Modeling Yields at the Zero Lower Bound: Are Shadow Rates the Solution?

Transcription

1 Modeling Yields at the Zero Lower Bound: Are Shadow Rates the Solution? Jens H. E. Christensen & Glenn D. Rudebusch Federal Reserve Bank of San Francisco Term Structure Modeling and the Lower Bound Problem Day 2: The Lower Bound Problem Lecture II.2 European University Institute Florence, September 8, 2015 The views expressed here are solely the responsibility of the authors and should not be interpreted as reflecting the views of the Federal Reserve Bank of San Francisco or the Board of Governors of the Federal Reserve System. 1 / 36

2 Overview of Paper 2 / 36 We compare the shadow-rate arbitrage-free Nelson-Siegel (AFNS) model derived in Christensen and Rudebusch (2015a) to a standard Gaussian AFNS model. We find that, near the zero lower bound, the shadow-rate model provides: Better fit; Better short rate forecasts; Compression of yield volatility. Thus, we recommend shadow-rate models as a useful modeling tool that provides improvements over Gaussian models in the current low-yield environment.

3 Black s Shadow-Rate Approach 3 / 36 To account for the zero lower bound, Black (1995) proposed using standard tools to model a shadow rate, s t, that may be negative, while the observed short rate is truncated: r t = max{s t, 0}. This nonlinearity is numerically straightforward but computation can be prohibitively burdensome. Krippner (2013) provides an option-based approximation to Black s shadow-rate concept. Christensen and Rudebusch (2015a) use that approach to generate shadow-rate arbitrage-free Nelson-Siegel models.

4 The Shadow-Rate AFNS Model Class (1) where ω(τ) is a deterministic function of model parameters. 4 / 36 In the shadow-rate AFNS class of models, the shadow rate is s t = L t + S t, and the Q-dynamics of the factors X t = (L t, S t, C t ) are dl t θ Q 1 L t ds t = 0 λ λ θ2 Q S t dt +ΣdW Q dc t 0 0 λ θ3 Q C t where Σ is a constant matrix. The instantaneous shadow forward rate is f t (τ) = L t + e λτ S t +λτe λτ C t + A f (τ), where A f (τ) is an analytical function of model parameters. The nonnegative instantaneous forward rate is ( ft (τ) ) 1 ( f t (τ) = f t (τ)φ +ω(τ) exp 1 [ ft (τ) ] 2 ), ω(τ) 2π 2 ω(τ) t,

5 The Shadow-Rate AFNS Model Class (2) 5 / 36 The yield-to-maturity is defined the usual way as y t (τ) = 1 τ = 1 τ t+τ t t+τ t f t (s)ds [ f t (s)φ ( ft (s) ) 1 +ω(s) exp ω(s) 2π ( 1 2 [ ft (s) ] 2 )] ds. ω(s) This is the measurement equation in the Kalman filter. Since yields are nonlinear functions of the state variables, we use the standard extended Kalman filter. Later, I perform a simulation study to analyze the efficiency of the extended Kalman filter for estimating shadow-rate models.

6 The Real-World Dynamics Christensen and Rudebusch (2012) study the performance of various AFNS models on a long sample of weekly U.S. Treasury yields ( ). They favor the following specification of the P-dynamics: dl t ds t = κ P 21 κ P 22 κ P 23 L t S t dt+σ dc t 0 0 κ P 33 C t where Σ is a diagonal matrix. This is the transition equation in the Kalman filter. θ P 2 θ P 3 dw L,P t dw S,P t dw C,P t We study this model throughout the paper, denoted the CR model, and its shadow-rate equivalent referred to as the B-CR model following notation in Kim and Singleton (2012)., Note: Unit root imposed to address finite-sample bias. 6 / 36

7 Kalman Filter Estimation with a Unit Root (1) 7 / 36 Normally, we start the Kalman filter using the unconditional distribution of the state variables. However, when we impose a unit-root property on the Nelson-Siegel level factor, the joint dynamics of the state variables are no longer stationary. Thus, we cannot start the Kalman filter at the unconditional distribution. Instead, we follow Duffee (1999) and derive a distribution for the starting point of the Kalman filter based on the yields observed at the first data point in each sample.

