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Term structure of economic management rate based on parameter analysis of estimation model of ordinary differential equation

Published Online: 15 Jul 2022
Volume & Issue: AHEAD OF PRINT
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Received: 17 Apr 2022
Accepted: 04 Jun 2022
Journal Details
License
Format
Journal
eISSN
2444-8656
First Published
01 Jan 2016
Publication timeframe
2 times per year
Languages
English
Introduction

For nearly half a century, the study of interest rate models has been an important study in economics and finance. Especially in the past 30 years, with the rapid development of financial derivatives, the research on interest rate structure. It has become an important player in the financial industry. There have been a lot of theoretical and empirical research in the research direction. The formation of this research hotspot certainly has its unique historical background and practical significance. The rise of interest rate research is primarily due to the breakdown of the Bretton Woods system. In 1972, the Bretton Woods system, and China following a major war to control its exchange rate, was hit hard by the U.S. economic downturn. Global trade, forcing the U.S. to finally stop the free exchange of dollars and gold, created Bretton. The forest exploded. Floating exchanges allow many countries to communicate very well. Since interest rates are an important tool to change prices and affect the supply and demand of money, changes in exchange rates will also lead to changes in interest rates, which directly promote the improvement of financial derivatives. Because the cost of financial derivatives is inseparable from the theoretical and methodological support of the benefit model, the research on embroidery models has attracted much attention and developed rapidly. During the research, scientists have come up with various ideas and models to explain real-time models in real-world industries. The purpose of introducing financial derivatives is to separate all risks from normal investment returns, reduce and minimize the uncertainty of investment behavior, which requires the interest rate structure model to be as consistent as possible with historical data. and improve the predictability of the model. However, due to the limitations of econometric methods, there are obvious deficiencies in the early methods and results. For example, in 1973, Black Scholes determined the difference between the derived value of a derivative product and the value of unpaid dividends, where value equals value. The constant volatility assumed in the equation is obviously inconsistent with the time variability of real volatility. It is the defects of these early asset pricing models that promote the further development of interest rate term structure model. From short-term interest rate model to forward interest rate model, from parametric model to nonparametric model, with the deepening of research, the original model has been improved in many aspects, and new methods and tools have also been successfully applied. Among them, the progress of time series analysis technology has played an important role in promoting the research of interest rate term structure. Currently, some derivative industries in China and the United States have gradually developed into mature industries, including some foreign markets[1]. China's financial system with financial and economic characteristics started late and is strictly constrained by China's financial management. However, with the in-depth progress of building a socialist market economic system, the openness of China's financial market and the market-oriented function of the financial system are constantly strengthened, and the interest rate control and exchange rate control have been loosened to a great extent, which has laid a solid foundation for the market-oriented process of China's financial industry. Therefore, how to correctly evaluate, learn from and absorb foreign mature experience and existing research results in the theory and method of interest rate term structure will provide a valuable reference for the effective allocation of resources and effective risk management in China's financial market. It is precisely for this need that we choose the term structure of interest rate as the research theme. We will not only complete all the details, compare and comment on the existing interest rate model theories and procedures, but also explain and analyze the factors and characteristics of the interest rate pattern in China's financial industry. While strengthening the dynamism, randomness and dynamism of the original flower pattern model, the characteristics of the flower pattern model in asset pricing, risk and performance are revealed. On the basis of improving the forecast of interest rate structure, this paper puts forward the resistance measures and investigation results on the impact of interest rate in our country. N Ramirez Velez and others found that if the financial market wants to play a good financing function, an effective and normal government bond market is essential [2]. The high liquidity of the Treasury bond market can be used to hedge the risk exposure in other markets and promote the issuance and secondary market transactions in other markets. Investors' high trust in the Treasury bond market and the high liquidity in the market make the Treasury bond market a risk-free interest rate that can be used as a benchmark to provide a basis for the pricing of other financial instruments. Yue, D. and others believe that, especially in the recent financial crisis, the high liquidity and almost non default characteristics of the Treasury bond market have been favored by investors [3]. The so-called bond interest rate curve refers to the nature of the change in the yield of bonds of different maturities before maturity, which reflects the cost of capital borrowing in the financial market. The interest rate term structure of zero coupon bonds is the change law of the yield to maturity of zero coupon bonds with different maturity periods, which reveals the relationship between the cash flow on the maturity date of bonds and the original investment. Since there is no default risk, the interest rate term structure of zero interest bond calculated according to national debt becomes the basic interest rate term structure, which is often used as the reference basis of discount rate in financial asset pricing and project investment decision-making model. There are three main reasons why the term structure of interest rate has become an important issue in the field of Finance and macro research. First, the term structure of interest rate itself contains a lot of economic information, which can be used to predict real economic activities, inflation level, household savings and consumption behavior (angetal, 2006; Fama, 1990; Mishkin, 1990); Second, the term structure of interest rate has an important impact on asset pricing, project investment and financing decisions. Fair value can be determined by reducing the future earnings of a financial asset or business over a period of zero interest rates; third, interest rate models can be studied from the perspective of financial policy and bank spreads over time by examining changes in interest rates Whether it affects long-term earnings growth based on the country's economy. From a financial and macroeconomic point of view, analyzing and constructing a loan interest rate structure model can not only analyze the reasons for the macroeconomic interest rate structure, but also integrate financial markets into macroeconomic theory and empirical research, so that policy decisions can be scientifically based. Brady, N. W. proved through research that in absolute terms, the bond balance in circulation in China's treasury bond market at the end of December 2011 was 7.38 trillion yuan, ranking third in the world after the United States and Japan. Moreover, China's economy currently ranks second in the world and is still growing rapidly. The importance of China's treasury bond market will increase [4]. Chinese government bonds were first issued by the Ministry of Finance in 1950, then suspended for historical reasons in 1958 and resumed in 1981. The main purpose of issuing bonds is to finance China's construction projects. For a long time, the scale of the national debt market is small, but in recent years, it has become one of the largest government bond markets in the market. Yu, C. and others found that in the early stage of the establishment of the national debt market, there was only the primary market [5]. The issuance of bonds is mainly driven by administrative forces and cannot be traded after issuance. In 1988, the Ministry of Finance launched a trial of over-the-counter circulation of government bonds in commercial banks in 61 cities, which can be regarded as the real beginning of the secondary trading market for government bonds. At the end of 1990, the stock exchange was established, and treasury bond trading gradually expanded to the whole country. Since then, OTC trading and exchange market coexist. However, since there is no registration and preservation institution for physical delivery of bonds, the transaction of bonds only exists in the local market. In 1993, Shanghai Stock Exchange introduced bond futures to mass investors, thus establishing a preliminary treasury bond futures market. However, in 1995, large-scale price manipulation and speculation occurred in the Treasury bond market and futures market, which led to the government's closure of the futures market and over-the-counter trading market (temporarily closed). Then, the Shanghai and Shenzhen stock exchange markets became the only legitimate bond trading markets. In 1996, the repurchase agreement (Repos) was listed on the Shanghai Stock Exchange. As banks are also participants in the exchange market, securities companies borrow funds from banks and speculate on stocks by signing repurchase agreements, which seriously interferes with the normal trading of the stock market. Since then, commercial banks have been prohibited from trading in the exchange market, and it is expressly prohibited to use repurchase agreements to finance other investment activities. Subsequently, In 1997, the central bank planned to establish a joint venture financial market, and the Chinese economy was divided into two stages. The inter-bank market is established by China's foreign exchange market and the price and trading platform of China's inter-bank financial center. The facilitation registration and cancellation agreement signed by my country at the end of 1996 is called the Guardian Agreement officially authorized by the government to clear inter-bank transactions. Since then, my country's secondary financial industry has formed a situation in which our businesses are in the same line: exchanges, banking business and the over-the-counter market of trading companies. Among them, the inter-bank market is the main market, accounting for 90% of the total business volume, while the retail market has always been small. From 1998 to 2001, the government allowed financial companies and insurance companies to participate in the banking market and the intergovernmental fund market. In April 2001, a market system was established to increase market liquidity. Since then, investors have agreed to participate in many markets. Since 2005, government securities have been issued on the stock exchange and the interbank market or all three at the same time.

Method

Model of interest rate loan provides relationship between last day and last day of debit card[6]. Generally, long-term contracts are more profitable than short-term loans, so the interest rate curve gradually increases, but sometimes short-term loans are more profitable because the yield is longer. Term bonds and curves are trending downward. In other cases, a bump or other shape appears in the middle of the curve. There are four basic economic theories that explain the structure of interest rates: expectations theory, liquidity theory, market segmentation theory, and preference theory. The main purpose of this theory is to explain the relationship between growth and development contracts in the context of equity, that is, differences in spending time pattern curves. Of the above four theories, the expectation of thought is the most classical and widely discussed in research[7].

In general, expectations assumptions assume that the interest rate structure of a loan is closely related to future interest rate expectations. The theory of expectation began to be concerned by the academic circles from Fisher (1986). Later, it was discussed by other scholars, such as Fisher (1930), Williams (1938), Lutz (1940) and Hicks (1946), and gradually improved and clarified. Now the expectation hypothesis plays an important role in the theoretical and empirical research of fixed income securities. In the initial discussion (Macaulay 1938; Hicks 1939; Lutz 1940), the expectation theory did not get any formal expression from the economic equilibrium model, but was just a hypothesis. For this reason, the expectation hypothesis of term structure has gradually evolved into a variety of expressions and mathematical expressions, but the core purpose is to understand the yield to maturity and return on holding period of bonds with different maturities. Through the unremitting efforts of scholars (Cox, Ingersoll and Ross 1981; Campbell 1986; McCulloch 1993), the expectation hypothesis has been extended to three basic theoretical models and has corresponding mathematical expressions.

According to Cochrane (2005), the three forms of expectation hypothesis are:

All bonds with different maturities have the same holding period yield. From the perspective of excess return, the equivalent expression is that the return on excess holding period of long-term bonds relative to short-term bonds is equal to zero. The rate of return of a single holding period can be expressed as shown in formula (1): Et(hprtt+1)=yt {E_t}\left( {hp{r_{t \uparrow t + 1}}} \right) = {y_t}

This expression of the expectation hypothesis is approximately equal to the concept of risk neutrality. The difference is that the model here adopts the logarithmic yield of bonds rather than the real horizontal yield. According to Jensen inequality, a variance adjustment item needs to be added to the real holding period yield level. This equation implies that investors will adjust the proportion of bonds with different maturities in the portfolio until the expected single-term yields of all bonds are equal.

The yield to maturity of phase τ bonds is the average of the yield to maturity (spot interest rate) of a single period in the future, which is shown in formula (2): yt(τ)=1τEt[yt(1)+yt+1(1)++yt+τ1(1)] y_t^{\left( \tau \right)} = {1 \over \tau }{E_t}\left[ {y_t^{\left( 1 \right)} + y_{t + 1}^{\left( 1 \right)} + \ldots + y_{t + \tau - 1}^{\left( 1 \right)}} \right]

This equation reflects the trade-off between investors' two investment strategies. One is to directly buy a phase τ bond. The other is to buy single-issue bonds with a maturity of 1, and invest after maturity. In an equilibrium state, the gains from these two methods should be equal. The forward interest rate is equal to the expected spot interest rate in the future. See formula (3): ftτ1τ=Et[yt+τ1(1)] f_t^{\tau - 1 \uparrow \tau } = {E_t}\left[ {y_{t + \tau - 1}^{\left( 1 \right)}} \right]

This equation also reflects the investment decision of whether investors lock in the future lending rate in advance. That is, you can buy forward contracts in the current period or lock the capital lending interest rate between and in advance through the purchase and sale of discount bond portfolio, and you can also borrow at through the spot interest rate. Both methods should give the same expected return.

The expectation theory of the above three expressions is regarded as pure expectations. Hypothesis, which ignores the possible risks of different trading strategies represented on both sides of the equation. Specifically, the volatility of the holding period yield of long-term bonds is higher than that of short-term bonds. The holding period yield of single-term bonds is equal to its yield to maturity, while the yield of long-term bonds is affected by the selling price at the end of the period; If the bonds are nominal bonds, rolling investment with short-term bonds is safer than direct long-term bond investment, because it can reflect the impact of inflation in time; The forward interest rate is currently known, while the future spot interest rate is not [8]. In order to overcome these risks, rational investors will actually increase a risk premium. The modern expectation hypothesis holds that the risk premium is constant, and a fixed risk premium is usually added to the right of the above three mathematical equations.

The scale of China's government bond market has developed rapidly since 2005. By the end of July 2014, the closing balance of China's government bonds had increased from nearly 2.6 trillion yuan to 10 trillion yuan, an increase of nearly four times. Figure 1 shows the main monthly trends in the structure of spot interest rates for treasury bonds in China's stock market and interbank market from January 2005 to December 2012[9]. Intuitively, first of all, we can see that the short, medium and long-term interest rates in China's securities exchange market and inter-bank market have fluctuated sharply from 2005 to 2009; After the financial crisis until the end of 2012, the interest rate level of all terms in the securities market tends to be stable, while the interest rate level in the inter-bank market still fluctuates greatly. Another obvious feature is that the term structure of interest rate has an upward slope most of the time, that is, the interest rate level gradually increases with the maturity, but there is also a downward slope, that is, the interest rate hangs upside down, or even a downward and upward change process.

Figure 1

Spot interest rate term structure of government bonds in China's securities market

Looking at the development of the single financial market in 2005–2006, China's stock market began to flourish at the beginning of the sub-crisis in 2005–2008. Since 2006, the central bank has raised deposit rates seven times in a row to increase market liquidity, which has led to the decline of bond prices and the rise of interest rates. Subsequently, the U.S. subprime mortgage crisis began to spread to the global economy. In October 2008, the central bank cut interest rates sharply to prevent a recession and a loss of liquidity in the Chinese market. As a result, bonds rose and interest rates fell sharply. In the stage of financial crisis, in addition to the monetary policy of the central bank, there is also the stimulus of fiscal policy, the most important of which is the package of economic stimulus policies with a total amount of 4 trillion. Due to the lag of policy stimulus, there was obvious inflation in China in 2010, and the rise of CPI and real estate market prices became a new hidden worry for economic development. Then the central bank raised deposit interest rates for five consecutive times, tightened liquidity and prevented excessive flow of credit funds to the real estate market. Accordingly, the interest rate level of the bond market has also changed, the most obvious of which is the short-term and medium-term government bonds in the inter-bank market [10]. The reason is largely related to the participants in the two markets. In the inter-bank market, the central bank and major financial institutions including commercial banks are the main direct participants. In addition to announcing the benchmark interest rates for bank deposits and loans, the central bank can also directly operate in the open market of the inter-bank market to exert influence on the interest rates in the inter-bank market. Banks and other financial institutions. The public is the main trader of Treasuries in the foreign exchange market. The central bank's monetary policy is unlikely to directly affect their demand for short- and medium-term bonds.

The conditional moment characteristics of the term structure of interest rates are investigated mainly from the perspective of autocorrelation and conditional variance. The autocorrelation characteristics are mainly measured by the autocorrelation coefficients of bond interest rates of different maturities. Conditional variance characterizes the volatility of bond yields, and is mainly measured by constructing a simple first-order VAR system, as shown in Table 1.

Half-life of the term structure of interest rates

the term 1 2 3 4 5 6 7 8 9 10
stock market interbank market 7.15 11.55 10.73 9.78 9.27 8.65 7.95 7.59 7.09 6.95
the term 7.08 10.08 8.83 7.35 6.66 6.78 6.87 7.56 7.23 7.59
stock market interbank market 11 12 13 14 15 16 17 18 19 20
the term 6.89 6.25 5.69 5.48 5.26 4.88 5.95 5.26 5.69 5.81
stock market interbank market 7.25 7.3 7.05 6.89 6.59 5.69 5.23 4.59 4.95 3.59
the term 21 22 23 24 25 26 27 28 29 30
stock market interbank market 3.68 3.58 3.14 3.36 3.26 3.244 3.01 2.7 2.07 1.58

The data show that the autocorrelation of yield to maturity in the securities market and the interbank market is generally high, but has a weakening trend with increasing maturity.

Experiment and discussion

In order to further understand the nature of the conditional variance of bond yields, a simple least squares regression was performed on the relationship between the conditional variance and the level of yield-to-maturity bonds, and the autocorrelation coefficient of the conditional variance of bond yields was calculated. The results are shown in Table 2. The results show that the regression coefficient of the conditional variance of the rate of return in the securities market to the level of the rate of return is positive, and most of the regression coefficients in the inter-bank market are negative, but the significance of the regression coefficient is not high. This shows that the volatility of Chinese government bond yields has no significant correlation with the level of yield to maturity. From the perspective of the autocorrelation coefficient, the autocorrelation of the conditional variance of the rate of return increases with the increase of the maturity period, but the size of the standard error shows that most of the autocorrelation coefficients are not significant. This shows that there is no significant autocorrelation in the risk of my country's government bond yields, at least for short-term bonds.

Properties of the Conditional Variance of Bond Yield to Maturity

stock market
the term 1 2 3 4 5
slope 0.325 0.0256 0.012 0.018 0.023
standard error 0.023 0.135 0.009 0.008 0.009
R2 0.023 0.008 0.025 0.059 0.076
autocorrelation coefficient −0.024 −0.016 0.093 0.239 0.321
standard error 0.036 0.035 0.088 0.069 0.098
interbank market
slope 0.008 −0.006 −0.008 −0.006 0.001
standard error 0.023 0.014 0.012 0.013 0.015
R2 0.001 0.002 0.009 0.006 0.002
autocorrelation coefficient 0.143 0.079 0.074 0.118 0.102
standard error 0.107 0.083 0.007 0.073 0.078

Assume the logarithm of a zero coupon bond with a V term, the yield to maturity, forward interest rate and holding period yield of the bond can be expressed as follows (4,5,6): yt(τ)=1τpt(τ) y_t^{\left( \tau \right)} = {1 \over \tau }p_t^{\left( \tau \right)} ft(τ1τ)=pt(τ1)pt(τ) f_t^{\left( {\tau - 1 \uparrow \tau } \right)} = p_t^{\left( {\tau - 1} \right)} - p_t^{\left( \tau \right)} hprtt+1(τ)=pt+1(τ1)pt(τ) hpr_{t \uparrow t + 1}^{\left( \tau \right)} = p_{t + 1}^{\left( {\tau - 1} \right)} - p_t^{\left( \tau \right)}

This chapter also selects the term structure data of interest rates in the securities market and inter-bank market from January 2005 to the end of December 2012. However, in order to compare with the data of the same period in the United States and ensure the robustness of the estimation results of the model, only the yield to maturity data with a maturity of 1–5 years is selected [11]. It should be noted that the variables used in this chapter are calculated according to the logarithmic price of zero coupon bonds. The data of the U.S. market are directly from Wharton Wrds financial database, while the data of the Chinese market are calculated according to the yield to maturity, and then take the logarithm. A4.1 to A4.3 in the appendix describe the time-varying characteristics of the forward interest rate and the holding period yield (2 to 5 years) of long-term bonds in China's securities and inter-bank market [12].

Firstly, the regression results of the excess return of bonds with different maturities on all forward interest rates are given, and then the coefficient γ of the linear combination of forward interest rates and the regression coefficient of the excess return of bonds with different maturities on CP single factor are obtained by using the two-step regression method. Finally, in order to test the prediction effect of CP single factor model, it is compared with Fama bliss's classical regression.

The term bond interest rate curve refers to the nature of the change in the yield to maturity of bonds of different maturities, which reflects the cost of borrowing capital in financial markets [13]. The interest rate structure of zero-coupon bonds is the changing law of zero-coupon bonds of different maturities before maturity, which reveals the relationship between the cash flow on the maturity date of bonds and the original investment. Because there is no default risk, the interest rate term structure of zero interest bond calculated according to national debt becomes the basic interest rate term structure, which is often used as the discount rate index in the financial asset pricing and project investment decision-making model. The reason why the term structure of interest rate has become an important issue in the field of finance research is mainly due to the following three reasons: first, the term structure of interest rate itself contains a lot of economic information, which can be used to predict real economic activities, inflation level, household savings and consumption behavior (Fama, 1990; Mishkin, 1990); Second, the term structure of interest rate has an important impact on asset pricing, project investment and financing decisions. The reasonable price can be judged by discounting the future cash flow of financial assets or projects with the term structure of zero interest bonds; Third, the monetary policy and transfer mechanism implemented by the central bank can be studied by analyzing how the term structure of loan interest rates affects long-term interest rate returns under the current economic situation [1415].

Conclusion

The interest rate term structure of bonds is also known as the yield-to-maturity curve, which depicts the characteristics of the bond's yield to maturity that varies according to the maturity. Laws that vary by time limit. Since government bonds have little risk of default, the term structure of interest rates for zero-yield bonds not only accurately depicts the relationship between the cash flows generated by the bond's maturity and the original investment, but also, as a discount rate, the value of other future occurrences or cash flow assets, For example stocks, measurable options, etc. This paper first analyzes these data and discusses the basic characteristics of the dynamic changes in the term structure of interest rates in China. The results of the analysis found that the entire term structure of China's securities market and inter-bank bond market. Influenced by the financial policy of the central bank, this feature is very obvious in the short-term interest rates in the interbank market: and the shape of the interest rate term structure of bonds is not all upward sloping, but the phases of frequent financial policy adjustments (such as 2007) Long-term bonds Backlash, with yields lower than mid- and long-term bond rates, does occur.

Figure 1

Spot interest rate term structure of government bonds in China's securities market
Spot interest rate term structure of government bonds in China's securities market

Properties of the Conditional Variance of Bond Yield to Maturity

stock market
the term 1 2 3 4 5
slope 0.325 0.0256 0.012 0.018 0.023
standard error 0.023 0.135 0.009 0.008 0.009
R2 0.023 0.008 0.025 0.059 0.076
autocorrelation coefficient −0.024 −0.016 0.093 0.239 0.321
standard error 0.036 0.035 0.088 0.069 0.098
interbank market
slope 0.008 −0.006 −0.008 −0.006 0.001
standard error 0.023 0.014 0.012 0.013 0.015
R2 0.001 0.002 0.009 0.006 0.002
autocorrelation coefficient 0.143 0.079 0.074 0.118 0.102
standard error 0.107 0.083 0.007 0.073 0.078

Half-life of the term structure of interest rates

the term 1 2 3 4 5 6 7 8 9 10
stock market interbank market 7.15 11.55 10.73 9.78 9.27 8.65 7.95 7.59 7.09 6.95
the term 7.08 10.08 8.83 7.35 6.66 6.78 6.87 7.56 7.23 7.59
stock market interbank market 11 12 13 14 15 16 17 18 19 20
the term 6.89 6.25 5.69 5.48 5.26 4.88 5.95 5.26 5.69 5.81
stock market interbank market 7.25 7.3 7.05 6.89 6.59 5.69 5.23 4.59 4.95 3.59
the term 21 22 23 24 25 26 27 28 29 30
stock market interbank market 3.68 3.58 3.14 3.36 3.26 3.244 3.01 2.7 2.07 1.58

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