Question

You are planning for your college education that begins next month. UT tuition, room and board will start off at $18,000 per year, with full payment each year required at the beginning of the year. Through a special legislative initiative, tuition, room and board will remain fixed over your 4 years. You will borrow the full amount of your college needs each year at 6% per year. When you graduate, you will get to wait a one-year grace period (during which interest will accumulate) until you begin repaying your loan. You will then repay your loan over 10 years in monthly payments, with the first payment at the end of the month after your grace period expires. How much will your monthly payment be?            

Answer 1

We can use Future value (FV) of an Annuity due formula to find out the Future value of the annual payments of $18,000 (as the payments are made at the starting of the year)

FV = PMT*(1+i) *{(1+i) ^n−1} / i

Where FV =?

PMT = Annual payment = $18,000

Annual interest rate = 6% per year

n = N = number of payments = 4 years

Therefore,

FV = $18,000 * (1+6%)* [(1+6%) ^4 -1]/6%

FV = $83,467.67 (this is the worth of annul payments till college ends but you will get one year grace period before the payment of loan get start)

Therefore future value of college expenses after one year grace period

= $83,467.67 * (1+6%)

= $88,475.73 (now this amount will be present value for loan repayments in monthly installments)

We can use PV of an Annuity formula to calculate the monthly payment of loan

PV = PMT * [1-(1+i) ^-n)]/i

Where PV = $88,475.73

PMT = Monthly payment =?

n = N = number of payments = 10 years *12 months =120 month

i = I/Y = interest rate per year = 6%, therefore monthly interest rate is 6%/12 = 0.5% or 0.005 per month

Therefore,

$88,475.73 = PMT* [1- (1+0.005)^-120]/0.005

PMT = $982.26

Therefore monthly payment on loan will be $982.26