Hans-Georg Zimmermann, Ralph Neuneier, Ralph Grothmann
This paper deals with a neural network architecture which establishes a portfolio management system similar to the Black / Litterman approach. This allocation scheme distributes funds across various securities or ﬁ- nancial markets while simultaneously complying with speciﬁc allocation constraints which meet the requirements of an investor. The portfolio optimization algorithm is modeled by a feedforward neural network. The underlying expected return forecasts are based on error correction neural networks (ECNN), which utilize the last model error as an auxiliary input to evaluate their own misspeciﬁcation. The portfolio optimization is implemented such that (i.) the allocations comply with investor’s constraints and that (ii.) the risk of the portfo- lio can be controlled. We demonstrate the proﬁtability of our approach by constructing internationally diversiﬁed portfolios across different ﬁnancial markets of the G7 contries. It turns out, that our approach is superior to a preset benchmark portfolio.
1 Introduction: Portfolio-Management
We integrate the portfolio optimization algorithm suggested by Black / Litterman  into a neural network architecture. Combining the mean-variance theory  with the capital asset pricing model (CAPM) , this approach utilizes excess returns of the CAPM equilibrium to deﬁne a neutral, well balanced benchmark portfolio. Deviations from the benchmark allocation are only allowed within preset boundaries. Hence, as an advantage, there are no unrealistic solutions (e. g. large short positions, huge portfolio changes). Moreover, there is no need of formulating return expectations for all assets.
In contrast to Black / Litterman, excess return forecasts are estimated by time-delay recur- rent error correction neural networks . Investment decisions which comply with given allocation constraints are derived from these predictions. The risk exposure of the portfolio is implicitly controlled by a parameter-optimizing task over time (sec. 3 and 5).
Our approach consists of the following three steps: (i.) Construction of forecast models
on the basis of error correction neural networks (ECNN) for all assets (sec. 2). To whom correspondence should be addressed: Georg.Zimmermann@mchp.siemens.de.
(sec. 3 and 4). By this, the proﬁtability of an asset with respect to all others is measured.
(ii.) Computation of excess returns (iii.) Optimization of the investment proportions Allocation constraints ensure, that the investment proportions
by a higher-level feedforward network on the basis of the excess returns. may deviate from a given
benchmark only within predeﬁned intervals (sec. 3 and 4).
Finally, we apply our neural network based portfolio management system to an asset allo- cation problem concerning the G7 countries (sec. 6).
2 Forecasting by Error Correction Neural Networks
Most dynamical systems are driven by a superposition of autonomous development and external inﬂuences . For discrete time grids, such a dynamics can be described by a
recurrent state transition
and an output equation (Eq. 1).
state transition eq. output eq.