The Information Content of the Yield Curve

Transcription

1 The Information Content of the Yield Curve by HANS-JüRG BüTTLER Swiss National Bank and University of Zurich Switzerland 0 Introduction 1 Basic Relationships 2 The CIR Model 3 Estimation: Pooled Time-series and Crosssection Data 4 Results 5 Conclusions

2 2 0 INTRODUCTION Goal: determine empirically the information content of the nominal yield curve of government bonds in Switzerland. Information Content: 1. Expected inflation rates. 2. Real spot interest rates. 3. Interest premium. 4. Expected nominal spot interest rates. Earlier attempts: Frankel [1982]: macroeconomic framework, nominal interest rate = real interest rate + expected inflation rate (Irving Fisher s hypothesis. Stanley Fischer [1975]: theoretical work on indexed bonds: nominal interest rate = real interest rate + expected inflation rate interest premium. The interest premium may have either sign.

3 Financial literature confirmed this result within quite different frameworks. To name a few: Fama and Farber [1979], Cox, Ingersoll and Ross [1981, 1985], Lucas [1982], Benninga and Protopapadakis [1985], Breeden [1986]. Empirical literature neglected the interest premium so far. Exceptions: Evans [1998] estimates the time-varying interest premium, but not the expected inflation rate (exogeneous. Remolona, Wickens and Gong [1998] estimate both the interest premium and the expected inflation rate (time series data, selected nominal and index-linked bonds, no specific yield curve model. 3

4 Our approach: No indexed bonds are issued in Switzerland. Search for model which explains nominal and real interest rates simultaneously. Two candidate models: 1. Overlapping generations model with cashin-advance constraint by Bakshi and Chen [1996]. 2. Three-factor model by Cox, Ingersoll and Ross [1981, 1985, henceforth CIR]. Choose the CIR model for reasons of tractability. Sample of pooled time-series and crosssection data. Cross-section data consists of the real instantaneous spot interest rate (not observed, the instantaneous spot inflation rate (drift, the consumer price level, and 4

5 the spot prices of a given number of nominal discount bonds in real terms with various terms (remaining years to maturity between 3 and 26 years. Maximize the likelihood of the given sample of pooled time-series and cross-section data with respect to the unknown CIR model parameters. Constraint: the theoretical pure discount bond prices should fit the actual pure discount bond prices observed on the trading day under consideration as well as possible. 5

6 6 1 BASIC RELATIONSHIPS Spot interest rate P t, T = exp R, c t, T T t (1-1 P t T R R, c t, T = ln P t, T T t Price of a pure discount bond Settlement date Maturity date Spot interest rate = n (nominal, r (real c Continuous compounding Instantaneous spot interest rate r, c t R, c t, t = lim t T ln P t, T T t (1-2 (1-3 r Instantaneous spot interest rate

7 7 ( T-year forward interest rate t, T, = exp F, c t, T, T (1-4 F, c t, T, = ln t, T, T (1-5 F The forward price of a pure discount bond, fixed at date t and paid at a later date T when the bond will be delivered. The bond matures at date ( T t. The ( T-year forward interest rate F, c t, T, = R, c t, t R, c t, T T t T (1-6

8 8 Instantaneous forward interest rate ƒ, c (t, T=F, c (t, T, T = lim T F, c (t, T, = R, c (t, T+ R, c(t, T T T t (1-7 ƒ The instantaneous forward interest rate = T = t ƒ, c (t, d = R, c (t, T T t (1-8 Spot price of pure discount bond P t, T = exp R, c t, T T t = exp = T ƒ, c (t, d = t (1-9 ƒ, c (t, T= ln P t, T T = P t, T T P t, T (1-10

