# Cross Hedging Stock Sector Risk with Index Futures by Considering the Global Equity Systematic Risk

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## Abstract

**:**

## 1. Introduction

## 2. Regime Switching Volatility Spillover GARCH (RSVSG) Model

## 3. Measurements of Hedging Performance, Minimum Variance Hedge Ratio (MVHR), and Volatility Spillover Ratio

## 4. Data Description and Empirical Results

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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1 | Because the state probability is time varying, ${\phi}_{{s}_{t}}$ and ${\omega}_{{s}_{t}}$ are also time varying after taking the weighted average using state probabilities. |

2 | Because all hedged portfolio returns are pretty small, the value of the expected utility is dominated by the second moment of the hedged portfolio return. Although it is not reported here, we find that hedging results are robust to the choice of the coefficient of absolute risk aversion for a wide range of $\kappa $ ($\kappa =1,4,20$). A hedging strategy with lower volatility has higher expected utility regardless the choice of the coefficient of absolute risk aversion. |

**Figure 4.**State-dependent volatilities of MSCI world index futures estimated with RSVSG for textiles.

Textiles | Communication and Internet | Transportation | Retailing | |
---|---|---|---|---|

Mean | 0.086 | 0.028 | −0.111 | 0.147 |

Maximum | 9.232 | 6.265 | 7.658 | 9.363 |

Minimum | −11.979 | −8.869 | −11.430 | −11.416 |

SD | 2.953 | 2.171 | 2.938 | 2.727 |

Skewness | −0.478 | −0.414 | −0.437 | −0.317 |

Kurtosis | 4.548 | 4.679 | 4.293 | 4.333 |

Jarque–Bera | 54.808 *** | 57.960 *** | 40.306 *** | 36.042 *** |

Automobile | Plastics and Chemicals | TAIFEX Futures | Taiwan 50 Futures | |

Mean | 0.246 | 0.114 | 0.085 | 0.106 |

Maximum | 12.810 | 8.512 | 6.392 | 6.540 |

Minimum | −14.360 | −11.618 | −10.015 | −10.328 |

SD | 3.867 | 2.786 | 2.560 | 2.588 |

Skewness | −0.279 | −0.478 | −0.546 | −0.330 |

Kurtosis | 3.940 | 4.605 | 3.897 | 3.493 |

Jarque–Bera | 19.770 *** | 57.683 *** | 33.023 *** | 11.215 *** |

Taiwan NFNE Futures | MSCI World Index Futures | |||

Mean | 0.090 | 0.192 | ||

Maximum | 7.668 | 7.941 | ||

Minimum | −9.659 | −10.044 | ||

SD | 2.579 | 2.173 | ||

Skewness | −0.490 | −0.577 | ||

Kurtosis | 4.113 | 5.134 | ||

Jarque–Bera | 36.371 *** | 97.375 *** |

**Table 2.**Estimates of unknown parameters of the regime switching volatility spillover GARCH (RSVSG). Data estimation period is from 20 May 2009 to 30 December 2015.

Textiles | Retailing | Transportation | Textiles | Retailing | Transportation | ||
---|---|---|---|---|---|---|---|

Transition Probability | Spillover Equation | ||||||

${p}_{0}$ | 2.829 | 1.525 | 0.653 | ${\gamma}_{W1}$ | 1.137 | 0.000 | 0.000 |

(0.399) *** | (0.576) *** | (0.430) * | (0.463) *** | (0.041) | (0.083) | ||

${q}_{0}$ | 1.954 | −0.352 | 1.332 | ${\alpha}_{W1}$ | 0.240 | 0.270 | 0.166 |

(0.566) *** | (0.453) | (0.468) *** | (0.098) *** | (0.126) ** | (0.082) *** | ||

Covariance Equation | ${\beta}_{W1}$ | 0.242 | 0.544 | 0.379 | |||

${\gamma}_{cc1}$^{3} | −1.270 | 0.152 | 1.165 | (0.154) * | (0.085) *** | (0.094) *** | |

(0.396) *** | (0.531) | (0.267) *** | ${\phi}_{1}$ | 0.768 | 0.600 | 0.495 | |

${\gamma}_{cc2}$ | 3.604 | 1.480 | 2.277 | (0.086) *** | (0.080) *** | (0.107) *** | |

