividends. Consequently, the return on wealth is not observable in the data. The nominal discount 5

factor m$ is equal to the real discount factor minus expected inflation π : t+1 t+1 m$ = m −π . (3) t+1 t+1 t+1 2.2 Economy dynamics To solve for the equilibrium asset prices we specify consumption and inflation dynamics. Consumption dynamics features time-varying consumption growth rate g and expected cont sumptiongrowthratex , time-varyingvolatilityofconsumptiongrowthσ2 andtime-varying t g,t volatility-of-volatility of consumption growth q : t x = ρ x +φ σ z , t+1 x t e g,t x,t+1 g = µ +x +σ z , t+1 g t g,t g,t+1 (4) √ σ2 = a +ρ σ2 + q z , g,t+1 σ σ g,t t σ,t+1 √ q = a +ρ q +φ q z , t+1 q q t q t q,t+1 where the parameters satisfy a > 0, a > 0, |ρ | < 1, |ρ | < 1 and φ > 0. The vector σ q σ q q of shocks (z , z , z , z ) follows i.i.d. normal distribution with zero mean and x,t+1 g,t+1 σ,t+1 q,t+1 unit variance and shocks are assumed to be uncorrelated among themselves. The second pair of equations in (4) is new compared to Bansal and Yaron (2004) and Bansal and Shaliastovich (2013). Stochastic volatility σ2 represents time-varying economic uncertainty in g,t+1 consumption growth with time-varying volatility-of-volatility (vol-of-vol) measured by q .3 t Since σ2 directly affects variation in x , the predictable component in consumption growth, g,t t we will refer to σ2 as the state variable that captures the long-run risk. The volatility-ofg,t volatility process q can be thought of as the volatility risk or, the short-run risk. As we will t see later, this terminology will be supported by our empirical findings. 3Recent studies provided empirical support in favor of time-varying consumption growth volatility, e.g., Bekaert and Liu (2004), Bansal and Yaron (2005), Lettau, Ludvigson, and Wachter (2008), Bekaert, Engstrom, and Xing (2009), among others. 6

In order for the real economy model (4) to have realistic implications for nominal bond risk premiums, we conjecture a fairly rich inflation process motivated by previous literature. Indeed, Bansal and Shaliastovich (2013) allow for expected inflation shocks to be correlated (negatively)withexpectedconsumptiongrowth,andZhou(2011)allowsforavol-of-volshock to affect inflation. We incorporate both of these features into expected inflation dynamics π : t+1 √ π = a +ρ π +φ z +φ σ z +φ q z , (5) t+1 π π t π π,t+1 πg g,t g,t+1 πσ t σ,t+1 where ρ is a persistence and aπ is the long-run mean of the inflation process. There are π 1−ρπ threeshocksthatdriveinflationprocess: (1)aconstantvolatilitypartφ withanautonomous π shock z ; (2) a stochastic volatility part φ σ that works through consumption growth π,t+1 πσ g,t √ channel z ; and (3) another stochastic volatility part φ q that works through the g,t+1 πσ t volatility channel z . Exogenous inflation shock z does not generate inflation risk σ,t+1 π,t+1 premium even in the presence of the time-varying volatility of this shock.4 In contrast, the second and the third shocks generate inflation risk premium because real side shocks (stochastic volatility of consumption growth and uncertainty) affect inflation. In addition, sinceφ andφ controlinflationexposurestothegrowthanduncertaintyrisks, thisprocess πg πσ implicitly violates inflation neutrality in the short run, but not in the long run.5 2.3 Pricing kernel In equilibrium, the log wealth-consumption ratio z is affine in expected consumption growth t x , stochastic volatility of consumption growth σ2, and the vol-of-vol factor q : t t t z = A +A x +A σ2 +A q . (6) t 0 x t σ t q t 4The inability of the expected inflation process with only one (autonomous) shock even with stochastic volatility to generate inflation risk premium is examined in Zhou (2009). 5Thereisnoviolationofinflationneutralityinthelongrunbecauseunconditionalexpectationofinflation process (5) is Eπ t = 1− aπ ρπ . 7

Campbell and Shiller (1988) show that the return on this asset can be approximated as follows: r = κ +κ z −z +g , (7) c,t+1 0 1 t+1 t t+1 where κ = ln((1 + expz¯)) − κ z¯, κ = exp(z¯)) , and z¯ is the average wealth-consumption 0 1 1 1+exp(z¯) ratio: z¯= A (z¯)+A (z¯)σ¯2 +A (z¯)q¯. (8) 0 σ q The equilibrium loadings for (6) are derived in Appendix A.1: 1− 1 ψ A = , x 1−κ ρ 1 x (cid:34) (cid:35) 1 (cid:18) θ (cid:19)2 A = θ− +(θκ A φ )2 , σ 1 x e (9) 2θ(1−κ ρ ) ψ 1 σ (cid:113) 1−κ ρ − (1−κ ρ )2 −θ2κ4φ2A2 1 q 1 q 1 q σ A = . q θ(κ φ )2 1 q As in Bansal and Yaron (2004), recursive preferences along with the early resolution of uncertainty are crucial in determining the sign of the equilibrium loadings of the state variables in our model. When the intertemporal elasticity of substitution ψ > 1, the intertemporal substitution effect dominates the wealth effect. In response to higher expected consumption growth, agents invest more and, consequently, wealth-consumption ratio increases. Therefore, the wealth-consumption ratio loading on the expected consumption growth is positive (A > 0)whereas loadings onthe volatilityand volatility-of-volatility ofconsumptiongrowth x arebothnegative(A < 0andA < 0)asintimesofhighvolatilityand/oruncertaintyagents σ q sell off risky assets driving the wealth-consumption ratio down.6 The persistence of expected growth shock ρ and time-varying volatility ρ magnify the effect of the changes in these x σ state variables on the valuation ratio since investors perceive such macroeconomic changes as long-lasting. Contrary to that, persistence of the volatility-of-volatility, ρ , roughly canq 6The solution for A represents one of a pair of roots of a quadratic equation, but we pick the one q presented in Eq. (9) as the more meaningful one. We elaborate on this choice in Section A.1. 8

cels out in the A loading. This provides further support for interpretation of q as a state q t variable that captures relatively short-run economic risks.7 Using the solution for the wealth-consumption ratio we show in Appendix A.2 that the conditional mean of the stochastic discount factor m is linear in the fundamental state t+1 variables and the innovation in m pins down the fundamental sources and compensations t+1 for risks in the economy: √ √ m −E [m ] = −λ σ z −λ σ z −λ q z −λ q z , (10) t+1 t t+1 g g,t g,t+1 x g,t x,t+1 σ t σ,t+1 q t q,t+1 where the quantities of risks are time-varying volatility and volatility-of-volatility of con- √ sumption growth, σ and q , respectively, and λ , λ , λ , λ represent the market prices g,t t g x σ q of risk of consumption growth, expected consumption growth, volatility, and volatility-ofvolatility: λ = γ, λ = (1−θ)κ A , g σ 1 σ (11) λ = (1−θ)κ A φ , λ = (1−θ)κ A φ . x 1 x e q 1 q q The market price of consumption risk λ is equal to the coefficient of relative risk aversion γ. g Other risk prices crucially depend on our preference assumptions. When agents have preference for early resolution of uncertainty (θ < 0), the market price of expected consumption risk is positive: λ > 0. In this case, positive shocks to consumption and expected consumpx tion cause risk premium to decrease as agents buy risky assets and drive wealth-consumption ratio up. On the contrary, market prices of risk of volatility and volatility-of-volatility are negative (λ < 0 and λ < 0): Consistent with the so-called leverage effect, in response to σ q either type of volatility positive shock agents sell risky assets and drive wealth-consumption ratiodownandvolatilityriskpremiumsup. Itisworthnotingthattheseeffectsarenotbased on the statistical linkages between return and volatility (as the endowment and volatility 7Bansal,Kiku,andYaron(2012)checkthattheirapproximatesolutionsareveryaccuratewhencompared against numerical solutions, used, e.g., in Binsbergen, Brandt, and Koijen (2012). 9

shocks are uncorrelated) but arise endogenously in the equilibrium. In the absence of the early resolution of uncertainty (γ = 1 and θ = 1) neither expected consumption, volatility, ψ or volatility-of-volatility compensate investors for these risks. 2.4 Asset prices 2.4.1 Real risk-free rate The real risk-free rate is the negative of the (log) price of the real one-period bond: 1 r = −p1 = −E [m ]− Var [m ]. (12) f,t t t t+1 2 t t+1 Given the solutions for the first and second moments of the pricing kernel (2) provided in Appendix A.2, the real risk-free rate is: r = −θlnδ +γµ −(θ−1)[κ +(κ −1)A +κ (A a +A a )] f,t g 0 1 0 1 σ σ q q +[γ −(θ−1)A (κ ρ −1)]x x 1 x t (cid:20) 1 1 (cid:21) (13) + −(θ−1)A (κ ρ −1)− (θ−1)2κ2A2φ2 − γ2 σ2 σ 1 σ 2 1 x e 2 g,t (cid:20) (cid:21) 1 + −(θ−1)A (κ ρ −1)− (θ−1)2κ2(A2 +A2φ2) q . q 1 q 2 1 σ q q t Note that the time variation in (13) crucially depends on the assumption of the preference for early resolution of uncertainty (θ < 0). Without it (θ = 1) the risk-free rate reduces to the standard power-utility case with no compensation for the fundamental economic risks: r = −lnδ+γµ − 1γ2σ2. Moreover, negative θ implies positive risk premium for the longf g 2 g run risk state variable x and negative risk premiums for both volatility factors (ignoring t Jensen’s correction). 10

2.4.2 Nominal risk-free rate The nominal risk-free rate is the negative of the (log) price of the nominal one-period bond. Thus, it is equal to the real risk-free rate plus inflation compensation. The closed form for the nominal risk-free rate is derived in Appendix A.4: 1 r$ = −θlnδ +γµ +a −(θ−1)[κ +(κ −1)A +κ (A a +A a )]− φ2 f,t g π 0 1 0 1 σ σ q q 2 π +[γ −(θ−1)A (κ ρ −1)]x x 1 x t (cid:20) (cid:21) 1 1 1 + −(θ−1)A (κ ρ −1)− γ2 − (θ−1)2(κ A φ )2 − φ2 −γφ σ2 (14) σ 1 σ 2 2 1 x e 2 πg πg g,t (cid:20) (cid:21) 1 1 + −(θ−1)A (κ ρ −1)− (θ−1)2κ2(A2 +A2φ2)− φ2 +(θ−1)κ A φ q q 1 q 2 1 σ q q 2 πσ 1 σ πσ t +ρ π . π t Since inflation is not an autonomous process, besides having a direct effect on the nominal rates, ρ π , it affects loadings on σ2 and q via additional terms, related to φ and φ π t t t πg πσ coefficients, respectively. This results in inflation non-neutrality, which means that inflation affects future real growth in the economy. 2.4.3 The n−period nominal bond price A general recursion for solving for the n−period nominal bond price is as follows:8 (cid:104) (cid:105) P$,n = E M$ P$,n−1 . (15) t t t+1 t+1 We assume that the (log) price of the n−period nominal bond p$,n follows the affine repret sentation of the real state variables x , σ2, q and inflation π : t t t t p$,n = B$,n +B$,nx +B$,nσ2 +B$,nq +B$,nπ . (16) t 0 1 t 2 t 3 t 4 t 8The solution for the n−period real bond price is provided in Appendix A.3. 11

We solve for the nominal bond state loadings B$,n, i = 0,...,4 using initial conditions i B$,0 = 0, i = 0,...,4 (since p$,0 = 0) and the above recursion.9 i t 2.4.4 Nominal bond risk premium Let rx$,n−1 be the bond excess return from t to t+1 for an n−period nominal bond holding t+1 one period. Then its expected value, or nominal bond risk premium, brp$,n, is given by the t (negative of) covariance between the nominal pricing kernel m$,n−1 and the nominal bond t+1 price p$,n: t (cid:104) (cid:105) brp$,n = −Cov m$ ,p$,n−1 t t t+1 t+1 (cid:104) (cid:105) = −(γ +φ )B$,n−1φ +(θ−1)κ A B$,n−1φ2 σ2 πg 4 πg 1 x 1 e g,t (cid:104) (cid:105) + ((θ−1)κ A −φ )(B$,n−1 +B$,n−1φ )+(θ−1)κ A B$,n−1φ2 q (17) 1 σ πσ 2 4 πσ 1 q 3 q t −B$,n−1φ2 4 π ≡ β$,n−1σ2 +β$,n−1q −B$,n−1φ2. 1 g,t 2 t 4 π The first two terms in (17) reflect consumption and volatility risk premiums amplified by the inflation shock parameters φ and φ while the third term captures the autonomous πg πσ inflationshockthroughφ . Theeffectofthelong-runriskcapturedbyA equilibriumloading π x on the wealth-consumption ratio amplifies the overall contribution of the consumption risk, σ . This effect is absent in Zhou (2011) and Mueller, Vedolin, and Zhou (2011), and thus, g,t makes it more difficult to explain the upward sloping term structure of the nominal yield curve. 2.4.5 Bond return predictability Bollerslev, Tauchen, and Zhou (2009) show that the equity variance risk premium – the difference in expectations of the equity variance under risk-neutral and physical measures – 9The solution is provided in Appendix A.5, equation (71). 12

isausefulpredictoroftime-variationinaggregatestockreturns. Motivatedbythisresult, we apply this measure to understand time variation in bond returns.10 Time-varying variance risk premium arises endogenously in the equilibrium and is given by: VRP = EQ(cid:2) σ2 (cid:3) −EP(cid:2) σ2 (cid:3) t t r,t+1 t r,t+1 (18) (cid:2) (cid:3) = (θ−1)κ A (1+κ2A2φ2)+A κ2φ2(A2 +A2φ2) q . 1 σ 1 x e q 1 q σ q q t A first and central observation here is that the time variation in the variance risk premium is solely due to the time variation in q state variable. If volatility-of-volatility is constant, t q = q, the variance risk premium reduces to a constant (θ − 1)κ A (1 + κ2A2φ2)q cont 1 σ 1 x e trary to empirical evidence that the variance risk premium is time-varying (e.g., BTZ). A second observation is that although consumption growth risk σ2 does not affect variance g,t risk premium directly it still has an indirect effect on it. If consumption volatility σ2 is g,t not stochastic then the wealth-consumption ratio equilibrium loadings A = 0 and A = 0 σ q by construction and variance risk premium is zero. A third observation is that if there is no recursive preference (θ = 1) then variance risk premium is zero by construction. Lastly, positivity of the variance risk premium is guaranteed by negative θ along with with negative A and A . σ q The common factor q in the nominal bond risk premium (17) and in the variance risk t premium (18) suggests that the latter should capture some time variation of the former. Thus, for developing intuition to what extent distinct economic forces might be driving bond premium, it is instructive to consider the following regression (ignoring a measurement error): brp$,n = a+bVRP , (19) t t 10Whilewedonotderivethebondvarianceriskpremium,inthisparticularmodelthesamestatevariables drive bond and equity variance risk premiums due to affine nature of the model. 13

