RISK AND RETURN

1. A stock earns the following returns over a five year period: R1 = 0.30, R2 = -0.20, R3 = -
0.12, R4 = 0.38, R5 = 0.42, R6 = 0.36. Calculate the following: (a) arithmetic mean return,
(b) cumulative wealth index, and (c) geometric mean return.

Solution:

R1 = 0.30, R2 = - 0.20, R3 = - 0.12, R4 = 0.38, R5 = 0.42, R6 = 0.36

(a) Arithmetic mean

0.30 – 0.20 - 0.12 + 0.38 + 0.42+0.36

= = 0.19 or 19 %
6

(b) Cumulative wealth index

CWI5 = 1(1.30) (0.80) (0.88) (1.38) (1.42) (1.36) = 2.439

(c) Geometric Mean

= [(1.30) (0.80) (0.88) (1.38) (1.42) (1.36)]1/6 – 1 = 0.1602 or 16.02 %

2. A stock earns the following returns over a five year period: R1 = 10 %, R2 = 16%, R3 = 24
%, R4 = - 2 %, R5 = 12 %, R6 = 15%. Calculate the following: (a) arithmetic mean return, (b)
cumulative wealth index, and (c) geometric mean return.

Solution:

R1 = 10 %, R2 = 16%, R3 = 24 %, R4 = - 2 %, R5 = 12 %, R6 = 15 %

(a) Arithmetic mean

10 + 16 + 24 - 2 + 12 + 15
= = 12.5 %
6

(b) Cumulative wealth index

CWI5 = 1(1.10) (1.16) (1.24) (0.98) (1.12) (1.15) = 1.997

(c) Geometric Mean

= [(1.10) (1.16) (1.24) (0.98) (1.12) (1.15)]1/6 – 1 = 0.1222 or 12.22 %

3. What is the expected return and standard deviation of returns for the stock described in 1?
Solution:
 The expected return and standard deviation of returns is calculated below

Return in % Deviation Square of deviation

Period
Ri (Ri-R) (Ri-R)2
1 30 11 121
2 -20 -39 1521
3 -12 -31 961
4 38 19 361
5 42 23 529
6 36 17 289
19 SUM= 3782
R=

Expected return = 19 %
Σ (Ri – R)2 3782
Variance = = = 756.4
n–1 6–1

Standard deviation = (756.4)1/2 = 27.50

4. What is the expected return and standard deviation of returns for the stock described in 2?

Solution:
The expected return and standard deviation of returns is calculated below.

Return in % Deviation Square of deviation

Period
Ri (Ri-R) (Ri-R)2
1 10 -2.5 6.25
2 16 3.5 12.25
3 24 11.5 132.25
4 -2 -14.5 210.25
5 12 -0.5 0.25
6 15 2.5 6.25
12.5 SUM= 367.5
R=

Expected return = 12.5 %

Σ (Ri – R)2 367.5
Variance = = = 73.5
n–1 6–1

Standard deviation = (73.5)1/2 = 8.57

5. The probability distribution of the rate of return on a stock is given below:

State of the Economy Probability of Occurrence Rate of Return

Boom 0.20 30 %
Normal 0.50 18 %
Recession 0.30 9%

What is the standard deviation of return?

Solution:

Probabilit Deviation
State of Return in
y of (Ri-R) Pi x (Ri – R)2
the % pi x Ri
occurrence
economy Ri
pi
Boom 0.2 30 6 12.3 30.26
Normal 0.5 18 9 0.3 0.05
Recession 0.3 9 2.7 -8.7 22.71

Expected return R = 17.7 SUM= 53.01

Standard deviation = [53.01]1/2 = 7.28

6. The probability distribution of the rate of return on a stock is given below:

State of the Economy Probability of Occurrence Rate of Return

Boom 0.60 45 %
Normal 0.20 16 %
Recession 0.20 - 20%

What is the standard deviation of return?

