Conclusion

In this paper, we extend the DCC model of Engle (2002a) with news impact in the conditional volatility and asymmetries in the dynamic correlation. We use three volatility models, GARCH, GJR-GARCH, and CARR model to go with the dynamic correlation models, DCC (symmetry) and ADCC (asymmetry). Therefore, we apply S&P 500 futures and 10-year T-bond futures to investigate whether asymmetries exist in conditional variances and correlations in the stock market and bond market of the U.S. The conditional volatilities of equity returns exhibit the asymmetric effects in the GJR-GARCH model while the little is found for bond returns. The performance of GJR-GARCH model is just better than OLS model, worse than the other dynamic models. We refer unfavorable performance in the GJR-GARCH model to its asymmetric effect in the stock return. Because of bad news has a great effect on the conditional variance that will increase the portfolio weights of 10-year T-bond. Furthermore, we examine the dynamic correlations of the S&P 500 futures and T-bond futures with DCC and ADCC model. Under consideration of asymmetric effect on conditional correlation, we find that the CARR-DCC and CARR-ADCC models are superior in the different target returns and risk aversions.

From the viewpoints of the investors, the above-mentioned models which mix rigorous mathematics and miraculous statistics are hard to understand for investors. The investors prefer the simplicity of investment strategy to the complexity of quantitative model. For that reason, the investors may choose the best quantitative model to allocate their assets and optimize their portfolio. In this paper, the investors may choose the CARR models as their quantitative methods in investment management since the CARR models lead to the better economic value of volatility. What is more, the economic value of volatility (switching fee) in this article is similar to “Two and Twenty” in hedge fund, this phrase indicates hedge

fund mangers charge a 2% of total asset value as a management fee, and an additional 20%

of profits earned.

Appendix A:

Proof of Optimal Portfolio Weights in a Minimum Variance Framework

( )

Substitute (4) into (3)

1 0

Appendix B:

Solution for Switching Fee

( ) ( ) ₍ ₎ ( )

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Table 1: Descriptive Statistics for Weekly Returns and Weekly Ranges Data of S&P 500 Index Futures and T-bond Futures, 1990/01/01-2008/04/25

The table reports the descriptive statistics for the data of weekly returns and ranges on S&P 500 index futures and T-bond futures used in this article. There are 955 weekly sample observations ranging from January 5, 1990 to April 25, 2008. All futures data are extracted form Thomson Datastream.

Panel A reports the sample moments of the data from 1990 to 2008. The data of weekly return on S&P 500 index futures and 10-year T-Bond futures are computed by

(

)

100 log× _e P_t^close P_t^close₋ . However, the data of weekly range on S&P 500 index futures and 10-Year T-bond futures are computed by^{100 log}^× ^e

(

^P^t^high ^P^t^low

)

. The data below are in weekly percentage unit, i.e., %.

The annualized value of mean and standard deviation are computed by Mean 52× and Std. Dev× 52, therefore, the annualized values of mean and standard deviation of S&P 500 index futures (10-Year T-bond futures) are 7.466% (1.414%) and 15.350% (6.120%).

The weekly return data of S&P 500 futures and 10-year T-bond futures are negative skewness (S=κ κ₃ ₂^{3 2} =μ σ₃ ³<0). In another word, the distribution of return is concentrated on the right of the figure and the left tail is longer. In addition, all the data of weekly return and range are positive excess kurtosis, the excess kurtosis statistic is defined as

2 4

4 2 3 4 3

K =κ κ − =μ σ − .

The Jarque-Bera statistic is used to test the null of whether the return and range data are unconditional normally distributed, based on the sample skewness and kurtosis. The Jarque-Bera is defined as ^Jarque⁻^Bera⁼ⁿ ⁶^×

(

^S²⁺^{1 4}^×^K²

)

^~^χ²⁽²⁾^.

Panel B and Panel C show the covariance and correlation matrices of S&P 500 futures and T-bond futures.

