**2. Related Work**

MVO assumes that investment decisions on getting a diversified portfolio depend on the two inputs: expected returns and the covariances of asset returns. However, as the estimation errors, mainly in expected return parameters, are amplified by optimization and then propagate into the solution of the optimization, extreme portfolio weights and a lack of diversification are commonly observed. This phenomenon has eventually ruined the out-of-sample performance of MVO [17,21]. To date, many efforts have been expended to handle the estimation risk on the parameter uncertainty. In order to reduce estimation error, the conventional regularization models have been applied for the MVO by [18,20]. They demonstrated superior portfolio performances when various types of norm regularities are combined into the optimization problem. Analogously, ref. [22] considered *L*1 and *L*2-norms for the mean-CVaR portfolio and [16] considered *L*1-norms for the multiple CVaR portfolio. Our paper extends this regularization literature to a multiple WCVaR Portfolio.

Robust portfolio optimization is another approach considering the estimation error and has been receiving increased attention [23]. Recently, ref. [24] proposed a robust portfolio optimization approach based on quantile statistics. Robust optimization also has been adopted on the other portfolio optimization problems. In [13], robust portfolio optimization using worst-case VaR was investigated, where only partial information on the distribution was known. In [15], the concept of WCVaR was introduced for the situation where the probability distributions are only partially known, and the properties of WCVaR are studied, such as coherency. Another approach is using a semi-nonparametric distribution, which may asymptotically capture the true density. This approach has been successfully tested for CVaR [25,26]. Our paper extends this robust portfolio optimization literature to a multiple *β* WCVaR portfolio.

Another direction to reduce the estimated error is to construct a risk-based portfolio that does not use expected returns. The minimum variance portfolio [27], risk parity portfolio [28,29], and maximum diversification portfolio [30] have been proposed as representative risk-based portfolio construction methods. The risk-based portfolio has the desirable property that the portfolio and its performance do not change greatly in response to changes to inputs [31,32]. Furthermore, an extension of each of them has been proposed such as minimum VaR and CVaR portfolio [33], risk and complex risk diversification portfolio [34,35] and higher order risk based portfolio [36]. Various empirical analyses and backtests of stock portfolios and asset allocations have shown better performance than mean-variance portfolios and market capitalization-weighted portfolios [37]. Our paper adds a minimum WCVaR-based portfolio to this risk-based portfolio construction literature.
