Problem 4: Recall the following portfolio allocation problem: an individual with initial wealth wo has to choose an allocation between a safe asset (with a zero rate of return) and a risky asset with a random rate of return = [−1,1], where E[6] > 0. If the individual invests x dollars in the risky asset, then final wealth is (wo-x) + x (1 + d) = w₁ +xd. The individual chooses x to maximize expected utility denoted by E[u(w₁ + xd)]. 1. Suppose the Bernoulli utility function is u(w) solute risk aversion is a constant. dwo ew. Show that ab- 2. Let x*(wo) denote the optimal amount of investment in the risky asset. Show that for the utility function of Part a., da = 0, i.e. the amount of investment in the risky asset is independent of initial wealth. Explain this result. 3. Now suppose that the utility function is u (w) = aw – ½bw², where a, b > 0. The parameters a and b are such that marginal utility of wealth is positive for all w. What can we say about in this case? Explain your result.
Problem 4: Recall the following portfolio allocation problem: an individual with initial wealth wo has to choose an allocation between a safe asset (with a zero rate of return) and a risky asset with a random rate of return = [−1,1], where E[6] > 0. If the individual invests x dollars in the risky asset, then final wealth is (wo-x) + x (1 + d) = w₁ +xd. The individual chooses x to maximize expected utility denoted by E[u(w₁ + xd)]. 1. Suppose the Bernoulli utility function is u(w) solute risk aversion is a constant. dwo ew. Show that ab- 2. Let x*(wo) denote the optimal amount of investment in the risky asset. Show that for the utility function of Part a., da = 0, i.e. the amount of investment in the risky asset is independent of initial wealth. Explain this result. 3. Now suppose that the utility function is u (w) = aw – ½bw², where a, b > 0. The parameters a and b are such that marginal utility of wealth is positive for all w. What can we say about in this case? Explain your result.
Chapter7: Uncertainty
Section: Chapter Questions
Problem 7.7P
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![Problem 4: Recall the following portfolio allocation problem: an individual
with initial wealth wo has to choose an allocation between a safe asset (with a
zero rate of return) and a risky asset with a random rate of return & € [−1, 1],
where E[8] > 0. If the individual invests & dollars in the risky asset, then final
wealth is (wo-x)+x(1+5) = w₁ +x6. The individual chooses to maximize
expected utility denoted by E[u(wo +xd)].
1. Suppose the Bernoulli utility function is u(w)
solute risk aversion is a constant.
= -e. Show that ab-
2. Let x* (wo) denote the optimal amount of investment in the risky asset.
Show that for the utility function of Part a., da = 0, i.e. the amount of
investment in the risky asset is independent of initial wealth. Explain this
result.
dwo
3. Now suppose that the utility function is u (w) = aw - bw², where a, b > 0.
The parameters a and b are such that marginal utility of wealth is positive
for all w. What can we say about in this case? Explain your result.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F82b46645-14e3-440f-8c2c-fd00e1b61743%2F948de21c-3a69-45bd-a008-69cd039a7ff9%2Fvi2k91j_processed.png&w=3840&q=75)
Transcribed Image Text:Problem 4: Recall the following portfolio allocation problem: an individual
with initial wealth wo has to choose an allocation between a safe asset (with a
zero rate of return) and a risky asset with a random rate of return & € [−1, 1],
where E[8] > 0. If the individual invests & dollars in the risky asset, then final
wealth is (wo-x)+x(1+5) = w₁ +x6. The individual chooses to maximize
expected utility denoted by E[u(wo +xd)].
1. Suppose the Bernoulli utility function is u(w)
solute risk aversion is a constant.
= -e. Show that ab-
2. Let x* (wo) denote the optimal amount of investment in the risky asset.
Show that for the utility function of Part a., da = 0, i.e. the amount of
investment in the risky asset is independent of initial wealth. Explain this
result.
dwo
3. Now suppose that the utility function is u (w) = aw - bw², where a, b > 0.
The parameters a and b are such that marginal utility of wealth is positive
for all w. What can we say about in this case? Explain your result.
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