Investments and Portfolio Analysis
Lecture 5 — Portfolio Allocation II
Dr. Jianan Liu
Fall, 2019
Today’s Agenda
I
Review
I
New topics:
1.
Combining risky assets
2.
N risky assets and one risk free assets
3.
Book Chapter 7
I
The midterm
Review of last lecture
I
Book Chapter 6
Utility function (value)
I
A common functional form of utility function:
U
=
E
(
R
p
)

1
2
A
σ
2
(
R
p
)
I
A
is the risk aversion parameter
I
Investors with
A
>
0 and finite are meanvariance investors
I
Meanvariance investor’s preference:
I
Given expected return, assets that have lower risk give
investors higher utility value
I
Given level of risks, assets that have higher expected return
give investors higher utility value
Calculation to pin down
A
I
Calculation to pin down
A
—one way is to find the exact
(
E
(
R
p
)
, σ
2
(
R
p
)) that makes investor
just
accepts the bet
I
Just
indicates that the investors are indifferent between
accepting the bets and not accepting the bets; that is,
U
= 0
Portfolios and weights
A combination of two risky assets—
A
and
B
—with weights
ω
A
and 1

ω
A
is denoted as portfolio
c
I
expected return on
c
:
E
(
R
c
) =
ω
A
E
(
R
A
) + (1

ω
A
)
E
(
R
B
)
I
variance of portfolio
c
:
Var
(
R
c
) =
ω
2
A
Var
(
R
A
)+(1

ω
A
)
2
Var
(
R
B
)+2
ω
A
(1

ω
A
)
Cov
(
R
A
,
R
B
)
I
covariance of
A
and
B
—
Cov
(
R
A
,
R
B
):
Cov
(
R
A
,
R
B
) =
Corr
(
R
A
,
R
B
)
×
p
Var
(
R
A
)
×
p
Var
(
R
B
)
Portfolio weights and balance sheet
There is a mapping between balance sheet and portfolio weights.
An investor has initial capital $
V
, and she invests in assets
A
and
B
I
when 0
≤
ω
A
≤
1,
Portfolio weights and balance sheet
I
when
ω
A
>
1, she is shorting
B
and buying
A
with shorting
proceeds
Portfolio weights and buying on margin
I
when
ω
A
>
1 and
B
is cash, it’s buying
A
on margin case
I
Recall, when initial margin is 50%, the investor buying
A
on
margin has balance sheet:
I
Her portfolio is consisted: asset
A
and cash(margin loans)
denoted as
B
, with weights:
ω
A
= $2
V
/
$
V
= 2
ω
B
=

($
V
)
/
$
V
=

1
Portfolio weights and shortselling
I
when
ω
A
<
0 and
B
is cash, it’s short selling
A
case
I
Recall, when initial margin is 50%, the investor short selling
A
has balance sheet:
I
Her portfolio is consisted of asset
A
and cash (proceeds and
deposits) denoted as
B
, with weights:
ω
A
=

$2
V
/
$
V
=

2
ω
B
= (3$
V
)
/
$
V
= 3
Capital allocation line—combining one risky asset
and one riskfree asset
I
Risky asset has expected return—
E
(
R
p
) and standard
deviation—
σ
(
R
p
), and the riskfree rate is
R
f
I
Opportunity sets—all possible combinations of
p
and riskfree
asset, which are on Capital Allocation Line (CAL)
Golden formulas of CAL
The weight on the risky asset is
y
. A combined portfolio
c
has:
I
Risk premium:
E
(
R
c
)

R
f
=
y
(
E
(
R
p
)

R
f
)
Golden formulas of CAL
The weight on the risky asset is
y
. A combined portfolio
c
has:
I
Risk premium:
E
(
R
c
)

R
f
=
y
(
E
(
R
p
)

R
f
)
I
standard deviation:
σ
(
R
c
) =
y
σ
(
R
p
)
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