Quiz 1 solution sketches 11:00 Lecture, Version A Note for multiple-choice questions: Choose the closest answer

PV of Perpetuity Usagi receives $10,000 per year, forever. The first payment is in one year, and payments are made every 3 months. (In other words, Usagi will receive $2,500 every 3 months starting in one year.) If her effective annual discount rate is 7%, what is the PV of this stream of payments?

PV of Perpetuity FV 9months = 2500/(1.07 1/4 – 1) = 146,554 PV = 146,554 / (1.07 3/4 ) = $139,303

Compounding Approximation Bubba Bo Bob will receive $1,000,000 per year, starting in one year from today. The SAIR is 10%, compounded every 4 seconds. Which of the following compounding frequencies will give you the answer closest to the PV of his stream of payments?

Compounding Approximation There is a very large number of times that interest is compounded per year. For compounding purposes, the result is very close to compounding an infinite number of times (i.e. continuous compounding)

Equal-Principal Payments Mikey borrows $6,000 today. He makes 6 monthly payments to pay back the loan, with each payment including equal amounts of principal. How much will the payment be 3 months from today if the SAIR is 12%, compounded monthly?

Equal-Principal Payments Amount owed in 2 months = $4,000 Interest incurred in 3 rd month = 4000*(.12/12) = $40 Principal reduction = $1,000 Total payment = $1,040

Growing Payments Gage borrows $500 today. He will pay back the money over the next 12 months. The SAIR is 24%, compounded monthly. The first payment of $X is paid today, with each subsequent payment 1% higher than the previous. Find X. 500 = X * 1/( ) * [1 – (1.01/1.02) 12 ] 500 = * X X = $44.84

Future Value on Specific Dates For the next two questions, assume the following information: Today is October 14, You invest $4,000 today. Find the FV on the following dates, given the SAIR and frequency of compounding.

Future Value on Specific Dates February 14, 2018, SAIR of 18%, compounded every 2 months 10/14/13  2/14/18 12*4 + 4 = 52 months (26 compounding periods).18/6 =.03 (interest rate every 2 months) FV = 4000 * (1.03) 26 = $8,626.37

Future Value on Specific Dates October 14, 2023, SAIR of 10%, compounded every 30 months 10/14/13  10/14/23 10 years = four 30-month blocks.1*2.5 =.25 (interest rate every 30 months) FV = 4000 * (1.25) 4 = $9,765.63

PV of Payments Roselyn will receive $800 in four months and $1200 in eight months. The effective annual discount rate is 15%. Find the total PV of the two payments. Rate every 4 months = (1.15) 1/3 – 1 = 4.769% PV = 800/ /( ) 2 PV = $1,856.83

PV of Perpetuity Theo will receive $1,000 every three years forever, starting seven years from now. The effective annual interest rate is 13%. Find the total PV of all payments. Rate every 3 years = (1.13) 3 – 1 = % FV 4 = 1000/ = $2, PV = /(1.13) 4 = $1,384.79

Semi-annual Payments Aubree borrows $50,000 today. She will make equal payments every 2 years for the next 30 years, with the first payment 2 years from today. If the effective annual interest rate is 5%, how much will each payment be to completely repay the loan over the next 30 years?

Semi-annual Payments 15 total payments (every 2 years for 30 years) Rate every 2 years = (1.05) 2 – 1 =.1025 Annuity formula: 50,000 = C/.1025 * [1 – 1/(1.1025) 15 ] 50,000 = * C C = $6,667.77