Theoretical Aspects Concerning the Use of the Markowitz Model in the Management of Financial Instruments Portfolios Lecturer Mădălina - Gabriela ANGHEL, PhD Student madalinagabriela_anghel@yahoo.com Artifex University of Bucharest Abstract Early attempts to develop a modern model for the assessment of performance of portfolios of instruments belong to American teacher Harry Markowitz. He has abandoned the classical approach of the analysis of financial investment (based solely on technical and fundamental analysis), pointing attention to performance analysis to the overview of a portfolio of financial instruments (analysis based on the report yield/risk of components in a portfolio). Key words: Markowitz model, profitability, risk, root-mean deviation, correlation coefficient JEL Classification: G11 In the case of modern portfolio theory, the investments are modeled statistically, taking into account the level of the profit expected and the degree of volatility of the financial instrument, the latter being considered as a carrier of risk specific to each instrument 1. The purpose of this theory is that each investor to identify the level of risk accepted and then to identify the portfolio with the highest yield for this level. Under conditions of certainty, Harry Markovitz has shown that the choice of the portofolio can be reduced to analysis of two sizes: "the rate of expected gain of the portfolio and dispersion or root-mean deviation, as a measure of the risk" 2. We can also affirm that the risk of a diversified portfolio dependents not only on the individual variations of titles rentability but also on adverse movements of all the assets. The most important discovery in Markowitz's work - "Portofolio selection. Efficient Diversification of Investments" - 1959 (work for which he was rewarded with Nobel Prize for economy in 1990), considered to be the basis of modern portfolio theory - is that according to which an investor can reduce ithe volatility of his portfolio (i.e., its risk) and may (at the same time) to grow its profitability. The Markowitz model is based on a number of assumptions which may be summarized as: 1 Anghelache, G.V.; Anghel, M.G. The using of the Markowitz model to identify the optimal portfolio, Romanian Statistical Review Reverential Session. 150 years of official statistics, no 6, 2009 2 Markowitz, H. Portofolio selection. Efficient Diversification of Investiments", Journal of Finance 7 Revista Română de Statistică Supliment Trim IV/2012 259
Investors consider each alternative of investment as being represented by the distribution of the hoped profit likelihood in a period of time; Investors maximize the utility anticipated within a period of time, and usefulness curve maximizes the marginal utility of their welfare. Investors estimate the risk on the basis of the change in profits expected; Investors make decisions only on the basis of the risk and hoped profit, so the usefulness curve is expressed as a function of the profit expected and a variance of the profit; For a given level of risk, investors prefer a huge profit; for a given level of expected profit, investors prefer to risk less. The practical use of Markowitz model makes it possible to determine the level of the individual dispersion of financial instruments profitability, both for a portfolio of simplified instruments(made of two financial securities ), and for a portfolio consisting of "n" financial instruments. Even if they are two or more securities in different markets, the construction of a portfolio involves browsing the following steps: Identification of the risk - revenue profile for each alternative of the combination of the securities in the portfolio; Determination of the combination of risky securities with the minimum variance depending on the degree of aversion of each investor; Determination of complete portfolio by combining the portfolio with its minimum variance with securities without risk that investor intends to introduce in his portfolio. Profitability and risk of a portfolio made up of two financial securities The simplest model of a portfolio that can be analyzed using the model developed by Markowitz is the one made up of two financial instruments. In this respect, we consider that an equity investor can choose to invest his savings in one of the two available financial securities - T 1 and T 2 or equally can build up a P portfolio distributing in this sense the amount he wishes to invest between the two previously mentioned securities 3. Mathematically, the investor's anticipation about the behavior of the two securities in the future period can be summarized as follows 4 : where: E i the mathematical