Portfolio optimization in the financial market withserially dependent returns under constraints.pdf The investment portfolio (IP) management is an area of both theoretical interest andpractical importance. The basis of the current classical theory of optimal portfolio allocationproblem is the single-period "mean variance" approach suggested by Markowitz[1] and the Merton dynamic IP model [2] in continuous time. At present, there exists avariety of models and approaches to the solution of the IP optimization problem, butmost of them are the complications and extensions of the Markowitz and Merton approachesto various versions of stochastic models of the prices of risky and risk-free securitiesand utility functions. The review of the main trends existing in the modern theoryof dynamic control of investments is given in [3].The most of the results presented in these works are limited to the cases without explicitconstraints on the trading volume amounts. However it's well-known that realisticinvestment models must include ones.In static framework one can take into consideration portfolio constraints that lead tothe linear or quadratic programming tasks. Taking into account the portfolio constraintsin dynamic models we come to the impractical for actual numerical implementationmodels, due to the "curse of dimensionality".We propose to use the model predictive control (also known as receding horizoncontrol) methodology in order to solve the problem. MPC proved to be an appropriateand effective technique to solve the dynamic control problems subject to input andstate/output constraints. The main concept of MPC is to solve an open-loop constrainedoptimization problem with receding horizon at each time instant and implement only theinitial optimizing control action of the solution [4].MPC have begun to be used with success in financial applications such as portfoliooptimization and dynamic hedging. Some of the recent works on this subject can befound, for instance, in [5]-[8]. In all these papers authors assume the hypothesis of seriallyindependent returns and consider the explicit form of the model describing the priceprocess of the risky assets (e.g. geometric Brownian motion, e.t.c.). The problem ofMPC for discrete-time systems with dependent random parameters is considered in [9].In that paper the evolution of parameters' vector is described by the linear stochasticdifference equation. The results are applied to the IP optimization.The main novelty of this paper is that the risky asset returns are assumed to be a sequenceof stochastic serially dependent variables for which the first and the second conditionaldistribution moments are known only. The optimal open-loop feedback controlstrategy is derived subject to hard constraints on the trading volume amounts. Predictivestrategies computation includes the decision of the sequence of quadratic programmingtasks. This approach thus leads to computationally tractable optimization problems.We present the numerical modeling results, based on stocks, traded on the RussianStock Exchanges MICEX, that give evidence of capacity and effectiveness of proposedapproach. Numerical examples based on real market have shown that our approach is atheoretically sound and computationally efficient method.1. Portfolio optimization problemConsider the investment portfolio consisting on the n risky assets and one risk-freeasset (e.g. a bank account or a government