9 Spot inflation rate and instantaneous forward inflation rate P y (t, T p(t p(t = exp R y, c t, T T t = exp = T ƒ y, c (t, d = t 9 (1-11 R y, c t, T = ln P y t, T T t ln p(t ln p(t = T t (1-12 P y t T p R y ƒ y c Purchasing power of money at the future date T in nominal terms at current prices as seen from date t. Settlement date Maturity date Price level of consumer goods. Spot inflation rate. Instantaneous forward inflation rate. Continuous compounding

10 10 Instantaneous spot inflation rate r y, c t R y, c t, t = lim t T ln p(t ln p(t T t = dp(t dt p(t (1-13 r y Instantaneous spot inflation rate ( T-year forward inflation rate y t, T, p(t p( exp F y, c t, T, T, (t T (1-14 F y, c t, T, = ln y t, T, T ln p( ln p(t = T (1-15 y F y ( T-year purchasing power of money. The ( T-year forward inflation rate F y, c t, T, = R y, c t, t R y, c t, T T t T (1-16

11 Instantaneous forward inflation rate ƒ y, c (t, T=F y, c (t, T, T = lim F y, c (t, T,, ( T T = R y, c (t, T+ dr y, c(t, T T t dt 11 (1-17 ƒ y The instantaneous forward inflation rate R y, c (t, T T t = = T = t ƒ y, c (t, d =ln p(t ln p(t (1-18 Instantaneous forward inflation rate and instantaneous spot inflation rate ƒ y, c (t, T=R y, c (t, T+ dr y, c(t, T T t dt dp(t = dt = r p(t y, c (T, t T. (1-19

12 Expected inflation rates t ƒ y, c (t, = t r y, c ( = t for t. dp( d p( 12 (1-20 t R y, c t, T T t = = T t ƒ y, c (t, d = t = t ln p(t p(t (1-21

13 13 Interest premium R n, k t, T = R r, k t, T + R y, k t, T k t, T (1-22 k Interest premium, k = m, c Fisher s equation in an uncertain world. In the CIR framework, the interest premium consists of the variance of the future consumer price level and the wealth premium (attitude towards risk, covariance between future real wealth and future inflation.

14 Calculation of the nominal yield curve Calculate the nominal spot interest rates or nominal yield curve, respectively, from a sample of observed prices of coupon-bearing government bonds. Input into all subsequent calculations. Known methods: Bootstrap method (most popular. Regression of bond prices on discount factors (Carleton and Cooper [1976]. Spline methods (McCulloch [1971, 1975], Vasicek and Fong [1982]. Assumed functional form (Nelson and Siegel [1987], Svensson [1995]. All these methods have many drawbacks as mentioned in Shea [1984, 1985]. Multi-objective goal attainment problem in Delabaen and Lorimier [1992], Lorimier [1995]. 14

15 Goal: in order to obtain a forward rate curve as smooth as possible, minimize the sum of squared differences in nominal instantaneous forward interest rates subject to the constraint that the theoretical bond prices fit the observed bond prices as well as possible (Büttler [2000]. Results of the DL methodology All experiments undertaken fit the theoretical term structure proposed by Vasicek [1977] perfectly well. The Delbaen-Lorimier methodology performs much better than the bootstrap method when the term structure is sufficiently bent. The Delbaen-Lorimier methodology is able to extract any term structure from a sample of bond prices. Example: wave-like term structure. 15

16 16 Instantaneous forward interest rate: ƒ n, c (t, T=a + b (T - t where a =0.01, b = , c = , d = Spot interest rate: R n, c (t, T=a + 1 b (T - t 2 2 d (T t 100 d (T - t sin c + sin 2 sin c + d (T - t d (T - t 2

17 17 Annually Compounded Spot and Forward Rates in Percent p. a. 5,5 5,0 4,5 4,0 3,5 3,0 2,5 2,0 Spot Wave Spot DL Forward Wave Forward DL Spot Bootstrap Yield to Maturity Spot NS Forward NS 1,5 1, Remaining Years 10 Wave-like Yield Curve

