Department of Economics
Financial Economics
University of California, Berkeley
Economics 136
November 9, 2003
Fall 2006
Economics 136: Financial Economics
Section Notes for Week 11
1 Capital Allocation Between a Risky Portfolio and a RiskFree
Asset
This material is in BKM chapter 7.
Assume for the moment that an investor must decide how to invest all of her wealth and has
only two options: a riskfree asset such as Treasury Bills (TBills) and a risky portfolio of
stocks (such as a mutual fund). Since all of her wealth must be invested, the decision that
she makes can be summarized by one parameter, the fraction of her wealth that she invests
in the risky portfolio,
w
. Since she must allocate all of her wealth to either the mutual fund
or TBill, the fraction of her wealth invested in TBill must be, 1

w
.
If we assume a functional form for the investors objective (or utility) function, then we can
determine the optimal fraction of wealth for the investor to put into the risky portfolio,
w
*
.
Let
R
P
denote the returns of the combined TBill and mutual fund portfolio and
R
MF
denote
the returns of the mutual fund. Assuming that people like high mean return and dislike high
return variance then they would like to solve the following optimization,
max
w
±
E
[
R
P
]

1
2
A
·
V ar
(
R
P
)
²
= max
w
±
R
f
+
w
(
E
[
R
MF
]

R
f
)

1
2
w
2
·
A
·
V ar
(
R
MF
)
²
Then the ﬁrst order condition (foc) is
0 =
E
[
R
MF
]

R
f

w
·
A
·
V ar
(
R
MF
)
w
*
=
E
[
R
MF
]

R
f
A
·
V ar
(
R
MF
)
Note:
w
*
gives the optimal fraction of wealth invested in the risky portfolio. The total
portfolio which invests
w
*
in the risky portfolio and 1

w
*
in the riskfree asset will result
in the optimal portfolio which has an expected return and standard deviation such that the
investor’s indiﬀerence curve is tangent to the CAL at this point. This is why it is optimal.
It gives the investor the highest possible utiity subject to the mean and standard deviation
of investment possibilities (the CAL is the frontier of these possibilities).
1
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View Full Document2 Finding the Optimal Risky Prtfolio
This material is in BKM chapter 8.
In the last section, we took the expected return and standard deviation of the risky portfolio
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 Fall '08
 SZEIDL
 Economics, rd, risky portfolio, Stock Fund

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