This preview shows pages 1–3. Sign up to view the full content.
Kata Bognar
kbognar@umich.edu
Economics 142
Probabilistic Microeconomics
UCLA
Fall 2008
Homework Assignment 3.
 Solutions 
1.
(Portfolio Problem) Casper Gutman is an expected utility maximizer and has
$100 [8]
to invest. His vNM utility function over money is
u
(
x
) = log
x.
There are two
assets available to him to buy, but not to sell:
•
a risk free treasury bill which costs
$10
and pays
$20
in the next period for
sure
•
a risky stock which costs
$5
and pays
$5
with probability
p
and
$20
with
probability
1

p
in the next period.
Gutman may buy any amount of these assets, even partial units.
(a)
For what values of
p
would Gutman only buy the risk free asset?
Answer:
To answer this question, you can refer to local risk neutrality.
The local risk neutrality tells us that whenever a gamble is favorable, even a
risk averse person would bet a small amount on the gamble. In the context
of portfolio choice problem, this means that whenever the expected value
of the excess return of the risky asset over the risk free asset is positive, a
risk averse person would invest some in the risky asset. This implies that
whenever the excess return has a negative expected value the person will not
invest in the risky asset. If you think about it, this makes perfect sense, if
there is no premium on investing in the risky asset then there is no ‘reason’
for this person to take up risk.
What is the excess return of the risky asset over the risk free asset? To
calculate that, ﬁrst you need to ﬁnd the return for a dollar investment for
both assets and then you need to ﬁnd the diﬀerence of those values. Using
the notation of the book, you want to calculate ˜
x
=
˜
R
1

R
0
.
(This is what we denoted by ˜
x
in class.) The next table summarizes what
we know about the assets.
money invested
state 1 return (
p
)
state 2 return (1

p
)
Y
0
10
20
20
˜
Y
1
5
5
20
R
0
1
2
2
˜
R
1
1
1
4
˜
x
0

1
2
Therefore the expected value of the excess return is
E
(˜
x
) =
p
(

1) + (1

p
)2
.
This is nonpositive whenever
p
≥
2
/
3, hence for all such values Gutman
only invests in the risk free asset.
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentAlternatively, you may actually solve part (c) ﬁrst and interpret the result
you get there. In part (c) you calculate the optimal amount of money to
spend on investing in the risky asset (the optimal number of the risky asset
to buy) as the function of the probability parameter. Then you should pick
probability values such that the optimal investment is not positive.
1
(b)
Assume that
p
= 1
/
2
.
Write down the maximization problem of Gutman.
How many units of the risky assets would he buy? How much money would
This is the end of the preview. Sign up
to
access the rest of the document.
 Fall '09
 Bognar
 Utility

Click to edit the document details