1
Finance School of Management
Chapter 12,Portfolio
Selection and Diversification
Objective
To understand the theory of personal
portfolio selection in theory
and in practice
2
Finance School of Management
Chapter 12 Contents
? The process of personal portfolio selection
? The trade-off between expected return and
risk
? Efficient diversification with many risky
assets
3
Finance School of Management
The Concept of ‘Portfolio’
?A person’s wealth portfolio includes
– Assets,stocks,bonds,shares in unincorporated
business,houses or apartments,pensions
benefits,insurance policies,etc,
– Liabilities,student loans,auto loans,home
mortgages,etc,
4
Finance School of Management
Portfolio Selection
?A study of how people should invest their
wealth optimally
?A process of trading off risk and expected
return to find the best portfolio of assets and
liabilities
?Narrow and broad definitions
5
Finance School of Management
Portfolio Selection
?Although there are some general rules
for portfolio selection that apply to
virtually everyone,there is no single
portfolio or portfolio strategy that is
best for everyone
6
Finance School of Management
The Life Cycle
? In portfolio selection,the best strategy depends on
an individual’s personal circumstances (family
status,occupation,income,wealth)
? Illustrations
– Investing in stock market,Chang (30,a security
analyst) / Obi (30,an English teacher)
– Buying insurance policies,Miriam (a parent with
dependent children) / Sanjiv (a single person with no
dependents)
7
Finance School of Management
Time Horizon
? In formulating a plan for portfolio selection,you
begin by determining your goals and time
horizons
– Planning horizon,the total length of time for which
one plans
– Decision horizon,the length of time between decisions
to revise the portfolio
– Trading horizon,the minimum time interval over
which investors can revise their portfolios / its
determination and impacts
8
Finance School of Management
Risk Tolerance
? A major determinant of portfolio choices
? It is influenced by such characteristics as
– age,family status,job status,wealth,and
– other attributes that affect a person’s ability to maintain
his standard of living in the face of adverse movements
in the market value of his investment portfolio
9
Finance School of Management
Professional Asset Managers
? Investment advisors &,finished products” from
a financial intermediary
? Specialization,information and cost advantages
10
Finance School of Management
The Trade-off Between Expected
Return and Risk
? The objective is to find the portfolio which offers
investors the highest expected rate of return for the
degree of risk they are willing to tolerate
? Two step process,
– find the optimal combinational of risky assets
– mix this optimal risk-asset with the riskless asset
11
Finance School of Management
Riskless Asset
? A security that offers a perfectly predictable rate
of return in terms of the unit of account selected
for the analysis and the length of the investor’s
decision horizon
? For example,if the U.S dollars is taken as the unit
of account and the trading horizon is half a year,
the riskless rate is the interest rate on U.S
Treasury bills maturing after half a year,
12
Finance School of Management
Rates of Return on Risky Assets
? Required return depends on the risk of the
investment,
– Greater the risk,greater the return
– Risk premium
13
Finance School of Management
S e c u ri ty P ri c e s
10
100
1000
10000
100000
0 5 10 15 20 25 30 35 40
Y e a rs
V
a
l
u
e
(
L
o
g
)
St o c k
Bon d
St o c k _ M u
Bon d _ M u
14
Finance School of Management
S e c u ri ty P ri c e s
10
100
1000
10000
100000
0 5 10 15 20 25 30 35 40
Y e a rs
V
a
l
u
e
(
L
o
g
)
St o c k
Bon d
St o c k _ M u
Bon d _ M u
15
Finance School of Management
P rob a b i l i s ti c S toc k P ri c e C h a n g e s Ov e r T i me
0, 0 0 0
0, 0 0 2
0, 0 0 4
0, 0 0 6
0, 0 0 8
0, 0 1 0
0, 0 1 2
0, 0 1 4
0, 0 1 6
0, 0 1 8
0, 0 2 0
0 200 400 600 800
Pri c e
P
r
o
b
a
b
i
l
i
t
y
D
e
n
s
i
t
y
St o c k _ Yea r _ 1
St o c k _ Yea r _ 2
St o c k _ Yea r _ 3
St o c k _ Yea r _ 4
St o c k _ Yea r _ 5
St o c k _ Yea r _ 6
St o c k _ Yea r _ 7
St o c k _ Yea r _ 8
St o c k _ Yea r _ 9
St o c k _ Yea r _ 1 0
16
Finance School of Management
P rob a b i l i s ti c B o n d P ri c e C h a n g e s o v e r T i me
0, 0 0 0
0, 0 0 5
0, 0 1 0
0, 0 1 5
0, 0 2 0
0, 0 2 5
0, 0 3 0
0, 0 3 5
0, 0 4 0
0, 0 4 5
0 100 200 300 400
Pri c e
P
r
o
b
a
b
i
l
i
t
y
D
e
n
s
i
t
y
Bon d _ Yea r _ 1
Bon d _ Yea r _ 2
Bon d _ Yea r _ 3
Bon d _ Yea r _ 4
Bon d _ Yea r _ 5
Bon d _ Yea r _ 6
Bon d _ Yea r _ 7
Bon d _ Yea r _ 8
Bon d _ Yea r _ 9
Bon d _ Yea r _ 1 0
17
Finance School of Management
Measuring Portfolio Return
– Ii, the initial investment in asset i
– wi,the proportion of the portfolio investing in asset i
– ri, the rate of return on asset i
– rp,the rate of return on the portfolio
??
?
????
n
i
ii
i
iip rwIrIr
1
1)1(
? ?
i i
w 1
18
Finance School of Management
Short Selling
– Ik<0, short selling (borrowing) asset k
1,0 ?? ?
? ki
ik wt h e nIIf
19
Finance School of Management
Mean and Variance of Portfolio Return
?? ???
? i
ii
n
i
iipp rwrEwrEr
1
)()(
? ??
i j
jiijjip ww ????
2
–, the expected value of ri
–, the standard deviation of ri
–, the correlation between ri and rj
i?
ij?
ir
20
Finance School of Management
Variance with 2 Securities
W 1 * S ig 1 W 2 * S ig 2
W 1 * S ig 1 1 R h o( 1,2 )
W 2 * S ig 2 R h o( 2,1 ) 1
2,12121
2
2
2
2
2
1
2
1
2 2 ?????? wwww
p ???
21
Finance School of Management
Variance with 3 Securities
W1 *Sig1 W2 *Sig2 W3 *Sig3
W1 *Sig1 1 Rho(1,2) Rho(1,3)
W2 *Sig2 Rho(2,1) 1 Rho(2,3)
W3 *Sig3 Rho(3,1) Rho(3,2) 1
3,232323,13131
2,12121
2
3
2
3
2
2
2
2
2
1
2
1
2
22
2
??????
???????
wwww
wwwwwp
?
?????
22
Finance School of Management
? Suppose you invest $6000 in Bristol-Myers at an expected
return of 15%,and $4000 in Ford Motor at an expected
return of 21%
? The standard deviation of the return on BM’s stock is 18.6%,
while the standard deviation of the return on FM is 28%
? The correlation between the returns is 0.4
%4.1721.40.15.60,?????pr
0 4 9 3.28.1 8 6.4.4.6.228.40.1 8 6.60,22222 ???????????p?
%4.220 4 9 3,??p?
A Portfolio of BM and FM
23
Finance School of Management
Portfolios of BM and FM
Bristol-Myers
Ford Motor
Standard Deviation (%)
Expected Return (%)
40% F M
60% BM
24
Finance School of Management
Combining the Riskless Asset and a
Single Risky Asset
? Let’s suppose that you have $100,000 to invest,You
are choosing between a riskless asset with a interest
of 6% per year and a risky asset with an expected
rate of return of 14% per year and a standard
deviation of 20%
? How much of your $100,000 should you invest in the
risky asset?
25
Finance School of Management
Mean and Standard Deviation
F 0 100% 0, 0 6 0, 0 0
G 25% 75% 0, 0 8 0, 0 5
H 50% 50% 0, 1 0 0, 1 0
J 75% 25% 0, 1 2 0, 1 5
S 100% 0 0, 1 4 0, 2 0
St a nd a r d
D evi a t i o nP o r t fo l i o
P r o p o r t i o n
I nves t ed i n
t he Ri s k y
As s et
P r o p o r t i o n
I nves t ed i n
t he Ri s k l es s
As s et
Ex p ect ed
Ra t e o f
Ret ur n
26
Finance School of Management
The Risk-Return Trade-off Line
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0 0.05 0.1 0.15 0.2 0.25 0.3
Standard Deviation
Ex
pe
cte
d R
etu
rn
S
J
H
F
G
27
Finance School of Management
Portfolio Efficiency
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0 0.05 0.1 0.15 0.2 0.25 0.3
Standard Deviation
Ex
pe
cte
d R
etu
rn
S
J
H
F
G R
inefficient
28
Finance School of Management
Combining the Riskless Asset and a
Single Risky Asset
– We know something special about the portfolio,
namely that security 2 is riskless,so ?2 = 0,and
?p becomes
? ? 112112122211 020 ???? wwwwwp ??????
11 )( wrrrr ffp ???
29
Finance School of Management
Combining the Riskless Asset and a
Single Risky Asset
– In summary
pffp rrrrwIf ?? ])([0 111 ????
pffp rrrre l s e ?? ])([ 11???
30
Finance School of Management
A P o rt foli o o f a R i s k y a n d a R i s k l e s s S e c u ri ty
- 0, 2 0
- 0, 1 5
- 0, 1 0
- 0, 0 5
0, 0 0
0, 0 5
0, 1 0
0, 1 5
0, 2 0
0, 2 5
0, 3 0
0, 0 0 0, 1 0 0, 2 0 0, 3 0 0, 4 0 0, 5 0
Vol a t i l i t y
R
e
t
u
r
n
31
Finance School of Management
C a p i tal Mark e t L i n e
0, 0 0
0, 0 5
0, 1 0
0, 1 5
0, 2 0
0, 2 5
0, 3 0
0, 0 0 0, 0 5 0, 1 0 0, 1 5 0, 2 0 0, 2 5 0, 3 0 0, 3 5 0, 4 0 0, 4 5 0, 5 0
Vol a t i l i t y
R
e
t
u
r
n
Long risky and short risk-free
Long both risky and risk-free
100% Risky
100% Risk-less
32
Finance School of Management
Risk Premium
?The slope measure the extra
expected return the market offers for each
extra risk a investor is willing to bear
p
f
fp
rr
rr ?
? 1
1 ???
11 )( ?frr ?