8 Kalman Filter Estimation with a Unit Root (2) The model states that zero-coupon yields are given by y t = A+BX t +ε t, ε t N(0, H). For the first set of yield observations, this equation reads y 1 = A+BX 0 +ε 0 BX 0 = y 1 A ε 0. Now, multiply from the left on both sides with B to obtain B BX 0 = B (y 1 A) B ε 0. Then, we can isolate X 0 by using the inverse of B B: X 0 = (B B) 1 B (y 1 A) (B B) 1 B ε 0. Here, ε 0 is normally distributed with zero mean and covariance matrix H. Thus, X 0 follows a normal distribution: X 0 N[(B B) 1 B (y 1 A),(B B) 1 B HB(B B) 1 ]. Thus, this is the normal distribution used to start the Kalman filter when unit-root properties are imposed. 8 / 36

9 Data and Model Estimation 9 / 36 Both models are estimated on a weekly (Fridays) sample of H.15 (3-m, 6-m) and Gürkaynak, Sack, and Wright (2007) (1-, 2-, 3-, 5-, 7-, 10-yr) zero-coupon Treasury bond yields from January 4, 1985, to October 31, We do weekly re-estimations with endpoints starting in January 6, 1995, to the end of the sample to generate real-time yield decompositions. This should allow for a comprehensive study of model performance both in the normal period (Jan. 6, 1995 to Dec. 12, 2008) and in the zero lower bound (ZLB) period (Dec. 19, 2008 to Oct. 31, 2014). Here, I have updated most results through August 14, 2015.

10 U.S. Interest Rates Reach Effective Zero Lower Bound Rate in percent year yield 5 year yield 1 year yield 3 month yield Since 2009 zero lower bound problems are apparent. Swanson and Williams (2014) find that medium- and long-term Treasury yields responded to economic news in 2009 and 2010 in a traditional way. Still, short maturities appeared constrained. However, problem is severe since August / 36

11 Estimated Parameters from B-CR Model 11 / 36 K P K,1 P K,2 P K,3 P θ P Σ K1, P σ (0.0001) K2, P σ (0.1474) (0.1337) (0.0904) (0.0364) (0.0002) K3, P σ (0.1200) (0.0087) (0.0004) The estimated parameters are very similar across the two models. Also, they are consistent with the results for AFNS models reported elsewhere, say, in Christensen et al. (2011).

12 Tests of Parameter Restrictions in AFNS Models Likelihood ratio test Independent factor AFNS model CR model 95 percentile of chi^2 distribution, df = 4 95 percentile of chi^2 distribution, df = 6 FOMC 12/ In AFNS models, we can test the significance of parameter restrictions with standard likelihood ratio tests. The CR model appears to be sufficiently flexible to capture the variation in the data and greater parsimony does not seem warranted. 12 / 36

13 Tests of Parameter Restrictions in B-AFNS Models Quasi likelihood ratio test Independent factor B AFNS model B CR model 95 percentile of chi^2 distribution, df = 4 95 percentile of chi^2 distribution, df = 6 FOMC 12/ In shadow-rate AFNS models, we test the significance of parameter restrictions with quasi likelihood ratio tests. The B-CR model appears to be sufficiently flexible to capture the variation in the data and again greater parsimony does not seem warranted. 13 / 36

14 How Good is the Option-Based Approximation? Average Absolute Simulation Error in Shadow Yields and Average Absolute Approximation Error in Observed Yields: Averaged over 7 dates last observation in each year ( ). Maturity in months Shadow yields Yields We assess the approximation provided by the option-based B-CR model via Monte Carlo simulations. We simulate 50,000 ten-year-long factor paths and calculate observed and shadow zero-coupon bond yields. Differences (in basis points) are small. Thus, we conclude that the option-based B-CR model provides a very close approximation to the Black model for our data. 14 / 36

15 Value of Option to Hold Currency Rate in basis points Five year Treasury yield Ten year Treasury yield Bear Stearns rescue 3/ FOMC 12/ FOMC 8/ The spread between the yield that respects the ZLB and the unconstrained yield of the shadow bond can be interpreted as the value of the option to hold currency. Since 2009 the option has had sustained material value. Thus, accounting for the ZLB should matter. 15 / 36

16 Forecast of Negative Short Rates 16 / 36 Probability percent line FOMC 12/ Any affine Gaussian dynamic term structure model places positive probabilities on future negative interest rates. Here, we exemplify this with three-month forecasts of the short rate from rolling estimations of the CR model.

