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International Journal of Information Technology & Decision Making Vol. 5, No. 3 (2006) 467–481 c World Scientific Publishing Company
YIELD CURVE ESTIMATION IN THE ILLIQUID MARKET: FRAMEWORK, MODELS AND EMPIRICAL STUDY
CHI XIE∗ , HUI CHEN† and XIANG YU‡ College of Business Management, Hunan University Changsha, Hunan, 410082, China ∗
[email protected] ‡
[email protected]
In this paper, we propose a framework to estimate the yield curve in the illiquid market. Within this framework, seven different curve-fitting models are compared from four aspects with the trading data of government bonds listed in the Shanghai Stock Exchange (SSE) of China. We find that the exponential spline model is optimal for this market. The characteristics and reasons underlying SSE interest rate fluctuations in the past two years are also analyzed. Keywords: Illiquid market; yield curve of the government bond; curve fitting.
1. Introduction Yield curve, which is also known as the term structure, plays an essential role in the risk management of interest rates. For instance, the slope of the yield curve can provide the information about expected changes in the interest rates. The risk indicators, such as the Fisher-Weil duration and convexity, are calculated based on the yield curve as well. Furthermore, in the Value-at-Risk estimation and the stress test, corespondent dots in the yield curve are employed to revaluate the portfolio so as to calculate the probability distribution of the changes during the defined periods. Yield curves have inspired generations of the researches and theories due to its importance as an indicator of interest-related risks. Up to present, the yield curve researches mainly consist of three methods: the equilibrium method, the arbitrage-free method, and the curve-fitting method. The equilibrium method hypothesizes that the whole economy can be described by several factors, provided that the whole economy is in a state of equilibrium, the partial differential equation of the zero coupon bonds’ prices is derived by maximizing the expected utility function of the investors. After solving these zero coupon bonds’ prices, the yield curve is derived. Many famous models including Dothan,4 † Corresponding
author. 467
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Cox, Ingersoll and Ross (CIR),3 and Longstaff–Schartz10 belong to this category. As to the arbitrage-free method, it is mainly used for the pricing of contingent claims. In the solving process, the yield curve acts as an input variable; hence, the dynamics of the factors are derived under the yield curve. Representative models include Ho–Lee,7 Hull–White,8 and Heath, Jarrow and Morton (HJM).6 Compared to the aforementioned, two methods, the curve-fitting method is rather static. This method focuses on the cross-section price data in the market which are used to derive the implied yield to maturity of the zero coupon bond. In practice, the theoretic bonds’ prices of the curve-fitting method can well match the observed prices. Hence, our focus of this paper would be curve-fitting which we regard as the method best reflect the risk free interest rate in the market. According to the approximation function, the curve-fitting method can be summarized as: the quadratic and cubic spline models by McCulloch,11,12 the exponential spline model by Vasicek and Fong,16 the B-spline model by Steely,14 the instantaneous forward rate parametric models proposed by Nelson and Siegel13 and modified by Svensson.15 On the basis of the foregoing researches, other economists also extend with modifications, such as the B-spline approximation function, which not only includes the discount function but also the spot rate and instantaneous functions (see Refs. 1, 9 and references therein). The roughness penalty functions are incorporated into the B-spline model to control the precision and smoothness of the estimated curve (see Refs. 2, 5 and references therein). However, few of the cited models consider the liquid differences of the bonds, since there are many liquid securities in developed countries. To be specific, large numbers of bonds exist in developed market. Therefore, the liquidity can be guaranteed by eliminating the securities of less liquidity with the filter algorithms in estimation. On the contrary, in developing countries, market scale is limited and liquidity is rather low. Meanwhile, large liquidity differences among the various bonds are observed. Under these circumstances, therefore, eliminating the bonds does not seem plausible any more. In order to solve the foregoing problem, we suggest a framework for the estimation of the yield curve in the illiquid market and the empirical evidences are studied as well. To be more specific, in Sec. 2, we present this framework and introduce seven different models for yield curve estimation; in Sec. 3, within the framework, we employ the government bonds listed in the Shanghai Stock Exchange (SSE) to compare the estimative ability of these models; in Sec. 4, the characteristics and reasons for the changes of the yield curve in the past two years are analyzed. Finally, our conclusions are made in the last part of this paper. 2. Estimation Framework and Fitting Model 2.1. Liquidity differences Yield curve, which plots a set of yield to maturities (interest rates) of the zero coupon bonds with the same risk, liquidity and tax treatment but with different terms to maturity, describes the relationships among short-, medium-, and long-term rates at a given point in time. In this paper, the government bonds are
