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Excess Returns and Beta:
Deriving the Security Market Line
W.L. Silber
I.
We showed that market forces combined with a search by investors for efficient
portfolios would produce the following relationships for each security
(i, j =
1
, .
.., n
) in a
portfolio:
(1)
m
f
m
i
m
f
i
R
R
X
R
R
s
s

=
∂
∂

/
This expression can be rewritten as
(2)
i
m
m
f
m
f
i
X
R
R
R
R
∂
∂

+
=
/
s
s
II.
Expression (2) says that the equilibrium expected return (
R
i
) on security
i
should
equal the riskfree rate (
R
f
) plus the market price of risk
( 29
m
f
m
R
R
s
/

times
i
m
X
∂
∂
/
s
.
What is
i
m
X
∂
∂
/
s
?
It is the increase in risk (
m
s
) associated with a small increase in
asset
i
, in other words, it is the risk contribution of security
i
to portfolio risk,
m
s
. We can
derive
i
m
X
∂
∂
/
s
by taking the derivative of the expression for the variance of a portfolio
with respect to X
i
.
The result, as shown in Garbade (p. 175, fn 12) is:
(3)
( 29
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 Spring '08
 Miyakawa

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