17 The Choice of Lower Bound 17 / 36 In general, the lower bound can be nonzero, i.e. r t = max{r min, s t }. In that case, the formula for the forward rate that respects the r min lower bound is: ( ft (τ) r ) min 1 f t (τ) = r min +(f t (τ) r min )Φ +ω(τ) exp ω(τ) 2π ( 1 2 where the shadow forward rate, f t (τ), and ω(τ) remain as before. [ ft (τ) r ] 2 ) min, ω(τ) We make a comprehensive comparison between fixing the lower bound, r min, at zero or leaving as a free parameter to be determined in the estimation of the B-CR model.

18 Sensitivity of Shadow Rate Rate in percent CR model B CR model, r(min) = 0 B CR model, r(min) free Rate in percent B AFNS(2) model B AFNS(3) model B AFGNS model Right chart shows the estimated shadow-rate paths from the B-CR model with and without the lower bound fixed at zero, and the estimated short-rate path from the CR model. Left chart shows the estimated shadow-rate paths from independent-factor B-AFNS(2), B-AFNS(3), and five-factor B-AFGNS models. Note that shadow rates are sensitive both to the choice of lower bound and to the number of state variables. 18 / 36

19 Importance of Choice of Lower Bound Parameter estimate FOMC 12/ Quasi likelihood ratio test FOMC 12/ percentile of chi^2 distribution, df = Right chart shows estimated values of the lower bound. Left chart shows quasi likelihood ratio tests of its significance. While a positive value for the lower bound has statistical support, we find no practical value of the added flexibility. Thus, consistent with theory and Treasury yield data, we recommend fixing it at zero. 19 / 36

20 Model Fit near the ZLB Model fit is almost identical in the normal period as one could expect. Some notable differences in favor of the shadow-rate models in the recent ZLB period. 20 / 36 Normal period (Jan. 6, 1995-Dec. 12, 2008) RMSE Maturity in months All yields CR B-CR, r min = B-CR, r min free ZLB period (Dec. 19, 2008-Oct. 31, 2014) RMSE Maturity in months All yields CR B-CR, r min = B-CR, r min free

21 Parameter Stability Parameter estimate CR model B CR model, r(min) = 0 B CR model, r(min) free FOMC 12/ Parameter estimate CR model B CR model, r(min) = 0 B CR model, r(min) free FOMC 12/ In addition to model fit, the estimated parameters are also hard to distinguish in the normal period κ P 33 and σ 11 are examples. However, we do see some wedges in the ZLB period. More importantly, some of the estimated parameters do change quite notably in the ZLB period within both models Fixing parameters at pre-crisis levels may not be appropriate. 21 / 36

22 Parameter Sensitivity of Short-Rate Projections Rate in percent B CR parameter, 12/ CR parameter, 12/ CR parameter, 10/ B CR parameter, 10/ B CR state (12/ ) = (0.0382, , ) CR state (12/ ) = (0.0382, , ) CR state (10/ ) = (0.0382, , ) B CR state (10/ ) = (0.0370, , ) Using pre-crisis versus current parameter estimates, we document both statistically and economically significant differences. Thus, real-time updating with rolling estimations is important. 22 / 36

23 Compression of Yield Volatility near the ZLB Rate in basis points CR model B CR model, r(min) = 0 B CR model, r(min) free Three month realized volatility of three month yield Rate in basis points CR model B CR model, r(min) = 0 B CR model, r(min) free Three month realized volatility of two year yield Shown are the predicted three-month conditional volatility of the three-month (right) and two-year (left) Treasury yield with a comparison to a measure of realized yield volatility based on daily yield changes. The B-CR model can replicate the compression in volatility of short- and medium-term yields since A nonzero lower bound can produce extremely low short-term yield volatilities. 23 / 36

