Fitting the term structure of interest rates for Taiwanese government bonds

Journal of Multinational Financial Management 9 (1999) 331 – 352 www.elsevier.com/locate/econbase

Fitting the term structure of interest rates for Taiwanese government bonds Bing-Huei Lin * Department of Business Administration, National Taiwan Uni6ersity of Science and Technology, 43 Keelung Road, Sec. 4, Taipei, Taiwan, ROC Received 15 July 1998; accepted 26 February 1999

Abstract The term structure of interest rates provides a basis for pricing fixed-income securities and interest rate derivative securities as well as other capital assets. Unfortunately, the term structure is not always directly observable because most of the substitutes for default-free bonds are not pure discount bonds. We use curve fitting techniques with the observed government coupon bond prices to estimate the term structure. In this paper, the B-spline approximation is used to estimate the Taiwanese Government Bond (TGB) term structure. We apply the B-spline functions to approximate the discount function, spot yield curve, and forward yield curve respectively. Among the three approaches, the discount fitting approach and the spot fitting approach are reasonable and reliable, but the spot fitting approach achieves the most suitable fit. Using this methodology, we can investigate term structure fitting problems, identify coupon effects, and analyze factors which drive term structure fluctuations in the TGB market. © 1999 Elsevier Science B.V. All rights reserved. Keywords: Term structure of interest rates; B-spline approximation; Discount fitting model; Spot fitting model; Forward fitting model JEL classification: E43; G15

* Corresponding author. Tel.: +886-2-27376748; fax: + 886-2-27376744. E-mail address: [email protected] (B.-H. Lin) 1042-444X/99/$ - see front matter © 1999 Elsevier Science B.V. All rights reserved. PII: S 1 0 4 2 - 4 4 4 X ( 9 9 ) 0 0 0 0 6 - 7

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1. Introduction The term structure of interest rates specifying the relationship between the prices (or yields) of default-free pure discount (zero-coupon) bonds and their time to maturity provides a basis for pricing fixed-income securities and interest rate derivative securities, as well as other capital assets. The term structure is thus important for portfolio management, financial engineering, and corporate finance in investment and financing decisions. There are three equivalent descriptions of the term structure of interest rates: the discount function which specifies zero-coupon bond (with a par value of $1) prices as a function of maturity, the spot yield curve which specifies zero-coupon bond yields (spot rates) as a function of maturity, and the forward yield curve which specifies zero-coupon bond forward yields (forward rates) as a function of maturity. The term structure of interest rates is not always directly observable because, with the exception of short-term treasury-bills, most of the substitutes for default-free bonds (government bonds) are not pure discount bonds. Thus the coupon bond price, which may contain coupon effects, may not provide a good substitute for calculating the term structure of interest rates. To estimate the term structure there are two categories of models: equilibrium models and empirical models. Equilibrium models, proposed by researchers such as Vasicek (1977), Dothan (1978), Brennan and Schwartz (1979), Cox et al. (1985), make use of the assumption that certain variables, such as the short-term risk-free rate, follow a stochastic process driving the term structure of interest rates. They then use an arbitrage pricing technique to span the whole term structure based on the stochastic variables. The resulting term structure of interest rates is a theoretical one consistent with arbitrage-free conditions in an efficient market, but it is hardly able to fit the actually observed data on bond yields and prices. Usually, the actual yield curves exhibit more varied shapes than those justified by the equilibrium models. In contrast to equilibrium models, empirical models, such as those developed by McCulloch (1971), Carleton and Cooper (1976), Schaefer (1981), Vasicek and Fong (1982), Chambers et al. (1984), Nelson and Siegel (1987), Steeley (1991), Pham (1998), use curve fitting techniques with the observed government coupon bond prices to estimate the spot yield curve (pure discount bond yield curve). Since a coupon bond is nothing more than a portfolio of pure discount bonds with maturity dates consistent with the coupon dates, the discount bond prices can be extracted from actual coupon bond prices. Regardless of the efficacy of their curve fitting techniques, the empirical models, by focusing on actually observed data, are able to describe a rich variety of realistic yield curve patterns. The resulting term structure of interest rates can be directly put into interest rate contingent claim pricing models, such as Ho and Lee (1986) Babbs (1990), Heath et al. (1992), Hull and White (1990) models, for pricing various interest rate contingent claims. The objective in empirical estimation of the term structure is to fit the data sufficiently well and, at the same time, obtain a sufficiently smooth and continuous function as the term structure of interest rates.

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In this article, we use a methodology, called B-spline approximation, suggested by Shea (1984) and successfully generalized and adopted by Steeley (1991), Lin and Paxson (1993), to estimate the Taiwanese Government Bond (TGB) term structure. We firstly apply the B-spline functions and use a linear regression approach to approximate the discount function (discount fitting model) as done by Steeley (1991), Lin and Paxson (1993). In addition, we also apply the B-spline functions and use a non-linear regression approach to approximate the spot yield curve (spot fitting model), as suggested by Vasicek and Fong (1982), Chambers et al. (1984). Both of the fitting models can provide a reasonable fitting for the term structure, but the spot fitting model is more satisfactory. Alternately we also apply the B-spline functions to approximate the forward yield curve (forward fitting model) directly. This is in an attempt to obtain a more smooth forward yield curve, which is stressed by Adams and Van Deventer (1994), Frishling and Yamamura (1996) whose aim was to obtain a maximized smoothness for the forward yield curve. Unfortunately, while the forward fitting model can provide the best fitted curve for the term structure, it can not provide a smooth, well-behaved forward yield curve. Using the methodologies included in this study, we can investigate the term structure fitting problems, identify coupon effects on the TGB prices, and analyze factors which drive the term structure fluctuations in the TGB market. The methodologies and results of this study provide important implications for financial institutions in pricing and trading fixed-income securities, and in hedging long-term interest risks, as well as in developing long-term interest rate derivative securities. Following the internationalization and liberalization of financial markets, the Taiwanese capital market has become one of the most important markets in the Asia – Pacific area. Although the Taiwanese government bond market is small and not liquid compared to other developed bond markets, it is increasingly important and worthy of study. The rest of this study is organized as follows: In Section 2 we review the empirical term structure estimation methodologies. In Section 3 we develop the B-spline approximation methodology to estimate the term structure. In Section 4 we conduct an empirical study for the TGB market. We then conclude in Section 5.

