## 1. Introduction

Real-life decision making can be a very difficult process. Therefore, we should have some supporting tools to make this process easier and more effective. An example of usually a complex decision-making process is investment portfolio selection, which an increasing number of people over the world are facing as they try to valorize their free funds.

To make a portfolio, the intuitive approach, sometimes supported by knowledge of basic quantitative characteristics of the investment (return, risk, etc.), is often applied. Then, the portfolio is especially made on the basis of human intuition, personal mood, or the mood of the crown on the capital market. Such a decision can be supported by a well-known psychological analysis developed by Le Bon [

1]. Such an approach is predominantly qualitative. Portfolios can be also made based on fundamental [

2] or technical [

3] analyses. Then, we receive the value of some quantitative indicator or “graphical” information based on a historical development of the asset prices. Portfolio selection reflecting only some qualitative/quantitative information or one-criterion perspective is too simplifying. Moreover, no mentioned approach enables selecting a portfolio, or exact determination of the assets’ shares in the portfolio. They provide only some information on selected investment instruments that support the following steps leading to a portfolio composition making.

To make a more complex and satisfactory investment decision, I propose using a decision-making theory that is often wrongly neglected on the capital market. Of course, a well-known Markowitz optimization model [

4] is sometimes applied. However, this model takes into account only one or two characteristics (return and risk). Multiple objective programming methods can reflect many qualitative and quantitative input information on the investment instruments or investor’s preferences. Therefore, these multi-criteria approaches are able to provide more complex and representative results. Moreover, they can also work with the outputs of the aforementioned analyses.

In my opinion, the most suitable multiple objective programming methods are interactive approaches reflecting a continuous information from a decision maker (DM). These methods allow a subsequent modification of the actual solution (portfolio) according to the DM’s (investor’s) preferences. Such an approach helps create a portfolio to the investor’s satisfaction. Interactive multiple objective programming methods were developed from the 1970s. The first methods were the Step Method (STEM) designed by Benayoun et al. [

5] and the Geoffrion–Dayer–Feinberg’s method (GDF) [

6]. In the 1980s, a stochastic form of the interactive multi-objective methods was introduced, e.g., [

7]. At the same time, a fuzzy form was also proposed, e.g., [

8]. Many fuzzy interactive methods work with α-cut, e.g., [

9]. To complete a brief overview of the interactive procedure, interactive goal programming methods have been also designed, e.g., [

10]. Interactive multiple objective programming methods have been evolving for the last few decades; below, we provide a more detailed overview.

Selecting a suitable method is affected by a particular decision-making situation or the DM’s abilities. In our case study, a portfolio from open unit trusts offered by CONSEQ Investment Management traded on the Czech capital market is being made. To make a satisfactory investment decision, the following requirements should, in my view, be fulfilled. The algorithm should not require any additional information difficult to determine by the investor (goal, threshold, α-cut, distribution of return, explicit relaxation substitution among objective values, etc.). The importance of the objectives must be adjustable by the investor. The strictly determined relaxation of some objectives should be acceptable. The algorithm should be user-friendly for its wider and easy applicability. A method fulfilling all these assumptions can solve the investment problem satisfactorily.

An improved form of extremely popular STEM, KSU-STEM, seems to be a good candidate. Although its algorithm is also not perfect, it has fewer shortcomings compared to other methods. Therefore, this approach deserves further research. In other words, I see current aspects of the algorithm for reflection or improvement. The first partial question is about a determination of the basal (worst) possible value of the objectives. An unnecessary pessimistic approach is modified. Second, the revealed original multi-objective mathematical model is transformed to the one-objective form by introducing a fuzzy goal principle. Such a concept is an efficient alternative to the current formulation. The fuzzy approach can very effectively take into account information about the preferred values of all objective functions (portfolio characteristics) simultaneously. Finally, a recalculation of weights of the unsatisfactory objectives within the interactive procedure is evaluated as redundant. This process can be replaced by simpler ways. The approaches with preserved original weights or with an integration of the weights to the newly determined fuzzy goals are proposed.

