1

Estimating the T erm Struct ure of I nterest R ates for Th

ai G overnment 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

ln ( )( )

P tr t

t( )( ) r t tP t e

( ) ln ( )d

f t P tdt

0( )

( )t

f s dsP t e

Theoretical Framework

4

Bond Price, Spot Rate and Forward Rate Relationship

0( )

( )t

f s dsP t e

( )( ) r t tP t eln ( )

( )P t

r tt

Discount function = price of zero-coupon bond P(t)

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

( ) ln ( )f t P tt

0( )

( )

tf s ds

r tt

( ) ( )dr

f t r t tdt

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 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

Time to Maturity

B-S

plin

e V

alue

B-Splines of degree 1

0 1( )

0iB t

1,

,i it t t

otherwise

Theoretical Framework

1 111

1 1

( ) ( ) ( ) , 1k k ki i ki i i

i k i i k i

t t t tB t B t B t k

t t t t

Recurrence relation

8

B-Splines of Degree One

Time to Maturity

B-S

plin

e V

alue

B-Splines of degree 1

Time to Maturity

B-S

plin

e V

alue

B-Splines of degree 1

Time to Maturity

B-S

plin

e V

alue

B-Splines of degree 1

Theoretical Framework

11 21

11 2

1 2 2 1

2

0 ,

,

,

0 ,

i

ii i

i i i i

ii i

i ii i i i i i i i

i

t t

t tt t t

t t t tB

t t t tt t t

t t t t t t t t

t t

11 21

21 2

2 2 1

2

0 ,

,

,

0 ,

i

ii i

i i i i

ii

i ii i i i

i

t t

t tt t t

t t t tB

t tt t t

t t t t

t t

Simplified to

9

B-Splines of Degree Two

Time to Maturity

B-S

plin

e V

alu

e

B-Splines of degree 2

Time to Maturity

B-S

plin

e V

alu

e

B-Splines of degree 2

Theoretical Framework

2

11 2 3

2 2

2 11 2

1 2 3 1 2 1 3 1

2 2 2

1 2

1 2 3 1 2 1 3 1 2 1 2

0 ,

,

,

i

ii i

i i i i i i

i ii i i

i i i i i i i i i i i i

i i i

i i i i i i i i i i i i i i i i

t t

t tt t t

t t t t t t

t t t tB t t t

t t t t t t t t t t t t

t t t t t t

t t t t t t t t t t t t t t t t

2 33 2

3

,

0 ,

i ii i

i

t t tt t

t t

10

B-Splines of higher degrees

is the pth spline of kth degree. and are the pre-specified knot values.

11

,

1( ) max ,0

p kp kkk

p ii p j p j i j i

B t t tt t

kpB

it jt

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)

44

33

,

1( ) max ,0

pp

p ii p j p j i j i

B t t tt t

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)

-3 -2 -1 0 5 10 15 20 25 300

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Time to Maturity

B-S

plin

e V

alue

B-Splines of degree 3

B1

B2

B3

B5B4 B6

Theoretical Framework

14

The Term Structure Fitting Using B-Splines

Approximation by curve S

λp are coefficients corresponding to the pth-spline that determines S(t)

Bond pricing Q represents bond price C is the cashflow matrix

1

( ) ( )P

p pp

S t B t

S B

Q CS

Q CB

Q D

Theoretical Framework

15

The Term Structure Fitting Using B-Splines

Bond pricing regression

Q represents bond price C is the cashflow matrix

Q D

Q D

* arg min Q D

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

1

( ) ( )uh

u u m mm

Q y t P t

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:

( )( ) r t tP t e

0( )

( )t

f s dsP t e

( ) (1 ) tP t r

Methodology

18

Discount Fitting Model

Bond price

Discount function

Discount fitting function

3

1

( ) ( )n

p pp

P t B t

3

1 1

( ) ( )uhn

u p u m p m up m

Q y t B t

1

( ) ( )uh

u u m mm

Q y t P t

3

1

(0) (0) 1n

p pp

P B

Restriction

Methodology

19

Spot Fitting Model

Bond price

Pure discount bond price

Spot function

Spot fitting function

1

( ) ( )uh

u u m mm

Q y t P t

( )( ) r t tP t e

3

1

( ) ( )n

p pp

r t B t

3

1 1

( )*exp ( )uh n

u u m m p p m um p

Q y t t B t

Methodology

20

Forward Fitting Model

Bond price

Pure discount bond price

Forward function

Forward fitting function

1

( ) ( )uh

u u m mm

Q y t P t

0( )

