Decomposing Real and Nominal Yield Curves

Transcription

1 Decomposing Real and Nominal Yield Curves Abrahams, Adrian, Crump, Moench Emanuel Moench Deutsche Bundesbank Frankfurt-Fudan Financial Research Forum September 25, 2015 The views expressed in this presentation are those of the authors and not necessarily those of the Bundesbank, the Eurosystem, the Federal Reserve Bank of New York or the Federal Reserve System. Emanuel Moench (Bundesbank) Decomposing Real and Nominal Yield Curves September / 34

2 Introduction Motivation Introduction Breakevens: market-based measure of inflation expectations closely watched by policy makers and market participants. Distortions: 1 Liquidity premia. 2 Inflation risk premia. Need to adjust for both to impute market-based expectations of future inflation, real short rates. Potentially need many factors to precisely fit both cross-sections, traditional estimation challenging. Emanuel Moench (Bundesbank) Decomposing Real and Nominal Yield Curves September / 34

3 Introduction What This Paper Does What We Do Joint affine model for TIPS and Treasuries explicitly taking into account TIPS relative illiquidity. Regression-based estimation approach (Adrian, Crump, Moench 2013). Model fits both cross-sections very precisely, easy to implement, forecasts inflation well out-of-sample. Use model to: 1 Study economic underpinnings of inflation risk premia. 2 Predict inflation with breakevens adjusted for liquidity, inflation risk. 3 Decompose conventional monetary policy effects on long-term real forwards. 4 Decompose effects of LSAPS Emanuel Moench (Bundesbank) Decomposing Real and Nominal Yield Curves September / 34

4 Introduction Estimating Affine Term Structure Models Affine Term Structure Models: Nontechnical Summary Yields of all maturities are linear (affine) functions of a small set of pricing factors: y (n) t = a n + b n,1 X 1,t b n,k X k,t + u t Parameters a and b are related across maturities n to ensure consistency of time series and cross section. This is achieved by imposing no-arbitrage. Price of an n-period bond today equals expected discounted price of (n 1) period bond tomorrow: [ ] P (n) t = E t P (n 1) + risk adjustment t+1 Emanuel Moench (Bundesbank) Decomposing Real and Nominal Yield Curves September / 34

5 Model and Estimation Holding Period Returns Excess Log Holding Period Returns - Nominal Bonds Using standard assumptions in affine DTSM, excess one-period holding returns are given by Nominal bonds: where rx (n) t+1 = log P(n 1) t+1 log P (n) t + log P (1) t = a n + c n X t + β nx t+1 a n = B n µ 1 2 B nσb n, c n = B n Φ, β n = B n. Emanuel Moench (Bundesbank) Decomposing Real and Nominal Yield Curves September / 34

6 Model and Estimation Holding Period Returns Excess Log Holding Period Returns - Real Bonds Similarly, real bonds (+ inflation): where a n,r = rx (n) t+1,r + π t+1 = log P (n 1) R,t+1 log P(n) R,t + log P(1) R,t = a n,r + c n,r X t + β n,rx t+1 ( (B n,r + π 1 ) µ + 1 ) 2 (B n,r + π 1 ) Σ (B n,r + π 1 ). c n,r = (B n,r + π 1 ) Φ, β n,r = (B n,r + π 1 ) Emanuel Moench (Bundesbank) Decomposing Real and Nominal Yield Curves September / 34

7 Model and Estimation Estimation Regression-Based Estimation Treating pricing factors as known, estimation can be done in three simple steps (see also Adrian, Crump, Moench JFE 2013) 1 Estimate physical dynamics by running VAR(1) on the factors. 2 Run time series regressions of excess bond returns on cst, contemporaneous and lagged pricing factors obtain betas and predictive slopes. 3 Run cross-sectional regression of predictive slopes on betas to obtain parameters governing factor dynamics under pricing measure. This estimation approach is instantaneous, robust, and allows for large number of factors. Emanuel Moench (Bundesbank) Decomposing Real and Nominal Yield Curves September / 34

8 Model and Estimation Estimation Data Liquidity factor: Simple average of 1 Nelson-Siegel-Svensson yield curve fitting error for TIPS. 2 Primary dealer transaction volume of TIPS vs. Treasuries. Gurkaynak, Sack, Wright NSS yield curves for Treasuries (2007) and TIPS (2010) Preferred spec: Liquidity factor, three PCs from nominal Treasuries, two from TIPS orthogonalized wrt nominal PCs and liquidity factor. Treasury returns for n = 6, 12, 18,..., 120, TIPS returns for n R = 24, 36, 42,..., 120 π : non-seasonally adjusted headline CPI. Emanuel Moench (Bundesbank) Decomposing Real and Nominal Yield Curves September / 34

9 Model and Estimation Liquidity Liquidity Factor Year Emanuel Moench (Bundesbank) Decomposing Real and Nominal Yield Curves September / 34

10 Model and Estimation Model Fit Model Fit - Time Series 7 Observed and Model Implied Treasury Yield of Maturity n = 120 Months 5 Observed and Model Implied TIPS Yield of Maturity n = 120 Months Observed Treasury Yield Fitted Treasury Yield Year 1 Observed TIPS Yield Fitted TIPS Yield Year Model fits both term structures with little error. Avg yield pricing errors < 2.5 bps for nominals and < 2 bps for TIPS. No serial correlation in return fitting errors. Emanuel Moench (Bundesbank) Decomposing Real and Nominal Yield Curves September / 34

11 Model and Estimation Risk Premia Breakeven Decomposition 5 10 Year Forward Decomposition Year Breakeven 5-10-Year Inflation Risk Premium 5-10-Year Liquidty Component of Breakeven 5-10-Year Expected Inflation 5-10-Year BCFF Survey Expected Inflation Year Dynamics of 5-10y forward breakevens mainly driven by IRP. Substantial liquidity premia, especially before 2004 and in crisis. Long-term infl. expectations essentially flat, consistent with surveys. Emanuel Moench (Bundesbank) Decomposing Real and Nominal Yield Curves September / 34

12 Model and Estimation Risk Premia Breakeven Decomposition - ctd y Forward Treasury and TIPS Term Premiums Nominal Term Premium Real Term Premium IRP Year Nominal term premia largely accounted for by real term premia. Inflation risk premium relatively small. Emanuel Moench (Bundesbank) Decomposing Real and Nominal Yield Curves September / 34

13 Model and Estimation Risk Premia Economics of Inflation Risk Premia SMOVE DISAG 5 10 Year Inflation Risk Premium and 3 month Swaption MOVE 5 10 Year Inflation Risk Premium and 4 quarter Blue Chip Disagreement month swaption MOVE (LHS) Blue Chip 4Q CPI Disagreement (LHS) 5 10 Year IRP Forward (RHS) 5 10 Year IRP Forward (RHS) Estimated IRP highly correlated with Swaption-impled Treasury volatility, Forecaster disagreement about future inflation, Business cycle indicators, flows into government bond funds. Emanuel Moench (Bundesbank) Decomposing Real and Nominal Yield Curves September / 34

14 Model and Estimation Risk Premia Inflation Forecasting 12m 24m 36m Panel A: In Sample Model Forecast Breakevens Random Walk Forecast Panel B: Out of Sample Model Forecast Breakevens Random Walk Forecast Risk-adjustment of breakevens strongly improves inflation prediction. Out-of-sample: 1999: :12 training sample, then recursively re-estimate from 2004: :01. OOS up to 20% reduction of RMSE s relative to unadjusted BEIs. Emanuel Moench (Bundesbank) Decomposing Real and Nominal Yield Curves September / 34

15 Model and Estimation Monetary Policy Analysis Decomposing the Effects of LSAPs What are the channels through which LSAPs work? Portfolio balance effect: LSAPs reduce long-term Treasuries held by private sector Term premia, yields decline.(e.g. Gagnon, Raskin, Remache, Sack 2011). Alternative channel: LSAPs affect short rate expectations, signal lower future short rates (e.g. Bauer&Rudebusch 2012). We use an event-study analysis as in GRRS for 15 announcement dates (Woodford 2012) and decompose daily yield changes into their various components. Emanuel Moench (Bundesbank) Decomposing Real and Nominal Yield Curves September / 34

16 Model and Estimation Monetary Policy Analysis Decomposing the Effects of LSAPs - ctd. Two-year Treasury y N TP N E[i] y R TP R E[r] Liq R BEI IRP E[π] Liq BEI All LSAP Events LSAP t-stat (-3.30) (-6.37) (0.17) (-3.71) (-6.11) (-0.32) (1.62) (1.71) (5.21) (0.44) (-1.62) Individual LSAP Events LSAP LSAP MEP LSAP Ten-year Treasury y N TP N E[i] y R TP R E[r] Liq R BEI IRP E[π] Liq BEI All LSAP Events LSAP t-stat (-4.08) (-4.93) (0.68) (-7.25) (-7.13) (1.43) (1.62) (2.12) (3.46) (-1.14) (-1.62) Individual LSAP Events LSAP LSAP MEP LSAP Emanuel Moench (Bundesbank) Decomposing Real and Nominal Yield Curves September / 34

17 Model and Estimation Monetary Policy Analysis Decomposing the Effects of LSAPs - ctd. LSAPs primarily work through large reduction of real term premia. Inflation risk premium rises somewhat. Expected inflation not significantly changed. Risk-adjusted expected future short rates rise slightly: no support for signaling channel. Emanuel Moench (Bundesbank) Decomposing Real and Nominal Yield Curves September / 34

18 Model and Estimation Monetary Policy Analysis Long-Term Real Forwards and Monetary Policy Hanson-Stein (2014), Nakamura-Steinsson (2014) find that long-term real forwards move substantially in response to monetary policy changes. Inconsistent with long-run money neutrality unless explained by term premia. We use our model to decompose reaction of long-term forwards to conventional monetary policy shocks into expectations and risk premium components. Identify policy shocks from fed funds futures as in Kuttner (2001), Bernanke and Kuttner (2005). Restrict sample to , prior to ZLB. Emanuel Moench (Bundesbank) Decomposing Real and Nominal Yield Curves September / 34

19 Model and Estimation Monetary Policy Analysis Long-Term Real Forwards and Monetary Policy - ctd. Response of Nominal Forwards Response of Real Forwards Change in Forward (%) Change in Forward (%) Maturity (months) Maturity (months) Response of Risk Neutral Real Forwards Response of Forward Real Term Premium Change in Forward (%) Change in Forward (%) Maturity (months) Maturity (months) Emanuel Moench (Bundesbank) Decomposing Real and Nominal Yield Curves September / 34

20 Model and Estimation Monetary Policy Analysis Long-Term Real Forwards and Monetary Policy - ctd. Long-term real forwards react to monetary policy shocks primarily because of real term premia. Risk-adjusted long-term real forward rates hardly affected. Corroborates conjecture of Hanson and Stein (2014), also in line with Nakamura/Steinsson (2014). Consistent with risk-taking channel of monetary policy. Emanuel Moench (Bundesbank) Decomposing Real and Nominal Yield Curves September / 34

21 Conclusion Conclusion Fast, transparent estimation of joint nominal/real ATSM with little fitting error. Decompose breakeven inflation into liquidity premium, inflation risk premium, and expected inflation. Use model to analyze pertinent monetary policy questions: LSAPs primarily reduce real term premia: no evidence of signaling channel. Conventional monetary policy also mainly moves real term premia: risk-taking channel of monetary policy. Emanuel Moench (Bundesbank) Decomposing Real and Nominal Yield Curves September / 34

