{"title": "Active Portfolio-Management based on Error Correction Neural Networks", "book": "Advances in Neural Information Processing Systems", "page_first": 1465, "page_last": 1472, "abstract": "", "full_text": "Keywords: portfolio management, \ufb01nancial forecasting, recurrent neural networks.\n\nActive Portfolio-Management\n\nbased on Error Correction Neural Networks\n\nHans Georg Zimmermann, Ralph Neuneier and Ralph Grothmann\n\nSiemens AG\n\nCorporate Technology\n\nD-81730 M\u00a8unchen, Germany\n\nAbstract\n\nThis paper deals with a neural network architecture which establishes a\nportfolio management system similar to the Black / Litterman approach.\nThis allocation scheme distributes funds across various securities or \ufb01-\nnancial markets while simultaneously complying with speci\ufb01c allocation\nconstraints which meet the requirements of an investor.\nThe portfolio optimization algorithm is modeled by a feedforward neural\nnetwork. The underlying expected return forecasts are based on error\ncorrection neural networks (ECNN), which utilize the last model error as\nan auxiliary input to evaluate their own misspeci\ufb01cation.\nThe portfolio optimization is implemented such that (i.) the allocations\ncomply with investor\u2019s constraints and that (ii.) the risk of the portfo-\nlio can be controlled. We demonstrate the pro\ufb01tability of our approach\nby constructing internationally diversi\ufb01ed portfolios across \u0001\u0003\u0002 different\n\ufb01nancial markets of the G7 contries. It turns out, that our approach is\nsuperior to a preset benchmark portfolio.\n\n1 Introduction: Portfolio-Management\n\nWe integrate the portfolio optimization algorithm suggested by Black / Litterman [1] into a\nneural network architecture. Combining the mean-variance theory [5] with the capital asset\npricing model (CAPM) [7], this approach utilizes excess returns of the CAPM equilibrium\nto de\ufb01ne a neutral, well balanced benchmark portfolio. Deviations from the benchmark\nallocation are only allowed within preset boundaries. Hence, as an advantage, there are no\nunrealistic solutions (e. g. large short positions, huge portfolio changes). Moreover, there\nis no need of formulating return expectations for all assets.\n\nIn contrast to Black / Litterman, excess return forecasts are estimated by time-delay recur-\nrent error correction neural networks [8]. Investment decisions which comply with given\nallocation constraints are derived from these predictions. The risk exposure of the portfolio\nis implicitly controlled by a parameter-optimizing task over time (sec. 3 and 5).\n\nOur approach consists of the following three steps: (i.) Construction of forecast models\n\n\u0004\u0006\u0005\b\u0007\n\t\n\t\u000b\t\f\u0007\r\u0004\u0006\u000e on the basis of error correction neural networks (ECNN) for all \u000f assets (sec. 2).\n\u0010 To whom correspondence should be addressed: Georg.Zimmermann@mchp.siemens.de.\n\n\f(sec. 3 and 4). By this, the pro\ufb01tability of an asset with respect to all others is measured.\n\n(ii.) Computation of excess returns \u0002\u0001\u0004\u0003\u0006\u0005\n(iii.) Optimization of the investment proportions \t\nAllocation constraints ensure, that the investment proportions\t\n\n\u0003 by a higher-level feedforward network\n\u0001 on the basis of the excess returns.\n\u0001 may deviate from a given\n\nbenchmark only within prede\ufb01ned intervals (sec. 3 and 4).\n\nFinally, we apply our neural network based portfolio management system to an asset allo-\ncation problem concerning the G7 countries (sec. 6).\n\n\u0001\b\u0007\n\n2 Forecasting by Error Correction Neural Networks\n\nMost dynamical systems are driven by a superposition of autonomous development and\nexternal in\ufb02uences [8]. For discrete time grids, such a dynamics can be described by a\n\nrecurrent state transition\n\f\u000b\u000e\n\n\u0005 and an output equation\u000f\u0010\u000b (Eq. 1).