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Table 1. The liquidity status of the bond market of SSE on Sepember 23, 2005. Trading Number
Trading Volume (hand)
696 9704 9905 9908 10004 10010 10103 10107 10110 10112 10115 10203 10210 10213 10214
960 1405 4189 61007 11950 139309 18545 7503 14139 4794 65329 4231 28543 27416 69104
Trading Value (Yuan) 1038504 1645197 4322622 62571369 12044579 140700147 19049848 8067293 14035490 4784823 66599550 4068532 28313124 24984443 69900886
Trading Number
Trading Volume (hand)
10215 10301 10303 10307 10308 10311 10403 10404 10405 10407 10408 10410 10411 10501 10502
31 4633 25122 14036 6027 5289 1869 79 53370 1237 2977 102 10862 7869 1402
Trading Value (Yuan) 31742 4670881 24332294 14149191 5918002 5537651 2011599 88519 52918976 1377003 3201826 114510 11075941 8662414 1393642
used, which means the risk and tax treatment are almost the same. In order to guarantee the reliability of the results, the bonds which are used in the estimation should have approximately the same liquidity. If the liquidity differences of the bonds are not remarkable, we can use all of the cross-section data to estimate the yield curve; however, if the liquidity differences are large enough, the influence must be taken into account. In general, liquidity differences commonly exist among bonds in the developing markets. To take an example, we take the trading data of the government bonds listed in the SSE of China on September 23, 2005. Table 1 lists the liquidity status of the 30 bonds according to the indicators of trading volume and value. Large liquidity differences can be observed in the market. The bond with trading number of 10010 is the most liquid, while the bond with trading number of 10215 has the lowest liquidity. The trading volume and trading value of No. 10010 is almost 4493 and 4432 times of No. 10215. Therefore, the empirical evidence of the developing bond markets shows that it is improper to ignore the liquidity differences among the bonds. Actually, if we do ignore the differences, the dots in the estimated yield curve might only represent the minority, not the whole market. 2.2. Framework For derivation convenience, we define current time as zero, the discount function (the price of the zero coupon bond) maturing at time t is d(t), the spot rate at time t is s(t), the instantaneous forward rate at time t is f (t). The spot rate and forward rate are in continuous compound rate. The foregoing three functions have the following relationships: t f (s)ds . (1) d(t) = exp(−s(t)t) = exp − 0
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Hence, when any of the three are provided, the other two functions can be derived according to formula (1). The yield curve is derived after the annual interest rate adjustment. If there are no frictions in the market and assuming an arbitrage-free condition, the theoretic prices of the bonds should equal the sum of the discounted cash flows. We assume that at time t, there are n bonds traded in the market. The price vector of the bonds is [P1 , P2 , . . . , Pn ]T (dirty prices including the accrued interest rates), the number i bond has j deterministic cash flows until maturity, and the cash flows vector and the payments time vector of the number i bond are [Ci1 , Ci2 , . . . , Cij ]T and [t1i , t2i , . . . , tji ]T , respectively. Under these assumptions, the theoretic price of the number i bond is P¯i = jh=1 Cih d(thi ). Furthermore, define the deviation between the theoretic price and the observed price of the number i bond as devi = Pi − P¯i . In this way, we can form a deviation vector of the n bonds, denoted by [dev1 , dev2 , . . . , devn ]T . Then, the linear or non-linear least squares is used to estimate the parameters of the following model: n wi devi2 , (2) L(n) = i=1
