Estimating Term Structure of U.S. Treasury Securities: An Interpolation Approach

Estimating Term Structure of U.S. Treasury Securities: An Interpolation Approach Feng Guo J. Huston McCulloch

Our Task Empirical TS are unobservable. Without a continuous spectrum of zero-coupon securities; In practice, the U.S. Treasury has instead issued a limited number of securities with discrete maturities; This paper extends term structure estimation literature by proposing a new method to recover a continuous term structure of interest rate.

Previous Research Spline-based interpolation or Curve-fitting models Fama-Bliss (1987), McCulloch (1975, 1993, 2000), Nelson-Siegel (1987), and Fisher et al. (1995) differ considerably in their respective interpolation functional forms and estimation techniques Advantages of the approach of interpolation Yield is a function of term to maturity only, independent of any preference form and general equilibrium setting; Simple, parsimonious functional forms; Allow easy transformation among discount function, zero-coupon yield, forward rate, and par bond yield; Easy to access, not based on historical data.

Motivation Two potential target functions: 1 Discount Function (McCulloch (1975) and McCulloch and Kwon (1993)) 2 log Discount Function (McCulloch and Kochin (2000), Nelson and Siegel (1987)) Discount Function log Discount Function Result in a linear bond price function (+) Hard to govern its behavior at long-end maturity (-) Result in a nonlinear bond price function (-) Easy to govern its behavior at long-end maturity (+)

Motivation (cont d) Two classes of potential interpolation functional forms 1 Cubic Spline (McCulloch and Kochin (2000)); 2 Exponential Decay (Nelson and Siegel (1987), Swensson (1994), Gürkaynak et al. (2006, 2008)); Cubic Spline Exponential Decay Nonparametric Parametric and parsimonious more flexibility on the shape of yield curve (+, -) Natural condition to control behavior at long-end (+) Relatively rigid forward rate curve (-, +) No extra condition for long-end behavior (+)

What We Do Propose a multiple exponential decay functional form to construct a negative log discount function; Three estimation methods are introduced to estimate the parameters of the objective function; Extend McCulloch cubic spline model to fit the same data set so as to compare with the multiple exponential function; Implement a procedure to pin down the number of parameters based upon tests for serial correlation and information criteria.

Outline 1 Introduction 2 Model Specification 3 Estimation and Results 4 Extension 5 Conclusion

Basic Definitions Back δ(m): Discount Function (P.V. of $1 at m) δ(0) = 1, δ( ) 0 y(m): (continuously compounded) Zero-coupon Yield log δ(m) δ(m) = exp ( y(m) m) = y(m) = m y p (m): (continuously compounded) Par Bond Yield p: Bond Price (with continuous coupons) (1) y p (m) = 1 δ(m) m 0 δ(s) ds (2) p = δ(m) + c m 0 δ(s) ds

Basic Definitions Forward Rate and Instantaneous Forward Rate: f (m 1, m 2 ) = 1 ( ) δ(m1 + m 2 ) log m 2 δ(m 1 ) = 1 [(m 1 + m 2 )y(m 1 + m 2 ) m 1 y(m 1 )] m 2 = f (m) f (m 1, 0) = y(m) + my (m) = ( log δ(m)) = δ (m) δ(m) (1) = δ(m) is an exponential decay curve; (3) = the rate of decay is f(m). (3) y(m) = 1 m m 0 f (s) ds, f (m 1, m 2 ) = 1 m 2 m 1 m2 m 1 f (s) ds (4)

Model Specification General Form φ(m) - δ(m) features an exponential decay curve; δ(m) - φ(m), log disc. fn., should be a linear combination of some basis functions Ψ(m) with k parameters and n maturities k φ(m i ) log δ(m i ) = β j Ψ j (m i ) i = 1,..., n j=1 s.t. δ(0) = 1 or φ(0) = 0, Ψ j (0) = 0; k n (5) 1.1 Discount Function to Maturity, 03/31/2010 0.9 0.7 δ(m) 0.5 0.3 0.1-0.1 0 10 20 30 40 50 60 70 80 90 100 Negative log Discount Function to Maturity, 03/31/2010 3.5 -logδ(m) 2.5 1.5 0.5-0.5 0 10 20 30 40 50 60 70 80 90 100 Years to Maturity

