Problem Set #1

1.Assume constant absolute risk aversion=. Using the double integration method from slide 10 of lecture 1, recover the utility function that has this absolute risk aversion?

2.Consider the basic investment problem in slide 13 of lecture 1, except that there are two risky assets with random returns R₁ and R₂, and two investment variables A₁ and A₂ to make decision over.

a)Demonstrate that the Risk Premium E[R_i-R_f]=-Cov[U'(w),R_i]/E[U'(w)]

b)Demonstrate that the differential return

E[R₂-R₁]=-Cov[U'(w)(R₂-R₁)]/E[U'(w)]

3. Assume there is a risk free investment and a risky investment as in Chapter 1. What is the expression fordA/dr_f?Is it most likely positive or negative?

4.What is the expression for the change in optimal utility if the risk free rate changes?

5. For the power utility derive the values of the parameter gamma that correspond to the cases of investment behavior as a function of changing wealth. There are six cases, three absolute, and three relative, are all cases possible?

6.(2011 mid-term question)

Assume the Markowitz model with no Risk free asset. Furthermore, consider the result

.

Show that

7.An Index portfolio has expected return=6% and std. deviation=10%.Portfolio J has an expected return of 4% a beta of .5 and std. deviation of 10%. Assuming the zero beta index has a standard deviation of 4%, what is

a)the expected return of the zero beta portfolio?

b)the systematic risk of asset j assuming there is no unsystematic risk?

c)the systematic risk of asset j assuming there is also a risk free rate?

8.Show that the solution of the Markowitz model is actually a linear combination of two portfolios, the minimum variance portfolio and the portfolio on the minimum variance set at the tangency point to a line drawn from the origin.(hint: you already know the global minimum variance portfolio's weights.

Problem Set #1

1.Assume constant absolute risk aversion=. Using the double integration method from slide 10 of lecture 1, recover the utility function that has this absolute risk aversion?

2.Consider the basic investment problem in slide 13 of lecture 1, except that there are two risky assets with random returns R₁ and R₂, and two investment variables A₁ and A₂ to make decision over.

a)Demonstrate that the Risk Premium E[R_i-R_f]=-Cov[U'(w),R_i]/E[U'(w)]

b)Demonstrate that the differential return

E[R₂-R₁]=-Cov[U'(w)(R₂-R₁)]/E[U'(w)]

3. Assume there is a risk free investment and a risky investment as in Chapter 1. What is the expression fordA/dr_f?Is it most likely positive or negative?

4.What is the expression for the change in optimal utility if the risk free rate changes?

5. For the power utility derive the values of the parameter gamma that correspond to the cases of investment behavior as a function of changing wealth. There are six cases, three absolute, and three relative, are all cases possible?

6.(2011 mid-term question)

Assume the Markowitz model with no Risk free asset. Furthermore, consider the result

.

Show that

7.An Index portfolio has expected return=6% and std. deviation=10%.Portfolio J has an expected return of 4% a beta of .5 and std. deviation of 10%. Assuming the zero beta index has a standard deviation of 4%, what is

a)the expected return of the zero beta portfolio?

b)the systematic risk of asset j assuming there is no unsystematic risk?

c)the systematic risk of asset j assuming there is also a risk free rate?

8.Show that the solution of the Markowitz model is actually a linear combination of two portfolios, the minimum variance portfolio and the portfolio on the minimum variance set at the tangency point to a line drawn from the origin.(hint: you already know the global minimum variance portfolio's weights.