Mortgage Convexity Samuel G. Hanson Harvard Business School shanson@hbs.edu This draft: January 2014 First draft: July 2012 Abstract Most home mortgages in the United States are …xed-rate loans with an embedded prepayment option. When long-term rates decline, the e¤ective duration of mortgage-backed securities (MBS) falls due to heightened re…nancing expectations. I show that these changes in MBS duration function as large-scale shocks to the quantity of interest rate risk that must be borne by professional bond investors. I develop a simple model in which the risk tolerance of bond investors is limited in the short run, so these ‡ uctuations in MBS duration generate signi…cant variation in bond risk premia. Speci…cally, bond risk premia are high when aggregate MBS duration is high. The model o¤ers an explanation for why long-term rates may appear to be “excessively sensitive” to movements in short rates and explains how changes in MBS duration act as a positive-feedback mechanism that ampli…es interest rate volatility. I …nd strong support for these predictions in the time series of US government bond returns. I thank seminar participants at the Boston University, Federal Reserve Board, the Copenhagen Business School FRIC’ Conference, as well as John Campbell, Anna Cieslak, Robin Greenwood, Arvind Krishnamurthy, 13 Lasse Pedersen, Erik Sta¤ord, Jeremy Stein, Larry Summers, Adi Sunderam, and Dimitri Vayanos for helpful comments. I am grateful to an anonymous referee for many helpful suggestions. I gratefully acknowledge funding from the Harvard Business School Division of Research. 1 Introduction A distinguishing feature of bond markets in the United States is the prominent role of …xed-rated home mortgages. Most …xed-rate mortgages issued in the US grant the borrower the right to prepay the loan at any time without penalty. When long-term interest rates decline, the option to prepay the mortgage and re…nance at the current interest rate moves towards the money. Because lenders expect to be repaid sooner, the e¤ective duration of outstanding mortgagebacked securities (MBS) falls— i.e., the sensitivity of MBS prices to changes in long-term yields declines. Conversely, as long-term interest rates rise, the prepayment option moves out of the money and the e¤ective duration of MBS rises. To quote a Wall Street adage, this means that a mortgage-backed security “goes up like a 2 year bond” when rates fall and “goes down like a 6 year”when rates rise. More formally, the negative relationship between price and yield is convex for a noncallable bond. By contrast, the price-yield relation for callable bonds such as MBS is typically concave— a feature that bond investors call “negative convexity.” Aggregate mortgage re…nancing activity varies signi…cantly over time. Because re…nancing entails large …nancial and non…nancial costs, households typically follow a trigger rule, re…nancing only when their option is su¢ ciently far in the money. Thus, depending on the past path of mortgage rates, there are times when many households move from being far from re…nancing to being close to re…nancing and vice versa. I argue that the resulting changes in aggregate MBS duration act as massive supply shocks to the total quantity of interest rate risk that must be borne by investors in the broader bond market. Because these shifts in bond market duration are large relative to bond investors’risk tolerance, they have a signi…cant impact on equilibrium term premium— i.e., on the expected return on long-term default-free bonds over short-term defaultfree bonds. In summary, time-variation in aggregate mortgage re…nancing has a signi…cant e¤ect on the pricing of interest rate risk throughout the US …xed income market and, in particular, has a large impact on US Treasury yields. Why do shifts in expected household mortgage re…nancing a¤ect the aggregate amount of interest rate risk that bond market investors must bear? First, household borrowers only gradually exercise their prepayment options following a decline in prevailing mortgage rates. Second, household borrowers do not alter their bond holdings to hedge their time-varying interest rate exposure. That is, households do not adjust their asset portfolios to o¤set the time-varying interest rate risk they are bearing on the liability side. The gradual response of mortgage prepayments to changes in mortgage rates and the lack of household hedging means that there are times when households are bearing more or less interest rate risk. Conversely, there are times when bond investors are bearing less or more risk. Thus, shifting re…nancing expectations generate a form of aggregate market congestion: there are times when most households are e¤ectively borrowing long term and other times when most are borrowing short term. If the risk-bearing capacity of professional bond investors is limited in the short run, then the term premium must adjust to induce investors to bear these risks. The sheer size of the MBS market within the US bond market plays a crucial role in this story. Bond markets have witnessed several MBS duration supply shocks that are larger than 1 the shift in duration induced by the Federal Reserve’ Quantitative Easing policies from 2008 to s 2012. For instance, there have been several occasions when the quarterly change in MBS duration was equivalent to a $1 trillion increase (in 2012 dollars) in the supply of 10-year Treasury notes, with a corresponding reduction in the supply of short-term T-bills. By way of comparison, the quarterly new-issue supply of 10-year Treasury notes in 2012 was roughly $65 billion. Thus, past shifts in MBS duration have arguably been very large relative to the risk tolerance of bond market arbitrageurs. I develop a simple model of this mortgage convexity mechanism. The model has the following key ingredients. First, expectations of future household mortgage re…nancing vary over time, which induces shifts in the e¤ective duration of household mortgage borrowing. Second, aggregate bond risks are priced by risk-averse, specialized bond investors as in Vayanos and Vila (2009). This ensures that demand curves for aggregate bond risks slope downward, so the term premium must adjust to induce investors to absorb the aggregate supply of bond duration. Speci…cally, the model predicts that measures of aggregate bond market duration, which derive most of their power from variation in MBS duration, should positively predict bond excess returns. Furthermore, since shocks to MBS duration are fairly transient, one would expect ‡ uctuations in MBS duration to lead to high frequency variation in bond risk premia. The model shows how negative MBS convexity may contribute to the excess sensitivity of long-term yields to movements in short rates (Gurkaynak, Sack, and Swanson (2005)) and, more generally, to the excess volatility of long-term yields (Shiller (1979)). Excess sensitivity re‡ ects what might be called a “MBS duration spiral.” An initial shock to the short rate directly raises long yields due to the expectations hypothesis. The rise in long yields raises the duration of MBS. The term premium must rise in order to induce risk-averse bond investors to bear the larger aggregate quantity of interest rate risk. The resulting rise in yields further raises MBS duration, which further raises the term premium, and so on. Thus, the fact that MBS duration is increasing in long yields gives rise to a positive-feedback channel that generates excess sensitivity and excess volatility. The model also suggests that this positive-feedback mechanism is strongest when the mortgage market is on a “re…nancing cli¤” i.e., when a small movement in long rates — will signi…cantly impact re…nancing behavior and, hence, MBS duration. The model therefore predicts that excess sensitivity and volatility should be most pronounced when the MBS market is more negatively convex. I also show that an MBS duration shock should have a larger e¤ect on the expected excess returns of long-term bonds than on those of intermediate-term bonds. This is a natural consequence of the fact that an MBS duration shock raises the current duration risk premium in bond markets. However, since these shocks are transient, shocks to MBS duration have a humpedshaped e¤ect on the yield curve and the forward rate curve. The e¤ects of a supply shock on yields— which equals the e¤ect on the bond’ average returns over its lifetime— will be greater s for intermediate term bonds than for long-term bonds when the supply shock is expected to be short-lived. That is, a shock to MBS duration increases the curvature of the yield curve. This suggests that transient shocks to MBS duration may account for some of the predictive power of the Cochrane and Piazzesi (2005) factor which picks up time-variation in the curvature of the 2 yield curve and is useful for forecasting transitory variation in bond returns. I …nd strong support for these predictions in US interest rate data between 1989 and 2011. Measures of MBS duration are strong predictors of excess government bond returns. And they appear to contain information that is not re‡ ected in traditional forecasting variables based on the current shape of the yield curve. My analysis of the time signature of these e¤ects indicates that shocks to MBS duration have a transitory impact on term premia which largely dissipates over the next 6 to 12 months. As predicted, shocks to MBS duration have a hump-shaped e¤ect on the yield and forward rate curves. I also …nd that the excess sensitivity of long rates to short rates is more pronounced when the MBS market is on a re…nancing cli¤— i.e., when a move in long rates has a larger impact on aggregate MBS duration. Lastly, I …nd that option-implied volatility of long yields is higher when the MBS market is more negatively convex. It is worth noting that I lack a good instrument for MBS duration that would allow me to cleanly identify the demand curve for interest rate risk. As a result, I rely on predictive regressions that provide indirect evidence consistent with the view that shifts in MBS duration trace out a downward-sloping demand for interest rate risk. These return forecasting regressions are analogous to a regression of prices (term premia) on quantities (duration supply). Thus, I take considerable care to address several natural concerns raised by this indirect approach. To begin, I control for an exhaustive set of factors that are thought to impact term premia to address concerns that my forecasting results are driven by an omitted variable. I then provide a host of further indirect evidence that is consistent with the MBS story, but that would not be predicted by alternative explanations for my …ndings. First, I show that the return forecasting power of MBS duration has grown signi…cantly over time as MBS markets have grown relative to the rest of the US bond market. Second, I show that shifts in MBS duration have far more forecasting power for US bond returns than for foreign bond returns. Third, I show that shifts in MBS duration impact the spreads between corporate bonds, interest rate swaps, and Treasuries in precisely the way that one would expect if MBS investors were using Treasuries and swaps— but not corporate bonds— to hedge variation in MBS duration. Collectively, these additional …ndings provide further support for my story emphasizing shifts in MBS duration. Bond investors often invoke shifts in MBS duration and the portfolio-hedging ‡ ows they trigger when explaining large movements in long-term interest rates. Many MBS investors delta-hedge the time-varying duration of MBS: if there is a drop in long rates that raises mortgage re…nancing expectations and lowers MBS duration, these investors buy more long-term Treasuries— …nanced by selling some short-term T-bills— in order to maintain their prior asset duration. Given the prominence that these dynamics receive in practitioner commentary, their relative absence from the literature on the term structure of interest rates is somewhat surprising. The handful of exceptions includes Fernald, Keane, and Mosser (1994), Kambu and Mosser (2001), Perli and Sack (2003), Duarte (2008), and Malkhozov, Mueller, Vedolin, and Venter (2013). Kambu and Mosser (2001), Perli and Sack (2003), and Duarte (2008) each argue that these hedging ‡ ows impact interest rate volatility, whereas the primary focus of my paper is to investigate the impact of MBS duration on the equilibrium term structure of yields and term 3 premia.1 In contemporaneous work, Malkhozov et all (2013) also explore the impact of MBS duration on excess returns. Their study places more emphasis on the implications of MBS hedging for interest rate volatility whereas I am primarily interested in understanding the pricing implications of these hedging ‡ ows. The ideas in this paper connect to several broader strands of prior research. First, the idea that supply and demand e¤ects can have important consequences in bond markets is central to a number of recent papers, including Vayanos and Vila (2009), Greenwood and Vayanos (2010, 2012), Krishnamurthy and Vissing-Jorgensen (2011, 2012), and Gagnon et al (2011). Important precursors to this recent work include Tobin (1958) and Modigliani and Sutch (1966). This literature has featured prominently in the “portfolio balance e¤ect”interpretation of the Federal Reserve’ recent Quantitative Easing policies, in which it is typically argued that Fed purchases of s long-term assets reduce bond risk premia. Second, there is a vast literature devoted to forecasting the excess returns on long-term bonds. Important contributions here include Fama and Bliss (1987), Campbell and Shiller (1991), and Cochrane and Piazzesi (2005). Third, a variety of papers have documented excess sensitivity including Cochrane and Piazzesi (2002), Gurkaynak, Sack, and Swanson (2005), and Hanson and Stein (2012). Fourth, a number of papers have argued that long-term yields are excessively volatile including Shiller (1979), Perli and Sack (2003), and Duarte (2008). Finally, a growing literature, including Gabaix, Krishnamurthy, and Vigneron (2007), Greenwood and Vayanos (2013), and Vayanos and Vila (2009), explores the implications of limited arbitrage in …xed income markets. The plan for the paper is as follows. Section 2 clari…es why shifts in MBS duration matter and develops a stylized model that generates several novel empirical predictions. In Section 3, I …nd strong support for these predictions in the time series of US government bond returns. Section 4 concludes. 2 2.1 Theoretical framework Shocks to MBS duration Suppose a large number of households announce that in six months time they will repay their existing mortgages at par and will take out new mortgages at the prevailing mortgage rate. And suppose for simplicity that this commitment to re…nance in six months is binding. Suddenly, from the perspective of investors who own the existing mortgages, the mortgages behave just like short-term bonds, so mortgage holders are bearing less interest rate risk. And over the following six months, households are bearing signi…cantly more interest rate risk— they will be worse o¤ if rates rise and better o¤ if rates fall. In order to hedge this interest rate exposure, households could initially sell some long-term Treasuries with the intention of buying them back Perli and Sack (2003) show that MBS hedging impacts swaption-implied yield volatility. Duarte (2008) also …nds that MBS hedging forecasts high future realized volatility which suggests that hedging ‡ ows do not simply impact interest rate option prices, but also impact the underlying bond yields. 1 4 in six months.2 However, due to a variety of frictions, costs, or a lack of …nancial sophistication, households do not hedge their time-varying exposure to interest rate risk. So, on net, there is a temporary reduction in the amount of interest rate risk that …xed income investors must bear in equilibrium. And if the risk-bearing capacity of bond investors is limited, the expected excess return on long-term bonds must fall in order to equate the supply and demand for bonds. In practice, households do not enter into binding agreements to re…nance. Instead, household re…nancing behavior is gradual, so shocks to long-term interest rates alter expected mortgage re…nancing. Critically, I assume households do not hedge their interest rate exposure— i.e., households do not alter their bond holdings to hedge the time-varying interest rate risk they are assuming on the liability side of their balance sheets. This plausible assumption is the key friction that generates time-varying risk sharing between household borrowers and bond investors and, thus, shifts in aggregate bond market duration. For instance, when future mortgage re…nancings are expected to be high, households are e¤ectively borrowing shorter term and, hence, are bearing more interest rate risk— i.e., households are exposed to the risk that they will re…nance at a less advantageous rate. Conversely, when expected re…nancings are low, households are e¤ectively borrowing longer term, so investors are bearing more interest rate risk. The total quantity of interest rate risk borne in the economy remains unchanged: there is a single long-term asset, namely housing, which is going to be …nanced long-term at some interest rate. Instead, what varies is the amount of interest rate risk that is borne by bond investors as opposed to household borrowers. Shifts in the extent of aggregate risk sharing between households and investors move the price of interest rate risk— i.e., the duration risk premium— in bond markets. If the total quantity of interest rate risk in the economy is …xed, why do shifts in MBS duration move bond prices? One natural answer, as in Gabaix, Krishnamurthy, and Vigneron (2007) is that, while this may be a wash from the perspective of some representative household (e.g., when the representative household is about to re…nance, she is taking more interest rate risk on the liability side but is exposed to less interest rate risk on her bond holdings), the change in the quantity of bond risk still looms large from the standpoint of a delegated investment manager who specializes in bond markets. If this manager is forced to invest a large fraction of her own wealth in the bond fund— e.g., to mitigate agency problems— then the manager will perceive an increase in the total quantity of risk. Thus, risks that are idiosyncratic from an aggregate perspective may be priced because of the large undiversi…ed exposures of specialized intermediaries who are the marginal buyers of bonds.3 It is crucial to distinguish here between the interest rate risk of MBS and MBS model risk. Interest rate risk derives from the fact that the value of MBS depends on the level of rates, assuming that household prepayment behavior is a known deterministic function of rates. By The household would invest the proceeds in short-term bills for 6 months. If households were to hedge in this way, their desire to short long-term bonds would perfectly o¤set the temporary decline in mortgage duration. 3 A second answer is to think about an overlapping generations setting where households borrow when young and lend when old. Since the young are bearing less interest rate risk when re…nancing expectations are low, this means that the old are bearing more risk which will impact equilibrium term premia. In this telling, time-varying risk-sharing between the old and young induces time-variation in bond risk premia. 2 5 contrast, model risk derives from the fact that household prepayment behavior is not a known function of interest rates. Instead, it is sometimes di¢ cult to predict how prepayments will respond to changes in rates. This paper focuses on the time-varying interest rate risk of the aggregate MBS market. The key insights are that household prepayment behavior is gradual and that households do not hedge their time-varying interest rate exposure. Thus, the total quantity of interest rate risk borne by professional bond investors can vary over time even if prepayment behavior is a deterministic function of the path of rates. As a result, the timevarying quantity of interest-rate risk emphasized here should be distinguished from MBS model risk as analyzed by Gabaix, Krishnamurthy, and Vigneron (2007). 2.2 Stylized model with short- and long-term bonds The main predictions can be illustrated using a simple dynamic model that features only shortand long-term bonds at each date t. The expected excess return on long-term bonds over shortterm bonds is the only endogenous variable at each date. The Internet Appendix extends the model to allow for multiple bond maturities. There is a set of risk-averse arbitrageurs (e.g., …xed income hedge funds, …xed income dealers, etc.) who price interest rate risk in bond markets. Assume these investors extend …xed-rate, prepayable mortgages to households and also own some noncallable bonds (e.g., Treasuries)— i.e., assume that the same arbitrageurs are marginal buyers in both MBS and Treasury markets. The key idea is that these arbitrageurs are specialists who are heavily exposed to risks speci…c to the bond market. The limited risk-bearing capacity of these investors ensures that, at least in the short term, demand curves for aggregate bond risk factors slope downward. In particular, the required return on long-term bonds must rise relative to short-term bonds to induce investors to bear more interest rate risk. I assume that, at each date t, arbitrageurs have mean-variance preferences over wealth at t + 1, with possibly time-varying risk aversion, t . This is the discrete-time analog of Vayanos and Vila (2009) who work in continuous time. The key idea is that arbitrageurs are concerned with their interim wealth and, hence, with interest rate risk— i.e., the risk that they will have to sell long-term bonds at a capital loss if interest rates rise tomorrow. Let rt+1 denote the log return on 1-period riskless bonds between t and t + 1. By de…nition (1) rt+1 = yt , i.e., the yield on 1-period bonds at time t. I assume an exogenous random process for the short rate. One can think of this as being pinned down by monetary policy or by a stochastic short-term storage technology that is in perfectly elastic supply. Thus, from the standpoint of (1) arbitrageurs at time t, the only relevant uncertainty is about rt+2 = yt+1 . The log return on (2) (2) (1) (2) 2-period bonds from t to t + 1 is rt+1 = 2yt yt+1 = 2yt rt+2 , so the excess return on 2-period (2) (2) (2) bonds relative to 1-period bonds is rxt+1 = rt+1 rt+1 = 2yt rt+1 rt+2 . Arbitrageurs choose 6 their long-term bond holdings bt to solve n o (2) (2) max bt Et [rxt+1 ] b2 ( t =2) V art [rxt+1 ] t bt (2) = max bt (2yt (1) bt rt+1 Et [rt+2 ]) b2 ( t =2) V art [rt+2 ] . t I assume that rt+1 and tV art [rt+2 ] follow exogenous stochastic processes. Thus, we have 1 (2) (2yt (2) bt (yt ) = ( t V art [rt+2 ]) rt+1 Et [rt+2 ]). (2) I model the MBS positive-feedback channel in a simple way. Let Qt denote the total dollar quantity of duration risk that needs to be held by …xed income arbitrageurs. In practical terms, we can identify Qt with the total number of “10-year Treasury duration equivalents” summed (2) (2) across all US …xed income markets. I allow Qt to depend on yt : I assume that Qt = Qt (yt ) (2) with Q0t (yt ) > 0 to capture the fact that MBS duration is increasing in long-term yields. As (2) discussed above, Q0t (yt ) > 0 is a property that holds locally due to time-variation in risk sharing between household mortgage borrowers and bond investors. And the vast size of the MBS market relative to the broader US bond market ensures that shifts in MBS duration have a signi…cant e¤ect on aggregate bond market duration. (2) (2) The equilibrium long-term yield must clear the market at each date, i.e., bt (yt ) = Qt (yt ), which delivers the following …xed-point condition: yt (2) = (rt+1 + Et [rt+2 ])=2 + ( t =2) V art [rt+2 ] Qt (yt ): (2) (3) (2) To ensure that this equilibrium is locally stable, I assume that 1 > ( t =2) V art [rt+2 ] Q0t (yt ): (2) This ensures that a small exogenous shock to yt leads to a larger increase in investor demand (2) (2) for bond duration than in duration supply (i.e., @bt (yt )=@yt > Q0t (y)). The expected excess return on long-term bonds relative to short-term bonds is E[rxt+1 ] = and the yield spread is yt (2) (2) tV art [rt+2 ] Qt (yt ), (2) (4) yt (1) = (Et [rt+2 ] rt+1 ) =2 + ( t =2) V art [rt+2 ] Qt (yt ): (2) (5) Naturally, in the limiting cases where arbitrageurs are risk neutral ( t = 0) or have to bear no (2) interest rate risk (V art [rt+2 ] Qt (yt ) = 0), the expectations hypothesis holds. 2.3 2.3.1 Predictions Forecasting excess bond returns The model predicts that measures of aggregate bond market duration, which derive most of their power from variation in MBS duration, should positively predict bond excess returns. 7 Proposition 1 Both the yield spread and aggregate bond market duration, Qt (yt ), will positively forecast excess bond returns. This remains true in a multivariate forecasting regression so long as t V art [rt+2 ] varies over time. Why isn’ the slope of the yield curve a su¢ cient statistic for forecasting excess bond returns? t In this simple model, the term spread contains information about both the expected path of future short rates and term premia, both of which vary over time. Duration measures pertain solely to the latter and, thus, can improve the forecasting power of regressions that include only the term spread. Formally, so long as t V art [rt+2 ] varies over time, both the yield spread and duration will positively predict excess returns in a multivariate forecasting regression. According to equation (2) (4), expected returns equal the product of the time-varying quantity of risk, Qt (yt ), and the time-varying price per unit of risk, t V art [rt+2 ]. Aggregate duration provides an accurate signal of the quantity of risk. Equation (5) shows that the yield spread contains information about both the quantity of risk and the price of risk, but it is a noisy indicator of expected returns because (2) it contains an expectations hypothesis component, (Et [rt+2 ] rt+1 ) =2. Thus, adding Qt (yt ) to a regression that contains the term spread can raise the forecasting power, particularly if the time-series variation in (Et [rt+2 ] rt+1 ) =2 is high. More generally, the aggregate supply of duration contains valuable information about term premia that cannot be easily recovered from the yield curve— e.g., duration may help to forecast excess bond returns even controlling for the term spread, the Cochrane Piassezi (2005) factor, or other simple yield-based proxies for term premia. Nonetheless, it is not my intention to argue that duration is an unspanned state variable in the sense of Du¤ee (2011) or Joslin, Priebsch, and Singleton (2013). In classic term structure approaches, if we know the true model, we can invert yields to obtain the full set of state variables. An unspanned state variable is a variable that is useful for forecasting returns but that has (almost) no impact on current yields and cannot be recovered in this way.4 Speci…cally, the Internet Appendix shows that I can nest all of my predictions in an a¢ ne term structure model similar to Vayanos and Vila (2009) in which MBS duration is a spanned state variable that impacts current yields. Thus, assuming a stationary data generating process, all the information about future bond returns would be contained in current yields, and duration would not add any further information. In practice, however, we do not know the true model generating bond yields. Furthermore, it seems likely that the true model evolves over time due to changes in the macroeconomy, the conduct of monetary policy, market structure, and the behavior of market participants. For instance, below I show that the impact of MBS duration on term premia and bond market spreads has grown over time, arguably because ‡ uctuations in duration have grown larger relative to investor risk tolerance. Thus, in practice, it is natural that duration is a useful summary statistic which contains additional information about future excess bond returns. As Du¤ee (2011) explains, this would arise if the evolution of short-rates under the risk-neutral (pricing) measure is independent of some variable— implying that it has no impact on current yields— even though that variable is relevant for forecasting future short rates under the objective measure. Such a situation could arise if a variable had o¤setting e¤ects on the evolution of future short rates and future term premia— e.g., some scary, bad news which raises future expected term premia but lowers future expected short rates. 4 (2) 8 2.3.2 Excess sensitivity of long-term rates to short rates The negative convexity of the US …xed income market naturally generates excess sensitivity of long-term yields to movements in short-term rates. To see this, consider the static case where y (2) = (r1 + E [r2 ]) =2 + ( =2) V ar [r2 ] Q y (2) and consider a change in r1 , holding …xed all other parameters including E [r2 ]. Since 1 > ( =2) V ar [r2 ] Q0 y (2) > 0, we have 1 P1 @y (2) ( =2) V ar [r2 ] Q0 y (2) = @r1 2 i=0 i = 1 1=2 1 > . 0 (y (2) ) ( =2) V ar [r2 ] Q 2 (6) This excess sensitivity stems from an “MBS duration spiral.” A small initial shock to the short rate of dr1 directly raises long-term yields by (1=2) dr1 due to the expectations hypothesis; the rise in yields extends the duration of MBS, which raises the term premium by (1=2) ( =2) V ar [r2 ] Q0 y (2) the resulting rise in yields further extends the duration of MBS, which further raises the term 2 premium by (1=2) ( =2) V ar [r2 ] Q0 y (2) dr1 , and so on.5 The same excess sensitivity point can be made in terms of forward rates. Working with forward rates allows for cleaner empirical tests since there is no direct expectations hypothesis term. The forward rate is f (2) = 2y (2) r1 = E [r2 ] + V ar [r2 ] Q( r1 + f (2) =2). Consider a change in r1 , holding …xed all other parameters including E [r2 ].6 We have @f (2) ( =2) V ar [r2 ] Q0 (y (2) ) = > 0: @r1 1 ( =2) V ar [r2 ] Q0 (y (2) ) (7) dr1 ; Proposition 2 If aggregate bond market duration is increasing in yields (Q0 y (2) > 0), longterm yields and forward rates will be excessively sensitive to movements in short-term rates. Furthermore, both long-term real and nominal rates will exhibit excess sensitivity. Thus, the model departs from Vayanos and Vila (2009) who assume that Q0 y (2) < 0— i.e., that arbitrageurs must hold a lower net-supply of long-term bonds when long-term yields are high. The assumption that Q0 y (2) < 0 implies that distant forwards underreact to movements in short rates. However, as discussed further below, the bulk of the empirical evidence suggests that distant forwards overreact, not underreact, to movements in short rates.7 5 Changes in expected future short rates will also lower risk premia in the model: @y (2) =@E [r2 ] = @y (2) =@r1 > 1=2— i.e., the MBS convexity e¤ect suggests that FOMC “forward guidance”may impact term premia. Intuitively, the ampli…cation mechanism is the same for either an independent change in r1 or E [r2 ]. 6 By considering su¢ ciently distant forwards, it is reasonable to assume that expected short rates in the distant future are insensitive to current short rates— i.e., that @E [r2 ] =@r1 0. Of course, even if @E [r2 ] =@r1 6= 0, we have @f (2) =@r1 > @E [r2 ] =@r1 , so the excess sensitivity result still carries through. 7 Vayanos and Vila (2009) assume that Q0 y (2) < 0 because it enables their baseline model— featuring an stochastic short rate, no independent supply shocks, and constant arbitrageur risk tolerance— to explain why the term spread forecasts excess bond returns. The intuition is that a rise in short rates— which directly raises long yields and lowers the term spread due to the expectations hypothesis— reduces the amount of long bonds that arbitrageurs must hold and, thus, the term premium. Thus, to jointly explain the forecasting power of the term 0 (2) spread and excess sensitivity, one may need to entertain a more complicated model where Qt (yt ) > 0 and, in addition to short rate shocks, there are independent shocks to the supply of duration or arbitrageur risk tolerance. 9 The model also generates a novel prediction about how the excess sensitivity of long-term forwards varies over time. Proposition 3 The excess sensitivity of long-term rates to short rates is more pronounced when the …xed income market is more negatively convex (@ 2 y (2) =@r1 @Q0 y (2) > 0). 2.3.3 Excess volatility of long-term interest rates The model also has implications for the volatility of long-term yields. Suppose we are at time t (2) (2) and consider the long-term yield at time t+1: yt+1 = (rt+2 +Et+1 [rt+3 ])=2+( =2) V ar [rt+3 ] Q(yt+1 ). Assume the only uncertainty is about rt+2 + Et+1 [rt+3 ], so that yt+1 Thus, we have V art [yt+1 ] (2) (2) Et [yt+1 ] (2) @y (2) (Et+1 j (2) @r1 y(2) =Et [yt+1 ] Et ) (rt+2 + rt+3 ) . (8) 1=2 1 ( =2) V ar [r2 ] Q0 (Et [yt+1 ]) (2) !2 V art [rt+2 + Et+1 [rt+3 ]] . (2) (9) Proposition 4 All else equal, long-term interest rate volatility, V art [yt+1 ], is increasing in the (2) negative convexity of the …xed income market, Q0 (Et [yt+1 ]). Excess volatility of long-term rates is a natural corollary of excess sensitivity. However, the negative convexity of the MBS market acts as a positive-feedback loop that ampli…es the e¤ects of any primitive shock— e.g., short-term rate, investor risk tolerance, or bond supply— that moves bond yields. Thus, the growth of the MBS market may have led to a secular increase in excess volatility. However, there is an additional time-series prediction: long rates should be particularly volatile at times when the MBS market is most negatively convex. Perli and Sack (2003) and Duarte (2008) develop a similar hypothesis and …nd support for it in recent US data. 2.3.4 Allowing for multiple bond maturities In the Internet Appendix, I extend the model with only short- and long-term bonds to allow for multiple bond maturities. The extension is a discrete time version of the no-arbitrage term structure model developed by Vayanos and Vila (2009) and Greenwood and Vayanos (2013). I add the MBS convexity e¤ect to this model in a simple fashion: I assume that the aggregate supply of duration which must be held by arbitrageurs rises when interest rates rise. However, I assume that these duration shocks dissipate quickly— i.e., duration shocks are transitory and are expected to mean-revert over time. These assumptions are a simple way to capture the dynamics of MBS duration explained in the paper. Solving the model, I obtain a discrete-time a¢ ne model of the term structure with two state variables, the current short rate and the current level of MBS duration, which depends on the past path of interest rates. The model generates the following predictions: 10 1. An MBS duration shock raises the current expected excess returns on long-term bonds over short-term bonds. Furthermore, since MBS duration is expected to quickly revert to its long-run mean, this e¤ect is short-lived in expectation. 2. A stronger MBS convexity e¤ect (i.e., a more negatively convex MBS universe) increases the sensitivity of long-term yields and forward rates to movements in short rates. 3. A stronger MBS convexity e¤ect increases the volatility of long-term yields and forwards. These three results echo those from the simple model presented above. However, the multiple maturity extension delivers several additional predictions that I can take to the data: 4. An MBS duration shock has a larger e¤ect on the expected excess returns on longer-term bonds than on those on intermediate-term bonds. This is a natural consequence of the fact that an MBS duration shock raises the current duration risk premium: the expected returns on long duration bonds move more than those on intermediate duration bonds. 5. However, since they are expected to quickly dissipate, shocks to MBS duration have a humped-shaped e¤ect on the yield curve and the forward rate curve— i.e., a shock to MBS duration increases the curvature of the yield curve. As in Greenwood and Vayanos (2013), the e¤ects of a supply shock on yields— which equals the e¤ect on the bond’ average s returns over its lifetime— will be greater for intermediate-term bonds than for long-term bonds when the supply shock is transient. These two predictions mean that it is impossible to infer whether there was, say, decline in the duration risk premium simply by asking whether the yield curve ‡ attened as is advocated by Krishnamurthy and Vissing-Jorgensen (2011), among others. And, notably, the last prediction suggests that transient shocks to MBS duration may account for some of the predictive power of the tent-shaped combination of forward rates identi…ed by Cochrane and Piassezi (2005). 2.3.5 The role of delta-hedging: “stock” versus “‡ ow” e¤ects One important simpli…cation in the current model is that there is only one class of investors who own MBS and price interest rate risk. In practice, there are two sets of intermediaries who own MBS. One set of intermediaries “delta-hedge”the embedded prepayment option and, thus, bear a constant amount of interest rate risk over time. Other investors do not delta-hedge and, thus, bear a time-varying amount of risk.8 Does it matter whether some MBS holders delta-hedge the prepayment option? In principle, the answer is “no”because the relevant hedging ‡ ows correspond one-for-one with changes in the aggregate quantity of duration risk. To see this, consider a modi…cation of the model feauturing a set of banks who extend all mortgages to households. Assume banks do not bear interest 8 Conversations with market participants suggest that the GSEs and commercial banks have historically been the most prominent delta-hedgers of MBS. 11 rate risk and instead delta-hedge their interest rate exposure. Speci…cally, as the prepayment option moves into-the-money, the banks buy noncallable bonds (e.g., Treasuries) to keep the total duration of their assets …xed (and equal to that of their liabilities). Finally, assume that the remaining supply of Treasuries is held by risk-averse bond arbitrageurs, and continue to assume that households do not hedge their time-varying interest rate exposure. Suppose there is a sudden drop in long rates that raises mortgage re…nancing expectations. From the perspective of banks, the existing mortgages now behave like short-term bonds. As a result, banks will signi…cantly reduce their delta hedges. That is, the banks will buy more longterm Treasuries— …nanced by selling some short-term T-bills— in order to maintain their previous asset duration. When borrowers actually re…nance, banks will unwind these transactions. Thus, banks have a temporarily elevated hedging demand for longer-term bonds. As a result, there is a temporary reduction in the amount of interest rate risk that bond arbitrageurs must bear, so the expected excess return on long-term bonds must fall in equilibrium.9 Clearly, the delta-hedging banks do not play an important role in this story. Hedging ‡ ows in the model with banks correspond one-for-one to changes in the aggregate quantity of duration risk born by investors in the model without banks. As a result, equilibrium bond prices in the modi…ed model will be the same as in the simpler model that omitted banks.10 3 3.1 Empirical analysis Measures of MBS duration and convexity The aggregate duration of the MBS market is not a simple function of the current mortgage rate. MBS duration also depends critically on the distribution of outstanding mortgage coupons and, in this way, re‡ ects the past path of mortgage rates. Furthermore, there are times when aggregate re…nancing and thus MBS duration are more or less sensitive to changes in long yields. Fortunately, the major brokerage …rms publish widely-followed estimates of the e¤ective duration of the US MBS market. Duration is the semi-elasticity of price with respect to yield— i.e., DU R = P 0 (y) =P is the percentage change in price for a small change in yield— and is the most widely used measure of interest rate risk. Bond convexity measures the curvature of the price-yield relationship— i.e., CON V = (1=2) P 00 (y) =P . Thus, the percentage change in price Instead of dynamically delta-hedging the interest rate exposure of MBS, investors can also statically hedge their exposures by purchasing interest rate options. Regardless of whether MBS investors pursue a dynamic or a static hedging strategy, other investors must take the other side of these trades. Thus, in the aggregate, …xed income investors must bear a time-varying amount of interest rate risk which impacts equilibrium term premia. 10 There are reasons to think that extent of hedging may play a non-trivial role in practice. The MBS market may not be perfectly integrated with the broader bond market, and the extent of hedging may re‡ the degree ect of integration. Suppose there are two types of MBS investors: (i) delta-hedging intermediaries (e.g., the GSEs and banks) who maintain a constant asset duration and (ii) “sleepers” who inelastically buy a …xed quantity of MBS irrespective of the duration risk they are taking (e.g., foreign o¢ cial holders). The duration that must be (2) absorbed by arbitrageurs is t QM BS (yt ). Thus, the fraction of MBS held by hedgers— an indicator of the degree t of market integration— will determine the extent to which shifts in MBS duration impact term premia. 9 12 following a y parallel shift in the yield curve is % ( P=P ) DU R y + CON V ( y)2 . (10) The duration of a negatively convex bond rises with yields. By contrast, the duration of a positively convex bond falls with yields. For callable bonds such as MBS, P (y) is calculated using a stochastic term structure model and a prepayment model that forecasts MBS cash ‡ ows as a function of future rates. Using the riskless yield curve and parameters governing interest rate volatility as inputs, one simulates a large number of scenarios for the future path of rates. One computes the expected cash ‡ ows in each scenario using the mortgage prepayment model and then discounts these expected cash ‡ ows using the implied zero-coupon curve in that scenario. The model-implied price is computed by taking the probability-weighted average of the discounted cash ‡ ows in each scenario.11 In the baseline results, I use estimates of MBS duration from Barclays Capital which are available from Datastream beginning in 1989. Aside from data availability, starting the analysis in the late 1980s is sensible since the MBS market only rose to prominence in the mid-1980s. The Barclays (formerly Lehman Brothers) Fixed Income Indices are the most widely followed set of bond indices in the US. However, as shown in the Internet Appendix, I obtain nearly identical results using the duration of Bank of America’ (formerly Merrill Lynch’ US Mortgage s s) Master index, which is available starting in 1991. The correlation between Barclays’and Bank of America’ MBS duration measures is 0:79. s The Barclays US MBS index covers mortgage-backed pass-through securities guaranteed by Ginnie Mae (GNMA), Fannie Mae (FNMA), and Freddie Mac (FHLMC), collectively known as “US Agency MBS.”The index is comprised of pass-through securities backed by “conventional” …xed-rate mortgages. The MBS index does not include non-Agency or private-label MBS (e.g., MBS backed by Jumbo, Alt-A, or subprime mortgages).12 I examine three related measures of duration: AGG DU Rt is the e¤ective duration of the Barclays Aggregate Index and measures the percentage change in the US bond market value following a parallel shift in the Treasury yield curve. The Aggregate Index is a proxy for the broad US …xed income market and includes Treasuries, Agency debentures, Agency MBS, investment grade corporates, and some ABS. Many bond portfolios are benchmarked relative to the Aggregate Index. M DU Rt BS is the e¤ective duration of the Barclays MBS Index and measures the percentage change in Agency MBS market value following a shift in the yield curve. To ensure the model-implied price equals the market price, analysts plug their discount function using an option-adjusted-spread (OAS)— the constant spread one must add to the riskless curve in all scenarios so the model-implied price equals the market price. E¤ective duration and convexity are computed at the current OAS. 12 Conventional mortgages satisfy several size, FICO, and loan-to-value requirements. For instance, mortgages in GSE MBS are subject to the conforming loan size limit set by Congress, typically have FICO scores over 620, and have a maximum LTV of 80%. The MBS Index is formed by grouping MBS pools into generic pools based on agency, program (e.g., 30-year or 15-year), mortgage coupon, and origination year. A generic pool is included in the Index if it has a contractual maturity greater than 1-year and more than $250 million outstanding. 11 13 M DU R_CN T RBt BS M VtM BS =M VtAGG gregate Index duration M DU Rt BS is the contribution of MBS to Ag- Duration contribution of MBS: M DU R_CN T RBt BS AGG DU Rt = Duration contribution of other sectors: (11) this way captures the fact that shifts in MBS duration have had a growScaling ing impact on aggregate bond market duration due to the growth of the MBS market. M DU R_CN T RBt BS proxies for the transient component of aggregate bond market duration due to MBS and constitutes my preferred forecasting variable. M DU Rt BS }| { z M BS M Vt M DU Rt BS AGG M Vt + z OT DU R_CN T RBt H }| { M BS M Vt OT 1 DU Rt H AGG M Vt . 3.2 Understanding MBS duration Before proceeding with the analysis, I explain several facts about MBS duration that play a key role in the story. First, due to the embedded re…nancing option, the average duration of MBS is low. Second, MBS duration rises as interest rates rise. This “negative convexity” property underlies the positive-feedback dynamic emphasized in the paper. Third, shocks to the MBS duration are transient, having a half-life of roughly 5:5 months. Finally, shifts in MBS duration are large relative to investors’ risk-bearing capacity. In combination, the last two properties mean that shocks to MBS duration generate signi…cant, short-lived shifts in bond risk premium. 3.2.1 Average MBS duration is low Panel A of Figure 1 plots my three duration measures over time. Table 1 presents summary statistics for the main variables in the paper. My monthly sample runs from January 1989 M to April 2011. DU Rt BS averages 3:35 years with a minimum of 0:58 years in May 2003 and a maximum of 4:83 years in May 1994. That is, the typical price-yield sensitivity of MBS is roughly equivalent to that of a 3:5 year, zero-coupon bond. The low average duration of MBS re‡ ects two factors. First, the self-amortizing nature of …xed-rate mortgages lowers their duration relative to non-amortizing bonds. Second, and more importantly, the prepayment option reduces the duration of MBS. A position in a callable bond is equivalent to a position in a similar noncallable bond plus a short position in an interest rate call option to repurchase the bond at par. Over the past 25 years the delta of the typical prepayment option has been substantial. The option is typically struck slightly out-of-the-money at origination, so the steady decline in rates since the mid-1980s means most prepayment options have remained near the money.13 13 The price of the callable bond is PC (y) = P (y) C (P (y)) where P (y) is the price of the underlying 14 3.2.2 Negative convexity of MBS MBS are typically negatively convex: their duration rises as interest rates rise due to a decline in expected prepayments. Empirically, a 100 bps increase in the 10-year Treasury yield has been M AGG associated with a 1:37 year increase in DU Rt BS and a 0:34 increase in DU Rt since 1989.14 MBS convexity has played a larger role in US bond markets over time for two reasons. First, the MBS market has grown signi…cantly. The MBS index grew from 25% of the Aggregate index in 1989 to 40% in 2008 but has fallen to 30% today due to the recent surge in Treasury borrowing. Second, advances in information technology and heightened competition in mortgage banking have reduced the costs of re…nancing. As a result, re…nancing now responds more aggressively to changes in primary mortgage rates, so the MBS market has become more negatively convex. For instance, prior to 2000, a 100 bps increase in the 10-year yield was associated with a 1:09 year increase in the MBS duration. Since 2000, a 100 bps increase in the 10-year yield was associated with a 1:57 year increase in the MBS duration.15 Another way to illustrate the negative convexity of the MBS market is to examine aggregate re…nancing behavior. MBS duration is low when expected future re…nancing is high. And current re…nancing activity is a strong signal of re…nancing over the near-term. Panel C of Figure 1 plots the Mortgage Bankers’ Association Re…nancing Index, which re‡ ects the raw number of mortgage applications classi…ed as re…nancings, versus a measure of “aggregate re…nancing incentives” Speci…cally, I show the Hodrick-Prescott (1997) …ltered version of the log-re…nancing . index, essentially the percentage deviation of re…nancings from an estimated trend. A borrower has strong incentives to re…nance when the primary rate is below her mortgage coupon, so my measure of aggregate re…nancing incentives is the di¤erence between the average coupon of MBS in the Barclays index and the 30-year primary mortgage rate from Freddie Mac’ survey, s ct yM;t . As shown in Panel C, there is a strong positive relationship between the two series. The corresponding time-series regression has a R2 of 0:65 and suggests that a 100 bps increase in re…nancing incentives boosts aggregate re…nancing by 85% relative to trend. 3.2.3 Shocks to MBS duration are transient Shocks to the bond market duration are transient. Speci…cally, the 1-month autocorrelation AGG of the duration of the Aggregate Index, DU Rt , is 0:88. The transient nature of aggregate noncallable bond and C (P ) is the price of the interest rate call. Since 0 < C 0 (P ) < 1, we have D u ra tio n o f c a lla b le 0 PC (y) N o n c a lla b le d u ra tio n e q u iva le nts = = (1 (1 C 0 (P (y))) C (P (y))) 0 D u ra tio n o f n o n c a lla b le P 0 (y) 00 D u ra tio n o f n o n c a lla b le < 00 P 0 (y) ; (D u ra tio n o f n o n c a lla b le ) 2 C o nve x ity o f c a lla b le 00 PC N o n c a lla b le d u ra tio n e q u iva le nts C o nve x ity o f n o n c a lla b le C a ll o p tio n >0 (y) P (y) C (P (y)) (P (y)) 0 2 : 00 Since P 00 (y) > 0 and C 00 (P ) > 0, the sign of PC (y) is ambiguous. However, because callable bonds become more 00 negatively convex as rates decline, we have PC (y) < 0 unless the prepayment option is deeply out-of-the-money. (10) 14 M Estimating DU Rt BS = a + b yt + "t using monthly di¤erences, I …nd b = 1:37 (R2 = 20:29). 