What Is a Term Structure of Interest Rates?

Term Structure of Interest Rates refers to the relationship between the yield of funds with different maturities (Yield) and maturity at a certain point in time. The term structure of interest rates reflects the relationship between the supply and demand of funds at different terms, reveals the overall level and direction of changes in market interest rates, and provides a reference for investors to engage in bond investment and strengthen government bond management.

Interest rate term structure

Term Structure of Interest Rates

1 What is the term structure of interest rates

2 Theory of term structure of interest rates

3 Term structure model and classification of interest rates

4 Empirical analysis of the term structure of interest rates in China

What is the term structure of interest rates

Strictly speaking, the term structure of interest rates

Four types of interest rate term structure theory:

Anticipation theory: Anticipation theory proposes the following proposition: The interest rate of long-term bonds is equal to the geometric mean of short-term interest rates that people expect during their validity period.

The key assumption of this theory is that bond investors have no particular preference for bonds with different maturities, so if the expected return of a bond is lower than other bonds with different maturities, investors will not hold this Bonds. Bonds with this characteristic are called complete substitutes. In practice, this means that if bonds of different maturities are a complete substitute, the expected returns on those bonds must be equal.

Expectation theory can explain facts

1. Over time, bond interest rates with different maturities have a tendency to move in the same direction. Historically, the short-term interest rate has the characteristic that if it rises today, it will tend to be higher in the future.

2. If the short-term interest rate is low, the yield curve tends to tilt upward, and if the short-term interest rate is high, the yield curve is usually reversed.

The theory of expectation has a fatal flaw. It cannot explain fact 3, that is, the yield curve usually slopes upward.

Segmented market theory: The segmented market theory treats bond markets with different maturities as completely independent and mutually segmented. The interest rate of each bond with a different maturity period depends on the supply and demand of the bond, and the expected returns on other maturity bonds have no effect on this. The key assumption: bonds with different maturities cannot be replaced at all.

The theory holds that, due to legal, preference, or other factors, neither investors nor bond issuers can realize the free transfer of funds between securities with different maturities at no cost. Therefore, the securities market is not a unified and indifferent market, but there are short-term markets, medium-term markets, and long-term markets.

Interest rates in different markets are determined by supply and demand in each market. When the intersection of the long-term bond supply curve and the demand curve is higher than the intersection of the short-term bond supply curve and the demand curve, the bond's yield curve slopes upward; on the contrary, the opposite is true.

Liquidity premium theory: Liquidity premium theory is the product of the combination of anticipation theory and market segmentation theory. It believes that the interest rate of long-term claims should be equal to the sum of the average value of short-term interest rates expected before the maturity of long-term claims and the liquidity premium that changes as bond supply and demand conditions change. The key assumption of the liquidity premium theory is that bonds with different maturities can be substituted for each other, which means that the expected rate of return of a certain bond will indeed affect the expected rate of return of other maturity bonds. However, the theory acknowledges that Investor preferences for bonds with different maturities. In other words, bonds with different maturities can replace each other, but they are not a complete substitute.

Deadline first theory: A more indirect method was adopted to modify the theory of expectations, but the conclusions reached were the same. It assumes that investors have a particular preference for bonds with certain maturities, that is, they are more willing to invest in bonds with such maturities.

Comprehensive overview of interest rate term structure theory:

The term structure theory of interest rates explains why there are differences in the spot interest rates of various government bonds, and this difference varies with the length of the term.

Expectation hypothesis

Expectation Hypothesis: The expectation hypothesis of the term structure of interest rates was first proposed by Irving Fisher (1896) and is the oldest term structure theory.

The expectation theory holds that the current interest rate of long-term bonds is a function of the expected interest rate of short-term bonds, and the relationship between long-term interest rates and short-term interest rates depends on the relationship between current short-term interest rates and future expected short-term interest rates. If E t ( r ( s )) is used to represent the expectation of the spot interest rate at the time t in the future, then the expected return on maturity can be expressed as:

Return to maturity expression

If the expected future short-term bond rate is equal to the current short-term bond rate, then the long-term bond rate is equal to the short-term bond rate, and the yield curve is a horizontal line; if the expected future short-term bond rate rises, the long-term bond rate must be Higher than the current short-term bond interest rate, the yield curve is an upward sloping curve; if the expected short-term bond interest rate is reduced, the longer the bond's maturity, the lower the interest rate, the lower the yield curve.

The main flaw of this theory is the strict assumption that people have certain expectations of future short-term bond rates. Second, the theory also assumes that the flow of funds between the long-term and short-term funds markets is completely free. Both assumptions are too idealistic, and are far from the actual financial markets.

Segmentation theory

Market segmentation theory: The expectation hypothesis provides an explanation of why the interest rates of bonds with different maturities are different. But the expectation theory has a basic assumption that expectations for future bond rates are certain. If expectations for future bond rates are uncertain, the expectation hypothesis will no longer hold. As long as the future interest rates of the bonds are uncertain, various bonds with different maturities cannot be completely replaced with each other, and funds cannot flow freely between the short- and long-term bond markets.

The market segmentation theory holds that the bond market can be divided into unrelated markets with different maturities, each with its own independent market equilibrium. Long-term borrowing activity determines the long-term bond interest rate, while short-term transactions determine the short-term interest rate independent of long-term bonds. According to this theory, the term structure of interest rates is determined by the equilibrium interest rates of different markets. The biggest flaw of the market segmentation theory lies in its clear claim that bond markets with different maturities are irrelevant. Because it can not explain the phenomenon of synchronous fluctuations in interest rates of bonds with different maturities, nor can it explain the apparent regular changes in the interest rates of the long-term bond market as the interest rate fluctuations of the short-term bond market.

Preference hypothesis

Liquidity Preference Hypothesis: Keynes first proposed the relationship between the degree of risk of different maturity bonds and the structure of interest rates. Hicks based on Keynes had a more complete theory of liquidity preference.

According to the liquidity preference theory, there is a certain degree of substitution between bonds with different maturities, which means that the expected return of a bond can indeed affect the yield of bonds with different maturities. But bonds with different maturities are not completely replaceable, because investors have different preferences for bonds with different maturities. Van Home believes that in addition to the expected information, the forward rate also includes risk factors, which may be a compensation for liquidity. Factors affecting deducted compensation for short-term bonds include: the availability of bonds with different maturities and the degree of investor preference for liquidity. In bond pricing, liquidity preferences cause price differences.

This theory assumes that most investors prefer to hold short-term securities. In order to attract investors to hold longer maturity bonds, they must be paid liquidity compensation, and the liquidity compensation increases with time, so the actual observed yield curve is always greater than expected by the expected hypothesis high. This theory also assumes that the investor is a risk-averse, and he will only make risky investments after receiving compensation. Even if investors expect the short-term interest rate to remain unchanged, the yield curve will also slope upward. If R (t, T) is the yield to maturity of the bond due at time T, E t ( r ( s )) is the expectation of the spot interest rate at the future time at time t, and L (s, T) is the time to mature at time T The instantaneous maturity premium of the bond at time s, according to the expected theory and liquidity preference theory, the yield to maturity is:

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From the three theories of the term structure of interest rates, the formation of the term structure of interest rates is mainly determined by the expectations of the direction of future interest rate changes.

Structural model

The term structure model of interest rates can be divided into single-factor models and multi-factor models according to the number of random factors included in the model.

The single factor model contains only one random factor, which means that the random factors at each point on the return curve are completely correlated. The multi-factor term structure model involves multiple random factors, indicating that the random factors at different points on the return curve have a certain degree of correlation. This classification method is simple and clear, and is widely accepted by the academic community. In addition to this classification method, it can also be classified according to the equilibrium basis of the term structure model of interest rates, that is, the no-arbitrage opportunity model and the general equilibrium model.

Compare

The general equilibrium model and the arbitrage-free opportunity model and their more important equilibrium models include the Vasicek model, the CIR model, and the double square root model. The stochastic differential equations of the instantaneous short-term interest rates of the three models are:

Hu and Lee Model

Hu and Lee model: d r ( t ) = ( t ) d t + a d w ( t ), is a normal number,

Hu and Lee Model

.

Black-Kalashinski model

Black-Karasinski model: dl n ( r ( t )) = [ ( t ) ( t ) l n ( r ( t ))] + ( t ) d w ( t ).

HJM model

HJM model: d f ( t , T ) = ( t , T ) d t + ( t , T , f ( t , T )) d w ( t ). Here w (t) is the standard Brownian motion.

The partial derivative in the Hu and Li model represents the slope of the initial forward interest rate curve f (0, t) at time t. It is this time-parameter function that makes the bond prices priced by the Hu and Li models consistent with the observed market bond prices. But this term structure model does not have the property of mean recovery, and the probability of negative interest rates is greater than zero. In the well-known Black and Karasinski (1991) log-normal interest rate term structure model, ( t ), ( t ), and ( t ) are deterministic functions of time parameters The selection of these parameters requires the model to accurately fit the term structure of the initial interest rate and the market volatility curve. Because the model contains the logarithm of interest rates, not only eliminates the possibility of negative interest rates, but it keeps interest rates away from zero interest rate values. (T, T) and ( t , T , f ( t , T )) in the Hess, Garrow, and Morton models (HJM) are the forward interest rate trend coefficients and diffusion coefficients at time T due.

Although the equilibrium model directly gives the dynamic evolution of short-term interest rates, it does not require that the price of zero-coupon bonds estimated according to the term structure model must conform to market prices. Why is there a difference between the model's estimated price and the market price of the bond? This is mainly because the factors affecting bond prices are not just short-term interest rates. Although the no-arbitrage opportunity model also gives the dynamic evolution process of the term structure of interest rates, it requires that the term structure given by the model must conform to the prevailing interest rate term structure of the market. Therefore, as long as the arbitrage-free maturity structure model is correctly given, the pricing of zero-coupon bonds according to the model must conform to the market price at that time, otherwise there will be arbitrage opportunities.

From the perspective of obtaining data from the two types of models, the equilibrium model mainly uses past historical data for statistical analysis, estimates the model's trend coefficient and volatility structure coefficient, and derives the dynamic evolution of the term structure of bond prices and interest rates. The no-arbitrage opportunity model requires information on the term structure of the spot interest rate, which is easy to obtain, and the no-arbitrage opportunity model can be adjusted in time based on the information on the term structure of the market interest rate. Therefore, the equilibrium model is very suitable for predicting the dynamic process of the term structure of bond prices and interest rates. Researchers can use the equilibrium model to understand the relationship between the shape of the term structure curve and the forecast of future economic conditions, but they cannot guarantee that the term structure model established using historical data can meet the actual evolution process later. The arbitrage-free opportunity model can be directly applied to market transactions, because the bond price and interest rate term structure of the theoretical model is consistent with the market's bond price and interest rate term structure.

From the perspective of the internal consistency of the two types of models, the parameters of the general equilibrium model are statistically analyzed and estimated from historical data accumulated over a long period of time. Therefore, the model's trend coefficient, fluctuation structure coefficient, and mean response value will not change every day. The parameter values can maintain a certain stability. Even if new market data is re-injected according to market changes, it will not have a significant impact on the size of trend parameters and fluctuation parameter values, so that the equilibrium model can maintain a certain consistency for a period of time. Sex. The no-arbitrage opportunity model needs to assume the trend variable, the volatility structure, and the average interest rate recovery, but at two different times, the parameters set by the model are unlikely to be consistent, unless the parameters adjusted by the market data itself exactly match Some consistency. Because the arbitrage-free opportunity model needs to be constantly adjusted according to changes in market conditions, that is, it is necessary to adjust parameters frequently, so that the model of zero coupon bonds can be used to fit the estimated price curve and market price curve, and the model of the interest rate term structure curve and market term structure curve To the best degree.

Comparison of single-factor model and multi-factor model The aforementioned equilibrium model and arbitrage-free opportunity model are both single-factor models. The single-factor model is simple in form, has a small number of parameters, is easy to estimate, and is relatively simple to apply.

Single factor

1) The single-factor model has poor flexibility, and it is difficult to reflect the dynamics of the actual yield curve and interest rate term structure of various possible zero-coupon bonds. Because the single-factor model includes only one factor that affects the dynamics of interest rates in the model, this is clearly inconsistent with reality. Economists have researched and found that at least three factors are needed to fully explain changes in interest rates. Studies by Litterman and Scheinkman show that a single factor (short-term interest rate) can only explain about 90% of changes in US Treasury rates. Jamshidian and Zhu used principal component analysis or factor analysis to analyze the historical data of the entire yield curve using data from the yen, the US dollar, and the German mark. The two principal component factors Only 85% to 90% of the change in the yield curve can be explained. One principal component factor can explain 68% to 76% of the total change in the yield curve, and three principal component factors can explain 93% to 94% of the total change in the yield curve. .

2) The single factor model implicitly assumes that all possible zero coupon bond rates are completely correlated.

3) The short-term bond pricing error using the single factor model is relatively small. However, if the single-factor model is used to price longer-term bonds, a larger error will occur. At this time, the multi-factor model is more suitable for pricing. In general, the error between the theoretical price and the actual market price estimated by the single factor model will exceed 1%, which is barely acceptable; however, if the single factor model is used to price derivative securities, the error will reach 20% At 30%, it is unacceptable.

The multi-factor model assumes that the dynamic evolution of the term structure of interest rates is driven by several factors. These factors can be the impact of the macro economy or the status of the yield curve itself, such as the level of income, the slope of the yield curve, and the curvature of the yield curve. They can also be short-term interest rates, fluctuations in short-term interest rates, and long-term interest rates. The main multi-factor models are Langnstaff and Swartz two-factor models, Brian and Swartz two-factor models, Schaefer, Anna Jacobson Schwartz ) 'S Schefa and Swartz models, Cheyne's three-factor model, and Barduz's three-factor model.

Multifactor

1. Because the multi-factor model includes a large number of parameters, the workload of establishing a multi-factor model is extremely heavy, and it is extremely difficult to estimate and calibrate the parameters.

2. The form of the model is complex and there are many parameters. It is often difficult, sometimes even impossible, to formulate clear calculation formulas for bond prices. Therefore, it is necessary to repeatedly execute the error minimization procedure when fitting the yield curve with a substitute function. .

3. When using a multi-factor model to price derivative securities, it is generally necessary to use numerical calculation methods to obtain the price of derivative products such as options. Only the Langstaff and Schwarz two-factor model can be launched with expiration time and exercise price The formula for calculating the price of an option.

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Empirical analysis

An Empirical Analysis of the Term Structure of Interest Rates in China

Introduction

In the field of investment in fixed income securities, the analysis of the term structure of interest rates is an important means. According to the formula for calculating the yield to maturity of bonds announced by the People's Bank of China, the actual yield maturity structure of China's government bonds can be obtained. The types of government bonds selected in the analysis of China's national debt maturity structure include 99 national bonds 5, 00 national bonds 7, 01 national bonds 2, 01 national bonds 14, 02 national bonds 6, 02 national bonds 7, and so on. The yield curve of these treasury bonds on February 28, 2003 is shown in Figure 1 below:

Such a yield curve cannot be explained clearly with the expectation hypothesis, nor can it be explained clearly with the liquidity preference theory. Liquidity preference theory assumes that investors are risk-averse, and they all prefer to hold short-term securities. Therefore, for investors to invest in long-term bonds, liquidity compensation must be paid to investors. This means that long-term interest rates are equal to the sum of short-term interest rates and liquidity compensation. Therefore, according to the theory of expectation or the theory of liquidity preference, only the upward slope, downward slope, and level of the yield term structure can be explained. But this phenomenon can be explained by market segmentation theory.

Segmentation theory

According to the market segmentation theory, the bond market is composed of unrelated markets with different maturities, and the interest rates of these markets are determined by independent market supply and demand. Therefore, bonds with different maturities cannot be completely replaced with each other, and funds will not flow freely between the short- and long-term bond markets. In this way, because the supply and demand of bonds with different maturities are different, the liquidity compensation obtained according to the maturity of the bonds will form an irregular sequence. This irregular liquidity compensation sequence combined with short-term interest rates will form an intermediate bulge yield term structure curve.

Select the 1-, 2-, and 4-week Treasury bond repurchase rates in the interbank government bond repo market from January 1998 to February 2003 to return to three Wasiseck models:

1 week model: dr (t): 2.011548 (0.022496-r (t)) + 0.010703 * dw (t) 2 week model: dr (t) = 1.570225 (0.021726-r (t)) + 0.008424 * dw (t ) 4-week model: dr (t) = 1.07l929 (O.019679-r (t)) + 0.005865 * dw (t) Zero-coupon bond yields simulated based on l-week, 2-week, and 4-week treasury bond repurchase rate models The term structure curve is shown in Figure 2:

In Figure 2, from top to bottom, the zero-term bond maturity structure simulated according to the regression models of the 1-, 2-, and 4-week government bond repurchase rates, respectively. The zero-coupon bond yield curve simulated according to the l-week model is slowly rising. The zero-coupon bond yield curve simulated according to the 2-week model is approximately a horizontal line, while the zero-coupon bond yield curve simulated according to the 4-week model is slowly decreasing. Yes, this represents three typical yield curves that fit the expectations theory. This may be that there are three different expectations among different investment groups in the Chinese government bond market, which is inconsistent with the expectation theory assuming that people have certain expectations for future short-term interest rates; it may also mean that there is market segmentation in China's government bond market and different markets. There are different expectations. From the regression model itself, the average recovery speed of the 1-week model and the fluctuation coefficient of short-term interest rates are the largest, indicating that the fluctuation of the national bond repurchase rate in 1 week is the most violent; the average recovery speed of the national bond repurchase rate and the fluctuation coefficient in 4 weeks are the smallest, indicating 4 The weekly Treasury bond repo rate has the slowest fluctuations.

The maturity structure model simulation and the actual national bond yield curve indicate that there is a market segmentation phenomenon in our national bond market. How to explain the market segmentation phenomenon in the Chinese government bond market? In China's bond market, the maturity structure of government bonds is too single. The proportion of short-term government bonds below one year and long-term government bonds over 10 years is too small. They are all medium-term national debt from 1 to 10 years. And different investors have different investment preferences for treasury bonds with different maturities. When no treasury bonds that meet their preferred investment maturity are found in the market, this investment demand will be transferred to treasury bonds with other maturities. This shift in demand will cause the investment demand for government bonds of certain maturities to be surprisingly high. The direct result is that the price of such bonds rises to a certain level, reducing its yield to maturity to lower than that of other bonds, and even It makes it difficult for liquidity compensation to make up for the decrease in the yield to maturity caused by the sharp increase in investment demand. In addition, the inconsistency of China's exchange market and inter-bank treasury bond market is also one of the reasons for market segmentation.

Overview

To solve this problem, manpower must be provided from several aspects. First, we must establish a unified government bond market, unify the existing interbank market and the exchange market, and remove barriers for investors to enter the market. In this way, the market competitiveness can be fully released, so that the level of national debt interest rates truly reflects the state of supply and demand of funds in the national bond market. Secondly, reform the current situation of the unreasonable maturity of the national debt. The long-, short-, and medium-term maturities of the national debt should be issued together to change the over-concentration of the time of issuance of the national debt. Using the practice of the United States, issuing national bonds every week is conducive to the formation of a complete national bond yield Rate curve.

curve

To better understand the yield of bonds, we introduce the concept of a yield curve. The yield curve is a curve formed by a combination of spot interest rates with different maturities. In practice, due to the tedious calculation of spot interest rates, there are also quite a few textbooks and operators who use the yield to maturity to characterize the term structure of interest rates.

basic type

From the shape point of view, the yield curve mainly includes four types. In the figure, (a) shows a rising interest rate curve, indicating that the longer the maturity of the bond, the higher the interest rate. This curve shape is called a "forward" interest rate curve. Figure (b) shows a declining interest rate curve, indicating that the longer the maturity of the bond, the lower the interest rate. This curve shape is called an "inverse" or "inverted" interest rate curve. Figure (c) shows a flat interest rate curve, indicating that the interest rates of bonds with different maturities are equal. This is usually a temporary phenomenon that occurs during the conversion of the positive interest rate curve and the inverse interest rate curve. Figure (d) shows the bulging interest rate curve, which shows that bonds with relatively short maturities have a positive relationship with maturity; bonds with relatively long maturities have an inverse relationship with maturity. From historical data, all four interest rate curves can be observed at different stages of the business cycle.