Question

1 You want to buy a home that cost 420,000 dollars, using a 30 year mortgage. If interest rates on the loan are 3.9% and interest is compounded monthly, what will be your monthly payment? (Please use at least 5 decimal places and do not use $ symbol in the answer)

2 If you save 8 hundred dollars per month in a bank account that earns a 8% interest rate (compounded monthly) for 30 years, how much will be in your account in 30 years time? (Please use at least 5 decimal places and do not use $ symbol in the answer)
3 What is the present value of an investment that pays out 40 dollars per quarter for the next 5 years, if the appropriate discount rate is 5%? (compound quarterly) (Please use at least 5 decimal places and do not use $ symbol in the answer)
4 You are offered a loan with an annual intereest rate of 2% . If this loan is compounded monthly (12 times a year), what is the effective annual rate (EAR) of the loan? (Please write in decimal format using 5 decimal places, for example if the answer was 4.234% please write .04234)

Answer 1

1. Calculation of monthly payment for a 30-year mortgage of $420,000 with interest rates of 3.9% compounded monthly using formula for mortgage payment:First, we need to find out the number of payments:30 years = 30 x 12 = 360 months.So, n = 360Then, we will calculate the monthly interest rate:i = (3.9% / 100) / 12 = 0.00325Using these values, the formula for mortgage payment is:PMT = (loan amount) x (monthly interest rate) / [1 - (1 + monthly interest rate)^(-n)]Now, we will substitute the values and compute:PMT = (420,000) x (0.00325) / [1 - (1 + 0.00325)^(-360)]PMT = 1,977.69002. Calculation of the amount saved in a bank account earning 8% interest (compounded monthly) when $800 per month is deposited for 30 years using the formula for future value:FV = Pmt x [(1 + i)n - 1] / iWhere, FV = future valuePmt = payment madei = interest raten = number of paymentsFirst, we will find out the number of payments:30 years = 30 x 12 = 360 months.So, n = 360Then, we will calculate the monthly interest rate:i = (8% / 100) / 12 = 0.0066667Using these values, the formula for future value is:FV = 800 x [(1 + 0.0066667)^360 - 1] / 0.0066667FV = 1,456,246.573. Calculation of present value of an investment that pays out $40 per quarter for the next 5 years, when the appropriate discount rate is 5% compounded quarterly using the formula for present value:PV = CF / [(1 + r/m)^(n x m)]Where, CF = cash flown = number of payment periodsm = number of compounding periods per payment periodr = interest rateFirst, we need to find out the number of payment periods:5 years = 5 x 4 = 20 quartersSo, n = 20Then, we will substitute the given values in the formula and calculate:PV = 40 / [(1 + 0.05/4)^(20 x 4)]PV = 605.866394. Calculation of effective annual rate (EAR) of the loan that has an annual interest rate of 2% and is compounded monthly (12 times a year) using the formula for effective annual rate (EAR):EAR = (1 + i / m)^m - 1Where, i = nominal annual interest ratem = number of times compounded in a yearFirst, we will calculate the monthly interest rate:i = 2% / 12 = 0.16667%Using these values, the formula for effective annual rate (EAR) is:EAR = (1 + 0.16667% / 12)^12 - 1EAR = 0.020171