Predictive control strategies for investment portfolio subject to constraints and trading costs

Consider an investment portfolio consisting of n risky assets and one risk-free asset (e.g., a bank account or a government bond). Let u_t (k) , (i = 1, n) denote the amount of money invested in the i-th asset at time k ; u₀(k) > 0 is the amount invested in a risk-free asset. Investor also can borrow the capital in case of need. The volume of the borrowing of the risk-free asset is equal to u_n+1(k) > 0. If u (k) < 0,(i = 1,n), then we use short position with the amount of shorting |u_i (k)|. Let r_i (k +1) denote the (simple) return of the i -th risky asset per period [k, k +1]. It is a stochastic unobservable at time k . By considering the self-finance strategies, the wealth process at the time k + 1 is given by V (k +1) = [1 + rJV (k) + i [r i (k +1) - rJu (k) - [r2 - rju„+1 (k), i=1 where r₁ is the riskless lending rate, r2 is the riskless borrowing rate, r1 < r2 . We impose the following constraints on the decision variables (a borrowing limit on the total wealth invested in the risky assets, and long- and short-sale restrictions on all risky assets): ur(k) < u (k) < u,(k)(i = Щ), (1) 0 < V(k) - iu (k) + u_n+1(k) < ur, (2) i=1 0 n+l(k) 0 is a given parameter representing the growth factor and the initial state is V (0) = V(0) . We use the model predictive control methodology in order to design feedback predictive control strategies for optimal dynamic allocation of a portfolio. We define the following objective with receding horizon (risk function) which is to be minimized at each time k : m J(k + m / k) = E{i [V(k + i / k) - V(k + i)] / V(k), r(k),..., r(k - N)} + i=1 m-1 +Ei {[u(k + i / k) - (I + Q[r(k + i)])u(k + i - 1/ k)]R(k, i) x i =0 x[u (k + i / k) - (I + Q[r(k + i)])u (k + i - 1/ k)]/ V (k), r(k),..., r(k - N +1)}, over the sequence of predictive control inputs u (k / k), u (k +1/ k),..., u (k + m - 1/ k) dependent on the portfolio wealth and the market information at the current time k , under constraints (1)-(3), where m is the prediction horizon. The first term represents the conditional mean-square error between the investment portfolio value and a reference (benchmark) portfolio, the second term penalizes for transaction costs associated with trading amount. At the time k , u (k) = u (k / k) is assumed to be control u (k) . To obtain the control at the next step k + 1, the procedure is repeated, and the control horizon is one step shifted. Our approach is direct in that it uses directly the observed historical data to construct an adaptive algorithm for online portfolio selection. The main features of our approach are (a) the ability to adapt to non-stationary market environments by dynamically incorporating new information into the decision process; (b) no stochastic assumptions are needed regarding the stock prices, and (c) the flexibility of dealing with portfolio constraints. We also present the numerical modeling results based on currency pairs traded on the international currency market FOREX that give evidence of capacity and effectiveness of proposed approach.

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