where the model-implied slope coefficient b is: Cov(brpn,VRP ) β$,n−1 b = t t = 2 (20) (cid:2) (cid:3) Var(VRP ) (θ−1)κ A (1+κ2A2φ2)+A κ2φ2(A2 +A2φ2) t 1 σ 1 x e q 1 q σ q q and R2 is: (cid:16) (cid:17)2 β$,n−1 Var(q ) b2Var(VRP ) 2 t R2 = t = . (21) Var(brp$ t ,n) (cid:16) β$,n−1 (cid:17)2 Var(σ2 )+ (cid:16) β$,n−1 (cid:17)2 Var(q ) 1 g,t 2 t As discussed above, the constant volatility-of-volatility (short-run risk) q implies no time t variation in the variance risk premium, and therefore R2 ≡ 0. The other extreme case, capturedbytheabsenceofthetime-varyingconsumptionvolatility(long-runrisk)σ2 implies g,t R2 = 1 and the empirical predictability pattern cannot be replicated. Thus, formula (21) illustrates that both types of risks, short- and long-run, affect variation in the bond risk premium. 3 Calibration In this section we discuss calibration of the nominal yield curve implied by our model (4) and inflation process (5). We consider two benchmark cases, Bansal and Yaron (2004, BY) and Bollerslev, Tauchen, and Zhou (2009, BTZ). Compared to BY, BTZ incorporate the timevarying vol-of-vol factor, but in the absence of the long-run risk channel. We differ from BTZ in two aspects: (1) we have the long-run risk state variable in the real side model; and (2) we model inflation process in order to derive implications for the nominal bond prices. We present all three models’ parameters (BY, BTZ, and ours) in Table 1. Panel A provides calibration values for the real economy dynamics. We set preference parameters δ = 0.997, γ = 8, and ψ = 1.5.11 Consumption growth parameters µ = 0.0015, g ρ = 0.979, φ = 0.001 are consistent with BY (and BTZ except for ρ = φ ≡ 0). Volatility x e x e persistence ρ = 0.978 is the same as in BY and BTZ, and a = (1−ρ )Eσ is set so that σ σ σ 11BY and BTZ use γ =10, but in our model slightly lower value of γ works reasonably well. 14

the unconditional expectation Eσ2 = 0.02342, which is slightly higher than in BY and BTZ t because we find that this value matches better the nominal yield curve in the model. We set the expected volatility-of-volatility level Eq = a (1−ρ )−1 = 10−9 so that a = 2−10 given q q q ρ = 0.8. In addition, φ = 10−4. Our choice of ρ and ρ is broadly consistent with the q q σ q estimates of Bollerslev, Xu, and Zhou (2013), who find that the long-run risk (proxied by σ2 ) is more persistent than the short-run risk (proxied by q ). Thus, the calibrated model g,t t is connected with the next empirical section where we show that these two types of risks in the bond premia are disentangled. Panel B provides calibration parameters for the expected inflation process. We set the average annualized annualized inflation rate Eπ = 2% and persistence parameter ρ = 0.95 π in accordance with the current Fed’s inflation target and Great Moderation period overall.12 Implied a on a monthly basis is equal to Eπ(1−ρ ) = 0.02/12×0.05 = 8×10−5. The total π π unconditional variance of the inflation process (5) is given by: 1 Var(π) = (cid:0) φ2 +φ2 Eσ2 +φ2 Eq (cid:1) . (22) 1−ρ2 π πg πσ π We calibrate variance-related parameters of (22) so that the total annualized unconditional inflationvolatilityis2%. Sinceρ = 0.95, Eσ2 = 0.02342, Eq = 10−9,theterminparentheses π in (22) on a monthly basis is: φ2 +φ2 ×0.02342 +φ2 ×10−9 = 0.022/12×(1−0.952) = π πg πσ 3.25×10−6. Further, we assume that the first (autonomous) shock contributes one half to the total variance while the other two shocks contribute equally to the remaining half of the total variance of the inflation process.13 Thus, the contribution of the first shock to the total inflation variance is 0.5×3.25×10−6 = 1.625×10−6, implying φ = 0.0013. The contribution π ofthesecondandthirdshocksareequaltoeachotherandto0.25×3.25×10−6 = 8.125×10−7. 12These numbers may be justified by the data after 1980s and especially after 2008, when Fed launched unprecedented measures of accommodative monetary policy, namely, quantitative easing. Our expected inflation rate is lower than the one in Bansal and Shaliastovich (2013), who set it at 3.61% (see their Table 5). 13Equal distribution of variance among the shocks results in slight overshooting of the model-implied interest rates levels relative to those in the sample. 15

Therefore, the implied φ = (8.125×10−7/0.02342) 1/2 = −0.0385. The negative sign of πg φ is motivated by previous empirical findings (Piazzesi and Shneider, 2007; Campbell, πg Sunderam, and Viceira, 2013; Bansal and Shaliastovich, 2013). In particular, Bansal and Shaliastovich(2013)useSPFsurveydataforone-yearaheadconsensusinflationforecastover 1969 - 2010 sample and a latent factor for the expected consumption growth to estimate relationship between the two. They find that expected inflation negatively affects future consumption growth thus suggesting non-neutrality of inflation. Last, the implied φ = πσ (8.125×10−7/10−9) 1/2 = 28.5. Figure 1 reports our calibration results. Both panels show the average nominal yield curve out to 10 years (blue solid line) in January 1991 - December 2010 sample period and the calibrated nominal yield curve (red dashed line) implied by our model (Panel A) and by our modified model in the absence of the long-run risk channel x (Panel B). It is obvious t from Panel A that our model matches very well the levels of the nominal yields and captures the slope the yield curve too. The 1-, 5-, and 10-year model-implied yields are 3.71%, 5.14% and 5.58% relative to observed yields of 4%, 4.95%, and 5.65% at corresponding maturities. Figure on Panel B shows that without long-run risk the model is not successful in fitting the nominal upward-sloping yield curve as it generates downward-sloping yield curve, even with the presence of time-varying economic uncertainty.14 To understand the effect of the longrun risk factor better, it is useful to write down the nominal yields as an affine combination of state variables: 1 (cid:104) (cid:105) y$,n = − B$,n +B$,nx +B$,nσ2 +B$,nq +B$,nπ , (23) t n 0 1 t 2 t 3 t 4 t where B$,n, B$,n, B$,n, B$,n are model-implied nominal bond price loadings provided in Ap- 1 2 3 4 pendix A.5. The equilibrium nominal yield loadings are plotted in Figure 2.15 In our model, 14Bansal and Shaliastovich (2013) fit the term structure of interest rates for the short- and intermediateterm yields (up to five years only), whereas our model quantitatively matches the level and slope of the nominal term structure from one- to ten-year interest rates. 15Nominal yield loadings are nominal bond price loadings with a negative sign. 16

nominal yields hedge expected consumption and inflation risks. As the top left panel of Figure 2 shows, nominal yields increase when expected consumption is high because −B$,n > 0, 1 and the effect is stronger for higher n. Intuitively, a negative shock to expected consumption drives bond prices up and yields down and a positive shock to expected consumption drives bond prices down and yields up. The same effect is obvious for expected inflation as −B$,n > 0 and this loading is also increasing with maturity (bottom right panel). The top 4 rightpanelshowstheeffectofconsumptionvolatilityshockonnominalyields. Corresponding loading −B$,n manifests negative correlation of expected inflation and consumption growth 3 that we discussed above. Initial effect of both positive consumption volatility and volatilityof-volatility shocks on yields is negative although this effect mean-reverts in the long-run. Given that the steady state values of consumption volatility and volatility-of-volatility processes are relatively small, long-run risk has the largest effect on nominal yields and helps to fit the upward-sloping nominal yields curve dramatically. Therefore, the slope of the term structure of interest rates appears to be tightly linked to the slow-moving predictable component in consumption growth. 4 Empirical Measurements The theoretical model outlined in Section 2 suggests that the interest-rate variance risk premium, the difference between the market’s risk-neutral expectation of future interest rate return variation and the objective expectation of the future interest rate variation, can serve as a useful predictor for the Treasury bond premiums. To measure the interest-rate variance risk premium and investigate this conjecture, we rely on the new nonparametric “model-free” variation concept derived from the interest rate swaptions market (for the risk-neutral expectation of the future interest rate return variation) and from the interest rate swap market (for the expectation of the future interest rate return variation under the physical measure). To that extent, we first provide an overview of the mechanics of 17

the interest rate swaptions (Section 4.1), then discuss the construction of swaption-implied variance (Section 4.2), and, finally, outline the construction of the realized interest rate swap variance (Section 4.3). 4.1 Interest rate swaptions Consider a forward start fixed versus floating interest rate swap with a start date T and m maturity date T . The fixed annuity payments are made on a pre-specified set of dates, n T < T < ··· < T , with the intervals equally spaced by δ, which equals six months in m+1 m+2 n the U.S. swaption market. The floating payments tied to the three-month LIBOR are made quarterly at T −δ/2, T , T −δ/2, T , ···, T −δ/2, and T .16 m+1 m+1 m+2 m+2 n n At time T , the value of the floating leg equals par, and the time–t value of the floating m leg is D(t,T ), where D(t,T ) is the time–t price of a zero-coupon bond maturing at time m m T . The time–t value of the fixed leg is equal to D(t,T ) + A (t), where A (t) ≡ m n m,n m,n (cid:80)n D(t,T ) is the present value of an annuity associated with the fixed leg of the j=m+1 j forward swap contract, also known as the “price value of the basis point” (PVBP) of a swap. The time-t forward swap rate, S (t), is the rate on the fixed leg that makes the present m,n value of the swap contract equal to zero at t: D(t,T )−D(t,T ) m n S (t) = . (24) m,n A (t) m,n This forward swap rate becomes the spot swap rate S (T ) at time T . m,n m m A swaption gives its holder the right but not the obligation to enter into an interest rate swap either as a fixed leg (payer swaption) or as a floating leg (receiver swaption) with a pre-specified fixed coupon rate. The underlying security of a swaption is a forward start interest rate swap contract. For example, let T be the expiration date of the swaption, K m be the coupon rate on the swap, and T be the final maturity date of the swap. The payer n 16We assume that both the fixed and floating legs pay $1 principal at T . n 18

swaption allows the holder to enter into a swap at time T with a remaining term of T −T m n m and to pay the fixed annuity of K. At time t, this swaption is usually called a (T −t) into m (T −T ) payer swaption, also known as a (T −t) by (T −T ) payer swaption, where n m m n m (T −t) is the option maturity and (T −T ) is the tenor of the underlying swap. Because m n m the value of the floating leg will be par at time T , the payer swaption is equivalent to a put m option on a bond with a coupon rate K and a remaining maturity of T − T , where the n m strike of this put option is $1. Similarly, the receiver swaption is equivalent to a call option on the same coupon bond with the strike price of $1. Let P (t;K) and R (t;K) denote the time-t value of a European payer and receiver m,n m,n swaption, respectively, expiring at T with strike K on a forward start swap for the time m period between T and T . At the option expiration date T , the payer swaption has a m n m payoff of [1−D(T ,T )−KA (T )]+ = A (T )[S (T )−K]+, m n m,n m m,n m m,n m where Eq. (24) evaluated at T is used. Therefore, the time-t (< T ) price of this payer m m swaption is given by (cid:110) (cid:111) P m,n (t;K) = EQ t e− (cid:82) t Tmr(s)dsA m,n (T m )[S m,n (T m )−K]+ = A (t)EAm,n(cid:8) [S (T )−K]+(cid:9) , (25) m,n t m,n m where Q is the risk-neutral measure and Am,n is the annuity measure with A (t) as the m,n numeraire. That is, the Radon-Nikodym derivative of the annuity measure with respect to the risk-neutral measure is dAm,n = e− (cid:82)Tmr(s)dsAm,n(Tm). Similarly, the time-t price of the dQ t Am,n(t) receiver swaption is given by R (t;K) = A (t)EAm,n(cid:8) [K −S (T )]+(cid:9) . (26) m,n m,n t m,n m We note from (25) and (26) that a swaption is tied to two sources of uncertainty: (i) the 19

forward swap rate S (t), and (ii) the swap’s PVBP realized at time T , A (T ). The m,n m m,n m change of measure from Q to Am,n allows us to focus on the risk of S (t) and facilitates m,n the pricing of swaptions. 4.2 Measure of swaption-implied variance Variance swaps on equities allow one to hedge the risk of the realized variance of stock returns. The variance contract on swap rates that we develop below allows us to hedge the risk of the realized variance of interest rate swap rates. At time t, the short leg promises to pay the long leg at T : m (cid:34) (cid:35) (cid:18) S (t+∆) (cid:19)2 (cid:18) S (t+2∆) (cid:19)2 (cid:18) S (T ) (cid:19)2 A (T ) ln m,n + ln m,n +···+ ln m,n m , (27) m,n m S (t) S (t+∆) S (T −∆) m,n m,n m,n m the product of the realized variance of the log forward swap rate logS (t) over [t,T ] and m,n m the PVBP A (T ). In return, the long leg pays the short leg A (T )×VP (t) at T , m,n m m,n m m,n m where VP (t) is determined at time t such that the value of the contract equals zero at m,n initiation. We refer to VP (t) as the variance price of the forward swap rate. m,n The variance contract on swap rates uses the sum of squared log changes to measure the realized variance of forward swap rates over [t,T ]. Similar to the payoff of a swaption, the m payoff of the variance contract on swap rates depends on the realized variance of forward swap rates as well as an annuity discount factor. This design makes it convenient to obtain the variance price VP (t) by a change of the risk-neutral measure to the corresponding m,n annuity measure. It also makes it easier to replicate the variance contract using swaptions given the similar payoff structures. The variance price VP (t) is the Am,n-expectation of m,n the quadratic variation of the forward swap rate S (t) over [t,T ]. m,n m Similar to the equity variance swap whose payoff can be replicated with a portfolio of out-of-the-money equity options, the time-varying payoff of the variance contract on swap rates can be replicated with a portfolio of out-of-the-money swaptions written on S (t). m,n 20