33
Finance School of Management
Achieving a Target Expected Return
? To find the portfolio corresponding to an expected
rate of return of 0.11 per year,we substitute 0.11 for
E(rp) and solve for w1
? Thus,the portfolio mix is 62.5% risky asset and
37.5% riskless asset
3 7 5.0
08.006.011.0
1
1
?
??
w
w
34
Finance School of Management
Portfolios of Two Correlated
Common Stock
? Two common stock with these statistics,
– mean return 1 = 0.15
– mean return 2 = 0.10
– standard deviation 1 = 0.20
– standard deviation 2 = 0.25
– correlation of returns = 0.90
– initial price 1 = $57.25
– initial price 2 = $72.625
35
Finance School of Management
2 -S h a res, I s One " B e tt e r?"
0
0, 0 2
0, 0 4
0, 0 6
0, 0 8
0, 1
0, 1 2
0, 1 4
0, 1 6
0 0, 0 5 0, 1 0, 1 5 0, 2 0, 2 5 0, 3
St a nda rd D e v i a t i on
E
x
p
e
c
t
e
d
R
e
t
u
r
n
36
Finance School of Management
S h a re P ri c e s
0
50
100
150
200
250
300
350
0 1 2 3 4 5 6 7 8 9 10
Y e a rs
V
a
l
u
e
(
a
d
j
u
s
t
e
d
f
o
r
S
p
l
i
t
s
)
Sha r e P_1
Sha r e P_2
37
Finance School of Management
P o rt foli o o f T w o S e c u ri ti e s
0, 0 0
0, 0 5
0, 1 0
0, 1 5
0, 2 0
0, 2 5
0, 1 5 0, 1 7 0, 1 9 0, 2 1 0, 2 3 0, 2 5 0, 2 7 0, 2 9
St a nda rd D e v i a t i on
E
x
p
e
c
t
e
d
R
e
t
u
r
n
S h a re 1
S h a re 2
Ef f ic ie n t
Sub -
o p t im a
l
M in im u m
Var ia n c e
38
Finance School of Management
Fragments of the Output Table
D a ta F o r tw o s e c u r i ti e s
T h i s d a t a h a s b e e n co n s t ru ct e d
t o p ro d u ce t h e m e a n -v a ri e n ce p a ra d o x
m u _ 1 1 5, 0 0 %
m u _ 2 1 0, 0 0 %
s i g _ 1 2 0, 0 0 %
s i g _ 2 2 5, 0 0 %
rh o 9 0, 0 0 %
w _ 1 w _ 2 Po rt _ Si g Po rt _ Mu
-2, 5 0 3, 5 0 0, 4 7 7 6 -0, 0 2 5 0
-2, 4 0 3, 4 0 0, 4 6 7 4 -0, 0 2 0 0
-2, 3 0 3, 3 0 0, 4 5 7 3 -0, 0 1 5 0
-2, 2 0 3, 2 0 0, 4 4 7 2 -0, 0 1 0 0
-2, 1 0 3, 1 0 0, 4 3 7 2 -0, 0 0 5 0
-2, 0 0 3, 0 0 0, 4 2 7 2 0, 0 0 0 0
-1, 9 0 2, 9 0 0, 4 1 7 3 0, 0 0 5 0
-1, 8 0 2, 8 0 0, 4 0 7 4 0, 0 1 0 0
-1, 7 0 2, 7 0 0, 3 9 7 6 0, 0 1 5 0
1, 3 0 -0, 3 0 0, 1 9 5 3 0, 1 6 5 0
1, 4 0 -0, 4 0 0, 1 9 4 9 0, 1 7 0 0
1, 5 0 -0, 5 0 0, 1 9 5 3 0, 1 7 5 0
1, 6 0 -0, 6 0 0, 1 9 6 2 0, 1 8 0 0
1, 7 0 -0, 7 0 0, 1 9 7 8 0, 1 8 5 0
1, 8 0 -0, 8 0 0, 2 0 0 0 0, 1 9 0 0
1, 9 0 -0, 9 0 0, 2 0 2 8 0, 1 9 5 0
2, 0 0 -1, 0 0 0, 2 0 6 2 0, 2 0 0 0
2, 1 0 -1, 1 0 0, 2 1 0 1 0, 2 0 5 0
2, 2 0 -1, 2 0 0, 2 1 4 5 0, 2 1 0 0
2, 3 0 -1, 3 0 0, 2 1 9 4 0, 2 1 5 0
2, 4 0 -1, 4 0 0, 2 2 4 7 0, 2 2 0 0
2, 5 0 -1, 5 0 0, 2 3 0 5 0, 2 2 5 0
-0, 3 0 1, 3 0 0, 2 7 2 3 0, 0 8 5 0
-0, 2 1, 2, 6 4 6, 0
-0, 1 0 1, 1 0 0, 2 5 7 1 0, 0 9 5 0
0, 0 0 1, 0 0 0, 2 5 0 0 0, 1 0 0 0
0, 1 0 0, 9 0 0, 2 4 3 2 0, 1 0 5 0
0, 2 0 0, 8 0 0, 2 3 6 6 0, 1 1 0 0
0, 3 0 0, 7 0 0, 2 3 0 5 0, 1 1 5 0
39
Finance School of Management
Sample of the Excel Formulae
w _ 1 w _ 2 P o r t _ S i g P o r t _ M u
- 2, 5 = 1 - A 1 4 = S Q R T ( w _ 1 ^ 2 * s i g _ 1 ^ 2 + 2 * w _ 1 * w _ 2 * s i g _ 1 * s i g _ 2 * r h o + w _ 2 ^ 2 * s i g _ 2 ^ 2 ) = w _ 1 * m u _ 1 + w _ 2 * m u _ 2
= A 1 4 + 0, 1 = 1 - A 1 5 = S Q R T ( w _ 1 ^ 2 * s i g _ 1 ^ 2 + 2 * w _ 1 * w _ 2 * s i g _ 1 * s i g _ 2 * r h o + w _ 2 ^ 2 * s i g _ 2 ^ 2 ) = w _ 1 * m u _ 1 + w _ 2 * m u _ 2
= A 1 5 + 0, 1 = 1 - A 1 6 = S Q R T ( w _ 1 ^ 2 * s i g _ 1 ^ 2 + 2 * w _ 1 * w _ 2 * s i g _ 1 * s i g _ 2 * r h o + w _ 2 ^ 2 * s i g _ 2 ^ 2 ) = w _ 1 * m u _ 1 + w _ 2 * m u _ 2
