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Kant Thamchamrassri

Nonparametric Econometrics Seminar. Estimating the T erm S tructure of I nterest R ates for Thai G overnment B onds : A B-Spline Approach. Kant Thamchamrassri. February 5, 2006. Introduction. Interest rate in modern financial theories

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Kant Thamchamrassri

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

  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) Spot rate = zero-coupon yield r(t) Forward rate f(t) Theoretical Framework

  5. Methods for Extractingthe 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 kth-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 Recurrence relation Theoretical Framework

  8. B-Splines of Degree One Simplified to Theoretical Framework

  9. B-Splines of Degree Two Theoretical Framework

  10. B-Splines of Higher Degrees • B-Splines of higher degrees • is the pth spline of kth degree. • and are the pre-specified knot values. Theoretical Framework

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

  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) B1 B2 B3 B4 B5 B6 Theoretical Framework

  14. The Term Structure Fitting Using B-Splines • Approximation by curve S • λpare coefficients corresponding to the pth-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 • tm is the time when the mth coupon or principal payment is made. • hu is the number of coupon and principal payments before the maturity date of bond u. • y(tm) is the cashflow paid by bond u at time tm. • P(tm) 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. Indices for Evaluation of Regression Equations • Mean integrated squared error (MISE) • is the yield curve derived from the B-spline approximation • is the ThaiBMA interpolated zero-coupon yield curve Methodology

  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 Restricted Discount Fitting (MISE) Spot Fitting (GCV) Empirical Results

  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 [-22.5 -15 -7.5 0 3 15 22.5 30 37.5] Conclusion

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