Chapter 06 - Risk & Return

8/7/2019 Chapter 06 - Risk & Return

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Risk & Return

Companies Per cent Return During year

1 2 3 4 5 6

WIPRO 37 24 -7 6 18 32

I-GATE 32 29 -12 1 15 30

Solutions:

To find the Covariance term :

Illustration1. You are evaluating an investment in two companies whose past ten years of returns are

(a) Calculate the standard deviation of each companys returns.(b) Calculate the correlation coefficient of the companies returns.(c) If you had placed 50% of your money in each, what would have been the standard dyearly return?

(d) What percentage investment in each would have resulted in the lowest risk?(e) Assume that a yearly risk-free return of 8% was available and that you had held onlyhave been the better to own?(f) Graph the risk and return of each fund. Given your answer to part (d), what was the(g) Use part (f) to determine:- How an average return of 10.8% would have been obtained.- How an average return of 17.8% would have been obtained.

(a) Find the average returns :

RWIPRO = (37 + 24 ++ 6)/10 = 15%RI-GATE = (32 + 29 ++ 10)/10 = 15%

Next, find theSD :

WIPRO (37 15)2 + (24 15)2 ++(6 15)2 /10

I-GATE = (37 15)2 + (29 15)2 ++(10 15)2/ 10

(b)


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Cov. = (37 15) (32 15) + (24 15) (29 15) +.+ (6 15) (10 15)/ 10

E(Rc) = R + c {E (Rp) - T}/ p

where c = (1 WT) p

8.0% + c {15.0 % - 8.0%}

The Correlation coefficient. 187.4 . = 0.94

(14.0) (14.3)

(c) p = (0.5)2(14.0)2 + (0.5)2(14.3)2 + 2(0.5)(0.5)(14.0)(14.3)(0.94) = 13.9%E(RP) = 0.5(15.0) + 0.5(15.0) = 15%

(d) Using the minimum variance equation and let W stand for WIPRO :WFST = 22 1 2 1,2 .

21 + 22 - 1 2 1,2WFST = . 14.32 (14.0)(14.3)(0.94) .

14.02 + 14. 32 2 (14.0)(14.3)(0.94) = 6706%

WI-GATE = 32.4%

(e) This part asks which of the funds provided the greater return per unit of risk.The risk-slope of the line :

15.0 8.0 = 0.5% per unit of 14.0

For I-GATE = 15.0 8.0 = 0.49% per unit of 14.3

They were very close, but WIPRO was better.

(f) Both funds have identical average returns. The minimum variance portfolio of W = 6would also have had a 15 % average return, but its risk would be lower than holding eitisolation. The minimum standard deviation was 13.9%

(g) Using 8% as the risk-free rate and the single efficient portfolio in part (f) as the optifollowing risk/return relationship was available :

= 8.0% + c[0.5036]


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Illustration 2

Groups

ABC 20% 1 15.00%

KLM 20% -1 15.00%

RST 20% 0 15.00%

(c) What should he do ? Indicate precise portfolio weighting.

Solutions:

(b)Return would remain at 20% but risk would fall to zero since r + -1.0

(c)Invest 50/50 in Group XYZ portfolio and group KLM portfolio.

Illustration 3

Consider the two stocks Wipro and TCS with a standard deviation 0.05 and 0.10 respectively.

The correlation coefficient for these two stocks is 0.8

To earn 10.8%, invest 60% risk-free and 40% it the optimal risky portfolio :10.8 % = 8% + (0.4)(13.9%)(0.5.36)

To earn 17.8%, borrow 40% on your equity and invest it with your equity in the optimalrisky portfolio :17.8% = 8.0% + W (13.9%)(0.5036)= 8.0% + (1.4)(13.9)(0.5036)

Mr. KK holds a well-diversified portfolio of stocks in XYZ Group. During the last 5 years returns onyear and had a standard deviation of 15.0%/ He is satisfied with the yearly availability of his portfoliwithout affecting overall returns. He approaches you for help in finding an appropriate diversificatioalternatives, you conclude : (i) future average returns and volatility of returns on his current portfolioexpected, (ii) to provide a quarter degree of diversification in his portfolio, investment could be made

ExpectedReturn

Correlationof Returnswith Group

XYZ

StandardDeviation

(a) If Bhatt invest 50% of his funds in ABC Group and leaves the remainder inXYZ Group, how would this affect both his expected return and his risk? Why?

