Lecture 3: Forecasting interest rates

Lecture 3: Forecasting interest rates Prof. Massimo Guidolin Advanced Financial Econometrics III Winter/Spring 2017

Overview The key point One open puzzle Cointegration approaches to forecasting interest rates Yield-curve based factor models Principal component modelling Excess bond return predictability Dynamic linear affine term structure models in forecasting The role of macroeconomic information A puzzle, predictability from past forward rates: the «tent» 2

The key point The time-t term structure contains substantial information about changes in the slope and curvature However, level is close to a random walk process Forecasts that imply substantial predictable variation in expected excess returns may point to overfitting Forecasting future, riskless interest rates is of obvious importance to traders, bond portfolio managers, and policy makers Adopt asset pricing approach: interest rates are functions of bond prices, their dynamics can be studied using tools of asset pricing o The theory is particularly powerful when applied to Treasury yields, since the underlying assets have fixed payoffs (unlike stocks) Using asset pricing a number of key insights on interest rate forecasting emerge Gaussian dynamic term structure models (TSM) are the tool to describe joint forecasts of future yields, returns, and risk premia These models impose no-arbitrage restrictions 3

One (possibly, open) puzzle Standard economic explanations of risk premia fail to explain the behavior of expected excess returns to bonds Unfortunately, standard economic explanations of risk premia fail to explain the behavior of expected excess returns to bonds o In the data, mean excess returns to long-term Treasuries are positive o Yet traditional measures of risk exposure imply that Treasury bonds are not assets that demand a risk premium o Point estimates of their consumption betas are negative and point estimates of their CAPM betas are approximately zero o Although expected excess returns to bonds vary over time, these variations are unrelated to interest rate volatility or straightforward measures of economic growth The figure displays zero-coupon bond yields btw. 3m and 10Y o Academics use zero-coupon yields interpolated from Treasuries o The interpolation is inherently noisy: Bekaert, Hodrick, and Marshall (1997, JFE) estimate that the std. deviation of measurement error is in the range of 7 to 9 bps of annualized yield for maturities > 1 year 4

A first peek at the data 5

Cointegration approaches Standard tests reveals that riskless yields are (near) unitroot, possibly cointegrated but that spreads are stationary A glance at the figure suggests that yields are cointegrated: spreads between yields on bonds of different maturities are meanreverting, but the overall level of yields is highly persistent Therefore term spreads tend to be stationary A robust conclusion of the literature is that standard tests cannot reject the hypothesis of a unit root in any of these yields o In an economic perspective it is easier to assume that yields are stationary and highly persistent rather than truly nonstationary o Unit root interest rates carry a positive prob. of falling below -100% Campbell and Shiller (1987, JPE) motivate a cointegration approach They make the simplifying assumption that the weak form of the expectations hypothesis holds so that maturitydependent constant 6

Cointegration approaches If investors have information not captured in the history of short rates, then spreads forecast changes in short rates The spread between the yield on an n-period bond and a 1-period yield is then Spreads are I(0) if one-period yields are I(1). Campbell and Shiller examine monthly observations of one-month and 20-year bond yields over the period 1959 to 1983 They cannot reject the hypotheses that yields are I(1) and spreads are I(0) and hence advocate an error-correction model (ECM): o Investors impound their information about future short-term rates in the prices of long-term bonds o If investors have information about future rates not captured in the history of short rates, then spreads forecast changes in short rates 7

Yield-curve based factor models The ECM approach does not recognize that period-t bond yields are determined based on all information that investors have about future interest rates, possibly beyond short-term rates This fact leads to a more parsimonious approach to modeling interest rates than an ECM Assume that all of the information that determines investors forecasts at t can be summarized by a p-dimensional state vector x t The yield on the LHS cannot be a function of anything other than the state vector, since only x t shows up on the RHS, Stack time-t yields on bonds with different maturities in a vector y t Assume there exists an inverse function s.t. The inverse function exists if yields contain the same information as x t the rank of f/ x t must be p 8

