利率期限結構因子與債券免疫之研究

行政院國家科學委員會補助專題研究計畫成果報告

利率期限結構因子與債券免疫之研究

On the Term Structure of Interest Rates Factors and Bond Immunization

計畫類別：5個別型計畫 □整合型計畫 計畫編號：NSC94-2416-H-011-011

執行期間： 94 年 8 月 1 日至 95 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 95 年 12 月 16 日

行政院國家科學委員會專題研究計畫成果報告 利率期限結構因子與債券免疫之研究

On the Term Structure of Interest Rates Factors and Bond Immunization 計畫編號：NSC94-2416-H-011-011

執行期限：94 年 8 月 1 日至 95 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 中文摘要

本研究主要目的是發展一模型，以即期利率 當成利率期限結構改變的未知驅動因子之替 代變數，並進行實證研究以瞭解與衡量無風 險債券投資組合的報酬。延續Elton, Gruber &

Michaelly (EGM 1990) 及 Navarro & Nave (2001)等模型的研究方法，我們選擇二或三個 最適的即期利率當成利率期限結構改變的未 知驅動因子之替代變數。當未知的驅動因子 之替代變數尋得後，利用此模型以存續期間 (duration)量度方式來衡量利率風險(包含價格 風險與再投資風險)。這些存續期間量度的觀 念類似Reitano (1990) 和 Ho (1992)所使用存

續期間向量中的元素。根據EGM 的方法，本

研究同時也提供了最適化決定存續期間向量 中元素的方法。此方法的好處是在不減損其 解釋能力下，顯著地減少了存續期間向量中 所需的元素，更符合債券組合管理的運用。

所以進一步的實證分析探討即期利率的利率 期限結構，可幫助我們建構更精確的債券評 價模型，免疫與管理債券投資組合。在實證 研究方面，本研究以美國公債市場資料進行 測試檢定，研究結果對學術理論與實務應用 應有相當之參考價值。

關鍵詞：利率期限結構、驅動因子、存續期 間、最適即期利率、即期利率模型

ABSTRACT

The purpose of this paper is to develop a three-factor model, using spot rates as proxies for the unknown driving factors of the term structure of interest rates, and to perform empirical tests on the measurement of risk exposures and explanation for default-free bond portfolio returns. By extending the framework of the Elton, Gruber and Michaely (EGM 1990), and Navarro and Nave (NN 2001), we can identify the three optimal spot rates as proxies for the state variables which represent in

parallel, slope, and curvature changes in the term structure of interest rates. Using the same data set of the monthly U.S. term structure of interest rates as the EGM, the empirical results demonstrate the robustness that the optimal three-spot-rate model has better predictability on unexpected changes in the term structure of interest rates, compared to the optimal two-spot-rate model proposed by the EGM.

Similar results were also obtained, as we apply our methodology to the updated data. Using the estimated durations to explain cross-sectional bond returns, the three-spot-rate model brings up encouraging outcomes. Coincidentally, the empirical results appear the three-spot-rate model predicts more accurately when non-parallel shifts or intense movements on the term structure occur.

Keywords: term structure of interest rates, driving factors, optimal spot rate, spot-rate model

2. INTRODUCTION

Numerous researchers have dedicated themselves on studying the term structure of interest rates over past 30 years. The term structure 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, interest rate derivatives, and other capital assets as well. It is thus important for portfolio management, financial engineering, and corporate finance in investment and financing decisions.

From 1970s, academics such as Vasicek (1977), Nelson and Schaefer (1983), and Cox, Ingersoll and Ross (CIR 1985) etc. utilized a small set of state variables that follow a stochastic process to construct their term structure of interest rates models. While these theoretical models were used for empirical tests,

a change in one or more spot rates is used as a proxy for the state variables. Using a small set of spot rates (as driving factors) to explain bond prices or returns is to assume that all spot rates can be constructed as functions of this set.

Previous studies such as Chen, Roll, and Ross (1986), and Elton, Gruber and Blake (1995), have demonstrated that no less than 5 factors are used for explaining stock returns, while for default-free bond the number of factors could be as low as two or three. Among them, the one-year rate is frequently employed for one-factor model (Babbel (1983), Nelson and Schaefer (1983)). Nelson and Schaefer (1983), however, applied long and intermediate rates (thirteen-year rate and the difference between thirteen-year rate and five-year rate) as proxies for two state variables to describe the behavior of the term structure of interest rates, this two-factor model has a better overall fitting for the term structure than one-factor models.