In particular, generalizing the algorithm used by CBOE in constructing VIX, we have (cid:40) (cid:41) (cid:90) (cid:90) 2 1 1 IV (t) ≡ P (t;K)dK + R (t;K)dK , (28) m,n A (t) K2 m,n K2 m,n m,n K>Sm,n(t) K<Sm,n(t) where T − t is the time-to-maturity for the variance swap. As observed from (28), this m replication portfolio contains positions in out-of-the-money swaptions with a weight that is inversely proportional to the squares of their strikes. A similar replication portfolio based on equity options has been employed in the literature to construct model-free implied volatility measures (Bollerslev, Tauchen, and Zhou, 2009; Carr and Wu, 2009). 4.3 Interest-rate realized variance In order to quantify the actual return variation of interest rates, we follow the methodology of Bollerslev, Tauchen, and Zhou (2009). Let pn be the logarithmic price of the n-period t zero-coupon bond at time t. The realized variation over the discrete time interval [t−1,t] can then be measured in a “model-free” way as follows: (cid:88) M (cid:20) i i−1 (cid:21)2 RV = pn(t−1+ )−pn(t−1+ ) → Return variation(t−1,t), (29) t M M i=1 where the convergence to the return variation relies on M → ∞,that is, on an increasing number of within-the-period price observations. Prior literature (Andersen, Bollerslev, Diebold, andEbens,2001;Andersen, Bollerslev, Diebold, andLabys,2001;Barndorff-Nielsen and Shephard, 2004; Wright and Zhou, 2009; Bollerslev, Tauchen, and Zhou, 2009) demonstrates that this “model-free” realized variance measure based on high-frequency intraday data results in more accurate ex post observations of the true (unobserved) return variation compared to more traditional sample variances based on daily or coarser frequency returns. In addition, RV measure provides a nonparametric empirical analog to σ2 that appears in t r,t the definition of the variance risk premium (18). 21

4.4 Interest-rate variance risk premium (IRVRP) The interest-rate variance risk premium is defined as the difference between the market’s risk-neutral expectation of the future swap rate variation over [t, T ] under the annuity m measure Am,n and the corresponding expectation under the physical measure over the same time interval IRVRP (t) ≡ IV (t)−RV (t). (30) m,n m,n m,n 5 Data and Estimates In this section we discuss the data, construction, and the estimates of the IRVRP, then describe Treasury yield data, and other control variables used in predictive regressions in Section 6. 5.1 Estimates of the interest-rate variance risk premium We use interest rate swaptions and swap market data for computing implied and realized variances of interest rates, respectively. We compute then the IRVRP as a difference between the implied variance and (the expectation of) the realized variance. 5.1.1 Construction of the implied variance of interest rates To construct the implied variance (28) we need two ingredients: First, we need to construct A (t) and S (t), the PVBP of a swap and the forward swap rate curves, respectively; m,n m,n Second, we need to obtain P (t;K) and R (t;K), payer and receiver swaption prices, m,n m,n respectively. We discuss the construction of these objects below. To construct A (t) and S (t) curves, we first obtain daily LIBOR rates with matum,n m,n rities of 3, 6, 9, and 12 months, as well as daily 2-, 3-, 4-, 5-, 7-, 10-, 15-, 20-, 25-, 30-, and 35-year spot swap rates between February 11, 2002 and January 31, 2013 from J.P. Morgan. February 11, 2002 is the first day on which we have the intraday swap quotes from Barclays 22

Capital that we need for constructing the proxy for the realized variance of interest rates. We bootstrap the swap rates to first obtain daily zero-coupon curves. Then we construct the PVBP curve A (t) and the forward swap rate curve S (t) up to 35 years according m,n m,n to (24).17 We focus on 1-, 3-, and 12-month expiry swaptions for the 10-year tenor in line with the frequencies of our predictive regressions. The market convention is to quote swaption prices in terms of their log-normal implied volatility based on Black (1976) formula.18 We combine daily observations of (European) swaption prices from J.P. Morgan and Barclays Capital, two of the largest inter-dealer brokers in interest rate derivatives markets. The swaption prices from J.P. Morgan are available starting from June 1, 1993 with five strikes, namely, at-the-money-forward (ATMF), ATMF ± 100, and ATMF ± 50 basis points. The swaption prices from Barclays are available from December 1, 2004 with thirteen strikes, namely, ATMF, ATMF ± 200, ATMF ± 150, ATMF ± 100, ATMF ± 75, ATMF ± 50, and ATMF ± 25 basis points. In our empirical analysis, we use swaption prices from J.P. Morgan from February 11, 2002 through December 1, 2004 and those from Barclays after December 1, 2004.19 To obtain swaption payer P (t;K) and receiver R (t;K) prices on a continuum of m,n m,n strikes as requested by Eq. (28), we follow Carr and Wu (2009) and Du and Kapadia (2012) and interpolate implied volatilities across the range of observed strikes and use implied volatility of the lowest (highest) available strike to replace those of the strikes below (above). We further generate 200 implied volatility points that are equally spaced over a strike range 17Wefirstuseastandardcubicsplinealgorithmtointerpolatetheswapratesatsemiannualintervalsfrom one year to 35 years. We then solve for the zero curve by bootstrapping the interpolated par curve with swap rates as par bond yields. The day count convention is 30/360 for the fixed leg, and Actual/360 for the floating leg. 18Many market participants think in terms of normal (or absolute or basis point) implied volatilities–the volatility parameter that, plugged into the normal pricing formula, matches a given price–as they are more uniform across the swaption grid and more stable over time than log-normal implied volatilities. 19All our empirical results remain unchanged if we use only the J.P. Morgan swaption data rather than a combination of the J.P. Morgan and Barclays Capital swaption data. Recall that we start our sample from February 11, 2002, the first day when we have intraday swap rates data available from Barclays Capital, the data that we use for constructing realized variance of interest rates. 23

with moneyness between 0.9×S (t) and 1.1×S (t), where S (t) is the current forward m,n m,n m,n swap rate on each day. This implied volatility/strike grid together with PVBP and forward swap curves allows us to compute the empirical counterpart for the implied variance (28). 5.1.2 Construction of the realized variance of interest rates Next, we construct the “model-free” realized variance of interest rates. To do this, we use intraday 10-year interest rate swap quotes from February 11, 2002 and January 31, 2013 obtained from Barclays Capital. The data cover the period from 8:20 to 15:00 New York time each day for a maximum of 80 five-minute observations per day.20 Ideally, the sampling frequency for the computation of the realized variance should go to infinity. However, in practice high-frequency data is affected by a number of microstructure issues such as price discreteness, bid-ask spreads, and nonsynchronous trading effects. A number of studies (Andersen, Bollerslev, Diebold, and Labys, 2000; Hansen and Lunde, 2006; Bollerslev, Tauchen, and Zhou, 2009), suggest that a five-minute sampling frequency provides a reasonable choice. Hence, we compute five-minute continuously compounded returns on a hypothetical 10-year zero-coupon bond as a difference between the 10-year swap rates at the five-minute intervals. We then square these differences to obtain the “model-free” proxy for the daily realized variance RV given by Eq. (29). t IRVRP given by Eqs. (18) and (30) is forward-looking and requests the expectation of the realized variance of interest rates. In contrast to the “model-free” measures of implied and realized variances constructed above, we need an explicit forecasting model for the RV . t While numerous forecasting models have been proposed in the literature (see, e.g., Andersen, Bollerslev, and Diebold (2006)), here we follow Bollerslev, Tauchen, and Zhou (2009) and rely on the heterogeneous autoregressive volatility model of realized volatility (HAR-RV), suggested by Andersen, Bollerslev, and Diebold (2007) and Corsi (2009). The model is simple enough to implement yet it appears to produce highly accurate volatility forecast 20In this set-up we follow Wright and Zhou (2009) who used this time period to estimate bond variance and jump risk premia in the U.S. Treasury futures market. 24

as the above authors have shown. It aims to parsimoniously capture the long memory behavior of volatility by incorporating daily, weekly, and monthly realized variances into the one-month ahead variance forecast: RV = α+β RV +β RV +β ,RV +(cid:15) , (31) t+22,mon D t W t,week M t,mon t+22,mon where RV = 1 (cid:80)21 RV and RV = 1 (cid:80)4 RV are the monthly and weekly t,mon 22 j=0 t−j t,week 5 j=0 t−j realized variances, respectively. 5.1.3 Empirical measure of the interest-rate variance risk premium The forward-looking IRVRP is computed then as a difference between the expectations of interest rate variances under the Q and P measures as defined in Eq. (30). Figures 3, 4, and 5 plot implied variance, expected variance, and resulting interest-rate variance risk premium. First result fromthese figuresis thatthe variance risk premium isalmost everywherepositive suggesting that market participants seek compensation for variance exposure. Second, market variance risk premium increased dramatically during the NBER recession, represented by the shaded blue bar on each chart.21 Such an increase general captures the spirit of increased uncertainty amid recessions: this is a reason that we loosely refer to the variance risk premium a as a compensation for uncertainty.22 In addition, swaption-based IRVRP increased notably amid European financial crisis in the second half of 2011. 5.2 Treasury yield data In our empirical exercise we use Fama-Bliss data set of monthly zero-coupon Treasury yield datafromCRSPtocomputeexcessreturnsfortwotofive-yearTreasurybondsfromFebruary 2002 to January 2013. We denote the h−period log return on a τ−year zero-coupon note with the log price p(τ) as r(τ) = p(τ−h) −p(τ), and the h−period log return in excess of the t t+h t+h t 21There was only one NBER recession in our sample period. 22Bloom (2009) provides a similar argument about the relationship between uncertainty and volatility. 25

h−period risk-free rate y(h) as rx(τ) = r(τ) −y(h). In our application we consider h = 1, 3, t t+h t+h t and 12 months.23 The summary statistics of the Treasury excess returns is presented in Panel A of Table 2. A notable difference between one-year and one-month returns is that the latter are much less persistent than the former. 5.3 Other predictive variables In addition to the variance risk premium, we include two other variables in our predictive regressions that are shown to capture variation in long-horizon bond returns. First, we use the classical Fama and Bliss (1987) predictor, the forward spread factor, defined as the spread between the forward rate of a particular maturity and the risk-free rate. Second, we use Cochrane and Piazzesi (2005) factor that is an affine combination of forward rates. Both variables are computed using Fama-Bliss data set, downloaded from CRSP and confined to our sample. Panel B of Table 2 summarizes statistics for the predictive variables. The forward spread is extremely persistent, with AR(1) coefficients between 0.92 and 0.97. IRVRP has persistence coefficient of 0.87, while Cochrane-Piazzesi factor is the least persitent factor, with AR(1) coefficient of 0.72. According to Panel A, 1-year Treasury excess returns have persistence similar to that of forward spread factors, while 1-month excess returns of Treasury bonds are less persistent and closer in persistence to IRVRP. 6 Empirical Results In this section we discuss how well the interest-rate variance risk premium – endogenous proxy for economic uncertainty in our model – predicts Treasury excess returns. To assess 23Note that maturities τ −1 month and τ −3 months, with τ =2,...,5 years are not in the Fama-Bliss dataset. Therefore,thepricesofthesesecuritiesareobtainedviaalinearinterpolationofadjacentmaturity securities’ prices. 26

its predictability content, we run the following regressions: rx(τ) = β(τ) +β(τ)(h)IRVRP(m,n) +β(τ)(h)FS(τ) +CP +(cid:15)(τ) , (32) t+h 0 1 t 2 t t t+h where rx(τ) is the h−period excess return on a τ−year Treasury synthetic zero-coupon t+h note, IRVRP(m,n) is the interest-rate variance risk premium that corresponds to the nt period swaption expiry on the m-period forward swap rate, FS(τ) is (τ)−maturity Famat Bliss forward spread, and CP is Cochrane-Piazzesi forward-rate factor. Excess returns are t computed using Fama-Bliss discount bond data set. For each bond maturity (τ = 2,3,4,5 years), and each return horizon (h = 1,3,12 months) we run univariate regressions as well as joint regressions on a subset of factors and also run a kitchen-sink-type regression using all three factors. IRVRP(m,n) is constructed in Section 5.1 using U.S. interest rate derivatives t data. In our empirical exercise we use forward swap tenor m = 10 years, and swaption expiries n = 1,3,12 months. The relatively short life period of the interest rate derivatives markets and intraday swap data availability limit the sample period of our regressions. 6.1 Predictability with the interest-rate variance risk premium We report empirical results for three different bond holding period return (HPR) horizons andforinterest-ratevarianceriskpremiumsbasedonthreedifferentswaptionexpiries. Thus, we have nine tables in total. Tables 3, 4, and 5 report regression results 1-, 3-, and 12-month HPR for the variance risk premium based on the 1-month expriry swaptions on the 10-year forward swap rate. Likewise, tables 6, 7, and 8 report regression results for 1-, 3-, and 12month HPR for the variance risk premium based on the 3-month expriry swaptions on the 10-year forward swap rate. Finally, tables 9, 10, and 11 report regression results for the same HPR horizons for the variance risk premium based on the 12-month expiry swaptions on the 10-year forward swap rate. In all these tables we present univariate results for the variance risk premium, forward spread, and Cochrane-Piazzesi factors, as well as joint regressions for 27

two and three Treasury excess return predictors. Four panels in each table correspond to 2-, 3-, 4-, and 5-year maturity of Treasury synthetic zero-coupon notes. 6.1.1 Results with 1-month expiry swaption IRVRP InthissectionwediscussresultsofTables3, 4, and5. AsTable3shows, excessreturnsforall maturities load positively and significantly on the 1-month expiry swaption-based variance risk premium, with solid Newey-West corrected t-statistics above or around 4. The adjusted R2 of these univariate regressions (column 2) varies from 43 to 25 percent and declines with bond maturity. Nevertheless, even for a 5-year maturity, swaption-based variance risk premium alone explains 25 percent of excess return variation. In addition, IRVRP remains significant in the presence of Fama-Bliss factor (column 5) and Cochrane-Piazzessi factor (column 6), and highly significant in the multi-variate regression with both factors (column 7), where all three factors are included. IRVRP in the predictive regressions appears to add a nontrivial forecasting power: for example, when it is added to CP factor, the regression’s adjusted R2 increases from 14 percent to 45 percent for 2-year returns; when it is added to Fama-Bliss factor, the adjusted R2 increases from 37 percent to 71 percent. Such an impressive marginal increase in R2 of about 30 percentage points due to added IRVRP declines with maturity. For 5-year excess returns, marginal increase due to IRVRP is 6 and 17 percentage points when added to the forward spread and Cochrane-Piazzesi factors, respectively (Panel D). Nevertheless, such a marginal increase in explanatory power appears to be economically significant for all maturities. As a comparison, Bollerslev, Tauchen, and Zhou (2009) find that the equity variance risk premium adds six percentage points in explanatory power in addition to the log of price-earning ratio that by itself explains seven percentagepointsofvariationinfuturequarterlystockreturns(seeTable4intheirpublished RFS version). As such, this table confirms our intuition that the interest-rate-derivativesbased variance risk premium appears to contain useful information for predicting Treasury excess returns beyond that contained in the standard predictors. Moreover, these results 28

