**1. Introduction**

The problem of finding the optimum portfolio for investors is known as a portfolio optimization problem. The portfolio optimization problem has been an important research theme, both academically and practically as it is a crucial part of managing risk and maximizing returns from a set of investments. The classical portfolio optimization approach is mean-variance optimization (MVO), which mainly concerns the expectation and variability of return (i.e., mean and variance [1]). Although the variance would be the most fundamental risk measure to be minimized, it has a crucial drawback: variance is a symmetric risk measure. Controlling the variance leads to a low deviation from the expected return with regard to both the downside and the upside.

Hence, asymmetric risk measures such as the Value-at-Risk (VaR) measure, which controls and manages the downside risk in terms of percentiles of the loss distribution of portfolio, have been proposed [2].

Instead of considering both the upside and downside of the expected return, the VaR risk measure focuses on only the downside of the expected return as the risk and represents the predicted maximum loss with a specified confidence level *β* (e.g., 99%). VaR became so popular that it was approved as a valid approach for calculating risk charges by bank regulators such as the Basel Accord II [3].

However, the VaR measure, if studied in the framework of coherent risk measures [4], lacks subadditivity, and, therefore, convexity in the case of general loss distributions. This drawback entails both inconsistencies with the well-accepted principle of portfolio

**Citation:** Nakagawa, K.; Ito, K. Taming Tail Risk: Regularized Multiple *β* Worst-Case CVaR Portfolio. *Symmetry* **2021**, *13*, 922. https://doi.org/10.3390/sym13060922

Academic Editor: Zhivorad Tomovski

Received: 20 April 2021 Accepted: 13 May 2021 Published: 21 May 2021

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**Copyright:** © 2021 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 (https:// creativecommons.org/licenses/by/ 4.0/).

diversification, i.e., diversification reduces risk. The VaR risk measure is non-convex and not smooth, making it difficult to optimize [5]. To reduce the computational burden of minimizing VaR, ref. [5] proposed a new mixed integer LP optimization based on the symmetric property of VaR. Besides, both variance and VaR ignore the magnitude of extreme or rare losses by their definition. Both risk measures cannot deal with extremely unlikely, but potentially catastrophic, events i.e., managing the tail risk [6].

The Conditional VaR (CVaR) risk measure responds to the aforementioned drawbacks of variance and VaR. CVaR is defined as the expected value of the portfolio loss that occurs beyond a certain probability level *β*. Obviously, CVaR is a more conservative risk measure than VaR. In [7], it was proven that the CVaR risk measure is a coherent risk measure that exhibits subadditivity and convexity. Additionally, the minimum CVaR portfolio that minimizes the CVaR results in a tractable optimization problem [8,9]. For example, when the portfolio loss is defined as the minus return of the portfolio, and a finite number of historical observations of returns are used in estimating CVaR, its minimization problem can be presented as a Linear programming (LP) optimization and can be solved efficiently. The minimum CVaR portfolio is a promising alternative to MVO for those reasons. In fact, the effectiveness of CVaR in portfolio construction designs has been demonstrated in a large number of recently published contributions, including index tracking and enhanced indexing [10–12].

However, there are three major challenges in the minimum CVaR portfolio. Firstly, when using CVaR as a risk measure, we need to determine the distfribution of asset returns, but it is difficult to actually grasp the distribution; therefore, we need to invest in a situation where the distribution is uncertain [13–15]. Secondly, the minimum CVaR portfolio is formulated with a single *β* and may output significantly different portfolios depending on how the *β* is selected [16]. In the context of MVO, this is called error maximization, which is the phenomenon that even small changes in the inputs can result in huge changes in the whole portfolio structure [17]. Thirdly, most portfolio optimization strategies do not account for transaction costs incurred by each rebalancing of the portfolio [18]. When buying and selling assets on the markets, commissions and other costs are incurred, such as globally defined transaction costs that are charged by the brokers or the financial institutions serving as intermediaries. Most of these transaction costs are incurred for portfolio turnovers. Transaction costs represent the most important feature to consider when selecting a real portfolio, given that they diminish net returns and reduce the amount of capital available for future investments [19].

The objective of this study is to propose a new tail risk-controlling portfolio construction method that addresses the above challenges and to confirm its performance. In this paper, we propose Regularized Multiple *β* Worst-case CVaR (RM-WCVaR) Portfolio Optimization. The characteristics of our portfolio are as follows. It makes CVaR robust with worst-case CVaR (WCVaR), which is an asymmetric risk measure and used in situations where the information on the underlying probability distribution is not exactly known [14,15]. Our portfolio is formulated with the multiple probability levels *β* of WC-VaR not to depend on a single *β* level. Finally, to control transaction costs, we add the *L*1-regularization term on the portfolio as stated in [18,20]. However, unlike these studies, we impose *L*1-norm penalty on portfolio turnovers rather than portfolio weights.

We also prove that the RM-WCVaR Portfolio Optimization problem is written as an LP optimization problem such as the single *β*-CVaR and WCVaR portfolio. We perform experiments on well-known benchmarks to evaluate the proposed portfolio. Compared with various portfolios, our portfolio demonstrates superior performance of having both higher risk-adjusted returns and lower maximum drawdown despite the lower turnover rate.

In the following sections, we first review the existing methods in Section 2. We formulate the VaR, CVaR, and WCVaR risk measures and the portfolio optimization with them in Section 3 and then, we propose the RM-WCVaR Portfolio in Section 4 and investigate the empirical effectiveness of the our portfolio in Section 5. Finally, we conclude in Section 6.