Elton, Gruber and Michaely (EGM 1990) point out one of the possible reason why immunization literature from the simplest model of Macaulay (1938) to the more complex two-factor models like Ingersoll (1983), Nelson and Schaefer (1983) and Brennan and Schwartz (1983), could not have better empirical results than a much simpler model is due to ignoring implied information in the term structure itself.

Although previous studies (Babbel (1983), Nelson and Schaefer (1983), and Lekkos (2001)) have used spot rates as proxies for state variables to describe the dynamics of the term structure of interest rates, essentially these spot rates were arbitrarily selected. Therefore, EGM propose a methodology to identify the optimal spot rates as proxies for state variables which drive the term structure movements for predicting unexpected changes in bond returns.

Using McCulloch’s estimates of spot rates with a series of times to maturity over the 30-year period from 1957 through 1986, EGM asserts that the four-year spot rate is the optimal proxy for the one-factor model (opt-1 model in their term), while the six-year rate and the eight-month rate are the optimal proxies for the two-factor model (opt-2 model). They also declare the superiority of the opt-2 model comparing with a series of benchmark models (including their opt-1 model) from previous

studies. Navarro and Nave (NN 1997, 2001) employ the same methodology with data collected from Spanish Public Debt Market and obtain similar results as the EGM’s. They identify the optimal spot rates within the Spanish term structure for the two-factor model as the three-year rate as well as seven-year rate.

Empirical studies of principal components analysis (Steeley (1990), Litterman and Scheinkman (1991); Knez (1994), Willner (1996) etc.) indicate the term structure movements could be classified into three main categories: level (parallel shift), steepness (slope) and curvature. The changes in the three components can explain up to 95% of total variance of changes in the term structure of interest rates. These results give us a motivation to describe the term structure movements in terms of the three specific factors and to investigate return exposures of bond and bond portfolios with respect to these factors. And the major task is to exact the unknown factors from the term structure itself rather than from other sources.

In this study we make an extension of EGM (1990) and NN’s (2001) framework to an optimal three-spot-rate (opt-3 in the EGM’s term) model as the optimal proxy for the three-factor model for more accurate prediction on the unexpected change of interest rates and better measurement of risk exposure along with the whole term structure. For comparative purposes, we firstly utilize the same data set of monthly term structure of spot rates as EGM quoted to test the superiority of the optimal three-spot-rate (opt-3) model developed by our study with the opt-2 model asserted by the EGM.

Then more updated monthly data also from the McCulloch’s website and the Treasury bond data from DataStream were collected for testing the robustness and fitness of these models for explaining cross-sectional bond returns. Having identified the optimal spot rates, we further estimate the smoothed coefficients of sensitivity for the spot-rate models and calculate their corresponding duration vectors (similar to Reitano (1990) and Ho (1992)). However there are little differences between the duration vectors and Ho’s (1992) key rate durations. The duration vectors of the spot-rate models are

estimated corresponding to the optimal spot rates rather than to a series of key rates as Ho (1992) stated to analyze risk exposures of option-embedded bonds. On the other hand, for the one-spot-rate model we can simply use modified Macaulay duration directly cited from DataStream. These estimated durations are used to describe cross-sectional bond returns.

The rest of this study is organized as follows: In Section 2 we develop the optimal

three-spot-rate model by extending the EGM and NN’s methodologies to identify the optimal spot rates as the state variables. In Section 3 we describe the data set used in the empirical study.

In Section 4 we conduct empirical examinations for comparing the different models proposed in section 2. We then conclude in Section 5.

2. METHODOLOGY

2.1 Extended EGM Framework (Searching for State Variables)

Instead of using a function of a small number of state variables that follow a stochastic process to describe the changes in default-free bond prices or bond returns, EGM (1990) proposed a method to explore the term structure itself for identifying the optimal spot rates with different maturities as the state variables. They empirically analyzed the term structure of spot rates and asserted that the optimal two-spot-rate (opt-2) model derived by their suggested methodology outperforms any competing models.

Originally EGM proposed one or two different spot rates as proxies of the unknown factors.

This paper extends the EGM’s framework from two spot rates to three spot rates as the unknown factors that explain the dynamics of the term structure of interest rates. Assume the changes in the spot rate with the i^th maturity are dependent on three factors, F1, F2 and F3 as follows:

t, i ,t i, ,t i, ,t i, i, t,

i β β dF β dF β dF ε

dr = ₀ + _{1 1} + _{2 2} + _{3 3} + (1) where:

dri ,t is the unexpected change in the i^th-period spot rate at time t;

,t ,t

,t,dF , dF

dF_{1 2} and ₃ denote the unanticipated changes in the unknown factors at time t;

εi ,t is the error term of the unexpected change in the i^th-period spot rate at time t, which is assumed to follow a normal distribution, N(0,σ²).

