Question

Assuming interest rates are positive, a dollar that is available today is worth more than a dollar in the future. Current dollars can be converted into future dollars by compounding, and future dollars can be transformed into current dollar equivalents by discounting.

     Part I. At the beginning of your third year of college you realize that you will need to borrow $10,000 to finance the remainder of your educational expenses. You approach your bank and find out you can borrow the $10,000 at 12-percent interest, but you do not have to start repaying the loan until you graduate in two years. However, the loan will accumulate the interest payments compounded annually at 12-percent per year during the two years you are still in school, and you will repay in equal annual payments over twenty years (a 20-year amortization). What will your annual payments be over the twenty years you are repaying the loan? What is the present value (using an interest rate of 12-percent) of the repayment schedule at the time you borrow the $10,000? Carefully show and explain your calculations.

     Part II. Just before you borrow the $10,000 from your bank you discover that you can obtain the same amount through a federal student loan program at an interest rate of 5-percent. If you borrow through the terms of this program you would repay the loan over 20 years in equal annual payments once you graduate and there will be no accumulation of interest during the two years you will still be in school. What will your annual payments be under this 5-percent loan? At the 12-percent market rate you face, what is the present value of the repayments on this federal student loan? Finally, what is the present value amount of the subsidy under the federal student loan relative to the private bank loan? Carefully show and explain your calculations.

Answer 1

The loan amount is $10000

Part 1
The interest is accumulated for period of 2 years at 12% interest rate

FV = PMT * (1+Interest Rate) ^ Duration

10000 * (1.12 ^ 2) = 12544

This is the loan amount which will be paid in 20 years with equal annual payment at 12% interest rate

PMT = (Principal * Interest Rate *(1+Interest Rate)^Duration) / ((1+Interest Rate)^ Duration) - 1

(12544 * 0.12 * (1.12 ^ 20)) / (1.12 ^ 20) - 1
=14520.37 / 8.6463
= 1679.38

=PMT(12%,20,-12544,,)
= 1679.38

The PV of this annual payment at the time of borrowing means the duration will be 22 years.
So the first payment will be discounted for 2 years and 2nd will be 3 years

PV = PMT / (1+Interest Rate) ^ Duration

1679.38 * ((1/(1.12^3)) + (1/(1.12^4)) + ............... + ((1/(1.12^21)) + ((1/(1.12^22))
= 10000.03

It is around $10000

Part 2

Now the interest rate is 5% and it has not been accumulated for the 2 year period.

FV of the loan after 2 years will be $10000.

=PMT(5%,20,-10000,,)
= 802.43

The annual payment will be $802.43

802.43 * ((1 / 1.12 ^3) + (1 / 1.12^4) + ............ + (1 / (1.12^22))
= 4778.15

The PV at the discount rate of 12% is $4778.15

The PV of the subsidy can be calculated subtracting the PV of 5% loan from 12% loan.

10000 - 4778.15 = 5221.85