2. Term structure fitting methodologies To estimate the term structure of interest rates, assume that there are n default-free coupon bonds included in the sample. The price of the coupon bond i, Bi is a linear combination of a series of pure discount bond prices, i.e. Ni

Bi = % di (tm )P(tm ),

(1)

m=1

where tm is the time when the m-th coupon or principal payment is made, Ni is the number of coupons and principal payments before the maturity date of bond i, di (tm ) is the cash flow paid by bond i at time tm, and P(tm ) is the pure discount

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bond price with face value $1 (called the discount function) with maturity at time tm. Once the discount function, P(t) is defined, the spot interest rate (the pure discount bond yield) is also defined by R(t) =

− InP(t) . t

(2)

The instantaneous forward interest rate for the period [t, t + Dt], Dt “ 0, is f(t) =

(P(t) . (t

(3)

To estimate P(t) in Eq. (1), the simplest and most straightforward way is to construct a regression model similar to that adopted by Carleton and Cooper (1976). By appropriately setting the time intervals and assigning coupon dates to each interval, one can obtain a linear regression model as in Eq. (1) plus an error term.1 We then can regress the observed bond prices on the promised payoff matrix and obtain the estimated coefficients as the discount function. However, this methodology does not satisfy most researchers because it cannot provide a smooth, continuous yield curve. Moreover, this method is only appropriate for estimating the discount function with shorter time to maturity.2 An alternative methodology to estimate a continuous yield curve is to use a spline function approximation technique. According to the Weierstrass Approximation Theorem, any continuous function can be arbitrarily closely approximated by a set of functions, for example polynomial functions, over a given interval. Based on this theory, we can approximate the discount function (assumed to be continuous) by using a set of spline functions which are dependent on time. Specifically, for the discount fitting model, let k

P(t) = % bj gi (t),

(4)

j=1

where gj (t) is the j-th approximation function which is dependent on time, and bj ’s are the coefficients to be estimated which are applied to the k approximation functions. Combining Eqs. (1) and (4), and introducing an error term, then k

Bi = % bj



j=1

Ni



% di (tm )gj (tm ) + oj.

(5)

m=1

In the discount fitting model, an additional restriction on the coefficients needs to be imposed, i.e. k

P(0) = % bj gj (0) =1.0.

(6)

j=1

1

The error term may be caused by transaction costs, coupon effects, market imperfections, etc. As discussed by Carleton and Cooper (1976), due to the paucity of data for longer times to maturity, it is easy to obtain a singular payoff matrix. This will cause the regression model to fail. 2

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With this restriction, we can be sure that the estimated discount function will take a value of unity at maturity. Moreover, we may also introduce constraints on the forward rates as done by Schaefer (1981) and examined by Shea (1984), to obtain a well-behaved forward yield curve. Schaefer (1981) constrained the slope of the discount function to be negative everywhere, while Shea (1984) reported that simple restrictions of fixed proportions between first derivatives were often sufficient. However, the addition of constraints to spline approximation functions is necessarily ad hoc, and the results of Shea (1984) demonstrate that non-negative constraints can dramatically alter the structure of the forward yield curve in places other than where negative rates are constrained away. Using B-spline approximation functions, following Steeley (1991), we do not impose these constraints other than the constraint in Eq. (6). Although the restriction can guarantee that the estimated term structure meets reality, it will influence the estimation efficiency. Having specified the function gj (t), Eq. (5) is a well-defined linear regression model. The crucial problem is how to choose the function gj (t) and the number of functions k. Unfortunately, there is no economic theory or rule for this purpose. The only rule is empirical. If the model can fit the observed data well and results in a smooth spot yield curve and a well-behaved forward yield curve, then it is assumed to be an appropriate model. In selecting the number of functions k, a higher degree of k might produce a greater degree of accuracy, but it will cause the model to be over-parameterized and will obtain insignificant coefficients. As a result, the estimation will be unreliable. In choosing the approximation functions, McCulloch (1971, 1975) used various quadratic and cubic piecewise polynomial functions, Schaefer (1981) used a set of Bernstein polynomial functions, Vasicek and Fong (1982) used the exponential spline, Steeley (1991) used the B-spline, and more recently Pham (1998) used the Chebyshev polynomial functions. McCulloch (1971) admitted that using piecewise quadratic functions to approximate the yield curve will generate a non-smooth forward yield curve.3 As an alternative to the above mathematical spline functions, Brown and Dybvig (1986), Brown and Schaefer (1994), De Munnik and Schotman (1994), Sercu and Wu (1997) used economic functions such as the Vasicek (1977) and the Cox et al. (1985) term structure models to fit the market yield curve. Although these functions can provide economic explanations, they fail to provide a rich variety of shapes to fit the versatile market yield curve. Vasicek and Fong (1982) claimed that since the polynomial function has a different curvature from the exponential function, which the discount function is supposed to be, a polynomial spline function tends to weave around the exponential function, resulting in highly unstable forward rates. Thus they used exponential spline functions to estimate the term structure of interest rates. In a different manner, Chambers et al. (1984) incorporated the exponential characteristic by suggesting that the spot yield curve rather than the discount function be approxi3

To obtain a smooth forward yield curve, one needs at least a cubic spline function to approximate the discount function. Because the forward interest rate is the first derivative of the discount function, to have a smooth forward yield curve, the discount function must be twice continuously differentiable.