Solution:

Probabilit
State of Return in
y of Deviation Pi x (Ri – R)2
the % pi x Ri
occurrence (Ri-R)
economy Ri
pi
Boom 0.6 45 27 18.8 212.06
Normal 0.2 16 3.2 -10.2 20.81
Recession 0.2 -20 -4 -46.2 426.89
Expected return R = 26.2 SUM= 659.76
Standard deviation = [659.76]1/2 = 25.69
7. The returns of two assets under four possible states of nature are given below:

State of nature Probability Return on asset 1 Return on asset 2

1 0.40 -6% 12%
2 0.10 18% 14%
3 0.20 20% 16%
4 0.30 25% 20%

a. What is the standard deviation of the return on asset 1 and on asset 2?

b. What is the covariance between the returns on assets 1 and 2?
c. What is the coefficient of correlation between the returns on assets 1 and 2?

Solution:

(a)
E (R1) =0.4(-6%) + 0.1(18%) + 0.2(20%) + 0.3(25%)
=10.9 %
E (R2) =0.4(12%) + 0.1(14%) + 0.2(16%) + 0.3(20%)
=15.4 %
σ(R1) = [.4(-6 –10.9)2 + 0.1 (18 –10.9)2 + 0.2 (20 –10.9)2 + 0.3 (25 –10.9)2]½
= 13.98%
σ(R2) = [.4(12 –15.4)2 + 0.1(14 –15.4)2 + 0.2 (16 – 15.4)2 + 0.3 (20 –15.4)2] ½
= 3.35 %

(b) The covariance between the returns on assets 1 and 2 is calculated below

State of Probability Return on Deviation Return on Deviation Product of

nature asset 1 of return asset 2 of the deviation
on asset 1 return on times
from its asset 2 probability
mean from its
mean
(1) (2) (3) (4) (5) (6) (2)x(4)x(6)
1 0.4 -6% -16.9% 12% -3.4% 22.98
2 0.1 18% 7.1% 14% -1.4% -0.99
3 0.2 20% 9.1% 16% 0.6% 1.09
4 0.3 25% 14.1% 20% 4.6% 19.45
Sum = 42.53

Thus the covariance between the returns of the two assets is 42.53.
 (c) The coefficient of correlation between the returns on assets 1 and 2 is:
Covariance12 42.53
= = 0.91
σ1 x σ2 13.98 x 3.35

8. The returns of 4 stocks, A, B, C, and D over a period of 5 years have been as follows:

1 2 3 4 5
A 8% 10% -6% -1% 9%
B 10% 6% -9% 4% 11%
C 9% 6% 3% 5% 8%
D 10% 8% 13% 7% 12%

Calculate the return on:

a. portfolio of one stock at a time

b. portfolios of two stocks at a time
c. portfolios of three stocks at a time.
d. a portfolio of all the four stocks.

Assume equiproportional investment.

Solution:

Expected rates of returns on equity stock A, B, C and D can be computed as follows:

A: 8 + 10 – 6 -1+ 9 = 4%
5

B: 10+ 6- 9+4 + 11 = 4.4%

5

C: 9 + 6 + 3 + 5+ 8 = 6.2%
5

D: 10 + 8 + 13 + 7 + 12 = 10.0%
5

(a) Return on portfolio consisting of stock A = 4%

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

proportions = 0.5 (4) + 0.5 (4.4)
= 4.2%

(c) Return on portfolio consisting of stocks A, B and C in equal

proportions = 1/3(4 ) + 1/3(4.4) + 1/3 (6.2)
 = 4.87%

(d) Return on portfolio consisting of stocks A, B, C and D in equal

proportions = 0.25(4) + 0.25(4.4) + 0.25(6.2) +0.25(10)
= 6.15%

9. A portfolio consists of 4 securities, 1, 2, 3, and 4. The proportions of these securities are:

w1=0.3, w2=0.2, w3=0.2, and w4=0.3. The standard deviations of returns on these securities
(in percentage terms) are: σ1=5, σ2=6, σ3=12, and σ4=8. The correlation coefficients among
security returns are: ρ12=0.2, ρ13=0.6, ρ14=0.3, ρ23=0.4, ρ24=0.6, and ρ34=0.5. What is the
standard deviation of portfolio return?