(0.414) *** | (0.765) ** | (0.859) *** | ${\omega}_{1}$ | 0.763 | 0.719 | 0.391 | |

${\gamma}_{cf1}$ | −0.418 | 0.377 | 1.203 | (0.071) *** | (0.074) *** | (0.112) *** | |

(0.293) * | (0.282) * | (0.210) *** | ${\gamma}_{W2}$ | 7.291 | 3.046 | 1.314 | |

${\gamma}_{cf2}$ | 2.313 | 0.657 | 2.317 | (2.446) *** | (5.129) | (0.649) *** | |

(0.302) *** | (0.585) | (0.199) *** | ${\alpha}_{W2}$ | 0.220 | 0.000 | 0.118 | |

${\gamma}_{ff1}$ | 0.014 | 0.001 | −0.002 | (0.153) * | (0.039) | (0.086) * | |

(0.059) | (0.040) | (0.076) | ${\beta}_{W2}$ | 0.049 | 1.000 | 0.874 | |

${\gamma}_{ff2}$ | 1.158 | 0.002 | 0.245 | (0.284) | (1.212) | (0.185) *** | |

(0.510) *** | (0.022) | (0.974) | ${\phi}_{2}$ | 0.544 | 0.775 | 0.701 | |

${\alpha}_{cc1}$ | 0.000 | 0.021 | 0.038 | (0.131) *** | (0.168) *** | (0.090) *** | |

(0.035) ** | (0.056) | (0.065) | ${\omega}_{2}$ | 0.739 | 0.844 | 0.841 | |

${\alpha}_{cc2}$ | −0.029 | −0.050 | 0.000 | (0.092) *** | (0.138) *** | (0.073) *** | |

(0.163) | (0.147) | (0.04) | |||||

${\alpha}_{ff1}$ | 0.000 | 0.080 | −0.076 | ||||

(0.049) | (0.071) | (0.087) | |||||

${\alpha}_{ff2}$ | −0.145 | 0.066 | 0.000 | ||||

(0.128) | (0.153) | (0.018) | |||||

${\beta}_{cc1}$ | 0.591 | 0.781 | 0.236 | ||||

(0.194) * | (0.075) *** | (0.210) | |||||

${\beta}_{cc2}$ | 0.428 | 1.365 | −0.835 | ||||

(0.269) *** | (0.155) *** | (0.332) *** | |||||

${\beta}_{ff1}$ | 0.863 | 0.845 | −0.169 | ||||

(0.050) * | (0.057) *** | (0.315) | |||||

${\beta}_{ff2}$ | −0.156 | 1.286 | 0.091 | ||||

(0.303) *** | (0.135) *** | (0.571) | |||||

$LL$^{2} | −1797.51 | −1781.97 | −1827.45 | ||||

Communication and Internet | Automobile | Plastics and Chemicals | Communication and Internet | Automobile | Plastics and Chemicals | ||

Transition Probability | Spillover Equation | ||||||

${p}_{0}$ | 1.624 | 2.208 | 1.463 | ${\gamma}_{W1}$ | 0.000 | 0.000 | 0.000 |

(1.122) * | (0.829) *** | (0.485) *** | (0.050) | (0.080) | (0.076) | ||

${q}_{0}$ | 0.001 | −0.013 | 0.026 | ${\alpha}_{W1}$ | 0.234 | 0.196 | 0.279 |

(0.036) | (0.098) | (0.202) | (0.114) ** | (0.224) | (0.198) * | ||

Covariance Equation | ${\beta}_{W1}$ | 0.579 | 0.593 | 0.512 | |||

${\gamma}_{cc1}$^{3} | (0.564) | 0.163 | 0.711 | (0.188) *** | (0.147) *** | (0.102) *** | |

(0.691) | (0.768) | (0.340) ** | ${\phi}_{1}$ | 0.436 | 0.863 | 0.760 | |

${\gamma}_{cc2}$ | 2.729 | 2.936 | 2.172 | (0.058) *** | (0.110) *** | (0.078) *** | |

(0.520) *** | (1.434) ** | (0.699) *** | ${\omega}_{1}$ | 0.781 | 0.688 | 0.725 | |

${\gamma}_{cf1}$ | 0.424 | 0.093 | 0.743 | (0.094) *** | (0.067) *** | (0.074) *** | |

