There is one period. Assume a representative agent with utility function U(c) = 1 - exp(-n). In parts c) through i) assume the following: 11.02 and 8 = .96. Consumption at t = 0 is C₁ = 24. At t = 1 one of two states 01 and 02 eventuate with probability ₁ = .5, and 72 = .5, respectively. There are two complex securities s¹ and s². ⚫s¹ has a payoff of 23 in 1 and 27 in 02. s2 has a payoff of 20 in 01 and 32 in 02. The security payoffs are used for consumption. Answer the following: (a) For what values of 7 is the utility function well behaved? (i.e. U' > 0 and U" < 0) (b) Derive an expression for the stochastic discount factor m₁+1? (c) What are the prices q and q² of complex securities s¹ and s² at t = 0? (d) What weights create Arrow-Debreu securities? (hint: What are the weights of the complex securities that create payoffs (1,0)' and (0,1)'?) (e) What are the state prices q₁ and q2? (hint: State prices are a function of prices from part c) and the the inverse of the payoff matrix from part d).) (f) What is the price of a risk free bond qb? (g) What are the risk neutral probabilities and N? (h) Using the risk neutral probabilities solve for q¹ and q². RN (i) Is greater than or less than the true probability ₁? Why? (hint: See discussion on page 266 in the textbook.)
There is one period. Assume a representative agent with utility function U(c) = 1 - exp(-n). In parts c) through i) assume the following: 11.02 and 8 = .96. Consumption at t = 0 is C₁ = 24. At t = 1 one of two states 01 and 02 eventuate with probability ₁ = .5, and 72 = .5, respectively. There are two complex securities s¹ and s². ⚫s¹ has a payoff of 23 in 1 and 27 in 02. s2 has a payoff of 20 in 01 and 32 in 02. The security payoffs are used for consumption. Answer the following: (a) For what values of 7 is the utility function well behaved? (i.e. U' > 0 and U" < 0) (b) Derive an expression for the stochastic discount factor m₁+1? (c) What are the prices q and q² of complex securities s¹ and s² at t = 0? (d) What weights create Arrow-Debreu securities? (hint: What are the weights of the complex securities that create payoffs (1,0)' and (0,1)'?) (e) What are the state prices q₁ and q2? (hint: State prices are a function of prices from part c) and the the inverse of the payoff matrix from part d).) (f) What is the price of a risk free bond qb? (g) What are the risk neutral probabilities and N? (h) Using the risk neutral probabilities solve for q¹ and q². RN (i) Is greater than or less than the true probability ₁? Why? (hint: See discussion on page 266 in the textbook.)
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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Transcribed Image Text:There is one period. Assume a representative agent with utility function U(c) = 1 -
exp(-n). In parts c) through i) assume the following:
11.02 and 8 = .96.
Consumption at t = 0 is C₁ = 24.
At t = 1 one of two states 01 and 02 eventuate with probability ₁ = .5, and 72 = .5,
respectively.
There are two complex securities s¹ and s².
⚫s¹ has a payoff of 23 in 1 and 27 in 02.
s2 has a payoff of 20 in 01 and 32 in 02.
The security payoffs are used for consumption.
Answer the following:
(a) For what values of 7 is the utility function well behaved? (i.e. U' > 0 and U" < 0)
(b) Derive an expression for the stochastic discount factor m₁+1?
(c) What are the prices q and q² of complex securities s¹ and s² at t = 0?
(d) What weights create Arrow-Debreu securities? (hint: What are the weights of the
complex securities that create payoffs (1,0)' and (0,1)'?)
(e) What are the state prices q₁ and q2? (hint: State prices are a function of prices from
part c) and the the inverse of the payoff matrix from part d).)
(f) What is the price of a risk free bond qb?
(g) What are the risk neutral probabilities and N?
(h) Using the risk neutral probabilities solve for q¹ and q².
RN
(i) Is greater than or less than the true probability ₁? Why? (hint: See discussion
on page 266 in the textbook.)
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