(b) If Bhatt invests 50% of his funds in KLM Group and leaves the remainder inXYZ Group, how would this affect both his expected return and his risk? Why ?

(a) Risk and return of ABC Portfolio are the same as those of XYZ portfolio and the correlation coeff

so there is no diversification gain.


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(a) What is the diversification gain from forming a portfolio that has equal proportions of each s

(b) What should be the weights of the two assets in a portfolio that achieves a diversification gain of

Solutions(a) The gain from diversification is :

0.075 0.0716

0.75 = 4.53%

(b) To obtain a diversification gain of 3%, the weighting of the portfolio should be 30% to 70%.

Illustration 4

Security RNL Security Reliance

Probability Return Probability Return

0.3 19% 0.2 22%

0.4 15% 0.3 6%

0.3 11% 0.3 14%

0.2 -5%

Ans :

Investment in securityRNL Investment in security reliance

Probability Return Probability Return Expected Return

0.2 22% 4.4

0.3 19% 5.70% 4.80% 0.3 6 1.8

0.4 15% 6 1 0.3 14 4.20.3 11% 3.3 4.8 0.2 5 -1

E(R) = 15.0%

S = 3.09

Illustration 5

DVYD considering an investment in one of two securities. Given the information that follows, whicbased on risk (as measured by the standard deviation) and return ?

ExpectedReturn

WeightedDeviation

E(R) =15.0%

2= 9.6%


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Return

Year A B C

2005 0.14 0.18 0.14

2006 0.16 0.16 0.16

2007 0.18 0.14 0.18

(a) What is the expected return on each of these securities over the three-year period?

(b) What is the standard deviation on each securitys return?

(c) What is the expected return on each portfolio ?

(d) For each portfolio, how would you characterize the correlation between the returns on its two asse

(e) What is the standard deviation of each portfolio ?

(f) Which portfolio do you recommend ? Why?

Solutions:

Illustration: 6

Rs 10 11 12

Fund company have been asked by a client for advice in selecting a portfolio of assets based on the fdata :

You have been asked to create portfolios by investing equal proportions (i.e., 50%) in each of two difsecurities. No probabilities have been supplied.

(a) E(RA) = E(R

B)= (R

C)=.16

(b) A

= (.00027).5 = 0.0164

B

= (.00027).5 = 0.0164

C

= (.00027).5 = 0.0164

(c) E(RAB

) = E(RAC

)= (RCBC

)=.16

(d) A and B are perfectly negatively correlated. A and C are perfectly positively correlated. B and

perfectly negatively correlated.(e)

AB= 0;

AB= 0.0164

Since A and C are identical, 2AB

= 0;

AB

= 0

(f) Choose either AB or BC. All three portfolios have E(Rp) = .16, but AB and BC have no risk, wh

Ac

= .0164. Therefore, AB BC provide the most reward for the least amount of risk.

You are considering purchasing the equity stock of B Company. The current price per share is Rs 10.the dividend a year hence to be Re1.00. You expect the price per share of B stock a year hence to havfollowing probability distribution.

Price a yearhence


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Probability 0.4 0.4 0.2

Solution:

(a) Expected price per share a year hence will be:

= 0.4 x Rs.10 + 0.4 x Rs.11 + 0.2 x Rs.12 = Rs.10.80

Probability distribution of the rate of return is

10% 20% 30%

0.4 0.4 0.2

Note that the rate of return is defined as:

(c )

The of the rate of return on Bs stock is calculated below:

---------------------------------------------------------------------------------------------------

---------------------------------------------------------------------------------------------------

10 0.4 4 -8

20 0.4 8 230 0.2 6 12

---------------------------------------------------------------------------------------------------

= 56 = 7.48%

Illustration: 7

Economics Condition

Stagnation Recession

Probability 0.3 0.3 0.2 0.2

a) What is the expected price per share a year hence?

b) What is the probability distribution of the rate of return on Bs equity stock?

(b)

Rate of retur

Probability (

Dividend + Terminal price-------------------------------- - 1

Initial price

The standard deviation of rate of return is : =pi(R

iR)2

Ri

pi

pIr

i(R

i-R)

R = piR

i

The stock of X Company performs well relative to other stocks during recessionary periods. The stoc

during growth periods. Both the stocks are currently selling for Rs 50 per share. The rupee returns (diwould be as follows:

HighGrowth

LowGrowth


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55 50 60 70

75 65 50 40

Calculate the expected return and standards deviation of :

Which of the above four options would you choose? Why?

