Unit III: Portfolio Analysis MBA & MBA – Banking and Finance (Term-IV) Course : Security Analysis and Portfolio Management Unit III: Portfolio Analysis

PORTFOLIO ANALYSIS Portfolios are combinations of securities Portfolios aim at diversification of risk Diversification involves spreading and minimization of risk Portfolio analysis deals with determination of future return and risk of portfolios

Harry Markowitz’s Approach to Portfolio Analysis Portfolio Return N RP = ∑ Xi Ri i=1 Where : RP = Expected Return to portfolio Xi = proportion of total portfolio invested in security i Ri = expected return to security i N = total number of securities in portfolio

Portfolio Risk The risk in a portfolio is less than the sum of the risks of the individual securities taken separately whenever the returns of the individual securities are not perfectly positively correlated. Thus, the smaller the correlation between the securities, the greater the benefits of diversification and hence less is the risk. Diversification depends upon the right kind of securities and not the number alone. Less is the correlation between the returns of the securities more is the diversification and less is the risk.

Two-security case σP = √ Xx2σx2 + Xy2σy2 + 2XxXy(rxyσxσy) Where: σP = portfolio standard deviation Xx = percentage of total portfolio value in stock X Xy = percentage of total portfolio value in stock Y σX = standard deviation of stock X σY = standard deviation of stock Y rXY= correlation coefficient of X and Y

Least Risky Portfolio Given the security A and security B XA = σ2B – rABσAσB σ2A + σ2B – 2 rABσAσB XB = 1 – XA In case r = -1, XA = σB (σA + σB) In this case risk will be equal to zero.

The Three-Security Case σP = √ X21σ21 + X22σ22 + X23σ23 + 2X1X2r12σ1σ2 + 2X2X3r23σ2σ3 + 2X1X3r13σ1σ3 Where: σP = portfolio standard deviation X1 = percentage of total portfolio value in stock 1 X2 = percentage of total portfolio value in stock 2 X3 = percentage of total portfolio value in stock 3 σ1 = standard deviation of stock 1 σ2 = standard deviation of stock 2 σ3 = standard deviation of stock 3 r12= correlation coefficient of 1 and 2 r23 = correlation coefficient of 2 and 3 r13 = correlation coefficient of 1 and 3

The N-security case N N σ2P = ∑ ∑ Xi Xj rij σi σj i=1 j=1 Where: σ2P = expected portfolio variance √ σ2P = portfolio standard deviation N N ∑ ∑ = N2 numbers are to be added together

Problem 1. Two shares P and Q, have the following expected returns, standard deviation and correlation: Stocks P Q Expected Return 18% 15% Standard Deviation 23% 19% Correlation coefficient = 0 Determine the minimum risk combination for a portfolio of P and Q If the correlation of returns of P and Q is -1.0, then what is the minimum risk portfolio of P and Q?

Problem 2. A person is considering investment in two shares A and B. the correlation coefficient between the returns of A and B is 0.1, other data is given below: Share Expected Return Std Deviation (%) (%) A 10 15 B 18 30 You are required to suggest a portfolio of shares A and B that should minimize the risk. Also determine the expected return and the minimum risk of such portfolio.

Problem 3. Two shares P and Q, have the following expected returns, standard deviation and correlation: E(rP) = 18% E(rQ) = 15% σP = 23% σQ = 19% Cor P,Q = 0 Determine the minimum risk combination for a portfolio of P and Q. If the correlation of returns of P and Q is -1.0, then what is the minimum risk portfolio of P and Q?

Problem 4. Consider the data given below: Particulars Stock ABC Stock XYZ Return (%) 12 or 16 22 or 10 Probability 0.5 each return 0.5 each return Find: Expected return and variance of Stock ABC and Stock XYZ Risk and Return of portfolio comprising 15% of stock ABC and 85% of stock XYZ. What should be the combination of securities if the required return on portfolio is 15%? iv) What should be risk of portfolio made of above securities which will offer a return of 15%?