Solution:

The standard deviation of portfolio return is:

= [0.32 x 52 + 0.22 x 62 + 0.22 x 122 + 0.32 x 82 + 2 x 0.3 x 0.2 x 0.2 x 5 x 6

+ 2 x 0.3 x 0.2 x 0.6 x 5 x 12 + 2 x 0.3 x 0.3 x 0.3 x 5 x 8
+ 2 x 0.2 x 0.2 x 0.4 x 6 x 12 + 2 x 0.2 x 0.3 x 0.6 x 6 x 8
+ 2 x 0.2 x 0.3 x 0.5 x 12 x 8]1/2

= 5.82 %

10. The following information is available.

Stock A Stock B
Expected return 24% 35%
Standard deviation 12% 18%
Coefficient of correlation 0.60

a. What is the covariance between stocks A and B ?

b. What is the expected return and risk of a portfolio in which A and B are equally
weighted?

Solution:

(a) Covariance (A,B) = PAB x σA x σB

= 0.6 x 12 x 18 = 129.6

(b) Expected return = 0.5 x 24 + 0.5 x 35 = 29.5 %

Risk (standard deviation) = [w2A 2A + w2B 2B + 2xwA wB Cov (A,B)]½
= [0.52 x 144 + 0.52 x 324 + 2 x 0.5 x 0.5x 129.6] ½
= 13.48 %
11. The following information is available.

Stock A Stock B
Expected return 12% 26 %
Standard deviation 15% 21 %
Coefficient of correlation 0.30

a. What is the covariance between stocks A and B?

b. What is the expected return and risk of a portfolio in which A and B are weighted 3:7?

Solution:

(a) Covariance (A,B) = PAB x σA x σB

= 0.3 x 15 x 21 = 94.5

(b) Expected return = 0.3 x 12 + 0.7 x 26 = 21.8 %

Risk (standard deviation) =[w2A 2A + w2B 2B + 2xwA wB Cov (A,B)]½

= [0.32x225+0.72 x441+2x0.3x0.7x94.5] ½

= 16.61 %

12. The following table gives the rate of return on stock of Apple Computers and on the market
portfolio for five years

Year Return on the stock Return

Apple Computers (%) Market Portfolio (%)
1 -13 -3
2 5 2
3 15 8
4 27 12
5 10 7

(i) What is the beta of the stock of Apple Computers?

(ii) Establish the characteristic line for the stock of Apple Computers.
Solution:

Year RA RM RA - RA RM - RM (RA - RA) (RM - RM) (RM - RM)2

1 -13 -3 -21.8 -8.2 178.76 67.24
2 5 2 -3.8 -3.2 12.16 10.24
3 15 8 6.2 2.8 17.36 7.84
4 27 12 18.2 6.8 123.76 46.24
5 10 7 1.2 1.8 2.16 3.24

Sum 44 26 334.2 134.8

Mean 8.8 5.2

134.8 334.2
 M
2
= = 33.7 Cov A,M = = 83.55
5-1 5-1

83.55
A = = 2.48
33.7

(ii) Alpha = R A – βA R M

= 8.8 – (2.48 x 5.2) = - 4.1

Equation of the characteristic line is

RA = - 4.1 + 2.48 RM
13. The rate of return on the stock of Sigma Technologies and on the market portfolio for 6
periods has been as follows:

Period Return on the stock Return on the

of Sigma Technologies (%) market portfolio (%)

1 16 14
2 12 10
3 -9 6
4 32 18
5 15 12
6 18 15

(i) What is the beta of the stock of Sigma Technologies.?