\n\u0004\u0012\u0011\n\n\u0007\u0014\u0013\n\nstate transition eq.\noutput eq.\n\n\u0007\"!\n\n\u0007\u0014#\n\n\u0007\u001c\u000f\u0017\u0016\n\n\u0007\u0015\u000f\u0017\u0016\n\u000b\u0010\u0018\n\nquanti\ufb01es the model\u2019s mis\ufb01t and serves\n\n, a neural network approach of Eq. 1 can be formulated as\n\nas an indicator of short-term effects or external shocks [8].\n\n\u000b , external in\ufb02uences\u0013\n\n. If the last model error\nis zero, we have a perfect description of the dynamics. However, due unknown\n\n\u000b\u000e\r\nThe state transition\n\n\u000b and\nis a mapping from the previous state\n\n\u000b\u000e\r\na comparison between the model output \u000f\u001a\u000b and observed data \u000f\u001b\u0016\n\u0007\u001c\u000f\u001d\u0016\n\u000b\u0010\u0018\nexternal in\ufb02uences \u0013\n\u000b or noise, our knowledge about the dynamics is often incomplete.\nUnder such conditions, the model error \u0011\n\u0007 \u001f\nof appropriate dimensions corresponding to \n\u0010$ , \u0013\n$ and\nUsing weight matrices\u001e\n\u000f\u0010$%\u0007&\u000f\u0017\u0016\n\u000b.-\n\u000b\u000e\r\n'\"(*)\u0017+\n\u001e,\n\n\u000f*\u000b\n\n1\u000b\nis recomputed by!\nIn Eq. 2, the output\u000f\u0002\u000b\n$ are adjusted by #\ndimensions in \n\noptimization task of appropriate sized weight matrices\u001e\n\t54\n>@?\nACB\nDEB\n\u000b\u000e7\n\n\u001f/\u0013\n\n\f\u000b and compared to the observation\u000f2\u0016\n\n. Different\n. The system identi\ufb01cation (Eq. 3) is a parameter\n\nFor an overview of algorithmic solution techniques see [6]. We solve the system identi\ufb01-\ncation task of Eq. 3 by \ufb01nite unfolding in time using shared weights. For details see [3, 8].\nFig. 1 depicts the resulting neural network solution of Eq. 3.\n\n\u0007\u0015\u000f\u0017\u0016\n\u000b\u0010\u0018\u0014\u0018\n\n\u000598\n\n\u000f*\u000b:\u0007\u0015\u000f\n\n\u000b\u0010;1<,=\n\n\u0007\"!\n\n\u0007\u0014#\n\n\u0007\u0014\u001f\n\nF:B\n\n[8]:\n\n\u000b.-\n\n\u00110!\n\n'\"(*)\u0017+\n\n(1)\n\n(2)\n\n(3)\n\nA\n\nzt\u22122\n\nD\n\nC\n\nt\u22121s\n\nD\n\nzt\u22121\n\nC\n\nts\n\nA\n\nzt\n\nA\n\nA\n\nD\n\nC\n\nst+1\n\nt+1y\n\nC\n\nst+2\n\nt+2y\n\nC\n\nst+3\n\nt+3y\n\n\u2212Id\nd\nyt\u22122\n\nB\nt\u22122u\n\n\u2212Id\n\nyd\nt\u22121\n\nB\nut\u22121\n\n\u2212Id\nyd\nt\n\nB\nut\n\nFigure 1. Error correction neural network (ECNN) using unfolding in time and overshooting. Note,\n\nwith target values of zero in order to optimize the error correction mechanism.\n\nis the \ufb01xed negative of an appropriate sized identity matrix, while MON0PRQ are output clusters\n\nthat HJILK\n\n\u0004\n\u0004\n\n\u0005\n\u0005\n\n\u000b\n\u000b\n\u0007\n\u000f\n\u000b\n\u000f\n\u000b\n\u0005\n\u0019\n\u0011\n\n\u000b\n\u0018\n\u0005\n\u000b\n\u0011\n\u000f\n\u000b\n\u000f\n\u000b\n\u000b\n\u0018\n\u0011\n$\n\u0018\n\n\u0005\n\u0005\n\u0011\n#\n\n\u000b\n\u0005\n!\n\u000b\n\u0002\n3\n6\n\u0016\n)\nG\n\f\u000b\u000e\n\n\u000f\u0017\u0016\n\n\u0011\u0002\u0001\u0004\u0003\n\nand \n\nThe ECNN (Fig. 1) is best to comprehend by analyzing the dependencies of \n\nthe externals \u0013\nin\ufb02uencing the state transition and \u0011\u0002\u0001\u0005\u0001\u0004\u0003\nthe targets\u000f\nhas an impact on \n\f\u000b\u000e\r\n\u000f\u0017\u0016\nof the internal expectations \u000f\n), external in\ufb02uences (coded in \u001f\nsystem (coded in \u001e\nwhich is also acting as an external input (coded in!\n\n\u0005 . The ECNN has two different inputs:\n[8]. At all future time steps \u0006\n-\b\u0007\n\u000b\u000e\n\n\u000b , \u0013\n\u000b , \n\u000b directly\n. Only the difference between\u000f\u0010\u000b and\n$ . A\n\u000b\u000e\n\nforecast of the ECNN is based on a modeling of the recursive structure of a dynamical\n) and the error correction mechanism\n\n$ and thus, the system offers forecasts \u000f\n\nUsing \ufb01nite unfolding in time, we have by de\ufb01nition an incomplete formulation of accu-\nmulated memory in the leftmost part of the network and thus, the autoregressive modeling\nis handicapped [9]. Due to the error correction, the ECNN has an explicit mechanism to\nhandle the initialization shock of the unfolding [8].