22 Appendix Market Prices of Risk λ 0 λ 1.1 λ 1.2 λ 1.3 λ 1.4 λ 1.5 λ 1.6 W Λ W λ1 X *** ** ** *** (-0.129) (-0.654) (-2.862) (-1.513) (-0.524) (2.486) (0.304) (0.014) (0.007) X ** *** 0.059*** *** *** (-0.031) (0.457) (-2.327) (-0.496) (-2.891) (2.632) (0.727) (0.004) (0.002) X *** ** ** *** *** (0.021) (0.075) (1.354) (-2.739) (-2.523) (-0.005) (-2.327) (0.002) (0.001) X (0.002) (0.649) (0.644) (0.195) (0.082) (0.184) (-1.365) (0.956) (0.913) X ** *** *** *** (0.006) (0.514) (2.381) (-0.789) (1.616) (-4.975) (-0.182) (0.000) (0.000) X ** (0.003) (1.078) (1.108) (1.162) (0.477) (-0.793) (-2.396) (0.471) (0.359) Level, slope, curvature, and second real factor risk priced. Real pricing factors drive time variation in price of level, slope, curvature risk. Exposure to first real PC and liquidity factor not priced in cross-section. Emanuel Moench (Bundesbank) Decomposing Real and Nominal Yield Curves September / 34

23 Appendix Related Literature Related Literature Real and Nominal Term Structures: Chen and Cheng (2005), Hördahl and Tristani (2008), Adrian and Wu (2009), Campbell, Shiller, Viceira (2009), Christensen, Lopez, Rudebusch (2010), Garcia and Werner (2010), Haubrich, Pennacchi, Ritchken (2012), Grishchenko and Huang (2012). TIPS Liquidity Premium: Sack and Elsasser (2004), Dudley, Roush and Ezer (2009), D Amico, Kim and Wei (2010), Pflueger and Viceira (2011). TIPS Optionality: Grishchenko, Vanden, Zhang (2011). Analysis of LSAPs/monetary policy shocks: Gagnon, Raskin, Remachem Sack (2011), Krishnamurthy and Vissing-Jorgensen (2011), Woodford (2012) Hanson and Stein (2012). Emanuel Moench (Bundesbank) Decomposing Real and Nominal Yield Curves September / 34

24 Estimation Affine Term Structure Models: Estimation Traditional estimation by state space methods. Numerical difficulties: High-dimensional nonlinear numerical optimization. Serial dependence of pricing errors. Standard errors. Alternative recent approaches: OLS+numerical: Joslin, Singleton, Zhu (2011), Hamilton-Wu (2011). Asymptotic LS and iteration: Diez de los Rios (2013). OLS only: Adrian-Crump-Moench (2013). Emanuel Moench (Bundesbank) Decomposing Real and Nominal Yield Curves September / 34

25 Model ATSM for Real and Nominal Bonds An ATSM for Real and Nominal Bonds: Assumptions K 1 vector of pricing factors: Nominal pricing kernel: M t = exp Prices of risk as in Duffee (2002): X t+1 µ = Φ (X t µ) + ν t+1 ( r t 12 λ tλ t λ tσ 1/2 ν t+1 ). λ t = Σ 1/2 (λ 0 + λ 1 X t ). Emanuel Moench (Bundesbank) Decomposing Real and Nominal Yield Curves September / 34

26 Model ATSM for Real and Nominal Bonds Imposing No-Arbitrage No-arbitrage implies: P (n) t = E Q t [exp ( r t... r t+n 1 ) 1]. [ P (n) t,r = E Q t exp ( r t... r t+n 1 ) Q ] t+n. Q t Solving one obtains exponentially affine pricing for Treasuries and TIPS: log P (n) t = A n + B nx t. log P (n) t,r = A n,r + B n,rx t. Inflation and nominal short rate linear function of pricing factors: r t = δ 0 + δ 1X t. π t = π 0 + π 1X t. Emanuel Moench (Bundesbank) Decomposing Real and Nominal Yield Curves September / 34

27 Model Pricing Pricing Functions Recursive pricing functions for Treasuries: A n = A n 1 + B n 1 µ B n 1ΣB n 1 δ 0 B n = B n 1 Φ δ 1 Recursive pricing functions for TIPS: A n,r = A n 1,R + (B n 1,R + π 1 ) µ (B n 1,R + π 1 ) Σ (B n 1,R + π 1 ) B n,r = (B n 1,R + π 1 ) Φ δ 1 where µ = (I Φ) µ λ 0, Φ = Φ λ 1. Emanuel Moench (Bundesbank) Decomposing Real and Nominal Yield Curves September / 34