The primary aim of this article is to determine a suitable method, more effective than commonly used concepts, as a support for a portfolio selection problem. Based on the algorithmic application abilities, the KSU-STEM approach is chosen as a perfect candidate for further interesting research deepening the ability to solve an investment problem. Therefore, the main aim is to improve its algorithm to solve the problem as satisfactorily as possible. The suitability of these revisions and improvements is tested on a real-life scenario making a portfolio of open unit trusts offered by CONSEQ Investment Management. The more general mission of this paper is to demonstrate a significant application power of (interactive) multiple objective programming methods in the portfolio selection problems in order to use them more in this area.

The structure of the article has the following form. After the introduction, the investment decision-making situation is outlined. The next section discusses the interactive multiple objective programming methods. Then, the algorithm of the STEM and KSU-STEM methods is described. Subsequently, the algorithm is revised using demonstrative investment examples. Finally, a portfolio from the open unit trusts is made by the improved KSU-STEM method. In conclusion, the article is summarized and some ideas for future research are outlined.

## 5. Selecting a Portfolio of CONSEQ Funds via Improved KSU-STEM

Finally, a revised version of KSU-STEM is applied to make a portfolio of the open unit trusts offered by CONSEQ Investment Management. As mentioned above, it is a longer-time risk-averse investment. The weights of three determined criteria (objectives) are calculated via a scoring method. The scores are assigned according to the preference of the risk-averse investment strategy specified in

Section 2. The scores and weights are in the following table (

Table 4).

For easier portfolio management, the number of open unit trusts is limited by minimum (15%) and maximum (40%) share in the portfolio. Further, the risk-averse investor limits a share of equity funds to 25% because these funds potentially generate the highest loss. All strategy aspects can be determined by a “more advanced” investor or with the assistance of an experienced investment counsel. The same applies to the application of the method itself.

The objective functions representing three portfolio characteristics are formulated in the weighted average form as follows

where

${r}_{i},{l}_{i},{c}_{i},i=1,2,\dots ,20,$ represent the return, risk, and cost of the

i-th open unit trust (indexed in order from

Table 1) and

${x}_{i},i=1,2,\dots ,20,$ represents a share of the

i-th fund in the portfolio generating the

n-component vector of variables x. The ideal (optimal) value of each objective is determined through Model (10) in the following form

where the feasible solution set

${X}^{1}$ includes the following conditions

where the binary variable

${y}_{i},i=1,2,\dots ,20,$ helps to ensure, within a set of constraints (34), a fixed possible interval for the share of each

i-th open unit trust in the portfolio. The second condition (35) represents the limit of equity fund share in the portfolio. Further, there is a standard condition for making a portfolio (36). Non-negativity and binarity conditions (37), (38) for the relevant variables must not be missed.

The basal values of the objectives are determined via the revised concept in Model (16). Then, both extremes of the objectives can be displayed in the following table (

Table 5).

Now, the initial investment portfolio is made via the multi-objective model (17)

which can be transformed to the maxmin form (20) in the particular shape

Mathematical models (33) and (40) are solved in the LINGO optimization software. The portfolio (as the optimal solution of (40)) has the following form (as already mentioned above): 40% Conseq Corporate Bond A, 25.03% Conseq Real Estate, 17.87% Conseq Invest Bond A, and 17.1% Conseq Invest Europe Equity B. The portfolio return (average monthly return) is 0.255%, the risk (measured as SRRI) is 2.513, and the cost (entry fee) is 3.553%. It is not surprising that the fund with the lowest level of risk dominates in the portfolio. The presence of a higher-risky equity fund is caused by its higher return, which is also quite an important portfolio characteristic for the investor. A weighted grade of membership (α) is 0.366. However, a real grade of membership for this solution is 0.415. The highest grade of membership is, of course, for a risk, (0.777). On the other side, the less important objective cost provides the lowest level of membership grade (0.415). This fact is understandable because higher cost does not matter much to the investor.