where the wi denotes the weight of the number i bond during the process of the minimization. Normally, to solve the problem, wi is assumed to 1/n or the functions related to the durations of the bonds. These two methods, however, do not reflect the liquidity differences of the bonds. In order to incorporate the liquidity factor into the estimation frameworks, define the trading volume vector and trading value vector of the n bonds as [v1 , v2 , . . . , vn ]T and [u1 , u2 , . . . , un ]T . The weight of the number i bond is defined as follows: v u i i exp − 1 + exp −1 vmax umax wi = n (3) v u . i i exp − 1 + exp −1 vmax umax i=1 In Eq. (3), we take the exponential function to get the weight coefficient, i.e. giving much more weight on the high liquid bond, while less weight on the low liquid bonds. Within this framework, the yield curve can represent the necessary returns of the whole market and will not be influenced much by minority investors. After the determination of the weight coefficients, the approximation function form to fit the foregoing mentioned three functions must be decided. In the following section, we will introduce seven different models. 2.3. Curve-fitting models 2.3.1. Cubic spline model Due to the limitation of the discontinuity of the forward rate curve estimation under the quadratic model, here we only introduce the cubic spline model. The
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model directly approaches the discount function by the cubic spline. Using mx to denote the longest term to maturity of the bond in the n bonds, we slice the interval [0 mx ] into k sub-interval. So we can get k + 1 nodes from t0 to tk . The cubic spline defines the discount function in the following way: d1 (t) = 1 + c1 t + b1 t2 + a1 t3 , t ∈ [t0 , t1 ], 3 d2 (t) = d1 (t) + (a2 − a1 )(t − t1 ) , t ∈ [t1 , t2 ], (4) .. . dk+1 (t) = dk (t) + (ak+1 − ak )(t − tk−1 )3 , t ∈ [tk−1 , tk ], Under formula (4) and constrain d(0) = 1, the discount function, the spot rate and forward rate are all continuous. It is known that both the numbers and positions of the nodes would influence the estimation result. We define the nodes position according to the criteria by McCulloch and assign all sub-intervals with the same number of bonds. As to model (4), we can directly use the weighted least squares to estimate the parameters. 2.3.2. Exponential spline model In order to avoid the limitation that the forward curve might fluctuate too heavily or be negative under the cubic spline model, Vasicek and Fong (1982) proposed the exponential spline, which is similar to the cubic spline. The exponential spline also fits the discount function directly and has the following form: t ∈ [t0 , t1 ], d1 (t) = f1 + c1 exp(−ut) + b1 exp(−2ut) + a1 exp(−3ut), 3 d2 (t) = d1 (t) + (a2 − a1 )[exp(−ut) − exp(−t1 u)] , t ∈ [t1 , t2 ], (5) .. . dk+1 (t) = dk (t) + (ak+1 − ak )[exp(−ut) − exp(−tk−1 u)]3 , t ∈ [tk−1 , tk ], s.t. f1 + c1 + b1 + a1 = 1. In Eq. (5), the relationship between the parameters and discount function is non-linear, the parameters must be estimated with some optimization algorithms. Actually, we estimate these parameters with the non-linear weighted least squares. 2.3.3. B-spline models and the modification Recently, some economists have modified the B-spline model following Steely’s initial formulation. Generally, the B-spline was used to approach the discount function. At present, not only the spot rate function but also the instantaneous functions are the target functions. At the same time, based on the B-spline, some roughness penalty functions (RPF) are added to the minimized function to control the precision and smoothness of the estimated curve. However, adding the RPF may lose some freedom in the process of estimation. Since the amount of the bonds traded
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in the illiquid market is very limited, we do not consider the RPF here. Compared with the cubic and exponential spline, B-spline is rather complex; the recursive formula of the cubic B-spline is defined as follows: s+4 s+4 3 1/(tj − ti ) max(t − tj , 0)3 , (6) gs (t) = i=s
j=s,j=i
where s ranges from 1 to 3 + k. Besides the k + 1 nodes from t0 to tk , three nodes must be added purposely at the two ends of the interval. In this way, the basis functions of the cubic B-spline are derived recursively according to the formula (6). Define the parameters vector of the basis function of the cubic B-spline as βs , we can take the cubic B-spline to approach the discount function, the spot rate function and instantaneous forward rate function. To summarize, the present researches mainly include three forms, i.e. the linear B-spline, the exponential B-spline and the integrated B-spline. These forms are defined as follows: d(t) =
3+k s=1
βs gs3 (t)
d(t) = exp − t
3+k
(7) βs gs3 (t)
(8)
s=1
3+k t d(t) = exp − βs gs3 (x)dx
(9)
0 s=1
In the above equations, the parameters of the linear B-spline can be estimated using the weighted least squares, while the parameters estimation of the exponential B-spline and integrated B-spline have to repeated employ optimization algorithms under the non-linear weighted least squares. 2.3.4. N–S and Svensson models Both the N–S model and Svensson model are parametric models, which fit the instantaneous forward rate curve. These models have the following forms: f (t) = β1 + β2 exp(−t/τ ) + β3 t exp(−t/τ )/τ,
(10)
f (t) = β1 + β2 exp(−t/τ1 ) + β3 t exp(−t/τ1 )/τ1 + β4 t exp(−t/τ2 )/τ2 .
