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"?