Question

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 $40,000. After doing a bit more research, he has found the following additional expenses involved in the purchase:

• Insurance and registration will cost $510 per year, payable at the start of each year
• Based on mileage estimates, petrol will cost $220 per fortnight, payable at the end of each fortnight
• Servicing will cost $380 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 4.6% 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 $80,000 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 4.6% 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 $386.00549595166 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 $80,000 and this weekly benefit of $386.00549595166, 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 $1608.9831851444 per month in advance for the coming 5 years.

(a) If Gillian can borrow/invest money at a rate of 4.1% 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

Price of the car = $ 40,000

Expenses involved in the purchase:

1. Insurance and registration costs = $510 per year, payable at the start of each year

2. Petrol costs= $220 per fortnight, payable at the end of each fortnight

3. Servicing costs = $380 per year, payable at the end of each year

Life time of Car = 5 years

Selling price of the car at the end of 5 years = Quarter of its Current Purchase price

Interest rate = 4.6 % p.a.

Solution a:

Present value of the running costs (Insurance & Registration, Petrol and Servicing):

Let us calculate these costs individually and sum them up in the end. Let us consider discounting rate as the bank interest rate i.e. 4.6 % p.a.

1. Insurance costs:

Insurance costs per year are $ 510, which are paid at the start of each year for 5 years.

The present value of these payments can be calculated using the Present Value of an Annuity Due Formula. Annuity refers to the recurring payments made at the beginning of each year for certain time periods.

Insurance payments are similar to Annuity Due as they are paid in a reccurring manner at the beginning of each year for 5 years, until Goeffry sells the car.

Present Value of Annuity Due is given as :

where PMT is the periodic payment = $ 510

r is the discounting rate = 4.6% p.a.

n is the number of periods = 5

Hence, Upon substituting the above values and solving, we get

PV of Insurance costs = 2,335.37

2. Present Value of Petrol costs:

Petrol cost payments are made at the end of each fortnight. The PV of these payments can be calculated using the PV of an Annuity Formula.

One year has approximately 26 fornights, Hence 5 years will account for 5* 26= 130 fortnights.

where PMT is the period payments value = 220

r is the interest rate = 0.046/130

n is the number of periods = 130

Upon substituting the above values in the formula and solcing, we get

Hence PV of Petrol costs = $ 27,947.34

3. Present Value of the servicing costs:

Servicing will cost $ 380 per year at the end of each year for 5 yearsand hence its present value can be calculated using PV of Annuity formula.

where PMT is the periodic payment = $ 380

r is the interest rate = 0.046

n is the number of years = 5

Upon substituting the above values in the formula and solving, we get

Hence the present value of the servicing costs = 1,663.55

The PV of the running costs = PV of Insurance + PV of Petrol costs + PV of Serviving costs

PV of Running costs = 2,335.37 + 27,947.34 + 1,663.55

PV of Running costs = $ 31,946.26

Solution b:

Present Value of the Sale Price can be calculated using PV formula

FV = Sale Price of car after 5 years

r = discounting rate = 0.046

n = 5 years

Hence, Present value of the Sale price is $ 7,986.23

Solution c:

Total cost of buying the car in today's dollars, when Sale price is also taken into account is:

= Purchase Price of the Car + Present Value of its Running costs - Present Value of its Sale Price

Purchase Price of the Car = $ 40,000 (Given)

PV of Running Costs = $ 31,946.26 (From Solution a)

PV of its Sale Price = $ 7986.23 (From Solution b)

Hence, Total Cost of buying the car is = $ 40,000 + $ 31,946.26 -$ 7986.23 = $ 68,829.45

Total cost of buying the car = $ 63,960.04

Solution d:

Assuming Geoffrey borrows $ 40,000 from the bank at an interest rate of 4.6 % p.a for 5 years, the equal monthly payments over the next 5 years can be calculated using the Annuity calculation formula:

Annuity referes to the recurring payments made at the end of each time period.

where P is the Loan amount = $ 40,000

r = interest rate = 4.6% p.a. = 0.046/12 per month

Tenure = 60 months (5*12)

Hence, equivalent monthly repayment over the next 5 years, where payments are made at the end of each month is $ 747.54