Question

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 $66,000 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 $313.69415778668 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 $66,000 and this weekly benefit of $313.69415778668, calculate the interest rate that represents the return on this investment, expressed as a nominal annual rate compounding weekly.

Answer 1

a)

Loan amount = $66,000

Loan Period = 5 years

Payment Frequency = Quarterly

Annual Effective Interest Rate = 3.2%

Annual Nominal Interest Rate for Quarterly 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 = 3.2%

Periods per year = number of compounding periods per year = 4

Annual Nominal Interest Rate = NOMINAL (3.2%, 4) = 3.162301%

Quarterly Interest Rate = Annual Nominal Interest Rate / 4 = 3.162301%/4 = 0.790575%

The Quarterly loan payment can be calculated using the PMT function in spreadsheet

PMT(rate, number of periods, present value, future value, when-due)

Where, rate = quarterly interest rate = 0.790575%

number of periods = no.of quarterly periods = 5*4 = 20

present value = loan amount = $66,000

future value = loan amount after the loan period = 0

when-due = when is the payment made each quarter = end = 0

Quarterly loan payment = PMT(0.790575%, 20, 66000, 0, 0) = $3,580.76

b)

Money owed by Geoffrey after a year can be calculated using the PV function in spreadsheet

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

Where, rate = quarterly interest rate = 0.790575%

number of periods = number of periods remaining in the loan = (5-1)*4 = 16

payment amount = Quarterly loan payment = $3,580.76

future value = loan amount after the loan period = 0

when-due = when is the payment made each quarter = end = 0

Money owed by Geoffrey after a year = PV(0.790575%, 16, -3580.76, 0, 0) = $53,618.20

c)

Total Payments made by Geoffrey in the first year = Quarterly payment * 4 = $3,580.76*4 = $14,323.05

Principal Payment made in first year = Principal balance at the start of the loan - principal balance after 1 year

= $66,000 - $53,618.20 = $12,381.80

Interest payment made in first year = total payments made in first year - principal payment made in first year

= $14,323.05 - $12,381.80 = $1,941.25

d)

Initial Cost = $66,000

Weekly benefit = $313.69415778668

No. of weekly payments = 5*52 = 260

Weekly return on this investment 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 weekly periods = 260

payment per period = weekly benefit = $313.69415778668

present value = Initial Cost = $66,000

future value = 0

when-due = when is the benefit received each week = end = 0

rate guess = a guess of the weekly return = 0.2%

Weekly return on this investment = RATE(260, 313.69415778668, -66000, 0, 0, 0.2%) = 0.168462%

This is a weekly rate. To convert this to a nominal annual rate we have to multiply this value by 52.

Nominal annual rate = 0.168462% *52 = 8.76%