Test 1 solution sketches Note for multiplechoice questions

Test 1 solution sketches Note for multiple-choice questions: Choose the closest answer

Loan calculations n Billy’s Pianos receives a loan of $180, 000 today. The stated annual interest rate is 8. 4%, compounded monthly. Payments are monthly, starting one month from today. The loan is amortized over 30 years.

Loan calculations n If Billy pays an equal amount of principal each month, how much will the first payment be? n n Monthly rate =. 0084 / 12 = 0. 7% Amount of principal paid each month = $180, 000 / 360 = $500 Amount of interest accrued in first month = $180, 000 *. 007 = $1, 260 First payment = 500 + 1, 260 = $1, 760

Loan calculations n If Billy makes equal month payments each month, how much will the first payment be? n n n 180, 000 = C /. 007 * [1 – 1 / (1. 007)360] 180, 000 = 131. 262 * C C = $1, 371. 31

Loan calculations n If Billy pays an equal amount of principal each month, how much will the last payment be? n n n Principal owed in 359 months = 180, 000 / 360 = 500 Interest owed = 500 *. 007 = 3. 50 Last payment = 500 + 3. 50 = $503. 50

Loan calculations n If Billy makes equal month payments each month, how much will the last payment be? n n Note: equal payments means first = last (so same answer as #2) 180, 000 = C /. 007 * [1 – 1 / (1. 007)360] 180, 000 = 131. 262 * C C = $1, 371. 31

Profitability Index n Carly Rae pays $50, 000 to open her dating service. She receives $2, 700 per year in cash flow, starting in two years. Annual discount rate is 5%. What is the profitability index? n n n PV of benefits = 2700 /. 05 * 1 / 1. 05 = 51, 429 PV of costs = 50, 000 PI = 51, 429 / 50, 000 = 1. 029

Effective Discount Rates n If the effective annual discount rate is 15%, then what is the effective discount rate for 8 months? n (1. 15)8/12 – 1 = 9. 76534%

PV of Annuity n Wolfgang will receive royalty payments of $500 every year, starting 5 years from today and ending 25 years from today. What is the present value of these payments if the effective annual discount rate is 15%? n n Annuity formula for 21 payments, discounted by 4 years due to 1 st payment in year 5 500/. 15 * [1 – 1 / 1. 1521] * 1 / 1. 154 = $1, 804. 59

Real payments n If the inflation rate this year is 5% and the nominal interest rate is 15%, then what is the real interest rate? n n (1 + real)(1 + inflation) = (1 + nominal) (1 + real)(1. 05) = 1. 15 1 + real = 1. 15 / 1. 05 = 1. 0952381 Real = 9. 52381%

Discounted vs. undiscounted payback periods n n Reba’s Rabbits invests $50, 000 today, and will earn $10, 000 each year starting one year from today. The effective annual discount rate is 9%. If Reba uses discounted cash flows, how many years is the payback period for this investment? n n 50000 = 10000/. 09 (1 – 1/1. 09 T) 61111 = (10000/. 09)/(1. 09 T) 1. 09 T = (10000/. 09)/61111 = 1. 81818 T = ln(1. 81818)/ln(1. 09) = 6. 93726 ≈ 7

Discounted vs. undiscounted payback periods n If Reba uses undiscounted cash flows, how many years is the payback period for this investment? n 50000 / 10000 = 5

Pyotr’s Beauty Products n Pyotr’s Beauty Products is considering buying a new device. This machine would cost $8, 000 today, and require maintenance costs of $600 every three years, starting in 2 years and ending in 11 years. The machine lasts 12 years, and the effective annual discount rate is 14%.

Part (a) n What is the present value of all costs of the machine over its life? n n Purchase cost today and maintenance costs in years 2, 5, 8, and 11 8000 + 600/(1. 142) + 600/(1. 145) + 600/(1. 148) + 600/(1. 1411) = $9, 125. 61

Part (b) n Pyotr pays $X per year for five years, starting today. These payments will have the same present value as the answer you got from part (a). Find X. n n n X + X/1. 14 + X/(1. 142) + X/(1. 143) + X/(1. 144) = 9125. 61 3. 91371 * X = 9125. 61 X = $2, 331. 70

Yield to Maturity n A bond has a face value of $750. It pays a coupon of 10% today, one year from today, and two years from today. Two years from today, the bond matures. If the current selling price of the bond is $800, what is the yield to maturity (expressed as an effective annual discount rate)?

Yield to Maturity n n 800 = 75 + 75/(1+r) + 825/(1+r)2 725(1+r)2 – 75(1+r) – 825 = 0 725 r 2 + 1375 r – 175 = 0 29 r 2 + 55 r – 7 = 0 Ignore negative root. r = 0. 119716 so r = 11. 97%. Or…

Yield to Maturity n n 800 = 75 + 75/(1+r) + 825/(1+r)2 725(1+r)2 – 75(1+r) – 825 = 0 Let x = 1+r 29 x 2 – 3 x – 33 = 0 Ignore negative root. x = 1. 1197 so r = 11. 97%

Balloon Payment n Michael is taking out a loan of $1, 000 today and he will pay $22, 000 per month for the next 10 years (120 payments, starting one month from today). The stated annual interest rate is 24%, compounded monthly. 13 years from today, Michael will make one additional payment to pay off the loan. How much will this payment be?

Balloon Payment n PV of monthly payments: n n PV of payment made in 13 years: n n 22000/. 02 * [1 – 1/(1. 02120)] = 997, 818. 55 1, 000 – 997, 818. 55 = 2, 181. 45 FV of payment made in 13 years: n 2, 181. 45 (1. 02)12*13 = $47, 904. 10