bond). Let ui(k) (i = 0, 1,2, ..., n) denote theamount of money invested in the ith asset at time k; u0(k)≥0 is the amount invested in arisk-free asset. Then the wealth process V(k) satisfies:01( ) ( ) ( ).niiV k u k u k==Σ + (1)Notice, that if ui(k)0 is positive weight coefficient. The performance criterion (8) is composed by alinear combination of a quadratic part, representing the quadratic error between theportfolio value and a benchmark, and a linear part, representing an expected errorbetween the portfolio value and a benchmark which is desired to overcome.2. Model predictive control strategies designTheorem. Let the wealth dynamics is given by (3) under constraints (4), (5). Thenthe MPC policy with receding horizon m, such that it minimizes the objective (8), foreach instant k is defined by the equation:u k = I U k (9)wherenI is n-dimensional identity matrix; 0n is n-dimensional zero matrix;U(k)=[uT(k/k), …, uT(k+m−1/k)]T is the set of predictive controls defined from the solvingof quadratic programming problem with criterionT ( / ) 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) T ẜY +ẜm =⎡⎣ẜx k G −ẜF ⎤⎦ẜU +ẜU k H k U k(10)under constraintsmin max U (k)≤S(k)U(k)≤U (k), (11)where 0 ( ) ( ) ( ) ,Tx k = ⎡⎣V k V k ⎤⎦min min 1 1 1 1 max max 1 1 1 1 ( ) [ ( ),0 ,...,0 ] , ( ) [ ( ),0 ,...,0 ] , T T T Tn n n nU k u k U k u k+ . + . + . + .= =min min minmin 1 0max max maxmax 1 0( ) ( ), ... ( ), ( ) ( ) ,( ) ( ), ... ( ), ( ) ( ) ,TnTnu k u k u k u k Vku k u k u k u k Vk=⎡⎣ − ⎤⎦=⎡⎣ − ⎤⎦H(k), G(k), F(k),S(k) are the block matrices of the form( ) ( ) ( , 1, ), ij Hk=⎡⎣H k⎤⎦ i j=m[ ] [ ] 1 2 1 2 ( ) ( ) ( ) ( ) , ( ) ( ) ( ) ... ( ) , m mG k=G k G k G k F k=F k F k F k1 1 ( ) diag( ( ),0 ,...,0 ) n n n nS k S k+ . + .=and the blocks satisfy the following recursive equations{ }( ) ( , 1) ( ) ( ) ( )( ) ( ) ( )/ ,TttTkH k Rkt B k tQm tBk tE B k tQm t Bk t= − + + − + ++ + − + F (12){ }( ) ( )( ) ( ) ( )( )( ) ( ) ( )/ , ,T T f ttfT T f tkH k B k t A Qm f Bk fE B k t A Q m f B k f t f−−= + − + ++ + − + < F (13)( ) ( ), , Ttf ft H k=H k t>f (14)( ) ( ) ( ) ( ), t Tt G k=A Q m−t B k+t (15)2 ( ) ( , ) ( ),mj ttj tF k R k j A B k t −==Σ + (16)1 0 ... 00 1 ... 0( ) ... ... ... ... ,0 0 ... 11 1 ... 1SS k⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎢⎣− − −⎥⎦(17)where[ ] 1 20η ( ) η ( ) ... η ( )diag(1 ,1 μ ), η( ), ,0 0 ... 0nk r k r k rA r B k k− − − ⎡ ⎤ = + + =⎢⎣ ⎥⎦(18)Bk+t =E B k+t k+t F Bk+t =B k+t k+t −Bk+t (19)1 1 ( ) ( 1) ,( 1, ), (0) , TQt =AQt− A+R t= m Q = R (20)[ ] 1 21 1, ( , )1 1R Rk i ρ(k,i) 1 1 .=⎡⎢⎣− −⎤⎥⎦= − (21)Remark. In virtue of the linear dependence of matrix on its argument, the conditionalmathematical expectations in expressions above are easily calculated.Proof. The system equations (3), (7) can be written in the matrix formx(k+1)=Ax(k)+B[η(k+1),k+1]u(k), (22)where x, A, B, u are defined by0 ( ) ( ) ( ) ,Tx k = ⎡⎣V k V k ⎤⎦0 A=diag(1+r,1+ μ ),[ ][ ]1 21 2η ( ) η ( ) ... η ( )η( ), ,0 0 ... 0( ) ( ) ( ) ... ( ) .nTnk r k r k rB k ku k u k u k u k= ⎡⎢⎣ − − − ⎤⎥⎦=Criterion (8) can be transformed into1 21( / ) ( ) ( ) (,)( )( 1/ ) ( , 1) ( 1/ )/ ( ), ,mTiTkJ k m k E x k i Rx k i R k i x k iu k i k Rk i u k i k x k=⎧+ = + + − + ⎨⎩⎫+ +− − +− ⎬⎭ΣF (23)with1 2 ( ) ( ) [η( 1), 1] ( ) [η( 2), 2] ( 1) ...