\n\n, we have no compensation\n\n, #\n\n\u000b\u000e\n\n).\n\n. This is called overshooting [8]. Overshooting provides additional information about\nthe system dynamics and regularizes the learning. Hence, the learning of false causalities\nmight be reduced and the generalization ability of the model should be improved [8]. Of\n\nThe autonomous part of the ECNN is extended into the future by the iteration of matrices\u001e\nand!\ncourse, we have to support the additional output clusters \u000f\n$ by target values. However,\n\ndue to shared weights, we have the same number of parameters [8].\n\n\u000b\u000e\n\n3 The Asset Allocation Strategy\n\n(\f\n\n\t\u001a\u0001\n\n\u0007 with\n\nhave a superior excess return should be enlarged, because they seem to be more valuable.\n\nFurther on, let us de\ufb01ne the cumulated excess return as a weighted sum of the excess returns\n\nis de\ufb01ned as the difference between the expected returns \u0004\n\n\u000e\u000b\n with investment proportions \t\u001b\u0001\n\nNow, we explain how the forecasts are transformed into an asset allocation vector\n\u0007\n\t\u000b\t\n\t\n\u0007\n\u0002 ). For simplicity, short sales\n(i. e. \tR\u0001\u000e\r\u0010\u000f ) are not allowed. We have to consider, that the allocation (i.) pays atten-\ntion to the uncertainty of the forecasts and (ii.) complies with given investment constraints.\nIn order to handle the uncertainty of the asset forecasts \u0004\n\u0001 , we utilize the concept of excess\nreturn. An excess returnL\u0001\n\u0001 and\n\u0003 of two assets \u0001 and\u0011 , i. e. L\u0001\n\u0003 . The investment proportions\t\u0017\u0001 of assets which\n\u0001.\u0007\nfor one asset \u0001 over all other assets\u0011 ,\n\u0001\u0004\u0003\n\u0005\u0013\u0012\n\nThe forbiddance of short sales (\t\u0017\u0001\u0017\u0014\u0018\u000f ) and the constraint, that investment proportions \t\u001b\u0001\nsum up to one (\f\n\n\u0001:\u0005\n\u0002 ), can be easily satis\ufb01ed by the transformation\n\tR\u0001:\u0005\nhave a mean value of \u001f5\u0001 .\nde\ufb01nes how much the allocation\t\u001d\u0001 may deviate from \u001f\nby a bias vector $\n\n\u0019\u001b\u001a\u001d\u001c\nis the benchmark allocation. The admissible spread \n\tR\u0001\"!\n\u000e%\n corresponding to the benchmark allocation:\n\u0001:\u0005&$*\u0001\n\nSince we have to level the excess returns around the mean of the intervals, Eq. 4 is adjusted\n\nThe market share constraints are given by the asset manager in form of intervals, which\n\n\u0003\u0015\u0014\u0016\u000f\n\n\u0007\u000b\t\n\t\u000b\t\n\u0007\n\n\u0007\n\t\n\t\u000b\t\n\u0007\n\n\t\u001a\u0001C\u0005\n\n\u001f5\u0001\n\n\u0007# \n\n\u0005\u0013\u0012\n\n\u0001\u0004\u0003\n\n(6)\n\n(7)\n\n(4)\n\n(5)\n\n\u0007\r\u0004\n\n\t\u001a\u0001\n\n\u0019\u001b\u001a\u001e\u001c\n\n\u001f5\u0001\n\n\u0001 :\n\n\u000b\n\u0005\n!\n\n\u000b\n\u0007\n\u000b\n\u0018\n\u0018\n\u0016\n\u000b\n\u000b\n\u0005\n$\n\u0005\n!\n\n\t\n\t\n\u0005\n\t\n\u0005\n\u0003\n\u0004\n\u0003\n\u0005\n\u0004\n\u0004\n\n\u000e\n6\n\u0003\n7\n\u0011\n\u0004\n\u0001\n\u0007\n\u0004\n\u0003\n\u0018\n\u0012\n\u0001\n\u0003\n\u0011\n\n\u0001\n\u0018\n\f\n\u000e\n\u0003\n7\n\u0005\n\u0011\n\n\u0003\n\u0018\n\u0005\n\u0011\n\u0012\n\u0005\n\u0004\n\u000e\n\u0018\n\u0003\n\u0001\n\t\n\u0001\n\u0007\n\u001f\n\u0001\n-\n \n\u0001\n\n\u0003\n\u0005\n\t\n$\n\u0005\n$\n\n-\n\u000e\n6\n\u0003\n7\n\u0011\n\u0004\n\u0001\n\u0007\n\u0004\n\u0003\n\u0018\n\u0003\n\fThe bias $*\u0001 forces the system to put funds into asset \u0001 , even if the cumulated excess return\ndoes not propose an investment. The vector $ can be computed before-hand by solving the\n\nsystem of nonlinear equations which results by setting the excess returns (Eq. 7) to zero:\n\n...