Advanced Time Value of Money Problems

Professor A. Spieler

Question 1 (mortgage problem)

(Try to work this question WITHOUT using Excel)

You purchase a house that costs $625,000 with an 8%, 30-year mortgage. You make a 20% down payment to avoid PMI insurance. 1. What is your monthly payment?

2. Amortize the first and second payments.

3. What is the mortgage balance after 5 years?

4. What percentage of the principal is paid off after 5 years?

5. Suppose after 5 years you refinance at 6% the remaining balance at a cost of

$10,000, for 30 years. What is your new monthly payment?

6. Further, suppose you maintain the same payments as in (1), i.e. pre-pay on the principal, how many YEARS until you payoff the mortgage?

Question 2 (2nd mortgage problem)

You are considering the purchase of a $500,000 home. You plan to take a 30-year fixed mortgage after making a 20% downpayment to avoid PMI. Payments are to be made monthly (at the end of the month) and the APR is 8%.

1. What is the monthly payment? 2. During what month does the principal portion first exceed the interest portion?

Are you surprised by your answer?

3. How long does it take to pay off your mortgage if you pay an additional $300

towards principal each payment?

4. How long does it take to pay off your mortgage if you pay an additional amount

each month equal to the current month’s principal?

Question 3 (College planning)

Your child was just born and you are planning for his/her college education. Based on your wonderful experience in Corporate Finance you decide to send your child to Hofstra University as well. You anticipate the annual tuition to be $60,000 per year for the four years of college. You plan on making equal deposits on your child’s birthday every year starting today, the day of your child’s birth. No deposits will be made after starting college. The first tuition payment is due in exactly 18 years from today (the day your child turns 18 – no deposit required, i.e. last deposit is on 17th birthday). Assume the annual expected return on your investments is 10% over this period.

(i) Calculate the annual deposit.

(ii) Calculate the amount needed if only equal annual deposits are made on birthday’s 5-10 inclusive.

(iii) Calculate the amount needed if two equal annual deposits are made on

birthday’s 5 and 13.

(iv) Answer part (i), now assume tuition rises 10% per year.

(v) Answer part (i) assuming first deposit will be made on your child’s 1st birthday. All other information is the same. What is the annual tuition payment? How does it compare to part (i)? Is your answer surprising?

Question 4 (Retirement planning)

You have just graduated Hofstra University at age 22. You hard work has paid off as you already have a job as an investment banker at Goldman Sachs waiting for you. You plan to work continuously until age 65 and retire exactly on that day. You expect to live until exactly 90 and enjoy your golden years and leave you heirs NOTHING. Assume your investments earn 8% per year. You plan to contribute $10,000 to your retirement fund every year on your birthday starting at age 23. Your last deposit will be at exactly age 65 and your first withdrawal will be at age 66. Your last withdrawal will be at the moment you die at age 90. Ignore all tax considerations for this problem.

(i) How much you will be able to spend each year in retirement?

FV (deposits) = PV (withdrawals) NOTE: This could be at any time period but t=65 is particularly convenient

(ii) How much will you be able to spend each year in retirement if you begin

deposits at age 30?

(iii) How much larger do your deposits have to be if deposits start at age 30 to

equal your answer in part (i)? Now let’s consider the effect of inflation. The values calculated above are nominal values. What is more important is real, i.e. inflation-adjusted, values? Assume inflation averages 4% per year. (iv) Re-calculate parts (i), (ii) and (iii) above. (v) Comment on above.