Equation (1) means that the unexpected changes in any spot rate in term structure can be well described by the three unknown factors. In selecting the three optimal spot rates (denoted as rx, ry, and rz) as proxies for the unknown factors, we propose the three-spot-rate model as follows:

, , ( , ,) (2 , , ,) ,

i t i x t i z t x t i y t z t x t i t

dr =a dr −b dr −dr −c dr −dr −dr +ε

where dr_xt,,dr_yt, ,anddr_zt,denote the unanticipated changes in the optimal spot rates (as proxies for the unknown factors) at time t. Equation (2) can be rewritten as:

, ( ) , ( 2 ) , ( ) , ,

i t i i i x t i y t i i z t i t

dr = a + +b c dr + − c dr + c b dr− +ε (2) where, the coefficients, (a_i+ +b c_{i i}), ( 2 )− c_i , and (c b_i− _i), are sensitivities related to the optimal long-term (rx), medium-term (ry), and short-term (rz) spot rates respectively.

In reality, rx, ry, and rz can be identified as the optimal long-term, medium-term, and short-term spot rates respectively along the term structure. Apart from the long-term spot rate, EGM suggest using the difference between the long-term and short-term rates (dr_zt, −dr_xt,), rather than the

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short-term spot rate only, as the second state variable in order to minimize the problem of multicollinearity. Furthermore, in accordance with the results of principle components analysis, we may look at equation (2) this way that we utilize dr_xt,,( dr_zt, −dr_xt,),and( 2dr_yt, −dr_zt, −dr_xt,)as proxies to capture the level (parallel shift), slope, and curvature changes on the term structure respectively. Here we used the term ( 2dr_yt, −dr_zt, −dr_xt,) to represent the curvature changes as Diebold, and Li (2006) identified.

For equation (2), the determination coefficient between the spot rate changes, dri and the changes in three components of the term structure drx,(dr_zt, −dr_xt,), and ( 2dr_yt, −dr_zt, −dr_xt,) is given by:

) (

) 1 (

, 2 ,

) , , ( ,

t i

t i z

y x

i Var dr

R Var ε

−

= (3) or alternatively represented as:

) ( ) ( )

( _{, , ,}

) , , (

, it it it

2 z y x

i Var dr Var dr Var

R ⋅ = − ε (4) In estimating the model as specified in equation (4), minimizing the residuals’ variance is equivalent to maximizing the left-hand side, R_i²_,(x,y,z) ⋅Var(dr_i,t), of the equation. In order to find the best proxies for the state variables, EGM proposed to maximize the weighted average of

) ( _,

) , , (

, it

2 z y x

i Var dr

R ⋅ over the choice of the appropriate proxies, rx, ry, and rz, and across all maturities.

That is we can identify the optimal spot rates (rx, ry, and rz) by maximizing the objective function, )

2 (

) , , ( ) ,

, ,

( i

i

z y x i z i

y

x wR Var dr

Max

∑

2.2 Discretization for Empirical Test

Having constructed the technique scheme for determining the optimal spot rates, we need to discretize the differential equation (2) for empirical purposes. According to equation (2), the unexpected change of interest rate can be discretized as follows:

, , ( , ,) (2 , , ,) ,

i t i x t i z t x t i y t z t x t i t

r a r b r r c r r r ε

Δ = Δ − Δ − Δ − Δ − Δ − Δ + (5) where Δrx,t, Δry,t, and Δrz,t denote the unexpected change over period [t − 1, t ] of the optimal long-term, medium-term, and short-term spot rate respectively. We then can conduct a multivariate regression with monthly term structure of interest rates based on equation (5). The coefficients,a , _i bi, and c are estimated from the in-sample data and used to predict the unexpected changes in _i interest rates for the out-of-sample period. Alternatively, the equation can be rearranged as follows:

, ( ) , ( 2 ) , ( ) , ,

i t i i i x t i y t i i z t i t

r a b c r c r c b r ε

Δ =  + + Δ  + −  Δ +  − Δ + (6)

The coefficients,(a_i + +b c_i _i), ( 2 )−  , andc_i (c_i− b_i), are sensitivities related to the optimal long-term (rx), medium-term (ry), and short-term (rz) spot rates respectively. Moreover, the unexpected changes in any spot rate (Δri,t) can be forecasted by the unexpected changes in the three selected optimal spot rates ( Δr_xt,,Δr_yt, ,andΔr_zt,). Although EGM employed two alternative models, random walk and pure expectation theory, to estimate the unexpected movements of spot

5

rates, their selection of optimal spot rates is mainly based on the former model which has also been utilized in most of previous studies (see Babbel (1983), and Nelson and Schaefer (1983)). Under the random walk assumption that the yield curve is expected to remain unchanged, any change in spot rates during each period is assumed to be unanticipated.

