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 $26,000. After doing a bit more research, he has found the following additional expenses involved in the purchase:

• Insurance and registration will cost $530 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 $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 4.5% per annum effective.

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.

Answer 1

Assume time in years is measured by t

Cost of Car (t=0) = $ 26,000

As he gets to invest money at 4.5%, we assume this to be the discount rate for the purchase and that the value of money will discount by this interest rate.

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Salvage value/value at which car can be sold (t=5) = $26,000 / 4 = $ 6,500

PV (sale)=FV/(1+r)^n =  6500 /((1+4.5%)^5) = $5215.93

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Calculation of other costs-

1. Petrol- $220 per fortnight for approximately.

Total number of days - 365*5 = 1825

Approximate Number of periods for which petrol costs will need to be paid (14 days in a fortnight) =130

Fortnightly interest rate = Yeary interest rates/number of periods in a year = 4.5%/26=0.173% (fortnightly)

PV = A /(1+(r/t))^(t*n)

We sum all present values to get the total present value of cost. The calculation will be shown in the excel.

where A - Monthly Payements

r/t - fortnightly interest rate

t*n - number of periods

Present value for the fortnights have been calculated as given in the formula in the excel-

Which gives The total present cost of petrol = $25,591

The formulae are shown here -

2. Servicing (t=1 to t=5 -  $400 each year)

Similar PV calculations for 5 periods are as shown in the excel. Formula for PV calculation remains same.

PV =FV/(1+r)^n

where r = 4.5%

Total Cost for Servicing = $ 1756

Formulae are shown here-

3. Insurance cost - $530 for (t = 0 to t = 5 assuming he sells the car post the insurance date)

PV calculations for 6 periods are as shown in the excel. Formula for PV calculation remains same.

PV =FV/(1+r)^n

where r = 4.5%

Total Cost for Insurance as shown in the excel below= $ 2,156

Formulae are as shown here (Note- Please take sum from G6-G11)-

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Total cost of running the car = Cost of buying car + (Cost of fuel + Cost of Insurance + Cost of Servicing)

= $ 26,000 +$ 1,756 +  $ 2,156 +$25,591

=$55,503

Considering the PV of Salvage Value = $5215.93

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Total cost of running and buying the car = $ 55,503 - $5,216 = $ 50, 287