Yield-curve based factor models If each element of x t has its own unique effect on the time-t yield curve, the yield curve can be inverted to infer x t o A necessary condition is that there are at least p yields in the vector y t o There are technical conditions associated with this result, but the intuition is that if each element of x t has its own unique effect on the time-t yield curve, the yield curve can be inverted to infer x t o Put differently, the time-t yield curve contains all information necessary to predict future values of x t This allows us to write that is determined by the mappings from factors to expected future one-period yields and excess bond returns A possible interpretation is that both x t and y t must follow firstorder Markov processes x t is Markov because it is defined as the set of information relevant to forming conditional expectations and if information at time t other than x t were helpful, then investors could use it 9

Yield-curve based factor models When yields follow a Markov process, the time-t dynamics (e.g., regimes) can be backed out of time-t yields o When forming forecasts as of time-t, the use of information other than time-t yields requires a compelling justification o Such information includes yields dated prior to t, measures of inflation, central bank policy announcements, and economic activity o All of information in these variables is embedded in time-t yield curve o However, if there is a good reason to believe that the mapping from current yields to expected future yields is unstable over time, while the mapping from, say, current inflation to expected future yields is stable, then it makes sense to use other data to forecast yields It is also important to recognize that a Markov process for yields does not imply that yields follow a first-order VAR Markov processes may be nonlinear, e.g., there may be regime shifts Even if a VAR(1) is a reasonable model of yields, we must decide how to compress the information in the cross-section of yields 10

Principal component forecasting Yields are commonly summarized using principal components o Since yields on bonds of different maturities are highly correlated, it does not make sense to estimate unrelated regressions for each yield A standard approach is to extract common factors from yields and apply a VAR to the factors o Forecasts of individual yields are determined by the cross-sectional mapping from the vector to the individual yield Following Litterman and Scheinkman (1991, JFI), researchers often summarize term structures by a small set of linear combinations The first few principal components of the covariance matrix of yields capture almost all of the variation in the term structure Std deviations of residuals from using three principal components range from 5 to 11 bps which is roughly the same range as the measurement error described by Bekaert et al. (1997, JFE) These first three principal components are commonly called level, slope, and curvature, respectively 11

Principal component forecasting Yields are commonly summarized using principal components 12

Principal component forecasting Yields are commonly summarized using principal components 13

Principal component forecasting The level factor tends to be unpredictable A VAR(1) applied to PCs allows us to use time-t principal components to predict time-(t+k) principal components Other models, for instance MSVAR, or possible or even worthy The mapping from PCs to yields translates these forecasts to expected future yields No statistical evidence that changes in level are forecastable 14

Principal component forecasting Slope and curvature are strongly forecastable At a quarterly horizon, about 20 (30) percent of the variation in slope (curvature) is predictable PCs other than the first three do not contribute much to forecasts of slope and curvature 15

Principal component forecasting Forecasts of future yields using current yields are necessarily also forecasts of expected log returns to bonds Duffee (2011) takes this approach and concludes that the model works well in pseudo out-of-sample forecasting Diebold and Li (2006, JoE) build a dynamic version of the term structure introduced by Nelson and Siegel (1987, JBus) o The cross-section of the term structure is summarized by the level, slope, and curvature factors of the Nelson-Siegel model o These three factors are assumed to follow a VAR(1) Diebold and Li find that the dynamics with the best pseudo out-ofsample properties are those in which level, slope, and curvature follow univariate AR(1) processes A popular forecasting approach uses a VAR that includes both compressed information from the term structure and compressed information from a large panel of macro variables Should we forecast future yields or future bond returns? 16

Excess bond return predictability If changes in long-term rates are unpredictable, then long bond excess returns must be predictable and related to term premia o A derivation in Campbell and Shiller (1987, JPE) shows that there is no difference o For reasonably long maturities, variations over time in the LHS are very close to variations in the 1 st PC, hence close to unforecastable o Therefore the RHS must also be unforecastable with time-t yields o The first term on the right side is a measure of the slope of the term structure, which varies widely over time o Because the sum on the RHS is unforecastable, the second term, excess returns to the bond, must also be strongly forecastable and positively correlated with the slope of the term structure This implication is confirmed with excess return regressions from CRSP constructed by subtracting the return to the shortestmaturity portfolio 17

