Question

Jim's 1998 mini-van is functional. He purchased it in 1998 for $11,000 from a bank which required he make a down payment of $2500 at a 3% annual interest rate. This was the bank's original offer:

Jim decided to take the bank's original offer at 3% interest. He paid for the vehicle once a month each month, paid $250 dollars each year in maintenance costs, and paid the car off in 5 years.

Jim returned to his original purchase and an alternative he was considering. The alternative required a down payment of only $2000. He could use his mini-van to take his friends to school and they could pay him a total of $600 a year in gas money. With the alternative, he would only receive $550 a year because of the reduced amount of spacing. Based solely on the down payment and gas money:

1. Should Jim have purchased the alternative instead of his mini-van?
2. Provide a benefits-cost ratio to support your decision.

Answer 1

Question 1: Jim's 1998 mini-van is functional. He purchased it in 1998 for $11,000 from a bank which required he make a down payment of $2500 at a 3% annual interest rate. This was the bank's original offer:

Part a) If, as an alternative, the bank decided to charge Jim at 2% every 6 months what would Jim’s nominal interest rate be?

Solution: Nominal interest rate is also defined as a stated interest rate. This interest works according to the simple interest and does not take into account the compounding periods. The nominal interest rate is calculated in the following way, where i is the nominal rate, r the effective annual rate, and n the number of compounding periods per year (for example, 12 for monthly compounding):

Nominal Rate, i = n × ((1 + r)^1/n - 1)

Substituting given values in the above formula, we get

i = 2 x ((1+0.02)^1/2 – 1)

Nominal Rate, i = 1.9901%

Part b) what is the effective interest rate per period of this purchase?

Solution: Effective interest rate is the one which caters the compounding periods during a payment plan. It is used to compare the annual interest between loans with different compounding periods like week, month, year etc. In general stated or nominal interest rate is less than the effective one. And the later depicts the true picture of financial payments.

Effective Rate = [{(1 + stated Interest rate/number of compounding period)^n} – 1]

Substituting given values in the above formula we get,

Effective Rate =[ {(1 + 0.02/2)^2}-1]

Effective Interest Rate = 2.01%

Question 2: Jim decided to take the bank's original offer at 3% interest. He paid for the vehicle once a month each month, paid $250 dollars each year in maintenance costs, and paid the car off in 5 years.

Part a) How much did Jim pay the bank each year?

Solution: Information given

Interest Rate= 3%

Loan tenure in years= 5

Loan Amount = 11000

Down payment = 2500

Here, we first have to calculate the EMI amount that is paid by Jim every month

The formula to calculate an EMI:

EMI = [P x I x (1+I)^N]/[(1+I)^N-1]

Where,

P = Loan amount or Principal

I = Interest rate per month (to calculate rate per month: 3/12)

N = the number of instalments

After substituting all the values in the above formula we get

EMI = [8500 x 0.250% x (1+0.250%)^60]/[(1+0.25%)^60-1]

EMI = 152.73 per month

Therefore the net amount paid by Jim each year to the bank = EMI x No of EMI per year

= 152.73 x 12

Total Amount Paid to Bank Each Year = 1832.81

Part b) How much did Jim pay in total for his mini-van?

The total amount paid by Jim = Down Payment +Total EMI payments + Annual maintenance charges

= 2500 + (152.73 x 60) + (250 x 5)

=2500 + 9164.03 + 1250

Total Amount Paid By Jim for His Mini Van =12914.03