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Lecture XII: Market Evaluation of Investment Risk
Part III: CAPM and APT
I.
Derivation of the Capital Asset Pricing Model
A. Some of the Theory
1. Tobin Revisited
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( 29 ( 29
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11
P
fM
P
M
PP
R
x
R
xR
x
xx
d d
dx
d
R
d
x dR
s
s
ss
=
+
=
 +
=
=
a.
The last term is the slope of the Expected Value
Standard Deviation frontier.
Note that we have
changed the space from Expected ValueVariance
frontier.
Most of the financial discussions are
actually in ES space, but this raises some
complications particularly with respect to covariance
terms.
b.
Next, we solve for x.
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1
P
fmM
Pm
fMm
PfM
R
x
RR
xR
x
RRRx
dx
d
RRR
= +

=  + ⇒=

⇒=

c.
Returning to the variance equation:
P
m
d
dx
s
s
d.
Combining both results:
P
P fMMf
d
d
R RR RR
sss

==

e.
Since the expected rate of return on the market
portfolio is greater than the risk free interest rate, the
slope of the capital market is positive.
2. Next, we develop the conditions for the asset not in the market
portfolio.
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2
2
2
22
2
2
2
1
2
1
21
A
j
j
jm
A
j
j
m
m
A
j
j
j
m
j
j
R
x
R
x
x
s
s
=
=++
=
+
 +
a.
In this case, we are creating a portfolio of the asset
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 Spring '11
 Moss

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