# Portfolio-optimization by the mean-variance-approach

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1 MaMaEuSch Management Mathematics for European Schools Portfolio-optimization by the mean-variance-approach Elke Korn Ralf Korn MaMaEuSch has been carried out with the partial support of the European Community in the framework of the Sokrates programme. The content does not necessarily reflect the position of the European Community, nor does it involve any responsibility on the part of the European Community. University of Kaiserslautern, Department of Mathematics

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3 0 CHAPTER 4 : Portfolio-optimization by the meanvariance-approach Overview Keywords - Economy: - Stocks - Risk-free bonds - Portfolio - Interest and rate of return / yield - Mean-variance-principle - Diversification Keywords Elementary mathematics: - Calculus of probabilities: Expected value and empirical variance - Finding optimal solutions (see Chapter ) - Calculation of interest - Power(s) and real power - The Eularian number e - Inequalities - Arithmetic average - Vectors and matrices Content - 4. Asset management service and portfolio-optimization - 4. Discussion: The portfolio-problem of the Windig Company Background: Stock terms and definitions, basic principles and history Basic principles of mathematics: Calculation of interest Continuation of discussion: Assessment of stock prices Basic principles of mathematics: Chance, expected value and empirical variance Continuation of discussion: Balance between risk and profit Basic principles of mathematics: The mean-variance-approach Continuation of discussion: Less risk, please! Optimization from a new standpoint Summary - 4. Portfolio-optimization: Critique on mean-variance-approach and current research aspects - 4. Further exercises Chapter 4 guidelines The main goal of this chapter is to introduce the problem of the optimal investment in securities within the mean-variance-approach according to H. Markowitz. Most of all, in case of an investment problem with two or three securities we should develop a graphical solution method (similar

4 0 to that in Chapter ) which can be employed during class instruction. At the same time one can in a natural way introduce calculus of probabilities as a mathematical model for chance and uncertainty of the future events. Some extensive preparation is required in order to cover all the material, depending on the students' state of knowledge. It is summarized in each section of this chapter. Thus in Section 4. we will by means of an example introduce the problem of optimal investment of capital, namely the portfolio-optimization. In sections 4./5/7/9 we will, while analyzing a portfolio-problem of a fictitious company, work on the main determinants of the portfolio-optimization, namely profit (modelled by means of expected value of the rate of return on investment) and risk (modelled by means of the empirical variance of the rate of return on investment). Moreover we will introduce the graphical solution method for the portfolio-problem in the case of two to three securities. In order to be able to handle these sections, economic terms such as rate of return, portfolio and stocks (see Section 4.3), as well as mathematical knowledge of calculation of interest (see Section 4.4), as well as calculus of probabilities (see Section 4.6), will be needed. Depending on the previous knowledge of the students, chapters 4.4 and 4.6 can be skipped. However Section 4.6 can be used as a short introduction to the calculus of probabilities, which is here specific to the modern application of "financial mathematics". In chapters 6 and 7 this introduction will be expanded, in fact in any case in which the applications of financial mathematics utilize the appropriate auxiliary means of the calculus of probabilities. Such an introduction is also appropriate because in general opinion only few things are as strongly associated with the terms uncertainty and coincidence as stock prices. Section 4.8 presents the theoretical principles of the mean-variance-approach according to Markowitz, firstly in case of an investment in two or three securities, and then in its generality, for which H. Markowitz was awarded a 990 Nobel Prize for contributions to economic science. The first two parts of the section can be, independently of the other sections in this chapter, discussed with students who possess the basic knowledge of probability theory. The last part of this Chapter can be most certainly worked out only with very advanced students, and serves as background information. A very important aspect of Section 4.8 is the diversification principle, which supports the philosophy of the investment in different goods. An outlook on the current mathematical methods of the portfolio-optimization will be provided in Section 4.. The introduction to the common probability distribution of random variables, as well as the terms covariance and correlation, is not always possible due to the short time frame of the class. One can therefore introduce a skimmed version of the mean-variance-approach in which it is possible to make a risk-free investment (cash, savings account) or go for a risky alternative (e.g. stocks, stock funds). In Section 4.6 one can discard the common distribution, covariance and correlation. However, one cannot then effectively present the diversification effect in Section 4.8. Due to the presence of stock market in the media, in certain parts of this and the following chapter we have included open exercises which involve TV, Internet and appropriate magazines and journals. 4. Asset management service and portfolio-optimization Good investment a fictitious experience report Design student Katerina Schmalenberger* (*imaginary) won million in the quiz show "Who wants to be a millionaire?" by means of careful planning and a bit of luck. Ms Schmalenberger does not want to hide the money under her mattress because first of all that is no safe place for it and secondly in that way it will not bring her any interest. Likewise she does not want to put the money on her savings account because there she will presently get interest of merely % per year, which is simply too little.