= A 1 6 + 0, 1 = 1 - A 1 7 = S Q R T ( w _ 1 ^ 2 * s i g _ 1 ^ 2 + 2 * w _ 1 * w _ 2 * s i g _ 1 * s i g _ 2 * r h o + w _ 2 ^ 2 * s i g _ 2 ^ 2 ) = w _ 1 * m u _ 1 + w _ 2 * m u _ 2
=S Q R T ( w _ 1 ^ 2 * s i g _ 1 ^ 2 + 2 * w _ 1 * w _ 2 * s i g _ 1 * s i g _ 2 * r h o + w _ 2 ^ 2 * s i g _ 2 ^ 2 )
= w _ 1 * m u _ 1 + w _ 2 * m u _ 2
40
Finance School of Management
Formulae for Minimum Variance
Portfolio
*
1
2
2212,1
2
1
212,1
2
1*
2
2
2212,1
2
1
212,1
2
2*
1
1
2
2
w
w
w
??
??
?
?
??
?
?
?????
????
?????
????
41
Finance School of Management
Portfolios of the Riskless Security
and Two Risky Securities
? The riskless security and two risky securities with
the following statistics,
– riskless rate of return rf = 0.06
– mean return 1 = 0.14
– mean return 2 = 0.08
– standard deviation 1 = 0.20
– standard deviation 2 = 0.15
– correlation of returns = 0
42
Finance School of Management
The Optimal Combination of the
Three Securities
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0 0.05 0.1 0.15 0.2 0.25 0.3
Standard Deviation
Ex
pe
cte
d R
etu
rn
S
R
T
E
43
Finance School of Management
Formulae for Tangent Portfolio
? ? ? ?
? ? ? ? ? ?
? ?
%77.30,%23.69
15.20.002.008.020.02.15.08.
15.20.002.15.08.
1
t a n
2
t a n
1
22
2
t a n
1
1
t a n
2
212,121
2
12
2
21
212,12
2
21t a n
1
??
????????
?????
?
??
???????
???
?
ww
w
ww
rrrrrrrr
rrrr
w
ffff
ff
?????
????
12154.0)( ?TrE 1 4 5 9 5.0?T?
44
Finance School of Management
Efficient Trade-off Line
? New efficient trade-off line
? Compare the old trade-off line connecting points F
and S
? Clearly the investor is better off
pp
T
fT
fp
rr
rr ??
?
4 2 1 6 5.06,??
?
??
ppr ?4.06,??
45
Finance School of Management
Achieving a Target Expected Return
? The investment criterion is to generate a 10%
expected rate of return
? Thus,the portfolio mix is 35% riskless asset and
65% tangent portfolio,namely 45% risky security 1
and 20% risky security 2
09487.14595.65.
65.
)06.12154(.06.010.0
???
?
???
p
T
T
w
w
?
46
Finance School of Management
Selecting the Preferred Portfolio
? It is important to note that in finding the optimal
combination of risky assets,we do not need to
know anything about investor preferences
? There is always a particular optimal portfolio of
risky assets that all risk-averse investors who
share the same forecasts of rates of return will
combine with the riskless asset to reach their
most-preferred portfolio
47
Finance School of Management
The Rationale for Portfolio Selection
Return
Risk
Low Risk
High Return
High Risk
High Return
Low Risk
Low Return
High Risk
Low Return
48
Finance School of Management
Efficient Frontier
? The jelly fish shape contains all possible combinations of risk and
return,The feasible set
? The red line constitutes the efficient frontier of portfolios of risky
assets,Highest return for given risk
? The tangent portfolio T is the optimal portfolio of risky assets that
all risk-averse investors will combine with the riskless asset
Standard Deviation
Expected Return T
49
Finance School of Management
0
5 10 15
Number of Securities
Po
rtfo
lio
sta
nd
ar
d
de
via
tio
n
Nondiversifiable (Market) risk
Diversifiable (Unique) risk
The effect of increasing the
number of securities in a portfolio
Approx,20%
Diversification of Risk
50
Finance School of Management
? ? ???
2
11
n
nn
ns t o c kp o r t
???
