Consider an investor with initial wealth yo, who maximizes his expected utility from final wealth, Eu()]. This investor can invest in two a risky securities, 1 and 2, with random return 1 and 2. Those risky returns are two binomial variables, perfectly correlated. More specifically, with probability p we have ₁ = r and 2 =r+6, and with probability 1-p we have ₁ = 0 and ₂ = -8, where r > 0 and 6 > 0. We assume that this investor has log preferences, that is u(y) = log(y). 1. For a given fraction, a, of the initial wealth, invested in risky security 2, what is the distribution of final wealth, i? 2. Determine the expression of E[u(1)] as a function of a. 3. Explain why there is an upper bound and a lower bound for a, and determine those bounds. 4. Determine the optimal fraction a*. 5. When p = 0.5, describe qualitatively the optimal investment strategy. Does it make sense?
Consider an investor with initial wealth yo, who maximizes his expected utility from final wealth, Eu()]. This investor can invest in two a risky securities, 1 and 2, with random return 1 and 2. Those risky returns are two binomial variables, perfectly correlated. More specifically, with probability p we have ₁ = r and 2 =r+6, and with probability 1-p we have ₁ = 0 and ₂ = -8, where r > 0 and 6 > 0. We assume that this investor has log preferences, that is u(y) = log(y). 1. For a given fraction, a, of the initial wealth, invested in risky security 2, what is the distribution of final wealth, i? 2. Determine the expression of E[u(1)] as a function of a. 3. Explain why there is an upper bound and a lower bound for a, and determine those bounds. 4. Determine the optimal fraction a*. 5. When p = 0.5, describe qualitatively the optimal investment strategy. Does it make sense?
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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![Consider an investor with initial wealth yo, who maximizes his expected utility from final
wealth, E[u()]. This investor can invest in two a risky securities, 1 and 2, with random
return ři and ř2. Those risky returns are two binomial variables, perfectly correlated. More
specifically, with probability p we have ř =r and r, = r+ô, and with probability 1- p we
have î = 0 and ř2 = -6, where r> 0 and ổ > 0.
We assume that this investor has log preferences, that is
u(y) = log(y).
1. For a given fraction, a, of the initial wealth, invested in risky security 2, what is the
distribution of final wealth, g1?
2. Determine the expression of Elu(1)] as a function of a.
3. Explain why there is an upper bound and a lower bound for a, and determine those
bounds.
4. Determine the optimal fraction a*.
5. When p =
0.5, describe qualitatively the optimal investment strategy. Does it make
sense?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F62b02545-3413-493c-8ddf-7a9affe3a8d1%2F674fff18-0fe9-4852-a881-5793f9445ad7%2Fg90rs_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider an investor with initial wealth yo, who maximizes his expected utility from final
wealth, E[u()]. This investor can invest in two a risky securities, 1 and 2, with random
return ři and ř2. Those risky returns are two binomial variables, perfectly correlated. More
specifically, with probability p we have ř =r and r, = r+ô, and with probability 1- p we
have î = 0 and ř2 = -6, where r> 0 and ổ > 0.
We assume that this investor has log preferences, that is
u(y) = log(y).
1. For a given fraction, a, of the initial wealth, invested in risky security 2, what is the
distribution of final wealth, g1?
2. Determine the expression of Elu(1)] as a function of a.
3. Explain why there is an upper bound and a lower bound for a, and determine those
bounds.
4. Determine the optimal fraction a*.
5. When p =
0.5, describe qualitatively the optimal investment strategy. Does it make
sense?
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