2.3 Durations on Optimal Spot Rates

The price of a bond P can be defined as the discounted value of a series of cash flows associated with the bond, Ci’s, with respect to its appropriate spot ratesr s

ti' :

∑

= ⁿ

i t

t i

i

ri

P C

1(1 ) (7) For the one-factor model, assuming that the unexpected changes in any spot rate depend on the changes in the unknown factor, F, the unexpected rate of bond price changes can be modeled as follows:

dF F MD

r r 1

dF r

1 C t P

1 P

dP _i

i i i

t t n

1

i t

t i

i ] *

) ( )

[( =−

∂

∂ + +

= − ∑

=

(8)

While the unknown factor (F) is substituted by the appropriate spot rater_ti , the term MD of equation (8) represents the modified Macaulay duration of the bond. Therefore, after having a model for describing the unexpected changes in any interest rater_ti , we can apply it to pricing fixed-income securities and measuring the price risk of bond portfolios. When we consider the optimal three-spot-rate model, equation (8) becomes:

) 1 (

1 z

z t t y y t t x x n t

i t

r dr r r dr P r r r dr P r r r P P

P

dP _s

s s

s s

s ∂

∂

∂ +∂

∂

∂

∂ +∂

∂

∂

∂

∂

= − ∑

=

(9)

According to equation (2-1), equation (9) can be discretized as follows:

z s y m x

l r D r D r

P D

P ≈− Δ − Δ − Δ

Δ (10)

where,D_l ,D_m,andD_s represents the bond duration corresponding to the optimal long-term (rx), medium-term (ry), and short-term (rz) spot rate, respectively, and

- -1 1

1 1 ⁱ

i

n t

l i i i i i t

i

D t a b c C r

P ₌

=

∑

- -1 1

1 -2 1 ⁱ

i

n t

m i i i t

i

D t c C r

P ₌

=

∑

- -1 1

1 - 1 ⁱ

i

n t

s i i i i t

i

D t c b C r

P ₌

=

∑

i i i 1

a + + = , -2b c c_i = , and (0 c b_i− _i) 0= ;

i i i 0

a + + = , -2b c c_i = , and (1 c b_i− _i) 0= ; or

6

i i i 0

a + + = , -2b c c_i = , and (0 c b_i− _i) 1= . (12)

This means the spot rate

ti

r is constant across time to maturities, and only parallel shift in the term structure is allowed. In this case, D_l ,D_m,or D_s, will be equivalent to the modified Macaulay duration of the bond with respect to the constant spot rate.

While we have calculated the unexpected rate of bond price changes corresponding to optimal spot rates, it is straight forward to calculate that the rate of value changes in a bond portfolio

)

(ΔV_P V_P induced by the unanticipated changes in the optimal spot rates:

z P s y P m x P l P

P D r D r D r

V

V ≈− Δ − Δ − Δ

Δ (13)

where D_l^P,D_m^P,andD_s^P are the bond portfolio durations estimated from the weighted average of the individual bond durations.

The relationship between the percentage change of a bond or bond portfolio to the duration vector is derived in equation (10) or (13) as Reitano (1990) introduced. Using the estimated durations with respect to the corresponding optimal spot rates, D_l ,D_m,andD_s,

) and

(or D_l^P,D_m^P, D_s^P to examine the fitness of the model for explaining cross-sectional bond (or bond portfolio) returns, we are able to compare the three-spot-rate model proposed in this paper with the two-spot-rate model, as well as the one-spot-rate model. As we omit the medium-term from the equation (10), the model reduces to the two-spot-rate model. Moreover, for one-spot-rate model, we can simply use the modified Macaulay duration to replace all these durations of corresponding optimal spot rates.