18 18 2 THE CIR MODEL If there are no indexed bonds traded in financial markets, then we must rely on a model which is able to explain nominal and real interest rates simultaneously. Two candidate models: CIR [1985] and Bakshi and Chen [1996]. Real instantaneous spot interest rate dr r, c (t= r r, c (t dt + r r, c (t dz 1 (t, (2-1 0,, r r c z 1 Continuously compounded real instantaneous spot interest rate Speed of adjustment Long-run equilibrium value Constant volatility parameter Gauss-Wiener process

19 19 Price of a real pure discount bond P r (t, T=A(t, T exp B(t, T r r, c (t, where A(t, T= + + T t 2 exp exp T t 1 +2 B(t, T= 2 exp T t exp T t 1 +2 = , = 2 2 (2-2 P r Price of a real pure discount bond Factor risk premium

20 Consumer price level and the instantaneous spot inflation rate dr y, c (t= 2 2 r y, c (t dt , 2, 2, r y, c (tdt dp(t p(t = r y, c(tdt + p 0 p 1, t r y c dr y, c r y, c (t dz 2 (t, r y, c (t dz 3 (t, (t, dp(t = 2 p r y, c (t p(tdt. 20 (2-3 Instantaneous spot inflation rate r; y c Drift of the instantaneous spot inflation rate z 2 p p z 3 Speed of adjustment Long-run equilibrium value Constant volatility parameter Gauss-Wiener process Consumer price level Constant volatility parameter Gauss-Wiener process Correlation coefficient between z 2 and z 3 Covariance operator

21 21 Price of a nominal pure discount bond P n (t, T=P r (t, T C(t, T 2 C(t, T= exp Dt, T r y, c (t 2 exp p + T t p + exp T t 1 +2 Dt, T = 2 exp T t p p + exp T t 1 +2 = p p 2 2 = (2-4 P n Price of a nominal pure discount bond

22 22 Yield curves in the CIR model R r, c t, T = B(t, T r r, c(t ln A(t, T T t R n, c t, T = R r, c t, T + 2 ln C(t, T + Dt, T r y, c (t T t (2-5 R r c R n c Real spot interest rate Nominal spot interest rate Take the limit as T t R r, c t, t = r r, c t R n, c t, t = r n, c t = r r, c t +1 p 2 r y, c (t (2-6 r r c r n c Real instantaneous spot interest rate Nominal instantaneous spot interest rate r; y c Drift of the instantaneous spot inflation rate Instantaneous interest premium in the CIR model c (t, t= 2 p r y, c (t= dt 1 t t r y, c (t=r y, c (t dp(t p(t

23 Disadvantages of the CIR model No correlation between real interest rates and inflation rate. Deflation rates are excluded. Monotonic adjustment. Advantage of the CIR model Closed-form solution. 23

24 3 ESTIMATION: POOLED TIME-SERIES AND CROSS- SECTION DATA 24 Sample of pooled time-series and cross-section data. Cross-section data consists of the real instantaneous spot interest rate (not observed, the drift of the instantaneous spot inflation rate (not observed, the consumer price level, and the spot prices of a given number of nominal discount bonds in real terms with various terms (remaining years to maturity between 3 and 26 years.

25 Stochastic differential equation for the price of a nominal pure discount bond in real terms. d ( = 1 B( r r, c (t ( dt B( ( r r, c (t dz 1 (t D( ( 2 r y, c (t dz 2 (t ( p r y, c (t dz 3 (t 25 (3-1 ( P n (t, T / p(t. The spot price of a nominal pure discount bond in real terms. Discrete time steps and bond-specific error term. j, t =1 B j r r, c, t1 j, t1 t B j j, t1 r r, c, t1 z 1, t D j j, t1 2 r y, c, t1 z 2, t j, t1 p r y, c, t1 z 3, t + 0 j, t1 z 3+j, t, j =1,, H. (3-2 0 H Bond-specific volatility parameter. Number of bonds selected for the crosssection data.

