Study on Static Term Structure of Interest Rates Based on Bezier Curve ZENG Shihong, XIA Liang Finance Department of Economics Management School, Beijing University of Technology, 100124 [email protected] Abstract: The term structure of interest rates static fitting curve refers to the use of different types of mathematical function to describe the yield curve of interest rate. The most popular method of static fitting is to use a B-spline curve to fit the yield curve of interest rate. However, this method is often subject to restrictions on the order. So, usually the fitting is on the third order. Using a special form B-spline curve what is Bezier curve, we get the term structure of interest rate curve of government bonds for China, the United States and Japan. We get a fitting method to change the orders of curve easily. Meanwhile, the complex curve fitting calculation, simplified to cluster analysis of the scattered points. Keywords: Bezier curves, B-spline, Interpolation, The term structure of interest rate, clustering analysis

1 Introduction Risk-free interest rate is the most basic variable of financial markets. More it is the one of the most important economic variable. Risk-free interest rate is essentially the prices of capital, reflecting the relationship between supply and demand of funds. The term structure of interest rates refers to the same level of risk, interest rates and the quantitative relationship between maturity, or a theoretical zero-coupon yield curve. It is the basis of the asset pricing, financial product design, hedging, arbitrage and investment and so on. The term structure of interest rates has always been important and finance in a very basic issue. The interest rate term structure model can be divided into two types of static models and dynamic models. Dynamic model are including three directions. They are equilibrium model, arbitrage-free model, dynamic model. Dynamic models commonly used stochastic differential equations describe the behavior of interest rates, and then using the method of Parameter estimation to get the dynamic changes of interest rate. It is a method needs more complex calculation. And require large sample. Static model of curve fitting, assuming that there is an interest rate function, and then select the cross-sectional data of bonds to estimate the parameters of the function [1]. From the History of view, the static model and dynamic model is almost parallel to development; from the focus point of view, dynamic model focuses on changes in assumptions, the static model is like to focus on improvements in goodness of fit. In the static method, due to the different nature of mathematical functions, the methods can be divided into two types witch are parametric and non-parametric method. Parameter method using the approximate functions, including polynomial functions, piecewise functions, piecewise linear function, exponential function, Nelson-Siegel model [2]. Spline methods using the cubic polynomial spline to fit the term structure of interest rates. Cubic polynomial spline law will be changed three bonds polynomial spline to smoothing spline, smoothing spline curve is the essence of B-spline curves [3]. Method using B-spline fitting FNZ model is the most popular method on today for central banks of some developed capitalist countries, such as the United States and Japan, to fit the term structure of interest rates [4]. Bai Xiaoying, Yang Feng-mei, Zhou Rongxi (2009) combined the method of linear programming and three bonds polynomial spline model. They use the linear programming to estimate the model [5]. Li Yiyi, Pan Wanbin, Miu Boqi (2009), using Akaike information criterion, select the knot of three bonds smoothing spline[6]. Overall, there are no obviously differents for these models on the goodness of fitting curves. The domestic generally is only one which is considered a bad method on fitting

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exponential spline curve. “The third order of spline curve and smoothing the third order of spline curve (B-spline) are no difference on goodness of fitting”[2]. On the fitting method, setting the objective function, and then select the appropriate knot is a general spline fitting method.

2 Data Table 1 China bond yield to maturity and remaining time 1 Securities Code yield remaining time 10301 1.54 0.191781 10704 2.7 0.345205 10503 0.97 0.372603 10004 2.03 0.690411 10511 1.88 0.857534 10311 1.54 0.939726 10605 1.45 1.452055 10612 2.71 1.676712 10407 1.82 1.70411 10110 1.52 1.789041 10618 0.75 1.871233 10112 1.55 1.884932 10410 1.86 1.956164 10203 1.98 2.353425 10705 3.17 2.367123 10505 2.53 2.454795 10509 2.67 2.706849 10513 2.83 2.958904 10601 2.52 3.216438 10606 2.62 3.454795 10613 2.04 3.723288 10308 2.88 3.769863 10620 2.91 3.964384 10701 2.93 4.158904 10707 1.4 4.452055 10501 3.35 5.219178 10603 3.51 6.29589 10616 2.92 6.79726 10703 3.4 7.282192 10710 3.77 7.542466 10213 3.66 7.780822 10512 3.89 10.93699 10107 3.84 11.64384 10619 3.37 11.93699 10303 3.92 13.35616 10504 3.98 15.43562 10609 3.7 16.55068 10713 4.52 17.69041 10706 4.27 27.44932

1

Material origin：http://www.sse.com.cn/sseportal/webapp/datapresent/SSEBondTotalPriceAct_1?searchDate=

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Table 2 U.S. Treasury interest rate and length2 Treasury interest rate length 0.01 0.083333 0.03 0.25 0.16 0.5 0.35 1 0.83 2 1.3 3 2.26 5 3.02 7 3.55 10 4.39 20 4.49 30 Table 3 Japan Treasury interest rate and length3 Treasury interest rate 0 0 0.6 0.2 1.1 1.4 0.5 1.4 1.7 1.7 1.5 1.3 2.1 2.1 2.2

length 0.25 0.5 1 2 3 4 5 6 7 8 9 10 15 20 30

3 Fitting the Bezier model Bezier curves are the special form of B-spline curve, B-spline curve is a general form of Bezier curve[7]. The continuity B-spline curve depends on the existence of derivative. When the order of spline is is greater than 3, it is difficult to get continuous of the derivative. It is generally used the third-order spline function [3]. The Bezier curve is characterized by a recursive function, it means that, without changing the case of other nature, the orders of Bezier curves can be changed. For example, there are 4 Control point. If we want to get 5 Control point from that curve, we just need to make

kn(1) =

n (1 + t )kn(1)−1

(1 − n) (t + 1)kn

Therefore, using Bezier to fit the curve function, the order of the curve can be completely under the shape of the curve. The function is determined without being restricted by the order. While the Bezier 2 3

：http://www.bloomberg.com/markets/rates/index.html ：http://www.bloomberg.com/markets/rates/japan.html

Material origin Material origin

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curve in high-order cases, the ability for control points to control the curve will become worse. However, with the increase in the order of the control points, the characteristics of polygonal curves became more and more close to the curve. It is a power to improve the flexibility of the curve control. Thus the goodness of fit improved. The difference between Bezier curves and B-spline curve is a curve, B-spline curve using the knot to determine the curve, Bezier curve use control-points to determine the curve. The method of B-spline curve focuses on the choice of the knot, while the knot must come form the sample set. On the third order of B-spline curves, each of three points determined a curve, but the other knot has no relationship with the curve. The Bezier curves, each section are also subject to all of the control points. How to select the control points is the key to fit Bezier curve. Through the scatter plot, we know that the curve of the interest rate should be an upward-sloping curve, so the control points should be above the curve. Therefore, we selected K + S points from sample to construct the characteristic polygon. In selecting these points we used Matlab7.0 to clustering the points[8]. Because the characteristic polygon should be above the curve, so we select the largest points from the clustering results in the interest rate. The points are control points of the curve. The vertical axis means bond yield to maturity, while the abscissa means remaining term government bonds. We draw three curves of Treasury interest rate. They are Chinese, the United States, Japanese, Among them, China's curve is based on the yield to maturity and bonds, the remaining duration. The United States and Japan, the curve is based on interest rates and maturity. China's curve is the third order of Bezier curve. The goodness of fit for United States and Japan are bad, so using 4-order Bezier curve fitting. From the figure we can see that the control-point of these curves is near to the curve itself, the shape can changed more flexible by the control of the control point.

Figure 1 China bond yield of term

Figure 2 U.S. bond yield of term

Figure 3 Japan bond yield of term

4 Conclusion China's interest rate curve increases with the increase of maturity,, which the requirements of liquidity preference theory is consistent. The longer maturity, investors demand higher returns, according to an analysis of different maturity, for short-term bonds (0 ~ 5 years), the momentum of rising interest rates is large, while the medium-term (5 ~ 8 years) were relatively stable, upward trend has slowed. A long-term bonds (8 ~ 20 years) is higher than the short-term Treasury .Form the Expectations Theory, It is means the investors expect the interest rates will rise in future. By market segmentation theory, that is a long-term bonds has little liquidity than short-term bonds, so, the investors should get a higher liquidity premium [2]. From Shape in the short-term bonds, as well as the medium-term the interest rate curve upward trend of slow, indicating the profitability of long-term bonds have further room for growth [9]. But it is noteworthy that the long-term rates are higher than the ultra-long (20 years) interest rate,

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according to liquidity preference theory, the mobility of ultra-long-term bonds is less than long-term bonds. The ultra-long interest rate should be given greater liquidity premium. But this premium is not shown. The value of ultra-long-term bonds have been seriously underestimated The short-term interest rates of U.S. government bonds have an obvious upward trend. But the upward trend long-term of moderating is not like China's, as a long-term bond interest rates lower than the long-term interest rates. It is said that the liquidity premium reflected in the long-term bonds interest rates, the long-term bonds in the U.S. bond market does not be significantly underestimated, like China. Recently, the international investors in the U.S. bond market, sell off the short-term bonds, and purchase long-term government bonds. The interest rates of U.S. Treasury short-term market fall rapidly. There had been "zero interest rate" phenomenon, which is like the curve short-term interest rates. Japanese government bonds interest rates far below to the two countries, the upward trend of short-term rates is not obvious, it is said that the market's preference for government bonds of various maturities is similar. United States and Japan have a certain amount of the short-term government bonds. Its duration is less than 1 year. Overall, the U.S. and Japanese bond market is relatively mature, the term structure of interest rates according to the theoretical hypotheses. Not only the long-term liquidity premium of Chinese bond market is not reflected, but also the short-term interest rates rose significantly, so China's bond market is still in the adjustments term.

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Acknowledgments The research was supported in part by The China Scholarship Council (CSC) ([06]3036).

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