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Ec 173A – FINANCIAL MARKETS
LECTURE NOTES
Foster, UCSD
21Jul08
TOPIC 6 – ASSET EQUILIBRIUM PRICING MODELS
A. The Capital Asset Pricing Model
1.
Introduction:
a)
One question from MPT may
bother
you.
1) Asset X in the CML diagram is
com
pletely dominated by market
portfolio M and by other single
assets.
[Fig. 1]
2) Yet, by definition, X is
included in M, so somebody
owns it!
Why?
b) The reason asset X is in the market,
yet in
the interior of the opportunity set of
efficient
portfolios, is that the RELEVANT
risk of a
single asset is NOT σ(r
x
).
What we said earlier about asset dominance as an equilibra
ting process must now be revised.
1) We know that investors diversify portfolios between a riskless asset and a market
portfolio M of risky assets.
As E(U) maximizers, they are concerned only
with over
all portfolio risk and return μ(r
p
) and σ(r
p
).
And furthermore, since they choose indi
vidual portfolios along the CML, they are only concerned with μ(r
m
) and σ(r
m
).
(This
is what we called the “Separation Theorem.”)
2) Hence, when an investor is contemplating purchase of asset X for addition to an
already diversified portfolio, the only thing that matters
is what X will do to overall
portfolio risk and return.
This is only weakly related to σ(r
x
).
But if σ(r
x
) isn't the
relevant risk of asset X, then what is?
c)
The CAPM provides an answer.
It was developed by William F. Sharpe in 1964 (Nobel
Prize 1990).
It adds 3 assumptions to those made by MPT:
#6
There are no taxes, brokerage fees, or other market imperfections.
#7
Total asset quantities are fixed, and all assets are marketable.
Hence, we are discus
sing secondary market transactions, not new issues of securities.
#8
Perfect competition  all investors are small and take security prices as given  they
are price takers.
(Not strictly true!)
2.
Derivation of the CAPM:
a)
Starting with market portfolio M, create enlarged portfolio M+ with proportion δ in asset
X and 1−δ in M.
For the new portfolio:
σ(r
m
)
σ(r
x
) σ(r
p
)
M
• Y
●X
μ(r
m
)
μ(r
x
)
r
f
μ(r
p
)
Fig. 1
)
,
(
)
1
(
2
)
(
)
1
(
)
(
)
(
)
(
)
1
(
)
(
)
(
2
2
2
2
2
m
x
m
x
m
m
s
m
r
r
r
r
r
r
r
r
σ
δ
μ
δμ

+

+
=

+
=
+
+
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ASSET PRICING MODELS
p. 2
b) To find the change in return and risk from adding X to M, take the following derivatives:
c)
Now the trick.
1) We know that in general equilibrium, supply = demand in all markets and excess
demand = 0.
In financial market equilibrium, excess demand for each asset = 0.
2) Now all of X WAS ALREADY INCLUDED in market portfolio M, so M
≡
M+.
Trying to add δ% of X to M is excess demand for X.
So we must evaluate the
derivatives above at δ = 0, which yields the following:
d) The extra return needed on the market portfolio M to compensate for the extra risk from
having δ% of X already in it is:
e)
But all investors diversify by choosing a portfolio along the CML, where their portfolio
risk premium = slope of CML = [μ(r
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This note was uploaded on 06/30/2009 for the course ECON 173A taught by Professor Foster during the Spring '09 term at UCSD.
 Spring '09
 Foster

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