2) The purchase of the car that Joe dreams about can be accomplished by making p
ID: 2792364 • Letter: 2
Question
2) The purchase of the car that Joe dreams about can be accomplished by making payments of $300 a month for six years, if the first payment is made on February 1,1995, and the last payment is made on January 1,2001. The financing company charges 6% nominal interest rate compounded monthly. Joe wants to be able purchase the car for cash on January 1, 1995, just after he graduates from college. Joe has a job and started making depositing of $275 each month into an account that pays 9% compounded monthly beginning with the first deposit on February 1,1990. The last deposit is to be made on January 1,1995.Will Joe have saved up enough money to purchase the car? If not, how much should Joe be saving each month if all other conditions remain the same?Explanation / Answer
We can use PV of an Annuity formula to find out the present value of the Car loan (purchase price of the car) on January 1, 1995
PV = PMT * [1-(1+i) ^-n)]/i
Where PV =?
PMT = Monthly payment = $300
Annual interest rate = 6% per year, compounded monthly, therefore i = 6%/12 = 0.5% per month
n = N = number of payments = 6 years *12 months = 72 months
Therefore,
PV = $300* [1- (1+0.005)^-72]/0.005
PV = $18,101.85
Now we can use FV of an Annuity due formula to find out the Future value of the monthly deposit of $275 starting from February 1, 1990 (as the payments are made at the starting of the month)
FV = PMT*(1+i) *{(1+i) ^n1} / i
Where FV =?
PMT = Monthly payment = $275
Annual interest rate = 9% per year, compounded monthly, therefore i = 9%/12 = 0.75% per month
n = N = number of payments = 5 years *12 months = 60 months
Therefore,
FV = $275 * (1+0.0075)* [(1+0.005) ^60 -1]/0.0075
FV = $20,897.20
As the future value of deposit is more than the present value of loan (purchase price of the car) on January 1, 1995 therefore Joe will have enough money to purchase the car.
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