利率期限結構之配適及其資訊內涵

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行政院國家科學委員會補助專題研究計畫成果報告

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※ 利率期限結構之配適及其資訊內涵 ※

※ Fitting and Information Content of the Term ※

※ Structure of Interest Rates ※

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計畫類別：□個別型計畫　　□整合型計畫計畫編號：NSC89－2416－H－011－001

執行期間： 88 年　8　月　1　日至　89　年　7　月　31　日計畫主持人：林丙輝

本成果報告包括以下應繳交之附件：

□赴國外出差或研習心得報告一份

□赴大陸地區出差或研習心得報告一份

□出席國際學術會議心得報告及發表之論文各一份

□國際合作研究計畫國外研究報告書一份

執行單位：國立台灣科技大學企業管理系

中　華　民　國　 89　　年　 10　　月　 20　　日

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行政院國家科學委員會專題研究計畫成果報告

利率期限結構之配適及其資訊內涵

Fitting and Infor mation Content of the Ter m Str uctur e of Inter est Rates 計畫編號：NSC 89-2416-H-011-001

執行期限：88 年 8 月 1 日至 89 年 7 月 31 日

計畫主持人：林丙輝國立台灣科技大學企業管理系

計畫參與人員：侯榮俊國立台灣科技大學企業管理系

中文摘要

本研究採用並比較各種經濟模型與統計模型，在配適台灣公債利率期限結構之問題。在統計模型方面係採用 B-spline 函數近似法，在經濟模型方面採用 Vasicek 與 CIR 模型。本研究結論為：以經濟模型配適利率期限結構，配適效果較統計模型為差，但其解釋與預測能力則較佳。

關鍵詞：利率期限結構、B-spline 函數近似法、利率期限結構模型

ABSTRACT

We apply empirical methodologies and use cross-sectional bond price data to estimate and analyze the TGB term structure of interest rates. We choose two economic models: the Vasicek model and the CIR model, and one mathematical model: the B- spline approximation function, as the discount bond function to extract the term structure from market coupon bond prices.

The empirical results show that the mathematical model can fit the term structure better than the economic models. But the economic models are able to explain the term structure dynamics. Thus the economic models can perform better in identifying miss-priced bonds and in predicting excess trading returns, than the mathematical model.

Keywor ds: term structure of interest rates, B-spline approximation, term structure models

1. INTORDUCTION

The term structure of interest rates

specifying the relationship between the prices of default-free 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. To estimate the term structure there are two categories of methodologies: The equilibrium methodology, based on models such as those proposed by Vasicek (1997), Brennan and Schwartz (1979), and Cox, Ingersoll and Ross (CIR, 1985) makes use of the assumption that certain state variables follow a stochastic process driving the term structure of interest rates. It then uses an arbitrage pricing technique to span the whole term structure. The resulting term structure is a theoretical one consistent with arbitrage-free conditions in an efficient market but is hardly able to fit the actually observed market data.

In contrast to the equilibrium methodology, the empirical methodology uses curve fitting techniques with the cross- sectional observed government coupon bond prices to estimate the term structure. To choose an appropriate function to fit the term structure, one can utilize economic models, such as the Vasicek and the CIR model of term structure of interest rates. One can also apply various spline functions to the observed government coupon bond prices for estimating the term structure. Regardless of the efficacy of curve-fitting techniques, the empirical methodologies, focusing on actually observed data, are able to describe a rich variety of yield curve patterns in reality.

There are many choices of approximation functions for the curve fitting technique in the literature. Models such as

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those developed by McCulloch (1971), Carleton and Cooper (1976), Schaefer (1981), Vasicek and Fong (1982), Nelson and Siegel (1987), Steeley (1991), Pham (1998) used various mathematical spline functions to approximate the term structure. Others such as Brown and Dybvig (1986), Brown and Schaefer (1994), Sercu and Wu (1997) used economic functions such as the Vasicek and the CIR term structure models to fit the market yield curve.

Presumably the pure mathematical models which use various flexible spline functions can fit the term structure arbitrarily well, while they are unable to provide economic meanings of the term structure. On the other hand, although the economic models can not provide flexible functions for fitting the term structure, they can provide economic implications for the term structure fitting.

2. FITTING METHODOLOGIES

Assume there are n coupon bonds in the sample to estimate the term structure. The price of the coupon bond

i, B i

is a linear combination of pure discount bond prices.

That is

B _i d _i t _m P t _m i n

m N

i

=

i

+ =

∑ = ⁽ ^{) (} ⁾ ^; ^{, , ,}

1 ε 1 2 Λ

;

where t

m

is the time when the m-th coupon or principal payment is made,

N i

is the number of remaining coupons and principal payments before the maturity date of bond

i, d i

(t

m

) is the cash flow paid by bond

i at time t m

, and

P(t m

) is the pure discount bond price with face value $1 with maturity at time

t m

. The error term

ε i

may be caused by transaction costs, coupon effects, market imperfections, and so on.

