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Formation of Optimum Portfolio using Single Index Model

Formation of Optimum Portfolio using Single Index Model. Short Sales Not Allowed Short Sales Allowed. Short Sales Not Allowed:. Why a stock does or does not enter into an optimal portfolio? The desirability of any stock is directly related to its excess return to beta ratio.

brett-mccarthy
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Formation of Optimum Portfolio using Single Index Model

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  1. Formation of Optimum Portfolio using Single Index Model Short Sales Not Allowed Short Sales Allowed www.pptmart.com

  2. Short Sales Not Allowed: • Why a stock does or does not enter into an optimal portfolio? • The desirability of any stock is directly related to its excess return to beta ratio. www.pptmart.com

  3. Ranking of Security: • We use the index “Excess return to beta” ratio. • i.e., www.pptmart.com

  4. The rules for determining which stocks are included in the optimum portfolio are as follows: • (1).Find the “Excess Return to Beta” Ratio And rank them from highest to lowest. (2) Invest in all stocks for which www.pptmart.com

  5. Calculating the Cut-off Rate C* • For a portfolio of i stocks Ci is given by • where σm2 = the variance in the market index • σej2 = unsystematic risk www.pptmart.com

  6. Setting the Cut-off Rate (C*): www.pptmart.com

  7. Table 1Data Required to Determine Optimal Portfolio RF = 5% www.pptmart.com

  8. Table 2Calculations for Determining Cut-off Rate with m2 = 10 www.pptmart.com

  9. There are 10 securities in the table. We have already ranked the securities according to (Ri –RF) /i and have used numbers that make the calculations easy to follow. The application of rule 2 involves the comparison of (Ri –RF) /i with C* . It appears that for securities 1 to 5 (Ri –RF) /i is greater than C* while for security 6 it is less than C*. Hence, an optimal portfolio consists of securities 1 to 5. www.pptmart.com

  10. Constructing the Optimum Portfolio:Securities are already selected.Percentage investment in each security. • The percentage invested in each security is www.pptmart.com

  11. Applying this formula to our example we have • Z1 = 2 / 100 (10 – 5.45) = 0.091 • Z2 = 3.75 / 100 (8 – 5.45) = 0.095625 • Z3 = 5 / 100 (7 – 5.45) = 0.0775 • Z4 = 20 / 100 (6 – 5.45) = 0.110 • Z5 = 2.5 / 100 (6 – 5.45) = 0.01375 • = 0.387875 • Dividing each Zi by the sum of the Zi, we find that we should invest 23.5% of our funds in security 1, 24.6% in security 2, 20 % in security 3, 28.4% in security 4, and 3.5% in security 5. www.pptmart.com

  12. Short-Sales Allowed: • When short sales are allowed, all stocks will either be held long or sold short. Thus, all stocks enter into the optimum portfolio and all stocks affect the cut-off point. Equation (1) still represents the cut-off point, but now the numerator and denominator of this equation are summed over all stocks. • C* = C10 www.pptmart.com

  13. www.pptmart.com

  14. Some Z will be positive, some will be negative • Positive Z or Positive Xi – Long Purchase • Negative Z Negative Xi – Short Sales • To calculate C* we must employ equation 1 with i set equal to the number of stocks under consideration. In this case we have a population of 10 stocks so that • C* = C10 = 4.52. www.pptmart.com

  15. Employing equation (2) • For each security we find • Z1 = 1/50 [10 – 4.52] = 0.110 • Z2 = 1.5/40 [8 – 4.52] = 0.131 • Z3 = 1/20 [7-4.52] = 0.124 • Z4 = 2/10 [6-4.52] = 0.296 • Z5 = 1/40[6-4.52] = 0.037 • Z6 = 1.5/30[4-4.52] = 0.026 • Z7 = 2/40 [3-4.52] = -0.076 • Z8 = 0.8/16[2.5-4.52] = -0.101 • Z9 = 1/20[2-4.52] = -0.126 • Z10 = 0.6/6[1.0-4.52] = -0.352 www.pptmart.com

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