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mated using an exponential function. Here we follow their approach to directly approximate the spot yield curve R(t), by letting (spot fitting model) k

R(t) = % bj gj (t).

(7)

j=1

Combining Eqs. (1), (2) and (7), and introducing an error term, then Ni



k

n

Bi = % di (tm ) exp −tm · % bj gj (tm ) + oi. j=1

m=1

(8)

With this methodology, restrictions on the coefficients which are required in the discount fitting model are no longer needed. Eq. (8) is not a linear model, thus a non-linear regression procedure is required, and the computation burden is much heavier than the discount fitting model. Since a smooth forward yield curve is a major concern for many studies, such as Adams and Van Deventer (1994), Frishling and Yamamura (1996), in fitting the term structure of interest rates, another alternative is to approximate the forward yield curve f(t) directly. That is (forward fitting model) k

f(t) = % bj gj (t).

(9)

j=1

Combining Eqs. (1), (3) and (9), and introducing an error term, then Ni

 &

Bi = % di (tm ) · exp − m=1

tm

0

k

n

% bj gj (s) ds + oi.

j=1

(10)

As in the spot fitting model, the forward fitting model has no restrictions on the coefficients needed. Although we expect this approach can fit the term structure better, estimation is computationally time-consuming. Moreover, with this approach the forward yield curve is expected to be most sensitive to the actual market data, and a flexible and versatile forward yield curve is expected.

3. The B-spline approximation In choosing spline approximation functions in Eqs. (5), (8) and (10), we use the B-spline functions to approximate the discount function, the spot yield curve, and the forward yield curve respectively. The B-spline functions were suggested by Shea (1984) and have been successfully used by Steeley (1991) to estimate the UK. Gilt-edged bond term structure, and Lin and Paxson (1995) to estimate the German government bond term structure. They all concluded that B-spline methodology can approximate the discount function appropriately and result in reliable and smooth spot and forward yield curves. Moreover, after examining various techniques used to term structure fitting, Deacon and Derry (1994) concluded that the B-spline approach is the most preferred methodology for practitioners. The B-spline function was defined by Powell (1981) as:

B.-H. Lin / J. of Multi. Fin. Manag. 9 (1999) 331–352 s+p+1

g ps (t) =

%

i=s



s+p+1

n

1 (t − ti ) j = s, j " i j 5

[max(t − tj, 0)]p,

s= 1, 2, ··· ,k;

337

=p+ m (11)

where g ps (t)is called the s-th p-order B-spline function. It is non-zero only if t is in the interval [ts, ts + p + 1 ], and zero otherwise. m is the number of sub-periods between time zero (t =0) and the longest maturity date of the sample bonds (called the approximation space). There are p + m B-spline functions required in this procedure. The two ends of any time interval [ts, ts + 1] are called knots. There are 2p + m +1 knots required within the time horizon. For example, in this article, we set p = 3 (cubic B-spline function4) and m= 3. We then need to specify ten knots and define six B-spline functions gs(t) (we omit the superscript p to simplify the presentation). The time to maturity of the sample bonds is between 0 to 15 years. We break it into three sub-periods: from 0 to 4, 4 to 8, and 8 to 15.5 To have all the six functions well-defined, we need to specify the knots beyond the two ends, 0 and 15 years. We then add − 3, − 2, −1 and 20, 25, 30 in the time horizon. That means we set t1, …, t10 equal to − 3, …, 306. It is these six B-spline functions which construct the basis for approximating the discount function. Having defined the B-spline functions, we can run Eqs. (5) and (8), and 10 to estimate the coefficients for the discount fitting model, the spot fitting model, and the forward fitting model. Concerning the error termi in Eqs. (5) and (8), and 10, we do not have any reason to expect it meets the classical regression assumption. Instead, as Vasicek and Fong (1982) pointed out, we have an a priori belief that the model is homoscedastic in yields while heteroscedastic in prices, because the bond price is almost certain, and equal to its face value when the time is near maturity. When the time is far from maturity, the bond price is more uncertain and the variance of bond prices is greater. Thus, we can expect that the variance of bond prices is time dependent. According to Schaefer and Schwartz (1987), the duration of the bond explains most of the variance of bond prices. It is then reasonable to assume that the error terms in Eqs. (5) and (8), and 10 are: E(o 2i ) =s 2hi ;

E(oi, oj ) =0

for

i" j,

(12)

where hi is the duration of bond i and hi =

1 Ni tmdi (tm ) % , Bi m = 1 (1 +ymi )tm

(13)

where ymi is the yield to maturity of bond i. 4

For the discount fitting model, a cubic function is necessary to obtain a continuous forward yield curve. For comparative purposes, we also apply the same cubic function for the spot fitting and the forward fitting models. 5 The criterion is that the number of sample bonds in each sub-period is approximately equal. 6 There is no economic theory or rule for this purpose. The selection procedure in this article is more or less ad hoc. We will describe this later.

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Once the continuous term structure is obtained, we can calculate the discretized yield curves. We calculate semi-annually compounded interest rates with maturity from 0.5 to 15 years. The semi-annually compound spot interest rate can be calculated by

 

2

1 R(t) = P(t)



t

(14)

−1,



and the equivalent forward interest rate for the period [t− 0.5, t] is F(t) =

P(t −0.5) P(t)

2

−1,

(15)

where t=0.5, 1.0, 1.5, …, 14.5, 15.0 years.