\n\n...\n\n\u0007\n\t\u000b\t\n\t\f\u0007\n...\n\u0007\n\t\u000b\t\n\t\f\u0007\n\n\t\u001a\u0001\n\u000e\u001d\u0011\n\nSince the allocation \u0011\n\n\u0007\u000b\t\n\t\u000b\t\n\u0007\n\nleads to a non-unique solution (Eq. 9) of the latter system (Eq. 8)\n\nrepresents the benchmark portfolio, the pre-condition\n\nmaximization task can be stated as a constraint optimization problem with\n\n-\u0003\u0002\n\n\u000f .\n\n. In the following, we choose\n\n\u001f5\u0001:\u0007\n\nfor any real number\n\n$*\u0001\b\u0005\u0001O)\nThe interval \t\n\u001f5\u0001\ncause the latter quanti\ufb01es the deviation of \t\nreturn of asset \u0001 at time \u0006 :\n\t\u001a\u0001\n\n\u0007\n\t\u000b\t\n\t\n\u0007\n\n\u00041\u0001\n\n\u0018E=\n\n de\ufb01nes constraints for the parameters\n\u0001 from the benchmark \u001f\n\nThis problem can be solved as a penalized maximization task\n\n\u000e be-\n\u0007\u001b\u0003\u000b\u0003\u001b\u0003\f\u0007\n\u0001 . Thus, the return\n\u000b as the actual\n\n\b\u0007\n\n(10)\n\n\u0001\b\u0005\n\n\u000b\u000e7\n\n(8)\n\n(9)\n\n(11)\n\n(12)\n\n\tR\u0001\"!\n\n\u0007# \n\n\u0007\n\t\f\u000b\n\n\tR\u0001\n\n\u001f5\u0001\r\u000b\u000f\u000e\u0011\u0010\n\nif\notherwise\n\n\u0013\u0016\u0015\u0018\u0017\u0016 \n\n\u000b\u000e7\n\n\u000b\u000e7\n\n\u0007\n\t\u000b\t\n\t\f\u0007\n\n\u0004\f\u0001\n\n\tR\u0001\n\nis de\ufb01ned as a type of\n\n-insensitive error function:\n\nwith\n\n\u000b\u000f\u000e\n\n\u000b\u000f\u0013\f\u000b\n\nOptimize the allocation parameters\n\nSummarizing, the construction of the allocation scheme consists of the following two steps:\n. (ii.)\n\n(i.) Train the error correction sub-networks and compute the excess returns \u0011\nshare constraints \u0011\n\n\u0013\f\u0015L\u0007# \n\u0001\u0004\u0003 using the forecast models with respect to the market\n\u001f5\u0001\n\n\u0007\u000b\t\n\t\u000b\t\n\u0007\n\t\n\t\n\t\u000b\u0007\n\n\u0007\n\t\u000b\t\n\t\n\n(13)\n\n\u0001:\u0007\n\n\tR\u0001\n\n:\n\n\u00041\u0001\n\nO)\n\n\u0018E=\n\n\u0010\u001a\u0019\n\nAs we will explain in sec. 5, Eq. 13 also controls the portfolio risk.\n\n4 Modeling the Asset Allocation Strategy by Neural Networks\n\nA neural network approach of the allocation scheme (sec. 3) is shown in Fig. 2.\n\nThe \ufb01rst layer of the portfolio optimization neural network (Fig. 2) collects the predictions\n\u0001 from the underlying ECNNs. The matrix entitled \u2019unfolding\u2019 computes the excess re-\n\u0001\u0003\u0002 assets as a contour plot. White spaces indicate weights with a value\nturns \n\u0002 . The layer entitled \u2019excess returns\u2019 is\nof \u000f , while grey equals \u0005\ndesigned as an output cluster, i. e. it is associated with an error function which computes\nerror signals for each training pattern. By this, we can identify inter-market dependencies,\nsince the neural network is forced to learn cross-market relationships.\n\n\u0002 and black stands for \u0007\n\nfor \u000f\n\n\u001f\n\u0005\n\u0005\n\u0011\n$\n\u0005\n$\n\u000e\n\u0018\n\u001f\n\u000e\n\u0005\n\t\n$\n\u0005\n$\n\u000e\n\u0018\n\u0003\n\u001f\n\u0005\n\u001f\n\u000e\n\u0018\n\f\n\u000e\n\u0001\n7\n\u0005\n\u001f\n\u0002\n\u0011\n\u001f\n\u0001\n\u0018\n\u0007\n\u0002\n\u0002\n\u0005\n \n\u0001\n\u0007\n-\n \n\u0001\n\u0012\n\u0001\nB\n\u0005\n\u0012\n\u0001\nB\n\u0004\n\u0001\nB\n\u0002\n3\n4\n6\n\u0005\n\u000e\n6\n\u0001\n7\n\u0005\nB\n\u000b\n\u0011\n\u0004\n\u0005\nB\n\u000b\n\u0004\n\u000e\nB\n\u000b\n\u0007\n\u0012\n>\n(\n\u001a\n\u0005\n\u0006\n\u0006\n\u0006\n\u0006\n\u0006\n\t\n\u001f\n\u0001\n\u0001\n\u0007\n\u001f\n\u0001\n-\n \n\u0001\n\u0001\n\u0003\n\u0002\n3\n4\n6\n\u0005\n\u000e\n6\n\u0001\n7\n\u0005\n\t\nB\n\u000b\n\u0011\n\u0004\n\u0005\nB\n\u000b\n\u0004\n\u000e\nB\n\u000b\n\u0007\n\u0012\n\u0018\n\u0007\n\n=\n>\n(\n\u001a\n\u0005\n\u0007\n\u000b\n\t\n\u0012\n\u000e\n\u0005\n\u0014\n\u000f\n\u0015\n\u0015\n\u0004\n\u0004\n\u0003\n\u0018\n\u0012\n\u001f\n\u0005\n\u0007\n \n\u0005\n\u001f\n\u000e\n\u0007\n \n\u000e\n\u0018\n\u0002\n3\n4\n6\n\u0005\n\u000e\n6\n\u0001\n7\n\u0005\nB\n\u000b\n\u0011\n\u0011\n\u0018\n-\n\u000e\n6\n\u0003\n7\n\u0005\n\u0012\n\u0001\n\u0003\n\u0011\n\u0004\n\u0001\nB\n\u000b\n\u0007\n\u0004\n\u0003\nB\n\u000b\n\u0018\n>\n(\n\u001a\n\u0005\n\u0003\n\u0004\n\u0001\n\u0003\n\u0005\n\fk market shares\n\nid (fixed)\n\nk asset allocations\n\nfolding (fixed)\n\nweigthed\n\nexcess returns\n\ni\n\nd\n\n \n(\nf\ni\nx\n\ne\n\nd\n)\n\nt\n\nn\nr\nu\ne\nr\n \ns\ns\ne\nc\nx\ne\n\n20\n\n40\n\n60\n\n80\n\n100\n\n120\n\n140\n\n160\n\n180\n\n200\n\nunfolding matrix\n\n1\n\n2\n\n3\n\n4\n\n5\n\n6\n\n7\n\n8\n\n9\n\n10 11 12 13 14 15 16 17 18 19 20 21\nassets\n\nk(k\u22121)\n\n2\n\nexcess returns\n\nln(m ), ..., ln(m )k\n\n1\n\nw = \n\n1w\n\n0\n\n0\n\nw\n\nk(k\u22121)\n\n2\n\nunfolding (fixed)\n\nk forecasts\n\n[ 1 0 ... 0 ]\n\ny\nt+6\n\nasset 1\n\ny\nt+6\n\n(...)\n\nasset 2\n\nt+6y\nasset k\u22121\n\n[ 0 ... 0 1 ]\n\nt+6y\nasset k\n\nECNN forecasts of k assets\n\nFigure 2. Arranged on the top of the ECNN sub-networks, a higher-level neural network models the\n. The diagonal matrix which\nportfolio optimization algorithm on basis of the excess returns\ncomputes the weighted excess returns includes the only tunable parameters\n. All others are \ufb01xed.\n\n\u0002\u0001\n\nH\u0003\u0005\u0004\n\nis weighted by a particular\n\nthe layer entitled \u2019weighted excess returns\u2019. Afterwards, the weighted excess returns are\nfolded using the transpose of the sparse matrix called \u2019unfolding\u2019 (see Fig. 2). By this, we\n\nNext, each excess return \n\u0001\u0004\u0003 via a diagonal connection to\nfor each asset \u0001 .\ncalculate the sum of the weighted excess returns \f\n\u0003\f\nAccording to the predictions of the excess returns \u0004\n\u0003 , the layer \u2019asset allocation\u2019 com-\n\u0007\u000b\t\n\t\n\t\u000b\u0007\nputes pro\ufb01table investment decisions. In case that all excess returns \u0011\n\u0007\n\t\n\t\u000b\t\n\u0007\n\u0007\u000b\t\n\t\u000b\t\n\u0007\nthe benchmark portfolio is reproduced by the offset \u0011\n\u0018\u0014\u0018\nOtherwise, funds are allocated within the preset investment boundaries \t\n\n ,\n\u0002 .\nwhile simultaneously complying with the constraints\t2\u0001\n\u0014\u0016\u000f and \f\n\tR\u0001\b\u0005\nIn order to prevent short selling, we assume\t\u0017\u0001\n\u0002 , i. e. \f\n\nfor each investment proportion. Further\non, we have to guarantee that the sum of the proportions invested in the securities equals\n\u0002 . Both constraints are satis\ufb01ed by the activation function of the \u2019asset\nallocation\u2019 layer\u2019, which implements the non-linear transformation of Eq. 5 using soft-\nmax. The return maximization task of Eq. 10 is also solved by this cluster by generating\nerror signals utilizing the prof-max error function (Eq. 14).\n\n\u0014\u0016\u000f\n\nare zero\n.\n\nO)\n\n\u0004\u0006\u0005\n\n\u0007\u000b\t\n\t\u000b\t\n\u0007\r\u0004\u0006\u000e\n\n\u0018E=\n\n(14)\n\n\u000b\u000e7\n\nThe layer \u2019market shares\u2019 takes care of the allocation constraints. The error function of\n\nEq. 15 is implemented to ensure, that the investments\t\u001b\u0001 do not violate preset constraints.