Panel A：S&P 500 Index Futures and 10-Year T-Bond Futures Weekly Data S&P 500 Index 10 Year Treasury Bond

Weekly Return (%) Weekly Range (%) Weekly Return (%) Weekly Range (%)

Panel B: Unconditional Covariance Matrices, 1990/01/01-2008/04/25

Weekly Return Weekly Range

S&P 500 T-bond S&P 500 T-bond S&P 500 4.526 0.029 S&P 500 2.990 0.106

T-bond 0.029 0.720 T-bond 0.106 0.288

Panel C: Unconditional Correlation Matrices, 1990/01/01-2008/04/25

Weekly Return Weekly Range

S&P 500 T-bond S&P 500 T-bond

S&P 500 1 0.016 S&P 500 1 0.114

T-bond 0.016 1 T-bond 0.114 1

Table 2: Estimation Results of Bivariate Return-based (GARCH) and Range-based (CARR) DCC and ADCC Model Using Weekly S&P 500 Index Futures and 10-Year T-bond Futures, 1990-2008.

Step 1 of GARCH (1,1) and CARR (1,1) Estimation

GARCH Model: h_{i t}_, =ω α ε_i+ _i _{i t}²_,₋₁+β_{i i t}h_,₋₁,ε_{i t}_, |Ι_t₋₁ ~N(0,h_{i t}_,) i=1, 2 CARR Model: λ_{i t}_, =ω α_i+ ℜ_i _{i t}_,₋₁+β λ_i _{i t}_{, 1}₋,ℜ Ι_{i t}_, | _t₋₁~ exp(1; )⋅ i=1, 2 where ε_{i t}_, and ℜ are the residual and range variable, respectively. _{i t}_, Step 2 of DCC and ADCC Estimation:

Dynamic Conditional Correlations:

(

)

^+A

(

¹ ^T¹

)

Asymmetric Dynamic Conditional Correlations:

(

)

^+A

(

¹ ^T¹

)

(

¹ ^T¹ ¹

)

where Z is the standard residual vector which is standardized by the volatility of _t GARCH and CARR model. _Q= ⎣ ⎦⎡ ⎤_q_ij and Q_t= ⎣ ⎦⎡q_{ij t}_, ⎤ are the unconditional and conditional covariance matrix of Z . The vector _t mt = Ι

[

Zt < o⁰

]

Zt and

1 _t ^T_t

m= T

∑

m m . Hence, conditional correlation ρ_12,t can easily be solved immediately. Panel A is the first-step estimation of the GARCH (1,1) and CARR (1,1) model. The results of estimation using GARCH and CARR models for S&P 500 Index futures and 10-year T-bond futures are displayed below. ^Q

( )

¹² ^{is the}

Ljung-Box Q statistic with 12 lags for the autocorrelation of time series data. Panel B is the second-step estimation of the DCC model. The values presented in parentheses are t-ratios for the coefficients and p-values for ^Q

( )

¹² ^.

Panel A: Step 1 of DCC Estimation

Volatilities Estimation of GARCH (Fitted by Return) and CARR (Fitted by Range) S&P 500 Index Futures 10 Year T-bond Futures GARCH(1,1) CARR(1,1) GARCH(1,1) CARR(1,1)

ωˆ ^0.028 Panel B: Step 2 of DCC and ADCC Estimation

Correlation Estimation of Return-based and Range-based DCC and ADCC Models S&P500 Index Futures and 10 Year T-bond Futures

GARCH-DCC GARCH-ADCC CARR-DCC CARR-ADCC

aˆ

Note: *** and ** represent significance at the 1% and 5% levels, respectively.