6 04 and how the investment will be made costs a yearly percentage charge fee, or there is another option, namely to pay the asset-based fees on the purchase price of the fund in the very beginning. Even if one builds pension funds by means of a retirement savings plan, one hands the others the work (or task) of dividing the money. Within the retirement savings plan the terms "long term investment" and "security" become very important. As a rule a type of insurance (e.g. in the event of death) is included. This means then that the pension insurer splits the money primarily across fixed-interest securities or funds or perhaps in a convenient insurance, and therefore invests less in stocks. Within an equity fund the associated capital investment company divides often very big capital, comprised of that of all buyers, from particular standpoints, into different stocks. In this way some investment companies buy only European stocks and moreover only those which have the biggest growing potential; or only Asian stocks, and in fact only those which currently bring most security. After this pre-selection there are then often only approximately 30 to 50 stocks left over which the capital will be divided. Based on business data and conditions for, e.g. security, the desired rate of return or the structuring of the fund, the portfolio, namely the investment, will be optimally divided by the managers. Seen algebraically, at the bottom line, the result is a highly multidimensional optimization task with (mostly non-linear) side conditions (see Chapter ), e.g. in the form Subject to maximize profit + profit + +profit 50 invested capital with weak fluctuation stock share 50% Determining optimal investment strategies: Portfolio-optimization The distribution of a capital over different investment options is called a portfolio. Determining the optimal investment strategy employed by the investor, meaning the decision on how many shares of which security they should hold in order to maximize their profit from the final capital X(T) in the planning horizon T, is in financial mathematics called portfolio-optimization. Here we have to pay attention that this optimization problem exhibits, aside from its ordinary quantitative and selective criteria ( How many shares of which security? ), a temporal dimension as well ( when? ). With this in mind one has to continuously make decisions. This is why we are dealing in general with a so-called dynamic optimization problem. In this chapter, however, we will observe only models and problems in which one can do without the temporal element of the decision problem. This is due to the fact that it will deal with a oneperiod-approach in which it will be decided on the distribution of the capital into different bonds only at the beginning of the investment period. This decision will not be revised anymore before the end of the investment period. The further handling of the portfolio-problem (in other words the problem of locating an optimal investment strategy) in its generality requires complicated algebraic methods such as the stochastic control theory. We will therefore here basically concentrate on presenting the solution to the simpler portfolio problems and only more closely study the oneperiod-model according to Markowitz. 4. Discussion: The portfolio-problem of the Windig Company The Windig Company* (*imaginary), located on the North Sea, has been manufacturing wind energy plants in middle watt range for ten years. Their hurricane-proof wind turbines have in the meantime had a big break with farms in isolated locations. The streak of successes of this mediocre company is not to be neglected and good labor in this field is very much in demand. Therefore

9 07 business, of the high dividend payment and strong stock price gain. Summed up these lie high above the proceeds of a risk-free investment such as fixed term deposit. As a matter of fact the profit from stock investment, which is as a rule long term (considered over 0 to 0 years), is despite of slumps which occur now and then still higher than that of the risk-free deposits (see Chart 4., however also Chart 6.). DAX (blue) 8000, , , , , ,00 000,00 000,00 0, Chart 4. Comparison of the development of the DAX-Index with a 7% return However generally one should regardless of how good the personal estimate of the selected stock(s) is always get it straight that a stock cannot promise certain profit and that it is still absolutely possible for one to lose part of the invested money. Therefore before each stock investment one should carefully check if one wants to risk it. Several important data from the history of the stock: - 60: Establishment of the first corporation in the world, the Vereinigte Ostindische Kompanie, in the Netherlands, for financing an overseas trade company - End of 7. century: Increase in stock dealing, particularly in England and France - 756: Dealing the first German stock, the Preußische Kolonialgesellschaft, in Berlin - 79: Founding of (the forerunner) of the New Yorker stock market, since 863 New York Stock Exchange - 844: In England it is legally possible to found corporations in all branches of acquisition - 884: The first American stock index, the Railroad Average, which contains the stocks of American railroad companies, is quoted - 897: The Dow-Jones-Stock Index is determined each trading day - 99: Black Friday ( stock market crash ) on at the New York stock market - 959: Issue of the first German people' share (PREUSSAG) - 96: Issue of the second German people's share (VW) - 987: Black Monday on , surprisingly rapid collapse on the international stock markets - 988: The German stock index (DAX) is introduced - 996: The people's share Deutsche Telekom Inc. goes public - 997: The fully electronic trading system Xetra is introduced by the Deutschen Börsen Inc : The DAX reaches the historical all-time high of 8064 points on Fixed-interest bonds As opposed to stocks, the value development of a fixed-interest bond is already determined in advance. As a rule one sets up a determined sum of money for a specific amount of time and then obtains interest which was stipulated in advance at regular intervals. At the end of the period