\n\n\t\u001a\u0001\n\n\u001f5\u0001\n\n\u000e\u0011\u0010\n\n\u000b\u000e7\n\n(15)\n\n\u0006\n\u0001\n\u0004\n\u0001\n\u0003\n\u0012\n\u0003\n\u0012\n\u0001\n\u0001\n\u0003\n\u0001\n\u0007\n\u0004\n\n\u0005\n\n\u000e\n\u0018\n$\n\u0005\n$\n\u000e\n\u0018\n\u0005\n\u0011\n\n)\n\u0011\n\u001f\n\u0005\n\u0018\n\u0011\n\u001f\n\u000e\n\u0018\n\u001f\n\u0001\n\u0007\n \n\u0001\n\u0007\n\u001f\n\u0001\n-\n \n\u0001\n\t\n\u0001\n\u0005\n\u0002\n3\n4\n6\n\u0005\n\u000e\n6\n\u0001\n7\n\u0005\n\u0004\n\u0001\nB\n\u000b\n\t\n\u0001\n\u0011\nB\n\u000b\nB\n\u000b\n\u0007\n\u0012\n>\n(\n\u001a\n\u0005\n\u0003\n\u0002\n3\n4\n6\n\u0005\n\u000e\n6\n\u0001\n7\n\u0005\n\t\n\u0007\n\t\n\u000b\n\u0007\n\u000b\n\n=\n>\n(\n\u001a\n\u0005\n\u0003\n\fThe \u2019market shares\u2019 cluster generates error signals for the penalized optimization problem\nstated in Eq. 11. By this, we implement a penalty for exceeding the allocation intervals\n\ncomputing the gradients in order to adapt the parameters\n\n . The error signals of Eq. 10 and Eq. 11 are subsequently used for\n\n\u0007\u0016 \n\n5 Risk Analysis of the Neural Portfolio-Management\n\n\u0003 .\n\nIn Tab. 1 we compare the mean-variance framework of Markowitz with our neural network\nbased portfolio optimization algorithm.\n\ninput:\n\noptimization:\n\noutput:\n\nMarkowitz:\n\nfor each decision:\nforecasts \u0004\n\n\u0011 ,\naccepted risk exposure\nwith \f\n\t\u0002\u0003\nvector \u0011\n\n\u0007\u0001\n\t\u001a\u0001\n\u0001\u0004\u0003\n\u0007\n\t\u000b\t\n\t\f\u0007\n\nfor each decision:\n\n\u00010\tR\u0001\n\nNeural Network:\nprediction models \u0004\nbenchmark allocation \u001f\ndeviation interval \u0006\u0001\n\u0007\n\t\u000b\t\n\t\u000b\u0007\r\u0004\n\nwith implicit risk control\nk decision schemes\n\n\tR\u0001\n\n\u00041\u0001\n\u0007\u000b\t\n\t\n\t\n\u0007\r\u0004\n\n\u0004\u0006\u0005\n\n,\n\n\u0001 ,\n\n\u0018E=\n\nTable 1. Comparison of the portfolio optimization algorithm of Markowitz with our approach.\n\nThe most crucial difference between the mean-variance framework and our approach is the\nhandling of the risk exposure (see Tab. 1). The Markowitz algorithm optimizes the expected\nrisk explicitly by quadratic programming. Assuming that it is not possible to forecast the\nexpected returns of the assets (often referred to as random walk hypothesis), the forecasts\n\n\u0001 are determined by an average of most recent observed returns, while the risk-covariance\nmatrix \t\n\nof the portfolio is determined by the volatility of the time series of the assets.\n\nis estimated by the historical volatility of the assets. Hence, the risk\n\nHowever, insisting on the existence of useful forecast models, we propose to derive the\ncovariance matrix from the forecast model residuals, i. e. the risk-matrix is determined\nby the covariances of the model errors. Now, the risk of the portfolio is due to the non-\nforecastability of the assets only. Since our allocation scheme is based on the model uncer-\ntainty, we refer to this approach as causal risk.\n\nB\u0004\u0003\u0004\u0003\u0004\u0003\u0004B\n\nUsing the covariances of the model errors as a measurement of risk still allows to apply the\nMarkowitz optimization scheme. Here, we propose to substitute the quadratic optimization\nproblem of the Markowitz approach by the objective function of Eq. 16.\n\n(16)\n\nO)\n\n\u000b\u000e7\n\n\u000b\b\u0007\n\u0007\u000b\t\n\t\u000b\t\f\u0007\n\n\u0001\u0004\u0003\nThe error function of Eq. 16 is optimized over time \u0006\neters\nit is possible to construct an asset allocations strategy which implicitly controls the risk\nexposure of the portfolio according to the certainty of the forecasts \u0004\ncan be extended by a time delay parameter\n4\t\b\nIf the predicted excess returns \u0011\n\n\u0018\u0007\u0006\n3 with respect to the param-\n\u0003 , which are used to evaluate the certainty of the excess return forecasts. By this,\n\u0001 . Note, that Eq. 16\n\u0001\u0004\u0003 are greater than\n\nzero, because the optimization algorithm emphasizes the particular asset in comparison to\n\nin order to focus on more recent events.