show that, in a univariate setting, variance risk premium has higher explanatory power for short-horizon Treasury excess returns at all maturities compared to Cochrane-Piazzesi factor and higher explanatory power than Fama-Bliss factor at a 2-year maturity. Regression results in Table 4 (3-month holding period Treasury excess returns) are broadly similar to Table 3. IRVRP is statistically significant at the 1% level in all four panels in the univariate regressions and not subsumed by Fama-Bliss and Cochrane-Piazzesi factors for Treasury 2, 3, and 4-year Treasury notes. However, IRVRP loses its significance in the presence of the Fama-Bliss factor for the 5-year (longest-maturity considered) Treasury note (Panel D) but it withstands its significance and is not subsumed by the Cochrane-Piazzesi factor. Additional insight from Panel D contends that the forward spread factor better captures short-horizon returns variation of the long-horizon bonds compared to Cochrane-Piazzesi factor, at least in our sample. The overall explanatory power of the IRVRP is smaller compared to that in Table 3, but it still adds some reasonable amount to either forward spread or Cochrane-Piazzesi factor in most of the reported regressions in this table. Regression results in Table 5 (one-year Treasury excess returns) are notably weaker than in either Table 3 or Table 4, with the IRVRP insignificant for all maturities and with the adjusted R2 of around 7 percent in Panel A (2-year Treasury notes) and almost negligible in Panel D (5-year Treasury notes). This result further supports the conjecture that the IRVRP’s content appears to be fairly short-lived but important for capturing variations in Treasury nominal securities over the short horizons.24 24In our sample forward spread and Cochrane-Piazzesi factors have considerably lower R2’s than in the original papers. This peculiarity is due to the sample period of our study that is confined by the availability oftheinterest-ratederivativesdata. Thissampleperiodisaffectedtoalargeextentbythezero-lowerbound period and QE-related events. Outside of the ZLB period and QE events, forward spread still retains its predictive power for one-year Treasury excess returns. 29

6.1.2 Results with 3-month expiry swaption IRVRP Tables 6, 7, and 8 present results with IRVRP constructed from the 3-month expiry of the interest rate swaptions. Table 6 highlights the strongest result in our paper. The IRVRP in univariate regressions is highly significant in explaining monthly returns of Treasury securities with Newey-West corrected t-stats around 6. Alone, IRVRP appears to explain from one half (2-year Treasury notes, Panel A) to roughly one third (5-year Treasury notes, Panel D) of excess return variation. IRVRP also explains notably more of return variation than the forward spread factor (50 versus 37 percent, Panel A) and considerably more than Cochrane-Piazzesi factor (14 percent, Panel A). Additional marginal explanatory power of 3-month IRVRP ranges across maturities from 40 to 8 percentage points (when added to the forward spread factor) and from 40 to 22 percentage points (when added to CP factor). Also, the statistical significance of the IRVRP is not subsumed by either of the two predictors considered at any maturity and remains significant at the 1 percent level of statistical significance. Table 7 is similar to Table 6. IRVRP is statistically significant at the 1 percent level for all maturities in univariate regressions and the adjusted R2 ranges from 38 to 10 percentage points based on the note maturity. It further adds nontrivial marginal explanatory power to other factors. Yet, Panel D shows that the IRVRP partly loses its significance to 10 percent level or slightly lower when added to the forward spread factor (Panel D, column 5) and in the joint regression (Panel D, column 7). Results of table 8 in general mirror those of Table 5. Overall, tables 6, 7, and 8 further strengthen our conjecture that the IRVRP is economically important for short-horizon returns generally for all considered Treasury note maturities. 6.1.3 Results with 12-month expiry swaption IRVRP Results based on the 12-month expiry swaptions (Tables 9, 10, and 11) generally repeat previous results. 12-month swaption expiry IRVRP is still highly significant at the 1 percent 30

level and is not subsumed by other predictors when applied to monthly and quarterly (shortrun) Treasury excess returns. It loses its explanatory power for annual (long-run) Treasury excess returns. This last set of reported results allows us to further argue that regardless of the swaption expiry, IRVRP is the robust factor in explaining short-run Treasury excess returns as opposed to long-run Treasury excess returns, where other economic forces appear to be at work. To summarize, empirical results discussed in Sections 6.1.1 - 6.1.3 convey, that based on the adjusted R2 from regressions, the 3-month swaption-expiry-based IRVRP appears to be the strongest candidate among the three IRVRPs (that correspond to 1-, 3-, and 12-month swaption expiries) that captures useful information from interest rate derivatives market for explaining and predicting the Treasury excess returns. 6.2 Some additional checks In addition to the interest-rate variance risk premium, we have used two alternative specifications of the variance risk premium in regressions (32). First, we replaced the IRVRP with the equity VRP since the latter one has been shown to be a useful predictor of future stock returns (Bollerslev, Tauchen, and Zhou, 2009). We found that, irrespective of the holding period horizon, equity variance risk premium is only marginally relevant for predicting Treasury excess returns. Mueller, Vedolin, and Zhou (2011) also found that equity variance risk premium (joint with CP factor) explains rather a small fraction of variation in Treasury bond returns. The difference between explanatory powers of equity and interest-rate variance risk premiums in predicting bond returns might in part be explained by the differences in traders’ clientele in the equity and derivatives markets as well as the persistence profile of the variance risk premiums in both markets: the interest-rate variance risk premium appears to be much more persistent (see Panel B of Table 2) than the equity variance risk premium.25 25In our sample, AR(1) coefficient for the equity VRP is 0.28. 31

Second, werunregressionswiththeIRVRPdefinedasimpliedvarianceminustherealized variance, with both concepts directly observable at time t. The results are not materially different although just a touch weaker (adjusted R2 is lower by two or three percentage points compared to the IRVRP that uses the P-expectation of the realized variance).26 7 Conclusion We study bond pricing implications in the context of the long-run risk asset-pricing model with two types of volatilities and inflation non-neutrality. The model is promising in explaining first and second moments of the Treasury market returns. The contribution of our paper to the fixed-income asset pricing literature is two-fold. First, our reasonably calibrated version of the model with long-run risks matches well the upward-sloping yield curve out to ten years, and the long-run risk appears to be the main driving force behind this result. Second, the interest-rate variance risk premium constructed from interest rate derivatives markets drives short-horizon (one- and three-month) Treasury excess returns, while other popular predictive variables, such as Fama-Bliss forward spread or Cochrane-Piazzessi forward-rate factor drive variation in longer-maturity Treasury excess returns. In our model time-varying bond risk premium is driven by two volatility factors (volatility of consumption and volatility-of-volatility of consumption) whereas variance risk premium loads entirely on the vol-of-vol factor. Since variance risk premium explains a significant part in variation in short-horizon Treasury excess returns, we interpret vol-of-vol factor as a short-run risk factor. Since the forward-rate-related factors appear to explain time-variation in long-horizon Treasury excess returns, we interpret these factors as related to the long-run risk factor, the second factor that drives the variation in bond risk premium in our model. Thus, our model and empirical findings provide useful insights on different volatility risks. These insights should be useful for market participants and monetary policy makers alike. 26Results discussed in this section are not reported but readily available upon request. 32

A Appendix A.1 Solution for the consumption-wealth ratio coefficients Euler equation imposes equilibrium restrictions on the asset prices: E[exp(m +r )] = 1. (33) t+1 t+1 This equation should hold for any asset, and for the wealth-consumption ration r as c,t+1 well. The solutions of A coefficients in Eq. (6) are obtained using Euler equation (33), return equation (7), and conjectured z dynamics (6). We solve for A , A , A , A - state 0 x σ2 q variables’ loadings in the price-consumption ratio z . We solve for A’s by pricing r using t c,t+1 Euler equation (33), wealth return equation (7) and assumed z dynamics in equation (6). Thus, Euler equation becomes: (cid:20) (cid:18) (cid:19)(cid:21) θ E [exp(m +r )] = E exp θlnδ − g +θr = 1. (34) t t+1 c,t+1 t t+1 c,t+1 ψ Using Jensen’s inequality, obtain: (cid:20) (cid:21) (cid:20) (cid:21) θ 1 θ E θlnδ − g +θr + Var θlnδ − g +θr = 0. (35) t t+1 c,t+1 t t+1 c,t+1 ψ 2 ψ Substituting out r , z , and z , obtain: c,t+1 t+1 t θ E [θlnδ − (µ +x +σ z )+θ(κ +κ (A +A x +A σ2 +A q )− t ψ g t g,t g,t+1 0 1 0 x t+1 σ g,t+1 q t+1 A −A x −A σ2 −A q +µ +x +σ z )]+ 0 x t σ g,t q t g t g,t g,t+1 (36) 1 θ Var [θlnδ − (µ +x +σ z )+θ(κ +κ (A +A x +A σ2 +A q )− 2 t ψ g t g,t g,t+1 0 1 0 x t+1 σ g,t+1 q t+1 A −A x −A σ2 −A q +µ +x +σ z )] = 0. 0 x t σ g,t q t g t g,t g,t+1 33

To solve for A , match terms in front of x : x t θ 1− 1 ψ − +θ(κ A ρ −A +1) = 0 ⇒ A = . (37) 1 x x x x ψ 1−κ ρ 1 x To solve for A , match terms in front of σ2 : σ g,t (cid:20) (cid:21) 1 θ (θκ A ρ −θA )σ2 + Var − σ2 z +θκ A φ σ z +θσ z = 1 σ σ σ g,t 2 t ψ g,t g,t+1 1 x e g,t x,t+1 g,t g,t+1 (cid:20)(cid:18) (cid:19) (cid:21) 1 θ θA (κ ρ −1)σ2 + Var θ− σ z +θκ A φ σ z = 0 ⇒ σ 1 σ g,t 2 t ψ g,t g,t+1 1 x e g,t x,t+1 1 (cid:34) (cid:18) θ (cid:19)2 (cid:35) (38) θA (κ ρ −1)+ θ− +(θκ A φ )2 = 0 ⇒ σ 1 σ 1 x e 2 ψ (cid:34) (cid:35) 1 (cid:18) θ (cid:19)2 A = θ− +(θκ A φ )2 . σ 1 x e 2θ(1−κ ρ ) ψ 1 σ To solve for A , set constant terms under the expectation in (36) equal to zero: 0 (cid:18) (cid:19) θ θlnδ +θ(κ +κ (A +A a +A a ))−A + θ− µ = 0 ⇒ 0 1 0 σ σ q q 0 g ψ (39) (cid:20) (cid:18) (cid:19) (cid:21) 1 1 A = lnδ +κ +κ (A a +A a )+ 1− µ . 0 0 1 σ σ q q g 1−κ ψ 1 To solve for A , match terms in front of q and set equal to zero: q t 1 √ √ (θκ A ρ −θA )q + Var [θκ A q z +θκ A (ρ q +φ q z )−θA q ] = 1 q q q t 2 t 1 σ t σt+1 1 q q t q t qt+1 q t 1 √ √ θA (κ ρ −1)q + Var(θκ A q z +θκ A φ q z ) = 0 =⇒ q 1 q t 2 1 σ t σt+1 1 q q t qt+1 (40) 1 1 (θκ φ )2A2 +θ(κ ρ −1)A + (θκ A )2 = 0 or, equivalently, 2 1 q q 1 q q 2 1 σ (θκ φ )2A2 +2θ(κ ρ −1)A +(θκ A )2 = 0. 1 q q 1 q q 1 σ The solution for A represents the solution to a quadratic equation and is given by: q (cid:112) 1−κ ρ ± (1−κ ρ )2 −(θκ2φ A )2 A± = 1 q 1 q 1 q σ . (41) q θ(κ φ )2 1 q 34

As Tauchen (2011) notes, a “positive” root A+ has an unfortunate property that q lim φ2A+ (cid:54)= 0, (42) q q φq→0 which is, essentially, a violation of the transversality condition in this setting: though uncertainty q vanishes with φ → 0, the effect of it on prices is not. Therefore, we choose A− t q q root as a viable solution for A : q (cid:113) 1−κ ρ − (1−κ ρ )2 −θ2κ4φ2A2 1 q 1 q 1 q σ A = . (43) q θ(κ φ )2 1 q To insure that determinant in (43) is positive, we also need to impose a constraint on the magnitude of the shock z : q,t+1 (1−κ ρ )2 φ2 ≤ 1 q . (44) q θ2κ4A2 1 σ 35

A.2 Solution for the pricing kernel Using the solutions for A(cid:48)s obtained in A.1, we solve for the expected value E (m ) and t t+1 variance Var (m ) of the pricing kernel: t t+1 θ E [m ] = θlnδ − E [g ]+(θ−1)E [r ] = t t+1 t t+1 t c,t+1 ψ θ = θlnδ − (µ +x )+(θ−1)E (κ +κ z +g −z ) g t t 0 1 t+1 t t ψ θ = θlnδ − (µ +x )+(θ−1)[κ +κ (A +A ρ x +A (a +ρ σ2 )+A (a +ρ q )) ψ g t 0 1 0 x x t σ σ σ g,t q q q t +µ +x −A −A x −A σ2 −A q ] g t 0 x t σ g,t q t (cid:18) (cid:19) θ = θlnδ + (θ−1)− µ +(θ−1)[κ +(κ −1)A +κ (A a +A a )] g 0 1 0 1 σ σ q q ψ (cid:124) (cid:123)(cid:122) (cid:125) −γ θ − x +(θ−1)[(A (κ ρ −1)+1)x +A (κ ρ −1)σ2 +A (κ ρ −1)q ] ψ t x 1 x t σ 1 σ g,t q 1 q t = θlnδ −γ(µ +x )+(θ−1)[κ +(κ −1)A +κ (A a +A a )] g t 0 1 0 1 σ σ q q +(θ−1)[A (κ ρ −1)x +A (κ ρ −1)σ2 +A (κ ρ −1)q ]. x 1 x t σ 1 σ g,t q 1 q t (45) The variance of the SDF Var [m ] is given by t t+1 (cid:20) (cid:21) θ Var [m ] = Var θlnδ − g +(θ−1)r t t+1 t t+1 c,t+1 ψ (cid:20) (cid:21) θ = Var − g +(θ−1)κ (A +A x +A σ2 +A q ) t ψ t+1 1 0 x t+1 σ t+1 q t+1 (cid:20) (cid:21) θ √ √ = Var − σ z +(θ−1)[κ (A φ σ z +A q z +A φ q z )+σ z ] t g,t g,t+1 1 x e g,t x,t+1 σ t σ,t+1 q q t q,t+1 g,t g,t+1 ψ (cid:20)(cid:18) (cid:19) (cid:21) θ √ √ = Var (θ−1)− σ z +(θ−1)κ (A φ σ z +A q z +A φ q z ) t g,t g,t+1 1 x e g,t x,t+1 σ t σ,t+1 q q t q,t+1 ψ (cid:2) (cid:3) = γ2σ2 +(θ−1)2κ2 A2φ2σ2 +(A2 +A2φ2)q . g,t 1 x e g,t σ q q t (46) 36

A.3 Solution for the n−period real bond price In this section we derive the (log) price of an n−period bond in closed form. A general recursion for solving for the n−period bond prices is as follows: Pn = E (cid:2) M Pn−1 (cid:3) . (47) t t t+1 t+1 Then the n−period log bond price 1 1 pn = E [m ]+ Var [m ]+E [pn−1]+ Var [pn−1]+Cov [m ,pn−1]. (48) t t t+1 2 t t+1 t t+1 2 t t+1 t t+1 t+1 The first two terms in (48) are given in (45) and (46). The last three terms in (48) are computed using a conjectured affine representation of the state variables: pn−1 = Bn−1 +Bn−1x +Bn−1σ2 +Bn−1q . (49) t+1 0 1 t+1 2 t+1 3 t+1 The expectation term for the bond price E (cid:2) pn−1 (cid:3) is given by: t t+1 E (cid:2) pn−1 (cid:3) =Bn−1 +Bn−1E [x ]+Bn−1E [σ2 ]+Bn−1E [q ] t t+1 0 1 t t+1 2 t t+1 3 t t+1 = Bn−1 +Bn−1ρ x +Bn−1(a +ρ σ2)+Bn−1(a +ρ q ) (50) 0 1 x t 2 σ σ t 3 q q t = (Bn−1 +Bn−1a +Bn−1a )+Bn−1ρ x +B2 ρ σ2 +Bn−1ρ q , 0 2 σ 3 q 1 x t n−1 σ t 3 q t and the variance term Var [pn−1] is: t t+1 (cid:16) (cid:17) Var (cid:2) pn−1 (cid:3) = E (cid:2) pn−1 −E (cid:2) pn−1 (cid:3)(cid:3)2 = (cid:0) Bn−1φ (cid:1)2 σ2 + (cid:0) Bn−1 (cid:1)2 + (cid:0) Bn−1φ (cid:1)2 q . (51) t t+1 t t+1 t t+1 1 e g,t 2 3 q t 37

Lastly, express the covariance term as a function of the state variables: Cov (cid:2) m ,pn−1 (cid:3) = E (cid:2)(cid:0) pn−1 −E (cid:2) pn−1 (cid:3)(cid:1) ×(m −E [m ]) (cid:3) t t+1 t+1 t t+1 t t+1 t+1 t t+1 √ √ = E (cid:2)(cid:0) Bn−1φ σ z +Bn−1 q z +Bn−1φ q z (cid:1) t 1 e g,t x,t+1 2 t σ,t+1 3 q t q,t+1 √ √ × ((1−γ)σ z +(θ−1)κ (A φ σ z +A q z +A φ q z ))] g,t g,t+1 1 x e g,t x,t+1 σ t σ,t+1 q q t q,t+1 (cid:2) (cid:3) = (θ−1)κ A Bn−1φ2σ2 +A Bn−1q +A Bn−1φ2q . 1 x 1 e g,t σ 2 t q 3 q t (52) Write down pn as a sum of (45), (46), (50), (51), and (52) and collect together constant t terms and loadings for state variables x , σ2, and q . This implies for coefficients: t t t Bn = c +Bn−1 +Bn−1a +Bn−1a 0 0 0 2 σ 3 q Bn = c +Bn−1ρ 1 1 1 x (53) 1 1 Bn = c +Bn−1ρ − (θ−1)2κ2A2φ2 + φ2 (cid:2) (θ−1)κ A +Bn−1 (cid:3)2 2 2 2 σ 2 1 x e 2 e 1 x 1 1 (cid:104) (cid:105) Bn = c +Bn−1ρ + (cid:0) Bn−1 (cid:1)2 + (cid:0) Bn−1φ (cid:1)2 +(θ−1)κ (cid:0) A Bn−1 +A Bn−1φ2 (cid:1) , 3 3 3 q 2 2 3 q 1 σ 2 q 3 q where c , i = 0,...,3 are given by: i c = θlnδ −γµ +(θ−1)[κ +(κ −1)A +κ (A a +A a )]; 0 g 0 1 0 1 σ σ q q c = −γ +(θ−1)A (κ ρ −1); 1 x 1 x (54) 1 1 c = γ2 + (θ−1)2κ2A2φ2 +(θ−1)A (κ ρ −1); 2 2 2 1 x e σ 1 σ 1 c = (θ−1)2κ2(A2 +A2φ2)+(θ−1)A (κ ρ −1). 3 2 1 σ q q q 1 q Note that c , i = 0,...,3, coefficients are the steady-state loadings for the real risk-free rate: i r = −[c c c c ]×[1 Ex Eσ2 Eq](cid:48), (55) f 0 1 2 3 38

A.4 Solution for the nominal risk-free rate Here we provide a derivation for the nominal risk-free rate, the (negative) of the (log) price of the one period nominal bond. We express the nominal risk-free rate in log terms, similar to the real risk free rate given in equation (12): 1 r$ = −E (cid:2) m$ (cid:3) − Var (cid:2) m$ (cid:3) f,t t t+1 2 t t+1 1 1 = −E [m −π ]− Var [m ]− Var [π ]+Cov [m ,π ] t t+1 t+1 t t+1 t t+1 t t+1 t+1 2 2 (56) 1 = r +E [π ]− Var [π ]+Cov [m ,π ] f,t t t+1 t t+1 t t+1 t+1 2 1 = r +a +ρ π − [φ2 +φ2 σ2 +φ2 q ]+Cov [m ,π ]. f,t π π t 2 π πg g,t πσ t t t+1 t+1 we need to compute the last term in (56) to complete the expression for the nominal risk-free rate in closed form: Cov [m ,π ] = E [[m −E [m ]]×[π −E [π ]]]. (57) t t+1 t+1 t t+1 t t+1 t+1 t t+1 The deviations of pricing kernel m and inflation π are given by: t+1 t+1 √ √ m −E [m ] = −γσ z +(θ−1)κ (A φ z +A q z +A φ q z ) t+1 t t+1 g,t g,t+1 1 x e x,t+1 σ t σ,t+1 q q t q,t+1 √ π −E [π ] = φ z +φ σ z +φ q z , t+1 t t+1 π π,t+1 πg g,t g,t+1 πσ t σ,t+1 (58) which implies for (57): E [[m −E [m ]]×[π −E [π ]]] = −γφ σ2 +(θ−1)κ A φ q . (59) t t+1 t t+1 t+1 t t+1 πg g,t 1 σ πσ t Combining together (13), (56), and (59), obtain the closed-form expression for the nominal 39

risk-free rate: 1 r$ = −θlnδ +γµ +a −(θ−1)[κ +(κ −1)A +κ (A a +A a )]− φ2 f,t g π 0 1 0 1 σ σ q q 2 π +[γ −(θ−1)A (κ ρ −1)]x x 1 x t (cid:20) (cid:21) 1 1 1 + −(θ−1)A (κ ρ −1)− γ2 − (θ−1)2(κ A φ )2 − φ2 −γφ σ2 (60) σ 1 σ 2 2 1 x e 2 πg πg g,t (cid:20) (cid:21) 1 1 + −(θ−1)A (κ ρ −1)− (θ−1)2κ2(A2 +A2φ2)− φ2 +(θ−1)κ A φ q q 1 q 2 1 σ q q 2 πσ 1 σ πσ t +ρ π . π t Similarly to the real risk-free rate, the steady-state nominal risk-free rate can be written as follows: r$ = −[c$ c$ c$ c$ c$]×[1 E E E ](cid:48), (61) f 0 1 2 3 4 x σ2 π where the c$, i = 0,...,4 loadings: i 1 c$ = c −a + φ2; 0 0 π 2 π c$ = c ; 1 1 1 c$ = c + φ2 +γφ ; (62) 2 2 2 πg πg 1 c$ = c + φ2 −(θ−1)κ A φ ; 3 3 2 πσ 1 σ πσ c$ = ρ . 4 π A.5 Solution for the n−period nominal bond price The n−period nominal log bond price p$,n is given by: t 1 1 p$,n = E [m$ ]+ Var [m$ ]+E [p$,n−1]+ Var [p$,n−1]+Cov [m$ ,p$,n−1]. (63) t t t+1 2 t t+1 t t+1 2 t t+1 t t+1 t+1 Assume that p$,n is affine as in the case of real bonds, with the additional state variable for t 40

inflation: p$,n = B$,n +B$,nx +B$,nσ2 +B$,nq +B$,nπ . (64) t 0 1 t 2 t 3 t 4 t We know the first and the second terms in (63) from nominal risk-free rate calculations. Compute the last three terms using a pricing conjecture (64): (cid:104) (cid:105) E p$,n−1 = B$,n−1 +B$,n−1ρ x +B$,n−1(a +ρ σ2 )+B$,n−1(a +ρ q )+B$,n−1(a +ρ π ) t t+1 0 1 x t 2 σ σ g,t 3 q q t 4 π π t (cid:104) (cid:105) = B$,n−1 +B$,n−1a +B$,n−1a +B$,n−1a 0 2 σ 3 q 4 π +B$,n−1ρ x +B$,n−1ρ σ2 +B$,n−1ρ q +B$,n−1ρ π . 1 x t 2 σ g,t 3 q t 4 π t (65) The shock to the nominal bond price is given by: (cid:104) (cid:105) √ √ p$,n−1 −E p$,n−1 = B$,n−1φ σ z +B$,n−1 q z +B$,n−1φ q z t+1 t t+1 1 e g,t x,t+1 2 t σ,t+1 3 q t q,t+1 (66) √ +B$,n−1[φ z +φ σ z +φ q z ]. 4 π π,t+1 πg g,t g,t+1 πσ t σ,t+1 Thus, the variance of the nominal bond price - the fourth term in (63) - is given by: (cid:104) (cid:104) (cid:105)(cid:105)2 (cid:104) (cid:105) Var[p$,n−1] = E p$,n−1 −E p$,n−1 = (B$,n−1φ )2 +(B$,n−1φ )2 σ2 t+1 t t+1 t t+1 1 e 4 πg g,t (cid:20) (cid:21) (67) (cid:16) (cid:17)2 (cid:16) (cid:17)2 (cid:16) (cid:17)2 + B$,n−1 +B$,n−1φ + B$,n−1φ q + B$,n−1φ . 2 4 πσ 3 q t 4 π Lastly, compute covariance between between nominal pricing kernel m$ and the nominal t+1 bond price p$,n−1: t+1 (cid:104) (cid:105) (cid:104) (cid:104) (cid:104) (cid:105)(cid:105)(cid:105) Cov m$ ,p$,n−1 = E (cid:2) m$ −E (cid:2) m$ (cid:3)(cid:3) × p$,n−1 −E p$,n−1 , (68) t t+1 t+1 t t+1 t t+1 t+1 t t+1 41

where the shock to the nominal pricing kernel in terms of state variables is: m$ −E (cid:2) m$ (cid:3) = m −E m −(π −E π ) t+1 t t+1 t+1 t t+1 t+1 t t+1 √ √ = −γσ z +(θ−1)κ (A φ σ z +A q z +A φ q z ) (69) g,t g,t+1 1 x e g,t x,t+1 σ t σ,t+1 q q t q,t+1 √ −φ z −φ σ z −φ q z , π π,t+1 πg g,t g,t+1 πσ t σ,t+1 (cid:104) (cid:105) and the shock to the nominal bond price, p$,n−1−E p$,n−1 , is given in (66). Thus, a final t+1 t t+1 expression for a covariance term in (63) is: (cid:104) (cid:105) (cid:104) (cid:105) Cov m$ ,p$,n−1 = (θ−1)κ A B$,n−1φ2 −(γ +φ )B$,n−1φ σ t t+1 t+1 1 x 1 e πg 4 πg g,t (cid:104) (cid:105) + ((θ−1)κ A −φ )(B$,n−1 +B$,n−1φ )+(θ−1)κ A B$,n−1φ2q q (70) 1 σ πσ 2 4 πσ 1 q 3 q t t −B$,n−1φ2. 4 π Combining together (56), (65), (67), and (70), obtain the solution for the nominal n−period bond price: (cid:104) (cid:105) 1 (cid:16) (cid:17)2 B$,n = c −a + B$,n−1 +B$,n−1a +B$,n−1a +B$,n−1a + φ2 B$,n−1 −1 0 0 π 0 2 σ 3 q 4 π 2 π 4 B$,n = c +B$,n−1ρ 1 1 1 x 1 1 (cid:104) (cid:105)2 B$,n = B$,n−1ρ +(θ−1)A (κ ρ −1)+ (γ +φ )2 + φ2 (θ−1)κ A +B$,n−1 2 2 σ σ 1 σ 2 πg 2 e 1 x 1 1 + (B$,n−1φ )2 −(γ +φ )B$,n−1φ 2 4 πg πg 4 πg B$,n = B$,n−1ρ +(θ−1)A (κ ρ −1)+ 1 (cid:104) (θ−1)κ A +B$,n−1 +φ (cid:0) B$,n−1 −1 (cid:1) (cid:105)2 3 3 q q 1 q 2 1 σ 2 φσ 1 (cid:104) (cid:105)2 + (θ−1)κ A +B$,n−1 φ2 2 1 q 3 q B$,n = φ (cid:0) B$,n−1 −1 (cid:1) . 4 π (71) 42

