The investor has two alternatives:

Investment theory, class #2
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Invest
δ
in the market portfolio
Invest
δ
in stock A
Additional return:
]
)
(
[
)
(
f
m
r
r
E
r
E
−
⋅
=
Δ
δ
New portfolio risk:
{
(
)
(
2
2
2
2
2
2
0
0
2
2
2
2
2
)
2
(
)
1
(
,
cov
1
2
)
1
(
m
m
m
f
rf
m
r
M
σ
δ
δ
σ
σ
δ
σ
δ
δ
σ
δ
σ
δ
σ
⋅
+
+
=
⋅
+
=
⇒
+
−
+
⋅
+
=
43
42
1
and since
δ
is very small, we can omit
2
δ
and get that the additional portfolio
risk is:
2
2
2
m
σ
δ
σ
⋅
=
Δ
The
compensation
on
risk
for
the
individual is therefore:
2
2
2
2
)
(
2
]
)
(
[
)
(
m
f
m
m
f
m
r
r
E
r
r
E
r
E
σ
σ
δ
δ
σ
−
=
⋅
−
⋅
=
Δ
Δ
Additional return:
]
)
(
[
)
(
f
A
r
r
E
r
E
−
⋅
=
Δ
δ
New portfolio risk:
( )
{
(
)
(
)
)
,
(
1
2
1
,
1
2
,
2
)
,
(
1
2
1
2
2
2
2
0
0
0
2
2
2
2
2
2
M
A
Cov
r
M
Cov
r
A
Cov
M
A
Cov
A
m
f
f
rf
A
m
⋅
⋅
⋅
+
⋅
+
⋅
=
⇒
⋅
⋅
⋅
−
⋅
⋅
⋅
−
⋅
⋅
⋅
+
−
+
⋅
+
⋅
=
δ
σ
δ
σ
σ
δ
δ
δ
δ
σ
δ
σ
δ
σ
σ
43
42
1
43
42
1
and since
δ
is very small, we can
omit
2
δ
and get that the additional
portfolio
risk
is:
)
,
(
2
2
M
A
COV
⋅
=
Δ
δ
σ
The compensation on risk for the
individual is therefore:
)
,
(
2
)
(
)
,
(
2
]
)
(
[
)
(
2
M
A
COV
r
r
E
M
A
COV
r
r
E
r
E
f
A
f
A
⋅
−
=
⋅
−
⋅
=
Δ
Δ
δ
δ
σ
Key: the two “compensation on risk” should equal, and we get:
)
,
(
2
)
(
2
)
(
)
(
2
2
M
A
Cov
r
r
E
r
r
E
r
E
f
A
M
f
M
⋅
−
=
−
=
Δ
Δ
σ
σ
If, for example, the RHS is larger than the LHS, the investor should sell short M
and buy A, and increase his return and not the risk.
From this we get:

Investment theory, class #2
Page 16 of 22
® All rights reserved to Tal Mofkadi
[
]
f
M
M
f
A
r
r
E
M
A
Cov
r
r
E
−
+
=
)
(
)
,
(
)
(
2
σ
and if we define
2
)
,
(
M
A
M
A
Cov
σ
β
=
, we get:
[
]
f
M
A
f
A
r
r
E
r
r
E
−
+
=
)
(
)
(
β
3.4
Calculating Beta
If we use a linear regression (OLS) of the form:
[
]
f
M
A
f
A
r
r
E
r
r
E
−
=
−
)
(
)
(
β
or alternatively:
(
)
{
{
)
(
)
(
1
M
r
A
r
E
r
E
A
f
A
β
β
β
α
+
=
−
. This is also known as the “market model”
Statistics tell us that in this model:
(
)
( )
(
)
( )
x
Var
Y
X
Cov
X
n
X
Y
X
n
Y
X
i
i
i
,
2
2
=
⋅
−
⋅
⋅
−
⋅
=
∑
∑
β
.
Technically, this is done using statistical program or by using Excel’s regression
capabilities or “Slope” worksheet function.
Will be seen in the tutorial!
3.4.1 The meaning of Beta
Beta is a measure of the additional risk of an asset to the overall market portfolio.
In other words, Beta is a proxy of an asset’s risk due to market “shocks”.
For example, if the beta of asset i is 0.5 this means that when the market portfolio
absorbs a shock of 1% (i.e. r
m
-r
f
is grater in 1%) asset i expect to absorb a shock of
0.5% (i.e. r
i
-r
f
is expected to be grater in 0.5%). This makes asset i less risky than
the market portfolio (why?) and therefore we should expect it to yield lower rate
of return.
Note that the beta of a portfolio is a weighted sum of its asset’s betas, that is:
∑
=
=
n
i
i
i
p
x
1
β
β

Investment theory, class #2
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® All rights reserved to Tal Mofkadi
3.5
Variance decomposition
For any asset, the total variance can be expressed as follows:
()
( )
( )
{
risk
Specific
2
risk
Systematic
2
2
risk
Specific
risk
Systematic
2
risk
Total
ε
σ
σ
β
ε
β
+
=
+
=
3
2
1
3
2
1
43
42
1
3
2
1
m
i
i
Var
m
Var
i
Var
3.6
The SML (security market line)
[
]
f
M
A
f
A
r
r
E
r
r
E
−
+
=
)
(
)
(
β
Note that:
1.
1
=
m
β
since
1
1
2
=
=
m
m
m
m
σ
σ
σ
β
2.
0
=
rf
β
since
0
0
0
2
=
⋅
⋅
=
m
m
rf
σ
σ
β
Thus we can graphically present the SML formula:
CML
Stock A
Standard deviation-
σ
Expected return
- E(r)
R
f
Specific risk
Market risk
Stupidity
Dreams

Investment theory, class #2