Table 3: Estimation Results of Bivariate Return-based DCC and ADCC Model with Asymmetry in Conditional Variance (GJR-GARCH) Using Weekly S&P 500 Index Futures and 10-Year T-bond Futures, 1990-2008

Step 1 of GJR-GARCH (1,1,1) Estimation:

GJR-GARCH Model:

Step 2 of GJR-GARCH-DCC and GJR-GARCH-ADCC Estimation:

GJR-GARCH-Dynamic Conditional Correlations:

GJR-GARCH-Asymmetric Dynamic Conditional Correlations:

(

)

^+A

(

¹ ^T¹

)

(

¹ ^T¹ ¹

)

where Z is the standard residual vector which is standardized by the volatility of _t GARCH model. _Q_{= ⎣ ⎦}_{⎡ ⎤}_q_ij and Q_t= ⎣ ⎦⎡q_{ij t}_, ⎤ are the unconditional and conditional covariance matrix of Z . The vector _t mt = Ι

[

Zt < o⁰

]

Zt and m=1T

∑

m m_t _t^T ^. Hence, conditional correlation ρ_12,t can easily be solved immediately. Panel A is the first-step estimation of the GJR-GARCH-DCC and GJR-GARCH-ADCC model. The results of estimation using GJR-GARCH model for S&P 500 Index futures and 10 year T-bond futures are displayed below. ^Q

( )

¹² is the Ljung-Box Q statistic with 12 lags for the autocorrelation of time series data. Panel B is the second-step estimation of the GJR-GARCH-DCC and GJR-GARCH-ADCC model.

The values presented in parentheses are t-ratios for the coefficients and p-values for

( )

¹²

Q .

Panel A: Step 1 of GJR-GARCH Estimation Volatilities Estimation of GJR-GARCH(1,1,1)

S&P 500 Index Futures 10 Year T-bond Futures

GJR-GARCH(1,1,1) GJR-GARCH(1,1,1)

ωˆ ^0.086 Panel B: Step 2 of GJR-DCC and GJR-ADCC Estimation Correlation Estimation of GJR-DCC and GJR-ADCC Models S&P500 Index Futures and 10 Year T-bond Futures

GJR-DCC GJR-ADCC

Note: *** and ** represent significance at the 1% and 5% levels, respectively.

Table 4: Comparison of the Volatility Values of Timing in the Minimum Variance Strategy Using Different Target Return in Symmetric Models, 1990-2008

This table reports the annualized expected returns (μ), volatility (σ), Sharpe rations (Sp), and switching fees (Δ_r) of OLS, GARCH-DCC, and CARR-DCC model with different target returns. The target returns are from 5 percent to 16 percent annually. Panel A shows the annualized means (μ), and volatility (σ) for each investment strategy. The estimated Sharpe ratios (Sp) of the OLS, GARCH-DCC, and CARR-DCC model are 0.498, 0.588, and 0.640, respectively. Panel B shows the average annualized switching fees (Δ^γ) with the varied constant relative risk aversion γ.

Panel A: Means, Volatilities and Sharpe Ratios of Optimal Portfolio OLS GARCH-DCC CARR-DCC

Target Return (%) μ σ S_p μ σ S_p μ σ S_p

5 5.000 1.740 0.498 5.102 1.647 0.588 5.190 1.652 0.640 6 6.000 3.749 0.498 6.220 3.550 0.588 6.410 3.559 0.640 7 7.000 5.759 0.498 7.338 5.452 0.588 7.630 5.466 0.640 8 8.000 7.768 0.498 8.456 7.354 0.588 8.850 7.373 0.640 9 9.000 9.778 0.498 9.574 9.257 0.588 10.070 9.280 0.640 10 10.000 11.787 0.498 10.692 11.159 0.588 11.290 11.187 0.640 11 11.000 13.796 0.498 11.810 13.061 0.588 12.510 13.094 0.640 12 12.000 15.806 0.498 12.928 14.964 0.588 13.730 15.002 0.640 13 13.000 17.815 0.498 14.045 16.866 0.588 14.950 16.909 0.640 14 14.000 19.824 0.498 15.163 18.768 0.588 16.170 18.816 0.640 15 15.000 21.834 0.498 16.281 20.671 0.588 17.390 20.723 0.640 16 16.000 23.843 0.498 17.399 22.573 0.588 18.610 22.630 0.640