(ii) Establish the characteristic line for the stock of Sigma Technologies

Solution:
(i

Year RA (%) RM (%) RA-RA RM-RM (RA-RA) (RM-RM)2

x(RM-RM)

1 36 28 8.8 2.4 21.12 5.76

2 24 20 -3.2 -5.6 17.92 31.36
3 -20 -8 -47.2 -33.6 1585.92 1128.96
4 46 52 18.8 26.4 496.32 696.96
5 50 36 22.8 10.4 237.12 108.16

 RA = 136 RM = 128 2358.4

Cov A,M =
RA = 27.2 RM = 25.6 5-1

 M2 = 1971.2
5–1

(ii) Alpha = 2358.4

RA – βA/ (5-1)
RM
A = -------------------
= = x 25.6)
27.2 – (1.196 1.196 = -3.42
Equation of the1971.2 / (5-1) line is
characteristic

RA = - 3.42 + 1.196 RM

14. The rate of return on the stock of Omega Electronics and on the market portfolio for 6
periods has been as follows:

Period Return on the stock Return on the

of Omega Electronics market portfolio
(%) (%)

1 18% 15%
2 10% 12%
3 -5% 5%
4 20% 14%
5 9% -2%
6 18% 16%

(i)What is the beta of the stock of Omega Electronics?

(ii) Establish the characteristic line for the stock of Omega Electronics.
Solution:

RM (%)
Period R0 (%) (R0 – R0) (RM – RM) (R0 –R0) (RM – RM) (RM - RM)2
1 18 15 6.33 5 31.65 25
2 10 12 -1.67 2 - 3.34 4
3 -5 5 -16.67 -5 83.35 25
4 20 14 8.33 4 33.32 16
5 9 -2 - 2.67 -12 32.04 144
6 18 16 6.33 6 37.98 36
R0 = 70 RM = 60 (R0-R0) (RM-RM) = 215 250
R0 =11.67 RM = 10
250 215
 =
M
2
= 50 CovO,M = = 43.0
5 5
43.0
0 = = 0.86
50.0

(ii) Alpha = RO – βA RM
= 11.67 – (0.86 x 10) = 3.07
Equation of the characteristic line is
RA = 3.07 + 0.86 RM

15. The risk-free return is 8 percent and the return on market portfolio is 16 percent. Stock X's
beta is 1.2; its dividends and earnings are expected to grow at the constant rate of 10
percent. If the previous dividend per share of stock X was Rs.3.00, what should be the
intrinsic value per share of stock X?

Solution:

The required rate of return on stock A is:

RX = RF + βX (RM – RF)
= 0.08 + 1.2 (0.16 – 0.08)
= 0. 176

Intrinsic value of share = D1 / (r- g) = Do (1+g) / ( r – g)

Given Do = Rs.3.00, g = 0.10, r = 0.176

3.00 (1.10)
 Intrinsic value per share of stock X =
0.176 – 0.10

= Rs. 43.42

16. The risk-free return is 7 percent and the return on market portfolio is 13 percent. Stock P's
beta is 0.8; its dividends and earnings are expected to grow at the constant rate of 5 percent.
If the previous dividend per share of stock P was Rs.1.00, what should be the intrinsic
value per share of stock P?

Solution:

The required rate of return on stock P is:

RP = RF + βP (RM – RF)
= 0.07 + 0.8 (0.13 – 0.07)
= 0. 118

Intrinsic value of share = D1 / (r- g) = Do (1+g) / ( r – g)

Given Do = Rs.1.00, g = 0.05, r = 0.118

1.00 (1.05)
Intrinsic value per share of stock P =
0.118 – 0.05

= Rs. 15.44
17. The risk-free return is 6 percent and the expected return on a market portfolio is 15 percent.
If the required return on a stock is 18 percent, what is its beta?

Solution:

The SML equation is RA = RF + βA (RM – RF)

Given RA = 18%. RF = 6%, RM = 15%, we have

0.18 = .06 + βA (0.15 – 0.06)

0.12
i.e. βA = = 1.33
0.09

Beta of stock = 1.33

18. The risk-free return is 9 percent and the expected return on a market portfolio is 12 percent.
If the required return on a stock is 14 percent, what is its beta?

Solution:

The SML equation is RA = RF + βA (RM – RF)

Given RA = 14%. RF = 9%, RM = 12%, we have

0.14 = .09 + βA (0.12 – 0.09)

0.05
i.e.βA = = 1.67
0.03

Beta of stock = 1.67

19. The risk-free return is 5 percent. The required return on a stock whose beta is 1.1 is 18
percent. What is the expected return on the market portfolio?