28 Model Estimation Estimation - Step 1 1.a) Compute time series average of pricing factors ˆµ. 1.b) Regress demeaned pricing factors onto their own lags ˆΦ. ˆν are the residuals from this regression and ˆΣ their sample variance covariance matrix. 1.c) Regress nominal short rate on constant and pricing factors: ˆδ 0, ˆδ 1. Emanuel Moench (Bundesbank) Decomposing Real and Nominal Yield Curves September / 34

29 Model Estimation Estimation - Step 2 2) Time series regressions of excess returns on cst, X t, and ˆν t+1 : rx (n) t+1 = a n 1 + c n 1 X t + β n 1X t+1 + e (n 1) t+1 rx (n 1) t+1,r + π t+1 = a n 1,R + c n 1,R X t + β n 1,RX t+1 + e (n 1) t+1,r. Stack estimates ˆβ n, ˆβ n,r, and ĉ n, ĉ n,r into matrices B and C. Stack coefficients â n ˆβ n ˆΣ ˆβ n, â n,r ˆβ n,r ˆΣ ˆβ n,r into the matrix A. Collect residuals ê (n) t+1 and ê(n) t+1,r variance-covariance matrix ˆΣ e. and compute their sample Emanuel Moench (Bundesbank) Decomposing Real and Nominal Yield Curves September / 34

30 Model Estimation Estimation - Step 3 3) Cross-sectional Generalized Least Squares regression for Φ and µ. We know from the recursive pricing formulas that [ ] β [ µ Φ] [ a + 1 = 2 β vec(σ) a R β R vec(σ) β R or B Ψ = [A C] Then, we can simply estimate Φ and µ as: ˆ Φ gls = (ˆB ols ˆ µ gls = (ˆB gls ) 1 ˆΣ e ˆB ols ˆB ols 1 ˆΣ e B Φ ols. ] c c R ) ( 1 1 ˆΣ e ˆB gls ˆB gls 1 ˆΣ e ˆα gls + 1 ) 2 ˆγ gls In the paper, we provide the asymptotic variance of λ 0 and λ 1 where µ = (I K Φ) µ X λ 0 and Φ = Φ λ 1 Emanuel Moench (Bundesbank) Decomposing Real and Nominal Yield Curves September / 34

31 Model Estimation Model-Implied Yields We can feed the estimated parameters back into the recursive pricing functions to obtain fitted yields of all nominal yields. If we want to compute the implied TIPS curve, we still need to estimate π 0 and π 1 : Given factors and all other parameters, TIPS yields are quadratic functions of π 1. Easy to find numerical solution. Use T n R observations on TIPS to pin down π 0 and π 1. Emanuel Moench (Bundesbank) Decomposing Real and Nominal Yield Curves September / 34

32 Model Further Results TIPS Factor Loadings X1 X2 X3 X4 X5 X6 Model Implied TIPS Factor Loadings X1 X2 X3 X4 X5 X6 Model Implied TIPS Return Factor Loadings Maturity Maturity Level factor largest driving force behind real yields. Liquidity and slope factors most important for excess returns on real yields. Emanuel Moench (Bundesbank) Decomposing Real and Nominal Yield Curves September / 34

33 Model Further Results UK Term Structure - Fit UK Nominal 10-year Yield UK Inflation-Indexed 10-year Yield 14 Observed Fitted 5 Observed Fitted Year Year specification fits both nominal and real UK yield curves well. Emanuel Moench (Bundesbank) Decomposing Real and Nominal Yield Curves September / 34

34 Model Further Results UK Term Structure - Breakeven Decomposition 10-year Term Premium Comparison 10-year Breakeven Decomposition Breakevens IRP E P [π] Nominal Term Premium Real Term Premium IRP Year Year UK inflation risk premium dropped markedly when BoE announced inflation target. Emanuel Moench (Bundesbank) Decomposing Real and Nominal Yield Curves September / 34