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7 PORTFOLIO THEORY
OHT 7.‹#›
LEARNING OBJECTIVES
• Calculating two-asset portfolio expected
returns and standard deviations
• Estimating measures of the extent of
interaction – covariance and correlation
coefficients
• Being able to describe dominance, identify
efficient portfolios and then apply utility
theory to obtain optimum portfolios
• Recognise the properties of the multi-asset
portfolio set and demonstrate the theory
behind the capital market line
Glen Arnold: Corporate Financial Management, Second edition
© Pearson Education Limited 2002
7 PORTFOLIO THEORY
OHT 7.‹#›
Holding period returns
One year:
Where:
s = semi-annual rate
R = annual rate
For three year holding period
Glen Arnold: Corporate Financial Management, Second edition
© Pearson Education Limited 2002
7 PORTFOLIO THEORY
EXAMPLE:
Initial share price = £1.00
Share price three years later = £1.20
Dividends: year 1 = 6p, year 2 = 7p, year 3 = 8p
Glen Arnold: Corporate Financial Management, Second edition
© Pearson Education Limited 2002
OHT 7.‹#›
7 PORTFOLIO THEORY
OHT 7.‹#›
EXPECTED RETURNS AND
STANDARD DEVIATION FOR
SHARES
Ace plc
A share costs 100p to purchase now and the estimates of returns for the
next year are as follows:
Event
Estimated
selling price, P1
Economic boom
Normal growth
Recession
114p
100p
86p
Estimated
dividend, D1
Return
Ri
Probability
6p
5p
4p
+20%
+5%
–10%
0.2
0.6
0.2
1.0
Glen Arnold: Corporate Financial Management, Second edition
© Pearson Education Limited 2002
7 PORTFOLIO THEORY
OHT 7.‹#›
THE EXPECTED RETURN
n
R =  R i pi
i= 1
where
R = expected return
R i = return if event i occurs
pi = pr obability of event i occurring
n
= number of events
Expected return, Ace plc
Event
Boom
Growth
Recession
Probability of
event pi
0.2
0.6
0.2
Return
Ri
+20
+5
–10
Expected returns
Glen Arnold: Corporate Financial Management, Second edition
© Pearson Education Limited 2002
Ri  pi
4
3
–2
5 or 5%
7 PORTFOLIO THEORY
OHT 7.‹#›
STANDARD DEVIATION
 =
n
2
 ( R – R ) pi
i
i =1
Standard deviation, Ace plc
Probability
pi
Return
Ri
Expected return
Ri
0.2
0.6
0.2
20%
5%
–10%
5%
5%
5%
Deviation
Ri – Ri
15
0
–15
2
Variance 
Standard deviation 
Glen Arnold: Corporate Financial Management, Second edition
© Pearson Education Limited 2002
Deviation
squared 
probability
2
(Ri – R i ) pi
45
0
45
90
9.49%
7 PORTFOLIO THEORY
OHT 7.‹#›
Returns for a share in Bravo plc
Event
Boom
Growth
Recession
Return
Probability
Ri
pi
–15%
+5%
+25%
0.2
0.6
0.2
1.0
Expected return on Bravo
(–15  0.2) + (5  0.6) + (25  0.2) = 5 per cent
Standard deviation, Bravo plc
Probability
pi
0.2
0.6
0.2
1.0
Return
Ri
–15%
+5%
+25%
Expected return
Ri
5%
5%
5%
Deviation
Ri – Ri
–20
0
+20
Variance  2
Standard deviation 
Glen Arnold: Corporate Financial Management, Second edition
© Pearson Education Limited 2002
Deviation
squared 
probability
2
(Ri – Ri ) pi
80
0
80
160
12.65%
7 PORTFOLIO THEORY
OHT 7.‹#›
COMBINATIONS OF INVESTMENTS
Hypothetical pattern of return for Ace plc
20
Return %
+
5
1
–
2
3
4
5
6
8
7
Time
(years)
–10
Hypothetical pattern of returns for Bravo plc
25
Return %
+
5
0
–
1
2
3
4
5
6
7
8
Time
(years)
–15
Glen Arnold: Corporate Financial Management, Second edition
© Pearson Education Limited 2002
