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Portfolio markowitz model


2. markowitz model

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2. markowitz model

  1. 1. THE MARKOWITZ MODEL We all agree that holding two stocks is less risky as compared to one stock. But building the optimal portfolio is very difficult. Markowitz provides an answer to it with the help of risk and return relationship.
  2. 2. Simple diversification  In case of simple diversification securities are selected at random and no analytical procedure is used.  The simple diversification reduces total risk. The reason behind this is that the unsystematic price fluctuations are not correlated with the market fluctuations.  As the portfolio size increases the total risk starts declining. It flattens out after a certain point. Beyond that limit risk cannot be reduced.
  3. 3. `  Problems of vast diversification  Purchase of poor performers  Information adequacy  High research cost  High transaction cost
  4. 4. Assumptions For a given level of risk, investors prefers higher return to lower return. Likewise for given level of return investors prefer low risk as compared to high risk.
  5. 5. The concept In developing the model, Markowitz has given up the single stock portfolio and introduced diversification. The single stock portfolio would be preferable if the investor is perfectly certain that his expectation of higher return would turn out to be real. But in this era of uncertainty most of the investors would like to join Markowitz rather than single stock. It can be shown with the help of example.
  6. 6. Stock ABC Stock XYZ Return % 11 or 17 20 or 8 Probability .5 each return .5 each return Expected return 14 14 Variance 9 36 Standard deviation 3 6
  7. 7. ABC expected return: .5 x 11+ .5x17= 14 XYZ expected return: .5 x 20+ .5x8= 14 ABC variance = .5(11-14)² + .5(17-14) ²= 9 XYZ variance= .5(20-14) ² + .5(8-14) ²= 36 ABC standard deviation= 3 XYZ standard deviation= 6
  8. 8. Now ABC and XYZ have same expected return of 14 % but XYZ stock is much more risky as compared to ABC because the standard deviation is much more high. Suppose the investor holds 2/3 of ABC and 1/3 of XYZ the return can be calculated as follows Rp=∑X₁ R₁ Rp= return form portfolio X₁= proportion of total security invested in security 1. R₁= expected return of security 1.
  9. 9. Let us calculate the expected return for both possibilities. possibility 1= 2/3 x 11 + 1/3 x 20 = 14 possibility 2= 2/3 x 17 + 1/3 x 8 = 14 In both the cases the investor stands to gain if the worst occurs, than by holding either of security individually. Holding two securities may reduce portfolio risk too. The portfolio risk can be calculated with the help of following formula.
  10. 10. __________________________ Ϭp= √ X₁²Ϭ₁² + X₂²Ϭ₂² + 2 X₁ X₂( r₁₂ Ϭ₁Ϭ₂) Ϭp= std. deviation of portfolio X₁= proportion of stock X₁ X₂= proportion of stock X₂ Ϭ₁= std. deviation of stock X₁ Ϭ₂= std. deviation of stock X₂ r₁₂= correlation coefficient of both stocks
  11. 11. r₁₂= covariance of X₁₂ Ϭ₁ Ϭ₂ Using the same example given in the return analysis , the portfolio return can be estimated Cov of X₁₂= 1/N ∑(R₁ - Ṝ₁) (R₂ - Ṝ₂) = ½ [(11-14)(20-14) + (17-14)(8-14)] = -18 Now r = -18/6x3= -1
  12. 12. In our example the correlation coefficient is -1.0. That means there is perfect negative correlation between the two and the return moves in opposite direction. If the correlation is +1 it means securities will move in same direction and if it is zero the return of both the securities is independent. Thus the correlation between two securities depend upon the covariance between the two securities and the standard deviation of each security.
  13. 13. ___________________________ Ϭp= √ X₁²Ϭ₁² + X₂Ϭ₂² + 2 X₁ X₂( r₁₂ Ϭ₁Ϭ₂) ______________________________________ = √ (2/3)² x 9 + (1/3)² x 36 + 2x 2/3 x 1/3 (-1x3x6) ______________ = √ 4+4-8 = 0 The portfolio risk is nil here.
  14. 14. The change in portfolio proportions can change the portfolio risk. Taking same example of ABC and XYZ stock, the portfolio std. deviation is calculated for different proportions. Stock ABC Stock XYZ Portfolio std. deviation 100 0 3 66.66 33.34 0 50.00 50.00 1.5 0 100 6
  15. 15. Markowitz efficient frontier The risk and return of all portfolios plotted in risk-return space would be dominated by efficient portfolios. Portfolio may be constructed from available securities. All the possible combination of expected return risk compose attainable set. The following example shows the expected return and risk of different portfolios.
  16. 16. PORTFOLIO EXPECTED RETURN % RISK A 17 13 B 15 8 C 10 3 D 7 2 E 7 4 F 7 8 G 10 12 H 9 8 J 6 7.5
  17. 17. The attainable set of portfolios are illustrated in fig. Each of the portfolios along the line or within the line ABCDEFGJ is possible. It is not possible for the investor to have portfolio of this perimeter because no combination of expected return and risk exists there. But there are some attractive options. Portfolio B is more attractive than portfolio F and H because it offers more return on same level of risk. Likewise, C is more attractive than portfolio G as for same return there is lower level of risk.
  18. 18. EFFICIENT FRONTIER 0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 Y-Values Y-Values
  19. 19. UTILITY ANALYSIS Utility is the satisfaction the investors enjoy from the portfolio return. An ordinary investor is assumed to receive greater utility from higher return and vice-versa. In a fair gamble which costs Re 1, the outcomes are A and B events. A event will yield Rs. 2. occurrence of B event is a dead loss i.e. 0. The chance of occurrence of both the events are 50-50. The expected value of investment is (1/2 x 2 + 1/2x 0) =Re 1. The expected value of gamble is exactly equal to cost. Hence it is a fair gamble.
  20. 20. Risk averter rejects a fair gamble because the disutility of the loss is greater for him than the utility of equivalent gain. Risk neutral investors means that he is indifferent to whether a fair gamble is undertaken or not. The risk seeking investor would select a fair gamble. The expected utility of investing is higher than the expected utility of not investing.
  21. 21. Leveraged portfolio In the above model, the investor is assumed to have certain amount of money to make investment for fixed period of time. There is no borrowing and lending opportunities. When the investors is not allowed to use the borrowed money, he is denied the opportunity of having financial leverage. Again the investor is assumed to be investing only on risky assets. Riskless assets are not included in the portfolio. To have a leveraged portfolio investor has to consider not only risky assets but also risk free assets. Secondly, he should be able to borrow and lend money at given rate of interest.
  22. 22. RISK FREE ASSET The features of risk free assets are: (i) Absence of default risk (ii) Full payment of principal and interest amount
  23. 23. Inclusion of risk free asset Now, the risk free asset is introduced and the investor can invest part of his money on risk free asset and the remaining amount on risky assets. It is also assumed that investor would be able to borrow money at risk free rate of interest. When risk free asset is included in portfolio, the feasible efficient set of portfolios is altered. But return from risk free asset is less as compared to risky assets so investor will make a combination of risk free and risky assets to maximize his return.
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