Solution :

2 (a) For Rs.1,000, 20 shares of Wipros stock can be acquired. The probability

Economic Condition Probability

High Growth 20 x 55 = 1,10.3

Low Growth 20 x 50 = 1, 0.3

Stagnation 20 x 60 = 1, 0.2

Recession 20 x 70 = 1, 0.2

Expected ret = (1,100 x 0.3) + (1,000 x 0.3) + (1,200 x 0.2) + (1,400 x 0.2

= 330 + 300 + 240 + 280

= Rs.1,150

= Rs.143.18

Economic condition Return (Rs) Probability

High growth 20 x 75 = 1, 0.3

Low growth 20 x 65 = 1, 0.3

Stagnation 20 x 50 = 1, 0.2

Recession 20 x 40 = 0.2

Return onWipro stock

Return onInfosys stock

a) Rs 1,000 in the equity stock of Wipro

b) Rs 1,000 in the equity stocks of Infosys

c) Rs 500 in the equity stock of Wipro and Rs.500in the equity stock of Infosys

d) Rs 700 in the equity stock of Wipro and Rs 300 in the equity of Infosys

Return (Rs)

Standard deviation of the return = [(1,100 1,150)2 x 0.3 + (1,000 1,150)2 x

0.3 + (1,200 1,150)2 x 0.2 + (1,400 1,150)2 x 0.2]1/2

(b) For Rs.1,000, 20 shares of Infosyss stock can be acquired. The probability distribution of the


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Expected return = (1,500 x 0.3) + (1,300 x 0.3) + (1,000 x 0.2) + (800 x 0.2)

= Rs.1,200

(c )

Return (Rs) Probabili

(10 x 55) + ( = 1,300 0.3

(10 x 50) + ( = 1,150 0.3

(10 x 60) + ( = 1,100 0.2

(10 x 70) + ( = 1,100 0.2

Expected ret = (1,300 x 0.3) + (1,150 x 0.3) + (1,100 x 0.2) + (1,100 x 0.2= Rs.1,175

Standard dev=

= Rs.84.41

d.

Return (Rs) Probability

(14 x 55) + ( = 1,220 0.3

(14 x 50) + ( = 1,090 0.3

(14 x 60) + ( = 1,140 0.2

(14 x 70) + ( = 1,220 0.2

Expected ret (1,220 x 0.3) + (1,090 x 0.3) + (1,140 x 0.2) + (1,220 x 0.2)

Standard dev== Rs.57.66

The expected return to standard deviation of various options are as follows :

Standard deviation of the return = [(1,500 1,200)2 x .3 + (1,300 1,200)2 x .3

+ (1,000 1,200)2 x .2 + (800 1,200)2 x .2]1/2 = Rs.264.58

For Rs.500, 10 shares of Wipros stock can be acquired; likewise for Rs.500, 10 sharbe acquired. The probability distribution of this option is:

[(1,300 1,175)2 x 0.3 + (1,150 1,175)2 x 0.3 + (1,100 11/2

For Rs.700, 14 shares of Wipros stock can be acquired; likewise for Rs.300, 6 shareprobability distribution of this option is:

[(1,220 1,165)2 x 0.3 + (1,090 1,165)2 x 0.3 + (1,140 1

1,165)2 x 0.2]1/2


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Option

A 1,150 143 8.04

B 1,200 265 4.53

C 1,175 84 13.99

D 1,165 58 20.09

Option `d is the most preferred option because it has the highest return to risk ratio.

Illustration 8

The return on four stocks X,Y,Z and A over a period of 6 years have been as follows :

1 2 3 4 5 6

X 10% 12% -8% 15% -2% 20%

Y 8% 4% 15% 12% 10% 6%

Z 7% 8% 12% 9% 6% 12%

A 9% 9% 11% 4% 8% 16%

Calculate the returns on :

Assume equiv. proportional investment.

Solution :

Expected rates of returns on equity stock X, Y, Z and A can be computed as follows:

X:

6

Y: 0.08 + 0.04 + 0.15 +.12 + 0.10 + 0.06 = 0.0917

6

Z: 0.07 + 0.08 + 0.12 + 0.09 + 0.06 + 0.12 = 0.0900

6

A: 0.09 + 0.09 + 0.11 + 0.04 + 0.08 + 0.16 = 0.095

Expectedreturn (Rs)

Standarddeviation

(Rs)

Expected /Standardreturndeviation

a) A portfolio of one stocks at a time

b) Portfolios of two stocks at a time

c) Portfolios of three stocks at a time

d) A portfolios of all four stocks

0.10 + 0.12 + (-0.08) + 0.15 + (-0.02) + 0.20 = 0.0783


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6

(a) Return on portfolio consisting of stock X

(b) Return on portfolio consisting of stock A and B in equal

proportions = 0.5 (0.0783) + 0.5 (0.0917)

= 0.09 =

(c ) Return on portfolio consisting of stocks X, Y and Z in equal

proportions = 1/3(0.0783 ) + 1/3(0.0917) + 1/3 (0.090)

= 0.09 =

(d) Return on portfolio consisting of stocks X, Y, Z and A in equal

proportions = 0.25(0.0783) + 0.25(0.0917) + 0.25(0.0900) +

0.08875 = =

Illustration 8:

The returns on the equity stocks of TCS limited and the market portfolios over a 12 year period are gi

Year

1 15 122 -6 1

3 18 14

4 30 24

5 12 16

6 25 30

7 2 -3

8 20 24

9 18 15

10 24 22

11 8 12

Solution

Return onauto TCSLtd. (%)

Return onmarketportfolio(%)

a) Calculate the beta for the stock of TCS limited

b) Established the characteristics line for the stock of TCS limited.

DefineRA

andRM

as the returns on the equity stock of ACE Limited a and Market portfolio respective

calculating the beta of the stock are shown below:


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Year

1 15 12 -0.09 -3.18 0.01 10.11

2 -6 1 -21.09 -14.18 444.79 201.073 18 14 2.91 -1.18 8.47 1.39

4 30 24 14.91 8.82 222.31 77.79

5 12 16 0-3.09 0.82 9.55 0.67

6 25 30 9.91 14.82 98.21 219.63

7 2 -3 -13.09 -18.18 171.35 330.51

8 20 24 4.91 8.82 24.11 77.79

9 18 15 2.91 -0.18 8.47 0.03

10 24 22 8.91 6.82 79.39 46.51

11 8 12 -7.09 -3.18 50.27 10.11

Alpha =

= 15.09 (0.96 x 15.18) =

Equation of the characteristic line is

RA

RM

RA-R

AR

M-R

M(R

A-R

A) (R

M-R

M)

RA = 15.09 RM = 15.18

(RA RA)2 = 1116.93 (R M RM) 2 = 975.61 (R A RA) (RM RM)

Beta of the equity stock of TCS Limited

(RA RA) (RM RM)

(RM RM) 2

= 935.86 = 0.96975.61

RA

AR

M

RA

= 0.52 + 0.96RM


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7 8 9 10

-5 21 18 6

0 18 27 10

= 14.0%

14.30%

shown below:

viation of your portfolio and the average

one of the two companies. Which would

single efficient portfolio of the two?


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187.40%

7.6 % and W = 32.4%er company in

al risky portfolio, the


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these stocks have average 20.0% pero and would like to reduce its risk

medium. After a lengthy review ofwill be the same as he has historicallyin stocks of the following groups :

icient is 1.0,


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ock ?

%?

31.75%

3.47

6.3541.47

=83.04

S=9.113

h investment is better,

WeightedDeviation


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ts?

ollowing

ferent

C are

ile AC has

You expecte the


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64 25.6

4 1.6144 28.8

56

(Ri- R)2 p

i(R

i-R)2

p

i(R

i-R)2

s of Y Company, on the other hand, do well

vidend plus price change) of these for the next year


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return on 20 shares is:


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ty

= Rs.1,165

es of Infosyss stock can

,175)2 x 0.2 + (1,100 1,175)2 x 0.2 ]

s of Infosyss stock can be acquired. The

,165)2 x 0.2 + (1,220


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7.83%

9.17%

9.00%

9.50%


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7.83%

8.50%

8.67%

0.25(0.095)

8.88%

ven below :

ly. The calculations relevant for


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0.29

299.06-3.43

131.51

-2.53

146.87

237.98

43.31

-0.52

60.77

22.55

0.52

RA-R

A/R

M-R

M

935.86


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Documents

Chapter 06 - Risk & Return