Consider a financial market consisting of n risky assets, indexed from 1 to n, and modeled by a probability space \(\left (\Omega, \mathfrak {F}, P\right)\). Let agents act their investment decisions over a one-period horizon with respect to the following set of feasible, or admissible, portfolios defined according to the budget constraint and no-short selling:

$$ \mathcal{X} = \left\{(x_{1}, \ldots, x_{n}) \in \mathbb{R}^{n} \colon \sum\limits_{i = 1}^{n} x_{i} = 1,\ x_{i} \geq 0, i = 1, \ldots, n\right\} $$

(1)

where *x*_{i} represents the proportion of capital to be allocated to asset *i*, with *i*=1,…,*n*.

The random rate of return of asset *i* at the end of the investment period, denoted as *R*_{i}, is assumed to have a continuous probability density function (pdf). Hence, the rate of return of a portfolio \(\mathbf {x} = \left (x_{1}, \ldots, x_{n}\right)^{T} \in \mathcal {X}\) is the random variable \(R(\mathbf {x}) = \sum \limits _{i = 1}^{n} R_{i} x_{i}\) with pdf *f*_{R(x)} induced by that of (*R*_{1},…,*R*_{n}). *R*(**x**) is measurable in (*R*_{1},…,*R*_{n}) with expected rate of return defined as

$$ \mathbb{E}(R(\mathbf{x})) = \sum\limits_{i = 1}^{n} \mathbb{E}(R_{i}) x_{i} $$

(2)

where \(\mathbb {E}(\cdot)\) denotes the expectation and \(\mathbb {E}(R_{i})\) is the expected rate of return of asset *i*. Given a level *z* for the rate of return, the cumulative distribution function (cdf) of *R*(**x**) is defined as \(F_{R(\mathbf {x})}(z) = P(R(\mathbf {x}) \leq z) = \int _{\{R(\mathbf {x}) \leq z\}} f_{R(\mathbf {x})}(r) dr\). It is also assumed that *F*_{R(x)} is continuous and strictly increasing with respect to \(z \in \mathbb {R}\)^{Footnote 1}. The portfolio loss distribution is defined as the negative of the portfolio return distribution, that is, *L*(**x**)=−*R*(**x**).

To identify the portfolio in \(\mathcal {X}\) that guarantees the “best” rate of return, a model for preferences under uncertainty needs to be defined. We adopt the so-called reward-risk approach that relates the portfolio selection problem to a multi-objective optimization problem in two steps. First, a set of objectives that the investor perceives as beneficial is identified, and second, a set of objectives he or she considers damaging is identified in relation to *R*(·). Then, a preference relation is defined based on these criteria as follows.

###
**Definition 1**

Let \(f_{1}, \ldots, f_{p} \colon \mathcal {X} \to \mathbb {R}\) and \(g_{1}, \ldots, g_{q} \colon \mathcal {X} \to \mathbb {R}\) be the reward-type and the risk-type objective functions, respectively. Then for all \(\mathbf {x}, \mathbf {y} \in \mathcal {X}\), we say that *R*(**x**) dominates (is preferred to) *R*(**y**) if and only if *f*_{i}(**x**)≥*f*_{i}(**y**) for all *i*=1,…,*p* and *g*_{j}(**x**)≤*g*_{j}(**y**) for all *j*=1,…,*q* with at least one strict inequality. Alternatively, we can say that portfolio **x** dominates portfolio **y**.

According to this definition, a portfolio \(\mathbf {x}^{*} \in \mathcal {X}\) is (Pareto) optimal if and only if it is non-dominated with respect to \(\mathcal {X}\), that is, another \(\overline {\mathbf {x}} \in \mathcal {X}\) that dominates **x**^{∗} does not exist. Thus, an efficient portfolio in the reward-risk model is a Pareto optimal solution of the following multi-objective problem:

$$ \begin{aligned} & {\text{minimize}} & & F(\mathbf{x}) = \left(-f_{1}(\mathbf{x}), \ldots, -f_{p}(\mathbf{x}), g_{1}(\mathbf{x}), \ldots, g_{q}(\mathbf{x})\right) \\ & \text{subject to} & & \mathbf{x} \in \mathcal{X}\,. \end{aligned} $$

(3)

The solutions to Problem (3) form the so-called efficient set, or Pareto optimal set, of which image in the reward-risk space is called efficient frontier, or Pareto front. Under certain smoothness assumptions, it can be induced from the Karush–Kuhn—Tucker conditions that, when the *p*+*q* objectives are continuous, the efficient frontier defines a piecewise continuous (*p*+*q*−1)-dimensional manifold in the decision space (Li and Zhang 2009). Therefore, the efficient frontier of a continuous bi-objective portfolio optimization problem is a piecewise continuous 1-D curve in \(\mathbb {R}^{2}\), while, in the case of a problem with three objectives, it is a piecewise continuous 2-D surface in \(\mathbb {R}^{3}\).