(0.282) * | (0.514) | (0.475) * | ${\gamma}_{W2}$ | 2.976 | 5.378 | 2.880 | |

${\gamma}_{cf2}$ | 1.868 | 0.785 | 1.173 | (9.570) | (18.033) | (8.560) | |

(1.070) *** | (0.211) *** | (0.376) *** | ${\alpha}_{W2}$ | 0.000 | 0.128 | 0.000 | |

${\gamma}_{ff1}$ | −0.001 | 0.000 | 0.000 | (0.035) | (0.316) | (0.040) | |

(0.031) | (0.030) | (0.115) | ${\beta}_{W2}$ | 1.000 | 0.872 | 1.000 | |

${\gamma}_{ff2}$ | 0.516 | 0.000 | 0.228 | (2.055) | (4.023) | (2.150) | |

(3.655) | (0.044) | (1.430) | ${\phi}_{2}$ | 0.485 | 1.048 | 0.949 | |

${\alpha}_{cc1}$ | −0.075 | 0.014 | 0.120 | (0.137) *** | (0.264) *** | (0.147) *** | |

(0.121) | (0.033) | (0.081) * | ${\omega}_{2}$ | 0.727 | 0.897 | 0.794 | |

${\alpha}_{cc2}$ | 0.485 | −0.010 | 0.185 | (0.204) *** | (0.116) *** | (0.122) *** | |

(0.273) ** | (0.104) | (0.121) * | |||||

${\alpha}_{ff1}$ | −0.130 | 0.063 | 0.072 | ||||

(0.120) | (0.083) | (0.091) | |||||

${\alpha}_{ff2}$ | 0.153 | −0.129 | 0.098 | ||||

(0.265) | (0.106) | (0.102) | |||||

${\beta}_{cc1}$ | 0.662 | 0.853 | 0.699 | ||||

(0.126) *** | (0.077) *** | (0.057) *** | |||||

${\beta}_{cc2}$ | −0.395 | 1.278 | 0.990 | ||||

(1.109) | (0.251) *** | (0.329) *** | |||||

${\beta}_{ff1}$ | 0.805 | 0.956 | 0.760 | ||||

(0.124) *** | (0.031) *** | (0.063) *** | |||||

${\beta}_{ff2}$ | 0.948 | 1.089 | 1.162 | ||||

(0.743) | (0.105) *** | (0.088) *** | |||||

$LL$^{2} | −1762.47 | −1933.01 | −1615.74 |

^{1}Figures in parentheses are standard errors and *, **, and *** indicate significance at the 10% level, 5% level, and 1% level, respectively;

^{2}LL stands for the log likelihood value;

^{3}State 1 is the low volatility state.

**Table 3.**Out-of-sample hedging effectiveness without regime switching and global volatility spillover effects estimated with bivariate BEKK GARCH model.

Variance of Hedged Portfolio Return | Percentage Variance Reduction ^{1} | Improvement of NFNE Futures over Other Futures ^{2} | Hedged Portfolio Returns | Expected Weekly Utility ^{3} | Utility Gain of NFNE Futures over Other Futures ^{4} | |
---|---|---|---|---|---|---|

Textiles | ||||||

Unhedged | 6.745 | 0.086 | ||||

TAIEX | 2.983 | 55.78% | 0.25% | −0.457 | −12.389 | 0.075 |

Taiwan 50 | 3.198 | 52.59% | 3.44% | −0.485 | −13.277 | 0.963 |

NFNE subindex | 2.966 | 56.03% | −0.450 | −12.314 | ||

Retailing | ||||||

Unhedged | 4.481 | 0.147 | ||||

TAIEX | 2.457 | 45.17% | −2.46% | −0.104 | −9.932 | −0.451 |

Taiwan 50 | 2.394 | 46.56% | −3.85% | −0.137 | −9.714 | −0.669 |

NFNE subindex | 2.567 | 42.71% | −0.114 | −10.383 | ||

Transportation | ||||||

Unhedged | 5.132 | −0.111 | ||||

TAIEX | 2.028 | 60.49% | 3.93% | −0.477 | −8.588 | 0.815 |

Taiwan 50 | 2.127 | 58.56% | 5.86% | −0.517 | −9.025 | 1.252 |

NFNE subindex | 1.826 | 64.42% | −0.468 | −7.773 | ||

Communication and Internet | ||||||

Unhedged | 3.166 | 0.028 | ||||

TAIEX | 1.485 | 53.10% | −3.81% | 0.006 | −5.933 | −0.474 |

Taiwan 50 | 1.238 | 60.91% | −11.62% | −0.035 | −4.985 | −1.422 |

NFNE subindex | 1.606 | 49.29% | 0.016 | −6.407 | ||

Automobile | ||||||

Unhedged | 6.921 | 0.246 | ||||

TAIEX | 2.026 | 70.73% | 2.34% | −0.394 | −8.497 | 0.672 |

Taiwan 50 | 1.950 | 71.82% | 1.25% | −0.439 | −8.241 | 0.416 |

NFNE subindex | 1.864 | 73.07% | −0.370 | −7.825 | ||

Plastics and Chemicals | ||||||

Unhedged | 4.404 | 0.114 | ||||

TAIEX | 1.096 | 75.12% | 11.85% | 0.124 | −4.258 | 2.095 |

Taiwan 50 | 1.054 | 76.06% | 10.91% | 0.068 | −4.150 | 1.987 |

NFNE subindex | 0.574 | 86.97% | 0.133 | −2.163 |

^{1}Percentage variance reductions are calculated as the differences of the variance of unhedged position and the estimated variance of alterative models over the variance of unhedged position, multiplied by 100;

^{2}Improvement of NFNE futures over other futures is defined as the difference of the percentage variance reduction of hedging with NFNE futures and the percentage variance reduction of hedging with TAIEX futures and Taiwan 50 futures estimated with a bivariate BEKK GARCH model;

^{3}Expected weekly utility is calculated based on Equation (12);

^{4}Utility gain of NFNE futures over other futures is defined as the difference of the expected utilities of hedging with NFNE futures over the expected utilities of hedging with TAIEX futures and Taiwan 50 futures estimated with a bivariate BEKK GARCH model;

^{5}Estimation of all models was conducted using data from 20 May 2009 to 30 December 2015; the data from 6 January 2016 to 28 December 2016 were used for out-of-sample analysis.

**Table 4.**Out-of-sample hedging effectiveness evaluated with variance reduction and utility gain under regime switching and global volatility spillover effects.

Variance of Hedged Portfolio Return | Percentage Variance Reduction ^{1} | Improvement of RSVSG over VSG and BEKK ^{2} | Hedged Portfolio Returns | Expected Weekly Utility ^{3} | Utility Gain of RSVSG over VSG and BEKK ^{4} | |
---|---|---|---|---|---|---|

Textiles | ||||||

Unhedged | 6.745 | 0.086 | ||||

BEKK | 2.966 | 56.03% | 0.53% | −0.450 | −12.314 | 0.135 |

VSG | 2.946 | 56.32% | 0.23% | −0.448 | −12.233 | 0.053 |

RSVSG | 2.930 | 56.56% | −0.458 | −12.180 | ||

Retailing | ||||||

Unhedged | 4.481 | 0.147 | ||||

BEKK | 2.567 | 42.71% | 4.36% | −0.114 | −10.383 | 0.791 |

VSG | 2.405 | 46.32% | 0.74% | −0.102 | −9.722 | 0.131 |

RSVSG | 2.372 | 47.06% | −0.104 | −9.591 | ||

Transportation | ||||||

Unhedged | 5.132 | −0.111 | ||||

BEKK | 1.826 | 64.42% | −0.43% | −0.468 | −7.773 | −0.094 |

VSG | 1.800 | 64.93% | −0.95% | −0.466 | −7.664 | −0.203 |

RSVSG | 1.849 | 63.98% | −0.473 | −7.867 | ||

Communication and Internet | ||||||

Unhedged | 3.166 | 0.028 | ||||

BEKK | 1.606 | 49.29% | 2.05% | 0.016 | −6.407 | 0.244 |

VSG | 1.554 | 50.93% | 0.40% | −0.005 | −6.220 | 0.057 |

RSVSG | 1.541 | 51.33% | 0.001 | −6.162 | ||

Automobile | ||||||

Unhedged | 6.921 | 0.246 | ||||

BEKK | 1.864 | 73.07% | 0.10% | −0.370 | −7.825 | 0.023 |

VSG | 1.977 | 71.43% | 1.74% | −0.386 | −8.294 | 0.492 |

RSVSG | 1.857 | 73.17% | −0.375 | −7.802 | ||

Plastics and Chemicals | ||||||

Unhedged | 4.404 | 0.114 | ||||

BEKK | 0.574 | 86.97% | −1.64% | 0.133 | −2.163 | −0.300 |

VSG | 0.710 | 83.88% | 1.44% | 0.106 | −2.733 | 0.270 |

RSVSG | 0.646 | 85.32% | 0.123 | −2.463 |

^{1}Percentage variance reductions are calculated as the differences of the variance of unhedged position and the estimated variance of alterative models over the variance of unhedged position, multiplied by 100;

^{2}Improvement of RSVSG over VSG and BEKK is defined as the difference of the percentage variance reduction of hedging with RSVSG and the percentage variance reduction of hedging with VSG and BEKK;

^{3}Expected weekly utility is calculated based on Equation (12);

^{4}Utility gains of RSVSG over VSG and BEKK are defined as the differences of the expected utilities of hedging with RSVSG over the expected utilities of hedging with VSG and BEKK;

^{5}Estimation of all models was conducted using data from 20 May 2009 to 30 December 2015; data from 6 January 2016 to 28 December 2016 were used for out-of-sample analysis.

**Table 5.**Out-of-sample hedging effectiveness evaluated with semivariance reduction and semi-utility gain under regime switching and global volatility spillover effects.

Semivariance of Hedged Portfolio Return | Percentage Semivariance Reduction ^{1} | Improvement of RSVSG over VSG and BEKK ^{2} | Hedged Portfolio Returns | Expected Weekly Semi-Utility ^{3} | Semi-Utility Gain of RSVSG over VSG and BEKK ^{4} | |
---|---|---|---|---|---|---|