Model Specification Ψ(m) Specify Ψ(m) as exponential functions so that φ(m) is a multi-exponential decay function k φ(m) log δ(m) = β j Ψ j (m) j=1 ) 1 exp ( mτj if j = 1,..., k 1 where Ψ j (m) = m if j = k (6)

e-fold life exp( m/τ) Timescale at which m = τ i for a quantity to decay to 1/e of its previous value, similar to Half Life ; Small value of τ j corresponds rapid decay in m and hence fits curvature at low maturities; vice versa. k determines the number of curvatures the function is able to feature. 1.1 1 τ 1 =0 0.9 τ 2 =1 0.8 0.7 1-1/e 0.6 0.5 0.4 Ψ(m; τ) τ 3 =5 τ 4 =15 τ 5 =35 0.3 0.2 τ 6 =100 0.1 τ 7 =Inf 40 0 0 5 10 15 20 25 30 35 Maturity

Comparison Semi-natural Cubic Spline Select ν + 1 knots {κ j } ν j=0, where κ 0 0 and κ ν m n ; Impose Natural restriction at the long end: φ(m) = 0, m [m n, ); Define the linear combination of the (ν + 2) functions: Ψ j (m) = θ j (m) θ j (m n) θ ν+2 (m n) θ ν+2(m), j = 1,..., ν + 1 (7) where θ 1 (m) = m, θ 2 (m) = m 2, θ 3 (m) = m 3 θ j (m) = max{0, (m κ j 3 ) 3 }, j = 4,..., ν + 2 (8)

Comparison Extended Nelson and Siegel Fit the forward rate curve by: ( ) ( ) ) f (m) = β 0 +β 1 exp ( )+β mτ1 2 exp ( )+β mτ1 mτ1 3 exp ( mτ2 mτ2 Equvalent to fit φ(m) by setting: Ψ 1 (m) = m ) Ψ 2 (m) = m exp ( mτ1 ) Ψ 3 (m) = 1 exp ( mτ1 ) ( )) Ψ 4 (m) = m exp ( mτ2 + τ 2 1 exp ( mτ2

Outline 1 Introduction 2 Model Specification 3 Estimation and Results 4 Extension 5 Conclusion

Select τ j Idea: depict the shapes implied by data Step 1: let τ k = so that β k governs the level of f (m) at long maturity (m ); Step 2: given n k 2, locate the place of each percentile, 1/k, according to the distribution of {m i } n i=1 ; Step 3: assign {τ j = m i j = 1,, k 1, i = floor (nj/k)}. Data

Select τ j Idea: depict the shapes implied by data Step 1: let τ k = so that β k governs the level of f (m) at long maturity (m ); Step 2: given n k 2, locate the place of each percentile, 1/k, according to the distribution of {m i } n i=1 ; Step 3: assign {τ j = m i j = 1,, k 1, i = floor (nj/k)}. Data k=2 k=3 k=4 0 τ 1 = m n m n 2 0 τ 1 = m n τ 2 = m2 n m n 3 0 τ 1 = m n τ 2 = m n τ 3 = m3 n m n 4 2 3 4

Estimation-I: Nonlinear Minimization Objective: Minimize the sum of squared deviation from the observed prices. n min {β j } k j=1 (p i ˆp i ) 2 (9) ˆp i = l c i 2 δ (l) }{{} coupon payments δ( ) = exp i=1 +δ(m i ) + ε i k β j Ψ j ( ) j=1 interpolation model

Estimation-II: Iterative Linear Least Squares Objective: Same as Estimation-I. Step 1: Estimate β 0 by OLS where y i is the observed YTM of the i th security; m i y i = k βj 0 Ψ j (m i ) + ε 0 i i = 1,..., n (10) j=1 Step 2: Given β q, evaluate the coupons and corresponding net prices by δ q ( ); ˆp net(q) i = p i c ( i 2 exp Ψ (l) ˆβ q) = ˆδ q (m i ) (11) l }{{} coupon payments

