Securities of Slovene companies are listed at the Ljubljana Stock Exchange. Market capitalisation at the Ljubljana Stock Exchange has been growing since 1996 due to new listings of equities. On the basis of financial data time series for listed equities, the financial investor can calculate a risk for each individual security with a selected risk measure and can determine an optimal portfolio, subject to selected constraints. In this paper, we shall consequently determine an optimal portfolio of equities for the financial investor, investing his assets only in selected equities listed at the Ljubljana Stock Exchange. Selecting an appropriate risk measure is especially important for a commercial bank in a risk management process. Commercial banks can use internal models in the risk management process and for the purpose of capital charges as well. An optimal portfolio will be calculated, using a non-linear mathematical model.
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The goal of this paper is to determine an optimal portfolio of equities for the financial investor, investing his assets only in selected equities, which are listed at the Ljubljana Stock Exchange. We are going to use standard deviation as a risk measure in the mathematical model.
In this paper, we will understand a commercial bank as a financial investor. Commercial banks have two possibilities in order to calculate capital charges for the market risk they are being exposed to. The first approach is the standardised approach, which has to be used by banks in case they do not have an internal model. If a bank uses an internal model for risk management purposes, it can use several risk measures in order to measure risk. Each risk measure has its strengths and its weaknesses. Consequently, the volume of risk calculated using a specific risk measure will vary among risk measures. Since the process of risk management in a commercial bank is based on a calculated value of risk measure, it is very important for a commercial bank to fully understand the interpretation of a selected risk measure. If the volume of risk varies using different risk measures, the decisions upon changes in the positions in a portfolio will be different.
Harry Markowitz (1952) introduced standard deviation as a risk measure. His work is the foundation of the portfolio theory. This is why we are going to calculate an optimal portfolio using standard deviation as a risk measure.
1. THEORETICAL BACKGROUND
By purchasing a certain security, the financial investor takes the risk that the actual rate of return on his investment will differ from the expected rate of return. The higher the risk of an individual security, the higher the required rate of return. Suppose the risk of an individual security is measured by the variance of its rate of return. If we mark the expected rate of return of a security by E(r), and its actual rate of return by r~, then the variance of the rate of return on this security VAR(r~) equals (Mramor, 1991, p. 45):
When the financial investor invests all his assets in one risky security, then a higher rate of return variance for this security also represents a higher level of risk for his entire assets. Assets with several different risky securities will, therefore, be less risky for an investor than assets containing only one risky security.
Let us now mark the rate of return on a risky security by r^sub i^ and the rate of return on the financial investor's entire assets by r. When the financial investor invests all his assets in n risky securities, then the expected rate of return on his assets equals the weighted arithmetic mean of the expected rates of return on all risky securities in the portfolio. If a^sub i^ is a share of all the assets invested in a risky security, the following applies:
under the condition that:
If the risk of financial assets is measured by the variance of its rate of return, the following applies (Mramor, 1991, p. …