**3. Preliminary**

In this section, we define VaR, CVaR, and WCVaR. After which, we formulate a minimum WCVaR portfolio optimization problem. Let *rj* be the return of stock *j* (1 ≤ *j* ≤ *n*) and *wj* be the portfolio weight for stock *j*. We denote *r* = (*<sup>r</sup>*1, ...,*rn*) and *w* = (*<sup>w</sup>*1, ..., *wn*). Here, *rj* is a random variable and follows the continuous joint probability density function *p*(*r*). *<sup>L</sup>*(*<sup>w</sup>*,*<sup>r</sup>*) refers to portfolio loss function and throughout this paper, we assume *<sup>L</sup>*(*<sup>w</sup>*,*<sup>r</sup>*) = − *<sup>w</sup>r*. The probability that the loss function is less than *α* is

$$\Phi(w,\alpha) = \int\_{L(w,r)\le\alpha} p(r)dr\tag{1}$$

When the portfolio weight *w* is fixed, <sup>Φ</sup>(*<sup>w</sup>*, *α*) which is the function of *α* is nondecreasing and is continuous from the right, but is generally non-continuous from the left. For simplicity, we assume that <sup>Φ</sup>(*<sup>w</sup>*, *α*) is a continuous function with respect to *α*. We can define VaR and CVaR as follows.

**Definition 1.**

$$\text{VaR}(w|\beta) := \text{a}(w|\beta) = \min(\text{a} : \Phi(w, \text{a}) \ge \beta) \tag{2}$$

**Definition 2.**

$$\begin{split} \mathbb{C}VaR(w|\beta) &:= \phi(w|\beta) \\ &= (1-\beta)^{-1} \int\_{L(w,r)\ge a(w|\beta)} L(w,r)p(r)dr \end{split} \tag{3}$$

Ref. [4] proposed the coherent risk measure, which characterizes the rationale of risk measure.

**Definition 3.** *The risk measure ρ that maps random loss X to a real number and satisfies the bellow four conditions is called a coherent risk measure.*

**Subadditivity:** *for all random losses X and Y , ρ*(*<sup>X</sup>* + *Y*) ≤ *ρ*(*X*) + *ρ*(*Y*) **Positive homogeneity:** *for positive constant a* ∈ R+, *ρ*(*aX*) = *aρ*(*X*) **Monotonicity:** *if X* ≤ *Y for each outcome, then ρ*(*X*) ≤ *ρ*(*Y*) **Translationinvariance:***forconstantm*∈R,*ρ*(*<sup>X</sup>*+*m*)*ρ*(*X*)+*m*

It is well known that CVaR is a coherent risk measure and VaR is not a coherent risk measure as it does not satisfy the Subadditivity [6].

 =

Next, we consider WCVaR. Rather than assuming exact knowledge of the return vector *r* distribution, we presume that the density function *p*(·) is only considered to belong to a certain set *P* of distributions, i.e., *p*(·) ∈ *P*.

The concept of the WCVaR is introduced in [15] as follows:
