Return and Risk -Two Assets

:- Ved Prakash panda

• The expected return from a portfolio of two or more securities is equal to the weighted average of the expected returns from the individual securities.

• Rp = WA (RA) + WB (RB) • Where, • Rp = Expected return from a portfolio of two securities • WA = Proportion of funds invested in Security A• WB = Proportion of funds invested in Security B• RA = Expected return of Security A• RB = Expected return of Security B • WA+ WB = 1

Example• Mr. RKV’s portfolio consists of six securities. The individual

returns of each of the security in the portfolio are given below.• Security Proportion(%) Return(%)• Wipro 10 18• RIL 25 12• SBI 8 22• ITC 30 15• BSNL 12 6• DLF 15 8• Calculate the weighted average of return of the securities

consisting the portfolio.

Solution• Security Weight (W) Return(%) (R) (WxR)• Wipro 0.10 18 1.80• RIL 0. 25 12 3.00• SBI 0.0 8 22 1.76• ITC 0.30 15 4.50• BSNL 0. 12 6 0.72• DLF 0. 15 8 1.20• 12.98• Portfolio return is 12.98%

Risk of portfolio (two assets)

2 2 2 2p 1 1 2 2 1 2 12 1 2= X + X + 2X X (r )Ã Ã Ã Ã

p = portfolio standard deviationX1 = percentage of total portfolio value in stock X1

X2 = percentage of total portfolio value in stock X2

1 = standard deviation of stock X1

2 = standard deviation of stock X2

r12 = correlation co-efficient of X1 and X2

1212

1 2

covariance of Xr =

σ σ

The correlation co-efficient is -1.0 , which indicates that there is a perfect negative correlation and the returns move in the opposite direction.

If the correlation is 1, perfect positive correlation exists between the securities and they tend to move in the same direction.

If the correlation co-efficient is zero, the securities’ returns are independent.

Thus, the correlation between two securities depends upon the covariance between the two securities and the standard deviation of each security.

CASE• The returns of Security of Wipro and Security of Infosys for

the past six years are given below• Year Wipro return(%) Infosys return(%)• 2003 9 10• 2004 5 -6• 2005 3 12• 2006 12 9• 2007 16 15

Calculation of mean return and standard deviation of security Wipro

• Year return(%) (R- R¯) (R- R¯) ²• 2003 10 2 4 • 2004 -6 14 196 • 2005 12 4 16 • 2006 9 1 1 • 2007 15 7 49• 40 266• Mean return= 40/5= 8• Standard deviation = √ 266 = 16.31

Calculation of mean return and standard deviation of security Infosys

• Year return(%) (R- R¯) (R- R¯) ²• 2003 9 0 0 • 2004 5 -4 16 • 2005 3 -6 36 • 2006 12 3 9 • 2007 16 7 49• 45 110• Mean return= 45/5= 9• Standard deviation = √ 110 = 10.49

Correlation coefficient• R12 = N ∑XY – (∑X) (∑Y)• √ N ∑X² - (∑X)² √ N ∑Y² - (∑Y)²

Correlation coefficient• A’s return % B’s return%• X X² Y Y² XY• 9 81 10 100 90• 5 25 -6 36 -30• 3 9 12 144 36• 12 144 9 81 108• 16 256 15 225 240• 45 515 40 586 444• • R12 = N ∑XY – (∑X) (∑Y)• √ N ∑X² - (∑X)² √ N ∑Y² - (∑Y)²• R12 = (5x444) – (45x40)• √ (5x515) - (45)² √ 5x586 - (40)²• = 2,220 – 1800 • √ 2575-2025 √ 2930 - 1600• = 420 • √ 550 √ 1330 • = 420 • 23.452 x 36.469• = 420 • 855.271• = 0.491

Verification: Calculation of covariance of Returns of securities A and B

• Year Wipro return(%) Infosys return(%) (RA- R¯) (RB- R¯) (RA- R¯) x(RB- R¯) • 2003 9 10 0 2 0• 2004 5 -6 -4 -14 56• 2005 3 12 -6 4 -24• 2006 12 9 3 1 3• 2007 16 15 7 7 49• Cov AB = 84

• Cor.AB = Cov. AB = 84/ ( 10.49 x 16.31) = 0.491• s.d Ax s.d B Cov. AB = Cor. AB x s.d A x s.d B = 0.491x 10.49x 16.31 = 84

Return of portfolio

• Wipro=80% , Infosys= 20%• Rp= (0.80x9)+ (0.20x8) = 7.2+ 1.6 = 8.8%

Risk of portfolio

• S.Dp = √ {(0.80) x (10.49)} + {(0.20)x (16.31)} + ( 2x 0.80x 0.20x 0.491x 10.49x 16.31)

• = √ (0.64x 110.04) + (0.04x266.02)+ 26.88• = √ 70.43 + 10.64 + 26.88• = √ 107.95• = 10.39