You have saved 3000 for a down payment on a new car The larg

You have saved $3,000 for a down payment on a new car. The largest monthly payment you can afford is $500. The loan will have a 11% APR based on end-of-month payments.

What is the most expensive car you can afford if you finance it for 48 months? Do not round intermediate calculations. Round your answer to the nearest cent.

$ What is the most expensive car you can afford if you finance it for 60 months? Do not round intermediate calculations. Round your answer to the nearest cent. $

Solution

Principal amount of loan can be calculated using formula for EMI as:

EMI = P × r × (1 + r)ⁿ /{(1 + r)ⁿ - 1 }

P = EMI/[r × (1 + r)ⁿ /{(1 + r)ⁿ - 1 }]

Where,

P = Principal

r = rate of interest = 11 % p.a. or 0.11/12 = 0.009166667 monthly

n = No. of periods = 48

Substituting all the values in above formula, we get:

P = $ 500 / [0.009166667 x (1 + 0.009166667)⁴⁸ / {(1 + 0.009166667) ⁴⁸ -1}]

   = $ 500 / [0.009166667 x (1.009166667)⁴⁸ / {(1.009166667) ⁴⁸ -1}]

= $ 500 / [0.009166667 x 1.54959805 / (1.54959805-1)]

= $ 500 / [0.009166667 x 1.54959805 / (0.54959805)]

   = $ 500 / (0.01420465/ 0.54959805)

   = $ 500/0.02584552

   = $ 19,345.71

Total cost of car = Principal of loan + down payment = $ 19,345.71 + $ 3,000 = $ 22,345.71

If n = 60 periods,

P = $ 500 / [0.009166667 x (1 + 0.009166667)⁶⁰ / {(1 + 0.009166667) ⁶⁰ -1}]

   = $ 500 / [0.009166667 x (1.009166667)⁶⁰ / {(1.009166667) ⁶⁰ -1}]

= $ 500 / [0.009166667 x 1.72891573/ (1.72891573-1)]

= $ 500 / [0.009166667 x 1.72891573/ (0.72891573)]

   = $ 500 / (0.015848394/ 0.72891573)

   = $ 500/0.021742423

   = $ 22,996.52

Total cost of car = Principal of loan + down payment = $ 22,996.52 + $ 3,000 = $ 25,996.52

For a loan period of 48 months, $ 22,345.71 car can be affordable where as for loan period of 60 months, $ 25,996.52 car can be affordable.