Problem 5. Consider two stocks A and B: Expected Return(%) Std Deviation(%) Stock A 14 22 Stock B 20 35 The returns on the stocks are perfectly negatively correlated. What is the expected return of a portfolio comprising of stocks A and B when the portfolio is constructed to drive the standard deviation of portfolio return to zero?

Problem 6. X owns a portfolio of two securities with the following expected returns, standard deviations and weight: Security Expected Standard Weight Return Deviation X 12% 15% 40% Y 15% 20% 60% Calculate the Portfolio Return What will be the maximum and minimum portfolio risk?

Problem 7. A portfolio consists of three securities P, Q and R with the following parameters: P Q R Corr. Expected return (%) 25 22 20 Standard deviation (%) 30 26 24 Correlation: PQ -0.50 QR +0.40 PR +0.60 If the securities are equally weighted, how much is the risk and return of the portfolio of these three securities?

Problem 8. A portfolio consists of 3 securities – 1, 2 and 3. The proportions of these securities are : W1 = 0.3, W2 = 0.5 and W3 = 0.2. The standard deviations of returns on these securities (in percentage terms) are : σ1 = 6, σ2 = 9 and σ3 = 10. The correlation coefficients among security returns are r12 = 0.4, r13 = 0.6 and r23 = 0.7. What is the standard deviation of portfolio return?

Problem 9. The following data is available of XYZ Investment Ltd. for investment in different securities: Security X Y Z Mean Return (%) 20 30 40 Std Deviation (%) 4 5 3 Correlation Coefficients Stocks X,Y 0.4 Stocks Y,Z -0.6 Stocks X,Z -1 You as a financial analyst, are required to calculate expected return and risk of each of the following portfolios The investor invests equally in each security

Contd. The investor invests 40% in X, 40% in Y and 20% in Z The investor invests 30% in X, 30% in Y and 40% in Z.

Problem 10. Stocks A, B and C display the following parameters: Stock A Stock B Stock C Expected Return 0.10 0.12 0.08 Standard Deviation 0.10 0.15 0.05 Proportion of Fund 0.40 0.40 0.20 Find portfolio return and risk if the correlation coefficient between the returns on stocks A and B, stocks B and C, stocks A and C are 0.3, 0.4 and 0.5 respectively.

THE SHARPE INDEX MODEL N Security Return: Ri = αi + ßiI Where: Ri = expected return on security I αi = intercept of a straight line or alpha coefficient ßi = slope of straight line or beta coefficient I = expected return on index (market) Portfolio Return: N RP = ∑ Xi (αi + ßiI) i=1 Xi = proportion of total portfolio invested in security i

Risk of Security: Systematic Risk = ß2 X (Variance of Index) = ß2 σ2I Unsystematic Risk = e2 = Total variance of security return – Systematic Risk Total Risk = ß2 σ2I + e2

Portfolio Variance (Risk) : N N σ2P =[ ( ∑ Xißi)2 σ2I ] + [ ∑ X2i e2i] Where: σ2P = variance of portfolio return σ2I = expected variance of return e2i = variation in security’s return not caused by its relationship to the index

Problem 1. An investor has 3 securities for consideration of investment about which following parameters are made available to you: Security X Y Z Proportion of funds Invested (%) 30 40 30 α (%) 3 1 -2 ß 1.2 0.7 1.1 Unsystematic Variance(%) 6 2 4 Assume that the return on the market index is 10% and variance of market return is 8%. Calculate portfolio return and risk using Sharpe’s Model.

Problem 2. An investor wants to build a portfolio with the following four stocks. With the given details, find out his portfolio return and portfolio variance. The investment is spread equally over the stocks. Company α ß Residual Variance A 0.17 0.93 45.15 B 2.48 1.37 132.25 C 1.47 1.73 196.28 D 2.52 1.17 51.98 Market return (Rm) = 11 Market return variance = 26

Problem 3. The following table gives data on four stocks: Stock Alpha Variance Variance Systematic Unsystematic A -0.06 5 4 B 0.1 2 6 C 0.00 3 1 D -0.14 3 2 The market is expected to have a 12 percent return over a forward period with a return variance of 6 percent. Calculate the expected return for a portfolio consisting of equal portion of stocks A, B, C and D.

