Problem 1 Consider a three period model with two assets, a safe and an unsafe asset. The safe asset always has a price of p, = 1 and returns dividends of r every period. The unsafe asset also returns dividends of r but its price has a random component. Its final price is P₁ =1+ where & ~N(0,0%). The unsafe asset is available in unit net supply in periods 1 and 2. All agents in this economy myopically maximize identical one period ahead utility and have mean-variance utility functions: Now E₁[U]=E[W]-yo 1+1 a.) Suppose there is a sophisticated investor, whom we will also refer to as the arbitrageur, with wealth at time 1 of w₁. Denote this investor's demand for the unsafe asset at time t by 2. Write down the sophisticated investor's wealth at time 3 as a function of parameters, time 2 variables, and the time 3 price. Using this equation, write down the expression that the investor maximizes at time 2 and solve the optimal time 2 demand 2 as a function of only time 2 variables and parameters. suppose there is also a noise trader, identical in every way to the sophisticated investor except t that he misperceives the expected price of the risky asset, and so maximizes expected utility given that distribution. Specifically, the noise trader believes that p₂ = E₁[P2]+p₁ +v and p₁ = E₂[P3] +P₂+ε where v~. - N(0,V₁(p2)) and P, N(p,) is independent and identically distributed. (E,[P,+1] is the true expectation and V₁[P1+1] is the true variance.) b.) Find the noise trader's optimal demand 2½ just as you found λ above. c.) If noise traders are present in measure μ and arbitrageurs in measure 1-μ, then what will be the time 2 price p2 of the unsafe asset (as a function only of parameters and the time 2 noise trader shock p₂? What is the mean and variance of this price at time 1?
Problem 1 Consider a three period model with two assets, a safe and an unsafe asset. The safe asset always has a price of p, = 1 and returns dividends of r every period. The unsafe asset also returns dividends of r but its price has a random component. Its final price is P₁ =1+ where & ~N(0,0%). The unsafe asset is available in unit net supply in periods 1 and 2. All agents in this economy myopically maximize identical one period ahead utility and have mean-variance utility functions: Now E₁[U]=E[W]-yo 1+1 a.) Suppose there is a sophisticated investor, whom we will also refer to as the arbitrageur, with wealth at time 1 of w₁. Denote this investor's demand for the unsafe asset at time t by 2. Write down the sophisticated investor's wealth at time 3 as a function of parameters, time 2 variables, and the time 3 price. Using this equation, write down the expression that the investor maximizes at time 2 and solve the optimal time 2 demand 2 as a function of only time 2 variables and parameters. suppose there is also a noise trader, identical in every way to the sophisticated investor except t that he misperceives the expected price of the risky asset, and so maximizes expected utility given that distribution. Specifically, the noise trader believes that p₂ = E₁[P2]+p₁ +v and p₁ = E₂[P3] +P₂+ε where v~. - N(0,V₁(p2)) and P, N(p,) is independent and identically distributed. (E,[P,+1] is the true expectation and V₁[P1+1] is the true variance.) b.) Find the noise trader's optimal demand 2½ just as you found λ above. c.) If noise traders are present in measure μ and arbitrageurs in measure 1-μ, then what will be the time 2 price p2 of the unsafe asset (as a function only of parameters and the time 2 noise trader shock p₂? What is the mean and variance of this price at time 1?
Essentials Of Investments
11th Edition
ISBN:9781260013924
Author:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Publisher:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Chapter1: Investments: Background And Issues
Section: Chapter Questions
Problem 1PS
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![Problem 1
Consider a three period model with two assets, a safe and an unsafe asset. The safe asset
always has a price of p, = 1 and returns dividends of r every period. The unsafe asset
also returns dividends of r but its price has a random component. Its final price is
P₁ =1+ where & ~N(0,0%). The unsafe asset is available in unit net supply in
periods 1 and 2. All agents in this economy myopically maximize identical one period
ahead utility and have mean-variance utility functions:
Now
E₁[U]=E[W]-yo
1+1
a.) Suppose there is a sophisticated investor, whom we will also refer to as the
arbitrageur, with wealth at time 1 of w₁. Denote this investor's demand for the
unsafe asset at time t by 2. Write down the sophisticated investor's wealth at
time 3 as a function of parameters, time 2 variables, and the time 3 price. Using
this equation, write down the expression that the investor maximizes at time 2 and
solve the optimal time 2 demand 2 as a function of only time 2 variables and
parameters.
suppose there is also a noise trader, identical in every way to the sophisticated
investor except
t that he misperceives the expected price of the risky asset, and so
maximizes expected utility given that distribution. Specifically, the noise trader believes
that p₂ = E₁[P2]+p₁ +v and p₁ = E₂[P3] +P₂+ε where v~.
- N(0,V₁(p2)) and
P, N(p,) is independent and identically distributed. (E,[P,+1] is the true
expectation and V₁[P1+1] is the true variance.)
b.) Find the noise trader's optimal demand 2½ just as you found λ above.
c.) If noise traders are present in measure μ and arbitrageurs in measure 1-μ, then
what will be the time 2 price p2 of the unsafe asset (as a function only of
parameters and the time 2 noise trader shock p₂? What is the mean and variance
of this price at time 1?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa6cfa530-7e74-4013-a9a7-79ba033e5858%2Ffced7e06-7d87-4b78-b385-833cbfb797e1%2Fsh2nr4h_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 1
Consider a three period model with two assets, a safe and an unsafe asset. The safe asset
always has a price of p, = 1 and returns dividends of r every period. The unsafe asset
also returns dividends of r but its price has a random component. Its final price is
P₁ =1+ where & ~N(0,0%). The unsafe asset is available in unit net supply in
periods 1 and 2. All agents in this economy myopically maximize identical one period
ahead utility and have mean-variance utility functions:
Now
E₁[U]=E[W]-yo
1+1
a.) Suppose there is a sophisticated investor, whom we will also refer to as the
arbitrageur, with wealth at time 1 of w₁. Denote this investor's demand for the
unsafe asset at time t by 2. Write down the sophisticated investor's wealth at
time 3 as a function of parameters, time 2 variables, and the time 3 price. Using
this equation, write down the expression that the investor maximizes at time 2 and
solve the optimal time 2 demand 2 as a function of only time 2 variables and
parameters.
suppose there is also a noise trader, identical in every way to the sophisticated
investor except
t that he misperceives the expected price of the risky asset, and so
maximizes expected utility given that distribution. Specifically, the noise trader believes
that p₂ = E₁[P2]+p₁ +v and p₁ = E₂[P3] +P₂+ε where v~.
- N(0,V₁(p2)) and
P, N(p,) is independent and identically distributed. (E,[P,+1] is the true
expectation and V₁[P1+1] is the true variance.)
b.) Find the noise trader's optimal demand 2½ just as you found λ above.
c.) If noise traders are present in measure μ and arbitrageurs in measure 1-μ, then
what will be the time 2 price p2 of the unsafe asset (as a function only of
parameters and the time 2 noise trader shock p₂? What is the mean and variance
of this price at time 1?
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