16. Suppose you are concerned about your company's prospects, so you purchase a put option on your company's stock (granted at time zero and exercisable at time one). You have no stock grant but have a wage of 0.70 payable (as before) at time one. The stock price can rise to $1.22 or fall to $0.92 at the end of the first year from its initial value of $1. The put has an exercise price of .95. What is your optimal dynamic portfolio? 17. For the put option analyzed in question 16, find a time zero portfolio of stock and cash/borrowing that has identical payoffs. That is, if you form a portfolio that puts $x into stock and $y into the riskless bond, its value replicates that of the put option: What does this imply about the market value of the put option? Force Quit MacBook Air (after 5 years). What is the 5% lower tail value at risk? What is the 7% upper tail of best portfolio outcomes? 7. Use a risk aversion parameter of A=5. What is the optimal buy-and-hold portfolio? 8. For the optimal buy-and-hold portfolio, plot the path of the average portfolio weight in stock over time. (That is, at each time t, find the average weight in stocks across the 1,000 simulations.) Provide intuition for the graph. 9. For A=5, which portfolio strategy would you prefer, the cubic one in question 4 or the buy- and-hold strategy in question 72 11. Using Trees For this question, use a 2-period (i.e., 2 years) tree approach. Let the stock rise by 22% or fall by 8% with equal probability in each year. (You may think of the stock as having an initial price of $1.) Let the riskless rate be 2% per period. Let the initial portfolio value be 1.0. We again use a mean-variance utility over log returns. You are free to borrow and short stock. 10. What is the expected return on the stock per period? What is its standard deviation? 11. Let A=5. What is your optimal dynamic portfolio? (What are wo, W1, and Wi,d?) 12. An investor is observed to place .60 in stock at time 1 following an initial decrease in the stock price over the first year. What is this investor's level of A? 13. Let's consider the more general case of non-iid returns, in this case persistence. For the first year, as before, the stock may rise by 22% or fall by 8% with equal probability. If the market rises during the first year, then during the second year the stock has an expected return of 10%, but with the same standard deviation as in the first year (and again, with equal probabilities of an up or down move). If the market falls during the first year, then during the second year the stock has an expected return of 4.5%, but again with the same standard deviation as in the first year (and again, with equal probabilities of an up or down move.) What is the optimal dynamic portfolio for A=5? Provide intuition for why it differs from the iid case of question 11? 14. Go back to the lid case by assuming the stock can rise by 22% or fall by 8% in each period. However, now assume that you receive a fixed labor income of 0.70 at time 1, but no labor income at time 2. For A=5, what is your optimal dynamic portfolio? Provide intuition for the impact of fixed labor income. 15. Re-answer question 14, but now assume that in addition to receiving a fixed labor income Force Qu 1. Simulating portfolio strategies For this question, in your simulations use h=1/12, and perform 1,000 simulations. Let u=0.05, =0.13, and rf.01. Let the initial portfolio value be 1.0. Borrowing and shorting are permitted. (Please do not use John's Monte Carlo engine for this. Use the approach from Module 4.) For your Normal Shocks, please use the file Final Normal shocks.xlsx. This gives a matrix that has 60 columns and 1,000 rows. This gives 1,000 simulations over 60 months. 1. Consider the constant weight portfolio strategy of putting 60% in stock and 40% in cash. Provide a histogram of the ending portfolio value (after 5 years). What is the 5% lower tail value at risk? What is the 7% upper tail of best portfolio outcomes? 2. Use a risk aversion parameter of A=5. (Remember to use log-returns for your utility function.) What is the optimal constant weight portfolio? 3. Suppose an investor holds a constant weight portfolio with stock weight 0.50. Can you infer her risk aversion parameter A? 4. Now, consider a time-varying portfolio strategy. To approximate a time-varying stock weight, let's use a cubic polynomial. Thus, let w(t)=ao+ait+azt'+azt, where the four parameters ao, ai, az, az, are chosen optimally by the investor. This would imply that the stock weight formed at the beginning of time t=jh is ao+a1(jh)+a2(j'h)2+a3(j'h), where; is the month number (0,...,59) and thus jh is in calendar time. For an A of 5, what are the investor's optimal choices of ao, ai, az, az? 5. For the optimal strategy determined in question 4, plot the weight in stock as a function of t. Provide intuition for the graph. 6. Now, consider a different kind of strategy: a buy-and-hold portfolio. That is, you put an initial weight of 60% in stock and 40% in cash. However, you never rebalance, and just leave the portfolio alone. (Your initial dollars in stock simply grow over time with the realized return on stock. Your initial dollars in cash simply grow over time with the riskless rate. Adding the two together will give you the future value of the entire portfolio.) Simulate the portfolio path over 5 years. Provide a histogram of the ending portfolio value 16. Suppose you are concerned about your company's prospects, so you purchase a put option on your company's stock (granted at time zero and exercisable at time one). You have no stock grant but have a wage of 0.70 payable (as before) at time one. The stock price can rise to $1.22 or fall to $0.92 at the end of the first year from its initial value of $1. The put has an exercise price of .95. What is your optimal dynamic portfolio? 