24 Yield Forecasts in Shadow-Rate Models n=1 y(x n T,τ). 24 / 36 Input: Estimated K P, Σ, θ P, λ, and filtered X t. The continuous-time P-dynamics are given by dx t = K P ( θ P X t )dt + ΣdW P t. The conditional mean of X T under the P-measure: E P t [X T ] = (I exp( K P (T t)))θ P + exp( K P (T t)) X t. The conditional covariance matrix of X T under the P-measure: V P t [X T ] = T t 0 e K P s Σ Σ e ( K P ) s ds. We simulate N = 1,000 draws directly at the desired horizon: ( ) XT n N Et P [X T ], Vt P [X T ], n = 1,..., The resulting conditional zero-coupon bond yield forecast is: Et P [y T (τ)] = 1 N N

25 Yield Forecasts in the Normal Period Three-month yield Six-month One-year Mean RMSE Mean RMSE RW CR model B-CR model, r min = B-CR model, r min free Two-year yield Six-month One-year Mean RMSE Mean RMSE RW CR model B-CR model, r min = B-CR model, r min free Ten-year yield Six-month One-year Mean RMSE Mean RMSE RW CR model B-CR model, r min = B-CR model, r min free / 36

26 Yield Forecasts in the ZLB Period Three-month yield Six-month One-year Mean RMSE Mean RMSE RW CR model B-CR model, r min = B-CR model, r min free Two-year yield Six-month One-year Mean RMSE Mean RMSE RW CR model B-CR model, r min = B-CR model, r min free Ten-year yield Six-month One-year Mean RMSE Mean RMSE RW CR model B-CR model, r min = B-CR model, r min free / 36

27 Short-Rate Projections in Shadow-Rate Models (1) 27 / 36 In affine models, the conditional expectation of X t is E P t [X t+τ ] = (I exp( K P τ))θ P + exp( K P τ)x t. In shadow-rate AFNS models, the instantaneous shadow rate is s t = L t + S t. Thus, the conditional expectation of the shadow-rate process is E P t [s t+τ ] = E P t [L t+τ + S t+τ ] = ( ) E P t [X t+τ ]. However, we are interested in the conditional expectation of the short rate: E P t [r t+τ ] = E P t [max(r min, s t )].

28 Short-Rate Projections in Shadow-Rate Models (2) We are interested in the conditional expectation of the short rate: E P t [r t+τ ] = E P t [max(r min, s t )]. The conditional covariance matrix of the state variables is: V P t [X t+τ ] = τ 0 e K Ps ΣΣ e (K P ) s ds. Hence, the conditional covariance of the shadow-rate process is Vt P [s t+τ] = ( ) 1 Vt P [X t+τ] 1. 0 Now, the conditional expectation of the short rate is given by Et P [r t+τ] = r t+τ f(r t+τ X t)dr t+τ = r min + (s t+τ r min )f(s t+τ X t)ds t+τ r ( min ) = r min +(Et P (Et P [s t+τ] r min ) [s t+τ] r min )N V P t [s t+τ] ( ) + 1 Vt P [s t+τ] exp 1 (Et P [s t+τ] r min ) 2. 2π 2 Vt P [s t+τ] 28 / 36

29 Forecast of Future Target Rates Normal period Six-month One-year Mean RMSE Mean RMSE RW KW model CR model B-CR model, r min = B-CR model, r min free ZLB period Six-month One-year Mean RMSE Mean RMSE RW KW model CR model B-CR model, r min = B-CR model, r min free Here, we study the models ability to forecast future policy target rates and include the Kim & Wright (2005, KW) model maintained at the Federal Reserve Board in the comparison. The B-CR model is competitive at forecasting future target rates in real time, both in the normal and the ZLB period. 29 / 36

30 Short-Rate Forecasts Also, the KW model is not reliable either since / 36 Rate in percent KW model CR model B CR model, r(min) = 0 B CR model, r(min) free Realized target rate FOMC 12/ This figure illustrates the real-time forecasts one year ahead of the short rate from the CR and B-CR models with a comparison to the KW model. The CR and B-CR models are indistinguishable prior to crisis and differences are minor until August However, since then the CR model has been a little erratic.

