Roll R.
Journal of Financial Economics
, 4, 129176 (1977).
ﬃ
ciency and the CAPM cannot
be tested separately, i.e., tests of CAPM is always a joint test of ME ±
CAPM.
Roll argues that if any expost mean variance e
ﬃ
cient portfolio is selected
as the market portfolio and betas are computed using this as the market
proxy, then the following must hold. Where
P
is the proxy for the market
portfolio,
r
p
is the expected return on the proxy for the market portfolio,
β
ip
is the beta for security i with the proxy market portfolio, and
r
zp
is
the minimum variance portfolio that has a zero beta with the market proxy
portfolio.
r
i
=
r
zp
+
β
ip
(
r
p
−
r
zp
)
(1)
To calculate (optimal) fund F, we need
max
tg
θ
=
P
n
i
=1
w
i
(
r
i
−
r
f
)
/
(
P
n
ij
=1
σ
ij
w
i
w
j
)
2
.
Note that
∂
∂
w
i
(
P
n
ij
=1
σ
ij
w
i
w
j
)=2
P
n
j
=1
σ
ij
w
j
.
Let
λ
=
P
n
i
=1
w
i
(
r
i
−
r
f
)
/
P
n
ij
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 Fall '10
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 Pricing, Capital Asset Pricing Model, Modern portfolio theory, Roll, Market Portfolio, ij wj, wi wj

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