We now specialize this general reward-risk framework by considering investors that focus on the mean rate of return (2) as the reward criterion and employ different measures to assess the downside risk of their investments.

### The mean-semi-variance model

One of the most well-known downside risk measures of a portfolio **x** is its semi-variance, already introduced in the seminal paper of (Markowitz 1959) and defined as

$$ V^{-}(R(\mathbf{x})) = \mathbb{E}\left(\left(\left[R(\mathbf{x}) - b\right]^{-}\right)^{2}\right) $$

(4)

where [ *u*]^{−}= max{0,−*u*} and *b* represents a given benchmark, for example, the portfolio expected rate of return, the Roy’s safety first criterion, or a Treasury rate of return. In our analysis, we set the target level *b* equal to 0 in order to estimate the variance of portfolio losses. As proved in Fishburn (1977), this downside risk measure is consistent with utility theory

An investor that operates her or his decisions according to the mean-semi-variance model is interested in maximizing the expected rate of return of her or his portfolio and, simultaneously, minimizing its semi-variance:

$$ \begin{aligned} & {\text{minimize}} & & \left(-\mathbb{E}\left(R(\mathbf{x})\right), V^{-}(R(\mathbf{x}))\right) \\ & \text{subject to} & & \mathbf{x} \in \mathcal{X}\,. \end{aligned} $$

(5)

### The mean-CVaR model

The CVaR of a portfolio **x** at the confidence level *α*, with *α*∈(0,1), represents the average of losses in the 100(1−*α*)*%* worst scenarios (Acerbi and Tasche 2002). According to our assumptions about the cdf of *R*(**x**) and the definition of the portfolio loss *L*(**x**) as the opposite of *R*(**x**), this downside risk measure is defined as

$$ CVaR_{\alpha}(L(\mathbf{x})) = \mathbb{E}\left(L(\mathbf{x}) \left| L(\mathbf{x}) \geq {VaR}_{\alpha}(L(\mathbf{x}))\right. \right), $$

(6)

where \({VaR}_{\alpha }(L(\mathbf {x})) = \sup \left \{z | P\left (L(\mathbf {x}) \geq z\right) \leq \alpha \right \} = F_{L(\mathbf {x})}^{-1}(1-\alpha)\) is the value-at-risk at the confidence level *α* of the portfolio **x**.

An agent whose preferences are dictated by the mean-CVaR model would select her or his portfolio weights according to the following bi-objective optimization problem:

$$ \begin{aligned} & {\text{minimize}} & & \left(-\mathbb{E}\left(R(\mathbf{x})\right), {CVaR}_{\alpha}(L(\mathbf{x}))\right) \\ & \text{subject to} & & \mathbf{x} \in \mathcal{X}\,. \end{aligned} $$

(7)

### The mean-semi-variance-CVaR model

Using only one criterion to model risk in a portfolio selection model may provide a restricted picture of the assessment process (Steuer et al. 2005). It is advisable to include multiple risk measures in order to cover the different facets of an investor’s preferences. In this study, we analyze the combined effects of CVaR and semi-variance. The related tri-objective optimization problem is

$$ \begin{aligned} & {\text{minimize}} & & \left(-\mathbb{E}\left(R(\mathbf{x})\right), V^{-}(R(\mathbf{x})), {CVaR}_{\alpha}(L(\mathbf{x}))\right) \\ & \text{subject to} & & \mathbf{x} \in \mathcal{X}, \end{aligned} $$

(8)

where *V*^{−} and *CVaR*_{α} are the same as the definitions in (4) and (6), respectively.

In this manner, the preferences are modeled as follows.

###
**Definition 2**

Portfolio **x** is preferred to portfolio **y** if and only if \(\mathbb {E}\left (R(\mathbf {x})\right) \geq \mathbb {E}\left (R(\mathbf {y})\right), {CVaR}_{\alpha }(L(\mathbf {x})) \leq {CVaR}_{\alpha }(L(\mathbf {y}))\) and *V*^{−}(*R*(**x**))≤*V*^{−}(*R*(**y**)), with at least one strict inequality.

This model may produce improved solutions when a mean-CVaR efficient portfolio has an excessively large semi-variance or when a mean-semi-variance efficient portfolio has an excessively large CVaR.

