The formation of the optimal portfolio in the transient period (Q2784985)

The authors consider the admissible set of portfolios consisting of two securities in the random perturbed medium \(\theta_{\varepsilon}\in \{\theta_1,\ldots,\theta_{m}\}\), \(P\{\theta_{\varepsilon}=\theta_{j}\}=p_{j}+\beta_{j}(\varepsilon)\), where \(\beta_{j}(\varepsilon)\to 0\) as \(\varepsilon\to 0, j=1,\ldots,m\). The expected return of the \(i\)-th type security has the form \(r_{i}^{\varepsilon}= \sum_{j=1}^{m}[p_{j}+b^1_{j}\alpha_1(\varepsilon)+ b^2_{j}\alpha_2(\varepsilon)+ \ldots]i_{ij}(\varepsilon)\), where \(i_{ij}(\varepsilon)\) is the return of the \(i\)-th type security under the condition \(\theta_{\varepsilon}=\theta_{j}\). Here \(i_{ij}(\varepsilon)=i_{ij}+i^1_{ij}\alpha_1(\varepsilon)+ i^2_{ij}\alpha_2(\varepsilon)+\ldots\), where \(\alpha_1(\varepsilon)\to 0\) as \(\varepsilon\to 0\) and \(\alpha_{l}(\varepsilon)=o(\alpha_{l-1}(\varepsilon))\) for any \(l\geq 2\). Let \(x_1, x_2=1-x_1,\) be a ~partition of the corresponding securities in the portfolio \(x=(x_1,x_2)\), and let \(r^{\varepsilon}_{x}=x_1 r^{\varepsilon}_{1}+x_2 r^{\varepsilon}_{2}\) be a portfolio return. Let \(x\) and \(y\) be portfolios from the admissible set, and \(r_{x}>r_{y}\). The authors obtain the necessary and sufficient condition in terms of \(\alpha_{1}(\varepsilon)\) for fulfillment of the relation \(r^{\varepsilon}_{y}>r^{\varepsilon}_{x}\). Let \(\sigma_{x}^{\varepsilon}\) be a portfolio risk. Let \(x\) and \(y\) be portfolios from the admissible set and let \(\sigma_{x}>\sigma_{y}\). The authors obtain the necessary and sufficient condition for fulfillment of the relation \(\sigma^{\varepsilon}_{y}>\sigma^{\varepsilon}_{x}\).