Term Structure Models with Negative Interest Rates Yoichi Ueno Bank of Japan Summer Workshop on Economic Theory August 6, 2016 NOTE: Views expressed in this paper are those of author and do not necessarily reflect those of the Bank of Japan or the Institute for Monetary and Economic Studies. 1
Background The total amount of fixed-rate sovereign debt trading at negative yields is $10.4 trillion ($7.3 trillion long term and $3.1 trillion short term) as of May 31 (Fitch, 2016). It had been assumed that nominal interest rates could not fall below zero as long as people could hold currency. Recent episodes, however, show that negative-yielding government bonds can coexist with currency. The power of arbitrage between government bonds and currency is not so strong as to forbid bonds yields falling below zero, although it is proposed that arbitrage still works to the extent that there exists a negative limit that nominal interest rates cannot go beyond (Viñals et al. (2016), Witmer and Yang (2016)). 2
Background (cont.) After the introduction of the negative IOER, government bond yields not only in shorter terms but also in longer terms have fallen below zero. Furthermore, government bond yields in various terms have become more deeply negative than the IOER. Switzerland Germany Japan 1.2% 0.6% -0.2% -0.4% -0.6% -0.8% -1.0% IOER -0.75% 3M 6M 1Y 2Y 3Y 5Y 7Y 10Y 1.0% 0.8% 0.6% 0.4% 0.2% -0.2% -0.4% -0.6% -0.8% IOER -0.1% IOER -0.2% IOER -0.3% IOER -0.4% 3M 6M 1Y 2Y 3Y 5Y 7Y 10Y 0.4% 0.2% -0.2% -0.4% IOER -0.1% 3M 6M 1Y 2Y 3Y 5Y 7Y 10Y 3
Background (cont.) The negative interest rate policy is conducted together with other unconventional monetary policy measures such as quantitative easing and/or forward guidance. This combination of unconventional policy measures causes some difficulties in evaluating the single effect of each policy measure. Besides, negative interest rates in nominal terms had been thought to be unreal, so theories and models to deal with negative interest rates are underdeveloped. 4
Contribution Develop a model to evaluate the effects of unconventional monetary policy measures including the negative interest rate policy on government bond term structures. Generalize two popular models, the Gaussian affine model and the Black model. The main difference between the two popular models is how they deal with non-negativity of nominal interest rates, or the power of arbitrage between government bonds and cash. Arbitrage between bonds and cash still works in the newly proposed model (Extended model). But, it is not so powerful as to prohibit bond yields becoming lower than the interest rate on cash or reserves. 5
Contribution (cont.) Propose an efficient and accurate solution method able to apply to both the Black model and the Extended model. Show that the Extended model is superior to the Gaussian affine model and the Black model by estimation results using government bond term structure data from Switzerland, Germany and Japan. Quantify each effect of forward guidance, quantitative easing and the negative policy interest rate. Find that the power of arbitrage between money or reserves and government bonds moves in tandem with basis swap spreads. 6
Model i t = s t 1 {st y t } + {φ t s t + (1 φ t )y t }1 {st <y t } s t = ρx t Q dx t = κ x (θ x x t )dt + σ x dw x,t Q dy t = σ y dw y,t φ t = φ t 1 {0 φt 1} + 1 {1 φt }, dφ t = σ φ dw Q φ,t. λ t = λ 0 + λ 1 x t dw t P = λ t dt + dw t Q 7