Problem 4. Consider a portfolio of four securities with the following characteristics: Security Weights αi βi Residual Variance 1 0.2 2.0 1.2 320 2 0.3 1.7 0.8 450 3 0.1 -0.8 1.6 270 4 0.4 1.3 180 Calculate the return and risk of the portfolio under Single Index model, if the return on market index is 16.4 per cent and the standard deviation of return on market index is 14 percent.

PROBLEM 5. The following table gives data on three stocks. The data are obtained from correlating returns on these stocks with the return on market index: Stock Alpha Variance Variance Systematic Unsystematic 1 -2.1 1.6 14 2 1.8 0.4 8 3 1.2 1.3 18 Which single stock would an investor prefer to own from a risk-return view point if the market index were expected to have a return of 15 per cent and a variance of return of 20 per cent?

Problem 6. The following table gives data on four stocks. The data are obtained from correlating returns on these stocks with the return on market index: Stock αi βi Variance Unsystematic A -1.5 1.25 24 B 2.15 1.47 47 C 1.70 0.69 36 D 0.83 0.88 30 The market index were expected to have a return of 17.5 per cent and a variance of return of 28 per cent. Which single stock would an investor prefer to own from a risk-return perspective?

PROBLEM Security Weights αi βi Variance Unsystematic 1 0.1 -0.28 0.91 7. Consider a portfolio of six securities with the following characteristics: Security Weights αi βi Variance Unsystematic 1 0.1 -0.28 0.91 23 2 0.15 0.76 0.87 60 3 0.20 2.52 1.17 52 4 0.10 -0.16 0.97 86 5 0.25 1.55 1.07 67 6 0.47 0.86 82

PROBLEM Assuming the return on market index is 14.5 per cent and the standard deviation of return on market index is 16 per cent, calculate the return and risk of the portfolio under Single Index model.

Exercise Q1. An investor has two securities X and Y, for investment purpose whose details are given below: Find the total return and risk of portfolio formed by 70% of security X and 30% of security Y. Security X Security Y Expected Return (%) 15 18 Risk in terms of Std. dev.(%) 4 7 Coefficient of Correlation 1

Exercise Q2. An investor is interested in investing his funds in securities of two companies, A Ltd. and B Ltd. , whose expected return and risk are given below : Find the total return and risk of portfolio formed by 60% of security A Ltd. and 40% of security B Ltd.. At what level of correlation, the risk of the portfolio will be minimum and what will be the risk of portfolio in the case? A Ltd. B Ltd. Expected Return (%) 16 12 Risk in terms of Std. dev.(%) 9 6

Exercise Q3. Mr. X is considering investment in securities P and Q , whose details are given below: If a portfolio of 30% of P and 70% of Q is formed, find the Expected return of the portfolio Minimum risk of the portfolio Maximum risk of the portfolio P Q Expected Return (%) 13 16 Risk in terms of Std. dev.(%) 4 7

Exercise Q4. An investor is considering investment in securities P and Q, whose details are given below If a portfolio of 60% of P and 40% of Q is formed, find the Expected return of the portfolio Risk of the portfolio if the correlation coefficient is 0.5 Risk of the portfolio when there is no correlation between the securities. P Q Expected Return (%) 14 19 Risk in terms of Std. dev.(%) 3 6

Exercise Q5. Given the following details of securities X and Y : A portfolio is formed by including 75% of X and 25% of Y. Show that the risk of portfolio depends on the correlation coefficient between the two securities. X Y Expected Return (%) 14 20 Risk in terms of Std. dev.(%) 3 7

Exercise Q6. Given the following parameters of securities X and Y : Find: Portfolio expected return and risk if the allocation of total investible fund in the securities X and Y is made in the ratio of 4:6 and the coefficient of correlation between the returns on the securities is -0.8. Optimum allocation of fund to be invested in X and Y if the coefficient of correlation between their returns is -1. Portfolio risk in case the portfolio expected return is 20% and coefficient of correlation is -0.6. Security Expected Return (%) Standard deviation (%) X 10 30 Y 60