Once the discount function,

P(t), is

defined, the spot interest rate is also defined by R t( )

= −

ln ( )

P t t

. The forward interest rate for the period [t, t +

∆ t ], ∆ t _→ 0

, is

f t

( )

= ∂ P t

( )

∂ . To estimate P(t), let t

P t b g _j t

j k

( )

= j

( )

∑ = 1

,

where g

j

(t) is the j-th approximation function which is dependent only on time to maturity,

and

b j ’s are the coefficients to be estimated

applied to the

k approximation functions.

Combining and introducing an error term, results in

B _i b _j d t g t

j k

i m j m

m N

i

= 

i

  

  +

= =

∑ ∑ 1 1

( ) ( )

ε .

Having specified the function

g j

(t), the equation is a well-defined regression model.

Concerning the error termε we assume that

i

E

(

ε _i ²

)

= σ ² h E _i

; (

ε ε _i

,

_j

)

=

0; for i ≠j, where h

i

is the duration of bond i.

3. THE ECONOMIC MODELS

One of the classical term structure of interest rate models is the Vasicek (1977) model. In the Vasicek model, the instantaneous short-term interest rate

r is

defined by the following process, which is called the O-U process

dr t ( ) = α β ( − r t dt ( )) + σ dW t ( )

^, where

α is the mean reversion coefficient,

β is the long-term mean, and σ is the

instantaneous volatility of the short-term interest rate.

dW(t) is the increment of a

standard Wiener process. The price of a zero- coupon bond at time

t, maturing at time T,

P r t T

( , , ) can be determined by

P r t T

( , , )

=

exp[

− A t T r

( , )

+ B t T

( , )], where

A t T e ^{T t}

( , )

( )

= − 1 ⁻ ^α ⁻

α

;

( )

B t T A t T T t A t T

( , ) ( , ) ( ) ( , )

=  − −

  

  − − −

β ξσ α

σ α

2 2

2 4

3

, and

ξ is the market price of interest rate risk,

which is assumed to be constant.

The major disadvantage of the Vasicek’s model is that the interest rate might evolve to a negative value, which is a clear violation of the arbitrage rule.

Alternatively, CIR (1985) assumed the short- term interest rate

r follows the following

process

) ( ) ( ))

( ( )

( t r t dt r t dW t

dr = α β − + σ

^.

The drift part is identical to the Vasicek assumption. The difference lies in the diffusion part in which the CIR assumption has a square root of

r in the process. This

process is much more difficult to deal with

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but it has an advantage that all future interest rates will be strictly non-negative. As can be seen in the process, the volatility of interest rate is positively related to the interest rate level. As the interest rate level approaches to zero, the volatility will also approach to zero.

As long as

β is positive, than interest rate

will be non-negative. This is a very desirable feature for nominal interest rates.

The price of a zero-coupon bond at time t, P r t T( , , ) can be determined as

)) ( ) , ( exp(

) , ( ) , ,

(

r t T G t T H t T r t

P = −

;

where

/

2

2 ) (

2 / ) )(

(

2 ) 1 )(

( ) 2 , (

σ αβ γ

γ ξ α

γ γ

ξ α

γ  

 

+

− +

= + ⁺ ⁺ _T ₋ ^T _t ⁻ ^t e T e

t G

;

γ γ

ξ

α ^γ

γ

2 ) 1 )(

(

) 1 (

) 2 ,

(

₍ ₎

) (

+

− +

+

= ^T ⁻ ^t _T ₋ − _t e T e

t

H

;

2

)

(

α ξ σ

γ = + +

.

4. THE B-SPLINE APPROXIMATION In choosing spline approximation functions we use the B-spline functions. The B-spline functions were suggested by Shea (1984) and have been successfully generalized and used by Steeley (1991) to estimate the U.K. Gilt-edged bond term structure. Lin (1999) also applied this methodology to estimate TGB term structure.

The B-spline function is defined as:

[ ]

g t

T T t T

s p

j i

j s j i s p

i s s p

j

( ) p

( ) max( , )

,

= −



  

 



 





 

 −

= ≠ + +

=

+ + ∏

∑ ¹ ¹ ¹ ⁰

^,

where g

_s ^p

( ) is called the s-th p-order B-

t

spline function..

5. THE DATA AND METHODOLOGY We studied the term structure of Taiwanese Government Bond (TGB) interest rates for the period from January 6, 1996 to September 6, 1999, using the prices of 37 TGBs. Weekly data obtained from weekly reports published by the Grand Cathay Security Company is used as the research sample. In total, there are 193 weekly data for each TGB in the sample.

To incorporate the coupon effects on the bond price, we added two variables to the

regression model：

i i i

N

m

m m i

i d t P t a Coupon b Payments B

i

+ ⋅ + ⋅ + ε

= ∑

=1

) ( ) (

where

Coupon represents the coupon rate _i

of bond

i, while Payments represents the _i

number of coupon payment per year for bond

i.