15 Beginning in 1997, Barclays’ estimates the e¤ective negative convexity of the MBS index. If I estimate (10) M DU Rt BS = a + b N CON VtM BS yt + "t , I obtain a = 0:02 and b = 0:97 (R2 = 0:75) precisely as one would expect. Furthermore, the estimated negative convexity has trended up over time. 15 duration dynamics is driven by the MBS component of the index. Speci…cally, the 1-month M autocorrelation of DU Rt BS is also 0:88, which implies that MBS duration shocks have a halflife of 5:5 (=ln (0:5) = ln (0:88)) months. By contrast, the duration of the non-MBS portion of the index is far more persistent with a 1-month autocorrelation of 0:96, implying a half-life for non-MBS duration shocks of 17:5 months. Why are shocks to MBS duration so transient? In the case of a downward shock to rates, the explanation is fairly mechanical. A decline in rates raises expected prepayments, thus lowering the amount of duration risk that holders of outstanding MBS face over the near term. As homeowners respond to this interest rate shock, prepayments rise. When a borrower prepays, she replaces a high coupon bond with a deep in-the-money prepayment option with a lower coupon bond with a slightly out-of-the-money prepayment option. Borrowers e¤ectively restrike the interest rate call options embedded in their mortgages. The net e¤ect of this re…nancing transaction raises the total quantity of duration that investors need to bear. However, because prepayments are gradual, there are temporary declines in aggregate MBS duration, which then predictably revert over time. By contrast, in the case of an upward shock to rates, we would not expect such rapid mean reversion in MBS duration. This is because upward shocks to rates are expected to slow re…nancing activity, so there is no mechanical restriking e¤ect. 3.2.4 Shifts in MBS duration are large relative to risk-bearing capacity Due to the vast size of the MBS market, shifts in MBS duration have a large e¤ect on the aggregate amount of interest rate risk born by investors. And shifts in MBS duration drive almost all high-frequency variation in aggregate bond market duration. This is shown in Panel B of Figure 1 which uses equation (11) to decompose 12-month changes in aggregate bond market duration into an MBS component and a non-MBS component. The MBS component accounts for AGG the vast bulk of the variation in aggregate duration. Indeed, the R2 from a regression of DU Rt M on DU R_CN T RBt BS is 0:61 in levels, 0:81 in 12-month changes, and 0:91 in 1-month changes. To get a sense of the dollar magnitudes, investors convert aggregate duration statistics into “10-year US Treasury equivalents.”If the market value of some asset class X is QX with e¤ective duration DX and the duration of a 10-year UST is DT 10 , then X represents QX (DX =DT 10 ) 10-year US Treasury equivalents. Panel D of Figure 1 plots the detrended component of MBS duration in 10-year Treasury equivalents (the trend is estimated using the Hodrick-Prescott (1997) …lter). I convert historical dollars to 2012 dollars using the CPI. The …gure shows that the resulting shifts in duration supply are massive. There are 26 sample months in which detrended MBS 10-year equivalents exceed +$500 billion or $500 billion. These are arguably very large shifts in the quantity of duration risk relative to investors’riskbearing capacity. We can compare the shifts in MBS duration with the Fed’ recent Quantitative s Easing (QE) operations. From 2008Q4 to 2010Q1 the Fed purchased $1; 250 billion MBS with an e¤ective duration of roughly 3 years. Since the duration of the 10-year Treasury note was roughly 8 years at this time, this amounted to a reduction of $469 billion ($469 = $1; 250 (3=8)) 10-year equivalents of duration that needed to be held by investors (assuming no further change 16 in interest rates). Thus, Panel D shows that numerous shifts in MBS duration since the late 1980s have been of a comparable scale to the Fed’ recent QE operations. Furthermore, these s shifts can occur quite rapidly— which is likely to be relevant if capital is slow-moving. 3.3 3.3.1 Forecasting excess bond returns Basic forecasting results I use data from Gürkaynak, Sack, and Wright (2007) on the nominal Treasury yield curve as updated regularly by the Federal Reserve Board. The log excess return on an n-year zero-coupon (n) (n) (n 1) (1) bonds is de…ned as rxt+1 = n yt (n 1) yt+1 yt . Panel A of Table 1 provides summary (n) (n) (1) statistics on 12-month excess returns (rxt+1 ) and yield spreads (yt yt ) for n = 5; 10, and 20. (m) I also summarize information on the instantaneous forward rates ft for m = 1; 2; 3; 4; 5. Table 2 uses the three bond duration measures to forecast excess returns on 10-year nominal US Treasuries. Speci…cally, Table 2 presents forecasting regressions of the form rxt+1 = a + b DU Rt + c0 xt + "t+1 : (10) (10) (12) For the sake of comparability with the recent literature, the regressions are estimated with monthly data, so each month I am forecasting excess returns over the following 12 months. To deal with the overlapping nature of returns, t-statistics are based on Newey-West (1987) standard errors allowing for serial correlation at up to 18 lags. I estimate these regressions with and without other forecasting variables identi…ed in the literature on bond risk premia. Speci…cally, I control for the term spread following Campbell and Shiller (1991) and the …rst …ve forward rates following Cochrane and Piazzesi (2005). I obtain similar results controlling for the 10-year forward rate spread following Fama and Bliss (1987) or if I control for longer-dated (e.g., 7, 10, and 20-year) forwards as in Cieslak and Povala (2011). AGG M Table 2 shows that Aggregate duration (DU Rt ), MBS duration (DU Rt BS ), and MBS duM ration contribution (DU R_CN T RBt BS ) are each very strong predictors of bond excess returns. MBS duration contribution is the strongest predictor, both in univariate speci…cations as well as in the multivariate regressions. Duration adds incremental forecasting power over and above (i) the term spread, (ii) the forward rate spread, and (iii) arbitrary combinations of forward rates. Speci…cally, as shown in Table 2, duration signi…cantly raises the R2 in these speci…cations, and the R2 s are already quite high. For instance, column (2) shows that using the …rst …ve forward rates as predictors delivers a forecasting R2 of 0:31. If we add MBS duration contribution to this multivariate speci…cation as in column (11), the forecasting R2 jumps to 0:46. Comparing columns (2) and (11), we see that the inclusion of duration measures roughly halves the absolute magnitude of the coe¢ cients on forward rates in the augmented CochranePiazzesi (2005) regression. I return to this observation below when I explain why MBS duration may help to account for some of the forecasting power of Cochrane and Piazzesi’ tent factor. s Figure 2 graphically illustrates the basic forecasting result. Speci…cally, Panel A shows a scatter plot of 12-month future excess returns versus MBS duration contribution. These plots 17 correspond to the multivariate estimates in column (10). Speci…cally, I plot the component of excess returns that is orthogonal to the term spread versus the component of duration that is orthogonal to the term spread. The strong positive relationship is evident in Panel A. Panel B plots 12-month future excess returns versus initial MBS duration contribution over time. The …gure shows that MBS duration adds signi…cant forecasting power because duration extensions in 1991, 1994, 1999, 2004, 2007, 2008, and 2011 were each followed by high bond excess returns. Conversely, duration contractions in 1993, 1995, 1998, 2002, 2003, and 2009 were followed by low excess returns. Thus, Figure 2 shows that the basic forecasting result is not driven by one subsample or by a handful of outlying events. Indeed, in untabulated regressions, I obtain strong results when I separately examine the 1989-1999 subsample or the 2000-2011 subsample. What is the economic magnitude of the estimated e¤ects? The coe¢ cient in column (10) M indicates that a 1 year increase in DU R_CN T RBt BS raises expected excess returns on 10year bonds by 14.629% over the following 12 months. A 1-standard deviation move in MBS duration contribution is 0:29 years, so this implies that a 1-standard deviation increase in duration raises expected excess returns by 4:20% = 14:629 0:29. Assuming this shift has no e¤ect on expected returns beyond 12 months (analysis below suggests this is a reasonable assumption), this corresponds to a rise in 10-year yields of 42 (= 420=10) bps today. Thus, the estimated e¤ects are highly economically signi…cant. However, given the size and speed of the MBS duration shocks relative to the scale of arbitrage capital, the e¤ects do not seem implausibly large. I return to this point below when I compare the implied price-impact from these regressions with estimates from the literature evaluating the Federal Reserve’ Quantitative Easing policies. s These forecasting regressions suggest a simple market-timing strategy for diversi…ed investors (e.g., endowments and pensions) who allocate capital between the bond market and other …nancial markets. Speci…cally, diversi…ed investors should overweight long-term bonds when MBS duration is high and underweight long-term bonds when MBS duration is low. Unlike the specialist bond arbitrageurs emphasized in the model, the changes in the aggregate quantity of interest rate risk perceived by these investors will be miniscule because bonds are only a small portion of their overall …nancial portfolios. Thus, such diversi…ed investors will perceive the resulting time-variation in expected returns as time– varying “alpha”as opposed to a time-varying bond risk premium. Naturally, the t-statistics from our forecasting regressions can be translated directly into a statement about the Sharpe ratio of strategy that buys (sells) long-term bonds to the extent M that DU R_CN T RBt BS is above (below) its unconditional average.16 Estimating a univariate regression with quarterly data yields a t-statistic of 2:91; implying a strategy Sharpe ratio of p 92 per quarter. Assuming that strategy returns are iid, this implies an annual 0:303 = 2:91 p Sharpe ratio of 0:606 = 4 0:303. Of course, the fact that this timing strategy appears to be so pro…table suggests that diversi…ed investors do not fully exploit it. This might be because various frictions combine to limit the speed at which arbitrage capital ‡ ows across markets as xt + "t+1 and consider returns on the strategy wt+1 = rt+1 (xt E [xt ]). We have q p 1 SR [wt+1 ] = [xt ] ["t+1 ] 2 [xt ] = t [bOLS ] T where t [bOLS ] = T 1 2 ["t+1 ] ( 2 [xt ]) is the population t-statistic for the OLS estimator of . 2 2 16 Suppose rt+1 = + p 18 in Du¢ e (2010). Or this might simply be because many diversi…ed investors are unaware of the high Sharpe ratio o¤ered by the strategy. 3.3.2 MBS duration accounts for all the forecasting power of aggregate duration Table 3 shows that all of the forecasting power of DU RAGG derives from the MBS component. OT M Speci…cally, I decompose DU RAGG into the sum of DU R_CN T RBt BS and DU R_CN T RBt H as in equation (11). As discussed above, the transient MBS component accounts for the vast majority of all high-frequency variation in aggregate bond market duration. Indeed, the 12-month OT M autocorrelations of DU RAGG , DU R_CN T RBt BS , and DU R_CN T RBt H are 0:16, 0:04, and 0:64. Thus, perhaps not surprisingly, the horseraces in columns (4), (8), and (12) of Table 3 show that all of the near-term forecasting power in aggregate bond market duration is attributable OT to MBS duration. A similar conclusion holds if, instead of DU R_CN T RBt H , I focus more narrowly on the duration of the Barclays Treasury Index. In summary, both the duration of the Aggregate Index and the MBS Index positively forecasts excess bond returns over the following 12 months. And the duration contribution of MBS— which scales MBS duration relative to the broader market— is the strongest forecaster. However, the measured duration of the Treasury and other non-MBS …xed income markets do not reliably forecast bond returns at the quarterly or annual frequencies considered here, at least not in my 1989-present sample. This is perhaps not surprising if one considers a preferred-habit model with slow-moving capital. In such a model, capital ‡ ows in response to persistent changes in bond risk premia which neutralizes some, but not all, of the e¤ects of persistent shocks to duration supply on expected returns. Thus, from a medium-frequency return forecasting perspective, it may be particularly useful to isolate the transient component of shocks to bond duration supply. And the preceding analysis suggests that MBS duration is an excellent proxy for this transient component of duration supply.17 3.3.3 Time signature of MBS duration e¤ects There are several reasons to expect variation in MBS duration to be associated with relatively high frequency variation in expected returns. First, if current MBS duration positively predicts bond excess returns over the following instant, we would not expect the current duration to reliably predict returns on 12-month forward basis since MBS duration is not itself very persistent. Second, there may be slow-moving capital e¤ects as in Du¢ e (2010) which implies that the shortrun demand curve for duration risk is more inelastic than the long-run demand curve. Speci…cally, suppose that, in the short-run, interest rate risk is priced by specialized bond market investors with a …xed risk tolerance. However, over longer horizons, diversi…ed investors may allocate …nancial capital between the bond market and other …nancial markets as in Du¢ e and Strulovici (2012). Thus, following a positive shock to duration supply, we might expect bond risk premia to Admittedly, since my measures of MBS duration re‡ forward-looking expectations of future MBS duration ect but my measures of Treasury duration are not forward-looking, this horserace between MBS and non-MBS duration is not a completely fair …ght. 17 19 jump. The anticipation of large future bond returns would draw in capital from other markets, raising the risk tolerance of bond investors and, thus, reducing the impact of supply on bond risk premia over time in the case of a permanent supply shock. To formally explore the time signature of my main forecasting result, I use MBS duration to forecast quarterly returns from one to eight quarters ahead. Figure 3 presents quarterly forecasting regressions of the form rxt+(j (10) 1)=4!t+j=4 M = a(j) + b(j) DU R_CN T RBt BS + c0(j) xt + "t+(j (10) 1)=4!t++j=4 (13) for j = 1; :::; 8. Figure 3 then plots the coe¢ cients b(j) versus the horizon j. This series of regressions provides a simple non-parametric way to trace out the “impulse response”of quarterly excess bond returns following a movement in MBS duration. Speci…cally, Panel A of Figure 3 plots the coe¢ cients b(j) from univariate regressions and Panel B plots the coe¢ cients from multivariate speci…cations that control for the term spread. Figure 3 shows that the forecasting power of bond duration is largely located in the following two quarters. The e¤ect decays meaningfully from three to four quarters out, and there is generally little predictive power beyond …ve quarters. To investigate the timing of these e¤ects more parametrically, I estimate …rst order Vector Autoregression (VAR) of including excess returns, MBS duration, and the term spread using quarterly data rxt+1=4 = a1 + b1 rxt + c1 DU Rt + d1 (yt yt+1=3 (10) (1) (10) (10) (10) (10) yt ) + "1;t+1=4 yt ) + "2;t+1=4 yt ) + "3;t+1=4 . (1) (1) (1) (14) DU Rt+1=4 = a2 + b2 rxt + c2 DU Rt + d2 (yt yt+1=4 = a3 + b3 rxt + c3 DU Rt + d3 (yt M M Using DU R_CN T RBt BS = M VtM BS =M VtAGG DU Rt BS as our measure of MBS duration, Figure 4 shows the resulting simple impulse response of quarterly excess bond returns to a shock to MBS duration. As expected, the e¤ect on excess bond returns from a shock to MBS duration is concentrated in the next 4 quarters. Furthermore, using a parametric VAR, paints a very similar picture of the timing of the e¤ects to the non-parametric approach show in Figure 3. 3.3.4 Results for multiple maturities If changes in DU Rt shift the duration risk premium in US bond markets, we would expect this to have a larger e¤ect on the expected returns of long-term bonds than on intermediate bonds. For instance, the coe¢ cient on DU Rt when forecasting 20-year excess bond returns should be larger than when forecasting 10-year excess bond returns. This is illustrated in the left-most plots in Figure 4 where I estimate rxt+1 = a(n) + b(n) DU Rt + "t+1 (n) (n) (15) 20 M for n = 2; ::::20. I then plot the coe¢ cients, b(n) , versus maturity n. I show results for DU Rt BS M in Panel A and DU R_CN T RBt BS in Panel B. As shown in Figure 4, the coe¢ cients b(n) are increasing in maturity n, consistent with the idea that shifts in MBS duration impact the duration risk premia in bond markets. How do shifts in MBS duration impact bond yields and the current shape of the yield curve? As explained above, due to their transitory nature, we would expect MBS duration supply shocks to have a hump-shaped e¤ect on the yield curve and forward rate curve. Although a transitory rise in the duration risk premium has the largest impact on the price and current expected excess returns of long-term bonds, it has the largest impact on the yield of intermediate term bonds. As in Greenwood and Vayanos (2013), the intuition is that the impact bond yields equals the e¤ect on a bond’ average returns over its lifetime. As a result, a temporary rise in duration s risk premium has a greater impact intermediate-term yields than on long-term yields when the supply shock is expected to be short-lived. These results are shown in the middle and right-most plots in Figure 4. Speci…cally, I plot the slope coe¢ cients versus maturity from estimating yt and ft (n) yt+1 = a(n) + b(n) DU Rt + "t+1 (n) (n) (n) ft+1 = a(n) + b(n) DU Rt + "t+1 (n) (n) for n = 0; 1; 2; :::; 20. In other words, I use DU Rt to forecast the decline in yields and forwards over the following 12-months for each maturity. Figure 4 shows that high level of current MBS duration has a humped shaped e¤ect on the yield and forward rate curves. Speci…cally, the expected decline in rates over the following 12-months is largest for intermediate yields and forwards, having its maximal e¤ect on 2 year yields. This suggests that transient shocks to MBS duration may account for some of the predictive power of the tent-shaped combination of forward rates identi…ed by Cochrane and Piassezi (2005). The Cochrane-Piazzesi (2005) factor picks up time-variation in the curvature of the yield curve and is useful for forecasting transitory variation in bond returns— i.e., variation at frequencies higher than a standard business cycle frequency. Thus, the variation in MBS duration may help explain why a tent-shaped combination of forwards can explain transitory variation in bond risk premia. Of course, any fast-moving state variable that a¤ects bond risk-premia should have a hump-shaped e¤ect on yields and forwards as basic matter of no-arbitrage bond pricing logic. However, the literature has generally struggled to produce economically plausible state variables that might induce high-frequency variation in bond risk-premium. I argue that MBS duration is one such variable, and, thus the fact that it …ghts with the Cochrane-Piazzesi (2005) factor as noted above is quite natural. In summary, the …ndings for multiple maturities are highly consistent with the model I develop in the Internet Appendix and prior work by Vayanos and Vila (2009) and Greenwood and Vayanos (2013), who argue that shifts in the maturity structure of borrowing— holding 21 constant total dollar borrowing— can signi…cantly impact term premia. And this indicates that the Fed’ Quantitative Easing policies can potentially impact the duration risk premium through s a broad “portfolio balance channel,” consistent with the …ndings of Gagnon et. al. (2011) and Greenwood and Vayanos (2013). 