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MBA & MBA – Banking and Finance
(Term-IV)
Course : Security Analysis and Portfolio Management
Unit III: Portfolio Selection
Portfolio Selection:
A) Markowitz Approach:
Find the set of portfolios that
satisfy the following 3 conditions:
1) Provides the minimum risk for every possible
level of return
2) The ‘Efficient' set
3) Investor selects from the ‘Efficient’ set the
single portfolio that meets his/her needs
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Efficient portfolio
Efficient portfolio is the portfolio that gives maximum return
for a given level of risk or minimum risk for a given level of
return.
Efficient Frontier:
Efficient frontier is the loci of all such efficient portfolios.
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•The question is : how should investors choose a
“best” option on the efficient frontier?
The following are the important parameters to be
considered:
1) Risk – return framework.
2) Utility functions, or indifference curves.
There can be various types of Indifference curves
representing investors’ perceptions towards riskreturn. Thus there can be following categories of
investors:
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1) Risk Lover Investor
U4
Rp
U3
U2
U1
Return
Risk
σp
Risk Lover Indifference curves
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2) a) Risk Averse Investor
U5
U4
U3
U2
Rp
U1
Return
Risk
σp
Risk Fearing Investor’s Indifference
Curves
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2) b) Less Risk Averse Investor
U4
U3
Rp
U2
U1
Return
Risk
σp
Less Risk Fearing Investor’s Indifference Curve
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Investor’s Equilibrium: Selection of Best portfolio.
Y is the point of optimal/best portfolio. At this point the efficient
frontier is tangent to the indifference curve.
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B) SHARPE METHOD OF
PORTFOLIO SELECTION
Single Index model by Sharpe helps in the construction of an
Optimal portfolio by providing a single number that measures
the desirability of including a stock in the Optimal Portfolio.
 2 m  10%
 Here, the desirability of any stock is directly related to its
excess return -to -beta ratio:

Ri R F
i
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Where
R i = expected return on stock i
R f = return on a riskless asset
βi = expected change in the rate of return on stock i associated
with a 1 % change in the market return.
If stocks are ranked by Excess return to beta (from highest to
lowest), the ranking represents the desirability of any stock ‘s
inclusion in a portfolio.
The number of stocks selected depends on a unique cut off rate
such that all stocks with higher ratios of Excess return to beta will
be included and all stocks with lower ratios excluded.
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STEPS to determine which stocks are included in the
Optimum Portfolio:
1. Calculate the excess return to beta ratio for each stock
under review .
2. Rank them from highest to lowest.
3. The optimum portfolio consists of investing in all stocks for
which
Ri  R F is greater than a particular cut off point C*
i
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Example: On the basis of following data find the
optimal portfolio. Also find proportion of the fund to
be invested in the securities of the optimal
portfolio.
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Security no. Mean return,R i Beta , βi
(1)
(2)
(3)
Unsystematic risk,
1
15
1
50
2
17
1.5
40
3
12
1
20
4
17
2
10
5
11
1
40
6
11
1.5
30
7
11
2
40
8
7
0.8
16
9
7
1
20
10
5.6
0.6
6
Also given RF = 5% and

2
m
 10%
R f = 5%
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Table : 2, Solution:
Security No.
Excess return,
R i - RF
Excess return to beta
Ri–RF
βi
1
10
10
2
12
8
3
7
7
4
12
6
5
6
6
6
6
4
7
6
3
8
2
2.5
9
2
2
10
0.6
1
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Establishing a Cut off Rate
All securities whose Excess return to beta ratio are above the cut off
rate are Selected
All securities whose ratios are below are rejected.
Finding the Cut off rate
For portfolio of i stocks ,C i is given by
i
 
2


1  

m
Ci 
( Ri  RF ) i
2
i 1
ei
2
m
Where :


