Portfolio Management

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Portfolio Management

  1. 1. PORTFOLIO MANAGEMENTWhat is portfolio management? An investor considering investment in securities is faced with the problem of choosing from among alarge number of securities. His choice depends upon the risk-return characteristics of individual securities.He would to choose the most desirable securities. Again he faced the problem of deciding which securitiesto hold and how much to invest in each. The investor faces an infinite number of possible portfolios orgroup of securities. The risk and return characteristics of portfolios differ from those of individual securitiescombining to form of portfolio. The investor tries to choose the optimal portfolio taking into consideration therisk-return characteristics of all possible portfolios.Phases of Portfolio Management Portfolio management is a process encompassing many activities aimed at optimizing the investmentof one’s funds. Five phases can be identified as this process:- 1. Security analysis 2. Portfolio Analysis 3. Portfolio Selection 4. Portfolio revision 5. Portfolio Evaluation
  2. 2. RISKMeaning of Risk A person making an investment expects to get some return from the investment in the future. But, asfuture is uncertain, so is the future expected return. It is this uncertainty associated with the return from aninvestment that introduces risk into an investment. Risk can be defined in terms of variability returns. “Risk is the potential for variability in returns.” Aninvestment whose returns are irly stable is considered to be high risk investment. The total variability in returns of a society represents the total risk of the security. Systematic risk andunsystematic risk are two components of total risk. Thus, Total risk = Systematic risk + Unsystematic risk.Systematic risk As the society is dynamic, changes occur in the economic, political and social systems constantly.These changes have an influence on the performance of companies and thereby on their stock prices. Butthese changes affect all companies and all securities in varying degrees. Thus the impact of economic, political and social changes is system- wide and that portion of totalvariability in security return caused by such system-wide factors is referred to as systematic risk. Systematic risk is further subdivided into interest rate risk, market risk, and purchasing power risk.Unsystematic risk When variability of returns occurs because of firms- specific factors, it is known as unsystematicrisk.
  3. 3. The unsystematic or unique risk affecting specific securities arises from two sources: (a) theoperating environment of the company, and (b) the financing pattern adopted by the company. These twotypes of unsystematic risk are referred to as business risk and financial risk respectively.Measurement of risk The variance and standard deviation measures the extent of variability of possible returns fromexpected return. Several other measures such as range, semi-variance and mean absolute deviation havebeen used to measure risk, but standard deviation has been the most popularly accepted measure. The systematic risk of a security is measured by a statistical measure called beta. The input data ofreturns for the calculation of beta are the historical data of returns of the individual security as well as thereturns of a representative stock market index two statistical methods may be used for the calculation ofbeta, namely the correlation method or the regression method. Using the correlation method, beta can be calculated from the historical data of returns by thefollowing formula: β=rim σi σ m/ σ 2 mWhere, Rim = Correlation coefficient between the returns of stock i and the returns of the market index σi = standard deviation of returns of stock i σm = standard deviation of returns of the market index σ2m = variance of the market returns
  4. 4. MEASUREMENT OF PORTFOLIO ANALYSISExpected return of a portfolio. As a first step in portfolio analysis, an investor needs to specify the list of securities eligible forselection or inclusion in the portfolioRisk of portfolio The variance (or risk) of a portfolio is not simply a weighted average of the variance of the individualsecurities in the portfolio. The relationship between each security in the portfolio with every other securityas measured by the covariance of return has also to be considered. The variance of a portfolio with onlytwo securities in it may be calculated with the following formula: σ2p = x21σ21 + x22σ22 + 2x1x2(r12σ1σ2)Reduction of portfolio risk through diversification The process of combining securities in a portfolio is known as diversification. The aim ofdiversification is to reduce total risk without sacrificing portfolio return.Security returns perfectly positively correlated. When security returns are perfectly positively correlated the correlation coefficient between the twosecurities will be +1. The return of the two securities then move up or down together. The portfolio varianceis calculated using the formula:
  5. 5. σ2p = x21σ21 + x22σ22 + 2x1x2r12σ1σ2Since r12=1, this may be written as σ2p = x21σ21 + x22σ22 - 2x1x2(r12σ1σ2)The right hand side of the equation has the same form as the expansion of the identity (a + b)2, namelya2+2ab+b2. Hence, it may be reduced as σ2p = (x1σ1 – x2σ2)2The standard deviation then becomes σp = x1σ1 + x2σ2This is simply the weighted average of the standard deviations of the individual securities.Security returns perfectly negatively correlated. When security returns are perfectly negatively correlated, the correlation coefficient between thembecomes -1. The two returns always move in exactly opposite directions. The portfolio variance may be calculated as: σ2p = x21σ21 + x22σ22 + 2x1x2r12σ1σ2Since r12=1, this may be rewritten as: σ2p = x21σ21 + x22σ22 - 2x1x2r12σ1σ2