3.3.5 Robustness checks Running a regression of returns on quantities does not, in general, allow one to cleanly identify a demand curve. Instead, to nail down the MBS duration supply channel, I would like to show that some component of MBS duration that is exogenous with respect to interest rates positively forecasts excess bond returns. Otherwise, one might worry that MBS duration is simply correlated with some omitted variable which is driving demand for bonds. Unfortunately, it is di¢ cult to construct a valid and powerful instrument for MBS duration. However, since the issues addressed here are of …rst order importance for our understanding of bond market dynamics and the determination of risk premia more generally, we should be willing to live with evidence that is admittedly somewhat indirect. Speci…cally, I am forced to rely on predictive regressions that provide indirect evidence consistent with the MBS hedging story. In the Internet Appendix, I address several concerns raised by this indirect approach. First, one might be concerned that variation in DU Rt does itself not drive the term premium but is simply correlated with an omitted variable that does. The Internet Appendix shows that the coe¢ cient on DU Rt is highly robust to the inclusion of additional controls that are thought to be associated with term premia. This should help to alleviate most natural concerns about omitted variable bias. Second, I address the standard econometric concerns that arise when estimating time-series forecasting regressions. These econometric issues are not a signi…cant concern in this context. Finally, one might worry that the results are somehow speci…c to the Barclays’indices I use throughout. The Internet Appendix shows that similar forecasting results obtain when using the Bank of America MBS indices instead of the Barclays’indices. 3.3.6 Comparison to prior estimates of the price impact of duration supply shocks I now compare the price-impact e¤ects implied by my regressions with estimates from the recent literature evaluating the Federal Reserve’ Large Scale Asset Purchase (LSAP) programs. To s draw this comparison, I assume that LSAP’ primarily work by impacting term premia following s Gagnon et al (2012) among others. To be clear, I am not arguing that mortgage re…nancing explains why LSAP policies do or do not work. However, both mortgage re…nancing waves and LSAPs are sources of duration supply shocks that have the potential to impact term premia. Thus, it is interesting to compare the price impact of supply shocks from LSAPs and mortgage re…nancing waves. Before proceeding, there are several reasons to expect di¤erent price-impact magnitudes for LSAP- and mortgage-re…nancing-induced duration shocks. First, it is likely that investor riskbearing capacity varies over time, so we should expect to …nd larger e¤ects for LSAPs to the extent they occurred when risk-bearing capacity was low. Further, the LSAP-induced shocks 22 would have been expected to last longer than mortgage duration shocks, suggesting that LSAPs should have a larger impact on yields due to their more persistent e¤ect on term premia. However, LSAPs were arguably far more “advertised” than mortgage duration shocks, suggesting that mortgage shocks might have a larger impact than LSAPs if, as in Du¢ e (2010), investors were more inattentive in the former case. Thus, it is not obvious whether we should expect to …nd larger e¤ects for MBS duration shocks or for LSAPs. Table 4 compares the estimated e¤ect of supply shocks measured in 10-year Treasury duration equivalents. Table 4 provides more detail on the LSAPs, the prior literature estimating their impact on 10-year yields, and the estimated e¤ects implied by the MBS duration e¤ect. I use my estimates to predict the e¤ect of LSAP announcements on 10-year Treasury yields. Speci…cally, (10) (10) (10) using speci…cations of the form yt yt+1 = a + b DU Rt + "t+1 as in Figure 4, I estimate M that a one standard deviation decline in DU Rt BS has lowered 10-year Treasury yields by 36 bps M over my 1989-present sample. Based on the relationship between DU Rt BS and MBS 10-year M equivalents in Figure 1, I estimate that a one standard deviation decline in DU Rt BS corresponds to a $503 billion reduction in 10-year equivalents in recent years.18 The …rst LSAP was announced in late 2008 and early 2009 amidst unusually strained market conditions (Krishnamurthy and Vissing-Jorgensen (2013)). Since LSAP1 represented a cumulative reduction in $750 billion 10-year equivalents, my estimates imply a 54 bps = 36 bps (750/503) decline in 10-year Treasury yields. This implied e¤ect is smaller than event-studybased estimates of the impact of LSAP1 on 10-year yields which average nearly 100 bps. However, this di¤erence makes sense since intermediary risk-bearing capacity was quite limited at the time of LSAP1. Table 4 shows that the implied e¤ect is slightly larger for univariate regressions using M DU R_CN T RBt BS or for multivariate regressions that control for the term spread. For LSAP2 and LSAP3, which were announced in late 2010 and late 2011 respectively, the e¤ects implied by my estimates are in line with the estimated LSAP announcement e¤ects in the prior literature. In summary, the price-impact of supply implied by my regressions is broadly in line with those documented in the recent event-study literature on LSAPs. To be clear, my results do not show that LSAPs have lowered the duration risk premium in bond markets or that they work at all.19 However, my evidence does provide an out-of-sample “proof of concept”for LSAP policies that seek to impact market-wide term premia. 3.3.7 A decomposition of aggregate MBS duration M I now decompose aggregate MBS duration. The decomposition shows that DU Rt BS re‡ ects both aggregate re…nancing incentives, prepayment burnout, and several other factors. I then M T I obtain this estimate by regressing the deviation of QM BS (DU Rt BS =DU Rt 10 ) from trend (in constant 2012 t M BS dollar) on DU Rt from 2008 to present. 19 Besides moving the duration risk premium, LSAP policies may impact yields throughout the bond market through a variety of distinct channels. This is a lively area of scholarly debate. See Krishnamurthy and VissingJorgensen (2011 and 2013). Furthermore, LSAPs may also have local e¤ects: comparing securities with similar durations, D’ Amico et al (2012) and Cahill et al (2013), …nd larger intra-day declines in yields on those securities that the Federal Reserve is actually purchasing. 18 23 show that both aggregate re…nancing incentives and these other terms appear to contribute to M the forecasting power of DU Rt BS . Let yM;t denote the primary mortgage rate, and take …rst-order expansion of the duration of mortgage class i (e.g., 6% FNMA 30-years originated 3 years ago) about its coupon cit : DU Rit = Dit CON Vit (yM;t cit ) + it , P P where it is the approximation error. Letting Xt = i wit Xit and t [Xit ; Yit ] = i wit (Xit Xt ) (Yit denote the market-weighted average and covariance, aggregate MBS duration is given by: DU Rt = Dt + CON Vt There are two main terms20 : Aggregate re…nancing incentives. The …rst term depends on the aggregate re…nancing incentive, (ct yM;t ). We can further decompose this term using deviations from time-series average of Dt and CON Vt Dt + CON Vt (ct yM;t ) = [D + CON V +[(Dt (ct yM;t )] CON V ) (ct yM;t )]: (ct yM;t ) + t Yt ) [CON Vit ; cit ] + t . D) + (CON Vt Since it is likely that Dt and CON Vt only vary slowly over time, this suggests that (ct yM;t ) should be a strong linear predictor of excess returns. Aggregate prepayment “burnout” t [CON Vit ; cit ] measures the extent to which low . callability is concentrated in high coupon MBS. Aggregate MBS duration is high when aggregate burnout is high. For instance, some analysts have argued that burnout has been elevated in recent years because borrowers who are still in higher coupon mortgages cannot re…nance due to negative home equity or tighter mortgage underwriting standards. This works largely as one might expect in the data. Speci…cally, there is a strong negative M relationship between DU Rt BS and (ct yM;t ): M DU Rt BS = 3:095 [18:22] 0:952 (ct [ 5:56] yM;t ) , R2 = 0:48. Adding a time trend (t) and interactions terms, I obtain M DU Rt BS = 3:610 [16:76] 0:835 (ct [ 5:18] yM;t ) 0:276 (ct [ 0:58] yM;t ) t 1:041 t, R2 = 0:58. [ 2:01] Thus, MBS duration has trended down by a little more than 1 year since 1989, and there is weak evidence that CON Vt has become slightly more negative over time. We can gain further insight into the residual t using a second-order Taylor expansion of DU Rit . This suggests that aggregate MBS duration is decreasing in the cross-sectional variation in mortgage outstanding coupons. 20 24 Consistent with the previous decomposition, columns 1-3 of Table 5 show that the aggregate re…nancing incentive, ct yM;t , is a strong negative predictor of excess bond returns. As above, my measure of aggregate re…nancing incentives is the weighted-average coupon on MBS in the Barclays Index minus the current 30-year primary mortgage rate from Freddie Mac. The forecasting power of this variable derives from comparing the primary mortgage rate (a variable that is almost a perfect linear combination of forwards) with the average coupon on outstanding MBS (a variable not spanned by forwards). Naturally, ct is itself a very slow-moving variable that re‡ ects historical path dependencies. Columns 7-9 of Table 4 show that I also obtain similar results if I forecast returns using a normalized version of the Mortgage Banking Association’ re…nancing s index, REF It . Speci…cally, the Hodrick-Prescott (1997) …ltered version of the log-re…nancing index is a strong negative predictor of returns. Thus, Table 5 shows that my main …ndings are not an artifact of using MBS duration— an admittedly complex model-based construct— and supports the broader hypothesis that aggregate mortgage re…nancing activity plays a key role in driving bond risk premia. Finally, columns 4-6 and 10-12 of Table 5 show forecasting horseraces between my preferred M measure of MBS duration (DU R_CN T RBt BS ) and re…nancing incentives (ct yM;t ) and re…nancing activity (REF It ), respectively. Consistent with the previous decomposition, these M horseraces suggest that much of the information that DU R_CN T RBt BS contains about future excess returns is also contained in ct yM;t (and REF It ). However, the horseraces suggest that model-based MBS duration estimates do contain additional information above and beyond these simpler measures. This is not surprising since MBS duration re‡ ects the forward-looking supM ply expectations of market participants. Furthermore, DU R_CN T RBt BS also captures e¤ects such as aggregate prepayment burnout, which are not re‡ ected in the simpler measures. 3.3.8 The growing impact of MBS duration on term premia Have shifts in MBS duration become a more or less important driver of term premia over time? My story suggests that these e¤ects should have grown more far important over time: the rapid growth of the MBS market means that these duration shocks have become far more signi…cant relative to the risk tolerance of bond investors. The signi…cant growth of the MBS market relative to the rest of the bond market is illustrated in Panel A of Figure 6. Fortunately, I can investigate the evolution of these e¤ects by examining data prior to 1989. Although I only have data on the model-based e¤ective duration of MBS beginning in 1989, Barclay’ publishes a cruder duration measure called Macaulay duration-to-worst dating back to s 21 1976. I use dur to denote the Macaulay analogs of the more commonly used modi…ed duration M measures (DU R).Thus, going back to 1976, I can ask whether durt BS and dur_cntrbM BS = t M M VtM BS =M VtAGG durt BS have predictive power for excess bond returns. While the modelbased e¤ective duration measures provide a more accurate measure of the sensitivity of MBS Macaulay duration-to-worst is the Macaulay duration corresponding to the yield-to-worst. Speci…cally, one assumes a deterministic set of future MBS cash‡ ows based on an assumed prepayment pro…le (typically the expected prepayment pro…le, which depends on current interest rates). The yield-to-worst and Macaulay durationto-worst are then simply the yield and Macauley duration assuming this set of future bond cash ‡ ows. 21 25 prices to changes in yields, Macaulay duration-to-worst contains similar information in the timeseries. These two measures are shown in Panel B of Figure 6. For instance, from 1989-present when both measures are available, the correlation between e¤ective duration and Macaulay duration-to-worst of MBS is 0:91 in levels and 0:91 in 12-month changes. I can also construct my measure of aggregate re…nancing incentives— i.e., the di¤erence between the average MBS coupon and current primary mortgage rates (ct yM;t ) going back to 1976. Since Panel A of Figure 6 shows that the MBS market grew signi…cantly relative to the rest of the bond market from 1976 to the early 1990s, examining the forecasting power of MBS duration over this early period is useful. Table 6 shows subsample forecasting results. I …nd that the three measures of MBS duration have no predictive power for excess bond returns over the 1976-1988 subsample, but have signi…cant predictive power from 1989-present (including both the 1989-1999 and 2000+ sub-samples). Figure 7 makes a similar point and shows the results from 60-month (20-quarter) rolling regressions of 3-month excess returns on MBS duration measures, controlling for the term spread. The results show an unmistakable upward trend in the predictive power of MBS duration. Over time, the coe¢ cients have grown and are being estimated more precisely, so the t-statistics have grown. The fact that these e¤ects have grown so much stronger over time, precisely as I would predict, provides further indirect evidence for my MBS hedging story. 