F2S: Mathematics of Finance—Problem Set 2 (Final Version)Name: ______

Annuities

Adopted from “Applied Finite Mathematics” Author: Sekhon, R. (Available at

In Problem Set 1 (Simple and Compound Interest), we did problems where an amount of money was deposited lump sum in an account and was left there for the entire time period. Now we will do problems where timely payments are made in an account. When a sequence of payments of some fixed amount are made in an account at equal intervals of time, we call that an annuity. Annuities are the subject of this section.

Ammoritization.Many situations call for the repayment of a loan by an annuity. When a loan is repaid by an annuity, it is said to be amortized. A portion of each payment is applied against the interest on the loan and the remainder is applied against the loan balance. We can think of the loan amount as the present value of an annuity. Examples include car loans and home loans. Here, P is the Present Value of the annuity, and R is the regular payment made m times per year for t years.

Present Value

There are other applications of the Present Value of Annuity, such as this one (which you will solve in Exercise #2): Suppose you have won a lottery that pays $1,000 per month for the next 20 years. But, you prefer to have the entire amount now. If the interest rate is 8%, how much will you accept?

Sinking Fund. If a payment of R dollars is made in an account m times a year at an interest r, then the final amount (Future Value) S after t years is:

Future Value

Work the following problems on separate paper, showing all setup and required work.

  1. Find the monthly payment for a car costing $25,000 if the loan is amortized over 4 years at an interest rate of 9%.
  1. Suppose you have won a lottery that pays $1,000 per month for the next 20 years. But, you prefer to have the entire amount now. If the interest rate is 8%, how much will you accept?
  2. A business needs $450,000 in 5 years. How much should be deposited each quarter in a sinking fund that earns 9% to have this amount in 5 years?
  3. How much money should be deposited at the end of each month in an account paying 7.5% for it to amount to $10,000 in 5 years?
  4. Dr. Hill wants to retire in 25 years and can save $650 every month. If he can achieve a rate of return of 7.8%, how much will he have at the end of 25 years?
  5. Sonya bought a used car for $15,500. Find the monthly payment if the loan is to be amortized over 3 years at a rate of 8.1%.
  6. I would like to trade in my gas-guzzling car for a new Toyotal Prius, which is priced at $22,549. Suppose that with my trade-in and some extra cash, I will have a 20% downpayment. I can secure a 60-month loan from my bank at an a.p.r. of 4.04%. What monthly payment will I be required to make?
  7. A company needs a piece of machinery that has a useful life of 5 years. It has an option of leasing it for $10,000 a year, or buying it for $40,000 cash. Considering the $10,000 a year option, what amount would the company need to have today, if the interest rate is 10%, in order to make the yearly leasing payment of $10,000 for 5 years? That is, compute the present value of the leasing option. [Many companies use this method in order to compare payment options.] After calculating the present value of the leasing option, which option is better?
  1. Jason's tuition at Stanford for the next year is $32,000. His parents have decided to pay the tuition by making nine monthly payments. If the interest rate is 6.2%, what is the monthly payment?
  2. I want to retire as a millionaire. Suppose that I have been saving diligently since my first job (let’s say 10 years ago). I have saved $300 per month. I plan to work until I reach my goal. When will I be able to retire, assuming 5.5% real rate of return (with inflation factored out), and assuming that I will continue to make the $300 per month payment?