Q2. (i) Consider a risk averse investor who must decide how much of his initial wealth w to put into a risky asset. The risky asset can have any of the positive or negative rates of return r with probability density function f(r). If ß is the amount of wealth to be put into the risky asset, final wealth under rate of return r will be (w − ß) + (1 + r)ß = w+ ßr. The investor's problem is to choose 3 to maximize the expected utility of wealth. We can write this formally as the following single-variable optimization problem max fu(w + Br)f(r)dr s.t.: 0 ≤ ß ≤w. B (1) Suppose we have interior optimal ß* falls in (0, w), which is determined by the following first order condition: fu' (w+Br)rf(r)dr = 0. (2)
Q2. (i) Consider a risk averse investor who must decide how much of his initial wealth w to put into a risky asset. The risky asset can have any of the positive or negative rates of return r with probability density function f(r). If ß is the amount of wealth to be put into the risky asset, final wealth under rate of return r will be (w − ß) + (1 + r)ß = w+ ßr. The investor's problem is to choose 3 to maximize the expected utility of wealth. We can write this formally as the following single-variable optimization problem max fu(w + Br)f(r)dr s.t.: 0 ≤ ß ≤w. B (1) Suppose we have interior optimal ß* falls in (0, w), which is determined by the following first order condition: fu' (w+Br)rf(r)dr = 0. (2)
Chapter7: Uncertainty
Section: Chapter Questions
Problem 7.7P
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![Q2. (i)
Consider a risk averse investor who must decide how much of his
initial wealth w to put into a risky asset. The risky asset can have any of the positive
or negative rates of return r with probability density function f(r). If ß is the amount
of wealth to be put into the risky asset, final wealth under rate of return r will be
(w B) + (1+r)ß = w+ ßr. The investor's problem is to choose 3 to maximize the
expected utility of wealth. We can write this formally as the following single-variable
optimization problem
(1)
Suppose we have interior optimal 3* falls in (0, w), which is determined by the following
first order condition:
Ju'(w + ßr)rf(r)dr = 0.
(2)
Define A(w)
as a measure of absolute risk aversion. Please show that if
u(.) displays decreasing absolute risk aversion (DARA), i.e. A(w) decreases with w, then
the risky asset must be a "normal" good, i.e. 3* increases with w. [Hint: Please read
Example 2.6 in Jehle and Reny.]
max fu(w + Br)f(r)dr s.t.: 0 ≤ B ≤w.
B
u" (w)
=
u' (w)
(ii) t. Can you provide one example for such DARA u(.)? Explain why this
u() exhibits DARA.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9c360332-0d73-4448-ad65-bf3146d8eea0%2F0af67ec8-1d2e-461a-9197-0a85546064bc%2F8ngatn_processed.png&w=3840&q=75)
Transcribed Image Text:Q2. (i)
Consider a risk averse investor who must decide how much of his
initial wealth w to put into a risky asset. The risky asset can have any of the positive
or negative rates of return r with probability density function f(r). If ß is the amount
of wealth to be put into the risky asset, final wealth under rate of return r will be
(w B) + (1+r)ß = w+ ßr. The investor's problem is to choose 3 to maximize the
expected utility of wealth. We can write this formally as the following single-variable
optimization problem
(1)
Suppose we have interior optimal 3* falls in (0, w), which is determined by the following
first order condition:
Ju'(w + ßr)rf(r)dr = 0.
(2)
Define A(w)
as a measure of absolute risk aversion. Please show that if
u(.) displays decreasing absolute risk aversion (DARA), i.e. A(w) decreases with w, then
the risky asset must be a "normal" good, i.e. 3* increases with w. [Hint: Please read
Example 2.6 in Jehle and Reny.]
max fu(w + Br)f(r)dr s.t.: 0 ≤ B ≤w.
B
u" (w)
=
u' (w)
(ii) t. Can you provide one example for such DARA u(.)? Explain why this
u() exhibits DARA.
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