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Risk and Return of Portfolio- Canvas.pptx

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Risk and Return of Portfolio- Canvas.pptx

  1. 1. RISK AND RETURN OF PORTFOLIO Dr. Deergha Sharma
  2. 2. Return of individual security Expected Return Where 𝑅=Expected rate of return P=Probability of return R= Rate of return N= number of years 𝑅 = 𝑡=1 𝑛 𝑃 ∗ 𝑅
  3. 3. Risk of Individual Security 𝜎2 = 𝑅 − 𝑅 2𝑃 Where 𝜎2=Variance R=Rate of return P=Probability of occurrence of return 𝑅= Expected rate of return
  4. 4. Return of Portfolio 𝑅𝑝 = Ʃ𝑤𝑥 ∗ 𝑅𝑥 + 𝑤𝑦 ∗𝑅𝑦 Where 𝑅𝑝=Expected return of a portfolio 𝑤𝑥= Proportion of fund invested in security x 𝑤𝑦= Proportion of fund invested in security y 𝑅𝑥 𝑅𝑦=Expected returns of security x and security y
  5. 5. Question ■ The rate of return and the possibilities of their occurrence for Alpha and Beta company scrips are given below. ■ Find the expected rates of return for both Alpha and Beta Scrips. ■ If an investor invests equally in both the scrips what would be the expected return. ■ If the proportion is changed to 25% and 75% and then to 75% and 25%, what would be the expected rate of return? Probability Return on alpha Scrip Return on Beta’s Scrip 0.05 -2.0 -3.0 0.20 9.0 6.0 0.50 12.0 11.0 0.20 15.0 14.0 0.05 26.0 19.0
  6. 6. Solution 𝑅 = 𝑡=1 𝑛 𝑃 ∗ 𝑅 = R1(P1)+R2(P2)+R3(P3)+R4(P4)+R5(P5) = -2.0(0.05)+9.0(0.20)+12(0.50)+15(0.2)+26(0.05) =12%(for Alpha security) = -3.0(0.05)+6(0.2)+11(0.5)+ 14(0.2)+19(0.05) =10.3% (for Beta security)
  7. 7. continue ■ If the investor invests equally 𝑅𝑝 = Ʃ𝑤𝑥 ∗ 𝑅𝑥 + 𝑤𝑦 ∗𝑅𝑦 = 0.5*12+0.5*10.3 = 6+5.15 = 11.15 ■ If 75% is put into Alpha and 25% into Beta = 0.75*12+0.25*10.3 = 9+2.575 = 11.575 ■ If 25% goes to Alpha security and 75% into Beta = 0.25*12+0.75*10.3 = 3+7.725= 10.725
  8. 8. Risk of portfolio 𝜎𝑝 2 = 𝑤𝑥 2 𝜎𝑥 2 + 𝑤𝑦 2 𝜎𝑦 2 + 2𝑤𝑥𝑤𝑦𝑟𝑥𝛾𝜎𝑥𝜎𝑦 Where 𝜎𝑃= standard deviation of portfolio consisting securities x and y 𝑤𝑥𝑤𝑦=Proportion of funds in securities x and y 𝜎𝑥𝜎𝑦= Standard deviation of returns of security x and security y 𝑟𝑥𝛾=Co-efficient of correlation between security x and security y
  9. 9. Co-efficient of correlation ■ The co-efficient of correlation indicates the similarity and dissimilarity in the Behaviour of two securities. The co-efficient can vary from (+1) to (-1) rxy=1 signifies perfect positive correlation between securities, and they tend to move in same direction. rxy=-1 signifies perfect negative correlation between securities, and they tend to move in opposite direction. rxy=0 signifies no correlation between securities, and security returns are independent.
  10. 10. Calculation of coefficient of correlation rxy=Covariance of x and y / 𝜎𝑥𝜎𝑦 ■ In absence of probability Where Covxy= 𝑅𝑥−𝑅𝑥 𝑅𝑦−𝑅𝑦 𝑛 ■ In presence of probability Covxy = 𝑖=1 𝑛 𝑃 𝑅𝑥 − 𝑅𝑥 𝑅 − 𝑅𝑦
  11. 11. Conditions ■ In absence of probability expected return will be calculated by: 𝑅 =ƩR/n ■ Standard deviation of each stock is calculated by: 𝜎 = 𝑅−𝑅 2 𝑛
  12. 12. Question ■ The risk and return characteristics of equity share of two companies are shown below: ■ An investor plans to invest 80% of its available funds in X Ltd. and 20% in Y Ltd. The coefficient of correlation between the returns of the shares of two companies is +1.Find out the expected returns and variance of the portfolio of shares of both companies. Particulars X Ltd. Y ltd. Expected Return 12% 20% Standard Deviation 3% 7%
  13. 13. Solution ■ Expected return of portfolio 𝑅𝑝 = Ʃ𝑤𝑥 ∗ 𝑅𝑥 + 𝑤𝑦 ∗𝑅𝑦 = (0.8*12)+(0.2*20) = 9.6+ 4.0 = 13.6% ■ The variance of the portfolio 𝜎𝑝 2 = 𝑤𝑥 2𝜎𝑥 2 + 𝑤𝑦 2𝜎2 + 2𝑤𝑥𝑤𝑦𝑟𝑥𝛾𝜎𝑥𝜎𝑦 = (32*.82)+(0.22+72)+2*0.8*0.2*1*3*7 = 5.76+ 1.96+6.72 = 14.44
