Risk Return & Portfolio Theory | CA Final SFM (1)

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Strategic Financial Management
CA Final
By CA Nikhil Jobanputra
Portfolio Theory Lecture 1
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Risk, Return Portfolio Theory
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Investment Objectives

• Maximization of Returns
• Minimization of Risk

Conflict in Objectives
High Returns
High Risk
Investing in Stock Market
Low Risk
Low Returns
Investing in Bond Market
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Investment Objectives

• Maximization of Returns
• Minimization of Risk

Conflict in Objectives
You cannot Achieve the objectives
Investing in Single Security
Need for Investment Portfolio
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Investment Portfolio
Analysis of Risk Returns
Portfolio Management
Single Security
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Concept 1Rate of Return on Single Security
D
P
Future Cash Flows
1
1
P

0
1
k
Expected Rate of Return
e
Present Value of Future Cash Flows
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Concept 1Rate of Return on Single Security
D
P
1
1
P

0
1
k
e
Where,
D

Expected
dividend
receivable
by
year
end
1
P

Expected
Market
Price
of
Equity
share
by
year
end
1
k

Expected
Rate
of
Return
by
the
Equity
Shareholder
e
(Also
known
as
Equity
Capitalisation
Rate)
(Or
Cost
of
Equity
for
the
company)
P

Current
Market
Price
of
Equity
share
0
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Q.1
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Question 1
Mr. X is to decide regarding whether he should
invest in equity share of NJ Ltd. Expected
dividend receivable by year end on share of NJ
Ltd. is 45. Expected Market Price of this
equity share by year end is 300. The expected
rate of return by Mr. X is 25 p.a. Determine the
current market price of equity share.
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D
P
45
300
1
1
P

P

0
0
1
k
1
0.25
e
345
P


276
0
1.25
Where,
D

Expected
dividend
receivable
by
year
end 45
1
P

Expected
Price
of
Equity
share
by
year
end 300
1
k

Expected
Rate
of
Return
by
the
Shareholder 0.25
e
P

Current
Market
Price
of
Equity
share
0
12
D
P
D
P
1
k
1
1
1
1
P


e
0
1
k
P
e
0
D
P
g
k

Growth Rate
1
1

1
e
P
P
P
0
1
0
g

P
D
P
P0
k
0
1
1

e
P
D
0
1
g
k

P
e
0
P
P
D
k
1
0
1


e
P
P
0
0
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Q.2
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Question 2
NJ Ltd. has been showing a consistent growth in
the share price as well as dividends in the
recent past. Such growth rate is about 10 per
annum. Price of this share prevailing today is
200 per share. The company is expected to declare
a dividend of 36 by end of the year. You are
required to determine the expected rate of return
for the shareholder at present.
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Expected Price of share by the end of year (P1)
200 10 220
Expected Dividend Receivable by year end (D1)
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?The expected total wealth of the shareholder by
end of the year
220 36 256

• If the investor makes an investment of 200
  today, the total
• returns in the year is expected to be 256
  200 56

?The expected Rate of Return for the shareholder
will be
56
0.28 or 28
200
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D
P
36
220
k
k
1
1

1

1
e
e
P
200
0
256
k

1
1.28 1
0.28 or 28
e
200
Where,
D

Expected
dividend
receivable
by
year
end
36
1
P

Expected
Market
Price
of
Equity
share
by
year
end
1
200 10 220
k

Equity Capitalisation Rate
e
P

Current
Market
Price
of
Equity
Share 200
0
17
D
36
1
k

g
k

0.10
k
0.18 0.1 0.28 or 28
e
e
e
P
200
0
Where,
D

Expected
dividend
receivable
by
year
end
36
1
k

Equity Capitalisation Rate
e
P

Current
Market
Price
of
Equity
Share 200
0
g

Growth Rate 10
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Concept 2Average Rate of Return on Single
Security

• When the information is available about the
  individual rate of return of a security over past
  few years
• The average rate of return of such security can
  be determined by Simple Arithmetic Mean

?
x
x

N
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Q.3
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Question 3
Determine the average rate of return based on the
following data
Year Expected Dividend () Expected Share Price ()
1 20 216
2 22 250
3 24 256
4 25 240
5 30 260
Presently the price of the share is 200.
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D
P
1
1
-
1
k

e
P
0
Where,
D

Expected
dividend
receivable
by
year
end
1
P

Expected
Market
Price
of
Equity
share
by
year
end
1
k

Expected
Rate
of
Return
by
the
Equity
Shareholder
e
P

Current
Market
Price
of
Equity
share
0
22
20
216
-
1
0.18 or 18

Year 1
-
k
e
200
22
250
k
Year 2
-

-
1
0.2593 or 25.93
e
216
24
256
Year 3
-

-
1
0.12 or 12
k
e
250
25
240
k
Year 4
-

-
1
0.0352 or 3.52
e
256
30
260

0.2083 or 20.83
Year 5
-
-
1
k
e
240
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Average Rate of Return
?
x
x

N
18 25.93 12 3.52 20.83

5
16.056
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Concept 3Expected Rate of Return on Single
Security (In a Probability Series)
?
x
x
.P
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Consider the example given below
(P)
(x)
x
Rate of Return
Probability
.P
15
0.20
3.0
18
0.30
5.4
20
0.30
6.0
25
0.20
5.0
?
x
.P

19.4
?
x
x
.P
19.4
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Measurement of Risk
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Measurement of Absolute Risk
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• The deviation in the returns can be considered as
  a basic cause of risk.
• The absolute risk of any security is measured by
  Standard Deviation (s) of its returns.
• The absolute risk of any security can also be
  measured by Variance (s2) of its returns.
• Standard Deviation (s) or Variance (s2) are
  indicators of total risk or absolute risk of any
  security.
• Variance of Returns (s2) is the squared value of
  Standard Deviation of returns (s).

