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Mean-Variance Analysis and CAPM. Eco 525: Financial Economics I. Slide 05-1. Lecture 05: Mean-Variance Analysis &. Capital Asset Pricing Model.
Typology: Summaries
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16:14 Lecture 05 16:14 Lecture 05
MeanMean-
-Variance Analysis and CAPM
Variance Analysis and CAPM
Slide 05- Slide 05
16:14 Lecture 05 16:14 Lecture 05
MeanMean-
-Variance Analysis and CAPM
Variance Analysis and CAPM
Slide 05- Slide 05
16:14 Lecture 05 16:14 Lecture 05
MeanMean-
-Variance Analysis and CAPM
Variance Analysis and CAPM
Slide 05- Slide 05
Simple CAPM with Quadratic Expected Utility
Simple CAPM with Quadratic Expected Utility
All agents are identical
Expected utility U(x
, x
) =
∑
π
u(x
, xs
)
⇒
m=
∂
u / E[
∂
u]
Quadratic u(x
,x
)=v
(x
) - (x
α
)
⇒
∂
u = [-2(x
α
),…, -2(x
α
)]
E[R
] – R
= - Cov[m,R
] / E[m]
= -R
Cov[
∂
u, R
] / E[
∂
u]
= -R
Cov[-2(x
α
), R
] / E[
∂
u]
= R
2Cov[x
,R
] / E[
∂
u]
Also holds for market portfolio
E[R
] – R
= R
2Cov[x
,R
]/E[
∂
u]
⇒
16:14 Lecture 05 16:14 Lecture 05
MeanMean-
-Variance Analysis and CAPM
Variance Analysis and CAPM
Slide 05- Slide 05
Simple CAPM with Quadratic Expected Utility
Simple CAPM with Quadratic Expected Utility
Homogenous agents + Exchange economy ⇒
x
1
= agg. endowment and is perfectly correlated with R
m
E[R
E[R
h h
]= ]=
R
R
f f
+
+
β β
h h
{ {E[R
E[R
m m
] ]-
-R
R
f f
} }
Market Securit
y
Lin
e
Market Securit
y Line
N.B.:
R
=R
(a+b
R
)/(a+b
R
) in this case (where b
<0)!
16:14 Lecture 05 16:14 Lecture 05
MeanMean-
-Variance Analysis and CAPM
Variance Analysis and CAPM
Slide 05- Slide 05
A
B
A
Β
A
B
A
B
16:14 Lecture 05 16:14 Lecture 05
MeanMean-
-Variance Analysis and CAPM
Variance Analysis and CAPM
Slide 05- Slide 05
Expected Portfolio Returns & Variance
Expected Portfolio Returns & Variance
recall that correlationcoefficient
∈
[-1,1]
μ
:=
E
[
r
] =
w
μ
, where each
μ
=
j
P
j
j
σ
2 p
:=
V ar
[
r
p
]
=
w
V w
= (
w
w
)
μ
σ
σ
σ
σ
¶ μ
w
w
¶
=
(
w
σ
w
σ
w
σ
w
σ
)
μ
w
w
¶
=
w
σ
w
σ
w
w
σ
≤
0
since
σ
≤ −
σ
σ
.
16:14 Lecture 05 16:14 Lecture 05
MeanMean-
-Variance Analysis and CAPM
Variance Analysis and CAPM
Slide 05- Slide 05
The Efficient Frontier: Two Perfectly Correlated Risky Assets
For
ρ
= 1:
Hence,
σ
σ
σ
μ
Lower part with … is irrelevant
E[r
2
]
E[r
1
]
σ
p
=
|
w
σ
−
w
)
σ
|
μ
p
=
w
μ
−
w
)
μ
w
=
σ
p
σ
σ
1
σ
2
16:14 Lecture 05 16:14 Lecture 05
MeanMean-
-Variance Analysis and CAPM
Variance Analysis and CAPM
Slide 05- Slide 05
Efficient Frontier: Two Perfectly Negative Correlated Risky Assets
For
ρ
= -1:
Hence,
σ
σ
E[r
2
]
E[r
1
]
σ
p
=
|
w
σ
−
(
−
w
)
σ
|
μ
p
=
w
μ
−
w
)
μ
σ
2
σ
1
σ
2
μ
σ
1
σ
1
σ
2
μ
slope:
μ
2
μ
1
σ
1
σ
2
σ
p
slope:
−
μ
2
μ
1
σ
1
σ
2
σ
p
w
=
σ
p
σ
σ
1
σ
2
16:14 Lecture 05 16:14 Lecture 05
MeanMean-
-Variance Analysis and CAPM
Variance Analysis and CAPM
Slide 05- Slide 05
The Efficient Frontier: One Risky and One Risk Free Asset
For
σ
= 0
σ
σ
σ
μ
E[r
2
]
E[r
1
]
16:14 Lecture 05 16:14 Lecture 05
MeanMean-
-Variance Analysis and CAPM
Variance Analysis and CAPM
Slide 05- Slide 05
The Efficient Frontier: One Risk Free and n Risky Assets
16:14 Lecture 05 16:14 Lecture 05
MeanMean-
-Variance Analysis and CAPM
Variance Analysis and CAPM
Slide 05- Slide 05
Optimal Portfolios of Two Investors with Different Risk Aversion
Optimal Portfolio: Two Fund Separation
Optimal Portfolio: Two Fund Separation
Price of Risk == highest Sharpe
ratio
16:14 Lecture 05 16:14 Lecture 05
MeanMean-
-Variance Analysis and CAPM
Variance Analysis and CAPM
Slide 05- Slide 05
Equilibrium leads to CAPM
Equilibrium leads to CAPM
Portfolio theory: only analysis of demand
Equilibrium Demand = Supply (market portfolio)
CAPM allows to derive
16:14 Lecture 05 16:14 Lecture 05
MeanMean-
-Variance Analysis and CAPM
Variance Analysis and CAPM
Slide 05- Slide 05
σ
p
M
r
M
σ
M
r
f
CML
The Capital Market Line
The Capital Market Line
16:14 Lecture 05 16:14 Lecture 05
MeanMean-
-Variance Analysis and CAPM
Variance Analysis and CAPM
Slide 05- Slide 05
Proof of the CAPM relationship
[old traditional derivation]
Refer
to
previous
figure.
Consider
a
portfolio
with
a
fraction 1 -
α
of wealth invested in an arbitrary security j
and a fraction
α
in the market portfolio
As
α
varies we trace a locus which
Tangency =
= slope of the locus at M =
= slope of CML =
jM
j
M
p
j
M
p
σ α α σ α σ α σ
μ
α
αμ
μ
)
1
(
2
)
1
(
)
1
(
2
2
2
2
2
−
−
=
−
=
d
μ
p
d
σ
p
|
α
=
μ
M
r
f
σ
M