References Adrian, T., and J. Rosenberg, 2008, “Stock Returns and Volatility: Pricing the Short-Run and Long-Run Components of Market Risk,” Journal of Finance, 63, 2997–3030. Andersen, T. G., T. Bollerslev, and F. X. Diebold, 2007, “Roughting it up: Including Jump Components in the Measurement, Modeling, and Forecasting of Return Volatility,” Review of Economics and Statistics, 89, 701–720. Andersen, T. G., T. Bollerslev, F. X. Diebold, and H. Ebens, 2001, “The Distribution of Realized Stock Return Volatility,” Journal of Financial Economics, 61, 43–76. Andersen, T. G., T. Bollerslev, F. X. Diebold, and P. Labys, 2000, “Great Realizations,” Risk, 50, 331–367. Andersen, T. G., T. Bollerslev, F. X. Diebold, and T. Labys, 2001, “The Distribution of Realized Exchange Rate Volatility,” Journal of the American Statistical Association, 96, 42–55. Andersen, T. G., T. Bollerslev, and P. C. F. X. Diebold, 2006, Handbook of Financial ForecastingAmsterdam, Elsevier, chap. Volatility and Correlation Forecasting. Bansal, R., D. Kiku, and A. Yaron, 2012, “An Empirical Evaluation of the Long-run Risks model for Asset Prices,” Critical Finance Review, 1, 183–221. Bansal, R., and I. Shaliastovich, 2013, “A Long-Run Risks Explanation of Predictability Puzzles in Bond and Currency Markets,” Review of Financial Studies, 26(1), 1–33. Bansal, R., and A. Yaron, 2004, “Risks for the Long Run: A Potential Resolution of Asset Pricing Puzzles,” Journal of Finance, 59(4), 1481–1509. , 2005, “Interpretable Asset Markets?,” European Economic Review, 49, 531–560. Barndorff-Nielsen,O.,andN.Shephard,2004,“PowerandBipowerVariationwithStochastic Volatility and Jumps,” Journal of Financial Econometrics, 2, 1–48. Bekaert, G., E. Engstrom, and Y. Xing, 2009, “Risk, uncertainty, and asset prices,” Journal of Financial Economics, 91(1), 59–82. Bekaert, G., and J. Liu, 2004, “Conditioning Information and Variance Bounds on Pricing Kernels,” review of Financial Studies, 17, 339–78. Binsbergen, J. H. V., M. Brandt, and R. S. Koijen, 2012, “On the Timing and Pricing of Dividends,” American Economic Review, 102(4), 1596–1618. Black, F., 1976, “The Pricing of Commodity Contracts,” Journal of Financial Economics, 3, 16–179. Bloom, N., 2009, “The Impact of Uncertainty Shocks,” Econometrica, 77(3), 623–685. Bollerslev, T., G. Tauchen, and H. Zhou, 2009, “Expected Stock Returns and Variance Risk Premia,” Review of Financial Studies, 22(11), 4463–4492. 43

Bollerslev, T., L. Xu, and H. Zhou, 2013, “Stock Return and Cash Flow Predictability: The Role of Volatility Risk,” Journal of Econometrics, forthcoming. Branger, N., P. Rodrigues, and C. Schlag, 2011, “Long-Run Risks in Economic Uncertainty and Implications for Equity Premium Puzzles,” Working paper, http://papers.ssrn. com/sol3/papers.cfm?abstract_id=2021070. Campbell, J. Y., and R. J. Shiller, 1988, “The Dividend-price Ratio and Expectations of Future Dividends and Discount Factors,” Review of Financial Studies, 1, 195–228. Campbell, J. Y., and R. J. Shiller, 1991, “Yield Spreads and Interest Rate Movements: A Bird’s Eye View,” Review of Economic Studies, 58, 495–514. Campbell, J. Y., A. Sunderam, and L. Viceira, 2013, “Inflation Bets or Deflation Hedges? The Changing Risks of Nominal Bonds,” Working Paper, Harvard University. Carr, P., and L. Wu, 2009, “Variance Risk Premiums,” Review of Financial Studies, 22, 1311–1341. Christoffersen, P., K. Jacobs, C. Ornthanalai, and Y. Wang, 2008, “Option Valuation with Long-Run and Short-Run Volatility Components,” Journal of Financial Economics, 90, 272–297. Cochrane, J., and M. Piazzesi, 2005, “Bond Risk Premia,” American Economic Review, 95, 138–160. Corsi,F.,2009,“ASimpleApproximateLong-MemoryModelofRealizedVolatility,”Journal of Financial Econometrics, 7(2), 174–196. Dai, Q., and K. Singleton, 2000, “Specification Analysis of Affine Term Structure Models,” Journal of Finance, LV, 1943–1978. Drechsler, I., 2013, “Uncertainty, Time-Varying Fear, and Asset Prices,” Journal of Finance, 68, 1843–1889. Du, J., and N. Kapadia, 2012, “The Tail in the Volatility Index,” Working Paper. Duffee, G. R., 2011, “Information in (and not in) the Term Structure,” Review of Financial Studies, 24(9), 2895–2934. Epstein, L., and S. Zin, 1989, “Substitution, Risk Aversion and the Temporal Behavior of Consumption and Asset Returns: A Theoretical Framework,” Econometrica, 57, 937–969. Fama, E., and R. Bliss, 1987, “The Information in Long-Maturity Forward Rates,” The American Economic Review, 77(4), 680–692. Ghysels, E., A. Le, S. Park, and H. Zhu, 2014, “Risk and Return Trade-off in the U.S. Treasury Market,” SSRN library: http://ssrn.com/abstract=2403483. Giacoletti, M., K. T. Laursen, and K. J. Singleton, 2015, “Learning, Dispersion of Beliefs, and Risk Premiums in an Arbitrage-free Term Structure Model,” Stanford University Graduate School of Business Research Paper No. 15-45. 44

Hansen, P. R., and A. Lunde, 2006, “Realized Variance and Market Microstructure Noise,” Journal of Business and Economic Statistics, 24, 127–161. Huang, J.-Z., and Z. Shi, 2012, “Determinants of Bond Risk Premia,” Working paper, Penn State University. Kreps, D. M., and E. L. Porteus, 1978, “Temporal Resolution of Uncertainty and Dynamic Choice Theory,” Econometrica, 46, 185–200. Lettau, M., S. Ludvigson, and J. Wachter, 2008, “The Declining Equity Premium: What Role Does Macroeconomic Risk Play?,” Review of Financial Studies, 21, 1653–2991687. Ludvigson, S., and S. Ng, 2009, “Macro Factors in Bond Risk Premia,” Review of Financial Studies, 22, 5027–5067. Mueller, P., A. Vedolin, and H. Zhou, 2011, “Short-run Bond Risk Premia,” Working paper, London School of Economics and Federal Reserve Board. Newey, W. K., and K. D. West, 1987, “A Simple Positive Semi-definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix,” Econometrica, 55(3), 703–708. Piazzesi, M., andM.Shneider, 2007, MBER Macroeconomics AnnualTheMITPress, vol.21, chap. Equilibrium Yield Curves, pp. 389–472. Tauchen, G., 2011, “Stochastic Volatility in General Equilibrium,” Quarterly Journal of Finance, 01(04), 707–731. Weil, P., 1989, “The Equity Premium Puzzle and the Riskfree Rate Puzzle,” Journal of Monetary Economics, 24, 401–421. Wright, J. H., 2011, “Term Premia and Inflation Uncertainty: Empirical Evidence from International Panel Dataset,” American Economic Review, 101, 1514–1534. Wright, J. H., and H. Zhou, 2009, “Bond Risk Premia and Realized Jump Risk,” Journal of Banking and Finance, 33(12), 2333 – 2345. Zhou, G., andY.Zhu, 2012, “VolatilityTrading: WhatistheRoleoftheLong-RunVolatility Component?,” Journal of Financial and Quantitative Analysis, 47(2), 273–307. , 2013, “The Long-run Risks Model: What differences Can an Extra Volatility Factor Make?,” Working paper, http://papers.ssrn.com/sol3/papers.cfm?abstract_ id=1403869. Zhou, H., 2009, “Variance Risk Premia, Asset Predictability Puzzles, and Macroeconomic Uncertainty,” Working paper, http://ssrn.com/abstract=1400049. , 2011, “Term Structure of Interest Rates with Inflation Uncertainty,” Technical appendix, Federal Reserve Board. 45

Table 1: Model Calibration This table reports the calibrated parameters used in previous studies and in our paper. Column“BY”referstothechoiceofparametersinBansalandYaron(2004),column“BTZ” – to that in Bollerslev, Tauchen, and Zhou (2009), and column “GSZ” refers to our choice of parameters. Type Parameters BY BTZ GSZ Panel A: Real economy dynamics δ 0.997 0.997 0.997 Preferences γ 10 10 8 ψ 1.5 1.5 1.5 µ 0.0015 0.0015 0.0015 g ρ 0.979 0 0.979 x Endowment φ 0.001 0 0.001 e a 1.20463e-05 1.20463e-05 1.20463e-05 σ ρ 0.978 0.978 0.978 σ a 2e-07 2e-10 q Uncertainty ρ 0.8 0.8 q φ 0.001 0.0001 q Panel B: Inflation dynamics Constant a 8.33333e-05 π Persistence ρ 0.95 π Autonomous φ 0.0013 π Consumption φ -0.0385 πg Uncertainty φ 28.5044 πσ Panel C: Campbell-Shiller constants κ 0.3251 0.3251 0.3251 0 κ 0.9 0.9 0.9 1 46

Table 2: Summary Statistics This table presents summary statistics for the data used in the study. Panel A presents a summary statistics for the annualized 1-month and 1-year Treasury excess returns for maturities 2 to 5 years; Panel B reports the summary statistics for the forward-rate based predictors and the 3-month interest rate swaption expiry based variance risk premium. In Panel B FS , j = 1, ...,4 refers to j Fama-Bliss j-year forward spreads, CP – to Cochrane-Piazzesi factor. IV is the implied variance derived from interest swaption market, RV is the expected value of the realized variance derived from the intraday quotes of the interest rate swap market. IRVRP is the interest-rate variance risk premium. Sample period is April 2002 to January 2013, at monthly frequency. Treasury excess returns, forward spreads, and Cochrane-Piazzesi factor are computed using Fama-Bliss Treasury Bond data set from CRSP. Interest rate swaptions data is from J.P. Morgan and Barclays Capital, and intraday interest rate swap quotes data is from Barclays Capital. Panel A: Summary statistics of Treasury excess returns 1-month excess returns 1-year excess returns 2yr 3yr 4yr 5yr 2yr 3yr 4yr 5yr Mean -1.37 -1.30 -1.24 -1.17 0.61 1.46 2.35 3.16 Max 0.93 1.69 3.61 4.84 3.03 6.37 9.34 11.79 Min -5.18 -5.50 -5.79 -6.03 -1.54 -2.55 -3.62 -4.63 Std. Dev. 1.71 1.84 2.04 2.24 1.10 2.09 2.96 3.68 Skewness -0.78 -0.64 -0.39 -0.23 0.18 0.10 0.02 -0.05 Kurtosis 2.19 2.12 2.19 2.32 2.40 2.51 2.35 2.28 AR(1) coeff 0.93 0.84 0.72 0.63 0.91 0.89 0.88 0.87 Panel B: Summary statistics for IRVRP and forward-rate based predictors FS FS FS FS CP IV RV IRVRP 2 3 4 5 Mean 0.37 0.91 1.47 1.95 1.89 3.16 1.16 2.01 Max 1.44 2.50 3.44 4.32 4.17 9.70 3.43 6.27 Min -0.70 -0.61 -0.54 -0.43 -0.76 0.66 0.52 -0.00 Std. Dev. 0.53 0.88 1.18 1.49 0.98 1.83 0.54 1.59 Skewness 0.07 0.07 -0.03 -0.15 -0.39 0.78 1.80 0.72 Kurtosis 2.19 1.81 1.75 1.69 2.88 3.48 6.93 2.47 AR(1) coeff 0.92 0.95 0.95 0.97 0.72 0.92 0.75 0.87 47

Table 3: 1m HPR Bond Predictability with 10y1m Interest-rate VRP This table presents regression results for the following regression: rx(τ) = β(τ) + β(τ) (h)IRVRP + t+h 0 1 t (cid:80)2 β(τ) (h)F +(cid:15)(τ) , where rx(τ) are excess returns on Treasury notes, h = 1 month and τ = 2, ..., 5 j=1 j t,j t+h t+h years. IRVRP is the interest-rate variance risk premium derived from interest rate derivatives markets, t F , j =1,2 is the forward spread (FS) and Cochrane-Piazzesi (CP) factors. t-statistics in parentheses are t,j calculatedusingNeweyandWest(1987)standarderrors. AdjustedR2aregiveninpercentagepoints. Treasury excess returns are computed using Fama-Bliss data set, obtained from CRSP. Interest rate swaptions and swaps data are from J.P. Morgan and Barclays Capital. Sample is from April 2002 to January 2013, monthly frequency. (1) (2) (3) (4) (5) (6) (7) PanelA:maturity=2years Const -0.026 -0.021 -0.027 -0.031 -0.032 -0.032 (-6.56) (-6.46) (-3.27) (-12.72) (-4.82) (-9.76) IRVRP 0.004 0.003 0.003 0.003 (4.65) (5.85) (4.40) (5.56) FS 1.983 1.732 1.688 (4.98) (7.87) (7.01) CP 0.689 0.361 0.090 (2.16) (1.43) (0.62) Adj. R2 42.62 37.35 14.16 71.00 45.88 70.95 PanelB:maturity=3years Const -0.025 -0.026 -0.027 -0.034 -0.032 -0.035 (-6.06) (-7.11) (-3.25) (-13.14) (-4.68) (-10.31) IRVRP 0.004 0.003 0.003 0.003 (4.49) (4.53) (4.16) (4.39) FS 1.407 1.194 1.167 (5.89) (8.22) (7.82) CP 0.730 0.412 0.082 (2.26) (1.58) (0.59) Adj. R2 36.07 44.63 13.82 67.11 39.76 66.97 PanelC:maturity=4years Const -0.024 -0.029 -0.027 -0.037 -0.032 -0.039 (-5.39) (-8.10) (-3.21) (-16.80) (-4.41) (-13.83) IRVRP 0.004 0.003 0.003 0.003 (3.96) (5.67) (3.58) (4.99) FS 1.159 1.018 0.989 (7.36) (9.25) (8.63) CP 0.765 0.460 0.151 (2.33) (1.68) (1.06) Adj. R2 27.74 44.58 12.25 61.16 31.43 61.26 PanelD:maturity=5years Const -0.024 -0.033 -0.028 -0.036 -0.032 -0.038 (-5.30) (-10.44) (-3.15) (-14.57) (-4.38) (-11.16) IRVRP 0.004 0.002 0.003 0.002 (4.40) (2.80) (3.97) (2.58) FS 1.066 0.905 0.876 (9.03) (7.94) (7.38) CP 0.826 0.510 0.167 (2.40) (1.74) (1.01) Adj. R2 24.92 49.30 11.73 55.03 28.63 55.07 48