2
i
i 1
i
2
ei
2
m
2
ei
= Variance in the market index
= stock’s unsystematic risk.
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
Table 3 : Determining Cut off Rate with
Security
NO.
Ri-RF (Ri-RF) βi
βi
2

ei


2
i
i
2
ei

2
m
( Ri  RF ) i
i 1

2
ei
= 10%



2
i
i 1
C
i
2
ei
1
10
2/10
2/100
2/10
2/100
1.67
2
8
4.5/10
5.625/100
6.5/10
7.625/100
3.69
3
7
3.5/10
5/100
10/10
12.625/100
4.42
4
6
24/10
40/100
34/10
52.625/100
5.43
5
6
1.5/10
2.5/100
35.5/10
55.125/100
5.45
6
4
3/10
7.5/100
38.5/10
62.625/100
5.3
7
3
3/10
10/100
41.5/10
72.625/100
5.02
8
2.5
1/10
4/100
42.5/10
76.625/100
4.91
9
2
1/10
5/100
43.5/10
81.625/100
4.75
10
1
0.6/10
6/100
44.1/10
87.625/100
4.52
16
Arriving at the optimal portfolio
With the help of Cut off rate ,we know which securities are to be included in the
Optimum portfolio, the next step is to calculate the percent of capital to be invested
in each security.
The percentage invested in each security is :
X
0
i

Z
Z
i
N
i 1
Where,

R

R

Z  
 
i
i
2
ei
i
i
F
i

 C *


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Z1=
2/100(10-5.45) =
0.091
Z2=
3.75 (8-5.45) =
100
0.095625
Z3=
5 ( 7 - 5.45 ) = 0.0775
100
Z4=
20 (6 - 5.45 ) =
100
0.110
Z5=
2.5 (6 - 5.45) =
100
0.01375
5
 Zi  0.387875
i 1
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Dividing each Z i by sum of Z i ,we would invest
 0.091 / 0.387875 = 23.5 % in security 1
0.095625 / 0.387875 = 24.6 % in security 2
0.0775 /0.387875 = 20 % in security 3
0.110 /0.387875 = 28.4 % in security 4
0.01375 /0.387875 = 3.5 % in security 5
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Problem No.1
Q.1 Vinoth received Rs.10 lakh from his pension fund. He
wants to invest in the stock market. The treasury bill
rate is 5% and the market return variance is 10. The
following table gives the details regarding the expected
return, beta and residual variance of the individual
security. What is the optimum portfolio assuming no
short sales?
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2
Security
Expected
Return
Beta
A
15
1.0
30
B
12
1.5
20
C
11
2.0
40
D
8
0.8
10
E
9
1.0
20
F
14
1.5
10
σei
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Problem No.2
Q.2 A portfolio manager has got the following
information about several stocks. He has to build an
optimum portfolio for his client without short sales.
The market index variance is 12 per cent and the risk
free rate of return is 7 per cent.
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2
Security
Expected
return
β
A
22
1.0
35
B
20
2.5
30
C
14
1.5
25
D
18
1.0
80
E
16
0.8
20
F
12
1.2
10
G
19
1.6
25
H
17
2.0
30
σei
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Problem No.3
• What is the optimum portfolio in choosing among the
following securities and assuming Rf= 5 percent and variance of
market return = 10 per cent?
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2
Security
Expected
Return
Beta
A
15
1.0
30
B
12
1.5
20
C
11
2.0
40
D
8
0.8
10
E
9
1.0
20
F
14
1.5
10
σei
25
Problem No.4
• You are attempting to construct an optimum portfolio. Over
your holding period you have forecast an expected return on
the market of 13.5% with a market variance of 256%. The
treasury security rate available is 7% (risk free). The following
securities are under review:
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Stock
Alpha
Beta
Boeing
3.72
.99
Residual
Variance
9.35
Bristol-Myers 0.60
1.27
5.92
BrowningFerris
Emerson
Electric
Mountain
States
Telephone
0.41
.96
9.79
-0.22
1.21
5.36
0.45
.75
4.52
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Problem No.5
Q.1 Mr. David is constructing a optimum portfolio. The market
return forecast says that it would be 13.5% for the next two
years with the market variance of 10%. The risk less rate of
return is 5%. The following securities are under review. Find out
the optimum portfolio.
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Company
α
β
Anil
Avil
Bow
Viril
Billy
3.72
0.60
0.41
-0.22
0.45
0.99
1.27
0.96
1.21
0.75
2
σei
9.35
5.92
9.79
5.39
4.52
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