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1. RISK AND RETURN
OF PORTFOLIO
Dr. Deergha Sharma

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Continue
■ Standard deviation of the portfolio
𝜎𝑃 = 𝜎𝑝
2
= √14.44
= 3.8

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. 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. 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. 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. 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. 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. 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. 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. 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. Risk and Return of Portfolio(Multi Security)

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. 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