The Correlation between Stock and Bond Markets
Abstract To analyze the correlation between stock and bond markets, this paper utilizes the Asymmetric Dynamic Conditional Correlation (ADCC) model proposed by Cappiello et al (2004). There is a very volatile correlation between two markets. The average coefficient of correlation is zero. Testing the dynamic correlations by using a set of macroeconomic information, the evidence shows that interest rate change, term structure and stock market volatility are the significant factors.
JEL Classification: E42, E44, G12, G18 Keywords: Dynamic Conditional Correlation, Stock-bond market correlation
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1. Introduction The correlations among different assets are important inputs for asset allocation, portfolio management and risk management. The correlation between equity returns and bond returns is one of the most important and widely studied. However, there has not been any consensus. The relations between stock and bond markets in the literature are concentrating on the aggregate level.1 The studies on aggregate level stock-bond market relation can be classified into two categories. The first one is about returns level. Stock and bond react to news (shocks) differently so that the correlation is time varying. There is also lead-lag relation in the two markets. The second one is about the uncertainty of returns. Volatility contains information so that there is volatility spillover and liquidity connection between two markets (See Baker and Wurgler (2005) and Gebhardt, Hvidkjaer and Swaminathan (2005)). The starting point of the research on the correlation of the two market returns is the present value model, i.e. the asset price is the discounted present value of future cash flow. For stock and bond, their prices can be written as:
ps = pb =
dividend rs − g coupon rb
(1.1) (1.2)
where the subscription s indicates stock; the subscription b indicates bond; p stands for price; dividend and coupon stand for future cash flow per period for stock and bond respectively; r stands for required return; g represents for growth rate. By using above two equations, there are some underlying assumptions. Bond is assumed to be perpetuity and stock is assumed to have constant growth rate of dividend. Through “over-simplifying”: rs=rb , g=0 and all earnings are paid out as dividend, we can get so-called Fed Model (Yardeni 1997; Abbott 2000):
Individual stocks and bonds are studied by Kwan (1996), Dopfel (2003) and Li (2002). The main findings are that there is average positive correlation between individual stocks and bonds, that individual stocks tend to lead individual bonds and that the AAA-rated bonds are relatively insensitive to firm specific information.
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( p / E) s =
1 rb
(1.3)
This model tells that the stock’s P/E ratio should be the reciprocal of the bond market’s yield to maturity. In other words, there is a negative relationship between the stock market’s P/E ratio and the level of the interest rates. The Fed Model has been criticized by Estrada (2006) from both theoretic view and empirical tests. In summary, first of all, the required return for stock is real term while the required return for bond is nominal term.2 Second, since stock and bond have different risk level so that it is implausible to assume rs=rb . 3 Third, the Fed Model is absolutely groundless when the interest rate is very low and one cannot judge equilibrium from the model since neither P/E ratio nor interest rate is able to be the benchmark for another. Fourth, the phenomenon of the comovement of the P/E ratio and the reciprocal of the interest rate is only valid in the 1968-2005 period for US markets. It is invalid in the longer period from 1871 to 2005 in US markets. In other 20 countries the Fed Model doesn’t hold either. Different from above Fed Model but also based on the present value model, Shiller and Beltratti (1992) found that the theoretical correlation between stock and bond long term returns is a mere 0.06.4 Campbell and Ammer (1993) found that about 70 percent of the variance of excess stock returns was attributable to the news about future risk premiums and 15 percent was attributable to news about future cash flows. Regarding bonds, bond returns mainly can be accounted by the news about inflation and news about future risk premium (credit risk). Overall, they found a moderate positive correlation of 0.20 between two markets in 1952-1987 sample period. A main conclusion of the Campbell and Ammer (1993) is that the real interest rate doesn’t move either market. In a world where there is risk free rate rf, we can re-write the pricing formula as:
Modigliani (1997) pointed firstly the inflation issue when considering the stock bond market relations. However, Modigliani and Cohn (1979) already pointed out the shortage in considering inflation when valuating firm’s debt. 3 However, Ritter (2002) is arguing that it is a conceptual mistake to think stock is riskier than bond. Contrary to prevailing 7% risk premium, there is only 1% risk premium between stock and bond if considering the holding period (stock returns show mean reversion), inflation and geometric average method. By and large, Ritter (2002) thinks the Fed Model is practical valid. 4 They use forecasted values of discount rates and dividend growth rates to infer “theoretical” prices of stocks and bonds.
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3
ps =
pb =
dividend r f + rstockpremium − rinf lation − g
coupon r f + rlongbondpremium
(1.4) (1.5)
then we can find different scenarios about both asset prices: Case 1: positive relation--- when rf increases, both prices go down; Case 2.1: only stock price changes and bond price is unchanging--- for example, when only stock’s future earning (can be either d or g in the formula) get improved; Case 2.2: only stock price changes---Another possibility is that the stock risk premium changes. When stock market is falling, investors may become more risk-aversion. Bonds look more attractive to investors and money goes to bond market, which is called “flight to quality”. On the other hand, investors are induced to those high returns when stock market is rising (less risk-aversion), which is called “flight from quality” (Li, 2002; Gulko, 2002; Stivers, Sun and Connolly, 2003; and De Goeij and Marquering, 2004); Case 3: only bond price moves and stock price is unchanging---when expected inflation rate goes up, the bond price decreases but stock’s cash flow is discounted on the real term so that it is unchanging; Case 4: negative relation--- when the whole economy is overheating (stock prices are up considering the future cash flow), the Federal Reserve may change targeted rate wanting to slow down the economy. It might increase the rate, which drags down the bond prices. Besides above 4 cases, there are other approaches for the relation between stock and bond markets as follows: Case 5: positive relation--- since both stock and bond markets are exposed to common macroeconomic conditions (except for the risk free rate in Case 1), there is average positive correlation between them. For example, when economic conditions are good, investors may purchase both assets to diversify their portfolio. The demand for two assets simultaneously may also due to the wealth effect in the Tobin’s general equilibrium framework (1969, 1982). Keim and Stambaugh (1986) and Campbell and Ammer (1993) find a low positive correlation between stock and bond returns. Case 6: positive relation (contagion) ---The correlations among countries’ financial market increase during crisis period is called “contagion”. Hartmann, Straetmann and
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Devries (2001), Gulko (2002), and Baur and Lucey (2006) found the stock-bond contagion phenomenon Base upon above several scenarios, it is easy to see the necessity of doing researches on the correlation between stock and bond markets. Before going into this research, the following items differentiate this paper from previous studies on the same topic. Stock price has time component, as firms grow, their stock prices will go up. This is obviously different from interest rate that basically follows random walk. It is inappropriate to compare them directly. Even taking P/E ratio instead of stock price, it is still inappropriate since interest rate is I(1) process generally and P/E ratio is I(0) process. Another shortcoming of the comparison of P/E ratio and interest rate is that interest rate is the required return, not the real return of the bond. P/E ratio also limits the data interval to be at least monthly. The third is that the correlation is usually static based on specific sample period, not conditional time varying correlation. This includes the most cited Shiller and Beltratti (1992), Campbell and Ammer (1993) and Bekart, Engstrom and Grenadier (2004).5 This paper improves the stock bond correlation research in following aspects. First, it measures real returns of two markets. Instead of using Yield to Maturity (YTM) of the long term bond, this paper uses total bond market index fund to proxy bond’s return. Second it studies the dynamic conditional correlation rather than static correlation. This paper uses Dynamic Conditional Correlation (DCC) model proposed by Engle (2003) and its extension by Cappiello et al (2004). Third, it not only explores the correlation between two markets, it also gives explanations on the dynamic conditional correlation so that we can find the drives for the time varying correlations. Specifically, this paper considers the interest rate, term structure and conditional volatility conditions in explaining the correlation. This has not been done in existing literatures.6
Besides GARCH type models, Pelletier (2003) adopts regime-switching approach where the transitions between regimes are modeled by a Markov Chain. Audrino and Barone-Adesi (2003) present rolling window average conditional correlation. 6 Since the paper is using daily data, other macroeconomic variables are not considered. This leaves misspecification problem in the later part. According to David and Veronesi (2004), state variables can include real interest rates, inflation, growth even oil prices, and the uncertainty in the economic factors. In addition, the interest rate is not strictly exogenous. This leaves identification problem in the model (for example, see Rigobon and Sack (2003)). A further step on this issue can be the ex ante correlation study. i.e. the residual correlation after explaining part by economic fundamental factors.
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The study on the correlation between two markets has threefold meaning. First, it provides information on the stock-bond correlation that would allow better asset allocation and risk hedging. Second, it provides a financial implication on dynamic asset allocation due to interest rate change and stock whole market volatility. The policy makers and practitioners get a picture on the whole financial system. Third, it reveals the investors’ behavior so that researchers can gain more understanding of the financial market integration and the risk aversion degree. This paper has two parts: the first part is to apply asymmetric Dynamic Conditional Correlation (aDCC) model to obtain the dynamic correlation between stock and bond markets. The results show that the correlation is very volatile, swinging between positive zone and negative zone. Overall, the average coefficient of correlation is zero. The second part is to analyze what affect the DCC. Stock market volatility is a remarkable factor. Interest rate change and term structure affect the correlations as well. The rest part of this paper is organized this way: section 2 is the methodology description; section 3 is data sample; section 4 gives empirical results; section 5 closes.
2. Asymmetric-DCC Model
Asset returns show conditional heteroskedasticity. Therefore a static correlation is not appropriate in describing relation between two markets. Multivariate GARCH model is necessary to obtain any dynamic relations. VECH is the mostly seen estimation method for multivariate GARCH(1,1) in the literature. But, there are two drawbacks of VECH method. The most noticeable one is that it cannot ensure the positive definiteness of covariance matrix. Another one is that the covariance is independent of conditional variances by VECH method. This is in conflict with the fact that correlation tends to increase as variability increases. These shortages are overcome by DCC estimation method7 (Engle, 2002). The DCC model is:
⎧ Rt = µ t + ε t ⎨ ⎩ε t Ψt −1 ~ N (0, H t )
(2.1)
There are only proposals in the internet about the explanations for the conditional correlations. Such proposals include, for example, Inghelbrecht (2006), Cajigas (2006) and Baele, Bekaert and Inghelbrecht(2006). This paper is independently developed by authors. 7 We have two choices in dealing with time varying correlation coefficient. One is to model correlation coefficient directly, which is the DCC; another one is to use Cholesky decomposition of covariance matrix.
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where Rt is asset return; µt is the mean of returns; εt is error term, which follows conditional normal distribution with zero mean but shows heteroschadasticity; Ψt-1 is the complete information set at t-1; Ht is the conditional variance. In expansion form for two assets 1 (stock) and asset 2 (bond), the variables can be described by:
⎡µ ⎤ ⎡ε ⎤ ⎡R ⎤ ⎡h Rt = ⎢ 1 ⎥ , µ t = ⎢ 1 ⎥ ε t = ⎢ 1 ⎥ , and H t = ⎢ 11 ⎣µ 2 ⎦ t ⎣ε 2 ⎦ t ⎣ R2 ⎦ t ⎣h21 h12 ⎤ h22 ⎥ t ⎦
Ht is modeled directly as a function of dynamic univariate variances and dynamic linear correlations. We can write Ht in the form: H t = Dt Pt Dt the diagonal of variances:
⎡ h Dt = ⎢ 11 ⎢ ⎣ ⎡ 1 Pt = ⎢ ⎣ ρ12 ⎤ ⎥ h 22 ⎥ t ⎦
(2.2)
where Pt is correlation matrix (not using Rt to avoid any confusing with returns) and Dt is
(2.3)
ρ12 ⎤
1 ⎥t ⎦
(2.4)
To model time varying correlation coefficient, Pt is transferred into:
Pt = [diag (Q)]t
−1 / 2
[Q]t [diag (Q)]t −1 / 2
⎡ 1 q12 ⎤ ⎢ q11 ⎢ q 22 ⎥ t ⎢ ⎦ ⎢ ⎣ ⎤ ⎥ ⎥ 1 ⎥ q 22 ⎥ t ⎦
(2.5)
or in an expansion form:
⎡ 1 ⎢ q pt = ⎢ 11 ⎢ ⎢ ⎣ ⎤ ⎥ ⎡q ⎥ 11 1 ⎥ ⎢q 21 ⎣ ⎥ q 22 ⎦ t
(2.6)
where q12,t follows a univariate GARCH(1,1) process: q12,t = (1 − α − β ) ρ 12 + αz1,t −1 z 2,t −1 + β q12,t −1 (2.7)
where ρ 12 is the unconditional correlation coefficients and z1,t is normalized error term in the mean equation: z1,t =
ε 1,t
h11,t
(2.8)
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z 2 ,t =
ε 2 ,t
h22,t
(2.9)
where conditional variances are got from univariate GARCH(1,1)-VECH process: h11,t = c11 + a11 * ε 1,t −1 + b11 * h11,t −1
2
(2.10) (2.11)
h22,t = c 22 + a33 * ε 2,t −1 + b33 * h22,t −1
2
The correlation coefficient from GARCH(1,1) DCC method is defined as:
ρ12,t =
q12,t q11,t q 22,t
(2.12)
It has been well documented that positive and negative innovations to returns have different impacts on conditional volatility. There are 3 types of asymmetric effects.8 This paper follows GJR model as used in paper by Kroner and Ng (1998). The model introduces asymmetric effect in following way:
⎡η1t ⎤ ⎢η ⎥ η it = max[0,−ε it ] and η t = ⎢ 2t ⎥ ⎢ M ⎥ ⎢ ⎥ ⎣η nt ⎦
(3.1)
There are alternatives for including asymmetric effect in DCC model (Cappiello et al 2004, Billio et al 2004, Hafner and Franses 2003). This paper follows the extension introduced by Cappiello et al (2004). The correlation matrix is modified into:
Qt = (Q - A' Q A - B' QB - G' NG) + A' z t −1 z t −1 ' A + B' Qt −1 B + G 'η t −1η t −1 ' G
(3.2)
where A, B, and G are diagonal parameter matrixes, N = E [η tη t '] . In estimation process, expectations are infeasible so that:
N=
1 T ∑ η tη t ' T t =1
(3.3)
The first one is to allow the conditional variance to respond differently to positive and negative innovations. There are many ways to model such asymmetric effect such as EGARCH model (Nelson (1991)), APARCH model (Ding, Granger and Engle (1993)) and GJR model (Glosten, Jagannathan and Runkle (1993)).8 The second type is to allow the shocks to enter variance equation non-linearly. This type is modeled in NGARCH model by Higgins and Bera (1992). The last type is to allow re-centering of “new impact curve” so that the point of no change in the variance is not necessarily centered at zero. Hentschell (1995) provides an overview of three types of asymmetric effect.
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Q=
1 T ∑ zt zt ' T t =1
(3.4)
zi,t is normalized error term in the mean equation as before. In univariate form:
q12,t = (1 − α − β − g ) ρ 12 + αz1,t −1 z 2,t −1 + βq12,t −1 + gη1,t −1η 2,t −1
(3.5)
Besides the asymmetric effect on conditional correlation, conditional variances are also modified:
h11,t = c11 + a11 * ε 1,t −1 + b11 * h11,t −1 + d11η1,t −1
2 2
(3.6)
h12,t = c12 + a 22 * ε 1,t −1 * ε 2,t −1 + b22 * h12,t −1 + d12η1,t −1η 2,t −1
(3.7)
3. Data
This paper follows the tradition, taking Standard and Poor’s 500 index (SP500) as whole stock market return proxy. In addition, this paper also considers segments in the equity market. Additional stock index series include: Standard and Poor’s small capitalization index (SPSML), Standard and Poor’s middle capitalization index (SPMID), and Nasdaq composite index (NASDAQ). Regarding the bond market, this paper considers two types proxy. One is the bond yield following literature convention; another one is the bond fund. Subject to the bond yield, 20 year treasury bond yield and 10 year treasury bond yield are used.9 Subject to the bond fund, two of largest bond index funds: PIMCO Total Return Fund (PTRAX) and Vanguard Total Bond Market Index Fund (VBMFX) are used. PTRAX represents the extreme of active management while VBMFX represents the extreme of passive management. Both of PTRAX and VBMFX are seeking to provide investment results that correspond to the total return of the bonds in the Lehman Brothers Aggregate Bond index.10 SPSML series began on August 16, 1995 and SPMID series began on August 21, 1991. PTRAX series began on 20-Jun-96 and VBMFX began on 4-Jun-90. The sample is
There are missing values from 2/18/2002 to 2/8/2006 for the 30 year bond yield. Therefore 30 year bond yield is not used although historical adjusting factors exist. 10 The Lehman Aggregate Bond Index comprises government securities, mortgage-backed securities, assetbacked securities and corporate securities to simulate the universe of bonds in the market. The maturities of the bonds in the index are over one year. The index constructed by the Lehman Brothers is considered to be the best overall bond index as it is used by over 90% of investors in the United States.
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from 20-Jun-96 to 23-Aug-2006 11so that it is balanced. The PTRAX and VBMFX series are obtained in http://finance.yahoo.com. The SP500, SPSML, SPMID and NASDAQ series are from ****. The yield of 10 year and 20 year bonds and short term interest rate represented by the 3 month T-bill annualized rate are obtained from http://www.federalreserve.gov/releases/H15/data.htm#top. Figure 1 gives a visual comparison of these series.
Table 1 gives statistical description of these series. Table 2 gives the unconditional correlation between SP500 and two types of bond market return series. It is remarkable that both bond index funds have high correlation. The Pearson Coefficient is 0.91. The 10 year bond yield change and 20 year bond yield change are also highly correlated. However, the changes in the yield to maturity of bonds are not coming along with bond index fund returns. The coefficients are even negative. The yield to maturity reflects the bond market conditions but it is not directly related to bond market returns so that this
paper takes the bond index fund returns as bond return’s proxy in the following parts.12 The SP500 has both negative Pearson Coefficients with two bond index funds (-
0.05 and -0.08 respectively). This is different from those that reports positive correlation between stock and bond markets (for example, Campbell and Ammer, 1993) The interest rate has unit root, which is verified by Augmented Dickey-Fuller test. (pvalue is 0.847). Phillips-Perron test gives same conclusion. That is why a first order difference of the interest rate is considered.13
4. Empirical Results
Table 3 shows the estimation results of asymmetric DCC (ADCC) model. We have 4 stock index (SP500, SPSML, SPMID and NASDAQ) and 2 bond index (PTRAX and VBMFX) so that there are totally 8 ADCC series. The eight ADCCs are depicted in the figure 2. ADF test shows that they are stationary (for example, the first two have p-values
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The end date is determined by the original work. It may be updated when the paper is submitted. For example, in the Engle (2002), it says “taking bond returns to be minus change in the 30 year benchmark yield to maturity”. 13 It is worthy of pointing that the Ang and Bekaert (2001) omit this econometric concern. They point out that the short term interest rate can explain stock returns.
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0.00278 and 0.0003943 respectively). By visualizing these dynamic correlations, we found that they follow very parallel time path and display similar turning points. Following part will focus on the ADCC between SP500 and two bond index series. Something worthy of noticing is that the asymmetric effect is significant for the PTRAX series but insignificant for VBMFX series. This may be due to the former one is representing active management while the latter one represents passive management. The question we encounter then is to inquire what are the factors that might contribute to the correlation coefficients to be time varying. Referring to David and Veronesi (2004), we think the dynamic conditional correlation should be depending on state variables that include real interest rates, inflation, growth even oil prices, and the uncertainty in the economic factors. This paper proposes hypothesis that the correlation is depending on the interest rate, return levels, conditional volatility of stock return risks and other type risks such as liquidity, inflation risk or different stages of bubble or business cycle. The unrestricted regression model is:
ρ DCC ,t = φ 0 + φ1 Rstockt + φ 2 Rbond ,t + φ3σ stock ,t + φ 4σ bond ,t
+ θ1 ∆i short ,t + θ 2 | ilong ,t − i short ,t | +ν
(4)
where ρ is dynamic conditional correlation between stock and bond markets; R is return level; σ is conditional standard deviation; ∆i is interest rate change (3 month T-bill rate); |ilong-ishort| is the term structure; v is the error term. Table 5 reports the regression estimation. I find the interest rate change, term structure and conditional volatility for both stock market returns are significant variables. Among these variables, stock market volatility is most influential. Interest rate change is positively related with the conditional correlation coefficients. 14 However, interest rate change is not significant anymore when the conditional stock market volatility exists in the regression. The term structure is always negatively significant. As investors are more uncertain about future economy conditions,
14
There is no endogeneity problem since there is no indication that the correlation between stock and bond market is one of Federal Reserve’s concerns to decide the target interest rates. It might consider the stock market return though. See Rigobon and Sack (2003).
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the difference between long term rate and short term rate enlarges, and the correlation between stock bond markets goes down. The correlation coefficients between stock-bond markets are negatively related with the volatility of the stock market. When the stock market volatility increases, the correlation decreases. This means that the “flight to/from quality” phenomenon is more prominent than contagion. Another important thing is that the stock market volatility is crucial in affecting the conditional correlation in the stock-bond market. The volatilities of the bond market returns however, are positively significant. Table 6 gives all eight ADCC series regression results. To know whether the coefficient is positive or negative, we run logistic regression and results are shown in table 7. The model is :
P(rho> 0) =
1 1+ e
−(φ0 +φ1Rstockt+φ2Rbond,t +φ3σ stock,t +φ4σbond,t +θ1∆ishort,t +θ2 |ilong,t −ishort,t |+ν )
(5)
where the dependent variable is the probability of the aDCC being positive. The results suggest that when the stock market is volatile, the correlation coefficient is more likely to be negative; when the bond market is volatile, the correlation coefficient is more likely to be positive; the bigger gap between long term rate and short term rate will lead to negative correlation coefficient.
5. Conclusions
The stock bond market relation is an important input in the asset allocation and risk management. Existing literature doesn’t give consistent conclusion on how the correlation is. This paper investigates the correlation between US stock and bond markets by using asymmetric dynamic conditional correlation. Results show that there is a very volatile correlation between two markets. The coefficient swings between positive zone and negative zone. The average coefficient is around zero, implying the two markets are almost independent. The dynamic conditional correlation between stock and bond markets is depending on some market conditions. Concretely speaking, the stock market volatility and term
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structure decreases the correlation coefficient. The bond market volatility, on the other hand, increases the correlation coefficient. Furthermore, increasing stock market volatility and difference between long term and short term interest rates will lead to negative correlation; increasing bond market volatility will lead to positive correlation. The contributions of this paper lie in three aspects. First, it takes the bond index fund to proxy bond market. This is better than the change of Yield to Maturity of long term debt. Second, this paper finds the dynamic conditional correlation of stock bond markets being volatile and close zero on average. Last, this paper gives the state variables on which the dynamic conditional correlation depends.
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Figure 1: The level comparison of two markets’ proxy series
sp500_2 1100 sp500_2 1000 900 800 700 600 500 400 300 200 100 01JA 1996 N 01JA 1998 01JA 2000 01JA 2002 01JA 2004 01JA 2006 N N N N N D TE A 01JA 2008 N nsdq_com p_10 S 500S L P M pt r ax_x100 S 500M D P I vbm x_x100 f
Note: In order to show the trend, we do scaling as following: the sp500_2 is SP500 index divided by 2; nsdq_comp_10 is NASDAQ index divided by 10; PTRAX_x100 is PTRAX times a factor of 100; VBMFX_x100 is VBMFX times a factor of 100;
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Table 1: Description of Variables: daily data, Jun 21 1996-Aug 23 2006
Series RETURN_SP500 RETURN_PTRAX RETURN_VBMFX RETURN_SP500SML RETURN_SP500MID RETURN_NSDQ TBSM3M D_TBSM3M CM10Y D_CM10Y CM20Y D_CM20Y
Obs 2562 2562 2562 2562 2562 2562 2562 2562 2562 2562 2562 2562
Mean 0.000262 0.00027 0.00024 0.000384 0.000441 0.000230 3.588755 -6.2E-05 5.12346 -0.00084 5.64895 -0.00087
StdError 0.011469 0.00277 0.002742 0.01203 0.01189 0.01817 1.695029 0.043455 0.89146 0.05780 0.74215 0.0521
Minimum -0.07113 -0.01716 -0.01298 -0.0633 -0.0733 -0.1016 0.8 -0.65 3.13 -0.21 4.13 -0.18
Maximum 0.055744 0.013296 0.01187 0.0545 0.0596 0.1325 6.24 0.48 7.06 0.28 7.32 0.25
Note: SP500 stands for stock market returns; PTRAX and VBMFX stand for bond market returns; TBSM3M is 3 month T-bill rate; CM10Y and CM20Y stand for 10 year and 20 year bond yield respectively. Prefix D represents first order difference. the data on Nov222001 are problematic for SMSPL, SMMID and NASDAQ I fill in with zero.
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Table 2: Pearson Correlation among Variables
return_sp500 Return_sp500 1 -0.05355 0.0067 Return_VBMfx -0.08101 <.0001 D_CM10Y 0.15119 <.0001 D_CM20Y 0.10377 <.0001 0.90938 <.0001 -0.85104 <.0001 -0.83379 <.0001 -0.85435 <.0001 -0.84526 <.0001 0.94854 <.0001 1 1 1 1 return_ptrax return_VBMfx D_CM10Y D_CM20Y
Return_ptrax
Note: SP500 stands for stock market returns; PTRAX and VBMFX stand for bond market returns; TBSM3M is 3 month T-bill rate; CM10Y and CM20Y stand for 10 year and 20 year bond yield respectively. Prefix D represents first order difference.
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Table 3: aDCC of pairs of variables Notes:
1. The DCC is estimated by separate two steps. The model is as follows: ⎧ r1 = µ1 + ε 1 ⎡ ⎪ ⎢ η1t = max[0,−ε 1t ] ⎪ ⎢ ⎪ ⎢ ε 1 ~ N (0, H 1 ) ⎪ 2 ⎢ ⎪ ⎣h1,t = c1 + a1 * ε 1,t −1 + b1 * h1,t −1 + d 1η1,t −1 ⎪ r2 = µ 2 + ε 2 ⎡ ⎪ ⎢ η 2t = max[0,−ε 2t ] ⎪ ⎢ ⎪ ⎢ ε 2 ~ N (0, H 2 ) ⎪ 2 ⎢ ⎪ ⎣h2,t = c 2 + a 2 * ε 2,t −1 + b2 * h2,t −1 + d 2η 2,t −1 ⎨ ε 1,t ε 2 ,t ⎪⎡ , z 2 ,t = ⎪⎢ z1,t = h1,t h2 , t ⎪⎢ ⎪⎢ z ~ N (0, q ), z ~ N (0, q ) 11 2 12 ⎪⎢ 1 ⎪⎢η1,t = max[0,− z1,t ],η 2,t = max[0,− z 2,t ] ⎪⎢ ⎪⎢qi , j ,t = (1 − α − β − g ) ρ i , j + αz i ,t −1 z j ,t −1 + βqi , j ,t −1 + gη1,t −1η 2,t −1 ⎣ ⎪ q12,t ⎪ ρ12,t = q11,t q 22,t ⎪ ⎩
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continued of Table 3
SP500 and PTRAX Variable µ1 C1 A1 B1 D1 µ2 C2 A2 B2 D2 α β g SP500 and VBMFX Variable µ1 C1 A1 B1 D1 µ2 C2 A2 B2 D2 α β g coeff 1.30E-04 1.72E-06 -0.01056 0.92252 0.15198 2.63E-04 1.02E-07 0.02064 0.95664 0.01908 0.0422 0.94337 0.0023 stderr 1.82E-04 4.46E-07 0.00843 0.01182 0.01842 4.91E-05 4.09E-08 0.00545 0.00927 0.00897 0.00804 0.01069 0.00546 t-stat 0.72 3.85 -1.25 78.06 8.25 5.35 2.49 3.78 103.21 2.13 5.25 88.21 0.42
coeff 1.30E-04 1.72E-06 -0.01056 0.92252 0.15198 2.47E-04 2.78E-06 0.04278 0.57408 0.02948 0.05045 0.929 0.00502
stderr 1.82E-04 4.46E-07 0.00843 0.01182 0.01842 5.62E-05 1.61E-06 0.03458 0.232 0.03638 0.00917 0.01435 0.00647
t-stat 0.72 3.85 -1.25 78.06 8.25 4.39 1.73 1.24 2.47 0.81 5.50 64.73 0.78
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continued of Table 3
SPSML and PTRAX Variable µ1 C1 A1 B1 D1 µ2 C2 A2 B2 D2 α β g SPSML and VBMFX Variable µ1 C1 A1 B1 D1 µ2 C2 A2 B2 D2 α β g
coeff 5.20E-04 3.66E-06 0.02041 0.86948 0.16816 2.63E-04 1.02E-07 0.02064 0.95664 0.01908 0.03116 0.95707 0.00347
stderr 1.97E-04 8.21E-07 0.0133 0.01645 0.02203 4.91E-05 4.09E-08 0.00545 0.00927 0.00897 0.00819 0.01241 0.00461
t-stat 2.64 4.46 1.53 52.87 7.63 5.35 2.49 3.78 103.21 2.13 3.80 77.12 0.75
coeff 5.20E-04 3.66E-06 0.02041 0.86948 0.16816 2.47E-04 2.78E-06 0.04278 0.57408 0.02948 0.03165 0.95585 0.00335
stderr 1.97E-04 8.21E-07 0.0133 0.01645 0.02203 5.62E-05 1.61E-06 0.03458 0.232 0.03638 0.00726 0.01092 0.00433
t-stat 2.64 4.46 1.53 52.87 7.63 4.39 1.73 1.24 2.47 0.81 4.36 87.50 0.77
19
continued of Table 3
SPMID and PTRAX Variable µ1 C1 A1 B1 D1 µ2 C2 A2 B2 D2 α β g SPMID and VBMFX Variable µ1 C1 A1 B1 D1 µ2 C2 A2 B2 D2 α β g
coeff 4.58E-04 2.50E-06 0.01574 0.89353 0.14462 2.63E-04 1.02E-07 0.02064 0.95664 0.01908 0.03338 0.95345 0.00323
stderr 1.89E-04 5.85E-07 0.01061 0.01381 0.0189 4.91E-05 4.09E-08 0.00545 0.00927 0.00897 0.00914 0.01409 0.00524
t-stat 2.42 4.27 1.48 64.69 7.65 5.35 2.49 3.78 103.21 2.13 3.65 67.66 0.62
coeff 4.58E-04 2.50E-06 0.01574 0.89353 0.14462 2.47E-04 2.78E-06 0.04278 0.57408 0.02948 0.03443 0.9505 0.00407
stderr 1.89E-04 5.85E-07 0.01061 0.01381 0.0189 5.62E-05 1.61E-06 0.03458 0.232 0.03638 0.00841 0.01301 0.00492
t-stat 2.42 4.27 1.48 64.69 7.65 4.39 1.73 1.24 2.47 0.81 4.10 73.05 0.83
20
continued of Table 3
NASDAQ and PTRAX Variable µ1 C1 A1 B1 D1 µ2 C2 A2 B2 D2 α β g NASDAQ and VBMFX Variable µ1 C1 A1 B1 D1 µ2 C2 A2 B2 D2 α β g
coeff 3.91E-04 1.73E-06 0.02732 0.92107 0.09191 2.63E-04 1.02E-07 0.02064 0.95664 0.01908 0.02349 0.97028 -4.57E-04
Stderr 2.44E-04 5.47E-07 0.00833 0.01005 0.0146 4.91E-05 4.09E-08 0.00545 0.00927 0.00897 0.00567 0.00765 0.00299
t-stat 1.60 3.16 3.28 91.62 6.29 5.35 2.49 3.78 103.21 2.13 4.14 126.81 -0.15
coeff 3.91E-04 1.73E-06 0.02732 0.92107 0.09191 2.47E-04 2.78E-06 0.04278 0.57408 0.02948 0.03112 0.9589 5.52E-04
stderr 2.44E-04 5.47E-07 0.00833 0.01005 0.0146 5.62E-05 1.61E-06 0.03458 0.232 0.03638 0.00663 0.0094 0.00414
t-stat 1.60 3.16 3.28 91.62 6.29 4.39 1.73 1.24 2.47 0.81 4.70 102.02 0.13
21
Figure 2: The DCC pairs of variables
0.6 0.4 0.2 -0.0 -0.2 -0.4 -0.6
RHO_R_SP500_R_PT
DCC of variables
200
400
600
800
1000
1200
1400
1600
1800
2000
2200
2400
0.64 0.48 0.32 0.16 0.00 -0.16 -0.32 -0.48 -0.64
RHO_R_S P 500_R_V B
DCC of variables
200
400
600
800
1000
1200
1400
1600
1800
2000
2200
2400
0.50
DCC of variables
0.25
0.00
-0.25
-0.50
RHO_R_SPSML_R_PT
200
400
600
800
1000
1200
1400
1600
1800
2000
2200
2400
0.48 0.32 0.16 0.00 -0.16 -0.32 -0.48 -0.64
RHO_R_SPSML_R_VB
DCC of variables
200
400
600
800
1000
1200
1400
1600
1800
2000
2200
2400
22
0.50
DCC of variables
0.25
0.00
-0.25
-0.50
RHO_R_S P MID_R_P T
200
400
600
800
1000
1200
1400
1600
1800
2000
2200
2400
0.50
DCC of variables
0.25
0.00
-0.25
-0.50
-0.75
RHO_R_SPMID_R_VB
200
400
600
800
1000
1200
1400
1600
1800
2000
2200
2400
0.6 0.4 0.2 -0.0 -0.2 -0.4 -0.6
RHO_R_NASDAQ_R_V
DCC of variables
200
400
600
800
1000
1200
1400
1600
1800
2000
2200
2400
0.4
DCC of variables
0.2
-0.0
-0.2
-0.4
-0.6
RHO_R_NASDAQ_R_P
200
400
600
800
1000
1200
1400
1600
1800
2000
2200
2400
23
Table 4: Basic Description of the ADCCs
Series RHO_R_SP500_R_PTRAX RHO_R_SP500_R_VBMFX RHO_R_SPSML_R_PTRAX RHO_R_SPSML_R_VBMFX RHO_R_SPMID_R_PTRAX RHO_R_SPMID_R_VBMFX RHO_R_NASDAQ_R_PRAX RHO_R_NASDAQ_R_VMFX
Obs 2562 2562 2562 2562 2562 2562 2562 2562
Mean -0.00267 -0.00846 -0.01215 -0.02199 -0.00037 -0.0034 -0.07047 -0.0698
StdError 0.255178 0.272786 0.212666 0.219826 0.223287 0.23241 0.190121 0.216661
Minimum -0.5518 -0.62696 -0.4916 -0.53924 -0.47498 -0.53553 -0.52354 -0.57709
Maximum 0.553515 0.615198 0.470911 0.460596 0.482356 0.488589 0.396196 0.506662
24
Table 5: Regression of the DCC coefficients
DCC between SP500 and PTRAX Φ0
-0.00086 0.117672075 -0.06589 17.00826 0.69829 1.78298 -0.16494 -0.10356 -932.32476 -23.29620 16046.31317 9.65158 0.297718763 2.39038 -0.07587874 -25.80626 0.209103 0.172513
0.07981 6.65944 0.52071 1.49807 -0.09831 -0.06677 -882.10209 -24.52697 20576.78328 13.90929 0.13143 1.25154 -0.07682 -29.86054 0.380592
0.08013 6.747
Φ1
Φ2
Φ3
-882.33517 -24.15217 20577.55998 13.98492
Φ4
θ1
θ2
-0.07695 -29.87057 0.380151
Adjusted R2
Note: The dependent variables are the DCC between bond market returns and SP500. The model is:
ρ DCC ,t = φ0 + φ1 Rstockt + φ 2 Rbond ,t + φ3σ stock ,t + φ 4σ bond ,t + θ1 ∆i short ,t + θ 2 | ilong ,t − i short ,t | +ν
The t-values are below the parameters.
25
Table 6: Regression of the DCC coefficients
Series 1 Series 2 Φ0
SP500 PTRAX 0.08013 6.747 -882.33517 -24.15217 VBMFX 0.00849 0.25157 -1004.51817 -29.12513 6.66459 -0.07120 -24.70913 0.321819 0.380151 PTRAX
SPSML VBMFX 0.00629 0.26205 -684.18228 -22.75033 7.61391 -0.07216 -29.27358 0.345103 0.10949 11.84210 -635.71964 -21.01787 11.87399 -0.07496 -32.61027 0.388110
SPMID PTRAX 0.12391 12.35921 -697.41074 -17.45619 10.84769 -0.07394 -29.85125 0.363106 VBMFX 0.02439 0.94470 -747.99420 -18.01149 7.38954 -0.07412 -28.63659 0.338751
NASDAQ PTRAX 0.01109 1.51884 -207.98224 -20.46725 17.38249 -0.08660 -47.91656 0.486874 0.325800 VBMFX -0.02855 -1.08866 -194.00637 -18.09454 5.78007 -0.08289 -34.91541
Φ3
Φ4
20577.55998 30523.01471 12051.09667 13.98492 -0.07695 -29.87057
24964.45088 12257.82732 26228.14867 16484.20743 20631.17505
θ2
Adjusted R
2
Note: see the notes in the table 5.
26
Table 7: Logistic regression of the sign of the aDCCs
Dependent variable: probability of Rho_SP500_PTRAX being positive Coeff Std Error T-Stat 1.2388 0.1486 8.33512 -10661.5 712.6148 -14.9611 130881.6 16697.66 7.83832 -0.6797 0.0371 -18.3235
Variable Constant H_R_SP500 H_R_PTRAX LONG_SHORT
Signif 0 0 0 0
Note: the model is P(rho > 0) =
1 1+ e
−(φ0 +φ1Rstockt+φ2Rbond,t +φ3σ stock,t +φ4σbond,t +θ1∆ishort,t +θ2 |ilong,t −ishort,t |+ν )
27