(11)
Every parameter in Eqs. (10) and (11) has its economic meaning, which are discussed in N–S (1987). Because, the relationships between the parameters and bond prices is non-linear, the non-linear weighted least squares is used during the estimation. So far, we have simply introduced the seven models. To summarize, the cubic spline, the exponential spline and the linear B-spline are used to approach the discount function; the exponential B-spline is used to directly approach the spot rate function; and the integrated B-spline, N–S and Svensson models are used to
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approach the instantaneous forward rate function. The theoretic foundation of the cubic spline, exponential spine and B-spline is the Weierstrass theorem; the differences between these models are that they use different basis functions. The empirical studies would further provide the evidence to determining which spline basis fit the curve best. Compared with the spline models, the theoretic foundations of the N–S and Svensson model are the stochastic differential equation. The parametric model can fit most shapes of the yield curve in the market, where the curves have high smoothness.
3. Empirical Study and Comparison 3.1. Comparative criteria Within the framework, we compare the superiority and inferiority of the seven models based on the following four criteria: (1) accurate pricing of the in-sample data and the smoothness of the estimated curve; (2) stability in the estimation process, which means that the initial values of the parameters will have little influence on the results; the forward curve will not fluctuate too heavily and the estimated forward rate will not be negative; (3) robustness, which means on most trading days, the optimal model and is superior to other models; and (4) accurate forecasting ability in the out-of-sample data. According to these four criteria, we do the comparison in the following way: (1) in-sample comparison: We randomly choose five trading days from the SSE, estimate the parameters under seven different models, compare the spot rate curve, the forward rate curve and calculate the indicators defined in the Table 2. (2) out-of-sample comparison: As for the data of the five chose trading days, we choose three bonds randomly from the trading data as out-of-sample and estimate the parameters with the remaining data. Next, we use the estimated model to price the three chosen bonds. In this way, we can calculate the deviation between the theoretic and observed price of the three bonds and calculate the defined indicators, where n is 3. (3) the stability comparison: During the process of non-linear least squares estimation, we analyze the sensitivity of the result to the initial values.
Table 2. Defined indicator. MSE Comparative Indicators
Pn
i=1
dev2i /n
MAE Pn
i=1
|devi |/n
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3.2. Data and comparisons In this paper, we chose five trading days randomly from the SSE market, i.e. August 22, 2003, April 9, 2004, October 28, 2004, March 16, 2005, and August 23, 2005. Approximately, a half year interval can be observed between each sample day. Within the whole sample period, the liquidity of the SSE grew substantially, and the number of the listed bonds increased from nineteen to thirty. However, the market is rather illiquid in comparison with the markets of developed countries. Moreover, we should note that the payment terms in Chinese market are quite different. The trading bonds listed in the SSE include the zero coupon bonds, the coupon bonds with annual and semi-annual interest. As for the maturities, they are unevenly distributed. Take the trading day of August 23, 2005 as an example, among the thirty bonds, seven bonds have maturities up to three years, nineteen bonds have maturities between three to seven years, and only four bonds have maturities greater than seven years. The unevenly distributed maturities of the different bonds make the estimation very difficult, and will make the market less likely to form an entire and smooth yield curve. 3.2.1. In-sample comparison We estimate the yield curve and forward curve under seven different models and calculate the indicators defined in Table 2, with the result listed in Table 3. In the estimation, there are totaly four nodes, which makes the number of the unknown parameters at most six. Here, we only present the graph of the yield curve and forward curve on October 28, 2004 in Fig. 1. Other graphs are available on request.
