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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
Recommended