### Scenario-based framework for portfolio optimization

A practical way to handle the random variables that represent assets and portfolio rates of return is to treat them as discrete. Therefore, let *Ω*={*ω*_{1},…,*ω*_{S}},*P*(*ω*_{s})=*p*_{s}, with *s*=1,…,*S* and \(\sum \limits _{s=1}^{S} p_{s} = 1, \mathcal {F}\) be a given *σ*-field and assume the following table of scenarios is available

$$ \left(\begin{array}{lll} \mathbf{r}_{1} & \ldots & \mathbf{r}_{S} \\ p_{1} & \ldots & p_{S} \end{array}\right) $$

(9)

where *S* represents the number of involved scenarios, **r**_{s}=(*r*_{1s},…,*r*_{ns})^{T} is the *n*-vector of rates of return for the *s*-th scenario, and *p*_{s} is the associated probability of occurrence, with *s*=1,…,*S*. Assume that *p*_{s}=1/*S* for all *s*. Thus, the rate of return of \(\mathbf {x} \in \mathcal {X}\) may assume *S* values calculated as \(r_{s}^{p}(\mathbf {x}) = \sum \limits _{i=1}^{n} r_{is} x_{i}\). Analogously, assuming that the distribution of the portfolio rate of return *R*(**x**) is such that \(P\left (R(\mathbf {x}) = r_{s}^{p}(\mathbf {x})\right) = 1/S\) with *s*=1,…,*S*, since the expected rate of return for the *i*-th security can be computed as the mean rate of return over the *S* scenarios (i.e., \(\mathbb {E}(R_{i}) = \frac {1}{S}\sum \limits _{s=1}^{S} r_{is}\)), the expected rate of return of portfolio **x** becomes

$$ \mathbb{E}(R(\mathbf{x})) = \sum\limits_{i=1}^{n} \mathbb{E}(r_{i}) x_{i} = \frac{1}{S} \sum\limits_{s=1}^{S} r_{s}^{p}(\mathbf{x})\,. $$

(10)

Now it is possible to reformulate the previously introduced downside risk-based portfolio allocation models in terms of the scenarios conveyed in (9).

Based on the findings of (Cumova and Nawrocki 2011), we estimate *V*^{−}(*R*(**x**)) by means of the so-called co-semi-variance matrix, henceforth denoted as \(C^{-} = \left (C^{-}_{ij}\right)\), which is defined element-wise as

$$ C^{-}_{ij} = \frac{1}{S} \sum\limits_{s=1}^{S} \left(r_{is}-b\right) \min\{r_{js}-b, 0\} $$

(11)

$$ C^{-}_{ji} = \frac{1}{S} \sum\limits_{s=1}^{S} \left(r_{js}-b\right) \min\{r_{is}-b, 0\} $$

(12)

for all *i,j*=1,…,*n*. It can be noted that *C*^{−} is an asymmetric matrix, since in general, assets *i* and *j* do not have the same below–target returns during the same period, that is, \(C^{-}_{ij} \neq C^{-}_{ji}\). The semi-variance of portfolio **x** can then be calculated as

$$ V^{-}(R(\mathbf{x})) = \sum\limits_{i = 1}^{n} \sum\limits_{j = 1}^{n} C^{-}_{ij} x_{i} x_{j}\,. $$

(13)

In this manner, Problem (5) translates into the maximization of a linear objective function, given by Eq. (10), and the simultaneous minimization of a quadratic objective function, given by Eq. (13), over the unit standard *n*-dimensional simplex. Regarding Problem (7), the CVaR expression (6) can be evaluated as an arithmetic mean over scenarios as follows. Let \(l_{s}^{p}(\mathbf {x}) = -r_{s}^{p}(\mathbf {x})\) be the portfolio loss in the *s*-th scenario, after sorting losses in ascending order, that is, \(l_{(1)}^{p}(\mathbf {x}) \leq l_{(2)}^{p}(\mathbf {x}) \leq \ldots \leq l_{(S)}^{p}(\mathbf {x})\), the *VaR*_{α} can then be defined as \(l_{\left (\lfloor \alpha S\rfloor \right)}^{p}(\mathbf {x})\) and *CVaR*_{α} can be estimated as

$$\begin{array}{*{20}l} {CVaR}_{\alpha}(L(\mathbf{x})) & = \frac{\sum\nolimits_{s=1}^{S} l_{(s)}^{p}(\mathbf{x}) \mathbb{I}\left(r_{(s)}^{p}(\mathbf{x}) \geq {VaR}_{\alpha}(\mathbf{x})\right)}{\sum\nolimits_{s=1}^{S} \mathbb{I}\left(l_{(s)}^{p}(\mathbf{x}) \geq {VaR}_{\alpha}(\mathbf{x})\right)} \\ & = \frac{1}{(1-\alpha) S} \left[\sum\limits_{s=\lceil\alpha S\rceil+1}^{S} l_{(s)}^{p}(\mathbf{x}) + (\lceil\alpha S\rceil - \alpha S) l_{\left(\lceil\alpha S\rceil\right)}^{p}(\mathbf{x})\right] \end{array} $$

(14)

where \(\mathbb {I}(\cdot)\) represents the indicator function. In this case, the investor maximizes the expected rate of return as given in Eq. (10) and minimizes the risk calculated according to Eq. (14).

Finally, Problem (8) simultaneously exploits Eqs. (10), (13), and (14) in the optimization process.