\n\nare reliable, then the weights\n\n\t\n\u001f\n\u0001\n\u0001\n\u0007\n\u001f\n\u0001\n-\n \n\u0001\n\u0012\n\u0001\n\u0001\n\u0001\n\u0003\n\u0007\n\u0007\n\u0001\n\u0007\n\u0001\n\u0011\n\u0013\n\u0018\n\u0007\n\u0002\n\u0017\n\u0001\n\u0017\n\u000f\n\f\n\u000e\n\u0001\n7\n\u0005\n\u0004\n=\n>\n(\n\u001a\n\u0002\n\u0010\n\u0001\nB\n\u0003\n\n\u0005\n\n\f\n\u000b\n\f\n\u0001\nB\n\u000b\n\u0011\n\u0004\n\u0005\n\u000e\n\u0007\n\u0012\n>\n(\n\u001a\n\u0005\n\t\n\u0005\n\t\n\u000e\n\u0018\n\t\n\u0001\n\u0011\n\u000e\n\u0007\n\u0012\n\u0018\n\u0004\n\n\u0001\n\u0003\n\n\u0001\nB\n\u0003\n7\n\u0005\n\u000e\n\u0002\n3\n4\n6\n\u0005\n\u000e\n6\n\u0001\n7\n\u0005\n\u0004\n\u0001\nB\n\u000b\n\t\n\u0001\n\u0005\n\u0011\n\u001f\n\u0001\n\u0018\n-\n\u000e\n6\n\u0001\n7\n\u0005\n\u0012\n\u0011\n\u0004\n\u0001\nB\n\u0004\n\u0003\nB\n\u000b\n=\n>\n(\n\u001a\n\u0005\n\u0005\n\u0002\n\u0012\n\u0001\n\t\n$\n\u0004\n\u0001\nB\n\u000b\n\u0007\n\u0004\n\u0003\nB\n\u000b\n\u0018\n\u0012\n\fother assets with less reliable forecasts. In contrast, unreliable predictions are ruled out by\npushing the associated weights\ntowards zero. Therefore, Eq. 16 implicitly controls the\nrisk exposure of the portfolio although it is formulated as a return maximization task.\n\nEq. 16 has to be optimized with respect to the allocation constraints \u001f\n\nthe de\ufb01nition of an active risk parameter\nfrom the benchmark portfolio\n\n\u0001 . This allows\n\n quantifying the readiness to deviate\n\n within the allocation constraints:\n\n\u0007\n\t\u000b\t\n\t\f\u0007\n\u0007\u0002\n\n\u001f5\u0001\n\n\tR\u0001\n\n(17)\n\nThe weights\nthen the benchmark is recovered, while\n\n\u0003 and the allocations \t\u001d\u0001 are now dependent on the risk level\n\n\u000f ,\n\u0002 allows deviations from the benchmark\nanalysis the risk sensitivity of the\n\n. If\n\nwithin the bounds \u0006\u0001 . Thus, the active risk parameter\n\nportfolios with respect to the quality of the forecast models.\n\n6 Empirical Study\n\nNow, we apply our approach to the \ufb01nancial markets of the G7 countries. We work on\nthe basis of monthly data in order to forecast the semi-annual development of the stock,\ncash and bond markets of the G7 countries Spain, France, Germany, Italy, Japan, UK and\nUSA. A separate ECNN is constructed for each market on the basis of country speci\ufb01c\neconomic data. Due to the recurrent modeling, we only calculated the relative change of\neach input. The transformed inputs are scaled such that they have a mean of zero and a\nvariance of one [8]. The complete data set (Sept. 1979 to May 1995) is divided into three\nsubsets: (i.) Training set (Sept. 1979 to Jan. 1992). (ii.) Validation set (Feb. 1992 to\nJune 1993), which is used to learn the allocation parameters\n(July 1993 to May 1995). Each ECNN was trained until convergence by using stochastical\nvario-eta learning, which includes re-normalization of the gradients in each step of the\nbackpropagation algorithm [9].\n\n\u0003 . (iii.) Generalization set\n\n\u0007\u000b\t\n\t\n\t\n\u0007\n\n\u0005\u0004\n which is calculated with respect to the market shares \u001f\n\nWe evaluate the performance of our approach by a comparison with the benchmark port-\nfolio\ncomparison of our strategy and the benchmark portfolio is drawn on the basis of the accu-\nmulated return of investment (Fig. 3). Our strategy is able to outperform the benchmark\n. A further enhancement of the port-\nportfolio\nfolio performance can only be achieved if one relaxes the market share constraints. This\nindicates, that the tight allocation boundaries, which prevent huge capital transactions from\nnon-pro\ufb01table to booming markets, narrow additional gains.\n\non the generalization set by nearly \u0002\n\n\u0001 . The\n\n\u000f\u0004\u0003\n\nIn Fig. 4 we compare the risk of our portfolio to the risk of the benchmark portfolio. Here,\nthe portfolio risk is de\ufb01ned analogous to the mean-variance framework. However, in con-\nof the\ntrast to this approach, the expected (co-)variances are replaced by the residuals\nunderlying forecast models. The risk level which is induced by our strategy is comparable\nto the benchmark (Fig. 4), while simultaneously increasing the portfolio return (Fig. 3).