Figure 1: Relationship between the nominal short rate and shadow rate Gaussian affine model Nominal short rate 45 Black model Nominal short rate Shadow rate Extended model Nominal short rate 45 Shadow rate 45 Shadow rate 8
Figure 2: Cumulative probability distribution function of nominal short rate 1.0 0.9 0.8 0.7 0.6 0.5 0.4 Extended Model φ=0.1 Extended Model φ=0.5 Extended Model φ=0.9 Black Gaussian Affine 0.3 0.2 0.1 0.0 Note: one-factor model, s t = 0, y t = 0, E t [s τ ] = 0.01,Var t [s τ ] = 0.02 2. 9
Approximation methods Government bond price, P τ, and yield, R τ, at maturity τ τ P τ E Q [exp ( i t dt)], 0 R τ log(p τ )/τ. Priebsch (2013): 2nd order approximation of bond yields; R τ 1 τ τ (EQ [ i t dt 0 τ ] 0.5Var Q [ i t dt 0 ]), 10
Approximation methods (cont.) New approximation method I τ I τ α 0 + α 1 i1 4 τ + α 2i3 4 τ, P τ E Q [exp( I )]. τ R τ log(e Q [exp( I )])/τ, τ The parameters α 0 α 1 α 2 are determined by minimizing the mean squared error as follows; s. t. min E Q [(I τ I ) 2 τ ] E Q [I τ ] = E Q [I ], τ Var Q [I τ ] = Var Q [I ]. τ 11
Approximation methods (cont.) Distribution of I τ /τ and its approximations 0.40 0.35 0.30 0.25 True Ueno Priebsch 0.20 0.15 0.10 0.05 0.00-6% -4% -2% 0% 2% 4% 6% 8% 10% Note: one-factor model, x 0 = 0, y = 0, θ = 0.01 κ = 0.1 σ = 0.2 φ = 0. Maturity is 10 year. 100,000 paths are generated. 12
Approximation methods (cont.) Maturity Shadow rate 1Y 5Y 10Y 30Y Exat 0.98829 0.92449 0.84104 0.58363 Priebsch(2013) 0.98829 0.92456 0.84178 0.59507 Rate of deviation 0.00050% 0.00723% 0.08848% 1.95929% 1% difference bps 0.050 0.145 0.884 6.468 This paper 0.98829 0.92449 0.84101 0.58294 Rate of deviation 0.00049% -0.00011% -0.00369% -0.11872% difference bps 0.049-0.002-0.037-0.396 0% Exact 0.99463 0.94622 0.87124 0.61258 Priebsch(2013) 0.99463 0.94628 0.87192 0.62391 Rate of deviation -0.00050% 0.00612% 0.07800% 1.84886% difference bps -0.050 0.122 0.780 6.107 This paper 0.99462 0.94622 0.87122 0.61189 Rate of deviation -0.00051% 0.00007% -0.00234% -0.11223% difference bps -0.051 0.001-0.023-0.374 Note: one-factor model, y = 0, θ = 0.01 κ = 0.1 σ = 0.2 φ = 0. 13
Data Switzerland 3M 10% 1.5% 6M 8% 1.0% 1Y 6% 0.5% 2Y 3Y 5Y 4% 2% -0.5% 7Y 0% -1.0% 10Y -2% 12/1 13/1 14/1 15/1 16/1-1.5% Source: Bloomberg 14
Data (cont.) Germany 3M 6M 1Y 2Y 3Y 5Y 7Y 10Y 10% 8% 6% 4% 2% 0% -2% 12/1 13/1 14/1 15/1 16/1 2.5% 2.0% 1.5% 1.0% 0.5% -0.5% -1.0% Source: Bloomberg 15
Data (cont.) Japan 3M 6M 1Y 2Y 3Y 5Y 7Y 10Y 10% 8% 6% 4% 2% 0% 1.5% 1.0% 0.5% -2% 12/1 13/1 14/1 15/1 16/1-0.5% Source: Bloomberg 16
Estimation method Estimate four kinds of models by (quasi-) maximum likelihood estimation; the Gaussian affine model, the Black model, two versions of the Extended model. The difference between the two versions is whether φ t is constant or variable. Use the Single-Stage Iteration Filter (SSIF) for the Black model and Extended model and the Kalman filter for the Gaussian affine model as in Joslin et al. (2011) and others. In the Black model and Extended model, the relationships between factors and bond yields are not linear, so non-linear filtering method should be used. The Extended Kalman filter is used in many studies (Xia and Wu (2016) and others). Tanizaki (1996), however, shows by Monte Carlo simulations that estimation biases arise in the Extended Kalman filter if there is high non-linearity in estimated systems and propose the usage of other non-linear filtering methods including SSIF. 17
