Determining a Correlation Between Financial Risk and Expected Return in Economics by Using Portfolio Optimization

Investing at the stock market is often considered as a way of gambling. That is because most people can’t manage the see the mathematics behind it. Someone with enough mathematical knowledge can see the algorithms in movements of stocks and find the correct function for it, basically optimize their portfolio. With the correct function one can foresee fate of their investments or when to invest on what. The aim of portfolio optimization is to find the set of efficient feasible portfolios. A portfolio is feasible if it satisfies a relevant set of relevant linear constraints; it is efficient if it provides less risk than any other feasible portfolio with the same expected return and more expected return than any other feasible portfolio with same risk. Thus, the research question of this Mathematics Extended Essay, “Is it possible to determine a correlation between financial risk and expected return in economics by using portfolio optimization?”arises. In order to answer the question first a scenario had to be selected. The scenario chosen for this extended essay was the Turkish Stock Market (more commonly known as its Turkish abbreviation IMKB) from January 2009 to September 2011. After that the statistical data needed for domains of functions is gathered from the Central Bank database. Then the objective function is created. To create the objective function, first the risk function had to be created. Portfolio risk is stated in terms of absolute deviation of rate of return. Risk function is a linear combination of the two semi-absolute deviations of return from the mean. (Spenza, 1993) The objective function is the function that minimizes the risk function depending on the nature of the investor, i.e. for a risk seeking investor higher interest rate investments would be chosen. To select the optimal portfolio, instead of Markowitz Model, a linear programming model (a model where functions are created as linear equations) is used because of computational difficulties caused by quadratic nature of the Markowitz Model. To solve the model LINDO optimization modeling software is used since the model contained 32 linear equations with 5 variables. At the end 12 different portfolios with different interest rates are produced for different types of investors