( )t

f s dsP t e

3

1

( ) ( )n

p pp

f t B t

3

1 10

( )*exp ( )mu

th n

u u m p p um p

Q y t B s ds

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

2

/

11

RSS mGCV

k nm

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

15 2

0

1

15MISE f t f t dt

2

1

1

15

n

i i ii

MISE f t f t t

f t

f t

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)

Degrees of B-splines

Fitting Models No. of

intervals 1 2 3 4 1 4.1652 0.6086 0.6398 0.6571 2 1.9095 0.6456 0.6504 0.6397 3 3.0853 0.6516 0.6402 0.6764 4 6.7737 0.7410 0.6720 0.7101

Non-restricted discount fitting

5 13.4286 0.7697 0.6788 0.7080 1 5.8474 0.6156 0.6482 0.6780 2 1.9105 0.6505 0.6669 0.6521 3 3.0855 0.6720 0.6485 0.6910 4 6.7741 0.7450 0.6812 0.7252

Restricted discount fitting

5 13.4325 0.7937 0.6901 0.7237 1 1.0697 0.6080 0.7835 24.9737 2 1.8126 0.5866 0.6166 0.6506 3 4.1256 0.6325 0.6482 0.6861 4 8.8449 0.6135 0.6638 0.6909

Spot fitting

5 17.9186 0.6366 0.6722 0.7161 1 0.5755 0.6149 1.2558 3.1312 2 0.5956 0.6290 0.7371 0.6560 3 0.5864 0.6093 0.6519 0.6910 4 0.6094 0.6371 0.6849 0.7163

Forward fitting

5 0.5883 0.6437 0.6925 0.7430

Empirical Results

26

Model Estimation, GCV (k = 3, n = 2)

Coefficient Standard Coefficient Standard Coefficient Standard Coefficient StandardCoefficients estimated deviation estimated deviation estimated deviation estimated deviation

λ 1 46.1719 1.1640 20.9119 10.5659 -9.9300 0.5426 34.5419 1.3728

λ 2 28.2470 0.3453 33.5065 0.2102 1.6937 0.1416 1.3276 0.4234

λ 3 19.0691 0.2291 21.7546 0.2013 1.5484 0.0597 1.8059 0.1956

λ 4 11.2992 0.3204 13.5150 0.3845 1.8390 0.0558 1.7731 0.1613

λ 5 8.0577 1.1845 8.5383 0.8324 1.9793 0.3786 2.9404 0.9888

Mean absolute error in priceStandard Error in price

0.006113

0.004267 0.004401 0.004572 0.004037

0.005407 0.005413 0.004916

Non-restricted discount fitting

Restricted discount fitting Spot fitting Forward fitting

Empirical Results

(%)

27

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

Empirical Results

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150.03

0.04

0.05

0.06

0.07

Time to Maturity (Years)

Yie

ldFitted Term Structures of Interest Rates Using Different Fitting Models

Non-Restricted Discount Fitting

Restricted Discount Fitting

Spot Fitting

Forward Fitting

28

Confidence Intervals for Estimated Coefficients (Spot Fitting, k = 3, n = 2)

Confidence interval 90% Confidence

level

95% Confidence level

99% Confidence

level

Coefficients Coefficient estimated

Lower Upper Lower Upper Lower Upper λ1 -9.9300* -10.8542 -9.0047 -10.9859 -8.8621 -11.4692 -8.5238 λ2 1.6937* 1.4603 1.9410 1.4258 1.9878 1.3298 2.0746 λ3 1.5484* 1.4471 1.6489 1.4290 1.6580 1.3923 1.6974 λ4 1.8390* 1.7507 1.9370 1.7313 1.9559 1.7145 1.9966 λ5 1.9793* 1.3817 2.6300 1.2961 2.7149 1.1116 2.8783

Note. * denotes statistical significance at 1% level.