Estimation results: Parameters Gaussian affine Switzerland Germany Japan Black Extended Fixed Extended Variable Gaussian affine Black Extended Fixed Extended Variable Gaussian affine Black Extended Fixed Extended Variable φ σ φ 1.0000 0.0000 0.0555 0.0000 1.0000 0.0000 0.0999 0.0039 1.0000 0.0000 0.0153 0.0000 0.0000 0.0000 0.0000 0.0504 0.0000 0.0000 0.0000 0.0511 0.0000 0.0000 0.0000 0.0164 Average of log likelihood 41.4711 41.5710 41.8567 41.9377 42.8840 42.7793 43.2187 43.3157 41.4112 42.4020 42.6489 42.7703 P-value 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 ( VS. Extended Fixed ) Pseudo P-value 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 ( VS. Extended Variable ) T 330 330 330 330 297 297 297 297 327 327 327 327 18
Estimation results: RMSE Switzerland bps 3M 6M 1Y 2Y 3Y 5Y 7Y 10Y Average Gaussian affine 7.86 7.44 11.33 5.77 6.88 6.27 4.48 5.75 6.97 Black 8.19 7.90 11.24 6.57 7.66 6.81 5.10 6.29 7.47 Extended Fixed 7.66 7.72 11.07 6.00 7.32 6.78 5.02 6.29 7.23 ExtendedVariable 7.64 7.37 11.15 6.00 7.08 6.37 4.54 5.85 7.00 Germany bps 3M 6M 1Y 2Y 3Y 5Y 7Y 10Y Average Gaussian Affine 5.19 5.59 8.79 5.72 6.66 6.82 6.87 6.20 6.48 Black 8.88 8.58 10.44 6.62 6.91 6.05 5.96 5.09 7.32 Extended Fixed 4.37 5.33 8.36 4.92 5.71 6.14 6.81 5.55 5.90 ExtendedVariable 4.78 5.50 8.51 4.57 5.35 5.42 6.03 5.14 5.66 Japan bps 3M 6M 1Y 2Y 3Y 5Y 7Y 10Y Average Gaussian Affine 3.99 3.15 7.50 6.49 6.50 7.48 12.63 10.74 7.31 Black 4.95 4.50 8.13 6.95 7.33 7.51 12.39 9.66 7.68 Extended Fixed 4.22 3.72 7.71 6.73 7.04 7.19 12.25 9.66 7.32 ExtendedVariable 3.96 3.46 7.62 6.26 6.83 7.08 12.13 9.57 7.11 19
Estimation results: Volatility Gaussian affine Switzerland Extended Variable 1.2 % 1.2 % 1.0 1.0 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.0 3M 6M 1Y 2Y 3Y 5Y 7Y 10Y 0.0 3M 6M 1Y 2Y 3Y 5Y 7Y 10Y Data during positive interest Data during zero interest Data during negative interest Model during positive interest Model during zero interest Model during negative interest 20
Estimation results: Volatility (cont.) Gaussian affine Germany Extended Variable 0.9 % 0.9 % 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0.0 3M 6M 1Y 2Y 3Y 5Y 7Y 10Y 0.0 3M 6M 1Y 2Y 3Y 5Y 7Y 10Y Data during positive interest Data during zero interest Data during negative interest Model during positive interest Model during zero interest Model during negative interest 21
Estimation results: Volatility (cont.) Japan 0.9 % Gaussian affine 0.9 % Extended Variable 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0.0 3M 6M 1Y 2Y 3Y 5Y 7Y 10Y 0.0 3M 6M 1Y 2Y 3Y 5Y 7Y 10Y Data during positive interest Data during zero interest Data during negative interest Model during positive interest Model during zero interest Model during negative interest 22
Estimation results: 10 year expected rate and term premium (cont.) Switzerland 10% 9% Term Premium Expected Rate 8% 7% 6% Market Rate Shadow Rate 5% 4% 3% 2% 1% 0% -1% -2% 89/1 91/1 93/1 95/1 97/1 99/1 01/1 03/1 05/1 07/1 09/1 11/1 13/1 15/1 23
Estimation results: 10 year expected rate and term premium (cont.) Germany 10% 9% Term Premium Expected Rate 8% 7% Market Rate Shadow Rate 6% 5% 4% 3% 2% 1% 0% -1% 91/10 93/10 95/10 97/10 99/10 01/10 03/10 05/10 07/10 09/10 11/10 13/10 15/10 24