Table 3. Calculation result of the defined indicators of the in-sample data. Comparative Day
20030822
20040409
20041028
20050316
20050823
MSE Cubic Exponential Linear B Exponential B Integrated B N–S Svensson
0.7872 0.5238 0.6306 0.7518 0.6009 1.5191 0.6882
3.1317 2.7320 2.8695 3.0212 2.9728 5.0482 4.1877
1.1198 1.0016 1.0662 1.6549 1.4708 1.6784 1.4021
0.1702 0.1492 0.1701 0.1710 0.1559 0.2474 0.2252
0.4937 0.4390 0.4501 0.4476 0.4690 0.7782 0.6682
MAE Cubic Exponential Linear B Exponential B Integrated B N–S Svensson
0.5847 0.5629 0.5869 0.6219 0.5394 0.7560 0.6061
1.3434 1.2163 1.2183 1.1826 1.1702 1.4956 1.4612
0.6868 0.5836 0.6520 0.9542 0.8906 1.0052 0.8917
0.3211 0.3005 0.3207 0.3468 0.3036 0.4054 0.3866
0.5851 0.5715 0.5619 0.5324 0.5665 0.7442 0.7170
Note: Bold number denotes the minimized indicator of the day.
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4
50
time
100
4 linear B spline
2 0
50
time
100
4 2 0
integrated B spline 0
50
time
100
150
4 2
exponential spline 0
50
time
100
150
6 4 2 0
150
6
475
6
0
150 interest rate
interest rate
0
6
0
interest rate
cubic spline
2 0
interest rate
interest rate
6
interest rate
interest rate
Yield Curve Estimation in the Illiquid Market
exponential B spline 0
50
time
100
150
100
150
6 4 NS
2 0
0
50
time
6 4
spot rate svensson extend
2 0
0
50
time
100
forward rate 150
Fig. 1. The yield and forward curve on October 28, 2004 under seven models.
According to the calculated results in Table 3, seven models have similar explanatory abilities to the in-sample data, which means that when the fitted error is small for one model, the other models can also fit the observed price well. For example, on August 22, 2003, March 16, 2005, and August 23, 2005, all these seven models fit the observed bonds prices very well. However, on April 9, 2004 and October 28, 2004, all these models have some difficulties explaining the observed bonds prices. One possible reason could be that some non-systematic risks are not well quantified in the models, or alternatively, the irrational behaviors in the market cause the observed prices to deviate a lot from the theoretic prices. Generally speaking, the spline models are superior to the parametric models in terms of the fitting ability. The N–S and Svensson model can generate rather smooth instantaneous forward rate curve which may lose some fitting abilities to fit the observed bonds prices. Further analysis is performed. In Table 3, the bold numbers denote the minimized indicators of the day. If a given model is superior to other models, the superior model should have more numbers which are bold and underlined. We see clearly from Table 3 that the exponential spline is the best model in term of the indicator of MSE: in all five trading days, the exponential spline has the smallest
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MSE compared with other models. In terms of the MAE, the exponential spline and integrated B-spline have similar ability to fit the data, and are superior to other models. In sum, the exponential spline has the strongest ability to explain the insample data, while the integrated B-spline is the second strongest. N–S model has the worst ability to fit these data, where most of the maximum numbers are found. In Fig. 1, the yield curve and forward curve on October 28, 2004 under seven different models are plotted. All the yield curves have an upward trend. The plots of the dots after one year do not show quite different. However, the yields within one year are of much difference. This actually shows that there are great deviations of the bond with maturity within one to three years between the theoretic prices and the observed prices. Comparing all the forward curves in Fig. 1, we can hardly find heavy fluctuations and negative values and all curves are in the same trend. Although, Fig. 1 only plotted the curves on one trading day out of the five days, the curves on the other four days have similar characteristics. The yield curves are all upward trend and the position of the curves changed in the whole sample period. To this point, we will further analyzed in Sec. 4. 3.2.2. Out-of-sample comparison In Table 4, we compare the forecasting ability of the five days under different models. On August 22, 2003, March 16, 2005, and August 23, 2005, all seven models can fit the observed bond prices pretty well, so all the models have quite good forecasting ability for the out-of-sample data. As for the indicators of the MSE and MAE, the exponential spline model has the best results, while the integrated B-spline model
Table 4. Calculation result of the defined indicator of the out-of-sample data. Comparative Day
20030822
20040409
20041028
20050316
20050823
MSE Cubic Exponential Linear B Exponential B Integrated B N–S Svensson
0.2283 0.0201 0.2315 0.2481 0.3023 0.4064 1.1550
4.9143 2.1404 2.8607 3.7157 6.3241 3.3664 3.4547
6.1502 5.8435 5.9751 5.9051 5.8214 6.8967 6.8521
0.2885 0.2541 0.2870 0.5408 0.2338 0.6952 0.3265
0.7019 0.3830 0.5564 0.8057 0.6783 1.3345 0.9297
MAE Cubic Exponential Linear B Exponential B Integrated B N–S Svensson
0.4392 0.1367 0.1571 0.1592 0.1721 0.6256 0.9591
1.8548 0.9957 1.2338 1.2414 1.5687 1.5403 1.5574
1.8044 1.7909 1.8057 1.7150 1.7161 2.0672 2.0569
0.3602 0.3832 0.3600 0.5274 0.3273 0.6764 0.4657
0.5879 0.4529 0.4805 0.7634 0.6251 1.0041 0.9382
Note: Bold number denotes the minimized indicator of the day.
