# Son accepted to college. Tuition will not increase for 4 years he is attending. First tuition pymt of 10,000 due in 6 months. Same payment due every 6 months unitl a total of 8 pymts made.The college offers a bank account with fixed APR of 4%(semiannually) for the next 4 yrs. You can with draw money every 6 months. How much money must you deposit today if you intend on not making any further deposits and will make all tuition pymts from this acct, leaving the acct empty when the last pymt is made?

### Solution:

The APR is 4%, so the semi-annual interest rate is 2%. If you deposit $10,000 in the bank after every six months, you will have $85,829.69 in the bank at the end of 4 years:

- After six months you will have $10,000
- After one year you will have 10,000×1.02+10,000 = $20,200
- After 1.5 years you will have $20,200×1.02+10,000 = $30,604
- After 2 years you will have $30,604×1.02+10,000 = $41,216.08
- After 2.5 years you will have $41,216.08×1.02+10,000 = $52,040.40
- After 3 years you will have $52,040.40×1.02+10,000 = $63,081.21
- After 3.5 years you will have $63,081.21×1.02+10,000 = $74,342.83
- After 4 years you will have $74,342.83×1.02+10,000 = $85,829.69

The present value of that $85,829.69, discounting at 2% is **$73,254.81** (i.e. $85,829.69/(1.02^8)), so if you put this amount in the bank now, and make the payments every six months, there will be nothing left in the account after four years.

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