Section 6.5: Financial Applications Involving Exponential Functions When you invest money, your money earns interest, which means that after a period of time you will have more money than you started with. The extra amount of money you have is called interest. The original amount of money that you invest is called the principal. There are two types of interest that we will learn about: Simple Interest: interest is calculated ONLY in terms of the original amount of money that you invested. That is, you only earn interest on the money that you invested (ie the principal not the accumulated interest). Compound Interest: interest is earned on two different things: (i) the amount of money that you invest and (ii) the interest that you earn on the money that you invested. 1
Part 1: Simple Interest Suppose someone invests $1000 into a savings account that earns an annual interest of 5%. Suppose this account has a simple interest rate. That means that interest is applied ONLY on the original amount each year. Determine how much money you would have after 3 years. Interest = Principal H Rate H Time I = Prt Total Amount = Principal + Interest A = P + I or A = P + Prt or A = P(1 + rt) Year (t) Total Amount at the End of the Year (A) Formula: A = P(1 + rt) 0 $1000 1 $1050 ($1000 + $1000(0.05)(1)) 2 $1100 ($1000 + $1000(0.05)(2)) 3 $1150 ($1000 + $1000(0.05)(3)) Graph the value of the investment over time. Notice that the data is linear. This is because we are earning the same amount of money each year. 2
Example: Kyle invested his summer earnings of $5000 at 8% simple interest, paid annually. a) Complete a table of values and graph the growth of the investment for 3 years using "time (years)" as the domain and "value of the investment" as the range. Year (t) Total Amount at the End of the Year (A) Formula: A = P(1 + rt) 0 1 2 3 b) What does the shape of the graph tell you about the type of growth? Why is the data discrete? c) What do the y intercept and slope represent for the investment? d) What is the value of the investment after 10 years? 3
Part 2: Compound Interest Suppose someone invests $1000 at 5% compound interest. This means that the principal ($1000) earns interest each year, and that the interest earned also earns more interest! Notice that compound interest is determined by applying the interest rate to the sum of the principal andany accumulated interest. As well, notice that the accumulated interest and the value of the investment do not grow by a constant amount as they do with simple interest. Enter these values in your calculator and perform an exponential regression. Write the equation of the exponential regression. What do you notice? 4
In a more general sense, compound interest can be represented by the formula... A = P(1 + i) n where P is the principal amount i is the interest rate per compounding period n is the number of compounding periods Notice that i is the interest rate per compounding period. In the example we looked at, the interest rate was 5% compounded annually (every 1 year). Thus, Compounding periods are usually daily, weekly, semimonthly, monthly, quarterly, semi annually or annually. The table below shows how many times interest is paid, and the interest rate for each of these options. 5
Example $5000 is invested at a rate of 6.0% per year for 6 years. Using A = P(1 + i) n, how much money would you have in the bank if the interest rate is compounded as follows: A) Yearly: Model: Ans: B) Semi annually: Model: Ans: 6
C) Quarterly: Model: Ans: D) Monthly: Model: Ans: 7
E) Daily: Model: Ans: Text book: Page 395 397, Questions 1, 8, 10, 14 8