**6. Conclusions**

Our study makes the following contributions. We propose a Regularized Multiple *β* WCVaR Portfolio, which solves three challenges in the minimum CVaR portfolio. We prove that the optimization problem reduces to a linear programming problem. We perform experiments on well-known benchmarks in finance to evaluate our proposed portfolio. Our portfolio shows superior performance in terms of having both higher risk-adjusted returns and lower maximum drawdown despite the lower turnover rate.

Directions of promising future work include (1) constructing a more sophisticated mixture distribution by assuming a probability distribution as in [15], rather than a simple empirical distribution in this study, (2) directly using a semi-nonparametric distribution capturing true CVaR as in [25,26] instead of WCVaR, and (3) obtaining a higher R/R by incorporating the expected return into our proposed portfolio.

**Author Contributions:** Conceptualization, K.N.; methodology, K.N.; software, K.I.; validation, K.N. and K.I.; formal analysis, K.N. and K.I.; writing—original draft preparation, K.N.; writing— review and editing, K.N. and K.I. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Data Availability Statement:** The data that support the findings of this study are openly available in https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data\_library.html.

**Acknowledgments:** The authors would like to thank the referees for their valuable comments and suggestions. Discussions with Masaya Abe, Senior Quants Analyst of Nomura Asset Management and Shuhei Noma, Nomura Asset Management have been insightful.

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