Textiles | ||||||

Short hedgers‘ positions (negative semivariance) | ||||||

Unhedged | 3.896 | 0.086 | ||||

BEKK | 2.085 | 46.49% | 0.80% | −0.450 | −8.789 | 0.117 |

VSG | 2.068 | 46.91% | 0.38% | −0.448 | −8.722 | 0.050 |

RSVSG | 2.054 | 47.29% | −0.458 | −8.672 | ||

Long hedgers‘ positions (positive semivariance) | ||||||

Unhedged | 2.779 | 0.086 | ||||

BEKK | 1.027 | 63.06% | 1.30% | −0.450 | −4.556 | 0.136 |

VSG | 1.010 | 63.67% | 0.69% | −0.448 | −4.487 | 0.067 |

RSVSG | 0.991 | 64.36% | −0.458 | −4.420 | ||

Retailing | ||||||

Short hedgers‘ positions (negative semivariance) | ||||||

Unhedged | 2.135 | 0.147 | ||||

BEKK | 1.323 | 38.05% | 7.26% | −0.114 | −5.404 | 0.630 |

VSG | 1.181 | 44.69% | 0.62% | −0.102 | −4.826 | 0.052 |

RSVSG | 1.168 | 45.31% | −0.104 | −4.774 | ||

Long hedgers‘ positions (positive semivariance) | ||||||

Unhedged | 2.265 | 0.147 | ||||

BEKK | 1.208 | 46.66% | 1.71% | −0.114 | −4.947 | 0.165 |

VSG | 1.174 | 48.18% | 0.20% | −0.102 | −4.798 | 0.016 |

RSVSG | 1.169 | 48.38% | −0.104 | −4.781 | ||

Transportation | ||||||

Short hedgers‘ positions (negative semivariance) | ||||||

Unhedged | 3.048 | −0.111 | ||||

BEKK | 1.318 | 56.78% | 0.16% | −0.468 | −5.738 | 0.015 |

VSG | 1.291 | 57.65% | −0.71% | −0.466 | −5.629 | −0.094 |

RSVSG | 1.313 | 56.94% | −0.473 | −5.723 | ||

Long hedgers‘ positions (positive semivariance) | ||||||

Unhedged | 2.066 | −0.111 | ||||

BEKK | 0.692 | 66.49% | 0.84% | −0.468 | −3.237 | 0.064 |

VSG | 0.689 | 66.65% | 0.69% | −0.466 | −3.222 | 0.049 |

RSVSG | 0.675 | 67.33% | −0.473 | −3.173 | ||

Communication and Internet | ||||||

Short hedgers‘ positions (negative semivariance) | ||||||

Unhedged | 1.516 | 0.028 | ||||

BEKK | 0.796 | 47.46% | 0.04% | 0.016 | −3.170 | −0.012 |

VSG | 0.818 | 46.01% | 1.49% | −0.005 | −3.279 | 0.097 |

RSVSG | 0.796 | 47.50% | 0.001 | −3.182 | ||

Long hedgers‘ positions (positive semivariance) | ||||||

Unhedged | 1.602 | 0.028 | ||||

BEKK | 0.778 | 51.40% | 3.55% | 0.016 | −3.098 | 0.213 |

VSG | 0.713 | 55.50% | −0.55% | −0.005 | −2.857 | −0.029 |

RSVSG | 0.722 | 54.95% | 0.001 | −2.885 | ||

Automobile | ||||||

Short hedgers‘ positions (negative semivariance) | ||||||

Unhedged | 4.012 | 0.246 | ||||

BEKK | 1.323 | 67.01% | 2.33% | −0.370 | −5.664 | 0.370 |

VSG | 1.412 | 64.81% | 4.54% | −0.386 | −6.033 | 0.739 |

RSVSG | 1.230 | 69.34% | −0.375 | −5.294 | ||

Long hedgers‘ positions (positive semivariance) | ||||||

Unhedged | 2.794 | 0.246 | ||||

BEKK | 0.642 | 77.04% | 0.68% | −0.370 | −2.936 | 0.072 |

VSG | 0.694 | 75.16% | 2.57% | −0.386 | −3.162 | 0.298 |

RSVSG | 0.622 | 77.72% | −0.375 | −2.865 | ||

Plastics and Chemicals | ||||||

Short hedgers‘ positions (negative semivariance) | ||||||

Unhedged | 2.044 | 0.114 | ||||

BEKK | 0.219 | 89.27% | 0.91% | 0.133 | −0.744 | 0.064 |

VSG | 0.241 | 88.19% | 1.99% | 0.106 | −0.860 | 0.179 |

RSVSG | 0.201 | 90.18% | 0.123 | −0.680 | ||

Long hedgers‘ positions (positive semivariance) | ||||||

Unhedged | 2.385 | 0.114 | ||||

BEKK | 0.361 | 84.85% | −0.93% | 0.133 | −1.312 | −0.099 |

VSG | 0.401 | 83.17% | 0.75% | 0.106 | −1.500 | 0.088 |

RSVSG | 0.383 | 83.92% | 0.123 | −1.411 |

^{1}Percentage variance reductions are calculated as the differences of the variance of unhedged position and the estimated variance of alterative models over the variance of unhedged position, multiplied by 100;

^{2}Improvement of RSVSG over VSG and BEKK is defined as the difference of the percentage variance reduction of hedging with RSVSG and the percentage variance reduction of hedging with VSG and BEKK;

^{3}Expected weekly utility is calculated based on Equation (12);

^{4}Utility gains of RSVSG over VSG and BEKK are defined as the differences of the expected utilities of hedging with RSVSG over the expected utilities of hedging with VSG and BEKK;

^{5}Estimation of all models was conducted using data from 20 May 2009 to 30 December 2015; data from 6 January 2016 to 28 December 2016 were used for out-of-sample analysis.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Hsu, W.-C.; Lee, H.-T.
Cross Hedging Stock Sector Risk with Index Futures by Considering the Global Equity Systematic Risk. *Int. J. Financial Stud.* **2018**, *6*, 44.
https://doi.org/10.3390/ijfs6020044

**AMA Style**

Hsu W-C, Lee H-T.
Cross Hedging Stock Sector Risk with Index Futures by Considering the Global Equity Systematic Risk. *International Journal of Financial Studies*. 2018; 6(2):44.
https://doi.org/10.3390/ijfs6020044

**Chicago/Turabian Style**

Hsu, Wen-Chung, and Hsiang-Tai Lee.
2018. "Cross Hedging Stock Sector Risk with Index Futures by Considering the Global Equity Systematic Risk" *International Journal of Financial Studies* 6, no. 2: 44.
https://doi.org/10.3390/ijfs6020044