26 26 Distribution of Wiener increments. z(t ((3 + H 1 = z 1, t z 2, t z 3, t z 4, t z 3+H, t (t ((3 + H (3 + H = t t t t t t t (3-3 z(t ((3 + H 1 0, (t ((3 + H 1 ((3 + H (3 + H (3-4 (, Gaussian or normal distribution with mean vector and covariance matrix.

27 27 Trend-adjusted increments y(t. y(t ((3 + H 1 = r r, c, t r r, c, t1 2 2 r y, c, t1 p t r y, c, t1 p t1 t r y, c, t 1, t 1 B 1 2, t 1 B 2 t t r r, c, t1 1, t1 t r r, c, t1 2, t1 t (3-5 H, t 1 B H r r, c, t1 H, t1 t Volatility matrix G(t. G(t ((3 + H (3 + H = G 11 (t (3 3 G 21 (t (H 3 0 (3 H G 22 (t (H H (3-6 First submatrix G 11 (t. G 11 (t (3 3 = r r, c, t r y, c, t p p t1 r y, c, t1 (3-6a

28 28 Second submatrix G 21 (t. G 21 (t H 3 = B 1 1, t1 r r, c, t1 B 2 2, t1 r r, c, t1 B H H, t1 r r, c, t1 D 1 1, t1 2 r y, c, t1 D 2 2, t1 2 r y, c, t1 D H H, t1 2 r y, c, t1 1, t1 p r y, c, t1 2, t1 p r y, c, t1 (3-6b Third submatrix G 22 (t. H, t1 p r y, c, t1 G 22 (t = (H H 0 1, t , t H, t1

29 Multivariate normal distribution of y(t. y(t ((3 + H 1 = G(t ((3 + H ((3 + H, S(t 0 ((3 + H 1 S(t=G(t (t G(t z(t ((3 + H 1 ((3 + H (3 + H Reduction to univariate normal distribution by means of transformed increments x(t. 29 (3-7 x(t ((3 + H 1 = Q(t 1 ((3 + H (3 + H 0 ((3 + H 1, I S(t=Q(t Q(t ((3 + H (3 + H y(t ((3 + H 1 (3-8 Q Upper-triangle matrix. Cholesky decomposition of S. I Identity matrix Determinant of Q(t. S(t = Q(t Q(t = Q(t Q(t = Q(t 2 0 Q(t = S(t 0 (3-9

30 Define 10-by-1 parameter vector = [,,,, 2, 2, 2, p,, 0 ] Logarithm of the likelihood, (, of a given sample y(t in terms of the parameters at time t. ( y(t = 3+H ln( H Σ 2 2 j =1 Q(t x j 2 ( y(t 30 (3-10 Covariance of y(t. y(t, y(t + s = G(t z(t, G(t+ s z(t + s = 0 ((3 + H (3 + H (3-11 Find parameter vector by constrained maximization of likelihood function. max Σ ( y(t, s. t. t k P n(, T k P n,obs (, T k k k =1,, K. (3-12

31 Data Forty weekly spaced trading days between 14 August 2000 and 14 May For each trading day, a sample consists of 52 weekly pooled time-series and cross-section data. Number of bonds selected for the cross-section data is H = 5. Their terms vary between 3 and 26 years. Number of bonds selected for the constraints is K = 20. Their terms vary between 0 and 27 years. The real instantaneous spot interest rate has been calculated from equation ( RESULTS 31

32 κ σ λ κ2 σp ρ Figure 1a: Parameter Estimates Parameter Values Date (Day.Month.Year θ θ2 σ2 r0 σ0 Figure 1b: Parameter Estimates Continued 0.10 Parameter Values Date (Day.Month.Year

33 Yields in Percent per Annum Nominal Real Date (Day.Month.Year Figure 2: Nominal and Real Yield Curves in Percent per Annum Remaining Years