Estimation results: 10 year expected rate and term premium (cont.) Japan 9% 8% 7% Term Premium Expected Rate 6% 5% Market Rate Shadow Rate 4% 3% 2% 1% 0% -1% -2% -3% -4% 89/4 91/4 93/4 95/4 97/4 99/4 01/4 03/4 05/4 07/4 09/4 11/4 13/4 15/4 25
Sensitivity of yield curves to the IOER Simple Example: E[i t ] = E[s t 1 {st y}] + y P(s t < y) 1.0 P(s t < y) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 E[s t ] = 0.0 E[s t + ] = 0.001 E[s t 1 st 1 ] =0 E[s t ] = 0.01 E[s t ] = 0.00 E[s t 1 st 1 ] = 0.00 IOER reduction E[i t ] y = 0 y = 1 Diff. E[s t ] = 1 0.40% -0.20% -0.60%P E[s t ] = 0.10% -0.85% -0.95%P 0.1 0.0 26
Sensitivity of yield curves to the negative interest rate policy (cont.) 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 Switzerland 2014/12 2015/6 2016/6 3M 6M 1Y 2Y 3Y 5Y 7Y 10Y Germany 2010/6 2011/12 2016/6 3M 6M 1Y 2Y 3Y 5Y 7Y 10Y 2014/12 0.03 2015/6-0.5% 0.12 2016/6-1.0% 0.33 2010/6 0.00 2011/12-1.1% 0.00 2016/6-1.0% 0.32 27
Sensitivity of yield curves to the negative interest rate policy (cont.) Japan 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 2009/6 2012/1 2016/6 3M 6M 1Y 2Y 3Y 5Y 7Y 10Y 2009/6 0.00 2012/1-1.5% 0.02 2016/6-3.7% 0.05 28
Relationship between quantitative easing and yield term premia: Switzerland 2Y 0.8% 0.6% 0.4% 0.2% -0.2% -0.4% Term Premium SNB total asset, CHF billion, RHS -0.6% 10/1 11/1 12/1 13/1 14/1 15/1 16/1 0 100 200 300 400 500 600 700 0.8% 0.6% 0.4% 0.2% -0.2% -0.4% Term Premium SNB securities, CHF billion, RHS -0.6% 10/1 11/1 12/1 13/1 14/1 15/1 16/1-3 -2-1 0 1 2 3 4 5 6 7 8 9 10Y 1.6% 1.4% 1.2% 1.0% 0.8% 0.6% 0.4% 0.2% -0.2% -0.4% -0.6% -0.8% -1.0% -1.2% -1.4% Term Premium SNB total asset, CHF billion, RHS 10/1 11/1 12/1 13/1 14/1 15/1 16/1 0 100 200 300 400 500 600 700 1.6% 1.4% 1.2% 1.0% 0.8% 0.6% 0.4% 0.2% -0.2% -0.4% -0.6% -0.8% -1.0% -1.2% -1.4% Term Premium SNB securities, CHF billion, RHS 10/1 11/1 12/1 13/1 14/1 15/1 16/1-3 -2-1 0 1 2 3 4 5 6 7 8 9 29
Relationship between quantitative easing and yield term premia: Germany 2Y 0.6% Term Premium 1500 0.6% Term Premium -1400 0.4% ECB total asset, EUR billion, RHS 2000 0.4% ECB Securities held for monetary policy purposes, EUR billion, RHS -900 0.2% 0.2% -400 2500 100-0.2% 3000-0.2% 600 1100-0.4% 10/1 11/1 12/1 13/1 14/1 15/1 16/1 3500-0.4% 10/1 11/1 12/1 13/1 14/1 15/1 16/1 10Y 1.5% 1.0% 0.5% -0.5% Term Premium ECB total asset, EUR billion, RHS -1.0% 10/1 11/1 12/1 13/1 14/1 15/1 16/1 1500 2000 2500 3000 3500 1.5% 1.0% 0.5% -0.5% Term Premium ECB Securities held for monetary policy purposes, EUR billion, RHS -1.0% 10/1 11/1 12/1 13/1 14/1 15/1 16/1-1400 -1200-1000 -800-600 -400-200 0 200 400 600 800 1000 1200 1400 30
Relationship between quantitative easing and yield term premia: Japan 2Y 0.04% 0.03% 0.02% 0.01% 0.00% -0.01% -0.02% Term Premium BOJ total asset, JPY trillion, RHS -0.03% 10/1 11/1 12/1 13/1 14/1 15/1 16/1-100 0 100 200 300 400 500 0.04% 0.03% 0.02% 0.01% 0.00% -0.01% -0.02% Term Premium BOJ Government Bond, JPY trillion, RHS -0.03% 10/1 11/1 12/1 13/1 14/1 15/1 16/1-100 -50 0 50 100 150 200 250 300 350 400 10Y 0.9% 0 0.9% 0 0.8% 0.7% 0.6% 100 0.8% 0.7% 0.6% 50 100 0.5% 0.4% 200 0.5% 0.4% 150 0.3% 0.3% 200 0.2% 0.1% -0.1% -0.2% Term Premium BOJ total asset, JPY trillion, RHS 300 400 0.2% 0.1% -0.1% -0.2% Term Premium BOJ Government Bond, JPY trillion, RHS 250 300 350-0.3% 10/1 11/1 12/1 13/1 14/1 15/1 16/1 500-0.3% 10/1 11/1 12/1 13/1 14/1 15/1 16/1 400 31