17. For the put option analyzed in question 16, find a time zero portfolio of stock and cash/borrowing that has identical payoffs. That is, if you form a portfolio that puts $x into stock and $y into the riskless bond, its value replicates that of the put option: What does this imply about the market value of the put option? Force Quit MacBook Air (after 5 years). What is the 5% lower tail value at risk? What is the 7% upper tail of best portfolio outcomes? 7. Use a risk aversion parameter of A=5. What is the optimal buy-and-hold portfolio? 8. For the optimal buy-and-hold portfolio, plot the path of the average portfolio weight in stock over time. (That is, at each time t, find the average weight in stocks across the 1,000 simulations.) Provide intuition for the graph. 9. For A=5, which portfolio strategy would you prefer, the cubic one in question 4 or the buy- and-hold strategy in question 72 11. Using Trees For this question, use a 2-period (i.e., 2 years) tree approach. Let the stock rise by 22% or fall by 8% with equal probability in each year. (You may think of the stock as having an initial price of $1.) Let the riskless rate be 2% per period. Let the initial portfolio value be 1.0. We again use a mean-variance utility over log returns. You are free to borrow and short stock. 10. What is the expected return on the stock per period? What is its standard deviation? 11. Let A=5. What is your optimal dynamic portfolio? (What are wo, W1, and Wi,d?) 12. An investor is observed to place .60 in stock at time 1 following an initial decrease in the stock price over the first year. What is this investor's level of A? 13. Let's consider the more general case of non-iid returns, in this case persistence. For the first year, as before, the stock may rise by 22% or fall by 8% with equal probability. If the market rises during the first year, then during the second year the stock has an expected return of 10%, but with the same standard deviation as in the first year (and again, with equal probabilities of an up or down move). If the market falls during the first year, then during the second year the stock has an expected return of 4.5%, but again with the same standard deviation as in the first year (and again, with equal probabilities of an up or down move.) What is the optimal dynamic portfolio for A=5? Provide intuition for why it differs from the iid case of question 11? 14. Go back to the lid case by assuming the stock can rise by 22% or fall by 8% in each period. However, now assume that you receive a fixed labor income of 0.70 at time 1, but no labor income at time 2. For A=5, what is your optimal dynamic portfolio? Provide intuition for the impact of fixed labor income. 15. Re-answer question 14, but now assume that in addition to receiving a fixed labor income Force Qu 1. Simulating portfolio strategies For this question, in your simulations use h=1/12, and perform 1,000 simulations. Let u=0.05, =0.13, and rf.01. Let the initial portfolio value be 1.0. Borrowing and shorting are permitted. (Please do not use John's Monte Carlo engine for this. Use the approach from Module 4.) For your Normal Shocks, please use the file Final Normal shocks.xlsx. This gives a matrix that has 60 columns and 1,000 rows. This gives 1,000 simulations over 60 months. 1. Consider the constant weight portfolio strategy of putting 60% in stock and 40% in cash. Provide a histogram of the ending portfolio value (after 5 years). What is the 5% lower tail value at risk? What is the 7% upper tail of best portfolio outcomes? 2. Use a risk aversion parameter of A=5. (Remember to use log-returns for your utility function.) What is the optimal constant weight portfolio? 3. Suppose an investor holds a constant weight portfolio with stock weight 0.50. Can you infer her risk aversion parameter A? 4. Now, consider a time-varying portfolio strategy. To approximate a time-varying stock weight, let's use a cubic polynomial. Thus, let w(t)=ao+ait+azt'+azt, where the four parameters ao, ai, az, az, are chosen optimally by the investor. This would imply that the stock weight formed at the beginning of time t=jh is ao+a1(jh)+a2(j'h)2+a3(j'h), where; is the month number (0,...,59) and thus jh is in calendar time. For an A of 5, what are the investor's optimal choices of ao, ai, az, az? 5. For the optimal strategy determined in question 4, plot the weight in stock as a function of t. Provide intuition for the graph. 6. Now, consider a different kind of strategy: a buy-and-hold portfolio. That is, you put an initial weight of 60% in stock and 40% in cash. However, you never rebalance, and just leave the portfolio alone. (Your initial dollars in stock simply grow over time with the realized return on stock. Your initial dollars in cash simply grow over time with the riskless rate. Adding the two together will give you the future value of the entire portfolio.) Simulate the portfolio path over 5 years. Provide a histogram of the ending portfolio value