hope of "i"security rate profitability; σ i the standard deviation of "i"security rate profitability; ρ ij the coefficient of correlation between 'i" and"j" securities rates profitability; Cov ij covariance between "i" and "j" securities rates profitability. A capital investor has the opportunity to form a portfolio combining the two securities in proportion of X 1 and X 2. In this case, the available total amount is invested in T 1 (the amount of the purchase of the first type of financial instrument) and T 2 (the amount of the purchase of the second type of financial instrument). In this case we can establish the following calculation relationship: X 1 + X 2 = 1 cu X 1, X 2 0 sau 0 X 1 1 şi 0 X 2 1 3 Anghelache G.V.; Anghelache, C. (2009) Risk and profitability basis of the financial analysis, Metalurgia International, vol. XIV, special issue no.12; 4 Roman, M. Financial and banking Statistics, ASE Publishing House, Bucharest, 2003 260 Revista Română de Statistică Supliment Trim IV/2012
Under the conditions mentioned above, you can determine the mathematical hope of portfolio rate yield P (E p ), using for this purpose the relationship: E p = X 1 E 1 + X 2 E 2 As it can be seen from the above relationship, hope return is the weighted average of the yields of securities, the average being the proportions. The second element which should be studied in order to characterize the efficiency of the portfolio considered is represented by the scattering of "P" (V p ) portofolio rate yield, which is actually a measure of the risk related to the investment portfolio. For this purpose we will use the following mathematical relationship: From the formulas above, we can deduce that the dispersion of the portfolio is significantly influenced by the following elements: the dispersion of each title included in the portfolio; the proportions in which are combined the two financial securities; the covariance between the two titles considered. To complete the analysis carried out on the basis of the above-mentioned relations, in the literature, it is advisable to study the existing correlation between the two securities included in the portfolio 5. Thus, we can see that, depending on the value of the correlation coefficient between the two securities - T 1 and T 2 can be identified three distinct cases, which can be summarized as follows: The value of the correlation coefficient is 1 (ρ 12 =1) In this case, it can be affirmed that the financial instruments T 1 and T 2 are perfectly and positively correlated, what signifies the anticipation for the return of these titles of some movements perfectly consistent over time, but with different amplitudes. In this situation, it is considered that the risk for the portfolio is in the highest degree, because the factors that influence the evolution of the two titles are similar and with an action of equal intensities. Also, it is noted that, in this case, changing the share of securities in the structure of portfolio does not bring significant improvements to the level of risk associated with it 6. For this value of correlation, relations on the basis of which an assessment of two financial securities portofolio can be made can be transcribed as: write: with =1 which means: In this case, it is noted that the standard deviation of the portfolio is equal to the average of the standard deviations of financial securities that compose it. 5 Anghel, M.G. (2009) Models of Estimation for the Profitability and the Risk of a Financial Security, The Romanian Statistics Review Supplement March; 6 Dragotă, V. Securities portfolio Management Second Edition', Economica Publishing House, Bucharest, 2009 Revista Română de Statistică Supliment Trim IV/2012 261
Bringing together the two equations and reporting to yield and to risk of the "P" portfolio, and we obtain the equation: as space of combining the securities in plane E-σ. It is known that: X 1 + X 2 = 1, respectively: X 2 = 1 X 1 In these conditions, the equation by which one can determine the mathematical hope of P(Ep) portfolio yield rate becomes: In this case, the yield of T1 security within the portfolio P can be determined using the formula: It also finds that, where mathematical hopes of return rates of the two securities are not equal (E 1 E 2 ) then the value of the standard deviation in the yield of portfolio can be calculated as follows 7 : The value of the correlation coefficient is -1 (ρ 12 = -1) Where the value of the correlation coefficient ρ 12 = -1, then T 1 and T 2 titles are perfectly and negatively correlated. In such a situation these anticipations relating relating to the yield of titles present perfect opposite fluctuations. It should