4. The data and results We studied the term structure of TGB interest rates for the period from 4 October, 1997 to 2 May, 1998. We used the prices of 25 government bonds offered by the Grand Cathay Security Company. The prices are quoted in yields to maturity of the bonds. We used the weekly (weekend, normally Saturday, occasionally Friday) yield as the research sample. Data is from weekly reports published by the Grand Cathay Security Company. In fact the TGB market is small and not as liquid as other developed markets. The original life of TGBs is from 3 to 15 years, and the issuing size is from 10 to 50 billion Taiwan dollars. Bonds issued before 1996 are paying coupons semi-annually. After 1996, the government issued bonds with annual coupon payments. In 1995, the government issued two zero-coupon bonds, with original duration equal to three years. To estimate the term structure of interest rates, we used the B-spline approximation technique as described in the previous section. The first step of the estimation procedure was to identify the parameters p, m, and the knots as specified in the previous section. In order to obtain a smooth forward yield curve for the discount fitting model, following Steeley (1991), we used cubic (p= 3) B-spline functions to approximate the discount functions. To decide m we used a trial-and-error procedure to compare the average of squared predicting errors (the actual price minus the model price) and the number of significant coefficients estimated with different values of m. Once m was defined, the within-the-sample knots could be identified simply by equalizing the number of samples for each sub-period between these knots. We found that when m = 3 (the within-the-sample knots were 0, 4, 8, and 15), the six (k = p =m) coefficients were all significant at the significance level 0.05. When m increased, although the average of squared predicting errors decreased by a trivial value, the number of significant coefficients estimated did not significantly increase. When k = 6, the average standard predicting error made by the model is only about 0.3 percent of the par value, a quite satisfactory level. As to the out-of-the-sample knots, we found that prediction errors were not significantly changed using different out-of-sample knots. By an ad hoc decision we set the

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out-of -sample knots equal to − 3, − 2, − 1,and 20, 25, 30. Although the B-spline functions as defined by the above procedures are for the discount fitting model, we also used them for the spot fitting and forward fitting models for consistency and comparison. Without losing generality, we report the results of the estimation for the case of May 2, 1998. Table 1 shows the results of the estimation using the three fitting models. All coefficients estimated are statistically significant at a significance level of 1%, implying that these models are adequate. As to the fitting error, it was 32, 28, and 27 basis points, in terms of percentage of par-value, for the discount fitting model, the spot fitting model, and the forward fitting model respectively. In terms of yields, the standard fitting error was 6, 5, and 5 basis points, respectively. From a practical point of view, these errors are quite significant. This may be due to other factors such as coupon effects on the bond prices. Checking the yield data of 2 May, 1998, in Fig. 1, we find that most of the bonds with coupons paying annually are consistently traded at lower yields than those with coupons paying semi-annually. To incorporate this information into the fitting models, we added the number of coupon payments per year as an additional variable in the fitting models. In addition, we also included the variable of coupon rate level in the fitting models to study the coupon effects. The results for the three models with coupon effects included are reported in Table 2. From Table 2, all but two of the coefficients estimated are statistically significant. For the spot fitting model and the forward fitting model, the coupon effects are significant. In the case of the spot fitting model, the coefficient for the number of payments is − 0.6496, reflecting that annual payment bonds were traded at higher prices (lower yields). As to the coupon rate level, the coefficient is 0.0591, implying higher coupon bonds were traded at higher prices (lower yield). In the case of the forward fitting model, the result is quite similar to that of the spot fitting model. In the case of the discount fitting model, the coupon effects are not significant. This may be because the discount fitting model had difficulty matching the curvature of

Fig. 1. Market yield curve (2 May, 1998): represents market yields for bonds paying coupon semi-annually; and  represents market yields for bonds paying coupon annually.

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Table 1 Results of model estimation (without coupon effects) for 2 May, 1998a,b,c Coefficients

Discount fitting model

Spot fitting model Standard deviation

b1 b2 b3 b4 b5 b6

7.5376* 9.3912* 12.3589* 11.3045* 7.2874* 9.4574*

0.0243 0.0532 0.1203 0.1415 0.3245 1.0426

Sample size Standard error in price Standard error in yield

25 0.3174

25 0.2837

25 0.2693

0.0006

0.0005

0.0005

0.4834* 0.6245* 1.1354* 1.1759* 1.5643* 0.8355*



a

i Discount fitting model: Bi =6j = 1bj N m = 1di (tm )gj (tm ) +oi.

b

6 i Spot fitting model: Bi =N m = 1di (tm )exp −tm ·j = 1bj gj (tm ) +oi.

c

tm i Forward fitting model: Bi =N 6j = 1bj gj (s) ds +oi. m = 1 di (tm ) · exp − 0



 

* Statistically significant at a significance level of 5%.

n

n

Standard deviation 0.0186 0.0232 0.0102 0.0163 0.0268 0.1277

Coefficient estimated

0.4015* 0.6225* 0.8429* 1.1056* 1.9605* 0.3059*

Standard deviation

0.0346 0.0337 0.0295 0.0503 0.0884 0.1215

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Coefficient estimated



Coefficient estimated

Forward fitting model

Table 2 Results of model estimation (with coupon effects) for 2 May, 1998a,b,c,d,e Coefficients