3. THE DATA

The data for empirical analysis of this study mainly come from two sources. In order to compare the result of the three-spot-rate model developed here with the optimal two-spot-rate model (opt-2) proposed by EGM (1990), we use the same dataset as the EGM (1990) which includes monthly US Treasury term structure of spot rates covering a 30-year period from 1957 through 1986. The data is available from the McCulloch’s website. Furthermore, the more recent data, for the 8.5-year period from July 1997 through December 2005, of monthly Treasury term structure of spot rates from the same source and the synchronous 8.5-year monthly Treasury bond data from the DataStream (Thomson Financial) are collected for testing the robustness of our models and examining the fitness of these models on describing cross-sectional bond returns.

The first data set constituting of the Part I of the empirical study, includes monthly term structure of spot rates with a series of different maturities over the 30-year period, from January 1957 through December 1986. Specifically, for each term structure of the first data set, we select a series of 31 spot rates with different times to maturity (beginning from one-month, and increasing by a monthly interval from one month to eighteen months, a quarterly interval from eighteen to twenty-four months, and an yearly interval from two to thirteen years) as EGM employed. The whole 30-year sample period were divided into 6 five-year sub-periods, where Period 1 is from January 1957 to December 1961; Period 2 is from January 1962 to December 1966; and so on.

Among them the data for the Period 1 and 4 are defined as in-sample data which were used for identifying the optimal spot rates for the spot-rate models, and the others are defined as out-of-sample data which were used for testing the fitness of the estimated models.

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The second data set constituting of the Part II of the empirical study contains of updated monthly term structure of spot rates covering the period from July 1997 through December 2005 is available from the McCulloch’s website as well. We also divide the data into two subsamples. The first 5-year period from July 1997 to June 2002 is defined as the in-sample data for identifying the optimal spot rates for the spot-rate models, and the rest 3.5-year period from July 2002 through December 2005 is the out-of-sample data for examining the predictability and explaining capability of the estimated models. Since the longest time to maturity of U.S. government bonds can be up to 30 years, we thus extend the maturity of the term structure data to 30 years for the Part II accordingly. As a result each term structure of spot rates for the second data set comprises of a series of 36 maturities (including 31 maturities described in the Part I plus five additional maturities, that are 14, 15, 20, 25, and 30 years).

In testing the models for explaining cross-sectional bond returns, only government bonds, which do not contain credit risks, are collected. The monthly bond data comprises of 3,334 in-sample and 4,035 out-of-sample observations (covering 102 months and 7,369 observations in total). Each observation includes a set of clean price, accrued interest, coupon, coupon dates, maturity, modified Macaulay duration, etc. which are used to calculate dirty price, monthly return, and durations of corresponding optimal spot rates of individual bond in each month. All the monthly term structure of spot rates and Treasury bond prices represent observations for the last business day in each month. The criteria for model evaluations are based on the mean square errors (MSE) on predicting the unexpected changes in interest rates for the out-of-sample data.

4. EMPIRICAL ANALYSIS

Based on the sample data set, the empirical analysis of this paper is divided into two parts accordingly.

Part I：January 1957 - December 1986 (360 months)

In this part we divided the whole 30-year sample period (1957-1986) into six five-year sub-periods as EGM defined. The dada for Period 1 (1957-1961) and Period 4 (1972-1976) are defined as in-sample data, were used to determine the optimal spot rates as proxies for the unobserved state variables of the factor models. Following the EGM’s methodology, the optimal spot rate for the one-spot-rate (opt-1) model can be identified as the four-year spot rate (4y), while the optimal spot rates for the two-spot-rate (opt-2) model are the six-year spot rate and the eight-month spot rate (8m, 6y). The results are exactly consistent with the EGM’s. When extending the EGM framework with equally weighted scheme for each maturity of spot rate and assumption that the innovation in spot rates follows a random walk, we can identify the six-month spot rate, three-year spot rate and nine-year spot rate, as the optimal proxies (6m, 3y, 9y), for the state variables of the three-spot-rate model (opt-3) to describe the unexpected movements on the term structure of interest rates.

The criteria employed for evaluating the performance of the estimated models is the mean and variance of mean square forecast errors (MSE) on unexpected changes in interest rates, and then using matched sample t-statistics to test the significance of the differences between the mean of models in pairs for the four out-of sample periods and the period combining all out-of-sample (240 months). Parameters of all models are estimated over the previous five-year periods and each maturity is weighted equally at yearly interval when calculating the mean and variance of mean square error of each model.