Table 4: 3m HPR Bond Predictability with 10y1m Interest-rate VRP This table presents regression results for the following regression: rx(τ) = β(τ) + β(τ) (h)IRVRP + t+h 0 1 t (cid:80)2 β(τ) (h)F +(cid:15)(τ) , where rx(τ) are excess returns on Treasury notes, h=3 months and τ =2, ...,5 j=1 j t,j t+h t+h years. IRVRP is the interest-rate variance risk premium derived from interest-rate derivatives markets, t F , j =1,2 is the forward spread (FS) and Cochrane-Piazzesi (CP) factors. t-statistics in parentheses are t,j calculatedusingNeweyandWest(1987)standarderrors. AdjustedR2aregiveninpercentagepoints. Treasury excess returns are computed using Fama-Bliss data set, obtained from CRSP. Interest rate swaptions and swaps data are from J.P. Morgan and Barclays Capital. Sample is from April 2002 to January 2013, monthly frequency. (1) (2) (3) (4) (5) (6) (7) PanelA:maturity=2years Const -0.020 -0.017 -0.022 -0.025 -0.026 -0.026 (-5.43) (-5.12) (-2.88) (-7.70) (-3.99) (-7.06) IRVRP 0.003 0.003 0.003 0.003 (4.31) (5.21) (4.05) (4.79) FS 1.608 1.396 1.346 (4.08) (4.91) (4.41) CP 0.618 0.328 0.112 (2.02) (1.31) (0.69) Adj. R2 32.40 28.06 12.37 53.39 35.14 53.33 PanelB:maturity=3years Const -0.018 -0.020 -0.021 -0.026 -0.024 -0.027 (-3.93) (-5.10) (-2.45) (-6.44) (-3.23) (-6.44) IRVRP 0.003 0.002 0.003 0.002 (3.75) (3.60) (3.37) (3.33) FS 1.288 1.124 1.103 (4.78) (4.95) (4.71) CP 0.646 0.375 0.069 (1.90) (1.25) (0.38) Adj. R2 19.18 32.12 8.61 42.61 21.31 42.17 PanelC:maturity=4years Const -0.015 -0.024 -0.019 -0.028 -0.022 -0.030 (-2.62) (-5.33) (-2.04) (-6.02) (-2.49) (-6.15) IRVRP 0.003 0.002 0.002 0.002 (2.58) (2.84) (2.19) (2.43) FS 1.157 1.066 1.046 (5.40) (5.26) (5.08) CP 0.658 0.426 0.110 (1.72) (1.20) (0.49) Adj. R2 9.41 30.20 5.46 34.15 10.98 33.70 PanelD:maturity=5years Const -0.014 -0.026 -0.018 -0.027 -0.021 -0.028 (-2.23) (-4.87) (-1.69) (-4.97) (-2.12) (-4.67) IRVRP 0.003 0.001 0.002 0.001 (2.83) (1.00) (2.44) (0.91) FS 1.068 0.993 0.980 (5.31) (4.51) (4.34) CP 0.710 0.452 0.083 (1.59) (1.07) (0.30) Adj. R2 8.10 29.17 4.41 29.28 9.20 28.69 49

Table 5: 12m HPR Bond Predictability with 10y1m Interest-rate VRP This table presents regression results for the following regression: rx(τ) = β(τ) + β(τ) (h)IRVRP + t+h 0 1 t (cid:80)2 β(τ) (h)F +(cid:15)(τ) , where rx(τ) are excess returns on Treasury notes, h=12 months and τ =2, ...,5 j=1 j t,j t+h t+h years. IRVRP is the interest-rate variance risk premium derived from interest rate derivatives markets, t F , j =1,2 is the forward spread (FS) and Cochrane-Piazzesi (CP) factors. t-statistics in parentheses are t,j calculatedusingNeweyandWest(1987)standarderrors. AdjustedR2aregiveninpercentagepoints. Treasury excess returns are computed using Fama-Bliss data set, obtained from CRSP. Interest rate swaptions and swaps data are from J.P. Morgan and Barclays Capital. Sample is from April 2002 to January 2013, monthly frequency. (1) (2) (3) (4) (5) (6) (7) PanelA:maturity=2years Const 0.004 0.007 0.007 0.006 0.007 0.007 (1.28) (2.96) (2.15) (1.80) (1.94) (1.86) IRVRP 0.000 0.001 0.001 0.001 (0.72) (1.20) (1.04) (1.25) FS -0.534 -0.592 -0.558 (-1.62) (-1.87) (-1.74) CP -0.107 -0.169 -0.081 (-0.81) (-1.22) (-0.54) Adj. R2 -0.07 6.59 -0.09 7.98 0.89 7.48 PanelB:maturity=3years Const 0.011 0.015 0.017 0.013 0.016 0.017 (1.88) (2.49) (2.66) (1.89) (2.46) (2.29) IRVRP 0.001 0.001 0.001 0.001 (0.67) (1.07) (1.00) (1.22) FS -0.242 -0.331 -0.262 (-0.60) (-0.85) (-0.68) CP -0.208 -0.320 -0.252 (-0.83) (-1.21) (-0.93) Adj. R2 -0.20 0.21 -0.02 0.84 0.80 1.00 PanelC:maturity=4years Const 0.020 0.018 0.026 0.017 0.026 0.024 (2.40) (1.86) (2.84) (1.56) (2.65) (2.05) IRVRP 0.001 0.000 0.001 0.001 (0.43) (0.30) (0.71) (0.63) FS 0.230 0.205 0.277 (0.52) (0.48) (0.67) CP -0.253 -0.374 -0.451 (-0.67) (-0.96) (-1.26) Adj. R2 -0.58 -0.02 -0.28 -0.87 -0.24 -0.00 PanelD:maturity=5years Const 0.025 0.021 0.032 0.021 0.030 0.027 (2.59) (1.77) (2.66) (1.65) (2.48) (2.03) IRVRP 0.002 0.001 0.002 0.001 (0.81) (0.32) (0.99) (0.52) FS 0.411 0.358 0.422 (0.98) (0.79) (0.96) CP -0.090 -0.300 -0.442 (-0.18) (-0.60) (-1.00) Adj. R2 0.50 2.14 -0.92 1.34 0.07 1.49 50

Table 6: 1m HPR Bond Predictability with 10y3m Interest-rate VRP This table presents regression results for the following regression: rx(τ) = β(τ) + β(τ) (h)IRVRP + t+h 0 1 t (cid:80)2 β(τ) (h)F +(cid:15)(τ) , where rx(τ) are excess returns on Treasury notes, h = 1 month and τ = 2, ..., 5 j=1 j t,j t+h t+h years. IRVRP is the interest-rate variance risk premium derived from interest rate derivatives markets, t F , j =1,2 is the forward spread (FS) and Cochrane-Piazzesi (CP) factors. t-statistics in parentheses are t,j calculatedusingNeweyandWest(1987)standarderrors. AdjustedR2aregiveninpercentagepoints. Treasury excess returns are computed using Fama-Bliss data set, obtained from CRSP. Interest rate swaptions and swaps data are from J.P. Morgan and Barclays Capital. Sample is from April 2002 to January 2013, monthly frequency. (1) (2) (3) (4) (5) (6) (7) PanelA:maturity=2years Const -0.029 -0.021 -0.027 -0.034 -0.035 -0.036 (-7.57) (-6.46) (-3.27) (-16.23) (-5.91) (-13.65) IRVRP 0.008 0.007 0.007 0.007 (6.44) (11.38) (6.48) (10.52) FS 1.983 1.699 1.639 (4.98) (9.03) (7.86) CP 0.689 0.382 0.121 (2.16) (1.64) (1.03) Adj. R2 50.21 37.35 14.16 77.53 54.10 77.73 PanelB:maturity=3years Const -0.028 -0.026 -0.027 -0.037 -0.035 -0.038 (-7.00) (-7.11) (-3.25) (-17.17) (-5.69) (-14.43) IRVRP 0.008 0.006 0.007 0.006 (6.12) (8.74) (6.10) (8.32) FS 1.407 1.185 1.153 (5.89) (10.23) (9.60) CP 0.730 0.429 0.097 (2.26) (1.77) (0.88) Adj. R2 42.75 44.63 13.82 73.63 47.02 73.62 PanelC:maturity=4years Const -0.027 -0.029 -0.027 -0.039 -0.035 -0.041 (-6.32) (-8.10) (-3.21) (-18.97) (-5.33) (-17.94) IRVRP 0.008 0.006 0.007 0.006 (5.55) (9.66) (5.42) (8.11) FS 1.159 0.984 0.948 (7.36) (10.79) (9.94) CP 0.765 0.471 0.181 (2.33) (1.83) (1.50) Adj. R2 33.67 44.58 12.25 64.52 37.76 64.86 PanelD:maturity=5years Const -0.027 -0.033 -0.028 -0.038 -0.036 -0.040 (-6.12) (-10.44) (-3.15) (-16.50) (-5.25) (-13.28) IRVRP 0.008 0.005 0.007 0.004 (5.80) (4.70) (5.71) (4.37) FS 1.066 0.877 0.843 (9.03) (8.81) (8.01) CP 0.826 0.526 0.176 (2.40) (1.90) (1.15) Adj. R2 29.44 49.30 11.73 57.95 33.61 58.07 51

Table 7: 3m HPR Bond Predictability with 10y3m Interest-rate VRP This table presents regression results for the following regression: rx(τ) = β(τ) + β(τ) (h)IRVRP + t+h 0 1 t (cid:80)2 β(τ) (h)F +(cid:15)(τ) , where rx(τ) are excess returns on Treasury notes, h=3 months and τ =2, ...,5 j=1 j t,j t+h t+h years. IRVRP is the interest-rate variance risk premium derived from interest rate derivatives markets, t F , j =1,2 is the forward spread (FS) and Cochrane-Piazzesi (CP) factors. t-statistics in parentheses are t,j calculatedusingNeweyandWest(1987)standarderrors. AdjustedR2aregiveninpercentagepoints. Treasury excess returns are computed using Fama-Bliss data set, obtained from CRSP. Interest rate swaptions and swaps data are from J.P. Morgan and Barclays Capital. Sample is from April 2002 to January 2013, monthly frequency. (1) (2) (3) (4) (5) (6) (7) PanelA:maturity=2years Const -0.023 -0.017 -0.022 -0.027 -0.028 -0.028 (-6.15) (-5.12) (-2.88) (-8.43) (-4.74) (-8.41) IRVRP 0.006 0.006 0.006 0.005 (5.68) (8.19) (5.62) (7.35) FS 1.608 1.361 1.305 (4.08) (5.06) (4.51) CP 0.618 0.328 0.123 (2.02) (1.39) (0.83) Adj. R2 38.11 28.06 12.37 58.00 40.99 58.07 PanelB:maturity=3years Const -0.021 -0.020 -0.021 -0.028 -0.026 -0.029 (-4.42) (-5.10) (-2.45) (-6.80) (-3.75) (-7.19) IRVRP 0.006 0.005 0.005 0.005 (4.66) (5.42) (4.41) (4.89) FS 1.288 1.109 1.089 (4.78) (5.20) (4.90) CP 0.646 0.371 0.064 (1.90) (1.30) (0.38) Adj. R2 23.00 32.12 8.61 45.95 25.18 45.53 PanelC:maturity=4years Const -0.018 -0.024 -0.019 -0.030 -0.024 -0.031 (-3.09) (-5.33) (-2.04) (-6.08) (-2.87) (-6.38) IRVRP 0.006 0.004 0.005 0.004 (3.44) (3.61) (3.14) (3.13) FS 1.157 1.034 1.015 (5.40) (5.30) (5.09) CP 0.658 0.404 0.110 (1.72) (1.18) (0.51) Adj. R2 12.70 30.20 5.46 35.61 14.11 35.17 PanelD:maturity=5years Const -0.017 -0.026 -0.018 -0.029 -0.024 -0.030 (-2.63) (-4.87) (-1.69) (-5.03) (-2.44) (-4.76) IRVRP 0.006 0.003 0.005 0.002 (3.45) (1.63) (3.18) (1.56) FS 1.068 0.959 0.948 (5.31) (4.57) (4.36) CP 0.710 0.437 0.066 (1.59) (1.07) (0.24) Adj. R2 10.43 29.17 4.41 30.17 11.46 29.57 52

Table 8: 12m HPR Bond Predictability with 10y3m Interest-rate VRP This table presents regression results for the following regression: rx(τ) = β(τ) + β(τ) (h)IRVRP + t+h 0 1 t (cid:80)2 β(τ) (h)F +(cid:15)(τ) , where rx(τ) are excess returns on Treasury notes, h=12 months and τ =2, ...,5 j=1 j t,j t+h t+h years. IRVRP is the interest-rate variance risk premium derived from interest rate derivatives markets, t F , j =1,2 is the forward spread (FS) and Cochrane-Piazzesi (CP) factors. t-statistics in parentheses are t,j calculatedusingNeweyandWest(1987)standarderrors. AdjustedR2aregiveninpercentagepoints. Treasury excess returns are computed using Fama-Bliss data set, obtained from CRSP. Interest rate swaptions and swaps data are from J.P. Morgan and Barclays Capital. Sample is from April 2002 to January 2013, monthly frequency. (1) (2) (3) (4) (5) (6) (7) PanelA:maturity=2years Const 0.004 0.007 0.007 0.006 0.006 0.007 (1.13) (2.96) (2.15) (1.59) (1.73) (1.69) IRVRP 0.001 0.001 0.001 0.001 (0.58) (1.13) (0.85) (1.16) FS -0.534 -0.602 -0.572 (-1.62) (-1.92) (-1.81) CP -0.107 -0.158 -0.072 (-0.81) (-1.16) (-0.49) Adj. R2 -0.27 6.59 -0.09 7.96 0.48 7.38 PanelB:maturity=3years Const 0.011 0.015 0.017 0.013 0.016 0.016 (1.64) (2.49) (2.66) (1.71) (2.21) (2.11) IRVRP 0.001 0.002 0.002 0.002 (0.62) (1.03) (0.90) (1.15) FS -0.242 -0.342 -0.275 (-0.60) (-0.89) (-0.73) CP -0.208 -0.310 -0.239 (-0.83) (-1.20) (-0.91) Adj. R2 -0.19 0.21 -0.02 0.95 0.72 1.03 PanelC:maturity=4years Const 0.018 0.018 0.026 0.016 0.024 0.023 (2.03) (1.86) (2.84) (1.44) (2.38) (1.94) IRVRP 0.002 0.001 0.003 0.002 (0.63) (0.52) (0.91) (0.82) FS 0.230 0.174 0.244 (0.52) (0.41) (0.60) CP -0.253 -0.399 -0.460 (-0.67) (-1.07) (-1.33) Adj. R2 -0.08 -0.02 -0.28 -0.58 0.48 0.41 PanelD:maturity=5years Const 0.023 0.021 0.032 0.020 0.028 0.026 (2.15) (1.77) (2.66) (1.51) (2.22) (1.91) IRVRP 0.004 0.002 0.005 0.003 (1.11) (0.72) (1.34) (0.93) FS 0.411 0.289 0.359 (0.98) (0.66) (0.83) CP -0.090 -0.348 -0.465 (-0.18) (-0.73) (-1.08) Adj. R2 1.76 2.14 -0.92 1.92 1.54 2.22 53

