1 Estimating the Term Structure of Interest Rates for Thai Government Bonds: A B-Spline Approach Kant Thamchamrassri February 5, 2006 Nonparametric Econometrics Seminar

2 Introduction Interest rate in modern financial theories  Fixed income market (bonds and derivative securities)  Other market securities (for time discounting)  Corporate investment decisions (alternative opportunities and cost of capital) The term structure of interest rates  Representing relationship between bond yields and maturities  Useful in pricing coupon bonds Introduction

3 Bond Pricing  Spot rate:  Forward rate: P(t) is the price at time t of a zero coupon bond of par value = 1 (also called discount factor) r(t) is the instantaneous spot rate at time t f(t) is the instantaneous forward rate at time t Theoretical Framework

4 Bond Price, Spot Rate and Forward Rate Relationship Discount function = price of zero-coupon bond P(t) Forward rate f(t) Spot rate = zero-coupon yield r(t) Theoretical Framework

5 Methods for Extracting the Term Structure Simple linear regression Polynomial splines Exponential splines Basis splines (B-splines) Nelson and Siegel (1985) and its variants Bootstrapping and cubic splines Theoretical Framework

6 Splines  Spline is a statistical technique and a form of a linear non-parametric interpolation method.  A k th -order spline is a piecewise polynomial approximation with k-degree polynomials.  A yield curve can be estimated using many polynomial splines connected at arbitrary selected points called knot points.  Some conditions are applied: continuity and differentiability Theoretical Framework

7 B-Splines of Degree Zero Theoretical Framework Recurrence relation

8 B-Splines of Degree One Theoretical Framework Simplified to

9 B-Splines of Degree Two Theoretical Framework

10 B-Splines of higher degrees  is the p th spline of k th degree.  and are the pre-specified knot values. Theoretical Framework B-Splines of Higher Degrees

11  Degree of polynomials (k)  Interval of approximation (n)  Number of basis functions (p) = n+k  Number of knots (n+1+2k) Theoretical Framework B-Splines of Degree Three (k=3)

12 B-Splines of Degree Three (k=3)  Knot specification [-3, -2, -1, 0, 5, 10, 15, 20, 25, 30] In-sample knots: 0, 5, 10, 15 Out-of-sample knots: -3, -2, -1, 20, 25, 30 Approximation horizon: [0, 15] Approximation intervals (n): 3 Number of knots (n+1+2k) = 10 Number of basis functions (p) = n+k = 6 Theoretical Framework

13 B-Spline Basis Functions (k=3) B1B1 B2B2 B3B3 B5B5 B4B4 B6B6 Theoretical Framework

14 The Term Structure Fitting Using B-Splines  Approximation by curve S λ p are coefficients corresponding to the p th -spline that determines S(t)  Bond pricing Q represents bond price C is the cashflow matrix Theoretical Framework

15 The Term Structure Fitting Using B-Splines  Bond pricing regression Q represents bond price C is the cashflow matrix Theoretical Framework

16 The Term Structure Fitting Methodology Bond pricing  the price of the coupon bond u is a linear combination of a series of pure discount bond prices  t m is the time when the m th coupon or principal payment is made.  h u is the number of coupon and principal payments before the maturity date of bond u.  y(t m ) is the cashflow paid by bond u at time t m.  P(t m ) is the pure discount bond price with a face value of 1 Methodology

17 The Term Structure Fitting Methodology Model formulation  P(t) is the price at time t of a zero-coupon bond (par value = 1)  Spot rate:  Forward rate: Methodology

18 Discount Fitting Model  Bond price  Discount function  Discount fitting function Restriction Methodology

19 Spot Fitting Model  Bond price  Pure discount bond price  Spot function  Spot fitting function Methodology

20 Forward Fitting Model  Bond price  Pure discount bond price  Forward function  Forward fitting function Methodology

21 Data & Estimation Setup Trading data on January 13, 2006 from the ThaiBMA  12 treasury-bills and 28 government bonds (LB series)  Input: time to maturity, coupon rate, weighted average yield, weighted average price  B-Splines of degree k = 1, 2, 3, 4  Approximation intervals n = 1, 2, 3, 4, 5  Knot specification Estimation horizon = 0 – 15 years Within-sample knots are integers (1 to 14) Out-of-sample interval length = horizon/n Methodology

22 Indices for Evaluation of Regression Equations Generalized cross validation (GCV)  RSS is residual sum of squares  k is the degree of B-spline polynomials  n is the number of approximation intervals  m is sample size Methodology

23 Mean integrated squared error (MISE)  is the yield curve derived from the B-spline approximation  is the ThaiBMA interpolated zero-coupon yield curve Methodology Indices for Evaluation of Regression Equations

24 Estimated Results Generalized cross validation (GCV) Mean integrated squared error (MISE) Comparison with the ThaiBMA Empirical Results

25 Minimum Values of Generalized Cross Validation (GCV) Empirical Results

26 Model Estimation, GCV (k = 3, n = 2) Empirical Results (%)

27 Fitted Term Structures of Interest Rates Using Different Fitting Models (k = 3, n = 2) Empirical Results

28 Confidence Intervals for Estimated Coefficients (Spot Fitting, k = 3, n = 2) Note. * denotes statistical significance at 1% level. Empirical Results

29 Confidence Intervals of Spot Fitting Model (k = 3, n = 2) Empirical Results

30 Minimum Values of Mean Integrated Squared Error (MISE) Empirical Results

31 Model Estimation, MISE (k = 3, n = 3) Empirical Results (%)

32 Fitted Term Structures of Interest Rates Using Different Fitting Models (k = 3, n = 3) Empirical Results

33 Confidence Intervals for Estimated Coefficients (Restricted Discount Fitting, k = 3, n = 2) Note. * denotes statistical significance at 1% level. Empirical Results

34 Confidence Intervals of Restricted Discount Fitting Model (k = 3, n = 2) Empirical Results

35 Fitted term structures: GCV, MISE in Comparison to the ThaiBMA Yield Curve Empirical Results

36 Confidence Intervals of Restricted Discount Fitting/ Spot Fitting with ThaiBMA Empirical Results Spot Fitting (GCV) Restricted Discount Fitting (MISE)

37 Conclusions Discount fitting can give unbounded term structures at very low maturities. Spot fitting is generally has lower GCV values than forward fitting (at k = 3). Suggested model: spot fitting Suggested B-splines  degree = 3  interval = 2  knot position [ ] Conclusion