Decomposition of the yield curves R τ log(p τ )/τ log (EQ τ [exp( i(x t, y t, φ t )dt)]) 0 τ f Q (x t, y t, φ t ) τ = E P [ i(x t, 0, φ t )dt 0 +{f Q (x t, y t, φ t ) f Q (x t, 0, φ t )}. τ ] + {f Q (x t, 0, φ t ) E P [ i(x t, 0, φ t )dt 0 ]} 32
Decomposition of the yield curves: Switzerland 2Y 10Y 0.4% 1.5% 0.2% 1.0% 0.5% -0.2% -0.4% -0.6% -0.5% -0.8% term premium -1.0% -1.0% -1.2% expectation part ioer effect 14/1 14/4 14/7 14/10 15/1 15/4 15/7 15/10 16/1 16/4-1.5% -2.0% term premium expectation part ioer effect 14/1 14/4 14/7 14/10 15/1 15/4 15/7 15/10 16/1 16/4 33
Decomposition of the yield curves: Germany 2Y 10Y 0.3% 0.2% 0.1% -0.1% 2.0% 1.5% 1.0% term premium expectation part ioer effect -0.2% -0.3% 0.5% -0.4% term premium -0.5% -0.6% expectation part ioer effect -0.5% -0.7% 14/1 14/4 14/7 14/10 15/1 15/4 15/7 15/10 16/1 16/4-1.0% 14/1 14/4 14/7 14/10 15/1 15/4 15/7 15/10 16/1 16/4 34
Decomposition of the yield curves: Japan 2Y 10Y 0.15% 0.10% 0.05% 0.00% -0.05% 1.0% 0.9% 0.8% 0.7% 0.6% 0.5% 0.4% term premium expectation part ioer effect -0.10% -0.15% -0.20% term premium expectation part 0.3% 0.2% 0.1% -0.25% ioer effect -0.30% 12/1 12/7 13/1 13/7 14/1 14/7 15/1 15/7 16/1-0.1% -0.2% -0.3% 12/1 12/7 13/1 13/7 14/1 14/7 15/1 15/7 16/1 35
What is behind the movements of the power of arbitrage? 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 φ 10Y CHFUSD Basis Swap (RHS) -0.9% -0.8% -0.7% -0.6% -0.5% -0.4% -0.3% -0.2% 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00-0.05 φ 10Y EURUSD Basis Swap (RHS) -0.6% -0.5% -0.4% -0.3% -0.2% -0.1% -0.05 11/9 12/9 13/9 14/9 15/9-0.1% -0.10 11/9 12/9 13/9 14/9 15/9 0.1% 0.06 0.05 0.04 0.03 0.02 0.01 0.00 φ 1Y JPYEUR Basis Swap (RHS) -0.4% -0.3% -0.2% -0.1% 0.06 0.05 0.04 0.03 0.02 0.01-0.8% -0.6% -0.4% -0.2% -0.01-0.02-0.03-0.04-0.05-0.06 03/1 05/1 07/1 09/1 11/1 13/1 15/1 0.1% 0.2% 0.3% 0.4% 0.00-0.01-0.02-0.03 φ 1Y JPYUSD Basis Swap (RHS) -0.04 03/1 05/1 07/1 09/1 11/1 13/1 15/1 0.2% 0.4% 36
A cross-currency basis swap A cross-currency basis swap is an agreement between two counterparties trading floating rate payments in their respective currencies. Without frictions in financial markets, a cross-currency swap should have a zero value with no spread on either side. But, there are relative funding costs in the different currencies over the lifetime of the swap. The market is charging a premium for transferring assets or liabilities from one currency to another. The cost is reflected as a spread on the floating leg in the foreign currency. U.S. investors can convert the cash flows from foreign government bonds in foreign currency to those in U.S. dollar through the basis swap market receiving premia. Even if the foreign government bond yields are negative, the large enough basis swap spreads can attract U.S. investors to investing the foreign government bonds. 37
Conclusion Develop a model to evaluate the effects of unconventional monetary policy measures including the negative interest rate policy on government bond term structures. Propose an efficient and accurate solution method applicable to both the Black model and the Extended model. Show that the Extended model is superior to the other models using data from Switzerland, Germany and Japan. Quantify each effect of FG, QE and the NIRP. Find that the power of arbitrage between money or reserves and government bonds moves in tandem with basis swap spreads. A two country term structure model is to be invented and empirically examined in future research. 38