3.3.9 Predicting foreign bond returns Do shifts in the duration of US mortgage backed securities impact term premia in foreign bond markets? This question is addressed in Table 7, which investigates whether the duration of US MBS forecasts the excess returns on long-term foreign bonds. What one should expect to …nd here depends on the degree of integration between the bond markets in di¤erent countries. At one extreme, if foreign bond markets were completely segmented from the US bond market, we would expect US MBS duration to have no forecasting power for foreign excess bond returns. At the other extreme, if there were a single integrated global bond market, we would expect term premia to move in lock-step across national bond markets. In between these two extremes, one would expect US MBS duration to have some predictive power for excess foreign bond returns, albeit less than for US excess bond returns. Following Campbell (1999, 2003), I work with IFS data and compute excess returns on coupon-bearing long-term foreign government bonds over short-term foreign bond using the Campbell, Shiller, and Schoenholtz (1983) approximation.22 I compute foreign government bond returns for ten developed countries: Belgium, Canada, Denmark, France, Germany, Italy, Japan, Sweden, Switzerland, and the UK. I then estimate univariate forecasting regressions of the form rxc;t+1 22 (n) (n) Dt yc;t (n) Dt = [1 (n) (Dt (n 1) 1)yc;t+1 (n) Yc;t ) n ]=[1 (1) yt F OR(n) = a + b DU Rt + "c;t+1 , (1) F OR(n) (16) (n) (n) yc;t (n) Yc;t ), This approximation says that log excess return on a n-year coupon bond over the short rate (yt ) is rxc;t+1 where (n) Yc;t ) (1 + (1 + (n) Yc;t 1 is the yield on a n-year coupon bond, = ln(1 + and ] is the approximate duration on an n-year coupon bond. 26 and multivariate regressions controlling for the US term spread and foreign country term spread rxc;t+1 F OR(n) = a + b DU Rt + c (yc;t U S(n) yt U S(1) ) + d (yc;t F OR(n) yt F OR(1) ) + "c;t+1 : F OR(n) (17) To assess the di¤erential forecasting power for foreign bond excess returns as compared to US F OR(n) U S(n) domestic bond returns, for each country I rerun the regressions with rxc;t+1 rxc;t+1 on the lefthand-side. I then report the coe¢ cient on DU Rt from this speci…cation as well as the associated M t-statistic, labeled as “Di¤ from USA” in the table. Panel A shows results for DU Rt BS and M Panel B shows results for DU R_CN T RBt BS . Table 7 shows that there is evidence of moderate cross-country spillovers. US MBS duration has some limited ability to forecast excess returns on long-term government bonds in Canada, Denmark, Japan, and the UK. However, in each case, the magnitude of the forecasting relationships for foreign bonds is much weaker, both economically and statistically, than the corresponding magnitude for US bonds. In short, the evidence is consistent with the view that the bond markets of, say, the US and the UK are partially integrated, so a duration supply shock in the US has some impact on bond risk premia in the UK. At the same time, the fact that the US return forecasting results are so much stronger than the foreign return forecasting results provides additional comfort that the results do not re‡ an omitted global risk factor. ect 3.4 Excess sensitivity of long-term yields and forwards to short rates The model suggests that distant forward rates should be excessively sensitive to movements in short-term interest rates (see Proposition 2 above). As shown in Hanson and Stein (2012), this general excess sensitivity result emerges very strongly in the data. This is consistent with Gurkaynak, Sack, and Swanson (2005) who …nd that distant forwards are excessively sensitive to macroeconomic announcements. And, as emphasized by Hanson and Stein (2012), one observes signi…cant excess sensitivity for distant real forward rates— i.e., forwards extracted from the TIPs yield curve. While this is puzzling from the standpoint of textbook macro-…nance theories, excess sensitivity follows naturally from models that emphasize supply and demand e¤ects in bonds markets. For instance, an extension in MBS duration increases the exposure of investors to movements in short-term real rates, necessitating a rise in the real term premium. Going further, the model predicts that distant forwards should be particularly sensitive to short rates when the MBS market is more negatively convex (see Proposition 3 above). When the market is more negatively convex, a given change in short rates has a larger e¤ect on the total quantity of interest rate risk and, hence, a larger impact on distant forward term premia. To test this prediction, I examine whether the high-frequency sensitivity of distant forwards varies with MBS convexity. In Table 8, I estimate daily regressions of the form ft X(10) =a+b yt $(2) + c N CON VtM BS + d ( yt $(2) N CON VtM BS ) + "t X(10) , (18) where N CON VtM BS = CON VtM BS (i.e., negative one times the convexity of the MBS index) for X = $ and T IP S— i.e., I examine changes in both 10-year nominal and 10-year real forwards. 27 I estimate these regressions from 1999-present to include data on real forwards from Gürkaynak, Sack, and Wright (2010).23 The theory predicts that d > 0: excess sensitivity should be more pronounced when the MBS market is more negatively convex. Consistent with this prediction, Table 8 shows that both distant nominal and real forwards are more sensitive to movements in short-term nominal rates when the MBS market is more negatively convex. Since N CON VtM BS is quite persistent at daily frequencies, t-statistics are based on Newey-West (1987) standard errors allowing for serial correlation up to 20 business days. From 1999-present, N CON VtM BS has averaged 1:528 with a standard deviation of 0:531. Thus, on average, a 100 bps increase in the 2-year nominal yield has been associated with a 54 bps (=0:322 + 0:140 1:528) increase in 10-year nominal forwards and a 30 bps increase in 10year real forwards. However, when N CON VtM BS is two standard deviations above average, the same 100 bps increase in the nominal short rates has been associated with a 68 bps increase in 10-year nominal forwards and a 42 bps increase in 10-year real forwards. Thus, variation in MBS convexity has been associated with meaningful variation in the sensitivity of distant forwards to movements in short rates. Furthermore, this is true whether (i) I look at N CON VtM BS or the scaled version of this variable, N CON VtM BS M VtM BS =M VtAGG or (ii) I examine sensitivity $(2) to movements in short-term nominal yields or short-term forwards (i.e., if I replace yt with $(2) ft in equation (18)). 3.5 Excess volatility of long-term interest rates The model suggests that, all else being equal, long-term interest rate volatility will be elevated when the MBS market is more negatively convex. Intuitively, the positive-feedback dynamic underlying MBS duration spirals is stronger when the mortgage market is on a re…nancing cli¤— i.e., when a small movement in rates will signi…cantly impact aggregate re…nancing behavior. To test this prediction, I follow the approach of Perli and Sack (2003), who relate weekly observations of option-implied interest rate variance to measures of MBS convexity. Speci…cally, p Perli and Sack (2003) assume that yt = 1 + xt "t where 2 follows an AR(1) process ";t 2 = 0 + 1 2 1 + ut , so that ";t ";t 2 y;t = (1 + xt ) 2 ";t = 0 (1 + xt ) + 1 1 + xt 1 + xt 1 2 y;t 1 + (1 + xt ) ut . There are three e¤ects: a volatility of volatility (1 + xt ) = (1 + xt 1 ) ity e¤ect is ignorable in a level e¤ect 0 (1 + xt ), a persistence e¤ect (the second term), and e¤ect (the third term). However, if xt is fairly persistent we will have 1. Furthermore, Perli and Sack (2003) …nd that the volatility of volatilpractice. Thus, using xt = N CON Vt , I focus solely on the level e¤ect 2 y;t = 0 (1 + N CON Vt ) + 1 2 y;t 1 + ut : 23 The convexity of Barclays MBS index is available for 1997-present. I obtain similar results for nominal forwards for this period. 28 Of course, there is a simultaneity problem here since N CON Vt is a function of 2 y;t . To deal with this, I substitute N CON Vt 1 for N CON Vt in the above. Following Perli and Sack (2003), I measure 2 y;t using the implied yield variance (the square of implied volatility) from 3-month by 10-year swaptions— i.e., the option to enter into a 10year swap at pre-speci…ed rate in 3 months. I …rst regress the implied variance from 3-month by 10-year swaptions on the implied variance from 2-year by 10-year swaptions and examine the residuals from this regression. As Perli and Sack (2003) argue, it is desirable to strip out structural/cyclical ‡ uctuations in interest rate volatility in this way because MBS convexity is expected to have fairly transient e¤ects on implied volatility. Using the resulting residualized measure, e2 y;t , I then estimate Data on MBS convexity is available beginning in 1997, so I estimate speci…cation (19) using weekly data from 1997-present. t-statistics are based on Newey-West (1987) standard errors allowing for serial correlation at a lag of up to 12 weeks. The results are shown in Table 9 and indicate that option-implied yield variance is higher when the MBS market is more negatively convex. This conclusion holds whether I look at N CON VtM BS or the scaled version of this variable, N CON VtM BS M VtM BS =M VtAGG , and when I control for lagged swap yields in equation (19). The estimates in column (2) imply that a one standard deviation increase in N CON V raises implied yield volatility by 0:10% p (= 0:020 0:519). The standard deviation of e y;t 1 is 0:48%, suggesting that movements in MBS convexity are associated with economically signi…cant shifts in interest rate volatility. e2 y;t 1 = 0 (1 + N CON Vt 1 ) + 2 1 e y;t 1 + ut . (19) 3.6 Impact of MBS duration on corporate and swap spreads In practice, MBS investors hedge the interest rate exposure of MBS using either Treasuries or interest rate swaps. Regardless of the speci…c hedging instrument, other investors must take the other side of these trades. Thus, in the aggregate, …xed income investors must bear a timevarying amount of interest rate risk, which impacts equilibrium term premia. In a simple model in which Treasuries and swaps are perfect substitutes, one would not expect these hedging ‡ ows to impact spreads between swaps and Treasuries. But di¤erent long-term …xed income assets with the same duration are not perfect substitutes. Once we relax the assumption of perfect substitutability, hedging ‡ ows triggered by shifts in MBS duration may impact yield spreads between duration-matched …xed income assets and give rise to predictable variation in the excess returns on various …xed income assets over duration-matched government bonds. In the Internet Appendix, I consider an stylized model with three long-term …xed income assets: government bonds, interest rate swaps, and high grade corporate bonds. I suppose that some fraction of MBS duration is hedged using interest rate swaps and the remainder is hedged using government bonds. While all long-term …xed income assets are exposed to interest rate risk, there are components of returns that are speci…c to government bonds, swaps, and corporate 29 bonds. For corporate bonds, this might re‡ changes in credit risk. And, for all three assets, we ect can think of these components as re‡ ecting shifts in supply and demand for speci…c assets. For instance, in the case of government bonds, this might re‡ idiosyncratic shifts in the demand ect as in a “‡ ight to quality” episode (e.g., Du¤ee (1996), Longsta¤ (2004), etc.) or idiosyncratic shifts in supply (e.g., Greenwood and Vayanos (2010) and Lou, Yan, and Zhang (2013)). In my stylized model, the expected excess returns on any long-term …xed income asset over short-term bonds is equal to a term premium earned by all long-term …xed income assets plus a risk premium that is speci…c to that asset. In particular, since long-term government bonds and swaps are both exposed to movements in the general level of interest rates, the impact of MBS duration on the overall term premium is independent of the fraction of MBS hedged with government bonds and swaps— just as in the simpler model developed above. However, MBS hedging ‡ ows now have an impact on the spread between long-term …xed income assets. For instance, the government-speci…c risk premium is high when MBS duration is high: since MBS investors hedge with government bonds, the government-speci…c risk premium must rise to induce arbitrageurs to hold more Treasuries. Since these hedging ‡ ows have no impact on the corporate-speci…c risk premium, they result in tighter than normal spreads between long-term corporate and government bonds. Furthermore, if a su¢ ciently large volume of MBS hedging takes place in swap markets, and the idiosyncratic movements in swaps are large relative to the idiosyncratic movements in Treasuries, we would also expect an increase in MBS duration to raise the spread between swaps and Treasuries. In summary, the analysis predicts that a high level of MBS duration should: (i) be associated with narrow current spreads between corporate bonds and duration-matched Treasuries; (ii) predict a future widening of spreads between corporate bonds and duration-matched Treasuries; and (iii) predict that the future returns on corporate bonds will underperform those on durationmatched Treasuries. And since swaps are used to hedge MBS, each of these predictions should be reversed for swaps. I …nd evidence consistent with each of these additional predictions in Table 10. In Panel A, I report regressions of the form SP READt = a1 + b1 M DU R_CN T RBt BS + c01 xt + vt : (20) Speci…cally, I regress 3-month changes in spreads on the contemporaneous change in MBS duration contribution and controls. I …nd that increases in MBS duration are associated with tighter spreads between corporate bonds and Treasuries and with wider swap spreads. This holds in univariate regressions as well as in multivariate regressions that control for changes in the term spread, the VIX, and the past returns on the stock market, which Collin-Dufresne, Goldstein, and Martin (2001) show are useful for explaining changes in corporate credit spreads. Spreads are in percentage points, so the coe¢ cient of 0:43 in column (1) suggests that a 1-year rise in M DU R_CN T RBt BS is associated with a 43 bps decline in investment grade spreads. Based on M the univariate estimates, a one standard deviation change in DU R_CN T RBt BS is associated with a 11 bps decline in all investment grade corporate spreads, a 4 bps decline in Aaa spreads, 30 a 19 bps decline in Baa spreads, and a 3 bps rise in 10-year swap spreads. In Panel B, I use the level of MBS duration to forecast spread changes over the next 12-months M SP READt+1 = a2 + b2 DU R_CN T RBt BS + c02 xt + ut+1 : (21) I …nd that a high level of MBS duration predicts that corporate spreads will widen over the next 12-months and that swap spread spreads will tighten. Again, this holds true in both univariate forecasting regressions as well as in multivariate regressions that control for the initial level of spreads and other conditioning variables. Finally, I forecast the excess returns on corporate bonds and swaps over duration-matched Treasuries over the following 12-months rxDU R t+1 M AT CH M = a3 + b3 DU R_CN T RBt BS + c03 xt + "t+1 : (22) By construction, these regressions provide almost identical information to those in Panel B. Speci…cally, a high level of MBS duration predicts that the returns on long-term corporate bonds will underperform those on duration-matched Treasuries, while swaps are expected to outperform duration-matched Treasuries. Finally, Perli and Sack (2003), Wooldridge (2001), and Reinhart and Sack (2000), argue that many MBS investors switched from hedging with Treasuries to hedging with interest rate swaps beginning in the late 1990s. Several episodes, including the ‡ ight-to-quality dislocations in the fall of 1998 and the jump in the convenience premium on Treasuries following the Treasury buyback announcement in early 2000, convinced many MBS investors that hedging with Treasuries exposed them to far greater basis risk than hedging with swaps. Interestingly, the time-series evidence is consistent with such a shift in hedging activity: in untabulated results, I …nd that the tendency for swap spreads to widen when MBS duration rises became signi…cantly more pronounced after 1997. In summary, the evidence is consistent with previous work, including Cortes (2003 and 2006) and Feldhutter and Lando (2008), which argues that MBS hedging plays a signi…cant role in explaining the level of US swap spreads. And this spread-based evidence supports the broader argument of this paper: shifts in MBS duration and the associated hedging ‡ ows function as large-scale supply shocks that have signi…cant e¤ects on bond market pricing. 4 Conclusion Changes in the e¤ective duration of mortgage-backed securities (MBS) function as large-scale duration supply shocks in US bond markets. As a result, bond term premia are high when aggregate MBS duration is high; and changes in MBS duration capture variation in bond risk premia that is not re‡ ected in traditional forecasting variables. The fact that these recurring and transient supply shocks have large e¤ects on bond prices highlights the critical role of limited and slow-moving arbitrage capital, even in markets as deep and liquid as the US bond market. 31 The negative convexity of MBS— the fact that MBS duration rises when interest rates rise— generates a positive feedback loop that helps explain the excess sensitivity of long rates to short rates and excess volatility of long rates. MBS convexity has the potential to amplify a variety of shocks within US bond markets. For instance, one might expect the e¤ective risk tolerance of …xed income arbitrageurs to decline following losses. If arbitrageurs are long duration and lose money when rates rise, negative MBS convexity could add another positive feedback loop to US bond markets. Speci…cally, a decline in arbitrageur risk tolerance would lead to a rise in yields; this would cause the duration of MBS to extend, raising required returns and causing yields to rise further; the resulting losses would further lower arbitrageur risk tolerance; and so on. Thus, MBS convexity may sometimes act as a destabilizing force within US bond markets— as it arguably did at the beginning of the three most recent Fed tightening cycles in 1994, 1999; and 2003. While it may be desirable for households to have access to mortgages that embed a no-penalty prepayment option, this may come at some cost in terms of …nancial stability. More speculatively, my analysis o¤ers a potential explanation for the …nding that failures of the expectations hypothesis are more pronounced in the US than in most other developed nations (see e.g., Campbell (2003)). Since the US is one of the few countries with long-term …xed rate mortgages that embed a true interest rate call, the existence of a signi…cant MBS convexity e¤ect suggests a novel solution to this puzzle. Finally, my analysis indicates that shocks to MBS duration impact term premia throughout the US bond market— for government debt, MBS, and corporate debt alike— as predicted by a model like Vayanos and Vila (2009) in the portfolio-balance tradition. This …nding is relevant for the ongoing debate about the e¤ectiveness of Quantiative Easing (QE) policies— the purchases of long-term bonds by the Federal Reserve and other global central banks. For instance, Krishnamurthy and Vissing-Jorgensen (2013) have argued that the markets for MBS, corporate bonds, and Treasuries are highly segmented from one another, so QE policies do not work through broad channels such a shifting the market-wide price of interest rate risk. 