Sample size Standard error in price Standard error in yield

Spot fitting model

Forward fitting model

Coefficient estimated

Standard deviation

Coefficient estimated

Standard deviation

Coefficient estimated

Standard deviation

7.4297* 9.3803* 12.2617* 11.2570* 7.3907* 9.0245* −0.6585 0.0855

0.0239 0.0411 0.0939 0.1530 0.3092 1.2281 0.3523 0.0663

0.4576* 0.5900* 1.0216* 1.2167* 1.5031* 0.8201* −0.6496* 0.0591*

0.0195 0.0171 0.0087 0.0152 0.0270 0.1055 0.1314 0.0214

0.3865* 0.6128* 0.8032* 1.1076* 1.8337* 0.6235* −0.6674* 0.0653*

0.0353 0.0405 0.0313 0.0415 0.0679 0.1437 0.1416 0.0301

25 0.2220

25 0.1695

25 0.1523

0.0005

0.0003

0.0003





a

i Discount fitting model: Bi =6j = 1 bj N m = 1di (tm )gj (tm ) +b7Paymentsi+b8Couponi+oi

b

6 i Spot fitting model: Bi =N m = 1 di (tm ) exp −tm · j = 1bj gj (tm ) +b7Paymentsi+b8Couponi+oi

c

tm i Forward fitting model: Bi =N 6j = 1 bj gj (s) ds +b7Paymentsi+b8Couponi+oi. m = 1 di (tm ) · exp − 0

d



 

n

n

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b1 b2 b3 b4 b5 b6 b7 b8

Discount fitting model

Paymentsi =1 if bond i pays coupon semi-annually, and Paymentsi =0 if bond i pays coupon annually. Couponi is bond i’s coupon in percentage of par value 100. * Statistically significant at a significance level of 5%.

e

341

342

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the yield curve. Checking the pricing errors of the three models, the error remains to be the highest for the discount fitting model. The fitting error declines sharply for the spot fitting model and the forward fitting model, with the pricing error approximately equal to 17 and 15 basis points in prices, respectively, and three basis points in yields for both models. Thus, we can identify coupon effects, the coupon payment effect, and the coupon rate effect on the TGB prices. Bonds with semi-annual coupon payments are traded at a higher yield than those with annual coupon payments. Presumably, this is because bonds with annual coupon payments are more recent issues and more actively traded, thus they are traded at a lower yield. Another reason is that bonds with semi-annual coupon payments are subject to higher reinvestment risks, thus they are traded at a higher yield. We also identify the negative coupon rate effect on TGB prices. Bonds with a higher coupon rate are traded at a lower yield, a contradiction to the normal tax effect on the bond prices. Presumably, this is because tax evasion is common in TGB trading, due to the differences in taxing interest income for individuals and institutions. For individuals, interest income from bond investment is taxed on a cash basis, while for institutions interest income from bond investment is taxed on a accrued basis. Moreover, there are no taxes on capital gains for securities trading in Taiwan. As a result, bonds held by individuals are transferred to the hands of institutions just before the coupon payment date. The individual, who does not receive the interest payment, thus pay no taxes on the income. The institution, on the other hand, holding the bond only for a few days, only has to pay a small amount of interest income taxes. As a result, bond trading in Taiwan is essentially tax-free, and the bond prices are not subject to tax effects. Next we used the three estimated models (the results of Table 2) to calculate spot and forward yield curves. Using Eqs. (14) and (15), we calculated discrete yield curves for 30 semi-annually compounded zero-coupon bond yields with times to maturity from 0.5 to 15 years. The results are shown in Fig. 2. In the discount fitting model and the spot fitting model, the estimated spot yield curves for both models look smooth and flat. However, the spot yield curve estimated from the spot fitting model exhibits a slightly flexible shape (especially in the two ends) compared to the one estimated from the discount fitting model. The forward yield curves for both the discount fitting model and the spot fitting model are also similar. Compared to the spot yield curves, the forward yield curves fluctuate more dramatically for longer maturity and swerve downwards over the long term. Regardless of the efficiency of the market prices, although the tendency of the forward yield curve to decline to negative values for longer maturity (over 15 years) makes the forward yield curve look quite bad, it actually reflects the actual market yield curve.7 Regarding the forward fitting model, the forward yield curve swings drastically and even swerves to significant negative values. This may be because the forward fitting model tends to over-fit the forward yield curve, thus resulting in a badly behaved forward yield curve. 7 From Fig. 1, we can find that the two bonds with longest maturity have their traded yields significantly below the yields of other long-term bonds.

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Fig. 2. Estimated term structure (2 May, 1998): Panel A, Discount fitting model; Panel B, Spot fitting model; Panel C: Forward fitting model.

Fig. 3 plots the pricing errors for the three fitting models. In general, these pricing errors scatter randomly around zero, and fluctuate more as the time to maturity increases. This may justify the assumption of our regression methodology with the correction for heteroscedasticity as in Eq. (12). Looking at Figs. 2 and 3 together, although the discount fitting model can provide the most smooth and well-behaved yield curves, it also has the most significant pricing errors compared to other fitting models. On the other hand, the forward fitting model can best fit the actual market data, but it can not provide well-behaved yield curves. The spot fitting model is relatively moderate with respect to fitting performance and reliability.

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Since the forward fitting model is not appropriate for term structure fitting, we compared the other two models in estimating the term structure for the sample period from 4 October, 1997 to 2 May, 1998. Table 3 shows the results for the discount fitting model. All the spline coefficients are significant at 5%, while most of all coefficients for coupon effects are not. The pricing error is on average 25.4 basis points in terms of percentage of par value, and on average five basis points in terms of yield. Compared to the spot fitting model, the discount fitting model does not fit the term structure satisfactorily. Table 4 shows the results of the spot fitting model. All spline coefficients and almost all coefficients for coupon effects are significant at 5%. The pricing error is on average 20 basis points in terms of percentage of par value, and on average three basis points in terms of yield. The fitting power is more satisfactory than the discount fitting model.

Fig. 3. Estimated term structure pricing error (2 May, 1998): Panel A, Discount fitting model; Panel B, Spot fitting model; Panel C, Forward fitting model.