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is the second best. The N–S model has the worst forecasting ability for the out-ofsample data, where large deviations between the observed prices and the forecasted prices can be observed.
3.2.3. Stability comparison In this section, we perform the sensitivity analysis of the estimation results to the initial values during the optimization process. The models which are estimated with non-linear squares are our focus. According to the foregoing introduction, the cubic spline and linear B-spline models can be estimated by weighted linear least squares. Therefore in terms of the stability, these two models are the best. As for other models, the initial values must be chosen to activate the optimization process. During the estimation, we find that the estimation results under the models of exponential spline, exponential B-spline and integrated B-spline are not sensitive to the initial values, which means that the initial values can be chosen randomly and the estimation results will converge to the same values. As with the N–S model and Svensson model, the estimation results are very sensitive to the initial values. If the initial values of some parameters are changed, the shapes of yield curve will change a lot. So relatively speaking, the stabilities of these two models are weak. To sum up the foregoing analysis, we conclude that the exponential spline is the optimal out of the seven models. It not only has a strong explanatory ability with respect to the in-sample data, but also has a very strong forecasting ability with respect to the out-of-sample data. In the process of estimation, this model can hardly be influenced by the initial values. The yield curve derived by this model not only shows smoothness, but also has pricing precision. This model is of great robustness, although not all of the calculated indicators in the trading days are the smallest. For most of the trading days, the exponential spline can produce the smallest errors. In Fig. 2, we present the graphs of the observed bond prices and the fitted prices under the exponential spline model. According to the figure, most of the prices on August 22, 2003, March 16, 2005, and August 23, 2005 can be fitted well. On October 28, 2004 the prices can also be well explained, only two bonds’ observed prices deviated a lot from their theoretic prices. Relatively speaking, the fitting effect on April 9, 2004 is not very satisfactory.
4. Analysis of the Yield Curve of SSE In Fig. 3, we plot the yield curves in the five trading days under the exponential spline model. From Fig. 3, we know that, in the past 2 years, the changes of the yield curve of the SSE have experienced two stages. The first stage is the period of the rapid increases of the medium-term and longterm yields. On August 22, 2003, although the yield curve is upward, the difference between the short term rate (three month) and long term rate (fourteen years) is
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120 110 price
110 price
20040409
20030822
100 90 80
100 90
0
10
20
80
30
0
10 number 20
number 120
120
20041028
100 90 80
20050316
110 price
price
110
30
100 90
0
10
number
20
30
80
0
10
number
20
30
120 * actual price o predicted price
100 90 80
20050823 0
10
number 20
30
Fig. 2. The fitted prices vs observed prices under the exponential spline.
5 4.5 4 3.5 interst rate
price
110
3 2.5 20030822 20040409 20041028 20050316 20050823
2 1.5 1 0.5 0
50
100
Fig. 3. The yield curve of the SSE during past 2 years.