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Specifically, the change in aggregate duration is the sum of the MBS contribution and the non-MBS contribution. Panel C plots detrended refinancing activity based on the Mortgage Bankers’ Association’s Refinancing Index versus aggregate refinancing incentives (the average coupon on outstanding MBS minus the primary mortgage rate). Panels D shows detrended MBS duration in 10-year Treasury equivalents (in billions of current dollars). Panel A: Effective duration in levels 5 Effective Duration (years) 4 3 2 1 0 1988m12 1.5 Effective Duration (years) 1 .5 0 -.5 -1 1988m12 1990m12 1992m12 1994m12 1996m12 1998m12 2000m12 2002m12 2004m12 2006m12 2008m12 2010m12 2012m12 1990m12 1992m12 1994m12 1996m12 1998m12 2000m12 2002m12 2004m12 2006m12 2008m12 2010m12 2012m12 Aggregate Duration MBS Contribution to Aggregate Duration MBS Duration Panel B: 12-month changes in effective duration Aggregate Duration Non-MBS Contribution MBS Contribution 36 Figure 1: Effective duration of the US fixed income market (continued) Panel C: Refinancing activity and refinancing incentives 2 1 0 -1 -2 1988m12 1000 10-yr Equivalents, 2012 $ Billion 500 0 -500 -1000 1988m12 1990m12 1992m12 1994m12 1996m12 1998m12 2000m12 2002m12 2004m12 2006m12 2008m12 2010m12 2012m12 1990m12 1992m12 1994m12 1996m12 1998m12 2000m12 2002m12 2004m12 2006m12 2008m12 2010m12 2012m12 Detrended Mortgage Refinancing Av. MBS Coupon - Current Mortgage Rate Panel D: Detrended Duration in 10-year US Treasury Equivalents (in billions of current $) 37 Figure 2: Basic forecasting result using MBS duration contribution: This is a graphical depiction of the forecasting regression of future 10-year excess returns on the term spread and _ , corresponding to column (10) of Table 2. Specifically, Panel A shows a scatter plot of the component of future 12-month excess returns that is orthogonal to the term spread versus the component of _ that is orthogonal to the term spread. Panel B shows the corresponding time-series plot. Panel A: Scatter plot 20 1994m12 1994m11 1995m1 2002m51994m10 2011m1 1997m9 2007m12 2002m3 2000m10 1999m12 2000m2 2011m4 2000m12011m2 1994m9 2010m12 1997m8 1992m10 2002m6 2000m9 2007m3 1992m4 1992m81992m6 1990m9 1997m3 1997m10 2007m1 1990m12 1994m6 2011m3 2002m21991m7 2002m4 2001m12 2000m3 2007m6 1995m2 1992m3 1997m11 1992m91997m5 2000m7 2007m2 1997m6 2009m8 1992m11 1997m4 1992m2 1992m5 1999m11 1990m10 2000m8 1997m1 1995m3 2007m7 1997m12 2001m9 2002m1 2000m5 1997m21991m81994m5 2007m4 1990m4 1990m8 2009m9 1992m7 1994m82000m4 2009m10 1991m6 1989m2 1992m12 1998m1 2007m11 2010m11 2007m5 2007m8 1991m9 1993m1 1989m32000m6 2004m6 1996m7 2007m9 2006m12 1994m7 2004m5 2001m82009m7 1996m12 1992m1 1990m11 2010m9 2008m3 2000m11 2009m6 1989m1 1991m5 2006m4 1997m7 2008m11991m10 2006m111991m2 1999m10 1990m3 2009m12 2010m101996m9 1991m1 1996m101996m8 1999m8 2008m4 2008m5 2001m11 1990m5 2008m9 2001m7 2009m11 2001m6 2006m5 1990m2 1995m4 2006m7 1990m1 2004m4 2008m8 1991m3 1998m4 2010m81998m2 1998m3 1996m11 2009m5 2006m8 2008m6 2001m10 1991m11 2004m7 2008m7 2001m5 1991m4 2006m102006m3 2010m5 2003m8 2008m2 1999m9 2005m11 2006m6 1996m6 1996m5 2003m10 2010m3 2006m9 2001m4 2004m8 2010m4 2010m7 2006m2 1989m5 2010m1 1989m4 2005m10 2001m1 2008m11 2010m2 2000m121990m7 1999m7 1990m6 1994m4 2001m2 2004m1 2007m10 1996m4 2003m31998m5 2002m7 2003m12 2002m11 2003m7 1993m2 1989m12 2003m11 1991m12 2004m12 1999m6 2004m11 1995m11 1995m72004m9 2006m1 2003m1 1989m6 2002m9 2010m6 2005m2 1995m10 2005m9 2003m9 2005m12 2003m2 2002m8 2002m10 2005m11989m7 1996m2 1989m11 1995m6 2009m2 2002m121998m6 1995m5 1999m5 2004m2 2009m4 2005m3 2009m11995m81996m3 2004m10 1998m7 1995m91989m9 1994m3 2001m3 1999m31989m8 2005m8 2005m7 1993m31989m10 2003m4 1995m12 2004m3 1996m1 1999m4 2005m4 1993m4 1998m8 2005m5 2009m3 1999m2 1994m2 2005m61993m5 2003m6 1993m7 2008m12 1998m10 2003m5 1998m11 1998m9 1993m6 1998m12 1999m1 1993m8 1994m1 1993m12 1993m11 1993m9 1993m10 Future Excess Return -10 0 10 2008m10 -20 -1 -.5 0 MBS Duration Contribution .5 1 Future Excess Return Fitted values Panel B: Time-series plot 20 1 MBS Duration Contribution Future Excess Return 10 .5 0 0 -10 -.5 -20 -1 1988m12 1990m12 1992m12 1994m12 1996m12 1998m12 2000m12 2002m12 2004m12 2006m12 2008m12 2010m12 Future Excess Return MBS Duration Contribution 38 Figure 3: Coefficients by quarterly forecast horizons. This figure plots the coefficients b(j) on the duration contribution of the MBS Index from estimating rxt  ( j1)/4t  j /4  a( j )  b( j )  DUR _ CNTRBt (10) MBS  c( j )xt   t(10)j1)/4t  j /4 ( for quarters j = 1, ..., 8, effectively tracing out a nonparametric version of the “impulse” to quarterly excess bond returns following a movement in MBS duration. Panel A plots the coefficients from univariate forecasting regressions. The multivariate specifications in Panel B controls for the term spread. Confidence intervals, based on Newey-West (1987) standard errors allowing for serial correlation at up to 6 months, are shown as dashed lines. Panel A: Univariate coefficients by quarterly forecast horizon 10 Coefficient (%) 5 0 -5 1 10 Coefficient (%) 5 0 -5 1 2 3 Horizon (quarters) 4 5 6 7 8 2 3 Horizon (quarters) 4 5 6 7 8 Panel B: Multivariate coefficients by quarterly forecast horizon 39 Figure 4: Impulse response of quarterly excess bond returns to an MBS duration shock. Using quarterly data, I estimate a first order VAR of the form rxt 1/ 4 (10)  a1  b1  DURt  a2  b2  DURt (1) (10)  c1  ( yt(10)  yt(1) )   1,t 1/ 4 (10)  c2  ( yt(10)  yt(1) )   2,t 1/ 4 (10)  c3  ( yt(10)  yt(1) )   3,t 1/ 4 DURt 1/4 (10) yt 1/4  yt 1/4  a3  b3  DURt For simplicity, I show results only for DUR = DUR_CNTRBMBS. Below I plot the (simple) impulse response from a shock to DUR_CNTRBMBS in quarter t = 0 on 10-year excess returns from quarters t = 1, ..., 8. Confidence intervals for the estimated impulse response are shown in gray. For simplicity, I use a a quarterly data set of non-overlapping observations samples in March, June, September, and December. VAR(1): Response of Excess Returns to an Impulse to MBS Duration 10 Excess Returns 5 0 -5 0 1 2 3 4 5 6 7 8 Quarters 95% CI impulse response function (irf) 40 Figure 5: Excess return, yield, and forward rate forecasting coefficients by bond maturity. This figure plots the coefficients b(n) from estimating rxt 1  a( n )  b( n )  DURt   t 1 (n) (n) versus maturity n, for n = 2,...,20. I then plot the coefficients b(n) versus maturity n from estimating yt( n )  yt(n1)  a( n )  b( n )  DURt   t(n1) f t ( n )  f t ( 1n )  a( n )  b( n )  DURt   t(n1) , for n = 0, 1, 2,...,20. For simplicy, I rescale the coefficients b(n) so that they reflect the impact of one standard deviation shift in DURt. Confidence intervals, based on Newey-West (1987) SEs allowing for serial correlation at up to 18 months, are shown as dashed lines. Panel A shows results for MBS duration, and Panel B shows results for MBS duration contribution. In each panel, the left-most plot shows the excess return forecasting results by maturity, the middle plot shows yield-change forecasting results by maturity, and the right-most plot shows forward-change forecasting results. Panel A: MBS duration Univariate return forecasting coefficients by maturity 10 .8 .6 .4 .2 0 -.2 -.4 0 2 4 6 8 Maturity (years) 10 12 14 16 18 20 0 2 4 6 8 Maturity (years) 10 12 14 16 18 20 Univariate yield forecasting coefficients by maturity .8 .6 .4 .2 0 -.2 -.4 Univariate forward forecasting coefficients by maturity 8 Coefficient (%) Coefficient (%) 6 4 2 0 Coefficient (%) 0 2 4 6 8 Maturity (years) 10 12 14 16 18 20 Panel B: MBS duration contribution Univariate return forecasting coefficients by maturity 10 .8 .8 8 Coefficient (%) Coefficient (%) .6 .4 .2 0 -.2 -.4 0 2 4 6 8 Maturity (years) 10 12 14 16 18 20 0 2 4 6 8 Maturity (years) 10 12 14 16 18 20 .6 Coefficient (%) .4 .2 0 -.2 -.4 0 2 4 6 8 Maturity (years) 10 12 14 16 18 20 Univariate yield forecasting coefficients by maturity Univariate forward forecasting coefficients by maturity 6 4 2 0 41 Figure 6: Growth of MBS market and MBS duration measures, 1976-present. Panel A shows the market value of the Barclays MBS index as a fraction of the market value of the Barclays Aggregate Index (a common proxy for broader the investment grade bond market) and as a fraction of the sum of the value of Barclay’s MBS and Treasury indices. Panel B compares the evolution of the effective duration of the MBS index from 1989-present to the Macaulay duration-to-worst of the MBS index from 1976-present. Panel A: The growth of the MBS market 60 40 % 20 0 1976m12 8 6 Years 4 2 0 1976m12 1978m12 1980m12 1982m12 1984m12 1986m12 1988m12 1990m12 1992m12 1994m12 1996m12 1998m12 2000m12 2002m12 2004m12 2006m12 2008m12 2010m12 2012m12 1978m12 1980m12 1982m12 1984m12 1986m12 1988m12 1990m12 1992m12 1994m12 1996m12 1998m12 2000m12 2002m12 2004m12 2006m12 2008m12 2010m12 2012m12 MBS % of Aggregate Index MBS % of MBS+Treasuries Panel B: MBS duration measures MBS Effective Duration MBS Macaulay Duration-to-Worst 42 Figure 7: Coefficients from 60-month rolling excess return forecasting regressions, 1976-present. This presents results from 60-month (20-quarter) rolling regressions of 3-month excess returns on MBS duration measures, controlling for the term spread rxt 1/ 4  a  b  durt  c  ( yt  yt )   t 1/ 4 . Panel A shows results for MBS Macaulay duration (dur = durMBS), Panel B shows results for MBS Macaulay duration contribution (dur = dur_cntrbMB), and Panel C shows results for aggregate refinancing incentives (dur = c - yM). The left-hand plot in each Panel shows the estimated regression coefficients on dur along with confidence intervals as dashed lines. The right-hand plot shows the associated t-statistics. I compute standard errors using Newey-West (1987) standard errors allowing for serial correlation at up to 6 months. (10) (10 ) (1) (10 ) Panel A: MBS Macaulay duration (dur = durMBS) Return forecasting coefficients 20 10 Return forecasting t-stat Coefficient (%) 10 t-statistic 0 5 0 -10 -5 1980 50 0 t-statistic 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 Panel B: MBS Macaulay duration contribution (dur = dur_cntrbMB) Return forecasting coefficients 10 Return forecasting t-stat Coefficient (%) 5 -50 0 -5 -100 1980 5 0 t-statistic -2 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 Panel C: Aggregate refinancing incentives (dur = c - yM) Return forecasting coefficients 2 Return forecasting t-stat 0 Coefficient (%) -5 -4 -6 -10 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 -8 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 43 Table 1: Summary statistics: This table presents means, medians, standard deviations, extreme values, and monthly time-series autocorrelations (denoted ) of variables between 1989:m1 and 2011:m4. Panel A present summary statistics for US Treasury excess returns, yield spreads, and forwards from Gürkaynak, Sack, and Wright (2007). These variables are all measured in percentage points. Panel B presents summary statistics for various measures of effective duration in years based on Barclays Capital bond indices. Panel C reports the corresponding summary statistics for effective convexity. The effective convexity measures are only available beginning in 1997:m4. N                           _   _ /     ∙ /       268 268 268 268 268 268 268 268 268 268 268 268 268 268 268 268 268 169 169 169 Mean 3.00 4.98 6.32 0.90 1.57 2.01 4.45 4.93 5.33 5.67 5.96 4.53 3.35 1.08 5.13 3.46 0.33 -0.18 -1.50 -0.52 Median 3.33 5.74 6.35 0.76 1.20 1.57 4.59 4.98 5.35 5.60 5.76 4.56 3.49 1.12 5.16 3.45 0.33 -0.19 -1.54 -0.54 SD 4.39 7.73 12.98 0.83 1.28 1.54 2.24 2.01 1.80 1.64 1.52 0.26 0.91 0.29 0.25 0.18 0.04 0.17 0.45 0.16 Min -8.49 -16.41 -29.35 -0.54 -0.58 -0.82 0.16 0.62 1.42 2.31 3.02 3.71 0.58 0.20 4.47 2.93 0.26 -0.52 -2.43 -0.85 Max 12.46 20.96 39.17 2.50 3.79 4.68 9.70 9.32 9.19 9.17 9.16 5.12 4.83 1.78 5.61 3.88 0.41 0.20 -0.44 -0.15 ρ 0.922 0.884 0.861 0.973 0.977 0.978 0.976 0.971 0.970 0.970 0.969 0.863 0.875 0.851 0.954 0.960 0.991 0.844 0.825 0.818 Panel A: US Treasury excess returns, yield spreads, and forwards (%) Panel B: Effective Duration (years) Panel C: Effective Convexity (years) 44 Table 2: Forecasting 10-year excess nominal bond returns, 1989-Present: Regressions of 12-month excess returns on 10-year Treasuries on the effective duration of the Aggregate Index, the effective duration of the MBS Index, and the effective duration contribution of the MBS index rxt 1  a  b  DURt  cxt   t 1 . (10) (10) The regressions are estimated with monthly data, so each month I am forecasting the excess return over the following 12 months. To deal with the overlapping nature of returns, t-statistics are based on Newey-West (1987) standard errors allowing for serial correlation at up to 18 lags. I estimate these regressions with and without other forecasting variables identified in the literature on bond risk premia. Specifically, I control for the term spread following Campbell and Shiller (1991) and the first five forward rates following Cochrane and Piazzesi (2005). (1) (2) (3) 6.725 [2.84] (4) 11.482 [5.44] (5) 7.749 [2.73] 2.905 [2.76] _ 2.178 [3.02] -37.854 [-6.77] 217.606 [7.00] -511.359 [-6.92] 533.181 [6.66] -201.349 [-6.26] Constant Observations R-squared 1.555 [1.04] 268 0.13 2.065 [0.48] 268 0.31 -25.513 [-2.34] 268 0.05 -51.686 [-5.08] 268 0.26 2.928 [3.88] -30.755 [-4.91] 171.892 [4.77] -403.066 [-4.80] 418.391 [4.72] -156.236 [-4.47] -34.283 [-2.56] 268 0.35 -4.740 [-1.18] 268 0.12 -14.948 [-4.58] 268 0.37 3.231 [4.67] -27.953 [-4.59] 146.201 [4.68] -340.160 [-4.74] 357.217 [4.60] -136.066 [-4.29] -6.634 [-1.27] 268 0.42 -7.158 [-1.85] 268 0.17 -15.515 [-4.23] 268 0.40 4.437 [6.48] 3.929 [4.59] 11.253 [3.70] 14.629 [5.78] 2.999 [4.57] -22.653 [-3.89] 114.797 [3.78] -268.889 [-3.78] 286.175 [3.65] -109.305 [-3.36] -11.768 [-1.95] 268 0.46 12.143 [4.44] (6) (7) (8) (9) (10) (11) 45 Table 3: Forecasting 10-year excess nominal bond returns, MBS vs. non-MBS duration: Regressions of 12-month excess returns on 10-year Treasuries on the effective duration of the Aggregate Index, the effective duration contribution of the MBS Index, and the effective duration contribution of the non-MBS index rxt 1  a  b  DURt (10)  cxt   t(10) . 1 The regressions are estimated with monthly data, so each month I am forecasting the excess return over the following 12 months. To deal with the overlapping nature of returns, t-statistics are based on Newey-West (1987) standard errors allowing for serial correlation at up to 18 lags. I estimate these regressions with and without other forecasting variables identified in the literature on bond risk premia. Specifically, I control for the term spread following Campbell and Shiller (1991) and the first five forward rates following Cochrane and Piazzesi (2005). (1) 6.725 [2.84] _ _ 11.253 [3.70] -14.620 [-2.10] 8.872 [3.25] -8.022 [-1.11] 2.928 [3.88] 2.999 [4.57] (2) (3) (4) (5) 11.482 [5.44] 14.629 [5.78] -13.676 [-1.92] 2.053 [2.74] 13.586 [6.63] -3.179 [-0.44] 2.912 [4.14] -30.755 [-4.91] 171.892 [4.77] -403.066 [-4.80] 418.391 [4.72] -156.236 [-4.47] Constant Observations R-squared -25.513 [-2.34] 268 0.05 -7.158 [-1.85] 268 0.17 55.500 [2.31] 268 0.12 23.133 [0.88] 268 0.20 -51.686 [-5.08] 268 0.26 -15.515 [-4.23] 268 0.40 49.011 [1.98] 268 0.23 -3.268 [-0.12] 268 0.41 -34.283 [-.56] 268 0.35 -22.653 [-3.89] 114.797 [3.78] -268.889 [-3.78] 286.175 [3.65] -109.305 [-3.36] -11.768 [-1.95] 268 0.46 -33.422 [-5.39] 183.721 [5.04] -432.257 [-4.87] 460.215 [4.87] -178.142 [-4.85] 48.946 [1.92] 268 0.39 -23.239 [-4.00] 117.115 [3.86] -274.699 [-3.96] 295.311 [3.99] -114.393 [-3.82] 10.278 [0.34] 268 0.47 (6) (7) (8) (9) 7.749 [2.73] 12.143 [4.44] -13.200 [-1.73] 10.178 [3.48] -5.577 [-0.68] (10) (11) (12) 46 Table 4: Comparison with literature on the Federal Reserve’s Large Scale Asset Purchase (LSAP) programs. This table compares the implied priceimpact from my regressions using MBS duration with that estimated by papers examining the Fed’s LSAP programs. I obtain the implied effect for each LSAP as: Implied Effect in bps Effect of 1 σ move in on in bps LSAP size in 10yr UST duration equivalents $b 10yr UST duration equivalents $b corresponding to 1 σ move in Estimated effect on 10-year yield (bps) Effect Implied by MBS Supply Shocks LSAP evaluation studies Event Description LSAP1 a Supply shock ($b) 10-yr UST dur equiv. 