Question

Today you bought a brand new 2022 Ford Mustang for $34,145. Unfortunately, cars depreciate in value quickly and the Ford Mustang depreciates at a rate of 8%.

a. Find the cars value after it has depreciated for 9 years.
b. A car is considered to be a classic car after it is 25 years old. Determine the cars value after it has depreciated for 25 years.
c. A classic cars value will raise as there are fewer and fewer of them left around. If the Mustang will start to grow at a rate of 14% each year, determine its value in the year 2059. (remember for the first 25 years, the value is going down, then after 25 years, its value starts to go back up again)

Answer 1

a) $16,122

b) $4,246

c) $112,357

Explanation:

What is depreciation

We should calculate the depreciation using the declining balance method
This uses the technique of computing the depreciation on the previous year's depreciated value

As an example
If the original cost is $100 and depreciation rate is 10%( 0.10 in decimal)

Depreciation Year 1 = 100 x 0.10 = $10

Depreciated value at end of year 1 = 100 - 10 = $90

For the second year, the depreciation amount is computed on $90
= 90 x 0.10 = $ 9

Depreciated amount at end of year 2 = 90 - 9 = $81

The depreciated value of an asset valued at $A at time of purchase at a depreciation rate of r (in decimal) for n years is given by the formula
A' = A(1 - r)ⁿ

We are given
A = $34,145
r = 8/100 = 0.08 therefore 1 - r = 1 - 0.08 = 0.92

a) Depreciated value after 9 years:
A' = 34145(0.92)⁹
= 16121.95

Rounded to the nearest dollar that would be $16,122 ANSWER

b) Depreciated Value after 25 years

A' = 34145(0.92)²⁵

= $4,246 rounded to the nearest dollar

c) Car's value after 50 years

At the end of 25 years, the Mustang has become a classic with a value of $4, 246

Then it starts to appreciate at the rate of 14% (0.14 in decimal)

The appreciated value each year after 25 years is given by
A'' = A'( 1 + r)ⁿ

where r is the rate of appreciation and n is the number of years

We are given r = 14% = 0.14

1 + r = 1.14

n = 25

A' = 4,246 the value after 25 years from purchase

Car value after 50 years = car value after 25 years appreciated for the next 25 years

= 4246(1.14)²⁵

= $112,357 rounded to the nearest $

So you can really make a killing after 25 years. As an interesting side note, the value of the car will reach the original value 16 years after 25 years or a total of 41 years after purchase