Chapter 10: The Mathematics of Money PART 2 Percent Increases and Decreases If a shirt is marked down 20% and it now costs $32, how much was it originally? Simple Interest If you invest a principle of $5000 and a bond pays 2% annual simple interest, how much does it pay you each year? Compound Interest If you invest a principle of $5000 in a bank account that offers 2% annual compound interest, how much is the account worth after 10 years? Fixed Deferred Annuity If you invest $500 each year into an account that offers 6% annual (compound) interest, how much will the account be worth in 10 years? Fixed Annuity A sequence of equal payments made or received over regular time intervals. When payments are made to make a lump sum at a point in the future, it is called a "deferred annuity" We are interested in the "future value" of each of the payments. When a lump sum is paid and a series of regular payments later, it is called an "installment loan". In this case, we are interested in the "present value" of each of the payments. Example: Suppose you decide to put $100 in a "college fund" account every month. The account earns 6% annual interest compounded monthly. How much will this account be worth in 18 years? What is the Future Value of the account?
Example: Suppose you decide to put $100 in a "college fund" account every month. The account earns 6% annual interest compounded monthly. How much will this account be worth in 18 years? What is the Future Value of the account? How much is the first $100 worth after 18 years? (216 months) What about the second installment? How much is it worth at the end of the 18 years? What about the third installment? What about the sum total of all 216 payments? Factor out the original $100 dollars?
We need to figure out a formula for the sum of all those terms with exponents.
Factor out the original $100 dollars? Fixed Deferred Annuity Formula Future Value F of a fixed deferred annuity consisting of T payments of $P dollars with a periodic interest rate of p% where L is the value of future value of the last payment.
Installment Loans: Installment loans have a present value (The amount of the loan you are given "now".) Each of the payments (made in the future) have a present value. The sum of those present values shoud be equal to the amount of the loan. Example: I take out a loan for $20,000. The bank offers 6 percent annual interest (compounded monthly). Suppose I make monthly payments of $F dollars for 10 years. How much are each of the payments? One month after I get the cash, I owe a payment of $F. What was the value of that payment one month earlier? Its present value? Find the present value of the second payment I make (after two months). Find the sum of all the present values of all the payments I make over 120 months. That should add up to $20,000.
The book writes the formula using "q" which is 1 / (1 + p) So how much is my monthly payment on the $20,000 loan?
1. There is a great sale this weekend at the local store. They are offering 20% off everything in the store. You have exactly $85.50 in your pocket. What is the most expensive item you can afford? You also have to pay 6% sales tax (applied to the sale price).