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As shown in Exhibit 1 through Exhibit 4, the MSE for the opt-3 model for the four out-of-sample periods (Period 2, 3, 5 and 6) are always the lowest one compared to the opt-2 and the opt-1 models. For each sub-period, the lager MSEs generally occur at the shorter maturities that are similar to the NN’s (2001) results. The opt-3 model overall has lower MSE than opt-2 and opt-1 model, especially in the short end (maturities less than 2 years). In formal testing, in the case of Period 2, as shown in Exhibit 1, the MSE of the opt-1 model is significantly higher than that of the opt-2 (opt-3) model at a significance level of 5%, with a t-statistic of 6.94 (8.68). While the MSE of the opt-2 model is significantly higher than that of the opt-3 model at a significance level of 10%.

In the cases of Period 3, 5, and 6, as shown in Exhibit 2, 3, 4, the evidence demonstrate the dominance of the opt-3 model over its counterparts, the opt-2 and the opt-1 model, is statistically significant.

Exhibit 5, showing the outcome of the period combining four out-of-samples at the same time, demonstrates that all three t-statistics (5.47, 3.40, and 5.90) have the differences significant at the 5% level in this 240-month sample period. These analyses show the robustness that the opt-3 model has the smallest MSE and the differences between opt-3 model vs. opt-1 and even vs. opt-2 models are significant. From the above results, the opt-3 model developed here has better predictability on the unexpected changes of interest rates and significantly surpasses opt-1 model and opt-2 model proposed by EGM. Next, the reliability of the three-spot-rate model with more current data was checked in the following section.

Part II：July 1997 - December 2005 (102 months)

The second part of the empirical study is to verify the robustness of model predictability using updated data, and further to test the fitness of the model for explaining cross-sectional bond returns.

The monthly term structure of spot rates for the more recent 8.5-year period, from July 1997 to December 2005, are collected from the McCulloch’s website as well. The Treasury bond prices data corresponding to the same period are collected from DataStream for calculating bond returns and durations.

4.1 Optimal spot rates

Each term structure data includes a series of spot rates of 36 different maturities. We also divided the 8.5-year data into to two subsamples. The first five-year period, from July 1997 to June 2002, is defined as the in-sample data for selecting the best spot rates as proxies for the unobserved state variables of different factor models through the EGM methodology. The rest 3.5-year period, from July 2002 to December 2005, is set as the out-of-sample data for analyzing the predictability of unexpected change of interest rates and testing the capability of the models for explaining cross-sectional bond returns.

Using the same methodology as in the Part 1, the optimal spot rate identified for the one-spot-rate model is the 8-year spot rate (8y). For the two-spot-rate model, they are the 3-year and 15-year spot rates (3y, 15y). While for the three-spot-rate model, the optimal proxies are the 1.75-year, 8-year, and 20-year spot rates (1.75y, 8y, 20y). These optimal spot rates are those with longer maturities compared to those in the Part 1 of the empirical study because corresponding to the longest time to maturity of the U.S. government bonds which can be up to 30 years, we extend the maturity of the term structure of spot rates to as long as 30 years accordingly.

We reconfirm the predictability of these models on unexpected changes in interest rates with out-of-sample data in the Part II, from July 2002 to December 2005. Exhibit 6 illustrates that the mean value of MSE of the three-spot-rate model is still the smallest one (0.0018), and the

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differences of the mean vale between each two models are all significant at the 5% level. The result once again supports the superiority of the three-spot-rate model relative to its two counterparties:

the two-spot-rate and one-spot-rate models.

4.2 Estimated consequent sensitivities

After optimal spot rates are selected, we can estimate the consequent sensitivities of the changes in these related proxy factors to each spot rate on the term structure based on the regression of equation (6). Exhibit 7 and 8 present the sensitivities estimated from in-sample data for the two-spot-rate and three-spot-rate model respectively. For the two-spot-rate model, long-term rate sensitivity (aˆ_i +bˆ_i) of the two-spot-rate model is an increasing concave curve from around -0.62 at one-month maturity to 1.07 at 30-year maturity. On the other hand, short-term rate sensitivity

)

(−bˆ_i exponentially decays from about 1.37 at one-month maturity to -0.3 at 30-year maturity. The curves of long-term rate and short-term rate sensitivities cross the x-axis at 3-year and 15-year maturities respectively. As the long-term rate sensitivity curve crosses the x-axis, which means that the 3-year spot is regressed on itself, resulting in the regression coefficient on the 3-year spot rate equal to 1, while zero on the 15-year spot rate. On the other hand, as the short-term rate sensitivity curve crosses the x-axis, the coefficient on the 3-year spot rate is zero while 1 for the 15-year spot rates. They are exactly the two optimal spot rates recommended by this study.