Estimation-II: Iterative Linear Least Squares (cont d) Step 3: Estimate β q+1 by the following regression: ( ) net(q) log ˆp i = } {{ } q m i ŷ i k j=1 β q+1 j Ψ j (m i )+ε q+1 i i = 1,..., n (12) { ŷ q+1 Step 4: Repeat Step 2 & 3 until max m } ŷm q ɛ. Results

Fitted Yield and Forward Rate Curves based on Estmation-II Estimate Real TS by TIPS (Treasury inflation-indexed securities) market quotes on Mar. 31, 2010, n = 29; Data All three model specifications are employed. 4 Multiple Exponential of Real YTM, k=5 4 Cubic Spline of Real YTM, k=5 3 3 2 2 Percent 1 0 Percent 1 0-1 -1-2 0 5 10 15 20 25 30 35 40 Years to Maturity -2 0 5 10 15 20 25 30 35 40 Years to Maturity Extended Nelson and Siegel of Real YTM 4 Percent 3 2 1 0 Zero Coupon YC Par Bond YC Forward Rate Curve Observed YTM -1-2 0 5 10 15 20 25 30 35 40 Years to Maturity

Question: How to pin down k in φ(m)? Empirical evidence: k n Evidence 1: WLS Durbin-Watson Test DW test for Real log disc. function, N=29, 03/31/2010 DW test for Nominal log disc. function, N=166, 03/31/2010 4 3 3.5 2.5 3 DW-Statistics 2.5 2 1.5 DW-Statistics 2 1.5 1 1 0.5 5% Critical Value L-Tail Mean Critical Value 5% Critical Value R-Tail DW Statistics 0.5 5% Critical Value L-Tail Mean Critical Value 5% Critical Value R-Tail DW Statistics 0 2 3 4 5 6 7 8 Number of Terms 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Number of Terms

Question: How to pin down k in φ(m)? (cont d) Evidence 2: Nonparametric Runs-Test -test the mutual independence of a two-valued sequence. H 0: values in x come in random order; H 1: they do not. Runs-test for Real log disc. fn., N=29, 03/31/2010 Runs-test for Nominal log disc. fn., N=166, 03/31/2010 1 0.9 0.8 0.8 0.7 0.6 P-Value 0.6 0.4 P-Value 0.5 0.4 0.3 0.2 0.2 0.1 0 2 3 4 5 6 7 8 Number of Terms 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 Number of Terms

Question: How to pin down k in φ(m)? (cont d) Evidence 3: Bayesian Information Criterion BIC for Real log disc. fn., N=29, 03/31/2010-8.8 BIC for Nominal log disc. fn., N=166, 03/31/2010-8 -9-8.5-9.2-9 -9.4-9.6-9.5 BIC -9.8 BIC -10-10 -10.5-10.2-11 -10.4-10.6-11.5-10.8 2 3 4 5 6 7 8 Number of Terms -12 2 4 6 8 10 12 14 16 18 20 22 24 26 28 Number of Terms

Question: How to pin down k in φ(m)? (cont d) A demonstration: set k = 2, 3, 4, 5, and 8 and fit the real TS on 3/31/2010 respectively. Percent Percent Multiple Exponential of Real YTM, k=2 5 4 3 2 1 0 0 5 10 15 20 25 30 35 40 Multiple Exponential of Real YTM, k=5 4 3 2 1 0-1 -2 0 5 10 15 20 25 30 35 40 Years to Maturity Percent Multiple Exponential of Real YTM, k=3 4 3 2 1 0-1 -2 0 5 10 15 20 25 30 35 40 Multiple Exponential of Real YTM, k=8 48 38 28 18 8-2 0 5 10 15 20 25 30 35 40 Years to Maturity Multiple Exponential of Real YTM, k=4 4 3 2 1 0-1 -2 0 5 10 15 20 25 30 35 40 Years to Maturity Zero Coupon YC Par Bond YC Forward Rate Curve Observed Yield Rate