Equation for Homogeneous
Diversification with n Stocks
? Investing equally in n stocks with the same standard
deviation and correlation
? The standard deviation of the portfolio is
Finance School of Management
Chapter 12,Portfolio
Selection and Diversification
Objective
To understand the theory of personal
portfolio selection in theory
and in practice
2
Finance School of Management
Chapter 12 Contents
? The process of personal portfolio selection
? The trade-off between expected return and
risk
? Efficient diversification with many risky
assets
3
Finance School of Management
The Concept of ‘Portfolio’
?A person’s wealth portfolio includes
– Assets,stocks,bonds,shares in unincorporated
business,houses or apartments,pensions
benefits,insurance policies,etc,
– Liabilities,student loans,auto loans,home
mortgages,etc,
4
Finance School of Management
Portfolio Selection
?A study of how people should invest their
wealth optimally
?A process of trading off risk and expected
return to find the best portfolio of assets and
liabilities
?Narrow and broad definitions
5
Finance School of Management
Portfolio Selection
?Although there are some general rules
for portfolio selection that apply to
virtually everyone,there is no single
portfolio or portfolio strategy that is
best for everyone
6
Finance School of Management
The Life Cycle
? In portfolio selection,the best strategy depends on
an individual’s personal circumstances (family
status,occupation,income,wealth)
? Illustrations
– Investing in stock market,Chang (30,a security
analyst) / Obi (30,an English teacher)
– Buying insurance policies,Miriam (a parent with
dependent children) / Sanjiv (a single person with no
dependents)
7
Finance School of Management
Time Horizon
? In formulating a plan for portfolio selection,you
begin by determining your goals and time
horizons
– Planning horizon,the total length of time for which
one plans
– Decision horizon,the length of time between decisions
to revise the portfolio
– Trading horizon,the minimum time interval over
which investors can revise their portfolios / its
determination and impacts
8
Finance School of Management
Risk Tolerance
? A major determinant of portfolio choices
? It is influenced by such characteristics as
– age,family status,job status,wealth,and
– other attributes that affect a person’s ability to maintain
his standard of living in the face of adverse movements
in the market value of his investment portfolio
9
Finance School of Management
Professional Asset Managers
? Investment advisors &,finished products” from
a financial intermediary
? Specialization,information and cost advantages
10
Finance School of Management
The Trade-off Between Expected
Return and Risk
? The objective is to find the portfolio which offers
investors the highest expected rate of return for the
degree of risk they are willing to tolerate
? Two step process,
– find the optimal combinational of risky assets
– mix this optimal risk-asset with the riskless asset
11
Finance School of Management
Riskless Asset
? A security that offers a perfectly predictable rate
of return in terms of the unit of account selected
for the analysis and the length of the investor’s
decision horizon
? For example,if the U.S dollars is taken as the unit
of account and the trading horizon is half a year,
the riskless rate is the interest rate on U.S
Treasury bills maturing after half a year,
12
Finance School of Management
Rates of Return on Risky Assets
? Required return depends on the risk of the
investment,
– Greater the risk,greater the return
– Risk premium
13
Finance School of Management
S e c u ri ty P ri c e s
10
100
1000
10000
100000
0 5 10 15 20 25 30 35 40
Y e a rs
V
a
l
u
e
(
L
o
g
)
St o c k
Bon d
St o c k _ M u
Bon d _ M u
14
Finance School of Management
S e c u ri ty P ri c e s
10
100
1000
10000
100000
0 5 10 15 20 25 30 35 40
Y e a rs
V
a
l
u
e
(
L
o
g
)
St o c k
Bon d
St o c k _ M u
Bon d _ M u
15
Finance School of Management
P rob a b i l i s ti c S toc k P ri c e C h a n g e s Ov e r T i me
0, 0 0 0
0, 0 0 2
0, 0 0 4
0, 0 0 6
0, 0 0 8
0, 0 1 0
0, 0 1 2
0, 0 1 4
0, 0 1 6
0, 0 1 8
0, 0 2 0
0 200 400 600 800
Pri c e
P
r
o
b
a
b
i
l
i
t
y
D
e
n
s
i
t
y
St o c k _ Yea r _ 1
St o c k _ Yea r _ 2
St o c k _ Yea r _ 3
St o c k _ Yea r _ 4
St o c k _ Yea r _ 5
St o c k _ Yea r _ 6
St o c k _ Yea r _ 7
St o c k _ Yea r _ 8
St o c k _ Yea r _ 9
St o c k _ Yea r _ 1 0
16
Finance School of Management
P rob a b i l i s ti c B o n d P ri c e C h a n g e s o v e r T i me
0, 0 0 0
0, 0 0 5
0, 0 1 0
0, 0 1 5
0, 0 2 0
0, 0 2 5
0, 0 3 0
0, 0 3 5
0, 0 4 0
0, 0 4 5
0 100 200 300 400
Pri c e
P
r
o
b
a
b
i
l
i
t
y
D
e
n
s
i
t
y
Bon d _ Yea r _ 1
Bon d _ Yea r _ 2
Bon d _ Yea r _ 3
Bon d _ Yea r _ 4
Bon d _ Yea r _ 5
Bon d _ Yea r _ 6
Bon d _ Yea r _ 7
Bon d _ Yea r _ 8
Bon d _ Yea r _ 9
Bon d _ Yea r _ 1 0
17
Finance School of Management
Measuring Portfolio Return
– Ii, the initial investment in asset i
– wi,the proportion of the portfolio investing in asset i
– ri, the rate of return on asset i
– rp,the rate of return on the portfolio
??