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Concept 4Standard Deviation of Returns of a
Security (In a Simple Individual Series)
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Steps for determining Standard Deviation (s)
1. Determine x
2. Determine dx x x
3. Determine dx2
4. Determine Variance of Returns (sx2)
5. Determine Standard Deviation of Returns (sx)
?
dx2
Variance of Returns (sx2)
N
Standard Deviation (sx)
? sx2
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Q.4
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Question 4 The following rates of returns have
been observed on stock X over past 8 years
Year Rate of Return ()
1 20
2 19
3 17
4 12
5 16
6 17
7 9
8 18
Determine standard deviation of returns on stock
X.
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?
x
x

x
x
Year
Rate of Return ()
d

d
2
N
1
20
4
16
128
x

2
19
3
9
8
3
17
1
1
16
4
12
-
4
16
5
16
0
0
6
17
1
1
7
9
-
7
49
8
18
2
4
?x
?
128
2
96
d
34
?
d2
Variance of Returns (s2)
N
96
s2

12
8
3.4641
Standard Deviation (s)
?s2
?12
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Concept 5Standard Deviation of Returns of a
Security (In a Probability Series)
Variance of Returns (s2) ? d2.P
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Q.5
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Question 5 The expected returns on stock X are
given as below
Rate of Return Probability
15 0.20
18 0.30
20 0.30
25 0.20
Determine the standard deviation of returns from
the above given data.
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(P)
(x)
x
Rate of Return
Probability
.P
15
0.20
3.0
18
0.30
5.4
20
0.30
6.0
25
0.20
5.0
?
x
.P

19.4
?
x
x
.P
19.4
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2
x
x
x
P
d

d
d
2
.P
15
0.20
-
4.4
19.36
3.872
18
0.30
-
1.4
1.96
0.588
0.108
20
0.30
0.6
0.36
6.272
25
0.20
5.6
31.36
?
2
d
.P 10.84
2
Variance of Returns (s2)
?
d
.P 10.84
?10.84
3.2924
Standard Deviation (s) ?s2
?Standard Deviation of Returns (s) 3.2924
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Concept 6Risk Associated with Investment
While comparing the performance of any security
with the other, standard deviation may not always
be sufficient for analysing the performance in
terms of Relative Risk.
In such situation, the performance of two
securities can be better compared through measure
of relative risk.
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Concept 7Co-efficient of Variation

• Co-efficient of Variation is a relative risk
  measure.
• Co-efficient of Variation (CV) relates risk of a
  stock in relation to returns of the same stock.

Standard Deviation
sx

CVx
x
Expected Rate of Return
Co-efficient of Variation measures risk in terms
of each percentage of returns.
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Example
Stock x has average return of 20 with standard
deviation of 6.
Its Co-efficient of Variation will be
sx
6

0.3
CVx
x
20
CV can be interpreted as, stock x has a risk of
0.3 for each percent of its return.
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Q.7
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Question 7 Consider the following cases
Particulars Case 1 Case 1 Case 2 Case 2 Case 3 Case 3 Case 4 Case 4
  X Y X Y X Y X Y
Rate of Return 15 15 19 15 20 15 18 24
Standard Deviation 2 3 3 3 2 3 1.6 3
You are required to analyse the above four cases
and observe whether standard deviation will be
effective tool for decision on selection of one
out of the two securities X and Y.
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Case 1
Both the securities X and Y have same rate of
return but risk of both the securities are
different. X has low risk as compared to Y as
represented by the standard deviations of both
the securities. Between the two, X will be
preferred.
Case 2
Both the securities X and Y have same degree of
risk, as represented by their standard
deviations. Security X offers higher returns as
compared to Y. Therefore, X will be preferred.
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Case 3
The risk involved in Security X is lower than
that of Security Y, as represented by their
standard deviations. Security X has higher
returns as compared to Y. Security X is preferred
from both risk and return view point.
Case 4
Security Y offers higher returns but also has
higher risk as compared to X. The standard
deviation of Security Y is higher than that of X,
indicating high risk. However, simply looking at
the risk and returns, it cannot be decided as to
which security shall be preferred. A relative
measure of risk in the form of coefficient of
variation will be required.
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Standard Deviation (s)
CV
Expected Rate of Return
sx
1.6
0.0889

CVx
Rx
18
sy
3

0.125
CVy
Ry
24
CV represents the degree of risk for each of
return. Assuming that the investor is risk
averse, Security X shall be preferred.
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Concept 8Determining Covariance and
Correlation between two stocks

• Correlation is a statistical measure that
  indicates the extent to which two or more
  variables fluctuate together.
• A positive correlation indicates the extent to
  which those variables increase or decrease in
  parallel
• A negative correlation indicates the extent to
  which one variable increases as the other
  decreases.