Excess bond return predictability The level of the term structure is unrelated to future excess returns A less-steep slope (larger value of the second principal component) corresponds to lower future excess returns Less clear are the links btw. other PCs and future excess returns o There is strong statistical evidence that greater curvature predicts lower excess returns o At the monthly horizon, 5% of the variation is predictable, 10% at the quarterly horizon 18

Dynamic, affine term structure models For tractability, assume that interest rates are linear and homoskedastic with Gaussian shocks and rule out arbitrage opportunities o Why a linear, homoscedastic model? o It is easy to find evidence of nonlinear, non-gaussian dynamics o Gray (1996, JFE) concludes that a model of time-varying mean reversion and time-varying GARCH effects fits the dynamics of the short-term interest rate o A zero bound imposes nonlinear dynamics on yields that bind Although they have fixed conditional variances, this is typically not a concern of the forecasting literature that focuses on predicting yields and excess returns rather than conditional second moments The short rate, r t, is a function of p state variables, x t, The state vector has first-order VAR(1) Markov dynamics Given a unique SDF, no-arbitrage implies 19

Dynamic, affine term structure models In a linear affine Gaussian homoscedastic model, the state follows a VAR(1) process and risk prices are linear in the state The SDF is assumed to have the log-linear form o t is a vector with time-varying prices of risk, a function of the state, o Bonds are priced using the equivalent martingale dynamics: At this point, bond prices can be written as (difference equation) o Zero-coupon yields are written as o Excess return: 20

Identification of linear affine TSM Restrictions and/or normalizations need to be imposed to identify a linear affine dynamic TSM As far estimation is concerned, see lecture notes from a.a. 2015 and earlier posted on the course web site One important problem is left: the state vector is arbitrary, in the sense that an observationally equivalent model is produced by scaling, rotating, and translating the state vector o Define such a transformation as o Dai and Singleton (2000, JF), an observationally equivalent model replaces x t with x * t, and replaces the parameters with Many ways to identify the state vector and thus parameters One way is to restrict the K matrix to a diagonal matrix, set μ to zero, and set the diagonal elements of Σ equal to one These define a rotation, translation, and scaling, respectively 21

Identification of linear affine TSM Two common identification restrictions are based on either some yields being true or on principal components Other approach equates x t with linear combinations of yields: consider any p d matrix L with rank p, d = number of maturities As long as LB is invertible, this defines a new state vector o A simple example is a diagonal L with p diagonal elements equal to one and the remainder equal to zero o This choice produces a state vector equal to p true yields, or yields uncontaminated by measurement error Other choice of L is based on PCs of yields: Then the factors correspond to level, slope, and so on Regardless of the choice of L, this kind of state vector rotation emphasizes that a TSM is really a model of one set of derivative instruments (yields) explaining another set of derivative instruments (more yields) 22

Do no arbitrage restrictions help? General consensus is that while restrictions on risk premia estimates helps forecasting, no arbitrage restrictions do not It seems natural to estimate unrestricted risk premia, λ 0 and λ 1, but Joslin, Singleton, and Zhu (2011, RFS) and Duffee (2011) show that this version of the no-arbitrage model is of no value in forecasting o Joslin et al.: no-arb, in the absence of restrictions on risk premia, have no bite when estimating conditional expectations of the state vector o These are determined by μ and K of the true, or physical measure o No-arb restrictions boil down to existence of equivalent martingale dynamics, and when risk premia are unrestricted, the parameters μ q and K q are unrelated to their physical-measure counterparts o The two measures share volatility parameters, in a Gaussian setting these parameters do not affect ML estimates of μ and K. o No-arb restrictions affect the mapping from the state to yields: A, B o Duffee argues these parameters can be estimated with very high precision even if no-arb restrictions are not imposed, since the measurement equation amounts to a regression of yields on other yields 23