  6. 6. The right hand side of the equation has the same form as the expansion of the identity (a - b)2, namelya2 - 2ab+b2. Hence, it may be reduced as σ2p = (x1σ1 – x2σ2)2The standard deviation then becomes σp = x1σ1 - x2σ2Security return uncorrelated. Hen the return of to securities are entirely uncorrelated, the correlation coefficient would be zero. Theformula of portfolio variance is: σ2p = x21σ21 + x22σ22 + 2x1x2r12σ1σ2Since r12=0, the last term in the equation become zero: the formula may be rewritten as: σ2p = x21σ 21 - x22σ22The standard deviation σp = √x21σ 21 - x22σ22
  7. 7. FEASIABLE SET OF PORTFOLIO With a limited number of securities an investor can create a very large number of securities andinvestor can create a very large number of portfolios by combining these securities in different proportions.These constitute the feasible set of portfolios in which the investor can possibly invest. This is known asAs each possible portfolio in the opportunity set of portfolios has an expected return and standard deviationassociated with it, each portfolio would be represented by a single point in the risk-return space enclosedwithin the two axes of the graph. The shaded area in the graph represents the set of all possible portfoliosthat can be constructed from a given set of securities. This opportunity set of portfolio takes a concave
  8. 8. shape because it consists of portfolios containing securities that are less than perfectly correlated witheach other. Thus, we find that portfolios lying in the North West boundary of the shaded area are more efficientthan all the portfolios in the interior of the shaded area. This boundary of the shaded area is called EfficientFrontier because it contains all the efficient portfolios in the opportunity set. The set of portfolios lyingbetween the global minimum variance portfolio and the maximum return portfolio on the efficient frontierrepresents the efficient set of portfolios. The efficient frontier is shown as under.The efficient frontier is a concave curve in the risk-return space that extends from the minimum varianceportfolio to the maximum return portfolio.
  9. 9. SELECTION OF THE OPTIMAL PORTFOLIO The portfolio selection problem is really the process of delineating the efficient portfolios and thenselecting the best portfolio from the set. The selection of the optimal portfolio thus depends on the investorsrisk aversion, or conversely on his risk tolerance. This can be graphically represented through a series ofrisk return utility curves or indifference curves. Each curve represents different combination of risk and returns all which are equally satisfactory tsothe concerned investor. The investor is indifferent between the successive points in the curve. Eachsuccessive curve moving upward to the left moving represents the high level of satisfaction or utility. Theinvestor goal would be maximize his utility by moving upto the higher utility curve. The optimal portfolio foran investor would be one at the point of tangency between the efficient frontier and his risk return utility orindifference curve.
  10. 10. Capital asset pricing method (CAPM) The capital asset pricing model was developed in mid-1960s by three researchers William Sharpe,John Linter and Jan Mossin independently. Consequently, the model is often referred to as sharpe-Lintor-mossin capital asset pricing model. The capital asset pricing model or CAPM is really an extension of the portfolio theory of morkwitz
  11. 11. Assumption The capital asset pricing model is based on certain explicit assumptions regarding the behavior ofinvestors. The assumptions are listed below: 1. Investor make there investment decisions on the basis of risk-return assessments measured in terms of expected returns and standard deviation of return. 2. The purchase or sale of a security can be undertaken in infinitely divisible unit. 3. Purchase and sale by a single investor cannot affect prices. This means that there is perfect competition where investors in total determine prices by their action. 4. There are no transaction costs. Given the fact that transaction costs are small, they are probably of minor importance in investment decision-making, and hence they are ignored. 5. There are no personal income taxes. Alternatively, the tax rate on dividend income and capital gains are the same, thereby making the investor indifferent to the form in which the return on the investment is received (dividends or capital gains). 6. The investor can lend or borrow any amount of fund desired at a rate of interest equal to the rate of risk less securities. 7. The investor can sell short any amount of any shares. 8. Investors share homogeneity of expectations. This implies that investors have identical expectations with regard to the decision period and decision inputs. Investors are presumed to have identical expectations regarding expected returns, variance of expected returns and covariance of all pairs of securities.
  12. 12. The capital market line All investors are assumed to have identical (homogenous) expectations. This portfolio of all risky securities is referred to as the market portfolio M. each security will be held in proportion which the market value of all risky securities in the market portfolio and a risky security.All these combination will lie along the straight line along the straight line representing the efficient frontier.This line formed by the action of all investors mixing the market portfolio with the risk free asset is known asthe capital market line (CML). All efficient portfolios of all investors will lie along this capital market line.