31 Short-Rate Expectations and Futures Rates (1) Rate in percent FOMC 12/ KW model CR model B CR model, r(min) = 0 B CR model, r(min) free Federal funds futures rate Rate in percent FOMC 12/ KW model CR model B CR model, r(min) = 0 B CR model, r(min) free Federal funds futures rate Illustration of the one-year (left) and two-year (right) short rate projections from the standard CR and shadow-rate B-CR models since 2007 with a comparison to the KW model. Also shown are 12- and 24-month federal funds futures rates that are used as a rough benchmark to assess the appropriateness of the variation in each model. 31 / 36

32 Short-Rate Expectations and Futures Rates (2) 32 / 36 One-year contract Two-year contract Mean RMSE Mean RMSE KW model CR model B-CR model, r min = B-CR, r min free Reported are the mean and root mean squared differences relative to the one- and two-year federal funds futures rates for the period from January 5, 2007, to August 14, The short-rate expectations from the B-CR model are clearly competitive relative to the other models.

33 Average Short-Rate Expectations and Term Premiums For a start, the term premium is defined as TP t (τ) = y t (τ) 1 τ In the CR model, it holds that 1 τ t+τ t ( Et P [r s]ds = θ2 P + TP t(τ) = A(τ) τ 1 κp 21 κ P 22 t+τ t ) L t + κp 21 κ P 22 κp 23 ( 1 e κ P 33 τ κ P 22 κp 33 κ P 33 τ E P t [r s ]ds. 1 e κp 22 τ κ P 22 τ P L t + 1 e κ 22 τ κ P 22 τ (S t θ2) P P 1 e κ 22 τ ) κ P 22 τ (C t θ3), P + κp 21 (1 1 e κp 22 τ ) ( 1 e λτ κ P 22 κ P 22 τ L t + λτ [ 1 e κ P 33 τ ( 1 e λτ + e λτ + λτ κ P 23 P 1 e κ 22 τ ) κ P 22 τ S t P 1 e κ 22 τ ]) κ P 22 τ C t κ P 22 κp 33 κ P 33 τ (1 1 e κp 22 τ ) κ P 22 τ θ2 P κ P ( 23 1 e κ P 33 τ P κ P 22 κp 33 κ P 33 τ 1 e κ 22 τ ) κ P 22 τ θ3. P In the B-CR model, 1 t+τ τ t Et P [r s ]ds has to be approximated by numerically integrating the formula for Et P [r s ]. 33 / 36

34 Average Expected Short Rates the Next Ten Years Rate in percent KW model CR model B CR model, r(min) = 0 B CR model, r(min) free SPF ten year forecast FOMC 12/ Shown is the average expected short rate the next ten years. Also shown is the expected annual average return over the next ten years from holding 3-m T-Bills according to the SPF. The KW model tracks survey expectations closely by construction Its policy expectations are relatively high. 34 / 36

35 Ten-Year Term Premiums Rate in percent KW model CR model B CR model, r(min) = 0 B CR model, r(min) free FOMC 12/ Due to the high policy expectations in the KW model, its term premiums remain low by historical standards. The B-CR model delivers more plausible estimates relative to where we are in the monetary policy cycle. 35 / 36

36 Conclusion 36 / 36 We compare the shadow-rate AFNS model derived in Christensen and Rudebusch (2015a) with a matching standard AFNS model. We find that, while providing equal performance in normal times, the shadow-rate model stands out near the ZLB: Better fit; More accurate forecasts of future short rates; Compression of yield volatility. Questions remain about the extent to which other models that account for the unique yield dynamics near the ZLB can improve performance even further. However, for now, we recommend shadow-rate models as a useful modeling tool that provides improvements over Gaussian models in the current low-yield environment.