7 PORTFOLIO THEORY
OHT 7.‹#›
Returns over one year from placing £571 in Ace and £429 in Bravo
Event
Returns
Ace
£
Returns
Bravo
£
Overall returns
on £1,000
Percentage
returns
571(1.2) = 685
429 – 429(0.15) = 365
1,050
5%
571(1.05) = 600
429(1.05) = 450
1,050
5%
Recession 571 – 571(0.1) = 514
429 (1.25) = 536
1,050
5%
Boom
Growth
Hypothetical pattern of returns for Ace, Bravo and the
two-asset portfolio
Bravo
25
20
Return %
+
Portfolio
5
0
–
–10
–15
1
2
3
4
5
6
7
8
Ace
Glen Arnold: Corporate Financial Management, Second edition
© Pearson Education Limited 2002
Time
(years)
7 PORTFOLIO THEORY
OHT 7.‹#›
PERFECT NEGATIVE CORRELATION
PERFECT POSITIVE CORRELATION
Annual returns on Ace and Clara
Event
i
Probability
pi
Returns
on Ace
%
Returns
on Clara
%
0.2
0.6
0.2
+20
+5
–10
+50
+15
–20
Boom
Growth
Recession
Returns over a one-year period from placing £500 in Ace and £500 in Clara
Event
i
Boom
Growth
Recession
Returns
Ace
£
Returns
Clara
£
Overall
return on
£1,000
600
525
450
750
575
400
1,350
1,100
850
Glen Arnold: Corporate Financial Management, Second edition
© Pearson Education Limited 2002
Percentage
return
35%
10%
–15%
7 PORTFOLIO THEORY
OHT 7.‹#›
Hypothetical patterns of returns for Ace and Clara
50
Clara
Portfolio
20
+
5
–
Ace
1
2
3
4
5
6
7
–10
–15
–20
Glen Arnold: Corporate Financial Management, Second edition
© Pearson Education Limited 2002
8
Time
(years)
7 PORTFOLIO THEORY
OHT 7.‹#›
INDEPENDENT INVESTMENTS
Expected returns for shares in X and shares in Y
Expected return for shares in X
Expected return for shares in Y
Return  Probability
–25  0.5 = –12.5
35  0.5 = 17.5
5.0%
Return  Probability
– 25 0.5 = –12.5
35 0.5 = 17.5
5.0%
Standard deviations for X or Y as single investments
Return
Ri
–25%
35%
Probability
pi
0.5
0.5
Expected return
Ri
5%
5%
Deviations
Ri – R
Deviations
squared 
probability
(Ri – R)2 pi
–30
30
Variance 
Standard deviation 
Glen Arnold: Corporate Financial Management, Second edition
© Pearson Education Limited 2002
450
450
900
30%
7 PORTFOLIO THEORY
OHT 7.‹#›
Exhibit 7.17 A mixed portfolio: 50 per cent of the fund invested in X and 50
per cent in Y, expected return
Possible
outcome
combinations
Joint
returns
Both firms
do badly
–25
0.5  0.5 = 0.25
–25  0.25 = –6.25
X does badly
Y does well
5
0.5  0.5 = 0.25
5  0.25 = 1.25
X does well
Y does badly
5
0.5  0.5 = 0.25
5  0.25 = 1.25
35
0.5  0.5 = 0.25
35  0.25 = 8.75
Both firms
do well
Return 
probability
Joint
probability
1.00 Expected return 5.00%
Standard deviation, mixed portfolio
Return
Probability Expected return
Ri
pi
R
–25
5
35
0.25
0.50
0.25
5
5
5
Deviations
Ri – R
Deviations
squared 
probability
(Ri – R)2 pi
–30
225
0
0
30
225
Variance  450
Standard deviation  21.21%
Glen Arnold: Corporate Financial Management, Second edition
© Pearson Education Limited 2002
7 PORTFOLIO THEORY
OHT 7.‹#›
A CORRELATION SCALE
So long as the returns of constituent assets of a
portfolio are not perfectly positively correlated,
diversification can reduce risk. The degree of
risk reduction depends on:
• the extent of statistical interdependence
between the returns of the different
investments: the more negative the better; and
• the number of securities over which to spread
the risk: the greater the number, the lower the
risk.
