Question

ONLY ANSWER QUESTION IN BOLD

Geoffrey is the owner of a small grocery store, and is considering buying a car to help him transport his wares. He has found a suitable used car online that he was able to negotiate to a price of $32,000. After doing a bit more research, he has found the following additional expenses involved in the purchase: Insurance and registration will cost $440 per year, payable at the start of each year Based on mileage estimates, petrol will cost $260 per fortnight, payable at the end of each fortnight Servicing will cost $400 per year, payable at the end of each year (as the car was recently serviced by the previous owner Assume that these are the only expenses involved with the purchase and operation of the car. Geoffrey believes that the car can be used for 5 years before it will no longer be reliable, at which point he expects to sell it for a quarter of its current purchase price. He also has a business account at the bank that lets him borrow or invest money at 3.2% per annum effective.

(a) Calculate the present value of the running costs (i.e. the insurance and registration, the petrol, and the servicing).

(b) Calculate the present value of the sale price.

(c) What is the total cost of buying and running the car in today's dollars? Your answer should also take into account the eventual sale price.

(d) What is the equivalent monthly repayment over the next 5 years, where payments are made at the end of each month?

Geoffrey decides not to buy the car mentioned earlier. Instead, he is now considering a food delivery service "You, bars, meats" that his friend Gillian has recently started. Gillian has agreed that for a single payment of $64,000 today to help her launch her business, she will provide all the delivery services that Geoffrey needs for his business for the next 5 years. Geoffrey is considering borrowing the full amount from his business account. Suppose that Geoffrey makes level quarterly repayments over the coming 5 years, the first payment being exactly 3 months from today. Again, the interest rate on Geoffrey's account is 3.2% p.a. effective.

(a) Calculate the size of the level quarterly repayment.

(b) How much money does Geoffrey owe on this loan after 1 year?

(c) How much interest does Geoffrey pay in the first year?

(d) Geoffrey believes that the overall benefit from this agreement amounts to $320.34696082538 per week in arrears (this would include money he would have spent on alternative delivery services, estimated additional profits from using Gillian's services, etc). By considering only the initial cost of $64,000 and this weekly benefit of $320.34696082538, calculate the interest rate that represents the return on this investment, expressed as a nominal annual rate compounding weekly.

Gillian has entered the agreement with Geoffrey described above. She estimates that the costs of the delivery services she has promised to Geoffrey (petrol, insurance, wear and tear, etc) amount to $1296.7416772338 per month in advance for the coming 5 years.

(a) If Gillian can borrow/invest money at a rate of 2.7% p.a. effective, what is the equivalent amount today of her future liabilities? Note that this calculation should not involve the payment she receives from Geoffrey today.

(b) The money she receives from Geoffrey can be considered a loan, with repayments being the value of the services she provides in return. What is the
interest rate, expressed as an effective annual rate, she is being charged on this "loan"?

Answer 1

Cost of delivery service per month = $1,296.7416772338

Cost Period = 5 years = 60 months

a)

Payment Frequency = Monthly

Annual Effective Interest Rate = 2.7%

Annual Nominal Interest Rate for monthly compounding can be calculated from Annual Effective Rate by using the NOMINAL function in spreadsheet

NOMINAL (Effective Rate, Periods per year)

Where, Effective Rate = Annual Effective Rate = 2.7%

Periods per year = number of compounding periods per year = 12

Annual Nominal Interest Rate = NOMINAL (2.7%, 12) = 2.667153%

Monthly Interest Rate = Annual Nominal Interest Rate / 12 = 2.667153%/12 = 0.222263%

Present Value of Gillian's Future liabilities can be calculated using the PV function in spreadsheet

PV(rate, number of periods, payment amount, future value, when-due)

Where, rate = monthly interest rate = 0.222263%

number of periods = cost period in months = 60

payment amount = Cost of delivery service per month = $1,296.7416772338

future value = 0

when-due = when is the payment made each month = beginning = 1

Money owed by Geoffrey after a year = PV(0.222263%, 60, -1296.7416772338, 0, 1) = $72,925.88

b)

Loan Amount = $64,000

Value of Services provided monthly = $1,296.7416772338

Loan Period = 5 years = 60 months

Monthly interest rate on this loan can be calculated using the RATE function in spreadsheet

RATE(number of periods, payment per period, present value, future value, when-due, rate guess)

Where, number of periods = no.of months of loan period = 60

payment per period = monthly value of services = $1,296.7416772338

present value = loan amount = $64,000

future value = 0

when-due = when is the payment made each month = beginning = 1

rate guess = a guess of the monthly interest rate = 0.6%

Monthly interest rate on this loan = RATE(60, 1296.7416772338, -64000, 0, 1, 0.6%) = 0.688334%

This is a monthly rate. To convert this to an effective annual rate (AER) we have to use the formula

1+Annual Effective Rate = (1+monthly interest rate)¹²

1+AER = (1+0.688334%)¹²

1+AER = 1.085800

Annual Effective Rate, AER = 1.085800 - 1 = 0.085800 = 8.58%