Table 3 Coefficients estimated by discount fitting modela Date

No. of bonds

Average

Coeff. 1

Coeff. 2

Coeff. 3

Coeff. 4

Coeff. 5

Coeff. 6

No. of payments

Coupon rate

In price

In yield

7.3939 7.3879 7.3819 7.3901 7.3835 7.3910 7.3914 7.3946 7.4030 7.4031 7.4060 7.3914 7.3943 7.3950 7.4085 7.4105 7.4113 7.4117 7.4135 7.4180 7.4202 7.4211 7.4212 7.4229 7.4187 7.4261 7.4260 7.4304 7.4321 7.4282 7.4297

9.4371 9.4512 9.4633 9.4452 9.4565 9.4416 9.4408 9.4382 9.4252 9.4249 9.4217 9.4480 9.4441 9.4430 9.4190 9.4152 9.4139 9.4132 9.4099 9.4027 9.3982 9.3968 9.3966 9.3916 9.3990 9.3859 9.3862 9.3791 9.3759 9.3828 9.3803

12.4021 12.2857 12.2262 12.3537 12.3527 12.4210 12.4221 12.3301 12.2858 12.2906 12.2330 12.1902 12.1565 12.1548 12.1806 12.1926 12.1933 12.1942 12.1995 12.1838 12.2057 12.2005 12.2048 12.2692 12.2644 12.2776 12.2726 12.2544 12.2675 12.2574 12.2617

11.3724 11.3619 11.5893 11.5544 11.6578 11.4910 11.4883 11.3737 11.3220 11.1953 11.1869 11.2985 11.3328 11.3401 11.1022 11.0877 11.0827 11.0960 11.0576 10.9708 11.0033 11.0324 11.0578 11.1400 11.2554 11.1852 11.1958 11.1674 11.2347 11.2531 11.2570

7.4381 7.0637 6.8139 7.0412 7.0049 7.1333 7.1395 7.2954 7.1189 7.1538 6.9195 6.6212 6.5348 6.5005 6.8720 6.8603 6.8734 6.7991 6.8925 6.9760 6.9346 6.9915 7.0394 7.2167 7.3958 7.4648 7.4147 7.4321 7.3325 7.4395 7.3907

8.4214 7.6239 8.0057 8.1504 8.0943 8.2382 c 8.1876 c 7.5500 7.9444 7.4249 7.9380 8.4183 8.3907 8.5802 7.4868 7.6354 7.6123 8.0897 7.6805 7.3616 7.8523 7.8684 8.1370 8.5469 8.6477 8.3905 8.7406 8.7410 9.1386 9.0367 9.0245

−0.6138 c −0.6428 c −0.6668 c −0.7204 c −0.7377 c −0.6624 c −0.6582 c −0.7347 c −0.7892 −0.8047 c −0.6814 −0.8575 −0.8043 −0.7939 −0.8673 −0.8452 −0.8456 −0.8375 −0.7336 −0.7938 −0.7913 −0.8167 −0.7849 −0.7842 c −0.7799 c −0.6211 c −0.6113 c −0.6621 c −0.6573 c −0.6898 c −0.6585 c

0.0821 c 0.1093 c 0.1221 c 0.1074 c 0.0992 c 0.0797 c 0.0792 c 0.0922 c 0.1101 c 0.1017 c 0.0713 c 0.1400 c 0.1274 c 0.1263 c 0.1252 c 0.1211 c 0.1208 c 0.1187 0.1066 c 0.1128 0.1081 0.1165 0.1102 c 0.1019 c 0.1029 c 0.0699 c 0.0696 c 0.0835 c 0.0811 c 0.0889 c 0.0855 c

0.2136 0.2984 0.3265 0.2749 0.3869 0.3544 0.3564 0.2362 0.2018 0.2729 0.1993 0.2638 0.2443 0.2428 0.2392 0.2139 0.2144 0.2090 0.2116 0.2026 0.1906 0.1801 0.2121 0.2674 0.2852 0.2825 0.2857 0.2727 0.2600 0.2540 0.2220

0.0004 0.0006 0.0007 0.0005 0.0007 0.0006 0.0006 0.0005 0.0005 0.0005 0.0005 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0004 0.0005 0.0005 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005 0.0005

0.2540

0.0005





i The fitting model is: Bi = 6j = 1 bj N m = 1 di (tm )gj (tm ) +b7Paymentsi+b8Couponi+oi.

c

Standard error

Statistically non-significant at a significance level of 5%.

345

a

21 21 21 21 21 21 21 22 22 22 22 23 23 23 23 23 23 23 23 23 24 24 24 24 25 25 25 25 25 25 25

Coupon effects

B.-H. Lin / J. of Multi. Fin. Manag. 9 (1999) 331–352

10/4/97 10/9/97 10/18/97 10/24/97 10/30/97 11/8/97 11/15/97 11/22/97 11/29/97 12/6/97 12/13/97 12/20/97 12/27/97 1/3/98 1/10/98 1/17/98 1/24/98 1/31/98 2/7/98 2/14/98 2/21/98 2/28/98 3/7/98 3/14/98 3/21/98 3/28/98 4/5/98 4/11/98 4/18/98 4/25/98 5/2/98

B-spline coefficients

Table 4 Coefficients estimated by spot fitting modela No. of bonds

Average a

21 21 21 21 21 21 21 22 22 22 22 23 23 23 23 23 23 23 23 23 24 24 24 24 25 25 25 25 25 25 25

Coupon effects

Coeff. 1

Coeff. 2

Coeff. 3

Coeff. 4

Coeff. 5

Coeff. 6

No. of payments

Coupon rate

In price

In yield

0.4548 0.5164 0.5530 0.4581 0.4422 0.4128 0.4150 0.4677 0.4364 0.4656 0.4485 0.4504 0.4807 0.4773 0.5118 0.5034 0.5006 0.5005 0.4784 0.4904 0.4832 0.4789 0.4762 0.4739 0.4498 0.4430 0.4445 0.4473 0.4402 0.4631 0.4576

