**Problem 2.**

$$\min\_{w,a,t\_{iq}} C\tag{13}$$

$$\text{s.t.} \mathfrak{a} + \left( \left| \mathcal{N}^{(i)} \right| (1 - \beta) \right)^{-1} \sum\_{q \in \mathcal{N}\_i} t\_{iq} \le \mathbb{C} \quad (i = 1, \ldots, l) \tag{14}$$

$$t\_{iq} \ge -w^\top r^{(i)}[q] - a \quad (i = 1, \dots, l, \ q \in \mathcal{N}\_i) \tag{15}$$

$$t\_{iq} \ge 0 \quad (i = 1, \dots, l, \ q \in \mathcal{N}\_i) \tag{16}$$

### **4. Regularized Multiple** *β* **WCVaR Portfolio Optimization**

In this section, we propose a RM-WCVaR portfolio that takes into account the multiple *β* WCVaR values and portfolio turnover. Our approach is similar in spirit to that of [38], given that our approach estimates the simultaneously approximating multiple conditional quantiles.

The intuition behind the formulation is to minimize the max margin among multiple *β* levels of WCVaR (Figure 1). Figure 1 illustrates it where *βk* = {0.97, 0.98, 0.99} and each *WCβk*is given as a solution to Problem 1.

Here, *WCβk* , *k* = 1, ... , *K* is the value of WCVaR obtained by solving Problem 2. Then, we minimized *C*, considering that *WCβk*is a main problem in this paper.