Empirical Results

29

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

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Time to Maturity (Years)

Yie

ldConfidence Intervals of Fitted Term Structure Using Spot Fitting Model

Spot Fitting

90% Confidence Interval

95% Confidence Interval

99% Confidence Interval

Empirical Results

30

Minimum Values of Mean Integrated Squared Error (MISE) Degrees of B-splines

Fitting Models No. of

intervals 1 2 3 4 1 598.7168 7.8305 8.1659 12.5315 2 6.9287 6.5993 7.1390 5.8159 3 6.2296 4.8726 5.7929 5.8060 4 6.4855 5.2441 5.7634 5.8075

Non-restricted discount fitting

5 6.8819 5.5633 5.7749 5.7998 1 18.2418 11.8095 11.6981 14.0616 2 4.4299 7.1090 5.3741 5.2986 3 5.1292 4.5609 5.1625 5.3254 4 6.5882 4.8556 5.2161 5.3259

Restricted discount fitting

5 6.7971 4.8999 5.2927 5.3016 1 10.8847 12.1679 28.5943 805.6486 2 9.0500 4.9496 5.3937 5.1979 3 12.2950 4.8651 5.3637 5.3449 4 21.8021 4.8851 5.2896 5.2878

Spot fitting

5 39.6497 4.8382 5.2997 5.1504 1 11.9066 11.6931 54.6733 195.7474 2 7.0404 7.3207 17.5182 5.3463 3 5.0594 5.0356 5.2257 5.3309 4 5.1150 5.2362 5.3070 5.3048

Forward fitting

5 5.1105 5.2570 5.2564 5.1738

Empirical Results

31

Model Estimation, MISE (k = 3, n = 3)

Coefficient Standard Coefficient Standard Coefficient Standard Coefficient StandardCoefficients estimated deviation estimated deviation estimated deviation estimated deviation

λ 1 20.3312 2.5008 14.7421 3.7146 -2.6445 0.3716 -4.8588 0.3406

λ 2 15.8184 0.2851 25.8484 0.1514 0.8933 0.1237 1.2564 0.1164

λ 3 17.191 0.1052 14.7316 0.1521 1.083 0.0462 1.0081 0.0428

λ 4 13.06 0.1388 11.2757 0.182 1.0809 0.0177 1.1851 0.0254

λ 5 9.512 0.2569 8.7929 0.2038 1.3274 0.0528 1.6465 0.0551

λ 6 7.7745 0.4664 5.618 0.7136 1.3475 0.2858 0.7464 0.1272

Mean absolute error in priceStandard Error in price

0.004945

0.004404 0.004639 0.004601 0.004578

0.005227 0.00501 0.004859

Non-restricted discount fitting

Restricted discount fitting Spot fitting Forward fitting

Empirical Results

(%)

32

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

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150.03

0.04

0.05

0.06

0.07

Time to Maturity (Years)

Yie

ldFitted Term Structures of Interest Rates Using Different Fitting Models

Non-Restricted Discount Fitting

Restricted Discount Fitting

Spot Fitting

Forward Fitting

Empirical Results

33

Confidence Intervals for Estimated Coefficients (Restricted Discount Fitting, k = 3, n = 2)

Confidence interval 90% Confidence

level

95% Confidence level

99% Confidence

level

Coefficients Coefficient estimated

Lower Upper Lower Upper Lower Upper λ1 14.7421* 8.2782 20.4095 6.6548 21.7498 2.8722 22.9186 λ2 25.8484* 25.5962 26.1076 25.5690 26.1544 25.4997 26.3236 λ3 14.7316* 14.4600 14.9820 14.4220 15.0243 14.3556 15.1203 λ4 11.2757* 10.9801 11.5577 10.9223 11.6237 10.7979 11.7636 λ5 8.7929* 8.4503 9.1338 8.4089 9.2183 8.2517 9.4354 λ6 5.6180* 4.5614 6.9330 4.3311 7.0500 3.6950 7.5496

Note. * denotes statistical significance at 1% level.

Empirical Results

34

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

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Time to Maturity (Years)

Yie

ldConfidence Intervals of Fitted Term Structure Using Restricted Discount Fitting Model

Restricted Discount Fitting

90% Confidence Interval

95% Confidence Interval

99% Confidence Interval

Empirical Results

35

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

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150.03

0.04

0.05

0.06

0.07

Time to Maturity (Years)

Yie

ldTerm Structures from GCV, MISE and ThaiBMA

GCV

MISE

ThaiBMA

Empirical Results

36

Confidence Intervals of Restricted Discount Fitting/ Spot Fitting with ThaiBMA

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Time to Maturity (Years)

Yie

ld

ThaiBMA curve and Confidence Intervals of Fitted Term Structure Using Spot Fitting Model

ThaiBMA

Spot Fitting

90% Confidence Interval95% Confidence Interval

99% Confidence Interval

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Time to Maturity (Years)

Yie

ld

ThaiBMA curve and Confidence Intervals of Fitted Term Structure Using Spot Fitting Model

ThaiBMA

Restricted Discount Fitting

90% Confidence Interval95% Confidence Interval

99% Confidence Interval

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

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

Conclusion