0.5480 0.5208 0.5005 0.5436 0.5439 0.5610 0.5598 0.5427 0.5736 0.5572 0.5730 0.5558 0.5447 0.5468 0.5407 0.5459 0.5473 0.5471 0.5620 0.5607 0.5663 0.5692 0.5712 0.5697 0.5818 0.5960 0.5951 0.6010 0.6080 0.5848 0.5900

0.9777 1.0082 1.0182 0.9848 0.9843 0.9694 0.9697 0.9982 1.0096 1.0123 1.0261 1.0357 1.0441 1.0440 1.0440 1.0416 1.0420 1.0422 1.0400 1.0474 1.0409 1.0421 1.0406 1.0222 1.0209 1.0170 1.0183 1.0238 1.0173 1.0240 1.0216

1.1967 1.1781 1.1212 1.1584 1.1382 1.1836 1.1832 1.1871 1.2068 1.2227 1.2277 1.2035 1.1858 1.1871 1.2275 1.2338 1.2340 1.2326 1.2443 1.2583 1.2574 1.2504 1.2486 1.2363 1.2185 1.2374 1.2336 1.2396 1.2299 1.2154 1.2167

1.4957 1.5855 1.6488 1.5726 1.5878 1.5434 1.5442 1.5350 1.5565 1.5567 1.5953 1.6489 1.6778 1.6770 1.6106 1.6069 1.6084 1.6197 1.6022 1.5891 1.5888 1.5842 1.5675 1.5319 1.4978 1.4814 1.4938 1.4887 1.5059 1.4944 1.5031

0.8971 0.9209 0.7781 0.9150 0.8829 0.9630 0.9556 0.9802 0.9616 1.0242 0.9540 0.8911 0.8420 0.8430 1.0201 1.0201 1.0143 0.9330 1.0152 1.0535 1.0082 0.9767 0.9694 0.9056 0.9060 0.9637 0.8847 0.8989 0.8245 0.8251 0.8201

−0.5755 −0.5566 −0.5523 −0.6474 −0.6757 −0.6151 −0.6193 −0.6841 −0.7637 −0.7421 −0.6160 −0.8312 −0.7239 −0.7225 −0.7946 −0.7899 −0.7841 −0.7997 −0.6590 −0.7339 −0.7589 −0.7780 −0.7686 −0.7802 −0.7873 −0.6163 −0.5999 −0.6600 −0.6438 −0.6885 −0.6496

0.0385 0.0224 c 0.0060 c 0.0515 0.0505 c 0.0536 c 0.0529 c 0.0356 0.0790 0.0499 c 0.0273 0.0884 0.0519 0.0535 0.0409 0.0443 0.0452 0.0467 0.0436 0.0470 0.0524 0.0622 0.0620 0.0594 0.0774 0.0523 c 0.0501 c 0.0670 0.0689 0.0596 0.0591

0.1716 0.2330 0.2442 0.2256 0.3418 0.3159 0.3193 0.1797 0.1618 0.2213 0.1499 0.2119 0.1801 0.1800 0.1603 0.1420 0.1395 0.1265 0.1593 0.1386 0.1275 0.1174 0.1514 0.2101 0.2439 0.2499 0.2496 0.2364 0.2276 0.2101 0.1695

0.0003 0.0004 0.0004 0.0004 0.0005 0.0005 0.0005 0.0003 0.0003 0.0004 0.0003 0.0005 0.0004 0.0004 0.0003 0.0002 0.0002 0.0002 0.0003 0.0002 0.0002 0.0002 0.0003 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0003

0.2002

0.0003



n

6 i The fitting model is: Bi = N m = 1 di (tm ) exp −tm · j = 1 bj gj (tm ) +b7Paymentsi+b8Couponi+oi.

c

Standard error

Statistically non-significant at a significance level of 5%.

B.-H. Lin / J. of Multi. Fin. Manag. 9 (1999) 331–352

10/4/97 10/9/97 10/18/97 10/24/97 10/30/97 11/8/97 11/15/97 11/22/97 11/29/97 12/6/97 12/13/97 12/20/97 12/27/97 1/3/98 1/10/98 1/17/98 1/24/98 1/31/98 2/7/98 2/14/98 2/21/98 2/28/98 3/7/98 3/14/98 3/21/98 3/28/98 4/5/98 4/11/98 4/18/98 4/25/98 5/2/98

B-spline coefficients

346

Date

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Fig. 4. Estimated term structure by discount fitting model (10/4/97 – 5/2/98): Panel A, Spot rate term structure; Panel B, Forward rate term structure.

Figs. 4 and 5 show term structure changes in the whole sample period for the discount fitting model and the spot fitting model using the coefficients estimated in Tables 3 and 4. Basically, the levels and the shapes of the term structure for the

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sample period are quite stable. The results for the entire sample period from 4 October, 1997 to 2 May 2, 1998 are quite similar to the case of 2 May, 1998 as reported in Fig. 2, implying the methodologies are quite consistent and reliable. Figs. 4 and 5 show that the yield curves obtained from the discount fitting mode are

Fig. 5. Estimated term structure by spot fitting model (10/4/97 – 5/2/98): Panel A, Spot rate term structure; Panel B, Forward rate term structure.