  14. 14. Continue ■ Standard deviation of the portfolio 𝜎𝑃 = 𝜎𝑝 2 = √14.44 = 3.8
  15. 15. Question ■ Stocks L and M have yielded the following returns for the past two years ■ What is the expected return on a portfolio made up of 60% of L and 40% of M? ■ Find out standard deviation of each stock. ■ What is the covariance and co-efficient of correlation between stocks L and M? ■ What is the portfolio risk of a portfolio made up of 60% of L and 40% of M? Years Return(L) Return(M) 2017 12% 14% 2018 18% 12%
  16. 16. Solution ■ Expected rate of return 𝑅 =ƩR/n ■ Expected rate of return of stock L= 12+18/2=15 ■ Expected rate of return of stock M= 14+12/2=13 ■ Portfolio Return 𝑅𝑝 = Ʃ𝑤𝑥 ∗ 𝑅𝑥 + 𝑤𝑦 ∗𝑅𝑦 = 0.6*15+ 0.4*13 = 9+ 5.2= 14.2
  17. 17. continue ■ Standard deviation of stock L 𝜎 = 𝑅 − 𝑅 2 𝑛 = 12−15 2− 18−15 2 2 = 3 ■ Standard deviation of stock M 𝜎 = 𝑅−𝑅 2 𝑛 = 14−13 2− 12−13 2 2 = 1
  18. 18. Continue ■ Covariance between stock L and M Covlm= 𝑅𝑥−𝑅𝑥 𝑅𝑦−𝑅𝑦 𝑛 = (-3) +(-3)/2 =-6/2 =-3 ■ Correlation coefficient rlm=Covariance of x and y / σxσy = -3/3*1 = -1
  19. 19. continue 𝜎𝑝 2 = 𝑤𝑥 2𝜎𝑥 2 + 𝑤𝑦 2𝜎𝑦 2 + 2𝑤𝑥𝑤𝑦𝑟𝑥𝛾𝜎𝑥𝜎𝑦 = (0.6)2*9+(0.4)2+2*0.6*0.4*-1*3*1 = 3.24+0.16+(-1.44) =1.96 𝜎 =√1.96 = 1.4
  20. 20. Question ■ A financial analyst is analyzing two investment alternatives, stock Z and stock Y. The estimated rates of return and their chances of occurrence for the next year are given below ■ Determine expected rates of return, variance, and standard deviation of Y and Z. ■ Is ‘Y’ comparatively riskless? ■ If the financial analyst wishes to invest half in Z and another half in Y, would it reduce the risk? Explain Probability of occurrence Security Y Rates of Return(%) Security Z Rates of Return(%) 0.20 22 5 0.60 14 15 0.20 -4 25
  21. 21. Solution ■ Expected rate of return of security Y 𝑅 = 𝑡=1 𝑛 𝑃 ∗ 𝑅 =0.2*22+ 0.6*14+0.2*(-4) = 4.4+8.4-0.8 =12 ■ Expected rate of return of security z 𝑅 = 𝑡=1 𝑛 𝑃 ∗ 𝑅 = 0.20*5+0.60*15+0.20*25 = 1+9+5 =15 ■ Variance and standard deviation of security Y 𝜎2 = 𝑅 − 𝑅 2𝑃 = (22-12)2*0.20+ (14-12)2*0.60+(-4-12)2*0.20 = 20+2.4+51.2 =73.6 =√73.6 𝜎 = 8.57
  22. 22. solution ■ Variance and standard deviation of security z 𝜎2 = 𝑅 − 𝑅 2𝑃 = (5-15)2*0.20+(15-15)2*0.60+(25-15)2*0.20 = 20+0+20 =40 =√40 𝜎 =6.32 ■ Since variance and standard deviation of security y is higher, it is riskier than security Z ryz= 𝐶𝑜𝑣 𝑦𝑧/𝜎y𝜎z 𝐶𝑜𝑣 𝑦𝑧 = 𝑖=1 𝑛 𝑃 𝑅𝑥 − 𝑅𝑥 𝑅 − 𝑅𝑦 = (22-12)(5-15)*0.2 + (14-12)(15-15)*0.60+ (-4-12)(25-15)*0.20 = -20+0-32 = -52 ryz= -52/6.3*8.6 = -52/54.18 = -0.95
  23. 23. Continue ■ Portfolio risk 𝜎𝑝 2 = 𝑤𝑥 2 𝜎𝑥 2 + 𝑤𝑦 2 𝜎𝑦 2 + 2𝑤𝑥𝑤𝑦𝑟𝑥𝛾𝜎𝑥𝜎𝑦 = (0.5)2*73.6+ (0.5)2*40+2*0.5*0.5*-0.95*8.6*6.32 = 18.4+10-25.81 =28.4-25.81 = √2.6 = 1.6 Combining Y and Z securities reduces the risk. This is because the securities have lower positive correlation coefficient.
  24. 24. Risk and Return of Portfolio(Multi Security)
  25. 25. Question ■ A portfolio consist of three securities with the following parameters: ■ If these securities are equally weighted, how much is the risk and return of the portfolio of these securities? Particulars Security (P) Security (Q) Security (R) Correlation Coefficient Expected Return(%) 25 22 20 Standard deviation(%) 30 26 24 Correlation Coefficient PQ -0.5 QR +0.4 PR +0.6
  26. 26. Solution ■ Expected portfolio return = (1/3*25)+(1/3*22)+(1/3*20) = 8.3+7.3+6.6 =22.2 ■ Risk of the portfolio = (1/3)2 (30)2+(1/3)2 (26)2 +(1/3)2 (24)2 +2(1/3)(1/3)(-0.5)(30)(26)+2(1/3) (1/3)(0.4)(26)(24)+2(1/3)(1/3)(0.6)(30)(24) = 100+75.11+64-86.67+55.47+96 = √303.91 = 17.43

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