34 Expected drift of the instantaneous spot inflation rate in the CIR framework. s r y, c (t= 2 + s r y, c (s 2 e 2 t s s r y, c (t= 2 2 s r y, c (s e 2 t s e 2 2 t s e 2 t s s r y, c (t, r y, c (t + =e 2 s r y, c (t, 0, t s (4-1 Term structure of expected spot inflation rates in the CIR framework. t R y, c (t, T= 2 + tr y, c (t 2 2 lim tr y, c (t, T= 2 T lim t T tr y, c (t, T= t r y, c 1 e 2 T t T t (t=r y, c (t, T t, (4-2

35 Expected Spot Inflation Rate Date (Day.Month.Year Future Years Figure 3: Expected Spot Inflation Rates in Percent per Annum

36 Expected three-month nominal spot interest rate in the CIR framework. s R n, c t, T = B(t, T sr r, c (t ln A(t, T T t + 2 ln C(t, T + Dt, T s r y, c (t T t (4-3 t s, T = t Expected real instantaneous spot interest rate in the CIR framework. 36 s r r, c (t= + s r r, c (s e t s s r r, c (t= 2 s r r, c (s e t s e 2 t s e t s 2 s r r, c (t, r r, c (t + =e s r r, c (t 0, t s, s r r, c (s=r r, c (s (4-4

37 37 Figure 4a: Expected Three-month Spot Interest Rates in Percent per Annum Expected Forward Observed Interest Rate Date (Day.Month.Year Future Years Figure 4b: Expected Three-month Spot Interest Rates in Percent per Annum Expected Forward Observed Interest Rate Date (Day.Month.Year Future Years

38 38 Figure 4c: Expected Three-month Spot Interest Rates in Percent per Annum 4.5 Expected Forward Observed Interest Rate Date (Day.Month.Year Future Years Figure 4d: Expected Three-month Spot Interest Rates in Percent per Annum Expected Forward Observed Interest Rate Date (Day.Month.Year Future Years

39 39 Figure 4e: Expected Three-month Spot Interest Rates in Percent per Annum Expected Forward Observed Interest Rate Date (Day.Month.Year Future Years Figure 4f: Expected Three-month Spot Interest Rates in Percent per Annum Expected Forward Observed Interest Rate Date (Day.Month.Year Future Years

40 40 Figure 4g: Expected Three-month Spot Interest Rates in Percent per Annum 3.5 Expected Forward Observed Interest Rate Date (Day.Month.Year Future Years Figure 4h: Expected Three-month Spot Interest Rates in Percent per Annum 3.5 Expected Forward Observed Interest Rate Date (Day.Month.Year Future Years

41 41 Are interest-rate forecasts unbiased? R obs n, c t, T = s R n, c t, T + u(s s, t s, T = t t s =1,7,14,, 91 days (4-5 Sample of trading dates. u(s Normal variate (i. i. d. Unbiased if 0 = 0 and 1 = 1.

42 Table 1: OLS Regressions Ind. Future Time Horizon in Days Var Const (0.11 (0.26 (0.29 (0.31 (0.32 (0.32 (0.30 Exp. L (0.03 (0.07 (0.08 (0.09 (0.09 (0.09 (0.09 R squ F-Ratio Accept yes yes no no no no no Ind. Future Time Horizon in Days Var Const (0.28 (0.26 (0.23 (0.21 (0.18 (0.16 (0.14 Exp. L (0.08 (0.07 (0.07 (0.06 (0.05 (0.05 (0.04 R squ F-Ratio Accept no no no no no no no Comments: The dependent variable is the observed three-month Libor. The size of the sample of settlement days is forty. We test the joint hypothesis that the coefficient of the constant term is equal to zero and that the coefficient of the expected three-month Libor is equal to one. The hypothesis is accepted if the F-ratio is less than the corresponding critical F-value. The critical one- 42

43 tailed F-value is equal to for a probability of 95%. The numbers presented in this table are cut off rather than rounded. 43