Panel B: Switching Fees with Different Relative Risk Aversions OLS to GARCH-DCC OLS to CARR-DCC GARCH to CARR-DCC Target Return (%) Δ₁ Δ 5 Δ ₁₀ Δ₁ Δ 5 Δ ₁₀ Δ₁ Δ 5 Δ₁₀

5 0.175 0.227 0.240 0.292 0.365 0.383 0.117 0.138 0.143 6 0.561 0.810 0.869 0.886 1.232 1.314 0.325 0.422 0.445 7 1.148 1.744 1.885 1.759 2.583 2.777 0.611 0.839 0.892 8 1.939 3.033 3.291 2.915 4.416 4.766 0.976 1.383 1.475 9 2.938 4.674 5.080 4.354 6.715 7.257 1.416 2.041 2.177 10 4.143 6.658 7.238 6.075 9.455 10.217 1.932 2.797 2.979 11 5.556 8.969 9.744 8.073 12.605 13.603 2.517 3.636 3.859 12 7.174 11.588 12.572 10.341 16.128 17.371 3.167 4.54 4.799 13 8.993 14.494 15.695 12.872 19.985 21.474 3.879 5.491 5.779 14 11.010 17.663 19.083 15.655 24.139 25.868 4.645 6.476 6.785 15 13.219 21.070 22.708 18.680 28.554 30.516 5.461 7.484 7.808 16 15.615 24.692 26.543 21.933 33.198 35.381 6.318 8.506 8.838

Table 5: Comparison of the Volatility Values of Timing in the Minimum Variance Strategy Using Different Target Return in Asymmetric Models, 1990-2008

This table reports the annualized expected returns (μ), volatility (σ), Sharpe rations (Sp), and switching fees (Δ_r) of OLS, GARCH-ADCC, and CARR-ADCC model with different target returns. The target returns are from 5 percent to 16 percent annually. Panel A shows the annualized means (μ) and volatility (σ) for each investment strategy. The estimated Sharpe ratios (Sp) of the OLS, GARCH-ADCC, and CARR-ADCC model are 0.498, 0.590, and 0.640, respectively. Panel B shows the average annualized switching fees (Δ^γ) with the varied constant relative risk aversion γ.

Panel A: Means, Volatilities and Sharpe Ratios of Optimal Portfolio OLS GARCH-ADCC CARR-ADCC

Target Return (%) μ σ S_p μ σ S_p μ σ S_p

5 5.000 1.740 0.498 5.099 1.634 0.590 5.191 1.652 0.640 6 6.000 3.749 0.498 6.213 3.521 0.590 6.412 3.560 0.640 7 7.000 5.759 0.498 7.327 5.408 0.590 7.633 5.469 0.640 8 8.000 7.768 0.498 8.441 7.294 0.590 8.853 7.377 0.640 9 9.000 9.778 0.498 9.555 9.181 0.590 10.074 9.285 0.640 10 10.000 11.787 0.498 10.669 11.068 0.590 11.295 11.193 0.640 11 11.000 13.796 0.498 11.783 12.955 0.590 12.516 13.101 0.640 12 12.000 15.806 0.498 12.897 14.842 0.590 13.736 15.009 0.640 13 13.000 17.815 0.498 14.011 16.728 0.590 14.957 16.917 0.640 14 14.000 19.824 0.498 15.125 18.615 0.590 16.178 18.825 0.640 15 15.000 21.834 0.498 16.239 20.502 0.590 17.399 20.733 0.640 16 16.000 23.843 0.498 17.353 22.389 0.590 18.620 22.641 0.640

Panel B: Switching Fees with Different Relative Risk Aversions OLS to Return ADCC OLS to GARCH-ADCC GARCH to CARR-ADCC Target Return (%) Δ₁ Δ 5 Δ ₁₀ Δ₁ Δ 5 Δ ₁₀ Δ₁ Δ 5 Δ ₁₀

5 0.172 0.225 0.238 0.293 0.366 0.383 0.121 0.141 0.145 6 0.557 0.808 0.868 0.886 1.232 1.314 0.329 0.424 0.446 7 1.144 1.746 1.889 1.760 2.583 2.777 0.616 0.837 0.888 8 1.939 3.042 3.303 2.915 4.415 4.764 0.976 1.373 1.461 9 2.941 4.694 5.103 4.354 6.712 7.254 1.413 2.018 2.151 10 4.153 6.690 7.275 6.074 9.451 10.211 1.921 2.761 2.936 11 5.574 9.016 9.797 8.070 12.598 13.595 2.496 3.582 3.798 12 7.202 11.652 12.643 10.337 16.118 17.360 3.135 4.466 4.717 13 9.032 14.576 15.784 12.866 19.972 21.460 3.834 5.396 5.676 14 11.062 17.764 19.192 15.647 24.123 25.852 4.585 6.359 6.660 15 13.286 21.191 22.838 18.670 28.536 30.497 5.384 7.345 7.659 16 15.697 24.833 26.694 21.921 33.177 35.358 6.224 8.344 8.664

Table 6: Comparison of the Volatility Values of Timing in the Minimum Variance Strategy Using Different Target Return in Asymmetric Models, 1990-2008

This table reports the annualized expected returns (μ), volatility (σ), Sharpe rations (Sp), and switching fees (Δ_r) of OLS, GJR-GARCH-DCC, and GJR-GARCH-ADCC model with different target returns. The target returns are from 5 percent to 16 percent annually. Panel A shows the annualized means (μ) and volatility (σ) for each investment strategy. The estimated Sharpe ratios (S_p) of the OLS, GJR-GARCH-DCC, and GJR-GARCH-DCC model are 0.498, 0.556, and 0.556, respectively. Panel B shows the average annualized switching fees (Δ^γ) with the varied constant relative risk aversion γ.

Panel A: Means, Volatilities and Sharpe Ratios of Optimal Portfolio OLS GJR-DCC GJR-ADCC

Target Return (%) μ σ Sp μ σ Sp μ σ Sp

5 5.000 1.740 0.498 5.045 1.640 0.556 5.045 1.639 0.556 6 6.000 3.749 0.498 6.098 3.534 0.556 6.097 3.531 0.556 7 7.000 5.759 0.498 7.150 5.428 0.556 7.149 5.423 0.556 8 8.000 7.768 0.498 8.203 7.322 0.556 8.200 7.315 0.556 9 9.000 9.778 0.498 9.255 9.216 0.556 9.252 9.207 0.556 10 10.000 11.787 0.498 10.308 11.110 0.556 10.304 11.099 0.556 11 11.000 13.796 0.498 11.360 13.004 0.556 11.356 12.991 0.556 12 12.000 15.806 0.498 12.412 14.898 0.556 12.408 14.883 0.556 13 13.000 17.815 0.498 13.465 16.792 0.556 13.460 16.775 0.556 14 14.000 19.824 0.498 14.517 18.687 0.556 14.512 18.667 0.556 15 15.000 21.834 0.498 15.570 20.581 0.556 15.563 20.559 0.556 16 16.000 23.843 0.498 16.622 22.475 0.556 16.615 22.451 0.556 Panel B: Switching Fees with Different Relative Risk Aversions OLS to GJR DCC OLS to GJR ADCC GJR DCC to ADCC 12 6.481 10.788 11.750 6.485 10.797 11.760 0.004 0.009 0.010 13 8.189 13.561 14.737 8.194 13.572 14.750 0.005 0.011 0.013 14 10.089 16.593 17.987 10.097 16.607 18.003 0.008 0.014 0.016 15 12.178 19.862 21.472 12.187 19.879 21.491 0.009 0.017 0.019 16 14.450 23.344 25.167 14.461 23.364 25.188 0.011 0.020 0.021

Table 7: Comparison of the Incremental Volatility Values of Timing in the Minimum Variance Strategy among the OLS, GJR-DCC, GJR-ADCC, Return-based DCC, Range-based DCC, Return-based ADCC, and Range-based ADCC model using 5%, 10% and 16% Target Return respectively, 1990-2008.