\n\n\u00121\u0001\n\nFig. 5 compares the allocations of German bonds and stocks across the generalization set: A\ntypical reciprocal investment behavior is depicted, e. g. enlarged positions in stocks often\noccur in parallel with smaller investments in bonds. This effect is slightly disturbed by\ninternational diversi\ufb01cation. Not all countries show such a coherent investment behavior.\n\n7 Conclusions and Future Work\n\nWe described a neural network approach which adapts the Black / Litterman portfolio op-\ntimization algorithm. Here, funds are allocated across various securities while simultane-\nously complying with allocation constraints. In contrast to the mean-variance theory, the\nrisk exposure of our approach focuses on the uncertainty of the underlying forecast models.\n\n\u0012\n\u0001\n\u0003\n\u0001\n\u0007\n \n\n!\n\t\n\u000f\n\u0007\n\u0002\n\u0001\n\u0005\n\t\n\u001f\n\u0005\n\u001f\n\u000e\n!\n\t\n \n\u0001\n\u0007\n\u001f\n\u0001\n-\n\n \n\u0001\n\n\u0003\n\u0012\n\u0001\nB\n\n\n\u0005\n\n\u0005\n\n\u0012\n\u0001\n\u0001\n\u0005\n\t\n\u001f\n\u0005\n\u001f\n<\n\u0001\n\u0011\n\u0006\n\u0018\n\fECNN\nBenchmark\n\n14\n\n12\n\n10\n\n8\n\n6\n\n4\n\n2\n\n0\n\nn\nr\nu\n\nt\n\ne\nr\n \n\nd\ne\n\nt\n\nl\n\na\nu\nm\nu\nc\nc\na\n\n\u22122\n\nJuly 1993\n\nDate\n\nFigure 3.\n\ngerman stocks\ngerman bonds\n\n0.1\n\n0.09\n\n0.08\n\n0.07\n\n0.06\n\n0.05\n\n0.04\n\n0.03\n\n0.02\n\nn\no\n\ni\nt\n\na\nc\no\nc\n\nl\nl\n\na\n\nportfolio risk\nbenchmark risk\n\nk\ns\ni\nr\n\nMay 1995\n\nJuly 1993\n\nMay 1995\n\ndate\n\nFigure 4.\n\nFig.3.\nComparison of accumulated\nreturn of investment (generalization set).\n\nFig.4.\n(generalization set).\n\nComparison of portfolio risk\n\nFig.5.\nand stocks (generalization set).\n\nInvestments in German bond\n\n0.01\nJuly 1993\n\nMay 1995\n\ndate\n\nFigure 5.\n\nThe underlying forecasts are generated by ECNNs, since our empirical results indicate, that\nthis is a very promising framework for \ufb01nancial modeling. Extending the ECNN by using\ntechniques like overshooting, variants-invariants separation or unfolding in space and time,\none is able to include additional prior knowledge of the dynamics into the model [8, 9].\n\nFuture work will include the handling of a larger universe of assets. In this case, one may\nextend the neural network by a bottleneck which selects the most promising assets.\n\nReferences\n\n[1] Black, F., Litterman, R.:Global Portfolio Optimization, Financial Analysts Journal, Sep. 1992.\n[2] Elton, E. J., Gruber, M. J.: Modern Portfolio Theory and Investment Analysis, J. Wiley & Sons.\n\u0003 ed., Macmillan, N. Y. 1998.\n[3] Haykin S.: Neural Networks. A Comprehensive Foundation.,\n[4] Lintner, J.:The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfo-\n\n\u0002\u0001\n\nlios and Capital Budgets, in: Review of Economics and Statistics, Feb. 1965.\n\n[5] Markowitz, H. M.: Portfolio Selection, in: Journal of Finance, Vol. 7, 1952, p. 77-91.\n[6] Pearlmatter, B.:Gradient Calculations for Dynamic Recurrent Neural Networks: A survey, In\n\nIEEE Transactions on Neural Networks, Vol. 6, 1995.\n\n[7] Sharpe, F.:A Simpli\ufb01ed Model for Portfolio Analysis, Management Science, Vol. 9, 1963.\n[8] Zimmermann, H. G., Neuneier, R., Grothmann, R.: Modeling of Dynamical Systems by Er-\nror Correction Neural Networks, in: Modeling and Forecasting Financial Data, Techniques of\nNonlinear Dynamics, Eds. Soo\ufb01, A. and Cao, L., Kluwer 2001.\n\n[9] Zimmermann, H.G., Neuneier, R.:Neural Network Architectures for the Modeling of Dynamical\n\nSystems, in: A Field Guide to Dynamical Recurrent Networks, Eds. Kremer, St. et al., IEEE.\n\n\f", "award": [], "sourceid": 1969, "authors": [{"given_name": "Hans-Georg", "family_name": "Zimmermann", "institution": null}, {"given_name": "Ralph", "family_name": "Neuneier", "institution": null}, {"given_name": "Ralph", "family_name": "Grothmann", "institution": null}]}