44 44 Observed Three-month Libor Observed Three-month Libor 4.0 Figure 5a: 1-day Ahead Forecast Observations Regression Constr. Reg Expected Three-month Libor 4.0 Figure 5c: 84-day Ahead Forecast Observations Regression Constr. Reg Expected Three-month Libor Observed Three-month Libor Observed Three-month Libor 4.0 Figure 5b: 7-day Ahead Forecast Observations Regression Constr. Reg Expected Three-month Libor 4.0 Figure 5d: 91-day Ahead Forecast Observations Regression Constr. Reg Expected Three-month Libor

45 Table 2: Restricted Least Squares Ind. Future Time Horizon in Days Var Const (0.00 (0.00 (0.00 (0.00 (0.00 (0.00 (0.00 Exp. L (0.00 (0.00 (0.00 (0.00 (0.00 (0.00 (0.00 P. R s t-ratio Accept yes yes yes yes yes yes yes Ind. Future Time Horizon in Days Var Const (0.00 (0.00 (0.00 (0.00 (0.00 (0.00 (0.00 Exp. L (0.00 (0.00 (0.00 (0.00 (0.00 (0.00 (0.00 P. R s t-ratio Accept yes yes yes yes yes yes yes Comments: The dependent variable is the observed three-month Libor. The size of the sample of settlement days is forty. We test the hypothesis that the coefficient of the expected three-month Libor is equal to one, given the restriction that the coefficient of the constant term is equal to zero. The hypothesis is accepted if the t-ratio is less than the corresponding critical t-value. The critical two-tailed t-value is equal to for a probability of 45

46 95%. The numbers presented in this table are cut off rather than rounded Premium Difference in Percentage Points Date (Day.Month.Year Future Years Figure 6: Premium Difference between FIML and MOGA Methods in Percentage Points 20 25

47 Interest Premium in Percent per Annum Date (Day.Month.Year Figure 7: Interest Premium in Percent per Annum Future Years

48 48 5 CONCLUSIONS Applying the CIR model, we determined empirically the term structure of real spot interest rates, the term structure of expected spot inflation rates and of interest premia from the nominal yield curve for forty consecutive weeks. The CIR model seems to be able, by and large, to explain the information content of the yield curve. We presented two indirect tests of the CIR model with regard to the future expected threemonth Libor. These tests are quite satisfactory. We presented empirical evidence that the interest premium has vanished under the new monetary regime of the SNB which was introduced at the beginning of the year 2000.

49 Figure 10a: A Simulation Path of the Consumer Price Index Consumer Price Index Simulated Observed Date (Day.Month.Year Figure 10b: A Simulation Path of the Instantaneous Inflation Rate Instantaneous Drift 26 Inflation Rate in Percent p. a Date (Day.Month.Year

50 50 Dates (Day.Month.Year Inflation Rates in Percent p. a Error Inst. Growth Rate 6 5 Inst. Growth Rate Error One-month Growth Rate One-month Growth Rate Error One-Year Growth Rate One-Year Growth Rate Simulated Drift 7 Figure 11: Simulated Drift versus Various Proxies

51 51 Consumer Price Index 103 CPI and its Backward-looking 13-month Moving Average. Period: 31-Jan Jul Observed Moving Average Date (Day.Month.Year

52 52 Date (Day.Month.Year Annually Compounded Inflation Rate in Percent p. a knot Derivative with Time Step of 1 Year 15 3-knot Derivative with Time Step of 1 Year 5-knot Derivative with Time Step of 1 Week Growth Rate with Time Step of 1 Year 18 Weekly Inflation Rates from Non-smoothed CPI. Period: 31-Jan Jul-2001

53 53 Date (Day.Month.Year Annually Compounded Inflation Rate in Percent p. a knot Derivative of Smoothed CPI with Time Step of 1 We Growth Rate of Non-smoothed CPI with Time Step of 1 Year 7 Weekly Inflation Rates from Smoothed CPI. Period: 31-Jan Jul-2001