In the three-spot-rate model, the curves of sensitivities are more complicated than the two-spot-rate model. Long-term rate sensitivity (aˆ_i +bˆ_i +cˆ_i) begins from about 0.26 at one-month maturity, decreases to -0.2 at 5-year maturity, and then rebounds to 1.5 at 30-year maturity.

Short-term rate sensitivity (cˆ_i −bˆ_i) steeply decreases from about 1.57 at one-month maturity to -0.64 at 12-year maturity, and then slowly goes up to 0.1 at 30-year maturity. On the other hand, medium-term rate sensitivity (−2cˆ_i) is a humped curve starting from -0.85 at one-month maturity to a maximum value of 1.04 at 7-year maturity, and then dropping to -0.61 at 30-year maturity.

These curves cross the x-axis at three overlapping points, 1.75-year, 8-year, and 20-year maturities respectively. These points coincide with the conditions described in equation (12) for the three-spot-rate model. That is while any two of three sensitivities are zero; the remainder should be equal to one. The three points moreover are the three optimal spot rates recommended by this study.

Since continuous curve of sensitivity is more convenient for estimating corresponding durations (i.e.D_l ,D_m,andD_s derived in equation (11)) of individual bond with any time to maturity. The estimated durations are used to explain cross-sectional bond returns as described in equation (10). There are many ways to perform the curve fitting (e.g. B-spline, Cubic-spline, and other mathematic function etc.), however we simply employ polynomial (quartic equation) to fit the sensitivity with a smoothed curve. The R-square of long-term and short-term spot rate sensitivities are 99.95%, and 99.93% for the polynomial curve of the two-spot-rate model. For the three-spot-rate model, the R-square of long-term, medium-term and short-term spot rate sensitivities are 99.74%, 99.71%, and 99.94%, respectively. Overall the R-squares of polynomial sensitivities are higher than 99.7%.

4.3 Explaining capability on bond returns

The superior predictability of the three-spot-rate model is reconfirmed through the above empirical analyses. Moreover, we distinguish its ability of forecasting cross-sectional bond returns among the alternative models. The three-spot-rate model should outperform the others, especially

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when non-parallel shifts or intense movements on the term structure take place. The smoothed sensitivities of the two-spot-rate and three-spot-rate models (as presented in Exhibit 7 and 8) are utilized to estimate their consequent duration vectors. For the one-spot-rate model, however, we can simply use modified Macaulay duration directly cited from DataStream.

The other data collected from DataStream are used to calculate monthly dirty price, return, and durations of related optimal spot rates of each Treasury bond over the whole 8.5-year period.

Because both bond returns (on yearly basis) and duration vector are estimated at the last trading day of each month, we run the regression of cross-sectional bond returns month by month against their corresponding one-month-lag duration vector (i.e. the end of previous month is the beginning of this month) as described in equation (10). That is, the independent variable is bond returns and the dependent variables are duration vector.

Exhibit 9, 10, and 11 demonstrate the regression results for the in-sample (60 months), out-of-sample (42 months) and all-sample (102 months) periods, respectively. The figure above each column of bar chart represents the number of months for each model that the adjusted R-square are higher than the percentage illustrated below the bars. For instance, the adjusted R-squares higher than 50% are 47 out of 60 months for the one-spot-rate model. On the other hand, there are 51 out of 60 months’ adjusted R-squares higher than 50 % for the three-spot-rate model.

For the in-sample case, Exhibit 9 provides the results that the three-spot-rate model has the largest mean adjusted R square as we expected, and the differences between the three-spot-rate model with the other two models are significant at 5% or at least 10% level. Generally, the three-spot-rate model for explaining cross-sectional bond returns outperforms the two-spot-rate and one-spot-rate (represented by modified Macaulay duration) models. The mean adjusted R-square of the one-spot-rate, two-spot-rate, and three-spot-rate models are 69.57%, 71.84%, and 73.98%

respectively. Overall different models for explaining cross-sectional bond returns during in-sample period are up to 74%.

The three-spot-rate model indeed increases the mean adjusted R-square at the out-of-sample period (Exhibit 10) as well, however, only marginally with the difference between the three-spot-rate model and the one-spot-rate model is only weakly significant (at 13% level). On the other hand, the two-spot-rate model is insignificant with the later. That is inconsistent with NN’s (2001) claiming. For out-of-sample period, the mean adjusted R-square of the one-spot-rate, two-spot-rate, and three-spot-rate models are 83.89%, 84.04%, and 85.85%, respectively. Even for the one-spot-rate model which use only the modified Macaulay duration, the mean adjusted R-square is as high as about 84% for the out-of-sample period. Compared to the in-sample period, these numbers inspired our curiosity to examine the historical data of different maturity of interest rates (as showed in Exhibit 12).

