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Suppose there are two assets, one is risk-free and one is risky. The risk-free asset has...
Suppose there are two assets, one is risk-free and one is risky. The risk-free asset has a sure r...
Suppose there are two assets, one is risk-free and one is risky. The risk-free asset has a sure rate of return rj, the risky asset has a random rate of return r. Suppose the utility function of an investor is U(x) =--. The initial wealth is wo, the dollar amount invested in the risky asset is θ. r is normally distributed with mean μ and variance σ2. Based on the maximum utility framework, find the optimal investment strategy 6. (25...
Exercise 2. Suppose that there is one risk free asset with return rf and one risky...
Exercise 2. Suppose that there is one risk free asset with return rf and one risky asset with normally distributed returns, r ~ N(u,02). Show that the CARA utility u(r) = -e-Ar gives the same optimal allocation of wealth to the risky asset as the mean-variance utility function we used in class. That is, show that E[r] – rf OCARA = AO2 Hint: Use the fact that if a random variable x is distributed normally with mean Mx and variance...
2. Consider an economy with 2 risky assets and one risk free asset. Two investors, A...
2. Consider an economy with 2 risky assets and one risk free asset. Two investors, A and B, have mean-variance utility functions (with different risk aversion coef- ficients). Let P denote investor A's optimal portfolio of risky and risk-free assets and let Q denote investor B's optimal portfolio of risky and risk-free assets. P and Q have expected returns and standard deviations given by P Q E[R] St. Dev. 0.2 0.45 0.1 0.25 (a) What is the risk-free interest rate...
The universe of available securities includes two risky stocks A and B, and a risk-free asset....
The universe of available securities includes two risky stocks A and B, and a risk-free asset. The data for the universe are as follows: Assets Expected Return Standard Deviation Stock A 6% 25% Stock B 12% 42% Risk free 5% 0 The correlation coefficient between A and B is -0.2. The investor maximizes a utility function U=E(r)−σ2 (i.e. she has a coefficient of risk aversion equal to 2). Assume that to maximize his utility when there is no available risk-free...
Suppose there is one risky asset and one risk free asset. Derive the optimal weights for...
Suppose there is one risky asset and one risk free asset. Derive the optimal weights for a mean-variance optimizer to hold of each.
2. (Understanding optimal portfolio choice) Consider two risky assets, the expected return of asset one is...
2. (Understanding optimal portfolio choice) Consider two risky assets, the expected return of asset one is μ-0.1, the expected return of asset two is μ2-0.15, the risk or standard deviation of asset one is σ1-0.1, the risk or standard deviation of asset two is σ2-02. The two assets also happen to have zero correlation. An investor plans to build a portfolio by investing w of his investment to asset one and the rest of his investment to asset two. Calculate...
4) Consider an agent with initial wealth W to be allocated between a safe (risk-free) asset...
4) Consider an agent with initial wealth W to be allocated between a safe (risk-free) asset and a risky asset. Let xs denote the quantity (dollars) of wealth invested in the safe asset. Let X, denote the quantity (dollars) of wealth invested in the risky asset. The safe asset has an interest rate of rs = 0. The risky asset has a probability n of earning an interest rate rr = -1 and a 1 – 1 probability of earning...
An investor invests 40% of her wealth in a risky asset with an expected rate of...
An investor invests 40% of her wealth in a risky asset with an expected rate of return of 15% and a standard deviation of 20%. The rest of her wealth is invested in the risk-free asset, which yields 6%. What are the expected return and standard deviation of her portfolio?
SECTION IT these questions are worth double in weight -41. (Double weight question) Suppose in security....
SECTION IT these questions are worth double in weight -41. (Double weight question) Suppose in security. The securities returns are: r on) Suppose investors form portfolios by combing one risky security with one risk free security. The Risk free -0.03 risky -0.15 and SD -0.25 Let wl denote the share of wealth in the risky asset utility function. wealth in the risky asset. Determine the optimal portfolio for an investor with the following (This is of the form E(u)-(expected return...
Suppose there are three assets: A, B, and C. Asset A’s expected return and standard deviation are 1 percent and 1 percent. Asset B has the same expected return and standard deviation as Asset A. However, the correlation coefficient of Assets A and B is −0
Suppose there are three assets: A, B, and C. Asset A’s expected return and
standard deviation are 1 percent and 1 percent. Asset B has the same expected
return and standard deviation as Asset A. However, the correlation coefficient of
Assets A and B is −0.25. Asset C’s return is independent of the other two assets.
The expected return and standard deviation of Asset C are 0.5 percent and 1
percent.
(a) Find a portfolio of the three assets that...
Suppose there are two assets, one is risk-free and one is risky. The risk-free asset has a sure rate of return rj, the risky asset has a random rate of return r. Suppose the utility function of an investor is U(x) =--. The initial wealth is wo, the dollar amount invested in the risky asset is θ. r is normally distributed with mean μ and variance σ2. Based on the maximum utility framework, find the optimal investment strategy 6. (25...
Exercise 2. Suppose that there is one risk free asset with return rf and one risky asset with normally distributed returns, r ~ N(u,02). Show that the CARA utility u(r) = -e-Ar gives the same optimal allocation of wealth to the risky asset as the mean-variance utility function we used in class. That is, show that E[r] – rf OCARA = AO2 Hint: Use the fact that if a random variable x is distributed normally with mean Mx and variance...
2. Consider an economy with 2 risky assets and one risk free asset. Two investors, A and B, have mean-variance utility functions (with different risk aversion coef- ficients). Let P denote investor A's optimal portfolio of risky and risk-free assets and let Q denote investor B's optimal portfolio of risky and risk-free assets. P and Q have expected returns and standard deviations given by P Q E[R] St. Dev. 0.2 0.45 0.1 0.25 (a) What is the risk-free interest rate...
2. (Understanding optimal portfolio choice) Consider two risky assets, the expected return of asset one is μ-0.1, the expected return of asset two is μ2-0.15, the risk or standard deviation of asset one is σ1-0.1, the risk or standard deviation of asset two is σ2-02. The two assets also happen to have zero correlation. An investor plans to build a portfolio by investing w of his investment to asset one and the rest of his investment to asset two. Calculate...
4) Consider an agent with initial wealth W to be allocated between a safe (risk-free) asset and a risky asset. Let xs denote the quantity (dollars) of wealth invested in the safe asset. Let X, denote the quantity (dollars) of wealth invested in the risky asset. The safe asset has an interest rate of rs = 0. The risky asset has a probability n of earning an interest rate rr = -1 and a 1 – 1 probability of earning...
SECTION IT these questions are worth double in weight -41. (Double weight question) Suppose in security. The securities returns are: r on) Suppose investors form portfolios by combing one risky security with one risk free security. The Risk free -0.03 risky -0.15 and SD -0.25 Let wl denote the share of wealth in the risky asset utility function. wealth in the risky asset. Determine the optimal portfolio for an investor with the following (This is of the form E(u)-(expected return...
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Learning Outcomes:
Upon the successful completion of this module, you should
understand the following concepts:
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