10 08 of validity of the bond one gets the invested capital back. The examples of fixed-interest bonds include federal savings bonds, long-term bonds, corporate bonds, state bonds or Pfandbrief. Moreover fixed deposits of a house bank or an account book can be in a sense considered a bond. One often connects the term fixed-interest bond with the term risk-free bond. This is only true insofar as one considers that the interest payments and the return value are determined in advance. Yet the risk-free bonds" are not entirely risk-free, particularly not if we are talking about business bonds of a ramshackle company or the government bond of an instabile and overdebted country. Under such circumstances the interest payments, and sometimes even the return on capital can be omitted. Thanks to the strict rules and thorough control by the supervision of banking, most of the fixed-interest bonds belonging to the bank are in fact almost risk-free. Discussion 3: - Which great corporations are there? - By means of magazines and Internet try to find out how many stocks were issued by certain corporations! Calculate, based on the current stock prices, how much money one has to invest in order to buy up all the stocks! Discuss the meaning of this value! - Inform yourself on the Internet, in magazines and manuals on the terms stock, bond and obligation. 4.4 Basic principles of mathematics : Calculation of interest Interest and compounded interest The bondholder of a fixed-interest bond regularly obtains agreed-upon interest which corresponds to the respective capital. If the original assets K 0 are invested as fixed deposit for a year with annual return of r %, at the end of the year we get the following closing capital: r r K ( r;) = K0 + K0 = K If this capital is fixed over several years, then we have to consider that normally the interest is credited on the account annually, so that there is equally interest on that interest in the aftermath. The interest on the interest is called compounded interest. Annual return: A capital of K 0, which is invested as fixed deposit with annual return of r % throughout n years finally results in the closing capital of: n r K ( r; n) = K Entirely in contrast to this is the concept of short-term financial investment. There is also the possibility of investing the fixed deposit for a month or even a few days. In this case one has to pay attention that the interest rate is valid for a full year and accordingly, in case of a short-term investment, only a part of it will be paid out. Return in case of a short-term financial investment ( Return of less than a year ): Opening capital of K 0, invested for m months results, in case of an annual return of r%, in closing capital of:

11 09 r m K ( r; m,) = K Opening capital of K 0, invested for t 360 (interest-)days, results, in case of an annual return of r%, in closing capital of: r t K ( r; t,) = K Incidentally in banking business one month is mostly converted into 30 days and one bank year consists of 360 days (in fact: interest days). Aside from the investment or saving interest which one quasi obtains as a reward for relinquishing the capital to the bank, there is one more contrarian perspective. After all at a bank one can invest money as well as lend it. In the latter case one has to pay the so-called loan interest to the bank. The situation becomes unfavourable for the borrower if they neither pay back the loan nor pay interest for the duration of the loan. This interest is added to the loan and demands, in case of a loan that is stretched over several years, the loan interest itself. The compounded interest effect increases the debt rapidly. The result are the same formulas as in the case of the fixed deposit because a loan is indeed the same as a fixed deposit, excepting the fact that one is on the other side of the transaction. ( Ex.4., Ex.4.) Continuous compounding In a fixed deposit over e.g. three months after the expiry one gets not only their capital back, but obtains the interest as well. If the interest rate has not changed in the meantime, this whole sum, in other words the capital plus interest, can be newly invested at the same interest rate over the same period of time. In this way the compounded interest effect is noticeable already after a short period of time. Return in case of a repeated short-term financial investment: Opening capital of K 0, j-times repeated, including the interest invested over respectively m months results in, in case of an annual return of r%, in closing capital of: j r m K ( r; m, j) = K Opening capital of K 0, j-times repeated, including interest invested over respectively t 360 (interest-) days results, in case of an annual return of r%, in closing capital of: j r t K ( r; t, j) = K ( Ex.4.3) Let us take a closer look at an extreme case: A daily allowance is invested 360 times consecutively at the same interest rate. In case of the capital of 5000, which is fixed as daily allowance at the annual interest rate of 4 % and is newly invested every day under the same conditions, we get 504,04 (rounded off) after a year: K ( 4;360,360) = = 504,

12 0 This final amount clearly exceeds the amount of 500, which is obtained if the money is fixed for a year at the same annual interest rate of 4 %. The interest which is paid out daily and added to the capital lead by means of the compounded interest to the higher final capital. One could now divide the time even more closely and introduce an hourly fixed deposit, minute or even second fixed deposit. We observe that the final capital moves towards a boundary level (i.e. ultimately hardly anything changes). One then says that the capital continuously pays interest. The capital of 5000 that continuously pays interest at an interest rate of 4 % results after a year in the closing capital of 504,05 (rounded off): K ( ) ; = 5000 e = 504, 05387, s whereby e is the Eularian number. This type of return is entirely common in banking circles and the interest rate is called continuous interest rate. The above formula is true for optional periods of time t [0, ], where t is measured in years (e.g. / years are t =,5). Continuous return: Opening capital of K 0, with a continuous interest rate of r%, results after period of time t in closing capital of: rs t K r ; t = K e 00. ( s ) 0 one year half a year continous Chart 4. Comparison between one-year, half-a-year and continuous return (r=0,) When borrowing one has to particularly pay attention to whether the interest is paid continuously or whether the interest is added to the loan quarterly or yearly. For this reason within the loan procedure as a rule what is always additionally specified is the effective interest rate. The effective interest rate is the interest rate which would during the yearly calculation of interest lead to the same closing payment as the offered interest calculation type (always based on the whole year). If for example the interest at the given annual interest of r% is added to the capital quarterly, one gets the effective annual interest r eff % by means of the equation one therefore compares ( ;3,4) ( ;) K r m = K r eff,

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