[η( ), ] ( 1), ( 1, )i i i x k i A x k A B k k u k A B k k u kB k i k iuk i i m− −+ = + + + + + + + + ++ + + + − =and matrices R1, R2(k,i) of the form (21).We can re-express (23) as follows{ } 1 2 ( / ) ( 1) ( 1) ( 1) ( 1) ( ) ( ) ( )/ ( ), ,T TkJ k+m k =E X k+ Δ X k+ −Δ k+ X k+ +U k Δ k U k x k F (24)with X(k+1)=Ψx(k)+Φ[Ξ(k+1),k+1]U(k), (25)where2( 1) ( / ) ( 1)( 2) ( 1/ ) ( 2)( 1) , ( ) , ( 1) , ,... ... ... ...( ) ( 1/ ) ( ) mx k u k k k Ax k u k k k AX k U k kx k m u k m k k m A⎡+⎤ ⎡ ⎤ ⎡η+ ⎤ ⎡ ⎤+ =⎢⎢ +⎥⎥ =⎢⎢ + ⎥⎥ Ξ + =⎢⎢η + ⎥⎥ Ψ=⎢⎢ ⎥⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢⎣ + ⎥⎦ ⎢⎣ +− ⎥⎦ ⎢⎣η + ⎥⎦ ⎢⎣ ⎥⎦2 221 2[η( 1), 1] 0 ... 0[η( 1), 1] [η( 2), 2] ... 0[ ( 1), 1] ,... ... ... ...[η( 1), 1] [η( 2), 2] ... [η( ), ]n nnm mB k kAB k k B k kk kA B k k A B k k B k m k m. ..− −⎡ + + ⎤⎢ + + + + ⎥Φ Ξ + + =⎢ ⎥⎢ ⎥⎢⎣ + + + + + + ⎥⎦R R kR R kkR R k mk R k R k R k m. . . .. . . .. . . .⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥Δ =⎢ ⎥Δ =⎢ ⎥⎢ ⎥ ⎢ ⎥⎢⎣ ⎥⎦ ⎢⎣ − ⎥⎦Δ + =Using (25) we can write (24) as follows{[ ] }{[ ] [ ] }11 21( / ) () ()2 ( ) ( 1) ( 1), 1 ( ), ( )( ) ( 1), 1 ( 1), 1 ( ) ( ) ( ).T TT TkT TkJ k m k x k x kx k k E k k x k U kU k E k k k k x k k U k+ = ΨΔΨ ++⎡⎣ ΨΔ − Δ +⎤⎦ Φ Ξ + + ++⎡⎣Φ Ξ + + Δ Φ Ξ + + + Δ⎤⎦F,F (26)Denote the following matrices{[ ] [ ] }{[ ] }{[ ] }112( ) ( 1), 1 ( 1), 1 ( ) ( ),( ) ( 1), 1 ( ), ,( ) ( 1) ( 1), 1 ( ), .TkTkkHk E k k k k xk kG k E k k x kF k k E k k x k= Φ Ξ + + ΔΦΞ + + +Δ= Ψ Δ Φ Ξ + += Δ + Φ Ξ + +,FFFIt can be shown that the blocks of the matrices H(k), G(k), F(k) satisfy the recursiveequations (12) - (20).Thus we have that the problem of minimizing the criterion (26) subject to (4) isequivalent to the quadratic program problem with criterion( / ) 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) T T Y k+m k=⎡⎣x k G k−F k⎤⎦U k+U k H k U ksubject to (11). Therefore we obtain the desired result, completing the proof of the theorem.3. Numerical examplesThis section tests the proposed approach on a simple example where we consider thesituation of an investor who has to allocate his wealth among five risky assets and onerisk-free asset. The updating of the portfolio based on the MPC is executed once everytrading day. We used five largest companies risky assets traded on the Russian StockExchanges MICEX: Gazprom, VTB, LUKOIL, Sberbank, NorNickel. We tested the resultson daily actual closing prices over a period of time from August 18, 2010 to January14, 2011. The risk-free asset considered here as bank account with risk-free rater1 =4% per annum.First we need to estimate the required model parameters (6) over the predictive horizonm.The practical difficulty with implementing obtained result is the choice of estimationparameter approach. There are essentially three ways to solve this problem. One approachconsidered in the statistics literature is to estimate the parameters using simpleaveraging. Second, the unknown parameters can be estimated using different modelspecifications describing the return asset evolution [10]. Third, one can use complexnonparametric methods, as implied in [9]. In reality, it is impossible to obtain precisionparameter estimates.In order to simplify our example, we used the following approach. We computed theexpected returns using 4-day simple averaging of past historical return data and assumethat the expected returns remain constant over the predictive horizon m. To obtain theexpected asset returns we use the adjusted procedure, updating the estimates at each decisiontime k. This procedure allows to adaptively track the behavior of risky asset returns.We obtained