be noted that, where the two titles are related strictly negative can be reached in a certain combination, the total elimination of the risk for the portfolio of securities. Also, in this situation, the relations of calculation of the standard deviation may be transformed as follows: write: which means: The standard deviation is always positive, so it makes the discussion for the sign of the expression that varies depending on the X 1 and X 2. For: This relationship, along with relationship E p = X 1 E 1 + X 2 E 2, allows the determination of the equation of connection between E p and σ p. We achieve: 7 Badea, L. Study on the applicability of the Markowitz model on the stock market in Romania, Theoretical and Applied Economics Review, no 6, 2006 262 Revista Română de Statistică Supliment Trim IV/2012
It is a linear relationship represented by a straight line. Part of this right, corresponding to: For: It s: and Doing similarly where the degree of correlation is equal to one, we get the linear equation l linking Ep and σ p. A part of this straight line corresponding to is the rule obtained combining portfolios T1 and T2. Finally, for we have σ p = 0. This result must be mentioned clearly, because it shows that from two risky securities it is possible choosing the rigorous proportions (0 X 1 and X 2 1), to build an unrisky portfolio. This result is possible if the securities T1 and T2 are perfectly and negatively correlated. The different value of coefficient correlation ± 1 (ρ 12 ± 1) If -1 < ρ 12 < +1 (including ρ 12 = 0) anticipated fluctuations for T 1 and T 2 are not absolutely dependent (positive and negative). It is the general case, there is a degree of correlation between the securities yield rates because they all follow more or less the general fluctuations of the economy. A low correlation coefficient may lead to a significant improvement of the risk value related to the investment portfolio. Also, a null value of this coefficient shall be deemed to be a potential source of decrease with 50% of risk of the present portofolio 8. In the general case (the correlation coefficient other than 1 and-1) for a portfolio of two securities shall be obtained the following expression of the risk: which means: As it can be seen, in this case, unlike previous situations, the expression of risk cannot be reduced to the form of a perfect square, making it more difficult the practical 8 Dragotă, V. Securities portfolio Management Second Edition', Economica Publishing House, Bucharest, 2009 Revista Română de Statistică Supliment Trim IV/2012 263
determnation of its the value. From this equation and from that of Ep (E p = X 1 E 1 + X 2 E 2 ), we establish the relationship linking E p and σ p. From the equation of E p we achieve X 1 = (E p -E 2 /(E 1 -E 2 ), value that we introduce into the equation of V p. Developing we achieve: The equation obtained in E-V plan is that of a parabole. In the E-σ plane the equation represents a hyperbola from which we keep a section, namely that corresponding to σ p positive values. An interesting aspect in the analysis of any portfolio of financial instruments is the assessment of the contribution of each security to the risk and the general efficiency of the portfolio from which it is part. From the formula that defines the security risk in a portfolio, can be formulated the following conclusions: the choice of a title for its inclusion in a portfolio will not be made depending on its individual characteristics (σ 1 ), but according to behaviour within the portfolio (cov 1p ). the risk of a security is not unique, it depends on the portfolio iin which it s included. References [1]. Anghelache, G.V.; Anghel, M.G. (2009) The using of the Markowitz model to identify the optimal portfolio, Romanian Statistical Review Reverential Session. 150 years of official statistics, no 6; [2]. Anghelache G.V.; Anghelache, C. (2009) Risk and profitability basis of the financial analysis, Metalurgia International, vol. XIV, special issue no.12; [3]. Anghel, M.G. (2009) Models of Estimation for the Profitability and the Risk of a Financial Security, The Romanian Statistics Review Supplement March; [4]. Badea, Leonardo (2006) Study on the applicability of the Markowitz model on the stock market in Romania, Theoretical and Applied Economics Review, no 6; [5]. Dragotă, V. (2009) Securities portfolio Management Second Edition', Economica Publishing House, Bucharest [6]. Markowitz, H. Portofolio selection. Efficient Diversification of Investiments", Journal of Finance 7; [7]. Roman, M. (2003) Financial and banking Statistics, ASE Publishing House, Bucharest. 264 Revista Română de Statistică Supliment Trim IV/2012