**5. Conclusions**

The introduction of the ESG index facilitated the diversity of capital investment and fund management. Based on this research, we have studied the essential nature of portfolios and the dependence structure of numerous securities (ETF). The most common and traditional way to reduce the risk and increase the risk-adjusted return is to manage the potential linear correlation among returns. However, to compete in the fierce battlefield of fund management, fund managers need to further investigate the dependence structure in not only constant but also time-varying cases; hence, more precise performance can be simulated in the hypothetical portfolios. Our research, using the constant and time-varying copula models, can help fund managers predict the potential benefits gained from the ESG index if they consider putting the ESG index into the current renewable energy index. These benefits are reasonable since they are similar in topic, and, as stated before, the ESG index has decent historical performance and satisfactory applicability to companies.

Accordingly, we segmen<sup>t</sup> the detailed dependence structures into four groups and compare their performances compared with merely investing in the renewable energy index (as the current renewable energy index fund/ETF). Overall, the results sugges<sup>t</sup> that we can trust the ESG index in hedging investment risk and increasing the profitability level in fund management. For example, in the case of investing in the ERIX index, the introduction of the ESG index can effectively reduce the potential value-at-risk level by as much as 50%; meanwhile, this index keeps the simulated revenue positive and

more stable. During huge losses, such as the 2008 and 2011 crises, the time varying CVaR graphs reveal that balancing the renewable energy index with the ESG index can be helpful in reducing large losses.

**Author Contributions:** Conceptualization, S.H.; investigation, G.L.; writing—original draft preparation, G.L.; writing—review and editing, S.H.; project administration, S.H.; funding acquisition, S.H. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by JSPS KAKENHI, gran<sup>t</sup> number 17H00983.

**Acknowledgments:** We are grateful to three anonymous referees for their helpful comments and suggestions.

**Conflicts of Interest:** The authors declare no conflict of interest.