–1
0
+1
Perfect
negative
correlation
Independent
Perfect
positive
correlation
Correlation scale
Glen Arnold: Corporate Financial Management, Second edition
© Pearson Education Limited 2002
7 PORTFOLIO THEORY
OHT 7.‹#›
THE EFFECTS OF DIVERSIFICATION
WHEN SECURITY RETURNS ARE NOT
PERFECTLY CORRELATED
Returns on shares A and B for alternative economic states
Event i
State of the economy
Probability
pi
Return on A
RA
Return on B
RB
0.3
0.4
0.3
20%
10%
0%
3%
35%
–5%
Boom
Growth
Recession
Company A: Expected return
Probability
pi
Return
RA
RA  pi
0.3
0.4
0.3
20
10
0
6
4
0
10%
Company A: Standard deviation
Probability
pi
0.3
0.4
0.3
Return
RA
Expected
return
RA
20
10
0
10
10
10
Deviation
(RA – RA)
Deviation
squared 
probability
(RA – RA)2 pi
10
30
0
0
–10
30
60
Variance 
Standard deviation  7.75%
Glen Arnold: Corporate Financial Management, Second edition
© Pearson Education Limited 2002
7 PORTFOLIO THEORY
OHT 7.‹#›
Company B: Expected return
Probability
pi
Return
RB
0.3
0.4
0.3
RB pi
3
35
–5
0.9
14.0
–1.5
13.4%
Company B: Standard deviation
Probability
Return
pi
RB
0.3
0.4
0.3
3
35
–5
Expected
return
RB
Deviation
(RB – RB)
Deviation
squared 
probability
(RB – RB)2 pi
13.4
10.4
32.45
13.4
21.6
186.62
13.4
–18.4
101.57

Variance 
320.64
Standard deviation  = 17.91%
Summary table: Expected returns and standard deviations for
Companies A and B
Expected return
Standard deviation
Company A
Company B
10%
13.4%
7.75%
17.91%
Glen Arnold: Corporate Financial Management, Second edition
© Pearson Education Limited 2002
7 PORTFOLIO THEORY
OHT 7.‹#›
A general rule in portfolio theory:
Portfolio returns are a weighted average of the
expected returns on the individual investment…
BUT…
Portfolio standard deviation is less than the
weighted average risk of the individual
investments, except for perfectly positively
correlated investments.
Expected return R %
Exhibit: 7.26 Return and standard deviation for shares in firms A and B
20
15
B
P
10
A
Q
5
5
10
15
20
Standard deviation %
Glen Arnold: Corporate Financial Management, Second edition
© Pearson Education Limited 2002
7 PORTFOLIO THEORY
OHT 7.‹#›
PORTFOLIO EXPECTED RETURNS
AND STANDARD DEVIATION
• 90 per cent of the portfolio funds are placed in A
• 10 per cent are placed in B
Expected returns, two-asset portfolio
• Proportion of funds in A = a = 0.90
• Proportion of funds in B = 1 – a = 0.10
Rp = aRA + (1 – a)RB
Rp = 0.90  10 + 0.10  13.4 = 10.34%
p =
a2  2A + (1 – a)2  2B + 2a (1 – a) cov (RA, RB)
where
p = portfolio standard deviation
A = variance of investment A
B = variance of investment B
cov (RA, RB) = covariance of A and B
Glen Arnold: Corporate Financial Management, Second edition
© Pearson Education Limited 2002
7 PORTFOLIO THEORY
OHT 7.‹#›
COVARIANCE
The covariance formula is:
n
cov (RA, RB) =  {(RA – RA)(RB – RB)pi}
i=1
Exhibit 7.27 Covariance
Event and
probability of
event pi
Returns
R R
Expected
returns
R R
Boom
Growth
Recession
20 3
10 35
0 –5
10 13.4
10 13.4
10 13.4
A
0.3
0.4
0.3
B
A
B
Deviations
RA – R R – R
A
10
0
–10
B
–10.4
21.6
–18.4
B
Deviation of A
deviation of B probability
(R – R )(R – R )pi
A
A
B
B
10  –10.4  0.3 = –31.2
0  21.6 0.4 =
0
–10  –18.4  0.3 = 55.2
Covariance of A and B, cov ( RA , R B ) = +24
Glen Arnold: Corporate Financial Management, Second edition
© Pearson Education Limited 2002
7 PORTFOLIO THEORY
OHT 7.‹#›
STANDARD DEVIATION
p =
a2  2A + (1 – a)2  2B + 2a (1 – a) cov (RA, RB)
p =
0.902  60 + 0.102  320.64 + 2  0.90  0.10  24
p =
48.6 + 3.206 + 4.32
p =
7.49%
Exhibit 7.28 Summary table: expected return and standard deviation