Table 9: 1m HPR Bond Predictability with 10y12m Interest-rate VRP This table presents regression results for the following regression: rx(τ) = β(τ) + β(τ) (h)IRVRP + t+h 0 1 t (cid:80)2 β(τ) (h)F +(cid:15)(τ) , where rx(τ) are excess returns on Treasury notes, h = 1 month and τ = 2, ..., 5 j=1 j t,j t+h t+h years. IRVRP is the interest-rate variance risk premium derived from interest rate derivatives markets, t F , j =1,2 is the forward spread (FS) and Cochrane-Piazzesi (CP) factors. t-statistics in parentheses are t,j calculatedusingNeweyandWest(1987)standarderrors. AdjustedR2aregiveninpercentagepoints. Treasury excess returns are computed using Fama-Bliss data set, obtained from CRSP. Interest rate swaptions and swaps data are from J.P. Morgan and Barclays Capital. Sample is from April 2002 to January 2013, monthly frequency. (1) (2) (3) (4) (5) (6) (7) PanelA:maturity=2years Const -0.021 -0.021 -0.027 -0.029 -0.034 -0.036 (-5.20) (-6.46) (-3.27) (-9.77) (-4.84) (-11.72) IRVRP 0.012 0.013 0.012 0.013 (4.23) (7.82) (4.65) (7.92) FS 1.983 2.094 1.885 (4.98) (8.05) (7.30) CP 0.689 0.697 0.381 (2.16) (2.61) (2.67) Adj. R2 23.40 37.35 14.16 65.55 38.27 69.50 PanelB:maturity=3years Const -0.020 -0.026 -0.027 -0.035 -0.034 -0.040 (-4.88) (-7.11) (-3.25) (-14.64) (-4.77) (-17.13) IRVRP 0.012 0.014 0.012 0.014 (4.07) (10.43) (4.52) (10.56) FS 1.407 1.510 1.396 (5.89) (11.05) (12.22) CP 0.730 0.739 0.301 (2.26) (2.70) (3.54) Adj. R2 19.72 44.63 13.82 71.44 34.19 73.43 PanelC:maturity=4years Const -0.019 -0.029 -0.027 -0.036 -0.034 -0.042 (-4.51) (-8.10) (-3.21) (-12.86) (-4.66) (-17.78) IRVRP 0.012 0.012 0.012 0.012 (3.83) (7.34) (4.28) (7.67) FS 1.159 1.152 1.059 (7.36) (11.79) (12.65) CP 0.765 0.773 0.396 (2.33) (2.75) (3.18) Adj. R2 15.95 44.58 12.25 60.55 28.75 63.48 PanelD:maturity=5years Const -0.019 -0.033 -0.028 -0.038 -0.035 -0.043 (-4.19) (-10.44) (-3.15) (-16.64) (-4.54) (-16.50) IRVRP 0.012 0.011 0.012 0.011 (3.61) (6.30) (4.08) (6.72) FS 1.066 1.032 0.961 (9.03) (10.96) (10.61) CP 0.826 0.835 0.312 (2.40) (2.78) (2.44) Adj. R2 13.12 49.30 11.73 59.62 25.33 60.86 54

Table 10: 3m HPR Bond Predictability with 10y12m Interest-rate VRP This table presents regression results for the following regression: rx(τ) = β(τ) + β(τ) (h)IRVRP + t+h 0 1 t (cid:80)2 β(τ) (h)F +(cid:15)(τ) , where rx(τ) are excess returns on Treasury notes, h=3 months and τ =2, ...,5 j=1 j t,j t+h t+h years. IRVRP is the interest-rate variance risk premium derived from interest rate derivatives markets, t F , j =1,2 is the forward spread (FS) and Cochrane-Piazzesi (CP) factors. t-statistics in parentheses are t,j calculatedusingNeweyandWest(1987)standarderrors. AdjustedR2aregiveninpercentagepoints. Treasury excess returns are computed using Fama-Bliss data set, obtained from CRSP. Interest rate swaptions and swaps data are from J.P. Morgan and Barclays Capital. Sample is from April 2002 to January 2013, monthly frequency. (1) (2) (3) (4) (5) (6) (7) PanelA:maturity=2years Const -0.016 -0.017 -0.022 -0.023 -0.027 -0.029 (-4.42) (-5.12) (-2.88) (-6.55) (-4.01) (-7.76) IRVRP 0.010 0.011 0.010 0.011 (3.99) (6.39) (4.28) (6.37) FS 1.608 1.679 1.503 (4.08) (5.22) (4.61) CP 0.618 0.588 0.333 (2.02) (2.21) (2.02) Adj. R2 17.86 28.06 12.37 48.86 29.24 51.96 PanelB:maturity=3years Const -0.014 -0.020 -0.021 -0.028 -0.026 -0.031 (-3.34) (-5.10) (-2.45) (-6.57) (-3.34) (-7.75) IRVRP 0.010 0.011 0.010 0.011 (3.48) (6.41) (3.68) (6.29) FS 1.288 1.357 1.280 (4.78) (6.03) (5.69) CP 0.646 0.617 0.211 (1.90) (2.02) (1.44) Adj. R2 11.09 32.12 8.61 47.02 19.02 47.44 PanelC:maturity=4years Const -0.012 -0.024 -0.019 -0.029 -0.024 -0.033 (-2.52) (-5.33) (-2.04) (-6.02) (-2.78) (-6.68) IRVRP 0.011 0.010 0.010 0.010 (3.08) (4.34) (3.23) (4.36) FS 1.157 1.140 1.086 (5.40) (6.10) (5.81) CP 0.658 0.627 0.245 (1.72) (1.80) (1.25) Adj. R2 7.60 30.20 5.46 37.22 12.59 37.47 PanelD:maturity=5years Const -0.011 -0.026 -0.018 -0.030 -0.024 -0.032 (-1.93) (-4.87) (-1.69) (-5.22) (-2.33) (-5.21) IRVRP 0.011 0.009 0.011 0.009 (2.70) (3.30) (2.82) (3.34) FS 1.068 1.032 1.002 (5.31) (5.45) (5.01) CP 0.710 0.679 0.135 (1.59) (1.63) (0.51) Adj. R2 5.74 29.17 4.41 33.00 9.76 32.56 55

Table 11: 12m HPR Bond Predictability with 10y12m Interest-rate VRP This table presents regression results for the following regression: rx(τ) = β(τ) + β(τ) (h)IRVRP + t+h 0 1 t (cid:80)2 β(τ) (h)F +(cid:15)(τ) , where rx(τ) are excess returns on Treasury notes, h=12 months and τ =2, ...,5 j=1 j t,j t+h t+h years. IRVRP is the interest-rate variance risk premium derived from interest rate derivatives markets, t F , j =1,2 is the forward spread (FS) and Cochrane-Piazzesi (CP) factors. t-statistics in parentheses are t,j calculatedusingNeweyandWest(1987)standarderrors. AdjustedR2aregiveninpercentagepoints. Treasury excess returns are computed using Fama-Bliss data set, obtained from CRSP. Interest rate swaptions and swaps data are from J.P. Morgan and Barclays Capital. Sample is from April 2002 to January 2013, monthly frequency. (1) (2) (3) (4) (5) (6) (7) PanelA:maturity=2years Const 0.005 0.007 0.007 0.007 0.007 0.007 (1.77) (2.96) (2.15) (2.31) (1.84) (1.80) IRVRP 0.001 0.001 0.001 0.001 (0.31) () (0.40) (0.36) (0.40) FS -0.534 -0.536 -0.528 (-1.62) (-1.64) (-1.59) CP -0.107 -0.109 -0.015 (-0.81) (-0.83) (-0.10) Adj. R2 -0.87 6.59 -0.09 5.84 -0.93 4.91 PanelB:maturity=3years Const 0.012 0.015 0.017 0.014 0.016 0.017 (2.40) (2.49) (2.66) (1.99) (2.28) (2.10) IRVRP 0.002 0.002 0.002 0.002 (0.55) (0.60) (0.62) (0.64) FS -0.242 -0.248 -0.193 (-0.60) (-0.62) (-0.48) CP -0.208 -0.215 -0.151 (-0.83) (-0.87) (-0.63) Adj. R2 -0.60 0.21 -0.02 -0.35 -0.57 -0.90 PanelC:maturity=4years Const 0.019 0.018 0.026 0.016 0.024 0.022 (2.80) (1.86) (2.84) (1.51) (2.37) (1.84) IRVRP 0.006 0.006 0.007 0.006 (1.13) (1.12) (1.22) (1.19) FS 0.230 0.189 0.273 (0.52) (0.44) (0.66) CP -0.253 -0.271 -0.370 (-0.67) (-0.75) (-1.23) Adj. R2 0.54 -0.02 -0.28 0.20 0.37 0.59 PanelD:maturity=5years Const 0.025 0.021 0.032 0.019 0.027 0.024 (3.14) (1.77) (2.66) (1.45) (2.17) (1.74) IRVRP 0.012 0.010 0.012 0.010 (1.75) (1.71) (1.80) (1.78) FS 0.411 0.334 0.415 (0.98) (0.82) (1.01) CP -0.090 -0.123 -0.358 (-0.18) (-0.27) (-0.90) Adj. R2 2.37 2.14 -0.92 3.39 1.52 3.22 56

6 5.5 5 4.5 4 3.5 3 0 1 2 3 4 5 6 7 8 9 10 Maturity (years) tnecreP Nominal yield curve data model (a) With LRR component 6 4 2 0 −2 −4 −6 0 1 2 3 4 5 6 7 8 9 10 Maturity (years) tnecreP Nominal yield curve data model (b) No LRR component Figure 1: The model-implied nominal yield curve The figure plots the average zero-coupon nominal Treasury yield curve as observed in the data using the sampleofJanuary1991-December2010monthlydataasthesolidbluelineinbothPanels(a)and(b). The figure also plots the model-implied yield curve with the long-run risk component (Panel (a)) and without the long-run risk component (Panel (b)) as the dashed red line. 57

Expected consumption loading Consumption vol loading 30 0 25 −10 −20 20 −30 15 −40 10 −50 5 −60 0 −70 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Consumption vol−of−vol loading Expected inflation loading 0 20 −10 15 −20 −30 10 −40 −50 5 −60 −70 0 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Figure 2: Equilibrium nominal bond yield loadings The figure plots the model-implied nominal bond yield loadings on expected consumption growth (top left panel),consumptionvolatility(toprightpanel),consumptionvolatility-of-volatility(bottomleftpanel),and expected inflation (bottom right panel). Maturity on horizontal axes is in months. 58

25 20 15 10 5 0 Jan−04 Jan−06 Jan−08 Jan−10 Jan−12 derauqstnecreP Impliedvariance,10y1mswaptions 8 6 4 2 0 Jan−04 Jan−06 Jan−08 Jan−10 Jan−12 derauqstnecreP P-measureexpectationofrealizedvarianceof10-yearswaprates 20 15 10 5 0 −5 Jan−04 Jan−06 Jan−08 Jan−10 Jan−12 derauqstnecreP Varianceriskpremium Figure 3: Interest-rate variance risk premium, 1-month horizon Thisfigureplotsimpliedvariance(toppanel),expectedvariance(middlepanel),andtheirdifference,variance risk premium (bottom panel). Implied variance is derived from the one-month expiry swaptions on the 10yearforwardswaprate. Realizedvarianceisderivedfromtheintradayten-yearswapratedata. Blueshaded barindicatesNBERrecession. Datasource: J.P.MorganandBarclaysCapital. SampleperiodisfromApril 2002 to February 2013. 59

15 10 5 0 Jan−04 Jan−06 Jan−08 Jan−10 Jan−12 derauqstnecreP Impliedvariance,10y3mswaptions 8 6 4 2 0 Jan−04 Jan−06 Jan−08 Jan−10 Jan−12 derauqstnecreP P-measureexpectationofrealizedvarianceof10-yearswaprates 10 5 0 −5 Jan−04 Jan−06 Jan−08 Jan−10 Jan−12 derauqstnecreP Varianceriskpremium Figure 4: Interest-rate variance risk premium, 3-month horizon Thisfigureplotsimpliedvariance(toppanel),expectedvariance(middlepanel),andtheirdifference,variance risk premium (bottom panel). Implied variance is derived from the three-month expiry swaptions on the 10-year forward swap rate. Realized variance is derived from the intraday ten-year swap rate data. Blue shaded bar indicates NBER recession. Data source: J.P. Morgan and Barclays Capital. Sample period is from April 2002 to February 2013. 60

8 6 4 2 0 Jan−04 Jan−06 Jan−08 Jan−10 Jan−12 derauqstnecreP Impliedvariance,10y12mswaptions 8 6 4 2 0 Jan−04 Jan−06 Jan−08 Jan−10 Jan−12 derauqstnecreP P-measureexpectationofrealizedvarianceof10-yearswaprates 6 4 2 0 −2 −4 −6 Jan−04 Jan−06 Jan−08 Jan−10 Jan−12 derauqstnecreP Varianceriskpremium Figure 5: Interest-rate variance risk premium, 12-month horizon Thisfigureplotsimpliedvariance(toppanel),expectedvariance(middlepanel),andtheirdifference,variance risk premium (bottom panel). Implied variance is derived from the twelve-month expiry swaptions on the 10-year forward swap rate. Realized variance is derived from the intraday ten-year interest rate swaps data. Blue shaded bar indicates NBER recession. Data source: J.P. Morgan and Barclays Capital. Sample period is from April 2002 to February 2013. 61