the second moments up to 6 order using the past 200 trading daysprior to the tracking period. These parameters are assumed to be stationary over the investmenthorizon and equal to the initial empirical estimates, based on backwards data.Our estimates were40.4814 -0.0906 -0.0351 0.0710 -0.0201 -0.05620.4569 0.0249 -0.0317 0.0618 -0.0432 -0.009210 0.3564 -0.0107 -0.0089 0.0611 -0.0179 -0.03280.5041 0.0152 -0.0276 0.0927 -0.0293 -00.5817 -0.0368 0.0138 0.0345 -0.0362− Θ = ..0212-0.0385⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢⎣ ⎥⎦The rows consist of the second moments of the risky assets returns Gazprom, VTB,LUKOIL, Sberbank, NorNickel.We set the tracking target to return 0.2 % per day (μ0=0.002). For our portfolio, weassumed an initial wealth of V(0)=V0(0)=1.The weight coefficients are set asR=diag(10−5, …,10−5), ρ(k,i)=0.2. We impose hard constraints on the tracking portfolioproblem with parameters γi'=0, γi''=3 (i=1, …,5), γ0'=3. In this example no short-sellingis allowed. We use Theorem 3.1 in order to define the optimal control portfolio strategy.For the on-line finite horizon problems MPC we used a horizon of m=10, and numericallysolved it in MATLAB by using the quadprog.m function.The results are summarized in three figures. Figure 1 plots portfolio (bold line) andbenchmark values (dotted line). In figure 2 we have investments in the risky assetsSberbank (solid line), Gazprom (dotted line). Figure 3 illustrates returns of risky assetsSberbank (solid line), Gazprom (dotted line).0 20 40 60 80 100TimeVV 00.9511.051.11.151.21.25Tracking from 18.08.2010 to 14.01.2011WealthFig. 1. 100 days performance of benchmark tracking, no short-selling is allowed(V- portfolio values, V0 - benchmark values).We find that on actual data the proposed approach is reasonable. The value of theportfolio is effectively tracked the benchmark and respected the constraints. Figure 2shows that the amounts of short-selling are significantly reduced. It is important to acknowledgethat, at least in this unsophisticated example, where we use simple averagingfor parameters estimation, the tracking performance appears to be rather efficient. Theobvious appeal of our approach is its simplicity and the fact that all of the models andstatistical techniques designed for the describing asset return evolution can be applieddirectly to estimate unknown parameters.0 20 40 60 80 100Timeu10.20.40.60.8Tracking from 18.08.2010 to 14.01.2011u2Wealth investments in assetsFig. 2. Asset allocation decision, no short-selling is allowed(u1 - Sberbank, u2 - Gazprom ).0 20 40 60 80 100Timeη10.020.040.06Tracking from 18.08.2010 to 14.01.2011η20-0.02-0.04-0.06Daily returnFig. 3. Risky assets returns (η1 - Sberbank, η2 - Gazprom).ConclusionIn this paper we studied a discrete-time portfolio selection problem subject to constraintson trading volume amounts. We propose to use the MPC methodology in orderto solve the problem. The optimal open-loop feedback portfolio control strategy is derived.We present the numerical modeling results, based on stocks, traded on the RussianStock Exchanges MICEX, that give evidence of capacity and effectiveness of proposedapproach.The main features of the model are (a) the flexibility of dealing with portfolio constraints,(b) the generality of stochastic return models that can be used with the method,and (c) the efficiency in numerical solution.

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