Expected return (%)
Standard deviation (%)
All invested in Company A
10
7.75
All invested in Company B
13.4
17.91
Invested in a portfolio
(90% in A, 10% in B)
10.34
7.49
Expected return R %
Exhibit 7.29 Expected returns and standard deviation for A and B and
a 90:10 portfolio of A and B
20
Portfolio (A=90%, B=10%)
15
B
10
A
5
5
15
10
Standard deviation %
20
Glen Arnold: Corporate Financial Management, Second edition
© Pearson Education Limited 2002
7 PORTFOLIO THEORY
OHT 7.‹#›
CORRELATION COEFFICIENT
cov (RARB)
RAB =
AB
24
RAB =
7.75  17.91
If RAB =
p =
cov (RARB)
 A B
= +0.1729
then cov (RARB) = RABAB
a22A + (1 – a)2 2B + 2a (1 – a) RABAB
Glen Arnold: Corporate Financial Management, Second edition
© Pearson Education Limited 2002
7 PORTFOLIO THEORY
OHT 7.‹#›
Exhibit 7.30 Perfect positive correlation
Returns on G
Returns on F
Exhibit 7.31 Perfect negative correlation
Returns on G
Returns on F
Exhibit 7.32 Zero correlation coefficient
Returns on G
Returns on F
Glen Arnold: Corporate Financial Management, Second edition
© Pearson Education Limited 2002
7 PORTFOLIO THEORY
OHT 7.‹#›
DOMINANCE AND THE EFFICIENT
FRONTIER
Exhibit 7.33 Returns on shares in Augustus and Brown
Event
(weather
for season)
Warm
Average
Wet
Probability
of event
Returns on
Augustus
pi
RA
20%
15%
10%
RB
–10%
22%
44%
15%
20%
0.2
0.6
0.2
Expected
return
Returns on
Brown
Exhibit 7.34 Standard deviation for Augustus and Brown
Probability
pi
Returns
on Augustus
RA
(RA – RA)2 pi
Returns
on Brown
0.2
0.6
0.2
20
15
10
5
0
5
–10
22
44
(RB – RB)2 pi
Variance, 
10
Variance, B
Standard deviation, 
3.162
Standard deviation, B
Glen Arnold: Corporate Financial Management, Second edition
© Pearson Education Limited 2002
180.0
2.4
115.2
297.6
17.25
7 PORTFOLIO THEORY
OHT 7.‹#›
Exhibit 7.35 Covariance
Probability
pi
Returns
RA RB
Expected
returns
RA RB
–10 15
0.2
20
0.6
15
22
0.2
10
44
Deviations
Deviation of A 
deviation of B  probability
(RA – RA)( RB – RB)pi
RA – RA RB – RB
20
5
–30
15
20
0
2
15
20
–5
24
5  –30  0.2 = –30
0
2  0.6 =
0
–5  24  0.2 = –24
Covariance (RA RB) –54
RAB =
RAB
cov (RA, RB)
AB
–54
=
= –0.99
3.162  17.25
Glen Arnold: Corporate Financial Management, Second edition
© Pearson Education Limited 2002
10
15
20
50
75
90
85
80
50
25
0
J
K
L
M
N
B
100
0
100
A
Portfolio
Brown
Augustus
weighting weighting
(%)
(%)
= 1.16
= 3.16
0.252  10 + 0.752  297.6 + 2  0.25  0.75  –54 =12.2
18.75
20
=17.25
= 7.06
0.52  10 + 0.52  297.6 + 2  0.5  0.5  –54
17.5
16.0
= 1.01
0.852  10 + 0.152  297.6 + 2  0.85  0.15  –54 = 0.39
0.92  10 + 0.12  297.6 + 2  0.9  0.1  –54
Standard deviation
0.82  10 + 0.22  297.6 + 2  0.8  0.2  –54
15.75
15.5
15
Expected
return (%)
Exhibit 7.36 Risk-return correlations: two-asset portfolios for Augustus and Brown
7 PORTFOLIO THEORY
OHT 7.‹#›
Glen Arnold: Corporate Financial Management, Second edition
© Pearson Education Limited 2002
7 PORTFOLIO THEORY
OHT 7.‹#›
Exhibit 7.37 Risk-return profile for alternative portfolios of
Augustus and Brown
21
B
20
Return Rp %
19
N
18
M
Efficiency frontier
17
L
16
K
15
J
A
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
p Standard deviation
Glen Arnold: Corporate Financial Management, Second edition
© Pearson Education Limited 2002
7 PORTFOLIO THEORY
OHT 7.‹#›
FINDING THE MINIMUM STANDARD
DEVIATION FOR COMBINATIONS OF
TWO SECURITIES
• If a fund is to be split between two securities, A and
B, and a is the fraction to be allocated to A, then
the value for a which results in the lowest standard
deviation is given by:
Glen Arnold: Corporate Financial Management, Second edition
© Pearson Education Limited 2002
