## 2 Mathematical Models
### DEA Model for \(\beta\)-Msv  
Let \(DMU=(\beta,MSV)\). Then, the \(\beta\)-MSV model based on the DEA model is as follows:  
\[\max\ \theta\] \[s.t.\ \varphi_{j}\left(\sum\limits_{i=1}^{n}\varepsilon_{i}(\Re_{i} )\right)\geq\varphi_{j}(Y_{0})+\theta.\tau_{j}(Y_{0}),\ \ j=1,...,J,\] \[MSV_{q}\left(\sum\limits_{i=1}^{n}\varepsilon_{i}(\Re_{i}) \right)\leq MSV_{q}(Y_{0})-\theta.\delta_{q}(Y_{0}),\ \text{q}=1,...,Q,\] \[\beta_{q}\left(\sum\limits_{i=1}^{n}\varepsilon_{i}(\Re_{i}) \right)\leq\beta_{q}(Y_{0})-\theta.\delta_{q}(Y_{0}),\ \text{q}=1,...,Q, \tag{10}\] \[\sum\limits_{i=1}^{n}\varepsilon_{i}=1,\ \ \varepsilon_{i}\geq 0,\ \ \ i=1,...,n.\]  
where\[\pi_{j}(Y_{0})=\max_{Y\in\chi}\bigl{(}\varphi_{j}(Y)\bigr{)}-\varphi_{j}(Y_{0}),\]  
\[\delta_{q}(Y_{0})=MSV_{q}(Y_{0})-\min_{Y\in\chi}MSV_{q}(Y).\]  
\[\hat{\delta}_{q}(Y_{0})=\beta_{q}(Y_{0})-\min_{Y\in\chi}\beta_{q}(Y).\]  
**Remark 2**: _In accordance with Remark (1), an efficient stock is characterized in this study by its capacity to yield the minimum risk for a particular level of return, or conversely, to provide a high return for a given level of risk. This characterization similarly holds true for efficient portfolios._  
**Proposition 1**: _Consider the optimization problem,_  
\[\min_{\omega\in\mathbb{R}^{n}}E[(\alpha+\omega\phi^{\prime})-]^{2}, \tag{11}\]  
_where the sentence \((\alpha+\omega\phi^{\prime})^{-}=\min((\alpha+\omega\phi^{\prime}),0)\). \(\phi^{\prime}\) is the transpose of the matrix \(\phi=(\phi_{1},...,\phi_{n})\), \(w\) is the asset weight vector, and \(\alpha\), \(\phi_{i}\), \(i=1,...,n\) are random variables by \(E[\alpha^{2}]<\infty\), \(E[\phi_{i}^{2}]<\infty\). If \(E[\phi_{i}]=0\), for all \(i\), then the problem (11) accepts a solution._  
Proof. : The proof of Proposition 1 can be found in [PERSON] et al. (2018). **Proposition 2**: _Suppose the initial capital is \(\gamma\) such that \(\sum_{i=1}^{n}\varepsilon_{i}=\gamma\) and \(\theta\leq 1\). Then, for any expected return of the portfolio, problem (10) has an optimal solution if it embraces feasible solutions._  
Proof. : We can rewrite the problem (10) as follows:  
\[min\frac{1}{T}\sum_{t=1}^{T}\left[\left(\sum_{i=1}^{n}\Re_{ti}\varepsilon_{i}- k\right)^{-}\right]^{2}\]  
\[s.t.\ \varphi_{j}\left(\sum\limits_{i=1}^{n}\varepsilon_{i}(\Re_{i} )\right)\geq\varphi_{j}(Y_{0})+\theta.\tau_{j}(Y_{0}),\ \ j=1,...,J,\]0  
\[s.t.\ \varphi_{j}\left(\sum\limits_{i=1}^{n}\varepsilon_{i}(\Re_{i} )\right)\geq\varphi_{j}(Y_{0})+\theta.\tau_{j}(Y_{0}),\ \ j=1,...,J,\]1  
\[s.t.\ \varphi_{j}\left(\sum\limits_{i=1}^{n}\varepsilon_{i}(\Re_{i} )\right)\geq\varphi_{j}(Y_{0})+\theta.\tau_{j}(Y_{0}),\ \ j=1,...,J,\]2  
\[s.t.\ \varphi_{j}\left(\sum\limits_{i=1}^{n}\varepsilon_{i}(\Re_{i} )\right)\geq\varphi_{j}(Y_{0})+\theta.\tau_{j}(Y_{0}),\ \ j=1,...,J,\]3  
where  
\[s.t.\ \varphi_{j}\left(\sum\limits_{i=1}^{n}\varepsilon_{i}(\Re_{i} )\right)\geq\varphi_{j}(Y_{0})+\theta.\tau_{j}(Y_{0}),\ \ j=1,...,J,\]4  
\[s.t.\ \varphi_{j}\left(\sum\limits_{i=1}^{n}\varepsilon_{i}(\Re_{i} )\right)\geq\varphi_{j}(Y_{0})+\theta.\tau_{j}(Y_{0}),\ \ j=1,...,J,\]5  
Since the maximum value of \(\theta\) occurs in \(0\leq\theta\leq 1\), constraint \(\frac{2}{1+\theta}-1\geq 0\) is guaranteed. Also, the following constraint,  
\[s.t.\ \varphi_{j}\left(\sum\limits_{i=1}^{n}\varepsilon_{i}(\Re_{i} )\right)\geq\varphi_{j}(Y_{0})+\theta.\tau_{j}(Y_{0}),\ \ j=1,...,J,\]6in problem (10), means the lowest amount of the MSV for the highest \(\theta\). Hence, the objective function of problem (12) is guaranteed. Now, let \(r_{i}\) and \(\mu_{i}\) be the return and expected return, respectively. Then, the objective function of problem (12) can be written as follows:  
\[s.t.\ \varphi_{j}\left(\sum\limits_{i=1}^{n}\varepsilon_{i}(\Re_{i} )\right)\geq\varphi_{j}(Y_{0})+\theta.\tau_{j}(Y_{0}),\ \ j=1,...,J,\]7  
We set \(\Re_{i}=r_{i}-\mu_{i}\).