  13. 13. The relationship between the return and risk of any efficient portfolio on the capital market line can beexpressed in the form of the following equation. R-e = Rf + [R-m – Rf / σm] σeWhere the subscript e denotes the efficient portfolio. Thus the expected return on an efficient portfolio is:(Expected return) = (price of time) + (price of risk) (amount of risk)The expected market line The CML shows the risk return relationship for all efficient portfolios. They would all lie along thecapital market line. All portfolios other than the efficient one will lie below the capital market line. The CMLdoes not describe the risk-return relationship of inefficient portfolios or individual securities. The capitalasset pricing model specifies the relationship between expected returns and risk for all securities and allportfolios, whether efficient or inefficient. We have seen earlier that the total risk of a security as measured by standard deviation is composedof the two components: systematic risk and unsystematic risk. As investment is diversified and more andmore securities are added to a portfolio, the unsystematic risk is reduced. For a very well diversifiedportfolio, unsystematic risk tends to become zero and the only relevant risk is systematic risk measured bybeta (β). Hence, it is argued that the current measure of security risk is beta. It follows that the expected return of a security or of a portfolio should be related to the risk of thatsecurity or portfolio as measured by β.
  14. 14. The security market line provides the relationship between the expected return and beta of a securityportfolio this relationship can be expressed in the form of the following equation: R-i = Rf + βi (R-m – Rf)CAPM The relationship between risk and return established by the security market line is known as thecapital asset pricing model. It is basically a simple linear relationship. The higher the value of beta, higherwould be the risk of the security and therefore, larger would be the return expected by the investors. Inother words, all securities all securities are expected to yield returns commensurate with their riskiness as
  15. 15. measured by β. This relationship is valid not only for individual securities, but is also valid for all portfolioswhether efficient or inefficient. The expected return on any security or portfolio can be determined from the CAPM formula if weknow the beta of that security portfolioSML and CML It is necessary to contrast SML with CML. Both postulate a linear (straight line) relationship betweenrisk and return. In CML the risk is defined as total risk and is measured by standard deviation, while in SMLthe risk is defined as systematic risk and is measured by β. Capital market line is valid only for efficientportfolios while security market line is valid for all portfolios and all individual securities as well. CML is thebasis of the capital market theory while SML is the basis of the basis of the capital asset pricing model.Leveraged betaThe beta of equity of a company or a project is likely to undergo changes with the change in the capitalstructure of the company. If the company is entirely equity based, its equity beta is likely to be lower than ifit under takes a borrowing. A number of factors like default risk, bankruptcy risk, and agency costscontribute to this phenomenon.For the sake of convenience, let us call the equity beta of a firm which is levered as levered beta and thatof a firm on an all equity structure as unlevered beta. βL = βU [(1-Lt)/(1-L)]QUESTIONSQ 1 A portfolio consists of two securities A and B in the proportion 0.6 and 0.4. The standard deviation are10 and 16 respectively. What is the standard deviation of the portfolio if the coefficient of correlation amongthem is (a) 0.5 (b) 0.2
  16. 16. Q 2 consider the two securities A and BParticulars Security A Security BExpected return 24% 8%Standard deviation 8% 10%Correlation coefficient 0.36Computer the expected value of the return and the standard deviationQ 3 Calculate the expected rate of the return of the following portfolio.Rf = 10%, Rm = 15%, β = 0.6Q 4 calculate the beta of the portfolio from the following detailsYear Return on security Market return1 10 122 6 53 13 184 -4 -85 13 106 14 167 4 78 18 159 24 3010 22 35Q 5 You have four securities A B C and D. the investments in these securities is as followsA 10000
  17. 17. B 20000C 16000D 14000The beta values are 0.80, 1.20, 1.40 and 1.75 respectively. (i) if the risk free rate is 4.25% and the marketreturn is 11% what is the expected return on the portfolio. (ii) if we sell the investment in security B andinvests it in RBI bonds yielding a return of 4.25% what is the expected return.Q 6 following information is givenRisk free rate 10%.Market return 15%Beta 1.2What is the return from this investment . if the return from this investment is 18% is the stock correctlyvaluedQ 7 ABC ltd. has an equity beta of 1.40. it has a debt equity mix of 30/70. The company is planning for therestructuring so that the debt equity mix becomes 40/60. If the taxz rate is 30% find the new equity betaQ 8 Following is the data regarding six securities A B C D E FReturn 8 8 12 4 9 8Risk (S.D.) 4 5 12 5 4 6 (i) Assuming three have to be selected which one have to be selected (ii) Assuming perfect correlation, show whether it is preferable to invest 75% in A and 25% in C or to invest 100% in E

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