150
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relatively small. According to the estimation result, this difference is only 1.5%. The short term bonds and long term bonds do not generate the effective liquidity premiums and risk premiums. The yield curve on August 22, 2003 shows that the slope is relatively small. When come to April 9, 2004, the medium-term rates and the long-term rates, especially the medium-term rates rapidly increase. The sudden increase causes the slope of the yield curve on October 28, 2004 to become much steeper. At this time, the spread between the three month rate and fourteen year rate is 3.5%. The relationship between these two rates is much more reasonable. The reason for the steeper yield curve is because in that period, the pressure of the expected inflation was relatively high, the investors need much more risk premiums. At the same time, in order to restrain the hasty increases of the commodity prices and inventory investments, the Peoples’ Bank of China (PBC) employed the monetary policy to continuously show the market a sign of contraction. For example, on September 21, 2003, the PBC adjusted the ratio of the residual from 6 to 7%. On April 25, 2004, the ratio of the residual was increased by 0.5%. The second stage of the changes of the yield curve is the yield decrease. Comparing the yield curve on October 28, 2004 with the yield curve on August 23, 2005, the short-term rates decrease a little, however, the medium-term and the long-term yields, especially the former, decrease a lot. The reasons behind this are mainly as follows: firstly, the persistent bear stock market made part of the investors transfer their capital to the bond market. So, the funds supply in the bond market were in abundance. This caused the decrease of the yields of all maturities. Secondly, the expectation of the appreciation of the RMB existed persistently during that period. Although on July 21, 2005 the PBC adjusted the exchange rate, the appreciation pressure did not disappeared. This appreciation expectation caused lots of the speculation funds gathered in the capital market. Because of the poor performance on the stock market, much more capital got into the bond market and kept the fund supply abundant. At the same time, the appreciation of RMB made it impossible to adjust the interest rate in short time. This could also result in the downward yield curve. Finally, the market scale of the bond market was rather small, and most of the trading bonds were in medium-term, i.e. within three years to seven years. Under this circumstance, the investors had no other choices but to buy the medium-term bonds. This resulted in the rapid decreases of the medium term rates in comparison with other term rates. At present, however, the relationships between the rates of the short- and long-term have became much more reasonable, for the pressure of the inflation is lessened. The spread between the three month and 14 years rate is 2.6%, and the risk premium is reasonable. 5. Conclusions In this paper, we have incorporated the indicators of trading volume and trading value into the estimation of the yield curve, and proposed a framework which aims at estimating the yield curve in an illiquid market. Within the framework, we have
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performed the empirical studies, comparisons and analysis. The findings can be as follows: (1) Within the proposed framework, the estimated curves can represent necessary returns of the market, without being greatly influenced by minority investors. (2) Within the proposed framework, seven different models are compared. The exponential spline is superior to other models for the SSE bond market because of both its out-of-sample forecasting ability and in-sample explaining ability. The stability and robustness of the exponential spline model are also relatively high. (3) For the past two years, changes of the yields of all maturities have been following a consistent trend, i.e. growing up first and dropping down next. The yield curve becomes increasingly steeper, and the relationship between the short term rates and long term rates becomes much more reasonable. Now, the yield curve can reflect the time value and risk value of market capital, and can be used as the basis of asset pricing, financial product designing, hedging and risk management. Also, the yield curve can reflect the monetary stance and inflation expectation to some extent. We should, however, be aware that the SSE provides a relatively small market scale, and the maturities of the bonds cluster in the medium-term. These facts are actually the limitation of the formation of an entire yield curve, which eventually hinders the investor’s decision to invest in the market. Therefore the market scale must be further extended. More shortand long-term bonds should be issued to enrich the investment categories for investors. Acknowledgment This paper is supported by SRFDP(20020532005) and EYTP. References 1. K. Adams and D. Deventer, Fitting yield curves and forward rate curves with maximum smoothness, Journal of Fixed Income 2 (1994) 52–62. 2. R. Bliss, Testing term structure estimation models, Advances in Futures and Option Research 9 (1997) 197–231. 3. J. Cox, J. Ingersoll and S. Ross, A theory of the term structure of interest rates, Econometrica 53 (1985) 385–408. 4. L. Dothan, On the term structure of interest rates, Journal of Financial Economics 6 (1978) 59–69. 5. M. Fisher, D. Nychka and D. Zervos, Fitting the term structure of interest rates with smoothing splines, Federal Reserve Board, Finance and Economics Discussion Series 1 (1995) 1–30. 6. D. Heath, R. Jarrow and A. Morton, Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation, Econometrica 60 (1992) 77–105. 7. T. Ho and S. Lee, Term structure movements and pricing interest rate contingent claims, Journal of Finance 41 (1986) 1011–1028.
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