Do no arbitrage restrictions help? Restrictions on risk premia increase the precision of estimates of physical dynamics The reason is that equivalent-martingale dynamics are estimated with high precision, and risk premia restrictions tighten the relation between physical and equivalent-martingale dynamics E.g., given the task of estimating λ 0 and λ 1, one is to set to zero any parameters that are statistically indistinguishable from zero o Other approaches rely on information criteria, Bayesian shrinkage, weighted averages btw. physical and risk-neutral values of zero o Christensen, Diebold, and Rudebusch (2011, JoE) propose a dynamic model of the three factors of Nelson and Siegel, subject to no arbitrage However, the main avenue seems clear: to bring macroeconomic information into the problem to derive the SDF from fundamentals Macro-finance models follow Ang and Piazzesi (2003, JME) by expanding the measurement equation of the yields-only framework Assume we observe some variables at time t, stacked in a vector f t 24

Macroeconomic factors in no-arbitrage models Usually it contains macro variables, but may contain survey data: o No-arb restrictions apply only to the vector A y and the matrix B y o The key assumption is that the same state vector that determine the cross-section of yields also determine the additional variables o A difficulty with macro-finance models is that the macro variables of interest, such as inflation, aggregate consumption, and output growth, are not spanned by the term structure o Aside from measurement error, regressions of the macro factors on p linear combinations of contemporaneous yields should produce R 2 s of 1! 25

Macroeconomic factors in no-arbitrage models o The hypothesis that the term structure contains all information relevant for forecasting future macro is overwhelmingly rejected o The term structure has less information about future inflation and IP growth than is contained in the single lag of the macro variable Duffee (2011, RFS) and Joslin, Priebsch, and Singleton (2014, JF) develop a restricted Gaussian no-arbitrage models in which no linear combinations of yields can serve as the model s state vector They show that hidden factors must exist The term structure of bond yields is not a first-order Markov process, i.e., investors have information about future yields and future excess returns that is not impounded into current yields Hidden factors can explain the low R 2 s as long as the factor(s) that are hidden are revealed in macroeconomic data o E.g., imagine that economic growth suddenly stalls o Investors anticipate that short-term rates will decline o But investors risk premia also rise because of the downturn 26

Macroeconomic factors in no-arbitrage models Unfortunately, the causal linkages btw. macroeconomic fundamentals and excess bond returns are modest at best o These effects move long-term bond yields in opposite directions and if they happen to equal each other in magnitude, the current term structure is unaffected o Nothing in the term structure predicts what happens next to either economic growth or bond yields o However, lower growth should predict higher excess returns At quarterly frequency, excess Treasury bond returns are countercyclical o On average, the nominal yield curve slopes up and hence expected excess returns to Treasury bonds are positive 27

The forecasting power of forward rates: a puzzle Oddly, past forward rates contain information on risk premia o Estimated correlations in the table imply that they should be negative Is there really information about future excess returns that is not captured by the current term structure? Cochrane and Piazzesi (2005, AER) find that 5-month forward rates constructed with yields on maturities of 1 through 5 years contain substantial information about excess returns over the next year They conclude that lags of forward rates contain information about the excess returns that is not in month t forward rates. Even if there are hidden factors, it is hard to understand why the information is hidden from the time-t term structure but not hidden in earlier term structures CP suggest measurement error in yields that is averaged away by including additional lags of yields 28

The forecasting power of forward rates: a puzzle For the 1964 through 2003 sample, including lagged forward rates raises R 2 from 5.5 to 11%; the null that the coefficients on the forward rates are all zero is overwhelmingly rejected However, over the full sample, the incremental explanatory power of the forward rates is more modest 29

Conclusion Finance theory provides some guidance when forming forecasts of future interest rates The Holy Grail of this literature is a dynamic model that is parsimonious owing to economically-motivated restrictions The requirement of no-arbitrage is motivated by economics, but by itself it is too weak to matter Economic restrictions with bite require, either directly or indirectly, that risk premia dynamics be tied down by economic principles No restrictions from workhorse models of asset pricing appear to be consistent with the observed behavior of Treasury bond yields Another open question is whether any variables contain information about future interest rates that is not already in the current term structure o Recent empirical work suggests that both lagged bond yields and certain macroeconomic variables have incremental information, but the robustness of these results is not yet known 30