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 $33,000. After doing a bit more research, he has found the following additional expenses involved in the purchase:

• Insurance and registration will cost $430 per year, payable at the start of each year
• Based on mileage estimates, petrol will cost $300 per fortnight, payable at the end of each fortnight
• Servicing will cost $370 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?

Answer 1

a) car price = 33000

ins.+reg = 430/year = 430*5 after 5 years

petrol cost = 300/fortnight = 300*2*12*5 for 5 years

servicing cost = 370/year = 370*5 for 5 years

life = 5 years

resale value = 33000/4 = 8250

rate of interest = 3.2% = .0.032

total running cost for 5 years = (430*5)+( 300*2*12*5)+(370*5) = 2150+36000+1850 = $40000

present value (PV) of running cost = 40000*1/1.17 = $34188.03

b) PV of resale value = 8250(1/1.17) = $7051.28

c) After 5 years total expenses = $40000+ 33000 - 8250 = 64750

PV of total expenses = 64750 *(1/1.17) = $ 55341.88

d) To calculate equivalent monthly repayment (P) over the next 5 years, where payments are made at the end of each month we will use formula - P = A/D

A = Total loan amount with respect to present value = 55341.88

D = discounting factor.

D = {(1+r)^n-1}/{r(1+r)^n}

n = payment times = 12*5 = 60

r = .032/12 (since there will be monthly repayments, the yearly rate of interest is divided by 12) = .003

D= (1.2 -1)/.003 = 66.7

therefore, P = 55341.88/66.7= $830 per month.