Solution:

The SML equation is: RX = RF + βX (RM – RF)

We are given 0.18 = 0.05 + 1.1 (RM – 0.05) i.e., 1.1 RM = 0.185 or RM = 0.1681

Therefore return on market portfolio = 16.81 %

20. The risk-free return is 10 percent. The required return on a stock whose beta is 0.50 is 14
percent. What is the expected return on the market portfolio?

Solution:

The SML equation is: RX = RF + βX (RM – RF)

We are given 0.14 = 0.10 + 0.50 (RM – 0.10) i.e., 0.5 RM = 0.09 or RM = 0.18

Therefore return on market portfolio = 18 %

21. The required return on the market portfolio is 15 percent. The beta of stock A is 1.5. The
required return on the stock is 20 percent. The expected dividend growth on stock A is 6
percent. The price per share of stock A is Rs.86. What is the expected dividend per share
of stock A next year?

What will be the combined effect of the following on the price per share of stock?

(a) The inflation premium increases by 3 percent.

(b) The decrease in the degree of risk-aversion reduces the differential between the return
on market portfolio and the risk-free return by one-fourth.
(c) The expected growth rate of dividend on stock A decrease to 3 percent.
(d) The beta of stock A falls to1.2

Solution:

RM = 15% βA = 1.5 RA =20 % g = 6 % Po = Rs.86

Po = D1 / (r - g)

Rs.86 = D1 / (0.20 - .06)

So D1 = Rs.12.04 and Do = D1 / (1+g) = 12.04 /(1.06) = Rs.11.36

RA = Rf + βA (RM – Rf)

0.20 = Rf + 1.5 (0.15 – Rf)

0.5Rf = 0.025

So Rf = 0.05 or 5%.
Original Revised

Rf 5% 8%
RM – Rf 10% 7.5%
g 6% 3%
βA 1.5 1.2
 Revised RA = 8 % + 1.2 (7.5%) = 17 %

Price per share of stock A, given the above changes is

11.36 (1.03)
= Rs. 83.58
0.17 – 0.03

22. The required return on the market portfolio is 16 percent. The beta of stock A is 1.6. The
required return on the stock is 22 percent. The expected dividend growth on stock A is 12
percent. The price per share of stock A is Rs.260. What is the expected dividend per share
of stock A next year?

What will be the combined effect of the following on the price per share of stock?

(a) The inflation premium increases by 5 percent.

(b) The decrease in the degree of risk-aversion reduces the differential between the return
on market portfolio and the risk-free return by one-half.
(c) The expected growth rate of dividend on stock A decrease to 10 percent.
(d) The beta of stock A falls to 1.1

Solution:

RM = 16% βA = 1.6 RA =22 % g = 12 % Po = Rs. 260

Po = D1 / (r - g)

Rs.260 = D1 / (0.22 - .12)

So D1 = Rs. 26 and Do = D1 / (1+g) = 26 /(1.12) = Rs.23.21

RA = Rf + βA (RM – Rf)

0.22 = Rf + 1.6 (0.16 – Rf)

0.6Rf = 0.036

So Rf = 0.06 or 6%.
Original Revised

Rf 6% 11%
RM – Rf 10% 5%
g 12 % 10 %
βA 1.6 1.1

Revised RA = 11% + 1.1 (5%) = 16.5 %

 Price per share of stock A, given the above changes is

23.21 (1.10)
= Rs. 392.78
0.165 – 0.10

23. The following information is given:

Expected return for the market = 15%

Standard deviation of the market return = 25%
Risk-free rate = 8%
Correlation coefficient between stock A and the market = 0.8
Correlation coefficient between stock B and the market = 0.6
Standard deviation for stock A = 30%
Standard deviation for stock B = 24%

(i) What is the beta for stock A?

Solution:

Cov (A,M) Cov (A,M)

AM = ; 0.8 =  Cov (A,M) = 600
A M 30 x 25

M2 = 252 = 625

Cov (A,M) 600

A = = = 0.96
M2 625

(ii) What is the expected return for stock A?

Solution:

E(RA) = Rf + A (E (RM) - Rf)

= 8% + 0.96 (7%) = 14.72%