?
????
n
i
ii
i
iip rwIrIr
1
1)1(
? ?
i i
w 1
18
Finance School of Management
Short Selling
– Ik<0, short selling (borrowing) asset k
1,0 ?? ?
? ki
ik wt h e nIIf
19
Finance School of Management
Mean and Variance of Portfolio Return
?? ???
? i
ii
n
i
iipp rwrEwrEr
1
)()(
? ??
i j
jiijjip ww ????
2
–, the expected value of ri
–, the standard deviation of ri
–, the correlation between ri and rj
i?
ij?
ir
20
Finance School of Management
Variance with 2 Securities
W 1 * S ig 1 W 2 * S ig 2
W 1 * S ig 1 1 R h o( 1,2 )
W 2 * S ig 2 R h o( 2,1 ) 1
2,12121
2
2
2
2
2
1
2
1
2 2 ?????? wwww
p ???
21
Finance School of Management
Variance with 3 Securities
W1 *Sig1 W2 *Sig2 W3 *Sig3
W1 *Sig1 1 Rho(1,2) Rho(1,3)
W2 *Sig2 Rho(2,1) 1 Rho(2,3)
W3 *Sig3 Rho(3,1) Rho(3,2) 1
3,232323,13131
2,12121
2
3
2
3
2
2
2
2
2
1
2
1
2
22
2
??????
???????
wwww
wwwwwp
?
?????
22
Finance School of Management
? Suppose you invest $6000 in Bristol-Myers at an expected
return of 15%,and $4000 in Ford Motor at an expected
return of 21%
? The standard deviation of the return on BM’s stock is 18.6%,
while the standard deviation of the return on FM is 28%
? The correlation between the returns is 0.4
%4.1721.40.15.60,?????pr
0 4 9 3.28.1 8 6.4.4.6.228.40.1 8 6.60,22222 ???????????p?
%4.220 4 9 3,??p?
A Portfolio of BM and FM
23
Finance School of Management
Portfolios of BM and FM
Bristol-Myers
Ford Motor
Standard Deviation (%)
Expected Return (%)
40% F M
60% BM
24
Finance School of Management
Combining the Riskless Asset and a
Single Risky Asset
? Let’s suppose that you have $100,000 to invest,You
are choosing between a riskless asset with a interest
of 6% per year and a risky asset with an expected
rate of return of 14% per year and a standard
deviation of 20%
? How much of your $100,000 should you invest in the
risky asset?
25
Finance School of Management
Mean and Standard Deviation
F 0 100% 0, 0 6 0, 0 0
G 25% 75% 0, 0 8 0, 0 5
H 50% 50% 0, 1 0 0, 1 0
J 75% 25% 0, 1 2 0, 1 5
S 100% 0 0, 1 4 0, 2 0
St a nd a r d
D evi a t i o nP o r t fo l i o
P r o p o r t i o n
I nves t ed i n
t he Ri s k y
As s et
P r o p o r t i o n
I nves t ed i n
t he Ri s k l es s
As s et
Ex p ect ed
Ra t e o f
Ret ur n
26
Finance School of Management
The Risk-Return Trade-off Line
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0 0.05 0.1 0.15 0.2 0.25 0.3
Standard Deviation
Ex
pe
cte
d R
etu
rn
S
J
H
F
G
27
Finance School of Management
Portfolio Efficiency
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0 0.05 0.1 0.15 0.2 0.25 0.3
Standard Deviation
Ex
pe
cte
d R
etu
rn
S
J
H
F
G R
inefficient
28
Finance School of Management
Combining the Riskless Asset and a
Single Risky Asset
– We know something special about the portfolio,
namely that security 2 is riskless,so ?2 = 0,and
?p becomes
? ? 112112122211 020 ???? wwwwwp ??????
11 )( wrrrr ffp ???
29
Finance School of Management
Combining the Riskless Asset and a
Single Risky Asset
– In summary
pffp rrrrwIf ?? ])([0 111 ????
pffp rrrre l s e ?? ])([ 11???
30
Finance School of Management
A P o rt foli o o f a R i s k y a n d a R i s k l e s s S e c u ri ty
- 0, 2 0
- 0, 1 5
- 0, 1 0
- 0, 0 5
0, 0 0
0, 0 5
0, 1 0
0, 1 5
0, 2 0
0, 2 5
0, 3 0
0, 0 0 0, 1 0 0, 2 0 0, 3 0 0, 4 0 0, 5 0
Vol a t i l i t y
R
e
t
u
r
n
31
Finance School of Management
C a p i tal Mark e t L i n e
0, 0 0
0, 0 5
0, 1 0
0, 1 5
0, 2 0
0, 2 5
0, 3 0
0, 0 0 0, 0 5 0, 1 0 0, 1 5 0, 2 0 0, 2 5 0, 3 0 0, 3 5 0, 4 0 0, 4 5 0, 5 0
Vol a t i l i t y
R
e
t
u
r
n
Long risky and short risk-free
Long both risky and risk-free
100% Risky
100% Risk-less
32
Finance School of Management
Risk Premium
?The slope measure the extra
expected return the market offers for each
extra risk a investor is willing to bear
p
f
fp
rr
rr ?
? 1
1 ???
11 )( ?frr ?