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relatively smooth compared to those obtained from the spot fitting model. According to the analysis of pricing errors, the spot fitting model, which also can provide reasonable and reliable yield curves, achieves a more satisfactory fit for the versatile market yield curve. At this stage we may conclude that the spot fitting model is the most appropriate and preferable approach for fitting the term structure of TGB interest rates. Hereafter we report only the results from the spot fitting model. From Panel A in Fig. 5, the spot yield curve moves from a lower level with a positive slope during October 1997, to a relatively higher level with a positive slope in a slightly humped shape in the long end during December 1997. It then evolves to be relatively flat in shape during April 1998. Panel B in Fig. 5 shows the movements of the estimated forward yield curve. As expected, the forward yield curve behaves quite reasonably, although it exaggerates the shape of the spot yield curve. That is also as expected, showing that the methodology is quite reliable. According to Fig. 5, loosely speaking, the movement in interest rates appears to be highly correlated with different times to maturity. We are now interested in knowing whether a single-factor model can effectively explain the term structure movements as concluded by Babbs (1990). We conducted a Factor Analysis on the spot rates exhibited in Fig. 5 which was based on the spot fitting model. Table 5 shows the five most significant factors for spot interest rates. The first factor with an eigenvalue of 21.26 accounts for 70.90% of total term structure variations. It is simply a weighted average of 19 shorter-term spot interest rates with maturity from 0.5 to 9.5 years, representing a factor driving short-term interest rates. The second factor is a weighted average of 11 longer-term spot interest rates with maturity from 10 to 15 years, and represents a factor driving long-term interest rates. The second factor with an eigenvalue of 7.90, accounts for 26.3% of total term structure variations. The first two factors account for 97.2% of the total term structure variation. Thus for spot rates, at least two factors are needed to specify the term structure fluctuations. The rationale of the Brennan and Schwartz (1979) term structure model, which used two factors (the short-term rate and the long-term rate) to specify the term structure fluctuations, is justified.

5. Conclusion We used the B-spline approximation methodology to fit the term structure of TGB interest rates. The TGB market is small and not as liquid as other developed markets, but it is becoming increasingly important in the Asia–Pacific area. Regardless of the efficiency of the market, we can fit the term structure of interest rates quite satisfactorily using market yield data. This has important implications for financial institutions in pricing and trading fixed-income securities, and in hedging long-term interest risks, as well as in developing long-term interest rate derivative securities. We used three approaches: the discount fitting, the spot fitting, and the forward fitting models to fit the term structure. Among the three different approaches, the

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Table 5 Factor analysis for spot interest rates Maturity

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 Eigenvalue Percentage Cumulative percentage

Factor loading Factor 1

Factor 2

Factor 3

Factor 4

Factor 5

0.76486* 0.95591* 0.97390* 0.97267* 0.98271* 0.99425* 0.99069* 0.96605* 0.93657* 0.90995* 0.89007* 0.88409* 0.88081* 0.87557* 0.86672* 0.84972* 0.82713* 0.78559* 0.73159* 0.65830 0.56141 0.47091 0.35240 0.25741 0.16988 0.09830 0.03062 −0.01133 −0.04761 −0.06466

0.12011 −0.11074 −0.20296 −0.19296 −0.12659 −0.01162 0.12520 0.24415 0.32313 0.37619 0.41063 0.41801 0.41550 0.41867 0.42910 0.45417 0.49816 0.56489 0.64237 0.72521* 0.81103* 0.87178* 0.93077* 0.96362* 0.98355* 0.99407* 0.99862* 0.99905* 0.99626* 0.99012*

0.63241 0.26197 0.02178 −0.08544 −0.10312 −0.08017 −0.00652 0.06217 0.11030 0.13596 0.14594 0.13199 0.12416 0.09023 0.07019 0.05355 0.03482 0.02841 0.02716 0.01880 0.02499 0.03726 0.02900 0.03888 0.04165 0.03489 0.03400 0.02727 0.01585 0.00004

0.02076 −0.02449 −0.05804 −0.06773 −0.07159 −0.06360 −0.04615 −0.02395 0.02205 0.06612 0.10977 0.14573 0.18244 0.21862 0.24261 0.26167 0.25718 0.25027 0.22536 0.19814 0.15691 0.11848 0.07948 0.03549 0.00990 −0.00059 −0.00747 0.00232 0.02826 0.06107

0.00578 −0.06630 −0.07922 −0.06729 −0.04712 −0.02232 0.01902 0.04844 0.07403 0.08478 0.07414 0.06657 0.04649 0.04000 0.02126 0.00257 −0.00630 −0.00740 −0.00347 0.00225 0.00471 0.01318 0.01640 0.01314 0.02075 0.02051 0.01954 0.01418 0.00262 −0.00493

21.26 70.90 70.90

7.90 26.30 97.20

0.48 1.60 98.80

0.28 0.90 99.80

0.04 0.10 99.90

* Largest factor loading for certain maturity.

discount fitting model failed to satisfactorily fit the term structure. The forward fitting model on the other hand, is too sensitive to actual market data, thus causing over-fitting problems, which resulted in an unreasonable forward yield curve. Compared to the other two models, the spot fitting model is the most reasonable and reliable model. We identified coupon effects, the coupon payment effect, and the coupon rate effect on TGB prices. Bonds with semi-annual coupon payments were traded at a higher yield than those with annual coupon payments. Presumably this is because

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bonds with annual coupon payments are more recent issues and more actively traded, thus they are traded at a lower yield. Another reason is that bonds with semi-annual coupon payments are subject to higher reinvestment risks, thus they are traded at a higher yield. We also identified the negative coupon rate effect on TGB prices. Bonds with a higher coupon rate are traded at a lower yield, a contradiction to the normal tax effect on the bond prices. Presumably this is because tax evasion is common in TGB trading, due to the differences in taxing interest income for individuals and institutions. Thus the bond prices are not subject to tax effects. Although for the sample period the term structure was quite stable both in levels and in shapes, through factor analysis, we identified at least two factors required to specify spot yield curve fluctuations.

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