Portfolio Theory - Principles of Diversification - Case Study Example

Portfolio Theory: Principles of Diversification By Portfolio Theory: Principles of Diversification Two Important Parameters The portfolio theory originated with the use of asset-pricing idea as an investment instrument. Investment instrument is an asset that can be bought and sold. The portfolio theory defines that an investor will buy a single risky fund plus a risk-free asset (Tullet, Mathew & Associates, n.d.). The combination depends on the investor’s risk appetite. Thus, the whole concept of portfolio theory relates to the fund used to buy risky assets. Let us assume, the entire financial market consists of three stocks, those of company X, company Y, and company Z in the following manner; X’s market capitalization is $1 billion; Y’s is $2 billion, and Z’s is $3 billion (Investopedia). Thus, total financial market value is $6 billion, and it is that the market portfolio would consist of 17 % X stock; 33% Y stock, and 50% Z stock. Both individual and corporate investors use the same concept when they build up a portfolio. The above discussion demonstrates that an asset is a weight on the portfolio. An investor, private or corporate selects a combination of securities based on the risk associated with the security. Thus, portfolio theory has two important parameters: weight of an asset in the portfolio and its risk (Tajirian n.d.). The concept risk relates to the return on investment. Let us consider a particular stock A. The stock A has predicted returns for different economic states as well as the probability of occurring these states. Theoretically three states are boom, average, and recession. Using formulas, one can calculate the expected return, E (rA), and the risk of return of stock A. The risk of return is expressed through standard deviation σ; standard deviation σ is the square root of variance σ 2. A portfolio consists of multiple financial instruments, each of them with predicted returns. Let us now say, we have three securities in a portfolio: stock B, stock C, and stock D. The portfolio return will be E (r portfolio) = WB x E (rB) + WC x E (rC) + WD x E (rD). The value of E (r Portfolio) will compensate the risk of each single security. The above stated situation is in the following example. Example # 1 (Sepand Jazzi): A portfolio consists of Gold Stock, Auto Stock with relative weight 75 % and 25 %. The return is shown below. Economic Probability Auto Stock Gold Stock Average Stock State Return Return Return Recession 33.33% - 8 % 20 % 0.085 Average 33.33% 5 % 3 % 0.045 Boom 33.33% 18% -20% 0.01 Convert predicted returns of two stocks to the return of one average stock. The formula is Average predicted return = Weight of Auto stock x Predicted return + Weight of Gold stock x Predicted return. Using formula, E(r) = Σ Ri x P (Ri), where, i = 1, 2, we calculate expected return of the portfolio. In this formula, Ri is return and P (Ri) is the probability of the return. The portfolio expected return is E (r portfolio) = 4 %. Using the predicted value of return of Average stock for each economic state and probability, we can calculate the risk of the portfolio. In this case, it is 3.89 %. This value is lower than individual risk values for Auto stock and Gold stock alone. Variance model: Concept and Application A portfolio selection concept is developed on the assumption that all investors are risk averse, and they chose mean variance efficient portfolios (Washington.edu n.d.). That is why; this approach is called mean variance model. In this model, it is also assumed that the returns from investment are normally distributed. This assumption allows using the expected return and standard deviation to illustrate the distribution of returns. Furthermore, the above concept allows building up a two assets or multiple assets portfolio; it constitutes the fundamental principle of diversification. The main purpose of diversification is to alleviate unsystematic risk. It is assumed that the risk associated to single investment or type of investment is possible to offset by the risk of another investment or type of investment. Thus, the principle of diversification states that the variability of multiple assets held together is less than variability of typical stock. The principle of diversification focuses on determining a mean-variance efficient portfolio from the security market. To understand the concept of mean-variance efficient portfolio, first one needs to understand the concept of capital market line (CML). In the following paragraph, the CML concept is described on σ – E (r) plane, in other words on the “Risk – Return” plane. Figure 1. CML equation, E (r i) = r f + [{E(rM ) – rf } / σM] x σ (Fama & French 2004). The axis σ is the level of risk or standard deviation of the portfolio, and E (r) is the expected return of the portfolio. The linear equation y = a + mx is changed according to the new coordinate system and expressed as E (r i) = r f + [{E(rM) – rf } / σM] x σ ……………………………………. Equation 1. In the equation 1, the intercept rf denotes risk free return, and the slope [{E(rM – rf) / σM] denotes market price of risk. On the σ – E ( r) plane, a curve is drawn (See Figure 1), which is called minimum variance frontier (Fama & French 2004). A tangent to this curve is drawn from the intercept rf. This straight line is called capital market line (CML); it is also the mean-variance