The risk-averse investor tries further to reduce any risk connected with the investment. The question is how to do it. At first, as the extreme objective values confirm, a risk reduction is possible. Second, it is interesting to find out whether it is possible to do a risk reduction without reducing the return. In many cases on the capital market, such an operation is not possible. However, this is not always the case, especially in a situation of not so value-flexible risk measure. When the portfolio cost is not too important a characteristic, some part of its value can be sacrificed in favor of the other criteria. So, let us try to do this change of the current portfolio. Under the condition of a not deteriorating return, the cost can be relaxed by 0.147 percentage point to reduce a portfolio risk. Then, Model (24) in the following particular form is solved

where the set

${X}^{2}$ is specified in the patterns in Models (12) and (27) as

The solution of Model (41) represents a revised portfolio consisting of the same mutual funds as the previous one. However, the shares are different. The portfolio composition is 37.32% Conseq Corporate Bond A, 32.68% Conseq Real Estate, 15% Conseq Invest Bond A, and 15% Conseq Invest Europe Equity B. The return of the portfolio is the same (0.255%), the risk is 2.45, and the cost is 3.692%. The weighted grade of the membership of this solution is slightly higher; in contrast, the real grade of membership is lower due to an increased cost. As the input data suggest, it is really possible to reduce the risk while maintaining (or increasing) return. The preference for a return increase would mean a lower risk reduction than a stable level of portfolio return. As expected, the share in the real estate fund increased due to its higher return at the expense of the shares of all other funds in the portfolio. If further cost increases would be acceptable, the process of improving the risk and return values could continue. However, the area for this change is significantly limited due to a maximum possible level of cost (4.3%). Another way to reduce risk is a reduction at the expanse of return under the same or better level of cost. This path leads through the replacement of the equity fund by a lower-risk fund with significantly lower return. In other way, while the risk is cannot be reduced much, the return can still decrease more significantly, as the extreme objective values show. Thus, such a change is not acceptable by the investor. In these circumstances, s/he decides to accept the last known fund portfolio.

#### Discussion

The introduced method accepts any investment strategy, e.g., ‘risk-seeking’ for potentially higher return. Then, the equity funds (generally with the highest risk, allowing a possible higher return) would be in a maximum feasible share in the portfolio. This approach could be applied to making a portfolio from another financial investment instruments (stocks, bonds, investment certificates, etc.). However, it can also be used in another field of practice: project management, production systems, service providing, etc. A partial evidence of the revised KSU-STEM applicability on other data can be found in [

32], which confirms the significant support of this method in selecting a portfolio of the funds offered by Česká spořitelna.

A non-traditional multi-objective approach of decision-making theory seems to be a very effective tool for a portfolio making. Compared to the one-criterion (fundamental, technical analysis, etc.), or ‘human intuition’ concepts, it provides a far more complex view to a portfolio selection. Moreover, the investor knows exactly what part of his budget is invested in what investment instrument. Unlike a well-known mean-variance, or more generally mean-risk, optimization approach, the proposed concept enables considering a wide range of quantitative, or qualitative investment criteria as well as all investor’s preferences. It is evident that an application power of the proposed concept grows with an increasing number of the criteria where the solution is very difficult to predict. This fact is eventually confirmed by the presented investment case. Finally, a significant applicability of the improved approach is also supported by an implementation of the selected concepts of fuzzy set theory.

Thus, a real-life case study selecting a portfolio of open unit trusts proves the application power of the proposed revised KSU-STEM method in the field of the capital market. The interactive procedure is very helpful to obtain a satisfactory portfolio composition. The approach is user-friendly. The interactive procedure is simplified by an easier re-setting of the weights. The revised determination of basal objective values avoids finding an unnecessarily pessimistic solution. Sharp conditions (26) or modified form (27) also contribute to more effectively finding a compromise solution. Then, the process is faster, which is also an important aspect today.