**Definition 4.**

$$\text{WCVaR}(w|\beta) := \sup\_{p(\cdot) \in P} \text{CVaR}(w|\beta) \tag{4}$$

Ref. [15] have shown that the WCVaR is a coherent risk measure as well as CVaR.

Hereafter, we assume that the return vector's distribution is only considered to belong to a set of distributions that includes all mixtures of any predetermined density distributions, i.e.,

$$P\_M = \{ \sum\_{i=1}^l \lambda\_i p^{(i)}; \lambda\_i \ge 0, \sum\_{i=1}^l \lambda\_i = 1, i = 1, \dots, l \} \tag{5}$$

where *<sup>p</sup>*(*i*)(·) denotes the *i*-th density distribution, and *l* denotes the number of the density distributions.

Since it is difficult to handle when the set *P* contains an infinite number of *<sup>p</sup>*(*i*)(·), we consider approximating *P* with a convex linear combination of a finite number of *<sup>p</sup>*(*i*)(·). In this study, the mixture of density distributions *PM* is represented by blocks of divided empirical distributions.

To compute the WCVaR, we define the auxiliary function *<sup>F</sup><sup>i</sup>*(*<sup>w</sup>*, *α*|*β*) as

$$F^{(i)}(w, a | \beta) = a + (1 - \beta)^{-1} \int\_{R^n} [-w^\top r - a]^+ p^{(i)}(r) dr\tag{6}$$

where *i* = 1, . . . , *l* and [*t*]+ := max(*<sup>t</sup>*, <sup>0</sup>). Then, the following lemma holds.

**Lemma 1** (Ref. [15])**.** *For an arbitrarily fixed w and β, WCVaR*(*w*|*β*) *with respect to PM is given by*

$$\text{WCC}R(w|\beta) = \min\_{a} \max\_{i \in L} F^{(i)}(w, a|\beta) \tag{7}$$

*where L* = {1, . . . , *l*}*.* *Moreover, denote*

$$F^L(w, \alpha | \beta) = \max\_{i \in L} F^{(i)}(w, \alpha | \beta) \tag{8}$$

*Minimizing the WCVaR*(*w*|*β*) *overall w* ∈ *X is equivalent to minimizing <sup>F</sup><sup>L</sup>*(*<sup>w</sup>*, *α*|*β*) *overall* (*<sup>w</sup>*, *α*) ∈ *X* × *R, in the sense that*

$$\min\_{w \in X} \mathcal{WC} VaR(w|\beta) = \min\_{(w,a) \in X \times R} F^L(w, a|\beta) \tag{9}$$

From now on, we discuss the computational aspect of minimization of WCVaR. Lemma 1 helps us to translate the minimization problem to a more tractable one. The WCVaR minimization is equivalent to the following problem:
