Predictive Control of Investment Portfolio on the Financial Market with Hidden Regime Switching and MS VAR Model of Returns

Abstract

We consider the problem of managing an investment portfolio in the financial market with switching of regimes taking into account constraints on the volume of investments and loans. It is assumed that the returns on risky assets are described by a vector autoregressive model with hidden regime switching (Markov Switching Vector Autoregression, MS VAR). The EM algorithm is used to estimate the parameters. The results of numerical modeling using real data of the Russian stock market are presented.

APPENDIX

Proof of the Theorem. The proof algorithm is based on the results obtained in [7]. Using the smoothing property of the conditional expectation, the criterion (2.20) can be represented in the form

$$ \eqalign {&J(k+m|k)\cr &\quad {}=E\Bigg \{V^{2} (k+1|k)R_{1} (k\!+\!1)-R_{2} (k\!+\!1)V(k\!+\!1|k)+u^{\mathrm {T}} (k|k)R(k\!+\!1)u(k|k) \cr &\quad \quad {}+E\bigg \{V^{2} (k\!+\!2|k)R_{1} (k\!+\!2)- R_{2}(k+2)V(k\!+\!2|k)+u^{\mathrm {T}} (k\!+\!1|k)R(k\!+\!2)u(k\!+\!1|k)+\ldots \cr &\quad \quad \quad {}+E\Big \{V^{2} (k+m|k)R_{1} (k\!+\!m)-R_{2}(k+m)V(k\!+\!m|k)\cr &\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad {}+u^{\mathrm {T}} (k\!+\!m\!-\!1|k) R(k\!+\!m)u(k\!+\!m\!-\!1|k)\cr &\quad \quad \quad \quad \quad \quad {}|\thinspace V(k\!+\!m-1),\theta (k\!+\!m\!-\!1)\Big \}\ldots |V(k\!+\!1),\theta (k\!+\!1)\bigg \} \ldots |V(k),\theta (k)\Bigg \}.}$$

(A.1)

Using (2.8)–(2.10), (2.11)–(2.14), and (2.7), we obtain

$$ \begin {gathered} \begin {aligned} &V(k+m-t\thinspace |\thinspace k) \\ &{} =\sum _{i_{m-t} =1}^{\nu }e_{i_{m-t} } \big [P\theta (k + m - t - 1) + \upsilon (k + m - t)\big ] \bigg [A^{(i_{m-t} )} V(k + m - t - 1\thinspace |\thinspace k) \\ &\enspace \enspace {}+\Big (B\left [\gamma ^{(i_{m-t} )} Y(k+m-t-1)+\sigma ^{(i_{m-t} )} W(k+m-t)\right ]+D^{(i_{m-t} )} \Big )u(k+m-t-1\thinspace |\thinspace k)\bigg ],\\ \end {aligned} \\ t={1,\ldots ,m -1 }.\end {gathered}$$

(A.2)

The successive calculation of the expectations in (A.1) with allowance for (A.2) and with the replacement of the parameters by their estimates (3.1)–(3.3) leads to the expression

$$ \begin {aligned} J(k+m\thinspace |\thinspace k)&=V^{2} (k)\sum _{i_{1} =1}^{\nu }\left (A^{(i_{1} )} \right )^{2} Q^{(i_{1} )} (k) -\sum _{i_{1} =1}^{\nu }Q_{2}^{(i_{1} )} (k)A^{(i_{1} )} V(k) \\ &\quad {}+\big [2V(k)G(k)-F(k)\big ]U(k)+U^{\mathrm {T}} (k)H(k)U(k), \end {aligned} $$

(A.3)

the matrices $G(k)$, $F(k)$, and $H(k)$ have the form (4.3)–(4.7), and the matrices $Q^{(i_{t} )} (k)$, $Q^{(i_{t},\ldots ,i_{s} )} (k)$, $Q_{2}^{(i_{1} )} (k)$, and $Q_{2}^{(i_{s} ,\ldots ,i_{s} )}(k)$ have the form (4.8)–(4.13). It is obvious that the problem of minimizing criterion (A.3) under constraints (2.17) is equivalent to the problem of minimizing criterion (4.1) where the terms independent of controls have been deleted under constraints (4.2). The proof of the Theorem is complete. $\quad \blacksquare$