7 PORTFOLIO THEORY
OHT 7.‹#›
INDIFFERENCE CURVES
Exhibit 7.38 Indifference curve for Mr Chisholm
X
Return %
14
Z
Indifference curve I 105
10
W
Y
16
20
Standard deviation %
Glen Arnold: Corporate Financial Management, Second edition
© Pearson Education Limited 2002
7 PORTFOLIO THEORY
OHT 7.‹#›
Exhibit 7.39 A map of indifference curves
North-west
I 129
I 121
I 110
Return %
I 107
S
14
10
I 105
T
Z
W
South-west
16
20
Standard deviation %
Glen Arnold: Corporate Financial Management, Second edition
© Pearson Education Limited 2002
7 PORTFOLIO THEORY
OHT 7.‹#›
Exhibit 7.40 Intersecting indifference curves
Return %
I105
I101
M
I101
I105
Standard deviation %
Glen Arnold: Corporate Financial Management, Second edition
© Pearson Education Limited 2002
7 PORTFOLIO THEORY
OHT 7.‹#›
Return %
Standard
deviation %
(a) Moderate risk aversion
Return %
Return %
Exhibit 7.41 Varying degrees of risk aversion as represented by
indifference curves
Standard
deviation %
(b) Low risk aversion
Standard
deviation %
(c) High risk aversion
Glen Arnold: Corporate Financial Management, Second edition
© Pearson Education Limited 2002
7 PORTFOLIO THEORY
OHT 7.‹#›
CHOOSING THE OPTIMAL
PORTFOLIO
Exhibit 7.42 Optimal combination of Augustus and Brown
I3
21
I2
20
I1
Return %
19
N
B
Efficiency frontier
18
M
17
L
16
K
15
J
1
A
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18
p Standard deviation %
Glen Arnold: Corporate Financial Management, Second edition
© Pearson Education Limited 2002
7 PORTFOLIO THEORY
OHT 7.‹#›
THE BOUNDARIES OF
DIVERSIFICATION
Exhibit 7.44 The boundaries of diversification
D
22
21
20
RCD = –1
19
RCD = +1
Return %
H
18
RCD = 0
17
E
G
16
F
15
C
Return %
1
2
3
4
5
6
7
p Standard deviation %
8
9
10
Glen Arnold: Corporate Financial Management, Second edition
© Pearson Education Limited 2002
7 PORTFOLIO THEORY
OHT 7.‹#›
EXTENSION TO A LARGE NUMBER
OF SECURITIES
Exhibit 7.47 A three-asset portfolio
A
Return
4
1
3
2
B
C
Standard deviation
Glen Arnold: Corporate Financial Management, Second edition
© Pearson Education Limited 2002
7 PORTFOLIO THEORY
OHT 7.‹#›
Exhibit 7.48 The opportunity set for multi-security portfolios and
portfolio selection for a highly risk-averse person and for a slightly
risk-averse person
IL3
IL2
Efficiency frontier
Return
IH3
IL1
V
IH2 I
H1
U
Inefficient region
Inefficient
region
Standard deviation
Glen Arnold: Corporate Financial Management, Second edition
© Pearson Education Limited 2002
7 PORTFOLIO THEORY
THE CAPITAL MARKET LINE
Exhibit 7.53 Combining risk-free and risky investments
Return
M
F
C
B
rf A
Standard deviation
Glen Arnold: Corporate Financial Management, Second edition
© Pearson Education Limited 2002
OHT 7.‹#›
7 PORTFOLIO THEORY
OHT 7.‹#›
Return
Exhibit 7.54 Indifference curves applied to combinations of the market
portfolio and the risk-free asset
M
Y
X
rf
Standard deviation
Glen Arnold: Corporate Financial Management, Second edition
© Pearson Education Limited 2002
7 PORTFOLIO THEORY
OHT 7.‹#›
Exhibit 7.55 The capital market line
Capital market line
N
T
Return
S
M
G
H
rf
Standard deviation
Glen Arnold: Corporate Financial Management, Second edition
© Pearson Education Limited 2002
7 PORTFOLIO THEORY
OHT 7.‹#›
Problems with portfolio theory:
• relies on past data to predict future risk and
return
• involves complicated calculations
• indifference curve generation is difficult
• few investment managers use computer
programs because of the nonsense results
they frequently produce
Glen Arnold: Corporate Financial Management, Second edition
© Pearson Education Limited 2002