33
Finance School of Management
Achieving a Target Expected Return
? To find the portfolio corresponding to an expected
rate of return of 0.11 per year,we substitute 0.11 for
E(rp) and solve for w1
? Thus,the portfolio mix is 62.5% risky asset and
37.5% riskless asset
3 7 5.0
08.006.011.0
1
1
?
??
w
w
34
Finance School of Management
Portfolios of Two Correlated
Common Stock
? Two common stock with these statistics,
– mean return 1 = 0.15
– mean return 2 = 0.10
– standard deviation 1 = 0.20
– standard deviation 2 = 0.25
– correlation of returns = 0.90
– initial price 1 = $57.25
– initial price 2 = $72.625
35
Finance School of Management
2 -S h a res, I s One " B e tt e r?"
0
0, 0 2
0, 0 4
0, 0 6
0, 0 8
0, 1
0, 1 2
0, 1 4
0, 1 6
0 0, 0 5 0, 1 0, 1 5 0, 2 0, 2 5 0, 3
St a nda rd D e v i a t i on
E
x
p
e
c
t
e
d
R
e
t
u
r
n
36
Finance School of Management
S h a re P ri c e s
0
50
100
150
200
250
300
350
0 1 2 3 4 5 6 7 8 9 10
Y e a rs
V
a
l
u
e
(
a
d
j
u
s
t
e
d
f
o
r
S
p
l
i
t
s
)
Sha r e P_1
Sha r e P_2
37
Finance School of Management
P o rt foli o o f T w o S e c u ri ti e s
0, 0 0
0, 0 5
0, 1 0
0, 1 5
0, 2 0
0, 2 5
0, 1 5 0, 1 7 0, 1 9 0, 2 1 0, 2 3 0, 2 5 0, 2 7 0, 2 9
St a nda rd D e v i a t i on
E
x
p
e
c
t
e
d
R
e
t
u
r
n
S h a re 1
S h a re 2
Ef f ic ie n t
Sub -
o p t im a
l
M in im u m
Var ia n c e
38
Finance School of Management
Fragments of the Output Table
D a ta F o r tw o s e c u r i ti e s
T h i s d a t a h a s b e e n co n s t ru ct e d
t o p ro d u ce t h e m e a n -v a ri e n ce p a ra d o x
m u _ 1 1 5, 0 0 %
m u _ 2 1 0, 0 0 %
s i g _ 1 2 0, 0 0 %
s i g _ 2 2 5, 0 0 %
rh o 9 0, 0 0 %
w _ 1 w _ 2 Po rt _ Si g Po rt _ Mu
-2, 5 0 3, 5 0 0, 4 7 7 6 -0, 0 2 5 0
-2, 4 0 3, 4 0 0, 4 6 7 4 -0, 0 2 0 0
-2, 3 0 3, 3 0 0, 4 5 7 3 -0, 0 1 5 0
-2, 2 0 3, 2 0 0, 4 4 7 2 -0, 0 1 0 0
-2, 1 0 3, 1 0 0, 4 3 7 2 -0, 0 0 5 0
-2, 0 0 3, 0 0 0, 4 2 7 2 0, 0 0 0 0
-1, 9 0 2, 9 0 0, 4 1 7 3 0, 0 0 5 0
-1, 8 0 2, 8 0 0, 4 0 7 4 0, 0 1 0 0
-1, 7 0 2, 7 0 0, 3 9 7 6 0, 0 1 5 0
1, 3 0 -0, 3 0 0, 1 9 5 3 0, 1 6 5 0
1, 4 0 -0, 4 0 0, 1 9 4 9 0, 1 7 0 0
1, 5 0 -0, 5 0 0, 1 9 5 3 0, 1 7 5 0
1, 6 0 -0, 6 0 0, 1 9 6 2 0, 1 8 0 0
1, 7 0 -0, 7 0 0, 1 9 7 8 0, 1 8 5 0
1, 8 0 -0, 8 0 0, 2 0 0 0 0, 1 9 0 0
1, 9 0 -0, 9 0 0, 2 0 2 8 0, 1 9 5 0
2, 0 0 -1, 0 0 0, 2 0 6 2 0, 2 0 0 0
2, 1 0 -1, 1 0 0, 2 1 0 1 0, 2 0 5 0
2, 2 0 -1, 2 0 0, 2 1 4 5 0, 2 1 0 0
2, 3 0 -1, 3 0 0, 2 1 9 4 0, 2 1 5 0
2, 4 0 -1, 4 0 0, 2 2 4 7 0, 2 2 0 0
2, 5 0 -1, 5 0 0, 2 3 0 5 0, 2 2 5 0
-0, 3 0 1, 3 0 0, 2 7 2 3 0, 0 8 5 0
-0, 2 1, 2, 6 4 6, 0
-0, 1 0 1, 1 0 0, 2 5 7 1 0, 0 9 5 0
0, 0 0 1, 0 0 0, 2 5 0 0 0, 1 0 0 0
0, 1 0 0, 9 0 0, 2 4 3 2 0, 1 0 5 0
0, 2 0 0, 8 0 0, 2 3 6 6 0, 1 1 0 0
0, 3 0 0, 7 0 0, 2 3 0 5 0, 1 1 5 0
39
Finance School of Management
Sample of the Excel Formulae
w _ 1 w _ 2 P o r t _ S i g P o r t _ M u
- 2, 5 = 1 - A 1 4 = S Q R T ( w _ 1 ^ 2 * s i g _ 1 ^ 2 + 2 * w _ 1 * w _ 2 * s i g _ 1 * s i g _ 2 * r h o + w _ 2 ^ 2 * s i g _ 2 ^ 2 ) = w _ 1 * m u _ 1 + w _ 2 * m u _ 2
= A 1 4 + 0, 1 = 1 - A 1 5 = S Q R T ( w _ 1 ^ 2 * s i g _ 1 ^ 2 + 2 * w _ 1 * w _ 2 * s i g _ 1 * s i g _ 2 * r h o + w _ 2 ^ 2 * s i g _ 2 ^ 2 ) = w _ 1 * m u _ 1 + w _ 2 * m u _ 2
= A 1 5 + 0, 1 = 1 - A 1 6 = S Q R T ( w _ 1 ^ 2 * s i g _ 1 ^ 2 + 2 * w _ 1 * w _ 2 * s i g _ 1 * s i g _ 2 * r h o + w _ 2 ^ 2 * s i g _ 2 ^ 2 ) = w _ 1 * m u _ 1 + w _ 2 * m u _ 2