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Covariance is a measure of how changes in one
variable are associated with changes in a second
variable. Specifically, covariance measures the
degree to which two variables are linearly
associated.
Note
Covariance between x and y (Covxy) is a product
of 3 factors
1. Standard deviation of x (sx)
2. Standard deviation of y (sy)
3. Correlation between x and y (Corxy)
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Covariance between two stocks x and y (In a
simple individual series)
?
dxdy
Covxy
N
Where Covxy indicates covariance between returns
of x and y
dx x x
dy y y
N Number of observations (Years)
Covxy sx .sy .Corxy
Covxy
Corxy
sx .sy
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Q.8
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Question 8 Consider the following data about
returns of two stocks X and Y
Year Returns on Stocks Returns on Stocks
  X Y
1 16 14
2 23 18
3 30 20
4 12 13
5 18 15

• You are required to determine the following
• Average rate of return of both the stocks
• Standard Deviation of returns of both stocks
• Covariance and correlation between two stocks

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Year
x
y
1
16
14
2
23
18
3
30
20
4
12
13
5
18
15
S
S
x 99
y 80
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Sx
99
19.8


x
5
N
Sy
80

y
16

5
N
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Year
x
y
d
d
d
2
d
2
d
d
x
y
x
y
x.
y
(x

x)
(y

y)
1
16
14

3.8

2
14.44
4
7.6
2
23
18
3.2
2
10.24
4
6.4
3
30
20
10.2
4
104.04
16
40.8
4
12
13

7.8

3
60.84
9
2.4
5
18
15

1.8

1
3.24
1
1.8
S
S
S
S
S
x 99
y 80
d
2

d
2
d
.
d
x
y
x
y
192.8
34
80
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Variance of Returns
Sdx2
192.8

38.56
sx2
N
5
Sdy2
34
sy2

6.8
N
5
Standard Deviation
6.2097
sx
? sx2
?38.56
2.6077
sy
? sy2
?6.8
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?
dxdy
80
Covariance between x and y gt
16
Covxy

N
5
Covxy
Correlation between x and y gt
Corxy
sx .sy
16
Corxy
?6.8
X
?38.56
16
16

Corxy
16.1928
?262.208
Corxy 0.9881
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Determining Covariance between returns of two
stocks in Probability Series
Covxy
?
dx .dy .P
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Q.9
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Question 9 Consider the following data regarding
two securities X and Y
Market Conditions Probability Returns on Stocks () Returns on Stocks ()
Market Conditions Probability X Y
Very Good 0.15 22 16
Good 0.40 18 14
Average 0.35 14 12
Bad 0.10 10 10

• You are required to determine the following
• Expected rate of return for Security X and Y
• Standard Deviation of returns for both the
  Securities
• Covariance and Correlation between returns of X
  and Y

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Market
Probability
x
y
x.P
y.P
Condition
Very Good
0.15
22
16
3.3
2.4
Good
0.40
18
14
7.2
5.6
Average
0.35
14
12
4.9
4.2
Bad
0.10
10
10
2
1
16.4
13.2
?
16.4
x
x
.P
?
13.2
y
y
.P
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Market
d
x

x
d
y

y
2
2
d
.
d
.P
d
.P
d
.P
x
y
x
y
x
y
Condition
Very Good
5.6
2.8
4.704
1.1769
2.352
1.6
0.8
1.024
0.256
0.512
Good

2.4

1.2
2.016
0.504
1.008
Average

6.4

3.2
4.096
10.24
2.048
Bad
S
S
S
d
2
.P
d
2
.P
d
.
d
.P
x
y
x
y
11.84
2.96
5.92
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Standard Deviation (sx and sy)
3.4409
sx
?sx2

?11.84
??dx2 .P
1.7205

??dy2 .P
sy
?sy2
?2.96
Covariance between x y (Covxy)
Covxy
?
dx .dy .P
5.92
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Correlation between x and y
Covxy
Corxy
sx .sy
5.92
5.92
5.92

1

Corxy
5.92
?35.0464
?2.96
X
?11.84
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Concept 9Determining Correlation between
returns of a Particular Stock and Market
To measure Covariance in a Probability Series
To measure Covariance in a Simple Individual
Series
?
dxdm
Covxm
?
dx .dm .P
Covxm
N
Covxm
Corxm
sx .sm
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End of Part 1
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Thank You !!
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Q.6
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Question 6 A stock, costing 120, pays no
dividends. The possible prices that the stock
might sell for at the end of the year with the
respective probabilities are
Price Probabilities
115 0.1
120 0.1
125 0.2
130 0.3
135 0.2
140 0.1

• Required
• Calculate the expected return.
• Calculate the standard deviation of returns.

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Q.8
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