2 2012 30 information technologies and control Bi-level Optimization for Portfolio Modelling Key Words: Financial portfolio; bi-level optimization; hierarchical theory. Abstract. The optimal resource allocation of sets of securities, available in the financial market can be done by solving a multi- criterion optimization task, aimed at maximization of portfolio return and minimization of portfolio risk. Here is proposed the utilization of hierarchical coordination for solving the bi-level optimization problem, which formalizes the investment process. A two-level hierarchical approach for solving the portfolio opti- mization problem is applied where the optimal Sharp ratio of risk versus return is solved at the upper level and as a result is determined by the investor’s preferences for taking risk on the basis of objective considerations. The optimal portfolio is evalu- ated at the lower level. Introduction The problem of portfolio optimization targets the op- timal resource allocation in the investment process [12]. The resource allocation is made by investing capital in financial assets (or goods), which give return to the investor after a period of time. For the investment process the target is to maximize the return while the investment risk has to be minimal [2,6,7,8,10,11]. The risk is equivalent to uncertainty. The term “risk” reflects the undetermined and non-predict- able future. The minimization of financial risk is with high priority during the investment and that is why the statistics and probability modelling are interested in it. The financial risk is always related with the portfolio management [20]. The difficulties of predicting the financial risk are related to the market behaviour, based on continuous dynamical changes. The investment models are based on mathematical analytical tools, which formalize both the behavior of the market players and future events in financial markets. In order to formalize the investment process, financial resource allocation has to be done. This requires a market analysis, which usually uses predefined assumptions. Usually, such assumptions concern uncertainty in ideal mathematical behaviour, constant and not changing environment influ- ences. In portfolio theory the decision maker makes deci- sions taking into account the risk of the investment. The portfolio optimization models are based mainly on probabil- ity theory. However, the probabilistic approach is not able to formalize the real market behaviour. Another uncertainty modeling approach of the financial market is the fuzzy set theory [4]. An essential contribution for the finance modelling and especially for risk assessment is the work of Markowitz [9] which concerns the individual investor. This theory is based on both optimization and probability theory. The investor’s goal is to maximize the return and to minimize the risk of the investment decisions. The investor’s return is formalized as the mean value of a random behaved function of the portfolio securities returns. The risk is formalized as a variance of these portfolio securities. The portfolio mod- elling is formalized by the above mathematical representa- tions of return and risk which define the portfolio optimi- zation problem. The portfolio solution depends on the level of risk which investor can bear in comparison with the level of portfolio return. Thus, for the practical utilization of the portfolio theory, the relation between return and risk is the main parameter for the investor. The portfolio risk is mini- mized according to two types of arguments: the portfolio content and the parameter of the investor’s risk preference. The market risk, which results in different values of the variances of the average return, is under consideration in the paper. The market risk is defined as a risk to the finan- cial portfolio, related to the dynamic changes of the market prices of equity, foreign exchange rates, interest rates, commodity prices [3]. The financial firms generally take a market risk to receive profits. Particularly, they try to take a risk they intend to have and they actively manage the market risk. Usually, the investment decision-making process is done by investors’ subjective assumptions about the the relationship between portfolio risk andreturn. In this paper decreasing the subjective influence in the investment pro- cess is proposed. This is achieved by calculating the un- known investor’s coefficient for undertaking risk based on the optimization problem. A bi-level optimization problem based on a hierarchical system’s modelling formalizes the portfolio investment. The parameter of the investor’s risk preference is evaluated at the upper level. After that, this parameter is used for optimal resource allocation of the portfolio optimization problem by minimizing risk and maxi- mizing return. In that manner, the process of portfolio re- source allocation is performed without subjective influence. Portfolio Optimization Problem The portfolio theory is developed to support decision making for investment allocation of financial assets selling (securities, bounds) at the stock exchange [1]. This alloca- tion is known as “investment” decision making. The inves- tor considers the asset as a matter of future income. The better combination of financial assets (securities) of the portfolio leads to better return for the investor. The port- folio contains a set of securities. The problem of portfolio optimization targets the optimal resource allocation in in- K. Stoilova