*Article* **Taming Tail Risk: Regularized Multiple** *β* **Worst-Case CVaR Portfolio**

**Kei Nakagawa 1,\* and Katsuya Ito 2**


**Abstract:** The importance of proper tail risk managemen<sup>t</sup> is a crucial component of the investment process and conditional Value at Risk (CVaR) is often used as a tail risk measure. CVaR is the asymmetric risk measure that controls and manages the downside risk of a portfolio while symmetric risk measures such as variance consider both upside and downside risk. In fact, minimum CVaR portfolio is a promising alternative to traditional mean-variance optimization. However, there are three major challenges in the minimum CVaR portfolio. Firstly, when using CVaR as a risk measure, we need to determine the distribution 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. Secondly, the minimum CVaR portfolio is formulated with a single *β* and may output significantly different portfolios depending on the *β*. Finally, most portfolio allocation strategies do not account for transaction costs incurred by each rebalancing of the portfolio. In order to improve these challenges, we propose a Regularized Multiple *β* Worst-case CVaR (RM-WCVaR) portfolio. The characteristics of this portfolio are as follows: it makes CVaR robust with worst-case CVaR which is still an asymmetric risk measure, it is stable among multiple *β*, and against changes in weights over time. We perform experiments on well-known benchmarks to evaluate the proposed portfolio.RM-WCVaR demonstrates superior performance of having both higher risk-adjusted returns and lower maximum drawdown.

**Keywords:** RM-WCVaR; tail risk; portfolio optimization
