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Exercise 2.6 (Investing wisely) Almost certainly, Albert Einstein did not say that "compound interest is the most powerful force in the universe." A company wants to maximize their cash holdings at the end of t time periods. They have an external inflow of dollars at the start of time period t , for t = 1,2,...,1. At the start of each time period, available cash can be allocated to any of K different investment vehicles (in afty available non-negative amounts). Money allocated to investment-vehicle k at the start of period t must be held in that investment k for all remaining time periods, and it generates income 4+1 .... ,, per dollar invested. It should be assumed that money obtained from cashing out the investment is incorporated into these parameters. For example, (vivl, 16,4,7,8,9,10, 11, 12) = (0.1,0.1,0.1, 1.1,0,0,0,0,0) can be interpreted as 1 dollar invested in investment vehicle #9 at the start of time period 4 yields 0.1 dollars of income for times periods 4-7, and with the original dollar returned in time period 7, and no returns at all in the remaining time periods 8-12. Note that at the start of time period t, the cash available is the external inflow of p, plus cash accumulated from all investment vehicles in prior periods that was not reiftvested. Finally, assume that cash held over in any time period earns interest of a percent. Formulate the problem, mathematically, as a linear-optimization problem. Then, model the problem with Python/Gurobi, make up some data, try some computations, and report on your results. Problem 4 (30 pts). Exercise 2.6 on Page 18 in Jon Lee's book. 1) Formulate the problem as a linear optimization problem and provide necessary explanation or justification of your model. 2) Solve your model using CVX and present the optimal solution and the optimal value of the instance generated by the following code. 1 I - 3; # number of time periods 2 K-3; number of investment vehicles 39 - 1; bank interest rate in percent 4 5 % external cash flow 6 P- (10, 1,8); 7 3 unit income vikt, s), k-1,...,K, t - 1,...,I, S - t, ...,I V- {! 10 V (1,:) - 1 [0.08 0.01 1.13; 11 0 0.02 1.1; 12 0 0 1.03]}} 13 V(2,:) - 110.06 0.03 1.07; 14 0 0.04 1.16; 15 0 0 1.04117 16 V(3,:) - 1 [0.03 0.01 1.08; 17 0 0.00 1.01; 1s 0 0.0 1.1] 17 Exercise 2.6 (Investing wisely) Almost certainly, Albert Einstein did not say that "compound interest is the most powerful force in the universe." A company wants to maximize their cash holdings at the end of t time periods. They have an external inflow of dollars at the start of time period t , for t = 1,2,...,1. At the start of each time period, available cash can be allocated to any of K different investment vehicles (in afty available non-negative amounts). Money allocated to investment-vehicle k at the start of period t must be held in that investment k for all remaining time periods, and it generates income 4+1 .... ,, per dollar invested. It should be assumed that money obtained from cashing out the investment is incorporated into these parameters. For example, (vivl, 16,4,7,8,9,10, 11, 12) = (0.1,0.1,0.1, 1.1,0,0,0,0,0) can be interpreted as 1 dollar invested in investment vehicle #9 at the start of time period 4 yields 0.1 dollars of income for times periods 4-7, and with the original dollar returned in time period 7, and no returns at all in the remaining time periods 8-12. Note that at the start of time period t, the cash available is the external inflow of p, plus cash accumulated from all investment vehicles in prior periods that was not reiftvested. Finally, assume that cash held over in any time period earns interest of a percent. Formulate the problem, mathematically, as a linear-optimization problem. Then, model the problem with Python/Gurobi, make up some data, try some computations, and report on your results. Problem 4 (30 pts). Exercise 2.6 on Page 18 in Jon Lee's book. 1) Formulate the problem as a linear optimization problem and provide necessary explanation or justification of your model. 2) Solve your model using CVX and present the optimal solution and the optimal value of the instance generated by the following code. 1 I - 3; # number of time periods 2 K-3; number of investment vehicles 39 - 1; bank interest rate in percent 4 5 % external cash flow 6 P- (10, 1,8); 7 3 unit income vikt, s), k-1,...,K, t - 1,...,I, S - t, ...,I V- {! 10 V (1,:) - 1 [0.08 0.01 1.13; 11 0 0.02 1.1; 12 0 0 1.03]}} 13 V(2,:) - 110.06 0.03 1.07; 14 0 0.04 1.16; 15 0 0 1.04117 16 V(3,:) - 1 [0.03 0.01 1.08; 17 0 0.00 1.01; 1s 0 0.0 1.1] 17