frontier with the risk free (less) assets (Fama & French 2004). The point, where CML touches the curve is called the market portfolio, and on the σ – E (r) plane it is denoted with M (The Capital Asset Pricing Model n.d.). Thus; σM is the market portfolio standard deviation, and rM its corresponding return. The curve is also called as an efficient frontier, and the straight line is called the efficient frontier with risk less assets. The curve and the straight line help an investor in building a diversified portfolio; risk frees assets are on a straight line and risky assets are inside the curve. The expected return of a portfolio is simply the weighted average of the securities in the portfolio, whereas the variance is not simply a weighted average of the standard deviation of the securities in the portfolio. For two securities (Pastor Lubos 2000) in a portfolio the return of the portfolio is expressed as E (Rp) = w1xR1 + w2xR2; the variance is expressed as σp2 = w12x σ12 + w22x σ22+2w1w2 Cov (R1,R2), where Cov (R1,R2) = covariance between returns on securities 1 and 2. The covariance between the return of any pair of security measures to what extent they move together or covary. This tendency is measured using correlation coefficient ρ; for securities 1 and 2 it is expressed as ρ1,2 = {Cov (R1,R2) / (σ1 x σ2)}. The correlation coefficient can only have values between +1 and – 1. The variance of two assets portfolio is expressed as σp2 = w12x σ12 + w22x σ22+2w1w2 x ρ1,2 x σ1 x σ2. Thus, variance of a portfolio depends on: (1) the composition of the portfolio, (2) the standard deviation of the returns for each security, (3) the correlation between the returns on the securities held in the portfolio. Based on the above definitions it is possible to build an efficient portfolio that is presented in the example 2. Example # 2. Data w1 w2 E(Rp) σp^2 σp E(R1) 8% 0 1 12.00% 0.0900 0.30 E(R2) 12% 0.2 0.8 11.20% 0.0496 0.22 σ1 20% 0.4 0.6 10.40% 0.0244 0.16 σ2 30% 0.6 0.4 9.60% 0.0144 0.12 ρ1,2 -0.5 0.8 0.2 8.80% 0.0196 0.14 1 0 8.00% 0.0400 0.20 E(R1) – Return of asset 1. σ1 – Standard deviation of asset 1 E(R2) – Return of asst 2. σ2 – Standard deviation of asset 2 w1 – Weight of asset 1 in the portfolio “p” E(Rp) – Return of the portfolio , “p” w2 – Weight of asset 2 in the portfolio “p” σp^2 - Variance of the portfolio “p” σp – Standard deviation of portfolio “p” Figure 2. Efficient frontier This example is derived with two assets in a portfolio with different weights. The figure (Figure 2) shows the behavior of return of the portfolio for a different combination of weights. The results of the analysis are shown in Figure 2. Portfolio “d” offers higher expected return and a lower risk than portfolio “e." Portfolios that offer highest expected return at a given level of risk are referred as efficient portfolio (Goetzman n.d) . From this perspective portfolios “e” and “f” are not efficient Portfolio Theory: Cost of Capital The portfolio theory is also used to evaluate the cost of capital (Bell South Telecommunications n.d). This model considers the risk of return and historical return of stock market. The formula for the evaluation of rate of return is Expected rate of return on a security = Rate of risk free investment + (Volatility of a security, relative to the asset class) x (market premium), or ri = rf + ß (rm-rf). This formula is derived from the equation 1. Example 3. Company ß = 1.4 Rf = 5 %, risk free return E [rM] = 13 %, historical stock market return. E[r] = 5 % + 1.4 (13 % - 4 %) = 16.2 % = Cost of capital Reference List Bell South Telecommunications n.d., Capital Asset Pricing Model Analysis of the Cost of Equity Capital, Alabama.gov., viewed on 19 April 2014, http://www.psc.alabama.gov/29054/BellSouth_DirectII/Billingsley/040120_29054_Exhibit_RSB-4.pdf Fama, E.f., & French, K.R. 2004, The Capital Asset Pricing Model: Theory and Evidence, Journal of Economic Prospective, volume 18, number 3, pages 25 – 46, umich.edu., viewed on 19 April, 2014, www-personal.umich.edu/~kathrynd/JEP.FamaandFrench.pdf Goetzman, W.N., n.d., An Introduction to Investment Theory, Yale School of Management, viewed on 20 April 2014, http://viking.som.yale.edu/will/finman540/classnotes/class2.html Investopedia. Market Portfolio, viewed on 19 April, 2014, www.investopedia.com/terms/m/market-portfolio.asp Pastor Lubos 2000, Portfolio Selection and Asset Pricing Model, The Journal of Finance, Vol. Lv, No 1, pages 179 – 221, chicagobooth.edu, viewed on 19 April, 2014 http://faculty.chicagobooth.edu/john.cochrane/teaching/empirical_asset_pricing/pastor_portfolio_JF.pdf Sepand Jazzi. (n.d.). Stock and Portfolio Variance and Standard Deviation [Video]. Retrieved from http://www.youtube.com/watch?v=q69sfKgsxEc Tajirian, A., n.d., Risk and Return: Portfolio Approach, morevalue.com, viewed on 20 April 2014, http://www.morevalue.com/i-reader/ftp/Ch12.PDF The Capital Asset Pricing Model n.d., nyu.edu., viewed on 19 April 2014, http://pages.stern.nyu.edu/~ashapiro/courses/B01.231103/FFL09.pdf Tullet, Mathew & Associates, n.d. History of Portfolio Theory, tma-invest.com, viewed on 20 April 2014, http://www.tma-invest.com/files/investment_research/History_of_Portfolio_Theory_and_Investment_Management_Breakthroughs.pdf Washington.edu n.d., Markowitz Mean-Variance Portfolio Theory, whashington.edu. viewed on 19 April 2014, http://www.math.washington.edu/~burke/crs/408/fin-proj/mark1.pdf Read More