Problem 1: Leasing at TST, Inc. Tri-State Technologies, Inc. (TST) is considering leasing a manufacturing facility for widgets. The cost of the lease is $1,000,000 per year. Under the proposed agreement, TST commits to a two-year lease. TST will operate the facility at full capacity in both years, producing 10,000 widgets each year. TST currently sells widgets on the wholesale market for $100 per unit, but the price is subject to random fluctuations. TST estimates that, during the first year, the average price will be $100x(1+R), where R, is a normally distributed random variable with a mean of zero and a standard deviation of 30%. Similarly, the average market price for the second year will change relative to the first year's price by a factor of (1+R2), where R, is a random variable independent of R, and has the same distribution as R. Assume that TST can sell all 10,000 widgets it produces in each year, and that all manufacturing costs are already included in lease payments. Q1. For arbitrary values of R, and R, write an algebraic expression for TST's total net profit for the 2- year period. In other words, use variables R, and R to generally represent the two yearly price fluctuations in the total profit expression, without computing any specific values. TST uses Monte Carlo simulation to obtain a probability distribution of the facility's net profit for the 2-year period. Below, we show the results of running the simulation for n=100,000 trials. Note that on this graph the continuous black line depicts the cumulative probability distribution, while the blue bars indicate the frequency distribution values. 20% 100.0% 18% Net Profit Distribution 98.5% 99.6%100.0% 95.4% Trials 100000 88.0% Mean $2,536 Median ($32,407) Minimum ($2.625.772) 73.79 Maximum $3,993,456 Mean Std. Error $2.145 90.0% 16% 80.0% 14% 70.0% 12% 52.27 60.0% Frequency 10% 50.0% Cumulative Distribution 8% 40.0% 28.976 6% 30.0% 4% 20.0% 2% 11.27 2.9% 0.0% 0.4% 2.5% 8.3% 17.7% 23.3% 21.5% 14.3% 7.4% 3.1% 1.1% 0.4% 0.0% 0.4% 10.0% (12, 16] (8, 12) (4.81 (0.41 14.01 (-8, 41 (-12, -81 (-20, -16] (-16, -12) 0% 0.0% (16, 201 >20 20