= A 1 6 + 0, 1 = 1 - A 1 7 = S Q R T ( w _ 1 ^ 2 * s i g _ 1 ^ 2 + 2 * w _ 1 * w _ 2 * s i g _ 1 * s i g _ 2 * r h o + w _ 2 ^ 2 * s i g _ 2 ^ 2 ) = w _ 1 * m u _ 1 + w _ 2 * m u _ 2
=S Q R T ( w _ 1 ^ 2 * s i g _ 1 ^ 2 + 2 * w _ 1 * w _ 2 * s i g _ 1 * s i g _ 2 * r h o + w _ 2 ^ 2 * s i g _ 2 ^ 2 )
= w _ 1 * m u _ 1 + w _ 2 * m u _ 2
40
Finance School of Management
Formulae for Minimum Variance
Portfolio
*
1
2
2212,1
2
1
212,1
2
1*
2
2
2212,1
2
1
212,1
2
2*
1
1
2
2
w
w
w
??
??
?
?
??
?
?
?????
????
?????
????
41
Finance School of Management
Portfolios of the Riskless Security
and Two Risky Securities
? The riskless security and two risky securities with
the following statistics,
– riskless rate of return rf = 0.06
– mean return 1 = 0.14
– mean return 2 = 0.08
– standard deviation 1 = 0.20
– standard deviation 2 = 0.15
– correlation of returns = 0
42
Finance School of Management
The Optimal Combination of the
Three Securities
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0 0.05 0.1 0.15 0.2 0.25 0.3
Standard Deviation
Ex
pe
cte
d R
etu
rn
S
R
T
E
43
Finance School of Management
Formulae for Tangent Portfolio
? ? ? ?
? ? ? ? ? ?
? ?
%77.30,%23.69
15.20.002.008.020.02.15.08.
15.20.002.15.08.
1
t a n
2
t a n
1
22
2
t a n
1
1
t a n
2
212,121
2
12
2
21
212,12
2
21t a n
1
??
????????
?????
?
??
???????
???
?
ww
w
ww
rrrrrrrr
rrrr
w
ffff
ff
?????
????
12154.0)( ?TrE 1 4 5 9 5.0?T?
44
Finance School of Management
Efficient Trade-off Line
? New efficient trade-off line
? Compare the old trade-off line connecting points F
and S
? Clearly the investor is better off
pp
T
fT
fp
rr
rr ??
?
4 2 1 6 5.06,??
?
??
ppr ?4.06,??
45
Finance School of Management
Achieving a Target Expected Return
? The investment criterion is to generate a 10%
expected rate of return
? Thus,the portfolio mix is 35% riskless asset and
65% tangent portfolio,namely 45% risky security 1
and 20% risky security 2
09487.14595.65.
65.
)06.12154(.06.010.0
???
?
???
p
T
T
w
w
?
46
Finance School of Management
Selecting the Preferred Portfolio
? It is important to note that in finding the optimal
combination of risky assets,we do not need to
know anything about investor preferences
? There is always a particular optimal portfolio of
risky assets that all risk-averse investors who
share the same forecasts of rates of return will
combine with the riskless asset to reach their
most-preferred portfolio
47
Finance School of Management
The Rationale for Portfolio Selection
Return
Risk
Low Risk
High Return
High Risk
High Return
Low Risk
Low Return
High Risk
Low Return
48
Finance School of Management
Efficient Frontier
? The jelly fish shape contains all possible combinations of risk and
return,The feasible set
? The red line constitutes the efficient frontier of portfolios of risky
assets,Highest return for given risk
? The tangent portfolio T is the optimal portfolio of risky assets that
all risk-averse investors will combine with the riskless asset
Standard Deviation
Expected Return T
49
Finance School of Management
0
5 10 15
Number of Securities
Po
rtfo
lio
sta
nd
ar
d
de
via
tio
n
Nondiversifiable (Market) risk
Diversifiable (Unique) risk
The effect of increasing the
number of securities in a portfolio
Approx,20%
Diversification of Risk
50
Finance School of Management
? ? ???
2
11
n
nn
ns t o c kp o r t
???
Equation for Homogeneous
Diversification with n Stocks
? Investing equally in n stocks with the same standard
deviation and correlation
? The standard deviation of the portfolio is
