IE 370 HW 2

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University of Louisville *

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370

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Industrial Engineering

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Feb 20, 2024

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Jake Broughton IE 370-01 02/13/24 Homework 2 1. A $250,000 bond having a bond rate of 6% payable annually is purchased for $235,000 and kept for 6 years, at which time it is sold. How much should the bond be sold for to yield at a 5% effective annual return on the investment? (15 points) P = Purchase price x (1 + Effective annual return)^# of years P = 235,000 x (1.05)^6 P = $314,922.48 2. Thomas borrows $7,500 at 10% per year compounded annually and will repay the loan in 3 equal annual payments starting 1 year after the loan is made. (20 points) a. What is the size of his annual payment? (5 points) b. What is the amount of his interest payment at the end of year 2? (10 points) c. What is the amount of his principal payment at the end of year 2? (5 points) a) PMT = [PV,FV,N,I] PMT = [7500,0,3,0.1] PMT = $3,015.86 b) FV = [PV,PMT,N,I] FV = $5,234.14 Interest in year 2 = (0.10)(5,234.14) = $523.41 c) Principal payment in year 2 = 3,015.86 – 523.41 = $2,492.45 3. $30,000 is deposited in a fund that pays 2% annual compound interest for the first 2 years, 3% annual compound interest for next 2 years, and 4% annual compound interest for the next 2 years. If uniform annual withdrawals occur over the 6-year period, what will be the magnitude of the annual withdrawals? (10 points) PV = (C x (1-(1+r)^(-n)))/r First 2 years: PV_1 = C x (1-(1+0.02)^-2)/0.02 = 1.98C Next 2 years: PV_2 = C x (1-(1+0.03)^-2)/0.03 = 1.9633C
Next 2 years: PV_3 = C x (1-(1+0.04)^-2)/0.04 = 1.923C PV = 1.98C + 1.9633C + 1.932C = 5.873C 5.873C = 30,000 C = $5,112.65 4. Katie plans to purchase a new car. She decides to borrow $25,000 from her friend at 8% per year compounded monthly for 4 years. She plans to repay the loan with 48 equal monthly payments. (25 points) a. How much is the monthly payment? (5 points) b. How much interest is in the 23rd payment? (10 points) c. What is the remaining balance immediately after she made her 37th payment? (5 points) d. Later, she became able to pay off the loan at the end of the 30th month. She has not yet made the payment due at that month. What is the payoff amount for her loan at that time? (5 points) a) 25,000 = (X)(1-(1+i))^-48/i I^12 = 0.08 => i^12/12 = 0.0067 25,000 = (X)(1-(1+0.0067))^-48/0.0067 X = 610.792 b) (1 – (1+0.0067)^-25/0.0067) x 610.792 + 610.792 = 14,627.171 In 23 rd payment: 14,627.171 x 0.0067 = 98.002 c) After 37 th payment: (1 – (1 + 0.0067^-11/0.0067) x 610.0792 = 6,456.071 d) (1 – (1+0.0067)^-18/0.0067) x 610.792 + 610.792 = 10,935.452 5. Zhang purchases a bond for $952. The bond matures in 3 years, and Zhang will redeem it at its face value of $1,000. Coupon payments are paid annually. If Zheng will earn a yield of 12% per year compounded yearly, what is the bond (coupon) rate? (15 points) Coupon rate = (Annual coupon payment)/(Face value of bond) x 100% Present value of bond = Face value of bond x 1/(1+YTM)^n + Annual coupon payment x (1 – 1/(1+YTM)^n)/(YTM) 952 = 1000 x 1/(1+0.12)^3 + Annual coupon payment x (1 – 1/(1+0.12)^3)/(0.12) Annual coupon payment = 952−1000 𝑥𝑥 1 /( 1+0 . 12 )^ 3 1−1 /( 1+0 . 12 )^ 3 0 . 12 = 48.76 Coupon rate = 48.76/1000 x 100% = 4.88%
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