750 Purchase of $300b Treasuries, $200b agencies, $1,250b MBS announced from 11/2008 - 3/2009. Purchase of $600b of long-term Treasuries announced in 11/2010 Announced 9/2011: Buy $400b LT USTs and sell $400b ST USTs Average study 83 Min study 55 Max study 107 _ Uni 54 Multi 68 Uni 69 Multi 79 LSAP2b LSAP3c 400 400 24 21 15 17 30 25 29 29 36 503 36 36 46 503 37 37 40 436 42 42 46 436 Effect of 1- move in DURt on 10-year yields (bps): 10-year UST equiv ($b) corresponding to 1- move in DURt a Li and Wei (2013) estimate that the $300 billion of Treasury purchases associated with LSAP1 reduced duration supply by $169 = 4.5 300 /8 billion 10-year Treasury equivalents. Thus, accounting for the purchases of $200 billion of agency debt and $1,250 billion MBS, I estimate that LSAP1 reduced the aggregate supply of duration by $750 4.5 300 4.5 200 3 1250 /8 billion 10-year Treasury equivalents. Gagnon et al (2011) examine movements in yields on eight LSAP1 announcement dates and estimate that LSAP1 reduced 10-year yields by 91 bps. Applying the coefficients from time-series regressions, Gagnon et al estimate an impact of LSAP1 of 61 bps. Krishnamurthy and Vissing-Jorgensen (2011) examine movement in yields on five LSAP1 announcement dates and estimate that LSAP1 reduced 10-year yields by 107 bps. Using a no arbitrage model, Li and Wei (2013) estimate that LSAP1 lowered yields by 99 bps. Jarrow and Li (2012) also estimate a no arbitrage model and estimate that LSAP1 and LSAP2 lowered 10-year yields by 70 bps. Assuming roughly 80% of this corresponds to LSAP1, this implies that LSAP1 lowered yields by 56 bps. Li and Wei (2013) estimate that LSAP2 reduced the supply of duration by $400 billion 10-year Treasury equivalents. Krishnamurthy and Vissing-Jorgensen (2011) examine movement in yields on three LSAP2 announcement dates and estimate that LSAP2 reduced 10-year yields by 30 bps. Using a no arbitrage model, Li and Wei (2013) estimate that LSAP2 lowered yields by 26 bps. Jarrow and Li (2012) also estimate a no arbitrage model and estimate that LSAP1 and LSAP2 lowered 10-year yields by 70 bps. Assuming roughly 20% of Jarrow and Li’s (2012) estimate corresponds to LSAP2, this implies that LSAP2 lowered yields by 14 bps. c b LSAP3 is often referred to as the “Maturity Extension Program” or MEP. Li and Wei (2013) estimate that LSAP3 reduced the supply of duration by $400 billion 10-year Treasury equivalents. Li and Wei (2013) estimate that LSAP3 lowered 10-year yields by 25 basis points. Using regressions similar to those in Greenwood and Vayanos (2013), Hamilton and Wu (2012) estimate that LSAP3 lowered 10-year yields by 17 bps. 47 Table 5: Horseraces between MBS duration and mortgage refinancing measures: Regressions of 12-month excess returns on 10-year nominal Treasuries on the effective duration contribution of the MBS index and either (i) aggregate refinancing incentives ( , ) computed as the difference between the average MBS coupon and the Freddie Mac primary market 30-year mortgage rate or (ii) the log-deviation of the MBA mortgage refinancing index from its Hodrick-Prescott (1997) trend ( ): rxt 1  a  b  DUR _ CNTRBt (10) MBS  c  REFIt  dxt   t(10) . 1 The regressions are estimated with monthly data, so each month I am forecasting the excess return over the following 12 months. To deal with the overlapping nature of returns, t-statistics are based on Newey-West (1987) standard errors allowing for serial correlation at up to 18 lags. I estimate these regressions with and without other forecasting variables identified in the literature on bond risk premia. Specifically, I control for the term spread following Campbell and Shiller (1991) and the first five forward rates following Cochrane and Piazzesi (2005). (1) , (2) -7.766 [-4.46] (3) -6.867 [-3.29] (4) 2.444 [0.72] (5) -4.702 [-1.44] (6) -5.158 [-1.51] (7) (8) (9) (10) (11) (12) -2.571 [-1.26] -4.670 [-3.33] _ 4.537 [4.70] -26.123 [-3.53] 131.748 [3.09] -318.861 [-3.34] 347.721 [3.59] -134.407 [-3.61] Constant Observations R-squared 4.303 [2.73] 268 0.05 -4.202 [-1.87] 268 0.42 -1.734 [-0.32] 268 0.50 -11.061 [-1.94] 268 0.19 -10.900 [-3.12] 268 0.45 15.470 [2.72] 7.686 [1.90] 4.038 [3.92] -23.254 [-3.91] 113.968 [3.50] -274.439 [-3.78] 299.820 [4.00] -116.017 [-3.89] -6.056 [-1.13] 268 0.51 5.273 [4.17] 256 0.17 4.624 [1.17] -6.772 [-6.31] -6.027 [-4.23] -2.264 [-1.12] 7.682 [2.04] -4.448 [-2.87] 7.378 [2.83] 3.298 [4.65] -4.432 [-2.59] 5.551 [2.37] 3.317 [4.55] -17.055 [2.27] 94.061 [2.43] -246.166 [2.85] 281.555 [3.13] -112.104 [3.15] -0.210 [0.13] 256 0.42 0.632 [0.13] 256 0.51 -2.978 [0.73] 256 0.21 -14.383 [-1.95] 74.597 [2.03] -196.714 [-2.42] 228.361 [2.66] -91.553 [-2.64] -8.102 [2.27] 256 0.46 -5.626 [-0.95] 256 0.52 48 Table 6: Forecasting 12-month ahead 10-year excess nominal bond returns, subsample results: 1976present: Regressions of 12-month excess returns on 10-year Treasuries on the effective duration of the MBS Index, the effective duration contribution of the MBS index, and aggregate refinancing incentives (10) (10 ) rxt 1  a  b  DURt  cxt   t 1 . For MBS duration, (effective) refers to option-adjusted effective duration which is available from (Macaulay) refers to Macaulay duration-to-worst which is available from 1976-present. 1989-present. A similar notation applies in the case of MBS duration contribution. The regressions are estimated with monthly data, so each month I am forecasting the excess return over the following 12 months. To deal with the overlapping nature of returns, t-statistics are based on Newey-West (1987) standard errors allowing for serial correlation at up to 18 lags. I estimate these regressions with and without controlling for the term-spreads. Panel A: Subsample results for MBS duration (2) (3) (4) (5) 1989-1999 2000+ 1989+ 1976+ 5.324 [3.33] 4.355 [9.44] 5.968 [4.57] 3.231 [4.67] Constant Observations R-squared -14.948 [-4.58] 3.062 [1.93] -19.279 [-2.65] 2.938 [4.30] -12.830 [-4.73] 2.935 [4.23] -27.100 [-4.08] 1.505 [0.76] 3.481 [3.34] -8.601 [-0.93] -0.962 [-0.27] 4.516 [1.76] 2.063 [0.11] 10.198 [3.05] 3.278 [2.44] -48.689 [-2.99] 132 0.41 1989-1999 4.383 [5.14] 2.333 [2.92] -17.803 [-4.01] 136 0.37 2000+ (1) 1989+ (effective) (6) 1976-1988 (7) 1989-1999 (8) 2000+ 4.437 [6.48] (Macaulay) 268 132 136 268 424 156 0.37 0.34 0.49 0.34 0.15 0.13 Panel B: Subsample results for MBS duration contribution 1989+ 1989-1999 2000+ 1989+ 1976+ 1976-1988 effective _ _ 14.629 [5.78] 19.993 [3.86] 11.886 [8.64] 14.551 [3.80] 6.733 [2.18] 2.579 [2.64] -8.244 [-1.63] 9.220 [1.71] 3.756 [1.58] -9.578 [-1.84] 156 0.15 , ) 1976-1988 -3.455 [-2.91] 8.205 [3.87] -15.428 [-5.32] 156 0.30 33.513 [2.96] 3.140 [2.84] -46.573 [-3.07] 132 0.42 1989-1999 -9.119 [-4.24] 6.066 [3.54] -5.710 [-1.70] 132 0.48 11.891 [5.76] 2.218 [2.89] -16.979 [-4.22] 136 0.36 2000+ -6.717 [-3.78] 3.828 [4.14] -3.209 [-1.21] 136 0.33 (Macaulay) 2.999 [4.57] Constant Observations R-squared -15.515 [-4.23] 3.004 [1.98] -21.018 [-3.04] 2.837 [4.07] -12.307 [-4.42] 2.319 [3.52] -20.371 [-3.39] 268 132 136 268 424 0.40 0.38 0.47 0.32 0.20 Panel C: Subsample results for mortgage refinancing incentive ( 1989+ 1976+ -7.766 [-4.46] 4.537 [4.70] -1.684 [-1.50] 4.531 [3.94] -4.479 [-1.79] 424 0.19 , Constant Observations R-squared -4.202 [-1.87] 268 0.42 49 Table 7: Using US MBS duration to forecast foreign bond excess returns, 1989-Present. This tables presents univariate forecasting regressions of the form rxc ,t 1 FOR ( n )  a  b  DURt US   cFOR1( n ) , and multivariate regressions of the form rxcFOR1( n )  a  b  DURt  c  ( yc ,t ( n )  ytUS (1) )  d  ( ycFOR ( n )  ytFOR (1) )   cFOR1( n ) . Using IFS data on ,t  ,t  ,t ,t  long-term government bond yields and short-term interest rates, I compute foreign government bond returns for the ten developed countries: Belgium, Canada, Denmark, France, Germany, Italy, Japan, Sweden, Switzerland, and the UK. Since the long-term yields are for coupon-bearing bonds, I follow Campbell (1999, 2003) and use the Campbell, Shiller, and Schoenholtz (1983) approximation to compute excess returns over short-term government bills (when available) or money market rates (when bills are not available). To deal with the overlapping nature of the regressions, t-statistics are based on Newey-West (1987) standard errors allowing for serial correlation at up to 18 lags. To assess the differential forecasting power between foreign bond and domestic bond excess returns, for each country I rerun the regressions with , 1 , 1 on the left-hand-side. I then report the coefficient on DURt from this specification as well as the associated t-statistic, and Panel B shows results for _ labeled as “Diff from USA” below. Panel A shows results for . JPN (14) -0.18 [-0.20] 1.04 [0.86] -1.03 [0.79] 2.52 [0.71] 268 0.03 -3.12 [-3.46] (15) 1.51 [1.21] (16) 2.13 [2.49] 2.10 [2.74] 2.15 [0.98] -10.59 [-3.95] 268 0.25 -0.79 [-0.88] USA (1) 2.42 [3.53] (2) 2.91 [5.81] 1.94 [3.13] (3) BEL (4) 0.77 [1.44] 1.37 [1.80] 1.74 [2.21] -4.82 [-2.16] 268 0.26 -2.10 [-4.88] (5) 0.44 [0.52] CAN (6) 0.69 [0.86] Panel A: Forecasting foreign bond excess returns using DEN FRA GER ITA (7) 0.83 [1.19] (8) 1.06 [1.78] 1.63 [2.28] 1.35 [2.29] -4.59 [-1.82] 268 0.22 -1.79 [-3.40] (9) 0.21 [0.31] (10) 0.74 [1.59] 1.52 [2.00] 1.05 [1.53] -3.60 [-1.62] 268 0.17 -2.22 [-4.98] (11) 0.20 [0.23] (12) 0.58 [0.88] 1.93 [2.75] 1.84 [2.51] -4.82 [-2.00] 268 0.30 -2.30 [-4.28] (13) -0.44 [-0.56] SWE (17) 1.03 [1.07] (18) 1.41 [1.73] 1.38 [1.56] 1.39 [1.95] -5.18 [-1.84] 268 0.11 -1.51 [-2.13] (19) SUI (20) 1.37 [2.85] 2.12 [3.19] 1.80 [3.31] -7.72 [-3.88] 268 0.37 -1.74 [-2.78] (21) -0.14 [-0.15] UK (22) 1.50 [1.99] 1.49 [2.12] 0.79 [1.18] -5.45 [-1.77] 268 0.22 -1.40 [-1.89] 0.92 [0.99] 1.90 [4.36] 0.56 [0.88] 1.79 [2.58] -6.63 [-4.17] 268 0.39 -1.07 [-2.66] Constant Observations R-squared Diff from USA -4.64 [-1.74] 268 0.14 -9.62 [-3.80] 268 0.29 1.11 [0.40] 268 0.00 -1.98 [-3.73] 1.19 [0.48] 268 0.02 -1.72 [-3.73] 0.00 [0.00] 268 0.01 -1.59 [-3.35] 1.81 [0.80] 268 0.00 -2.21 [-4.65] 1.83 [0.69] 268 0.00 -2.22 [-3.70] 4.05 [1.58] 268 0.00 -2.86 [-3.09] -2.75 [-0.78] 268 0.05 -0.91 [-0.87] -0.09 [-0.03] 268 0.01 -1.39 [-1.93] 2.73 [0.98] 268 0.00 -2.56 [-3.30] -0.53 [-0.19] 268 0.02 -1.50 [-1.85] USA (1) _ 8.76 [4.31] (2) 10.05 [5.70] 1.90 [3.43] (3) BEL (4) 2.27 [1.44] 1.36 [1.74] 1.70 [2.07] -4.58 [-1.82] 268 0.26 -7.48 [-5.17] (5) 2.14 [0.98] Panel B: Forecasting foreign bond excess returns using CAN DEN FRA GER (6) 5.31 [4.00] 0.69 [0.89] 1.57 [2.12] -5.88 [-3.27] 268 0.37 -4.66 [-4.81] (7) 3.53 [1.45] (8) 3.61 [2.03] 1.61 [2.29] 1.31 [2.12] -4.86 [-1.83] 268 0.22 -6.19 [-3.92] (9) 1.88 [0.96] (10) 2.56 [1.50] 1.52 [1.98] 0.96 [1.35] -3.81 [-1.35] 268 0.17 -7.42 [-4.80] (11) 2.69 [1.34] (12) 3.22 [1.78] 1.97 [2.76] 1.78 [2.36] -6.34 [-2.35] 268 0.32 -6.69 [-4.09] (13) 2.82 [1.45] _ ITA (14) -1.10 [-0.44] 1.01 [0.83] -0.98 [-0.75] 3.12 [0.75] 268 0.03 -11.57 [-3.65] (15) JPN (16) 6.32 [2.84] 2.01 [2.53] 1.58 [0.69] -9.53 [-3.03] 268 0.23 -3.96 [-1.46] 5.57 [2.77] SWE (17) 4.31 [1.69] (18) 4.49 [1.86] 1.36 [1.52] 1.27 [1.66] -5.15 [-1.45] 268 0.11 -5.70 [-2.99] (19) SUI (20) 4.46 [-3.05] 2.13 [3.17] 1.61 [3.01] -7.77 [-3.84] 268 0.38 -5.56 [3.12] (21) 2.15 [1.10] UK (22) 5.39 [2.37] 1.55 [1.96] 0.71 [1.06] -6.30 [-1.83] 268 0.23 -4.72 [-2.26] 4.07 [1.76] -2.14 [-0.94] Constant Observations R-squared Diff from USA -6.01 [-2.28] 268 0.19 -10.66 [-4.05] 268 0.33 0.26 [0.10] 268 0.01 -6.63 [-5.29] 0.47 [0.22] 268 0.03 -5.94 [-5.57] -1.05 [-0.35] 268 0.02 -5.23 [-4.06] 0.47 [0.19] 268 0.01 -6.88 [-5.38] -0.41 [-0.18] 268 0.02 -6.07 [-4.70] 4.89 [1.58] 268 0.01 -10.91 [-3.88] -3.70 [-1.79] 268 0.07 -3.20 [-1.33] -1.30 [-0.41] 268 0.02 -4.45 [-3.04] -0.06 [0.03] 268 0.01 -6.61 [4.06] -1.84 [-0.75] 268 0.04 -4.70 [2.30] 50 Table 8: Response of US long-term forwards to changes in short-term rates, 1999-present. Daily regressions of changes in nominal and real forward rates on short-term nominal rates, allowing for a differential response depending on MBS negative convexity. MBS negative convexity is simply negative one times MBS convexity. Specifically, I estimate f t X (10)  a  b  yt$( 2 )  c  NCONVt MBS  d  ( yt$( 2 )  NCONVt MBS )   tX (10) , and f t X (10)  a  b  f t $( 2 )  c  NCONVt MBS  d  ( f t $( 2 )  NCONVt MBS )   tX (10) , for X = $ and TIPS. t-statistics, based on Newey-West (1987) standard errors allowing for serial correlation at a lag of up to 20 business days, are shown in brackets. $ (1) y $(2) TIPS (2) 0.124 [1.86] $ (3) 0.475 [3.99] TIPS (4) 0.142 [1.98] $ (5) TIPS (6) $ (7) TIPS (8) 0.322 [2.67] f$(2) NCONVMBS NCONVMBS ∙ (MVMBS/MVAGG) y$(2)×NCONVMBS y$(2)×NCONVMBS ∙ (MVMBS/MVAGG) f $(2)×NCONVMBS f $(2)×NCONVMBS ∙ (MVMBS/MVAGG)  Constant R-squared 0.140 [1.74] 0.114 [2.51] 0.107 [0.48] 0.278 [1.96] 0.000 [0.06] -0.001 [-0.58] 0.002 [0.23] 0.000 [0.06] 0.264 [3.33] 0.000 [0.13] 0.074 [1.41] -0.001 [-0.65] 0.364 [4.97] 0.084 [1.66] 0.001 [0.09] -0.001 [-0.16] 0.155 [2.94] 0.135 [4.00] 0.247 [1.77] 0.354 [3.74] 0.000 [0.26] 0.19 0.000 [0.01] 0.21 0.000 [0.19] 0.12 -0.001 [-0.27] 0.21 -0.001 [-0.28] 0.12 0.000 [0.02] 0.31 0.000 [0.19] 0.19 -0.001 [-0.16] 0.31 51 Table 9: Implied variance of US long-term yields and MBS convexity, 1997-present. Weekly regressions of swaption-implied yield variance on MBS negative convexity: MBS    t2 (y (10) )  a  b   t21 (y (10) )  c  yt(10)  d  NCONVt 1  ut . 1 MBS negative convexity is simply negative one times MBS convexity. I follow the empirical approach of Perli and Sack (2003) as outlined in the main text. Specifically, the left-hand side variable, Δ , is the residual from a regression of 10-year by 3-month forward swaption-implied variance (the square of implied volatility) on 10-year by 2-year forward implied yield variance. t-statistics, based on Newey-West (1987) standard errors allowing for serial correlation at a lag of up to 12 weeks, are shown in brackets. (1) Δ , 0.941 [49.81] (2) 0.941 [50.40] 0.004 [1.07] 0.013 [2.14] ∙ (3) 0.939 [49.61] (4) 0.939 [49.86] 0.006 [1.64] 0.020 [2.29] 0.048 [2.14] 0.078 [2.70] -0.072 [-2.31] 785 0.89 / -0.021 [-2.28] 785 0.89 -0.051 [-1.62] 785 0.89 Constant Observations R-squared -0.026 [-2.42] 785 0.89 52 Table 10: The impact on MBS duration on corporate bond and swap spreads, 1990-Present: Panel A shows regressions of 3-month changes in spreads over Treasuries on contemporaneous 3-month changes in MBS duration: SPREADt  a1  b1  DUR _ CNTRBt MBS   c1xt  vt . Panel B uses the current level of MBS duration to forecast changes in spreads over the next 12 months: MBS SPREADt 1  a2  b2  DUR _ CNTRBt  c2 x t  ut 1. Panel C uses the current level of MBS duration to forecast excess returns over duration-matched Treasuries over the following 12 months: DUR  MATCH MBS rxt 1  c3 x t   t 1  a3  b3  DUR _ CNTRBt To deal with the overlapping nature of the regressions, t-statistics are based on Newey-West (1987) standard errors allowing for serial correlation at up to 6 lags in Panel A and up to 18 lags in Panels B and C. In Panel A, I control for changes in the term spread, the VIX, and the past returns on the stock market, all of which Collin-Dufresne, Goldstein, and Martin (2001) show are useful for explaining changes in corporate credit spreads. In Panel B and C, I control for the initial level of the term spread, the VIX, and the initial level of the spread for the each asset. Corporate bond spreads in Panel A and B for investment grade corporates, Aaa-rated corporates, and Baa-rated are the (option-adjusted) spreads on Barclays indices over duration-matched Treasuries. The 10-year swap spread is from Bloomberg. The returns in Panel C are the excess returns on these indices over duration-matched Treasuries. For swaps, I do not have an exact measure of excess returns over duration-matched Treasuries. Thus, I use the Campbell, Shiller, and Schoenholtz (1983) approximation. I compute this excess return using the yield on 10-year swaps and the 10-year par Treasury yield from Gurknayak, Sack, and Wright (2007). IG corporate spread (1) (2) Aaa corporate spread (3) (4) Baa corporate spread (5) (6) 10-yr swap spread (7) (8) Panel A: Contemporaneous 3-month changes in spreads over duration-matched Treasuries Δ Δ Δ _ -0.43 [-3.56] -0.16 [-2.21] 0.05 [0.52] 0.03 [2.78] -0.01 [-2.40] 0.03 [0.79] 265 0.51 -4.0 -0.14 [-1.92] 0.04 [0.50] 0.04 [0.46] 0.03 [2.40] 0.00 [1.14] 0.00 [0.17] 265 0.32 0.9 -0.72 [-3.52] -0.35 [-2.75] 0.06 [0.48] 0.03 [2.70] -0.02 [-3.84] 0.04 [0.94] 265 0.50 -9.0 0.13 [3.40] 0.17 [5.22] -0.08 [-2.98] 0.00 [1.98] 0.00 [1.21] -0.01 [-1.10] 265 0.22 4.3 Constant Observations R-squared bps effect 1-DUR 0.01 [0.19] 265 0.06 -11.2 0.00 [0.06] 265 0.01 -3.5 0.01 [0.13] 265 0.10 -18.8 -0.01 [-0.97] 265 0.09 3.3 53 Table 10: The impact on MBS duration on corporate bond and swap spreads, 1990-Present (continued): IG corporate spread (1) (2) Aaa corporate spread (3) (4) Baa corporate spread (5) (6) 10-yr swap spread (7) (8) Panel B: Forecasting future 12-month changes in spreads over duration-matched Treasuries _ 0.85 [2.68] 0.63 [3.72] -0.12 [-1.44] 0.00 [0.15] -0.47 [-1.89] Constant Observations R-squared bps effect 1-DUR -0.87 [-2.52] 256 0.07 -24.7 0.24 [0.68] 256 0.35 -18.4 -0.33 [-1.70] 256 0.02 -8.9 0.31 [1.46] 0.30 [2.81] -0.08 [-1.65] 0.01 [0.81] -0.82 [-4.53] 0.34 [1.64] 256 0.44 -8.6 -1.24 [-2.61] 256 0.08 -34.5 1.18 [2.93] 0.92 [4.07] -0.11 [-1.23] 0.01 [0.76] -0.61 [-2.88] 0.04 [0.11] 256 0.37 -26.8 0.15 [1.55] 256 0.06 4.9 -0.17 [-2.00] -0.16 [-2.87] -0.09 [-4.32] 0.00 [0.44] -0.47 [-4.29] 0.56 [4.25] 256 0.40 4.7 Panel C: Forecasting 12-month excess returns over duration-matched Treasuries _ -4.65 [-2.75] -3.59 [-3.87] 0.51 [1.11] 0.03 [0.25] 2.49 [1.74] -0.57 [-0.30] 256 0.32 -2.28 [-1.99] -1.58 [-2.31] 0.41 [1.74] 0.05 [0.87] 0.15 [0.16] -0.11 [-0.11] 256 0.13 -6.53 [-3.17] -5.10 [-4.66] 0.50 [1.05] -0.07 [-0.71] 3.44 [3.09] 0.29 [0.13] 256 0.38 1.20 [2.08] 0.49 [1.02] 0.22 [1.91] -0.01 [1.01] 3.86 [4.48] -2.43 [-2.52] 256 0.32 Constant Observations R-squared 5.21 [2.82] 256 0.07 2.38 [2.41] 256 0.06 7.37 [2.97] 256 0.08 -1.10 [-2.26] 256 0.05 54