H
AAS
S
CHOOL
OF
B
USINESS
U
NIVERSITY
OF
C
ALIFORNIA
AT
B
ERKELEY
UGBA 103
© A
VINASH
V
ERMA
P
ORTFOLIO
T
HEORY
: T
WO
S
ECURITY
P
ORTFOLIOS
[
R
EAD
THIS
NOTE
TOGETHER
WITH
THE
E
XCEL
F
ILE
CALLED
“2
SEC
PORTFOLIOS
.
XLS
”
]
1.
The main question that we are concerned with in
portfolio theory
is that of
asset
allocation:
how much should we invest in what asset or security
1
given our twin goals
of maximizing the expected return on the portfolio and minimizing the portfolio risk?
We shall get basic insights into the problem by looking at it in the simplified context of
two securities before we go on to the case of many securities. Let us start analyzing
twosecurity portfolios by recalling from the previous lecture note (Uncertainty and
Risk in Finance) the formula for the expected return on a portfolio:
E R
x E R
p
i
i
i
n
(
~
)
(
~
)
=
=
∑
1
.
Setting
n=2
, and denoting
E R
p
(
~
)
more concisely as
E
p
, and
E R
(
~
)
1
and
E R
(
~
)
2
as
E
1
and
E
2
, we get:
E
x E
x E
p
=
+
1
1
2
2
.
[1]
2.
The previous note concluded with the formula for the variance of the returns on the
portfolio:
σ
p
i
j
ij
j
n
i
n
x x
2
1
1
=
=
=
∑
∑
.
Setting
n=2
, and expanding the double sum in terms of the matrix in the last section of
the previous note, we get:
SECURITY 1
SECURITY 2
SECURITY 1
x x
x
1
1
11
1
2
1
2
=
x x
x x
1
2
12
1
2
21
=
SECURITY 2
x x
x x
2
1
21
1
2
12
=
x x
x
2
2
22
2
2
2
2
=
Summing the four boxes would lead to (the two “offdiagonal” boxes are identical):
p
x
x
x x
2
1
2
1
2
2
2
2
2
1
2
12
2
=
+
+
,
Observing that
2
1
12
12
ρ
=
, we can restate the equation above as:
2
1
12
2
1
2
2
2
2
2
1
2
1
2
2
x
x
x
x
p
+
+
=
.
[2]
3.
We shall analyze twosecurity portfolios by plotting the
reward
as measured by the
expected return
on the vertical axis against the
risk
as measured by the
standard
1
A security is a homogenized asset. A house is an asset, because one house differs from another, and the
difference is relevant to financial valuation. For the purposes of financial valuation, one share of
Microsoft Common Stock is exactly the same as any other share. Thus, Microsoft Common stock is a
security. We shall be using the words “asset” and “security” interchangeably distinguishing between the
two only if the distinction matters in the specific context.
1
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AAS
S
CHOOL
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B
USINESS
U
NIVERSITY
OF
C
ALIFORNIA
AT
B
ERKELEY
UGBA 103
© A
VINASH
V
ERMA
deviation
of returns
on the horizontal axis. Since we are interested in exploring how
combining two securities in a portfolio might affect the risk of the portfolio, we shall
keep
fixed
all
attributes of the two securities
other than
the correlation between them,
which will be gradually varied from its highest possible value of +1 to its lowest
possible value of –1. We can think of changes in correlation as follows: Suppose there
are two pools of securities such that securities in each pool are identical except for the
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 Spring '08
 MCCULLOUGH
 Variance, Modern portfolio theory, HAAS SCHOOL OF BUSINESS

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