Outline 1 Introduction 2 Model Specification 3 Estimation and Results 4 Extension 5 Conclusion

Constrained Fit (Est.-III) Objective: Screen out the ill-priced securities from the pool. Selected Fit: Based on unconstrained estimations, select security i s.t. p i ˆp i and do the unconstrained estimation again using these underpriced securities only. Constrained Nonlinear Minimization: impose price constraints on the nonlinear minimization problem min {β j } k j=1 s.t. p i ˆp i n (p i ˆp i ) 2 i=1 for i = 1,..., n (13)

Constrained Fit (Est.-III) (cont d) Iterative Constrained Quadratic Programming: similar to an iterative LS with extra constraints min β ε ε 2 s.t. ε 0 (14) where ε i = log pi n n log ˆp i ( = log p i ) c i 2 δ(l) l ( log 1 + c i 2 ) + Ψ(m i )β then follow the similar iterative procedure as in Estimation-II. Results

Outline 1 Introduction 2 Model Specification 3 Estimation and Results 4 Extension 5 Conclusion

Highlights of Current Work Propose new functional form: Multiple Exponential Decay smoothes through idiosyncratic variations associated with the forward rate curve; governs the shape of curves by adjusting τ. Introduce new estimation method: Constrained Fit serve different curve-fitting purposes. Implement new procedure to control goodness of fit DW test Runs-test BIC

Future Work Complete daily estimates, and build up a criterion to evaluate all combinations of interpolation models and estimation methods Propose better term structure estimation techniques; Apply the estimated term structure to study macroeconomic fluctuations.

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Appendix 1: Data Source The Wall Street Journal daily market quotes Back 1 Back 2

Appendix 1: Data Source (cont d) Real Yield Curve is derived from TIPS market data; Nominal Yield Curve is derived from T-Notes & Bonds market data; Market quotes are daily released by the Wall Street Journal; Exclude less-than-one-year TIPS bonds from the data set; Back 1 Back 2

Appendix 2: Fitted φ(m) based on both Estmation-I and II Real φ(m) is derived from TIPS (Treasury inflation-indexed securities) market quotes on Mar. 31, 2010, n = 29; Nominal φ(m) is derived from T-Notes & Bonds market quotes on the same day, n = 166. φ(m) 74 64 54 44 34 24 14 4 Multiple Exponential Interpolation for U.S. Real Term Structure, k=5 Est.-I (t = 101.0s) Est.-II (t = 0.096s) Observed φ(m) -6 0 5 10 15 20 25 30 35 40 φ(m) 215 190 165 140 115 90 65 40 15 Multiple Exponential Interpolation for U.S. Nominal Term Structure, k=17 Est.-I (t = 5.24s) Est.-II (t = 0.15s) Observed φ(m) -10 0 5 10 15 20 25 30 35 40 Years to Maturity Back

Appendix 3: Fitted Real φ(m) and yield curves based on both Est.-II and III Estimate Real log discount function and yield curves on Mar. 31, 2010 by both Unconstrained Fit (Est-II) and Constrained Fit (Est-III), k = 4. φ(m) 77 65 53 41 29 17 Multiple Exponential of Real φ(m), 03/31/2010 5 Est-II Est-III (Selected Fit) Est-III (QuadProg Fit) Observed φ(m) -7 0 5 10 15 20 25 30 35 40 Percent Multiple Exponential of Real YTM by Est-II 4 3 2 1 0-1 Zero Coupon YC Par Bond YC Forward Curve Observed YTM -2 0 5 10 15 20 25 30 35 40 Multiple Exponential of Real YTM by Est-III (Selected) 4 Multiple Exponential of Real YTM by Est-III (QuadProg) 4 Percent 3 2 1 0-1 Zero Coupon YC Par Bond YC Forward Curve Observed YTM -2 0 5 10 15 20 25 30 35 40 Years to Maturity Percent 3 2 1 0-1 Zero Coupon YC Par Bond YC Forward Curve Observed YTM -2 0 5 10 15 20 25 30 35 40 Years to Maturity Back