This table reports the incremental time-varying values of volatility. In this paper, we propose the asymmetric effect on the DCC model for better performance. Panel A shows the volatility value with 5% target return. In Panel A, the range ADCC model has no superior as a dynamic forecasting model though there is no significant difference between range DCC and range ADCC model. Panel B and Panel C with target return of 10% and 15% respectively show the opposite results of incremental value of volatility on model selection. In Panel B as well as in Panel C, the return-based ADCC is better than the return-based DCC model, and the range-based DCC is superior to the range ADCC model.

Panel A: Incremental Switching Fees with Target Return 5%

OLS GJR-DCC GJR-ADCC GARCH-ADCC GARCH-DCC CARR-DCC CARR-ADCC

CRRA Δ1 Δ ₅ Δ ₁₀ Δ₁ Δ₅ Δ₁₀ Δ₁ Δ₅ Δ₁₀ Δ₁ Δ ₅ Δ₁₀ Δ₁ Δ₅ Δ₁₀ Δ₁ Δ₅ Δ₁₀ Δ₁ Δ ₅ Δ ₁₀ OLS ^0.000 0.000 0.000 0.116 0.167 0.179 0.116 0.167 0.179 0.172 0.225 0.238 0.175 0.227 0.240 0.292 0.365 0.383 0.293 0.366 0.383

GJR-DCC ^0.000 ^0.000 ^0.000 ^0.000 ^0.000 ^0.000 ^0.056 0.058 0.059 0.059 0.060 0.061 0.116 0.167 0.179 0.177 0.199 0.204

GJR-ADCC ^0.000 ^0.000 ^0.000 ^0.056 ^0.058 ^0.059 ^0.059 ^0.060 ^0.061 ^0.176 ^0.198 ^0.204 ^0.177 0.199 0.204

GARCH-ADCC ^0.000 ^0.000^0.000 ^0.003 ^0.002 ^0.002 ^0.120 ^0.140 ^0.145 ^0.121 ^0.141 ^0.145 GARCH-DCC 0.000 0.000 0.000 0.117 0.138 0.143 0.118 0.139 0.143

CARR-DCC 0.000 0.000 0.000 0.001 0.001 0.000

CARR-ADCC 0.000 0.000 0.000

Panel B: Incremental Switching Fees with Target Return 10%

OLS GJR-DCC GJR-ADCC GARCH-DCC GARCH-ADCC CARR-ADCC CARR-DCC

CRRA Δ₁ Δ ₅ Δ ₁₀ Δ₁ Δ₅ Δ₁₀ Δ₁ Δ₅ Δ₁₀ Δ₁ Δ ₅ Δ₁₀ Δ₁ Δ₅ Δ₁₀ Δ₁ Δ₅ Δ₁₀ Δ₁ Δ ₅ Δ ₁₀ OLS 0.000 0.000 0.000 3.660 6.108 6.674 3.661 6.112 6.679 4.143 6.658 7.238 4.153 6.690 7.275 6.074 9.451 10.211 6.075 9.455 10.217

GJR-DCC ^0.000 ^0.000 ^0.000 ^0.001 ^0.004 ^0.005 ^0.483 0.550 0.564 0.493 0.582 0.601 3.660 6.108 6.674 2.415 3.347 3.543

GJR-ADCC ^0.000 ^0.000 ^0.000 ^0.482 ^0.546 ^0.559 ^0.492 ^0.578 ^0.596 ^2.413 ^3.339 ^3.532 ^2.414 3.343 3.538

GARCH-ADCC ^0.000 ^0.000^0.000 ^0.010 ^0.032 ^0.037 ^1.931 ^2.793 ^2.973 ^1.932 ^2.797 ^2.979 GARCH-DCC 0.000 0.000 0.000 1.921 2.761 2.936 1.922 2.765 2.942