6. THE EMPIRICAL ANALYSIS Model Estimations

Most of the coefficients estimated were statistically significant implying that the estimation is adequate. The fitting errors of the B-spline model were significantly lower than those of the Vasicek and CIR models.

Thus the B-spline model consistently fits the market bond prices better than either the Vasicek model or the CIR model. In addition, most of the number of coupons effects were negatively significant for all cases, reflecting that annual payment bonds were traded at lower yields. On the other hand, none of the coupon rate effects are significant, implying TGB bond prices are not subject to coupon rate effects.

Having obtained the continuous term structure of interest rates, we calculated discrete yield curves for 30 semi-annually compounded zero-coupon bond yields with times to maturity from 0.5 to 15 years.

Generally speaking, yield curves obtained from the Vasicek model exhibit a smooth shape compared to other models. Yield curves estimated from the B-spline model are more flexible than other models.

The Regr ession Analysis

If the theoretical model has economic implications and can explain the market term structure of interest rates, then the model residuals will have a relationship with the abnormal returns of bond holding portfolio.

In order to test whether the model residuals have this information content, we define the abnormal return of bond holdings as follows:

) ,

(

_, ₁

,

, _t = _i _t − _t _i _t Φ _t Φ _t −

i HR E HR

AR

;

where

AR _i _, _t

denotes the abnormal return at time

t for holding bond i, and the expected

holding return E

_t

(

HR _{i t} _, Φ Φ _t

,

_t ₋ 1

) can be

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defined as (the benchmark 1):

1 , 1 , , 1

,

, )

(

−

− −

+

= − Φ Φ

t i t i t i t

t t i

t B

Coupon B

HR B

E )

) )

, which denotes the theoretical one period holding return of

i, B ) _i _, _t

denotes the model price of the bond. Alternatively we can define the expected holding return as the normal capital asset pricing model (the benchmark 2):

( )

E HR _t ( _{i t} _, Φ Φ _t , _t ₋ 1 ) = α _{i t} _, ₋ 1 + β _{i t} _, HR _{m t} _, − α _{m t} _, ₋ 1

, where

α i t , − 1

is the yield of bond i,

α m t , − 1

is yield of the market portfolio, HR

_{m t} _, ₋ ₁

is the holding return of the market portfolio.

β i t ,

is the sensitivity of the risk premium of bond

i to market risk premium, which is defined as

the relative of the duration of bond

i to the

duration of the market portfolio. For empirical purposes, we set the market portfolio

α _{m t} , − 1 = α _t − 1

and HR

_{m t} _, ₋ ₁ = HR _t ₋ ₁

, where

α t − 1

is the yield of the average portfolio and HR

_t ₋ ₁

is the holding return of the average portfolio.

Having defined the abnormal return, we conduct the following regression analysis:

t i t i

t i t

i e

b B a

AR _,

1 ,

, = + +

−

ε ) −

;

where

ε

ˆ

_i _, _t ₋ ₁

denotes the estimated model residual. The null hypothesis of the regression analysis is

H ₀

:

b <

0.

Generally speaking, in the case of benchmark 1, model residuals are significantly correlated with abnormal returns for most of the bonds. In the case of benchmark 2, the relationship is much weaker. Among the 3 models, the residuals of B-spline model exhibit the weakest relationship with abnormal returns. This evidence implies that economic models have more information content concerning the term structure movements. While the mathematical model can fit the market price more closely, it has less economic meaning concerning the term structure movements.

The Tr ading Rules Analysis

The trading rule is when the bond is

under-priced in the market, then buy the bond and hold for one period. When the bond is over-priced in the market, then we sell the bond and hold the short position for one period. The cumulative abnormal return for the above trading strategies can be calculated.

The results show the cumulative abnormal returns for the trading strategies during the whole sample period are significant. In the case of benchmark 2, the cumulative abnormal return is lower than the case of benchmark 1.

7. REFERENCES

Brennan, M.J. and E.J. Schwartz (1979), A continuous time approach to the pricing of bonds,

Journal of Banking and Finance, 3,

133-155.

Cox, J.C., J.E. Ingersoll, and S.A. Ross (1985), A theory of the term structure of interest rates, Econometrica, 53, 385-407.

Lin, B.H. (1999), Fitting the term structure of interest rates for Taiwanese government bonds,

Journal of Multinational Financial Management, Vol. 9, pp. 331-352.

McCulloch, J.H. (1975), The tax-adjusted yield curve, Journal of Finance, 31, 811-830.

Pham, T.M. (1998), Estimation of the term structure of interest rates: an international perspective,

Journal of Multinational Financial Management, 8, 265-283.

Sercu, P. and X. Wu (1997), “The information content in bond model residuals:

an empirical study on the Belgian bond market”, Journal of Banking and Finance, 21, 685-720.

Steeley, J.M. (1991), Estimating the Gilt- edged term structure: basis splines and confidence intervals,

利率期限結構之配適及其資訊內涵

行政院國家科學委員會補助專題研究計畫成果報告

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