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**Unit III: Portfolio Selection**

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

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Portfolio Selection: A) Markowitz Approach: Find the set of portfolios that satisfy the following 3 conditions: 1) Provides the minimum risk for every possible level of return 2) The ‘Efficient' set 3) Investor selects from the ‘Efficient’ set the single portfolio that meets his/her needs

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Efficient portfolio Efficient portfolio is the portfolio that gives maximum return for a given level of risk or minimum risk for a given level of return. Efficient Frontier: Efficient frontier is the loci of all such efficient portfolios.

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The question is : how should investors choose a “best” option on the efficient frontier? The following are the important parameters to be considered: 1) Risk – return framework. 2) Utility functions, or indifference curves. There can be various types of Indifference curves representing investors’ perceptions towards risk-return. Thus there can be following categories of investors:

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**σp Rp Risk Lover Indifference curves 1) Risk Lover Investor Return**

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**σp Rp Risk Fearing Investor’s Indifference Curves**

2) a) Risk Averse Investor U4 U5 U3 U2 Rp U1 Return Risk σp Risk Fearing Investor’s Indifference Curves

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**σp Rp Less Risk Fearing Investor’s Indifference Curve**

2) b) Less Risk Averse Investor U4 U3 Rp U2 U1 Return Risk σp Less Risk Fearing Investor’s Indifference Curve

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**Investor’s Equilibrium: Selection of Best portfolio.**

Y is the point of optimal/best portfolio. At this point the efficient frontier is tangent to the indifference curve.

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**B) SHARPE METHOD OF PORTFOLIO SELECTION**

Single Index model by Sharpe helps in the construction of an Optimal portfolio by providing a single number that measures the desirability of including a stock in the Optimal Portfolio. Here, the desirability of any stock is directly related to its excess return -to -beta ratio:

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Where R i = expected return on stock i R f = return on a riskless asset βi = expected change in the rate of return on stock i associated with a 1 % change in the market return. If stocks are ranked by Excess return to beta (from highest to lowest), the ranking represents the desirability of any stock ‘s inclusion in a portfolio. The number of stocks selected depends on a unique cut off rate such that all stocks with higher ratios of Excess return to beta will be included and all stocks with lower ratios excluded.

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**STEPS to determine which stocks are included in the Optimum Portfolio:**

Calculate the excess return to beta ratio for each stock under review . 2. Rank them from highest to lowest. 3. The optimum portfolio consists of investing in all stocks for which is greater than a particular cut off point C*

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**Example: On the basis of following data find the optimal portfolio**

Example: On the basis of following data find the optimal portfolio. Also find proportion of the fund to be invested in the securities of the optimal portfolio.

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Security no. (1) Mean return,R i (2) Beta , βi (3) Unsystematic risk, 1 15 R f = 5% 2 17 1.5 40 3 12 20 4 10 5 11 6 30 7 8 0.8 16 9 5.6 0.6 Also given RF = 5% and

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**Table : 2, Solution: 1 10 2 12 8 3 7 4 6 5 2.5 9 0.6 R i – R F**

Security No. Excess return, R i - RF Excess return to beta R i – R F βi 1 10 2 12 8 3 7 4 6 5 2.5 9 0.6

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**Establishing a Cut off Rate**

All securities whose Excess return to beta ratio are above the cut off rate are Selected All securities whose ratios are below are rejected. Finding the Cut off rate For portfolio of i stocks ,C i is given by Where : = Variance in the market index = stock’s unsystematic risk.

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**Table 3 : Determining Cut off Rate with = 10%**

Security NO. Ri-RF βi (Ri-RF) βi C 1 10 2/10 2/100 1.67 2 8 4.5/10 5.625/100 6.5/10 7.625/100 3.69 3 7 3.5/10 5/100 10/10 12.625/100 4.42 4 6 24/10 40/100 34/10 52.625/100 5.43 5 1.5/10 2.5/100 35.5/10 55.125/100 5.45 3/10 7.5/100 38.5/10 62.625/100 5.3 10/100 41.5/10 72.625/100 5.02 2.5 1/10 4/100 42.5/10 76.625/100 4.91 9 43.5/10 81.625/100 4.75 0.6/10 6/100 44.1/10 87.625/100 4.52

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**Arriving at the optimal portfolio**

With the help of Cut off rate ,we know which securities are to be included in the Optimum portfolio, the next step is to calculate the percent of capital to be invested in each security. The percentage invested in each security is : Where,

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Z 1 = 2/100( ) = 0.091 Z 2 = (8-5.45) = 100 Z 3 = 5 ( ) = 0.0775 Z 4 = 20 ( ) = 0.110 Z 5 = 2.5 ( ) =

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**Dividing each Z i by sum of Z i ,we would invest**

0.091 / = 23.5 % in security 1 / = 24.6 % in security 2 / = 20 % in security 3 0.110 / = 28.4 % in security 4 / = 3.5 % in security 5

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Problem No.1 Q.1 Vinoth received Rs.10 lakh from his pension fund. He wants to invest in the stock market. The treasury bill rate is 5% and the market return variance is 10. The following table gives the details regarding the expected return, beta and residual variance of the individual security. What is the optimum portfolio assuming no short sales?

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**Security Expected Return Beta σei A 15 1.0 30 B 12 1.5 20 C 11 2.0 40**

8 0.8 10 E 9 F 14

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Problem No.2 Q.2 A portfolio manager has got the following information about several stocks. He has to build an optimum portfolio for his client without short sales. The market index variance is 12 per cent and the risk free rate of return is 7 per cent.

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**Security Expected return β σei A 22 1.0 35 B 20 2.5 30 C 14 1.5 25 D**

18 80 E 16 0.8 F 12 1.2 10 G 19 1.6 H 17 2.0

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Problem No.3 What is the optimum portfolio in choosing among the following securities and assuming Rf= 5 percent and variance of market return = 10 per cent?

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**Security Expected Return Beta σei A 15 1.0 30 B 12 1.5 20 C 11 2.0 40**

8 0.8 10 E 9 F 14

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Problem No.4 You are attempting to construct an optimum portfolio. Over your holding period you have forecast an expected return on the market of 13.5% with a market variance of 256%. The treasury security rate available is 7% (risk free). The following securities are under review:

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Stock Alpha Beta Residual Variance Boeing 3.72 .99 9.35 Bristol-Myers 0.60 1.27 5.92 Browning-Ferris 0.41 .96 9.79 Emerson Electric -0.22 1.21 5.36 Mountain States Telephone 0.45 .75 4.52

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Problem No.5 Q.1 Mr. David is constructing a optimum portfolio. The market return forecast says that it would be 13.5% for the next two years with the market variance of 10%. The risk less rate of return is 5%. The following securities are under review. Find out the optimum portfolio.

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**Company α β σei Anil 3.72 0.99 9.35 Avil 0.60 1.27 5.92 Bow 0.41 0.96**

9.79 Viril -0.22 1.21 5.39 Billy 0.45 0.75 4.52

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