CARR-DCC 0.000 0.000 0.000 0.001 0.004 0.006

CARR-ADCC 0.000 0.000 0.000

Panel C: Incremental Switching Fees with Target Return 16%

OLS GJR-DCC GJR-ADCC GARCH-DCC GARCH-ADCC CARR-ADCC CARR-DCC

CRRA Δ1 Δ ₅ Δ ₁₀ Δ₁ Δ₅ Δ₁₀ Δ₁ Δ₅ Δ₁₀ Δ₁ Δ ₅ Δ₁₀ Δ₁ Δ₅ Δ₁₀ Δ₁ Δ₅ Δ₁₀ Δ₁ Δ ₅ Δ ₁₀ OLS ^0.000 0.000 0.000 14.450 23.344 25.167 14.461 23.364 25.188 15.615 24.692 26.543 15.697 24.833 26.694 21.921 33.177 35.358 21.933 33.198 35.381

GJR-DCC ^0.000^0.000^0.000^0.011 ^0.020 0.021 1.165 1.348 1.376 1.247 1.489 1.527 14.450 23.344 25.167 7.483 9.854 10.214

GJR-ADCC ^0.000^0.000^0.000^1.154 ^1.328 1.355 1.236 1.469 1.506 7.460 9.813 10.170 7.472 9.834 10.193

GARCH-ADCC ^0.000^0.000^0.000^0.082 ^0.141 0.151 6.306 8.485 8.815 6.318 8.506 8.838

GARCH-DCC 0.000 0.000 0.000 6.224 8.344 8.664 6.236 8.365 8.687

CARR-DCC 0.000 0.000 0.000 0.012 0.021 0.023

CARR-ADCC 0.000 0.000 0.000

Panel A: Close Price

S&P 500 Index Futures 10 Year Treasury Bond Futures

200

S&P500 Index Futures (%) 10 Year Treasury Bond Futures (%)

-16

S&P500 Index Futures (%) 10 Year Treasury Bond Futures (%)

Panel D: Open Interests and Trading Volumes (Futures Contract) S&P500 Index Futures 10 Year Treasury Bond Futures

0 1000000 2000000 3000000 4000000

90 92 94 96 98 00 02 04 06

OPEN INTEREST_SP500 VOLUME TRADED_SP500

0 1000000 2000000 3000000 4000000 5000000 6000000 7000000 8000000

90 92 94 96 98 00 02 04 06 OPEN INTEREST_TBOND VOLUME TRADED_TBOND

Figure 1: S&P 500 Index and 10 Year Treasury Bond Weekly Closing Prices, Returns, Ranges, Open Interests and Trading Volumes, 1990/01/05-2008/04/25. This figure shows the weekly close prices, returns, ranges, open interests and trading volumes of S&P 500 index futures and 10 year Treasury bond futures over the sample period.

Panel A: Volatility Estimates for the GARCH Model

Panel B: Volatility Estimates for the GJR-GARCH Model

Panel C: Volatility Estimates for the CARR Model

0 1 2 3 4 5 6 7 8

90 92 94 96 98 00 02 04 06

S&P 500 T-bond

Figure 2: Volatility Estimates for the GARCH, GJR-GARCH and CARR Model

Panel A: Correlation and Covariance Estimates of Return-based DCC (GARCH-DCC) and Range-based DCC (CARR-DCC) Model

Correlation Estimates of DCC Model

-.8 Covariance Estimates of DCC Model

-3

Panel B: Correlation and Covariance Estimates of Return-based ADCC (GARCH-ADCC) and Range-based ADCC (CARR-ADCC) Model

Correlation Estimates of ADCC Model

-.8 Covariance Estimates of ADCC Model

-3

Panel C: Correlation and Covariance Estimates of GJR-GARCH-DCC and GJR-GARCH-ADCC Model Correlation Estimates of GJR-GARCH-DCC and ADCC Model

-.8

GJR GARCH DCC GJR GARCH ADCC

Covariance Estimates of GJR-GARCH-DCC and ADCC Model

-3

GJR GARCH DCC GJR GARCH ADCC

Figure 3: Correlation and Covariance Estimates for the DCC and ADCC Fitted by GARCH, CARR and GJR-GARCH, respectively.

Panel A: Portfolio Weights Derived by the GARCH-DCC Model