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The relation between the term structure of interest rates and
the stock-bond correlation in US
Cristian Danciu S1836099
Abstract
This paper examines the correlation between stock and bond returns. Furthermore, it analyzes the impact of the term structure of interest rates on the relationship between stock and bond markets. The results indicate that the correlation coefficient significantly varies through time. Moreover, I find that the long-term component of the term structure of interest rates has a negative influence on the stock-bond relationship. In contrast to this, both the short-term component and medium-term component have a positive impact on the correlation coefficient between stock and bond returns.
Keywords: stock-bond correlation, dynamic conditional correlation, term structure of interest rates
2 1. Introduction
This paper analyzes the correlation between stock and bond returns. Understanding the nature of the relation between stock and bond markets is important for several reasons. In the first place every individual or corporate investor that manages an asset portfolio needs to neutralize risk by means of diversification. Knowing not only the correlation between stocks and bonds but also the correlation between each asset in the investment portfolio is of crucial importance when performing an asset allocation. In the case of an asset portfolio that contains only stocks and bonds, the most important information needed in order to properly diversify it is the correlation between stocks and bonds. Markowitz (1952) was the first one to emphasize the importance of the relation between stock and bond returns in the process of diversification. When using this relation as an input for asset allocation, portfolio management and risk management increases the performance of the portfolio.
Secondly, the policymakers look at both the bond and stock markets. A monetary policy that does not take into account the behavior of the stock market when managing the bond yields, is not a rational one. Furthermore, every central bank behaves like a public investor as well. Consequently, the information about correlation between stock and bond returns is useful in the investment process.
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In addition to this, Chiang and Li (2008) utilize the Vanguard Total Bond Market Index Fund. This fund utilize an index that comprises government securities (Treasury and agency), mortgage-backed securities, asset-backed securities, corporate securities and international dollar-denominated issues in order to simulate the universe of bonds in the market.
In spite of this intensive research on the variation of this relation, little is known about the determinants of the dynamic nature. Only few of the above mentioned studies studies have analyzed the economic and financial variables that determine the values of the correlation coefficient. Moreover, no research has analyzed the impact of the term structure of interest rates on the relation between stock and bond markets. Taking into account the various natures of the determinants, the term structure of interest rates may have a significant impact on the dynamic of the correlation between stock and bond markets.
I use the following logic as the main reason for researching the influence of the term structure of interest rates on the stock and bond correlation coefficient. The term structure of interest rates presents the characteristics of the bond market. A change in the shape of the term structure will determine changes in the bond returns. Consequently this will determine a change in the correlation between stock and bond markets.
The term structure represents the relation between the interest rates and the time to maturity for a borrower in a given currency. The shape of the term structure of interest rates is usually an upward sloping asymptotical one. So the longer the maturity, the higher the yield but with decreasing marginal growth. However the shape of the term structure can vary over time due to specific events or economic decisions taken. Estimating the shape of the term structure of interest rates is of crucial importance in the economic environment since the decisions that are to be taken by both the authorities and individuals depend on the term structure.
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First, understanding this link is significant in performing a proper asset allocation and risk management. Furthermore, both the term structure and the correlation between stock and bond returns are important for the central banks in implementing a monetary policy. For example, when adjusting the level of interest rates, central banks closely analyze the currency market and stock market. Otherwise, there will be irrational decisions that could affect both the monetary and financial equilibrium.
Furthermore, the stock market returns depend on the level of interest rates. If the stock market returns decrease below the level of interest rates offered by the banks in the system, then the investors would move their funds to the banks. On the other hand, if the opposite happens, the banks will suffer of lack of funds. Even if the policymakers focus more on the term structure, they could also use information contained in stock and bond markets in order to estimate the level of economy growth and inflation. Knowing if different shapes of the term structure determine the change in the correlation between stock and bond returns helps both individuals and financial institution to get a deeper look into the financial mechanism. Analyzing together the correlation coefficient and the term structure of interest rates could have a positive effect on the financial environment from the decisions point of view.
There is a vast literature focusing on the correlation between stock and bond returns and also on the estimation of the term structure of interest rates. However, none of the existing papers bring together these two concepts from the cause – effect point of view.
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recent stock crashes in order to test the hypothesis that the correlation between stocks and bonds is positive before a crash and negative after. The analysis period is from 1945 to 2000, focusing on the last 15 years due to the higher frequency of stock crashes. Similar to my research, he uses S&P 500 index as a proxy for stock returns while for the bond market returns he focuses on Treasury Bonds. From the methodology point of view, this paper uses the event study method (Kritzman [1994]). This research concludes that during stock market crashes both the contagion and decoupling phenomena are present.
Ilmanen (2003) investigates the correlation between stock and bond returns over a very long period, from 1926 to 2001. First of all, he proves the dynamic nature of the stock-bond relation. This research uses the dividend discount model in order to figure out the drivers of the stock and bond returns. In doing so, he concludes that stocks and bonds are both subject to discount rate uncertainty. When analyzing the historical correlation between stock and bond returns, Ilmanen takes into account four key drivers of this relation: business cycle or growth outlook, inflation environment, volatility conditions, monetary policy stance.This paper concludes that stock-bond correlation are more likely to be negative when inflation is low, when growth is slow, equities are vulnerable and volatility high. He underlines the importance of stock-bond relation in government valuation. As a result, negative stock-bond correlation gives bonds great hedging properties. Similar to Gulko (2002), Ilmanen (2003) also investigates the decoupling or “flight-to-quality” phenomenon. He finds three periods of decoupling- near 1930, near 1960 and near 2000. So, economic growth and volatility shocks tend to push stock and bond prices in opposite directions causing the above mentioned financial phenomenon. With regards to inflation, when it has a low level, discount rates are more stable and growth uncertainty dominates making the correlation between stocks and bonds lower.
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period of analysis is from 1986 to 2000. Connolly et al. (2004) proves the time-variation of the stock-bond correlation by calculating the 22-trading day correlation coefficient. Similar to Gulko (2002) findings, during the stock market crash of October 1987 the correlation coefficient between stock and bond returns has a negative value. Moreover, this paper is interested in the impact of market uncertainty on stock-bond relationship during periods of sustained negative correlation like the above mentioned one. First, they find a negative relation between their uncertainty measures and the future correlation between stock and bonds. Second, they conclude that bond returns tend to be high (low), relative to stock returns, during days when implied volatility increases (decreases) substantially and during days when stock turnover is unexpectedly high (low).
Following these three studies, Andersson, Krylova and Vähämaa (2004) analyze the impact of macroeconomic expectations and perceived stock market uncertainty on the time-varying correlation between stock and bond returns. Similar to Gulko (2002) and Ilmanen (2003), they analyze decoupling or ”flight-to-quality” phenomenon between stocks and bonds. In contrast to the previous two researches, this paper focuses on two markets. They analyze the determinants of the stock-bond relationship on both the U.S. and German market. The stock returns are calculated using S&P 500 index for U.S. and DAX index for Germany. The bond returns for both U.S. and German market are obtained from the benchmark 10-year government bond indices. However the analysis period differs for the two countries. The sample period starts from January 1991 to April 2004 for U.S. and for Germany from January 1994 to April 2004. As a proxy for macroeconomic expectations this paper uses monthly data on inflation and growth expectations, they focus on the expected growth rates of consumer price indices (CPI) and real gross domestic product (GDP). My research is similar to this paper from two points of view. First, I also use a DCC GARCH (1, 1) model in order to calculate the correlation coefficient between stocks and bonds. This way I find supporting evidence of the dynamic nature of the stock-bond relationship. Second, the impact of the determinants is examined using a simple regression. Actually, the work of Andersson, Krylova and Vähämaa (2004) is the starting point of my paper.
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they were unable to find any relationship between economic growth expectations and stock-bond correlations.
Saleem (2008) investigates the nature of the stock-bond relationship in the emergingmarkets such as Russia. First, he uses a Constant conditional correlation GARCH model in order to test for the constant nature of the stock-bond relation. However, the correlation coefficients are not statistically significant. These results are a motivation to use a DCC GARCH model that proves the dynamic nature of the correlation between stock and bond returns. Engle (2000) and Engle and Sheppard (2001) were the ones to describe the theoretical and empirical properties of DCC models. They used this model in order to estimate the conditional covariance of 100 assets using S&P 500 Sector Indices and Dow Jones Industrial Average stocks.
I focus my research on the United States since the current financial crisis that caused some important changes in the economic environment originated from that market. In particular, I analyze if the numerous changes in interest rates have an impact on the correlation between stock and bond returns. Therefore, the purpose of this paper is to analyze whether changes in the shape of the yield curve affect the above-mentioned relationship or not. I structure my research in three main parts. First, I am calculating and analyzing the value of the correlation coefficient between stock and bond returns. In order to do this I use a DCC GARCH (1, 1) proposed by Engel (2000) and Engle and Sheppard (2001). Next, in the second step I investigate the term structure of interest rates. I use the Nelson and Siegel model so that I can calculate the coefficients corresponding to the long-term, short-term and medium-term component. Finally, I combine these two results for the main purpose of this research.
In the last step, I estimate a regression in order to examine the impact of the term structure components on the value of the correlation coefficient between stock and bond markets. This regression has as a dependent variable the correlation coefficient, while the independent variables are the coefficients of the long-term, short-term and medium-term components of the term structure of interest rates.
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Furthermore this could be the starting point of theories and papers that bring together two important financial concepts.
The remainder of the paper is structured as follows. In Section 2 I analyze the data set used in this research. Section 3 describes the methodology I utilize. I examine the empirical result of my study in Section 4 while Section 5 concludes.
2. Data
The data set comprises of daily observations on US stock and bond returns from January 1990 to April 2009. As a proxy for the stock returns I use the S&P 500 index while for the bond returns the procedure of creating a proxy is as follows. With regards to the S&P 500 observations, the data was taken from Datastream, while the yield curve rates were provided by the U.S. Treasury Department. I use the daily treasury yield curve rates in order to construct an equally weighted bond index, where each maturity has the same weight in the index. I utilize this methodology so that the new bond index that I constructed captures a wide range of maturities. This way I capture the complete maturity spectrum of the bond market in the U.S. Furthermore I wanted to research the implication of a new proxy for the bond returns. It is interesting to conclude whether a equally weighted index is better or not than the other proxies used in the previous researches.
The U.S. Treasury Department has information on 11 maturities: 1, 3, 6 months as well as 1, 2, 3, 5, 7, 10, 20, 30 years. The U.S. Treasury Department uses the interpolation method in order to calculate the rates corresponding to the above mentioned maturities. However, the 1 month yields were not available until the 30th of July 2001. Furthermore, the 20 year yield was introduced on 1st of October 1993 and the 30 year yield rate was not available from 19th of February 2002 to 8th of February 2006.
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Table 1 presents the descriptive statistics of the two indices returns used in my research. Table 1. Descriptive statistics for S&P 500 and Bond Indices
Mean Max Min Standard
deviation
Skewness Kurtosis Jarque Berra Probability of JB S&P 500 0.0002 0.1042 -0.0946 0.0117 -0.2772 12.5386 18277.51 0.0000 Bond Index -0.0003 0.1733 -0.1587 0.0142 -0.0416 22.6872 77599.62 0.0000
*Note: The descriptive statistics are for a sample of daily observations from January 1990 to April 2009.The bond index is an average weighted index using eleven maturities.
The stock returns vary from a maximum of 0.1042 to a minimum of -0.0946 while the bond index returns varies from a minimum of -0.1587 to a maximum of 0.1733. The mean of S&P index is positive and higher than the negative mean of the Bond Index even if the maximum of the Bond Index is slightly higher than the one of the other index. Furthermore the minimum of the Bond Index is lower, meaning that this index is more volatile than the stock index. With regards to the normal distribution, both indices are leptokurtotic and negatively skewed. In addition to this, after analyzing the values of JB and the associated probabilities, I conclude that the two series of returns are not normally distributed.
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Table 2. Descriptive statistics for S&P 500 and Bond Indices – five sub-samples
Mean Max Min Std. deviation
1st sub-sample S&P 500 0.0002 0.0366 -0.0372 0.0079 Bond Index -0.0004 0.1360 -0.0457 0.0061 Correlation Coefficient -0.1797 0.0919 -0.4297 0.1150 2nd sub-sample S&P 500 0.0007 0.0307 -0.0313 0.0070 Bond Index 0.0002 0.0595 -0.0380 0.0076 Correlation Coefficient -0.3901 -0.0083 -0.5698 0.11435 3rd sub-sample S&P 500 0.0002 0.0498 -0.0711 0.0131 Bond Index -0.0002 0.0329 -0.0561 0.0087 Correlation Coefficient 0.0570 0.4176 -0.4730 0.2239
4th sub-sample S&P 500 -2.91e-05 0.0557 -0.0504 0.0118
Bond Index -0.0002 0.0709 -0.0970 0.0108 Correlation Coefficient 0.2322 0.6508 -0.2063 0.2092 5th sub-sample S&P 500 -0.0004 0.1042 -0.0946 0.0159 Bond Index -0.0008 0.1732 -0.1587 0.0219 Correlation Coefficient 0.1854 0.6021 -0.2676 0.2530
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Analyzing Table 2 that describes the data of the new sub-samples I state that the volatility problem has been overcome. The volatility of the Bond Index is lower in all the sub-periods except the second and fifth sub-period where it is slightly higher than the volatility of the S&P 500 index. This could be due to the fact that on 1st of October 1993 the US Treasury Department includes the 20 years yield. In the case of the fifth period, the higher bond volatility is determined by the introduction of the 30 years yield on 8th of February 2006.
As I stated before, the higher Bond Index volatility is determined by the non continuous nature of the 1 month, 20 years and 30 years yields. Another way to solve this problem would have been the elimination of the discontinue yields from the calculation of the Bond Index. However, this would have narrowed the bond universe and would have not reflected the real properties of the bond market.
The descriptive statistics from the five sub-periods efficiently provide an indication of the dynamic nature of the correlation coefficient between stock and bond returns. The mean of the correlation coefficient is negative in the first two sub-periods while in the last three sub-periods takes positive values. As it will be further seen in Figure 1, the coefficient between stock and bond returns is very volatile. This is also proved by the values of the standard deviations. In the first two sub-periods the standard deviation of the correlation coefficient is slightly higher than 11% while the in last three sub-periods the value of the standard deviation is slightly higher than 20%.
12 3. Methodology
My research uses two models: the Dynamic Conditional Correlation GARCH (1, 1) model and the Nelson and Siegel three-factor base model. First, I use the DCC GARCH (1, 1) model proposed by Engel (2000), Engle, and Sheppard (2001) to model the time-varying co-movements between stock and bond returns. The initial assumption of this model is that the stock market returns from the k series are multivariate normally distributed with zero expected value and conditional variance-covariance matrix Ht, so the model can be specified as follows:
(1)
with εt | Ft-1 ~ Ht), where rt is the (k × l) vector of returns, εt is a (k × 1) vector of zero
mean return innovations conditional on the information Ft-1, available at time t-1 and for the
bi-variate case. Engle (2000) expresses the conditional variance-covariance matrix as:
, (2)
where Dt stands for the (k × k) diagonal matrix of the conditional volatility of the returns on each
asset included in the sample and Rt represents the (k × k) time varying correlation matrix. The
implementation of the DCC GARCH model involves two steps. First, the model estimates the mean equation for each asset in the sample embedded in a univariate GARCH model of its conditional variance. Therefore, Engle (2000) defines Dt in the following way:
, (3)
where hiit is the conditional variance of each asset. Furthermore, hiit is assumed to follow a
univariate GARCH (p, q) process, given by the following expression:
(4)
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In the second step, the standardized mean return innovations are assumed to follow a multivariate GARCH (m, n) process to picture the development of the time varying correlation matrix, Rt that can be described in the following equation:
, (5)
where is a (k × k) symmetric positive definite
matrix with , Q* is the (k × k) unconditional variance matrix of and α and β are non-negative scalar parameters that satisfy α+ β<1. In what concerns the conditional correlation coefficient between two assets i and j, this is and can be obtained using the following equation:
DCC GARCH model can be estimated using the quasi-maximum likelihood method (QMLE). Engle (2000) describes this method in the following way:
(7)
Taking into account the fact that then the log-likelihood function can be rewritten as is given below:
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important feature of this model is the fact that the Bank of International Settlements works with this model in order to describe the term structure of interest rates for different countries.
The spot rate curve, as specified by the Nelson and Siegel model is described by the following equation:
, (9)
where τ represents the time to maturity. Furthermore λt is set to a predetermined value. As
Diebold and Li(2006) I fix λt equal to the value of 0.0609.
This model has several properties that make it popular and frequently used in financial practice. First of all, it uses a small number of parameters that ease the estimation procedure. Moreover, these parameters allow the model to represent a variety of monotonic, humped and S-type shapes usually encountered in real data. In the second place, the model estimates forward and yield curves which have as a starting point the short rate value of β1,t + β2,t that can be easily
computed. Next, the model levels off at a constant infinite-maturity of β3,t.. These properties can
be described as follows:
and (10) Finally, Nelson and Siegel model clearly specifies the short, medium and long-term components. This can be described as the contribution of each element to the final shape of the yield curve. The long-term component has the value of 1 and is the component of β1,t. So this
component has a constant value for each maturity. is the component of β2,t and it describes the short-term element. It starts at the value of 1 and then decreases exponentially with the maturity. The medium-term component is and it creates a hump-shape by starting at the value of zero, increases for medium-term maturities and decays at zero with long-term maturities. In what concerns λt, this decay parameter determines at which
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In the final step of my research, I use the results obtained in the previous two steps. I utilize the correlation coefficients obtained after applying the DCC GARCH (1, 1) model and the values of β1,t , β2,t and β3,t resulted from the Nelson-Siegel model. In order to analyze the impact of the
term structure of interest rates on the correlation between stock and bond markets I estimate the following regression:
, (11) where ρij,t is the time-varying correlation coefficient , β1,t , β2,t and β3,t are the time-varying
coefficients that describe the term structure of interest rates and is the error term. I am interested in the values of in order to determine if the term structure of interest rates is a valid determinant for the correlation between stock and bond returns. Furthermore, I analyze these values from the economical point of view. I estimate this equation using the Least Squares method.
4. Results
I use a three-step analysis of the results of my research. First, I present the descriptive statistics of the correlation coefficient between stock and bond markets. This coefficient is a result of the implementation of a DCC GARCH (1, 1) model. Similar to Cappiello, Engle and Sheppard (2003), Ilmanen (2003), Connolly, Stivers and Sun (2004), Li (2004) and Saleem (2008) I use the assumption of a dynamic correlation coefficient.
Table 3. Descriptive statistics for the correlation coefficient between stock and bond returns
Mean Max Min Standard
deviation
Skewness Kurtosis Jarque Berra Probability of JB Corr. coefficient -0.0189 0.6508 -0.5698 0.3028 0.1972 2.0431 214.4585 0.0000
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The correlation coefficient varies from -0.5698 to 0.6508 providing support for the hypothesis of a dynamic relation between stock and bond returns. Furthermore, after analyzing the values of the skewness and kurtosis I conclude that the series is platokurtotic and positively skewed. Even if these values are quite close to the normal ones (0 for the skewness and 3 for the kurtosis), the series of correlation coefficient is not normally distributed. I emphasize the same property from the value of 214.4585 of JB and the 0.0000 for the associated Probability.
The properties of the correlation coefficient series are graphically described in Figure 1. Figure 1.Evolution of the correlation coefficient between stock and bond returns
*Note: The correlation coefficient between stock and bond returns is calculated using a DCC GARCH(1, 1) model. The sample contains 4805 daily observations from January 1990 to April 2009.
Figure 1 emphasizes the dynamic nature of the correlation between stock and bond returns. This is consistent with Saleem (2008), who first uses the Constant conditional correlation GARCH model to calculate the coefficient.
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Furthermore, the correlation coefficient is most of the time positive even if the mean value is negative.
Andersson, Krylova and Vähämaa (2004) find the same property, underlying the fact that the positive values of stock-bond correlation have a higher frequency. However, the value of the coefficient can drop significantly around market crashes.
As seen in Figure 1, the correlation coefficient has negative value during 1997, determined by the East Asian currency crisis. Furthermore, the drop in the value of the correlation coefficient in 2000 is due to the Internet bubble. This variation in the value of the correlation coefficient is consistent with Gulko (2002). This research observes that around the stock market crashes, the value of the correlation coefficient between stock and bond returns has a negative value. The drop in the value of the coefficient is called decoupling. The impact of the current financial crisis is graphically presented in the drop of the coefficient value from the last period. Due to the drop in the stock market returns in the autumn of 2008, the value of the correlation coefficient is moving towards a negative value dropping in an accelerated manner.
I use the Nelson and Siegel model in order to estimate the term structure of interest rates. β1,t is
the long-term component of the Nelson and Siegel model, β2,t is the short-term component and β3,t is the medium-term component.
In the second step, I analyze the descriptive statistics of β1,t , β2,t and β3,t.
Table 4. Descriptive statistics for β1,t , β2,t and β3,t
Mean Max Min Standard
deviation
Skewness Kurtosis Jarque Berra Probability of JB β1,t 0.0547 0.0910 0.0175 0.0152 0.0835 2.3343 94.2946 0.0000 β2,t 0.4941 2.3345 -0.5987 63.5622 1.2053 3.4176 1198.41 0.0000 β3,t -0.0602 0.6082 -2.6422 0.7046 -1.2119 3.4885 1224.03 0.0000
*Note: The variable coefficients β1,t , β2,t and β3,t were calculated using the Nelson and Siegel model(1987) which I
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on β2,t is the short-term while the medium-term component is on β3,t. The descriptive statistics are for daily
observations from January 1990 to April 2009.
The value of β1,t, varies from 0.0175 to 0.0910. This series is not normally distributed due to the
value of 0.0835 and 2.3343 that the skewness and kurtosis take. However even if it is slightly positively skewed and platokurtotic, the series of β1,t is very close to a normal distribution.
After analyzing the values of JB and probability of JB, I conclude that this variable is not normally distributed. With regards to β2,t, it has a much wider variation interval that includes also
negative values. This second variable is also not normally distributed but positively skewed and leptokurtotic. The values of JB and probability of JB underline the same property of β2,t.
Finally, in contrast to β2,t, β3,t is negatively skewed but also leptokurtotic. However, it has a
negative mean and a different variation interval. So β3,t is also not normally distributed due to the
fact that JB and Probabilty of JB have the values of 1224.03 and respectively 0.0000. The properties of these three variables are graphically described in Figure 2.
I underline the opposite variation of β2,t and β3,t. The first shock is due to the inclusion of the 20
19 Figure 2.Evolution of β1,t , β2,t and β3,t
*Note: β1,t , β2,t and β3,t are calculated after using the Nelson-Siegel three-factor model (1987) in order to estimate
the term structure of interest rates. The sample contains 4805 daily observations from January 1990 to April 2009.
Concerning the evolution of β1,t, this variable does not present any major peak or slump.
Furhermore as seen in Table 4 the values taken by this variable have a quite narrow variation interval.
The values of β2,t and β3,t have an opposite variation. This means that taken alone, some of this
two values is zero or very close to the null value. However, β2,t and β3,t are the coefficients of the
short-term and medium-term component that have different values. So β2,t and β3,t actually
determines the contribution of the short-term and respectively medium-term component of the Nelson and Siegel model to the final value of the interest rate.
From this point of view, the long term component has actually the value of 1. So the starting point of the interest rates value is β1,t. Furthermore, the value of β2,t determines a positive
contribution of the short-term component to the value of the interest rates. In constrast to β2,t, the
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interest rates. So the shape of the term structure is determined in positive way by the long-term and short-term component while the medium-term component has a negative influence.
In the analysis of data part I expressed my concerns about a possible breakpoint due to the discontinuous nature of 1 month, 20 year and 30 year yields. Figure 2 provides evidence of the stuctural break on the variation of β2,t and β3,t. As I mentioned before I will use the Chow test in
order to further investigate this issue.
In the final step of the result analysis, I pay special attention to the impact of the term structure of interest rates on the correlation between stock and bond returns. In order to investigate this I utilize both the results from the DCC GARCH (1, 1) and from the Nelson and Siegel model. In other words I estimate regression (11) where the correlation coefficient is the dependent variable and β1,t , β2,t and β3,t are the independent variables. Table 5 presents the estimation output of the
above-mentioned regression.
Table 5. Regression of correlation coefficient against β1,t , β2,t and β3,t
The table presents the estimation output of regression (11) where was calculated using a DCC GARCH(1,1) model and β1,t , β2,t and β3,t were obtained after implementing the Nelson
and Siegel model (1987). The analysis is performed on a sample of 4805 daily observations from January 1990 to April 2009.
Variable Coefficient Standard Error t-Statistic Probability
a0 0.7163 0.0149 47.9785 0.0000* β1,t -12.9237 0.0025 -52.3318 0.0000* β2,t -0.2395 0.0008 -3.0471 0.0023* β3,t -0.1506 0.0007 -2.1614 0.0307* R -squared 0.5512 Adj. R-squared 0.5509
*Statistically significant at a 5% confidence level
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even if its value is between 0 and 1. Furthermore, the coefficient of β1,t has a negative value of
-12.9237. This means that the long-term component of the Nelson and Siegel model influences the correlation between stock and bonds in a negative way.
The coefficient of β2,t is also negative but in contrast to β1,t is much smaller and close to zero.
However, it is statistically significant at a 5% level. Analyzing its value from an economical point of view, I emphasize the fact that the short-term component from the Nelson and Siegel model influences in a negative way the relation between stock and bond markets. Finally, analyzing the value coefficient of β3,t, similar to the other two it is also negative. Furthermore,
the coefficient of β3,t is also statistically significant. So all the components of the Nelson and
Siegel model do influence the value of the correlation coefficient between stock and bond returns.
The value of R squared is 0.5512 meaning that the independent variables explain 55.12% of the variation in the value of the correlation coefficient. This result shows a significant negative impact of the term structure of interest rates on the relation between stock and bond markets. I conclude that the long-term component of the term structure of interest rates has the biggest impact on the correlation coefficient. Next, the short-term component also influences the value of the coefficient in a negative way. Finally, the medium-term component has also a negative influence on the relation between stock and bond returns.
Finally, I investigate the issue of the breakpoint in the variation of β2,t and β3,t. In order to do so,
I use the Chow breakpoint test. The main idea of this test is to split the data in two or three subsamples and to fit the regression separately for each of them. Further, if there are any significant differences then there are structural changes in the relationship. Therefore, I can use this test to research the impact of the term structure of interest rates before and after the introduction or deletion of a certain yield rate.
22 Table 6 presents the result of the Chow breakpoint test.
Table 6. Chow Breakpoint Test
F-statistic 171.8831 Probability 0.0000
Loglikelihood ratio 643.5873 Probability 0.0000
*Note: The breakpoint date is 30th of July 2001
Analyzing the value of F-statistic from Table 6 I conclude that I can estimate two models regression due to the structural break in my dataset. As I stated before the introduction and deletion of certain yield rates caused this problem. However, finding two different regressions for the two different subsamples would solve this issue.
I estimate regression (11) for the two data sub-samples that I created using as a breakpoint date July 30, 2001.
Table 7. Regression of correlation coefficient against β1,t , β2,t and β3,t -first sub-sample
The table presents the estimation output of regression (11) where was calculated using a DCC GARCH(1,1) model and β1,t , β2,t and β3,t were obtained after implementing the Nelson
and Siegel model (1987). The analysis is performed on a sample of 2899 daily observations from January 1990 to July 2001 .
Variable Coefficient Standard Error t-Statistic Probability
a0 0.4632 0.0258 17.9264 0.0000* β1,t -7.6621 0.4129 -18.5561 0.0000* β2,t 4.8611 0.2438 19.9313 0.0000* β3,t 4.3583 0.2152 20.2522 0.0000* R -squared 0.3583 Adj. R-squared 0.3576
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First I estimate regression (11) until the introduction of the 1-month yield on the 30th of July 2001. In contrast to the estimation after using the whole data set, the estimation on the first sub-sample has interesting different results. First, the coefficients of β2,t and β3,t have a positive value.
This means that the short-term and medium-term component influence in a positive way the correlation between stock and bond returns. The other coefficients have the same sign but different values. However, the components of the Nelson and Siegel model explain only 35.83 % of the variation of the correlation coefficient between stock and bond markets. Furthermore, all the coefficients are statistically significant.
Table 8. Regression of correlation coefficient against β1,t , β2,t and β3,t - second sub-sample
The table presents the estimation output of regression (11) where was calculated using a DCC GARCH(1,1) model and β1,t , β2,t and β3,t were obtained after implementing the Nelson
and Siegel model (1987). The analysis is performed on a sample of 1906 daily observations from July 2001 to April 2009.
Variable Coefficient Standard Error t-Statistic Probability
a0 0.8338 0.0321 25.9422 0.0000* β1,t -16.1869 0.7000 -23.1231 0.0000* β2,t 6.2049 0.6616 9.3781 0.0000* β3,t 1.8205 0.2352 7.7389 0.0000* R -squared 0.4004 Adj. R-squared 0.3994
*Note: The breakpoint date is 1st of August 2001
Next, I estimate regression (11) for the second subsample, after the introduction of the 1-month yield in the data set. The estimation output is similar to the one on the first sub-sample. Therefore, in contrast to the estimation on the whole data set, β2,t and β3,t have a positive
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After splitting the data set into two sub-samples, I use two different models as indicated by the Chow test. These two models differ from the original one in the signs of β2,t and β3,t. In the
second sub-sample β2,t has a greater influence on the correlation coefficient. Also the impact of β1,t is significantly bigger than before the introduction of the 1-month yield. This is the effect of the 1-month yield on the correlation between stocks and bonds.
However, after analyzing the output of regression 11 for both the whole sample and the two sub-samples I decide to test for the assumption of multicollinearity. Furthermore, after analyzing Figure 2 I underlined the opposite variation of β2,t and β3,t . This might be a graphical proof of a
strong negative correlation between the short-term coefficient and the medium-term coefficient of the Nelson and Siegel model. I utilize the correlation matrix in order to check the values of the correlation coefficients between β1,t, β2,t and β3,t. Table 9 presents the correlation matrix for
regression 11 when applied for the whole data sample. Table 9. Correlation matrix of β1,t, β2,t and β3,t
β1,t β2,t β3,t
β1,t 1 0.462 -0.432
β2,t 0.462 1 -0.997
β3,t -0.432 -0.997 1
*Note: β1,t, β2,t and β3,t are the long-term, short-term and respectively medium-term component of the term structure
of interest rates. The three components are calculated using the Nelson and Siegel model.
The values of the correlation coefficient between β2,t and β3,t indicate a strong negative
correlation between the short-term component and the medium-term component. A correlation coefficient of -0.997 indicate an almost perfect correlation. There are several solutions for the multicollinearity problem. First of all, this could be ignored if the model is consistent as a whole. This first solution is not appropriate for this model since the evolution of the Nelson and Siegel components indicate a serious problem. A second solution is to drop one of the components. A final way to deal with the multicollinearity problem is to transform the correlated variables into a ratio and to include only the ratio and not the individual variables in the regression.
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I estimate regression 12 as a solution for the multicollinearity problem.
ρ ij,t = a0 + a1ls + ε t , (12) where ρ ij,t is the correlation coefficient between the stock and bond returns, ls is the difference
between the 10 years yield and the 3 months yield and εt is the error term. In other words I compute the difference between a long-term rate and the short-term rate. I choose the 10 year and the 3 month yield because they are continuously used by the U.S. Treasury Department. Furthermore, it represents the biggest difference between a long-term rate and a short-term rate. Next I present the estimation output of regression (12) over the whole data sample.
Table 10. Regression of correlation coefficient against the long-short difference
The table present the estimation output of regression (12) ρ ij,t = a0 + a1ls + ε t where ρ ij,t was calculated using a
DCC GARCH model and ls is the difference between the 10 year yield rate and the 3 month yield rate. I perform the analysis on a sample of 1906 daily observations from July 2001 to April 2009.
Variable Coefficient Standard Error t-Statistic Probability
a0 -0.0377 0.0073 -5.1688 0.0000*
a1 0.0116 0.0036 3.2056 0.0014*
R -squared 0.0021
Adj. R-squared 0.0019
*Note:Statistically significant at a 5% confidence level
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addition to this, I find that the term structure of interest rates has a significant impact on the stock-bond relationship.
In particular, after trying to explain the variation of the correlation coefficient with the components obtained from the Nelson and Siegel model I find that difference between long-term and short-term yield influence the relation between stock and bond returns.
5. Conclusions
This paper examines the correlation between stock and bond returns. My empirical findings demonstrate that the correlation coefficient between stocks and bonds vary considerably over time. I find that the correlation coefficient may change significantly and turn from positive to negative in a very short period. During stock market crashes, the value of the correlation coefficient becomes negative. However, the positive values have a higher frequency. This high volatility should be carefully analyzed when performing an asset allocation and risk management process.
Furthermore, I model the term structure of interest rates using the Nelson and Siegel model. I then utilize the model’s parameters in order to investigate the impact of the term structure of interest rates on the relationship between stock and bond markets. I find a high variation in the values of parameters β2,t and β3,t over time. This could have as a determinant the introduction of
the 1-month, 20-years and 30-years yield rates on different points in time from 1990 until April 2009. A possible solution for this problem is the utilization of two different models that could describe the impact of the term structure of interest rates on the correlation coefficient between stock and bonds.
I use two models - one before the introduction of 1-month yield and one after the introduction 30th of July 2001 is the breakpoint. The two models for the new sub-samples present different sign of the influence of β2,t and β3,t on the correlation between stock and bond returns. Therefore,
I solve the structural change problem by using two different regression.
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way the relation between stock and bond returns. However, if I use two different models I find that the long-term component has a negative influence on the stock-bond relationship while the short-term component and the medium-term component influence the correlation coefficient between stock and bond returns in a positive way.
The problem of multicollinearity determines me to modify the initial regression. Consequently, I research the impact of the long-short difference on the correlation coefficient between stock and bond returns. I find a significant impact of this new determinant. However, I conclude that the relation between stock and bond markets is a complex one and that further researches need to be performed in order to find a complete range of determinants. In contrast to the previous researches, I investigate the bond market as the main determinant of the stock-bond relationship. Therefore, my research could be the starting point of a new research area that could consider the bond market as an important determinant of the correlation between stock and bond returns.
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