We found that drastic movements on the term structure occured over the in-sample period from July 1997 to June 2002. The long-term (30 years) and short-term (3 months) interest rates crossed each other at least twice over the five years period. Further examining the time series data we find the medium-term (2 years) and short-term (3 months) interest rates intersect with each other more frequently. On the other hand, the movements on the term structure of interest rates tend to be parallel in shape, although with changing slopes during the out-of-sample period, from July 2002 to December 2005. This could be one of the reasons why by the three-spot-rate model we do enhance the explaining power of monthly bond returns except weak significance for the out-of-sample period. Exhibit 11 illustrates the regression results for the overall sample period (102 months) combining in-sample with out-of-sample periods. While we consider the overall 8.5-year period, the three-spot-rate model has the maximum mean adjusted R-square (78.87%) compared to other

11

models and the difference between the three-spot-rate model with the others are all significant at the 5% level.

5. CONCLUSIONS

A three-factor model directly utilizing spot rates as proxies for the unknown factors driving the term structure of interest rates, consistent with empirical phenomena in describing the term structure changes (parallel, slope, and curvature movements) was developed, and empirically tested on the measurement of risk exposures and explanation for default-free bond portfolio returns. Based on the framework of the Elton, Gruber and Michaely (EGM 1990), and Navarro and Nave (NN 2001), the optimal spot rates for the three-spot-rate model to describe the unexpected change of the term structure can be identified as the 6-month, 3-year, and 9-year spot rates (6m, 3y, 9y) from in-sample period of Part I data set, and 1.75-year, 8-year, and 20-year spot rates (1.75y, 8y, 20y) from in-sample period of Part II data set, respectively. Empirical results over different periods (including Part I and Part II) demonstrate the robustness that the optimal three-spot-rate (opt-3) model has the smallest mean square forecast errors (MSE) and significantly surpass the one-spot-rate (opt-1) and two-spot-rate (opt-2) models proposed by EGM. That is, the opt-3 model developed here has superior predictability of the unexpected change of interest rates than the opt-2 and opt-1 models.

Another purpose of the empirical analysis in Part II is to test the fitness of these models for explaining cross-sectional bond returns. The three-spot-rate model also has highest mean adjusted R square than other models do, although the difference between the three-spot-rate model and one-spot-rate model is only weakly significant at 13% level during out-of-sample period. One of possible explanation is the term structure becomes more and more flat during out-of-sample period.

In the period, even only utilizing modified Macaulay duration to represent the one-spot-rate model can have explaining power for cross-sectional bond returns as high as around 84%. Therefore, the three-spot-rate model is not able to take advantage over this period. But when we consider the overall 8.5 years, the three-spot-rate model has the maximum mean adjusted R-square (78.87%) comparing with other models and the difference between three-spot-rate model with all competing model are significant at the 5% level again.

In closing, the EGM framework could be reliable for selecting optimal spot rates as proxies for the unknown driving factors of the term structure of interest rates. We propose a three-spot-rate model and estimate the state variables through EGM’s methodology. The concise three-spot-rate model determined from information in the term structure of interest rate itself can provide the good results and capture similar characteristics (parallel, slope, and curvature) as principal components analysis and the parsimonious model (Nelson and Siegel (1987)). Empirical results show that the superiority of the model is consistent with a variety of horizons. The three-spot-rate model is used to predict the unexpected change of the whole term structure and to explain cross-sectional bond returns, with encouraging results. Coincidentally, the empirical results appear the three-spot-rate model predicts more accurately when non-parallel shifts or intense term structure movements occur.

REFERENCES

Babbel, D. F. “Duration and the Term Structure of Interest Rate Volatility.” in: G. G. Kaufman, G.

O. Bierwag, and A. Toevs, eds.: Innovations in Bond Portfolio Management: Duration Analysis and Immunization (JAI Press, Greenwich, CT) (1983), pp. 239-265.

Brennan, M. and E. Schwartz. “Duration, Bond Pricing, and Portfolio Management.” in: G. G.

Kaufman, G. O. Bierwag, and A. Toevs, eds.: Innovations in Bond Portfolio Management: