Dynamic stock–bond return correlations and financial market uncertainty

An exception is d’Addona and Kind (2006). They test the G7 countries by using economic fundamentals to explain the correlation between stock and bond returns. Their study is based on monthly data. Our study, as discussed at a later point, includes daily and weekly observations. As a result, the state variables used to explain the stock–bond relation will be different.

The BEKK method is a GARCH-type model developed by Baba et al. (1991), a paper that has been later published as Engle and Kroner (1995).

Scruggs and Glabadanidis (2003) also propose a time-varying model to emphasize a similar issue.

Using the same maturity of each government bond index allows us to make comparisons across different markets. Yet, government bonds have very low default risk. The results could be quite different if the bonds were of lower credit quality.

The unconditional stock–bond return correlations for CA, FR, GR, IT, the UK, and the US are: −0.08 (t = −5.90), −0.11 (t = −8.090), −0.12 (t = −8.63). 0.092 (6.59), −0.095 (−6.79), and −0.17 (−12.58), respectively. The correlation table for each country is available upon request.

DeGoeij and Marquering (2004) apply a GARCH-BEW model to examine the stock–bond relation. However, the BEW method cannot ensure the positive definiteness of the covariance matrix. D’Addona and Kind (2006) provide a comprehensive study of the stock–bond correlation by comparing rolling regression, BEKK-GARCH, and CCC-GARCH models using monthly data for the period January 1980 through March 1997. Yet, the BEKK method in modeling a multivariate GARCH approach often involves computational complexity, especially when the variables involved get larger. A main drawback of the BEKK and the constant correlation coefficient (CCC) GARCH (Bollerslev 1990) models is that the correlation coefficient in a multivariate setting is assumed to be invariant over time.

Chiang et al. (2007a) and Yu et al. (2010) have applied Engle’s (2002) model to analyze equity market correlations.

The variances of the normalized residuals z _S,t and z _B,t equal q _S,t and q _B,t , respectively. Both q _S,t and q _B,t have an expected value of 1.

Taking the U.S. market in Table 1 as an example, although both a = 0.0396 and b = 0.9556 are statistically significant, the AR(1) term appears to play a dominant role in explaining the evolution of time-varying correlation. This phenomenon applies to all the markets under investigation. The parameter κ is insignificant except for the U.K., suggesting that there is no significant evidence in favor of the asymmetric hypothesis of having a positive shock vs. a negative shock. We then estimate the model by excluding this parameter from the insignificant countries.

To save space, we do not report the statistics derived from weekly data. The estimates are available upon request.

On February 27, 2007, the Dow Jones industrial average in the U.S. market tumbled 416.02 points, to 12,216.24, the biggest point loss (−3.3 %) since September 17, 2001, when the 30-share index was down nearly 685 points. The S&P 500 index fell 50.33 points (−3.5 %) to 1,399.04. It was the worst one-day percentage loss since March 2003. In Canada, the Toronto composite index was down 2.7 %. In Europe, the FTSE100 dropped 2.3 %, the DAX30 slipped 2.96 %, and the CAC40 lost 3 %. In Asia, China’s Shanghai stock index plunged 8.8 %, Japan’s market fell 2.9 %, and Hong Kong fell 2.5 % on February 27, 2007, its biggest one-day drop in a decade. This outbreak of investor gloom was due to weak corporate profits and expectations of reversals in market spreads worldwide. On the other hand, Treasury prices rallied as investors sought safety.

As we shall show in the next section (in Table 4), the end of the first sample period is mainly determined by a logistic smooth transition model that shows the structural change. We find that the transition mid-points for most countries are detected at the start of 1999. We use the U.S. market as a benchmark because of the relatively large size of its capital market.

There is no clear-cut beginning for the crisis period. We choose September 7, 2008, because on that date, the Federal Housing Finance Agency (FHFA) placed Fannie Mae and Freddie Mac in conservatorship. As conservator, the FHFA has full powers to control the assets and operations of the firms (Jickling 2008).

To save space, we report only the statistics for daily data. Both weekly and monthly data show similar results. The statistics are available upon request.

Favero (2009) reports that the Baa–Aaa spread and the volatility in the VIX strongly co-move and have the same cyclical properties during NBER-dated recessions. As discussed below, we include both the Baa–Aaa spread and VIX volatility in our explanatory variables.

Since the dependent variable is bound to interval [−1, +1], we apply a Fisher transformation \(\left( {\hat{\hat{\rho }}_{SB,t}^{*} = \frac{1}{2}\ln \left[ { \frac{{1 + \hat{\rho }_{sB,t}^{*} }}{{1 - \hat{\rho }_{SB,t}^{*} }} } \right]} \right)\) on the correlation coefficient first and then conduct the regression estimation.

Connolly et al. (2005) find that bond returns tend to be high (low) relative to stock returns during the time period when implied stock market volatility is high (low).

Tang and Yan (2010) use the credit risk spread, which is the difference between the yield on a corporate bond and a government bond, to measure the risk of investing in bonds. In practice, the credit spread can be measured by the difference between the yield on Moody’s Aaa seasoned corporate bonds and the 10-year Treasury bond; an increase in the credit spread will impair the bondholder by causing a higher yield to maturity and a lower bond price. Because we have used a few proxy variables in the measure of bond risk, to avoid the multi-collinearity problem, we do not include this measure of credit risk in our estimations.

Fama and French (1993) define default risk as the difference between the return on a market portfolio of long-term composite corporate bonds (Ibbotson Associates) and the return on long-term government bonds.

The TED spread exceeded 300 basis points in September and early October 2008, after the bankruptcies of several big banks and investment companies in the U.S. market that constituted part of the global financial crisis. On October 10, 2008, the TED spread hit a record high of 458 basis points (the U.S. 3-month Treasury bill was 0.24 % and the corresponding LIBOR was 4.818 %; the difference was 4.58 %), signifying a severe default risk and credit crunch in interbank lending.

Krugman (2009) notes that the “TED was a good indicator of fear in the banking system.” This view is consistent with an earlier report by de Aenlle (1992), who noted that “as the TED spread continues to narrow, confidence grows. That, in turn, means lower interest rates and, much of the time, a higher stock market.”

The interaction between the VIX and bond market news can be seen in an episode on August 8, 2011. As Standard & Poor's announced that it had downgraded the U.S. credit rating from AAA to AA+, the Dow Jones industrial average sank 634.76 points, or 5.6 %, falling to 10,810. The S&P 500 lost 79.92 points, or 6.7 %, falling to 1,120. And the Nasdaq Composite dropped 174.72 points, or 6.9 %, falling to 2,358. In London, the FTSE closed at 5,068.95, off 178.04 points, while the German DAX lost 312.89, to close at 5,923.27. The VIX "fear" index jumped 44 %, to 45.98. Ironically, bond prices rose, and the yield on the benchmark 10-year U.S. Treasury bill fell to 2.34 % from 2.56 %. (see Sweet 2011. http://​money.​cnn.​com/​2011/​08/​08/​markets/​markets_​newyork/​index.​htm, August 30, 2011).

Although the literature has suggested including uncertainty among the macroeconomic fundamentals such as the variability of GDP and inflation, we did not employ these variables because of their inappropriateness in measuring daily observations. Recent studies find very little evidence to support including these variables (Baele et al. 2010).

Footnote 34 presents a possible explanation for the endogenous behavior of the coefficient of the VIX. The prevailing stock index, TOTM, is used as a wealth effect, serving as a control variable.

Since the independent variables in Eq. (7) contain various measures of uncertainty from the stock market (or the bond market), some degree of muticollinearity may be present. Econometric theory suggests that if two variables contain a similar information set (near muticollinearity is present), the estimated standard errors become large. As a result, the usual t-tests will lead to the conclusion that parameter values are not significant. Taking the U.S. market as an example, if the VIX and the conditional variance are highly correlated, either or both coefficients will be insignificant. The evidence from Tables 6–9 in most cases indicates that both coefficients are statistically significant. This indicates that the muticollinearity problem is not serious, should it exist. An econometric treatment for dealing with the muticollinearity problem is to drop an insignificant variable. In our analysis, we don’t think this is necessary. Moreover, dropping a relevant variable may cause estimates of the parameters of the remaining variables to be biased (Kennedy 2008, 197).

When replacing the VIX by the VDAX, the implied volatility from the German DAX, we obtain a comparable result (not reported to save space). The estimations using the VDAX are available upon request.

Our finding is slightly different from the result documented by Connolly et al. (2005), since in their study of the U.S. market, the VIX is used to measure stock market volatility; the conditional variance was left out of their model. In our model, the VIX is considered as an external influence on the non-U.S. markets. The foreign influence on the U.S. market using the VDAX yields a similar result. The coefficient of the VDAX on the stock–bond correlation is −0.0305 (t = −15.31).

Kwan (1996) establishes a micro-level linkage with a firm’s valuation in that an increase in the default spread is interpreted as a threat to the expected future cash flows of the issuing firm, which also affects the firm’s stock price. His study assumes that stocks and bonds are issued by the same firm. Hence, specific information about the firm should have an impact on both the firm's outstanding stocks and its outstanding bonds, leading to a co-movement between individual stock and bond prices. This study is different from Kwan’s, since we conduct a macro-level investigation and examine the relationship of returns between 10-year government bonds and a stock index in response to default risk.

On December 3, 2008, the default risk spread hit a record high of a 3.5 % annual rate against an average level of 0.94 % for the whole sample period.

As we shall see in the estimation of the crisis period.

If we have a linear function that \(\hat{\hat{\rho }}_{SB,t}^{*} = \varphi_{0} + \varphi_{1} \sigma_{S,t}^{2} + \varphi_{2} \sigma_{B,t}^{2} + \varphi_{3} VIX_{t} + \varphi_{4} (DEFT)_{t} + \varphi_{5} (TED)_{t} + \varphi_{6} TOTM_{t} + \varepsilon_{t} ,\quad ({\text{A}}1)\) and assume \(\varphi_{3} = \nu_{0} + \nu_{1} \, DEFT_{t} + e_{t} ,\quad ({\text{A2}}).\) Substituting A(2) into A(1) will yield a nonlinear component in Eq. (7).

The reason for selecting the coefficient of the implied volatility of stock returns (VIX), rather than that of the domestic conditional variance (\(\sigma_{s,t}^{2}\)), is that the VIX appears to be an external force that is consistently interacting with the DEFT in the same time zone.

We also estimate the regression model that replaces the VIX index with the VDAX index. Evidence form Äijö (2008) shows that a large proportion of the forecast variance of the SMI (Swiss market index) and the STOXX (Euro STOXX 50 index) can be explained by the DAX. The VDAX can be viewed as a proxy for the VSTOXX, European volatility. Our test indicates (not reported) that all of the statistics using the VDAX produce very similar results in terms of signs and significance levels. This is not surprising, since we find that the correlation between the VIX and the VDAX for the sample period is as high as 0.87.

We re-estimate the regression model by employing weekly data. The evidence shows that both daily and weekly data produce very comparable statistical outcomes. The estimates of variables, in general, maintain similar qualitative results as we check the sign, significance level, and explanatory power. To save space, we do not report the results. However, the table is available upon request.

An exception in the stock market is the estimated coefficient of \(\sigma_{s,t}^{2}\) for Italy, which exhibits a positive sign in the first period. A negative sign is found in the bond market in the second period for the U.S. and Canada. During this period (1/02/1999–9/06/2008), the VIX was relatively low and so were interest rates. The volatility of the bond market appears to play a significant role in explaining the stock–bond correlations.

This period coincides with the high-tech bubble.

It is referred to as the VIX variable in markets outside the U.S. and the VDAX in the U.S. market.