Section 4B: The Power of Compounding Definitions The principal is the amount of your initial investment. This is the amount on which interest is paid. Simple interest is interest paid only on the original principal, and not on any interest added at later dates. Compound interest is interest paid both on the original principal and on all interest that has been added to the original principal. Example. Suppose we want to invest $250. We can either invest in a savings bond which returns 10% per year (as simple interest) or in a savings account which also returns 10% per year (as compound interest). To what value will your initial investment grow if you invest in the savings bond for one year? 2 years? To what value will your initial investment grow if you invest in a savings account for one year? 2 years? 1
Definition When interest is compounded just once a year, the interest rate is the called the annual percentage rate (or APR). Example. You are investing $300 in a bank account which has a compounded APR of 7%. Use the table below to determine the amount in the bank after three years. After N years Interest Balance N = 0 N = 1 N = 2 N = 3 The Compound Interest Formula (For Interest Paid Once a Year) A = P (1 + AP R) Y, where Notes: A = accumulated balance after Y years, P = starting principal (initial deposit), AP R = annual percentage rate (as a decimal), and Y = number of years. 1. The starting principal, P, is often called the present value (PV), because we usually begin a calculation with the amount of money in an account at present. 2. The accumulated balance, A, is often called the future value (FV), because it is the amount that will be accumulated at some time in the future. 3. When using this formula, you must express the APR as a decimal rather than as a percentage. 2
Example. Use the formula for annual compound interest to solve each of the following problems: If you invest $500 at a 7% APR, how much money would you have after 3 years? If you invest $500 for 3 years accruing compound interest annually, and end up with $800 in your account, what was the APR? If you end up with $1000 dollars after investing for 4 years at a 8% APR, what was the initial investment? If you invest $700 at 5% APR for 3 years, how much interest do you earn? 3
Compounding More Often If a bank account accrues interest more than once a year, you must first find the periodic interest rate, which is the APR divided by the number of compounding periods per year. Example. Restate how a bank would apply interest in the following situations. A bank gives 6% interest compounded monthly. The bank gives 6% 12 A bank gives 4% interest compounded semi-annually. = 0.5% of interest every month A bank gives 7% interest compounded quarterly. A bank gives 2% interest compounded daily. Example. Suppose a bank accrues interest at 10% APR compounded quarterly (every three months). Fill in the table below to determine the amount in the account after one full year with a principal of $250. Compare this to the results from the first page. After N Quarters Interest Balance 0 quarters 1 quarter (3 months) 2 quarters (6 months) 3 quarters (9 months) 4 quarters (12 months) 4
Compound Interest Formula (For Interest Paid n Times per Year) where A = P (1 + periodic interest rate) total number of compoundings = P A = accumulated balance after Y years, P = starting principal (initial deposit), AP R = annual percentage rate (as a decimal), ( 1 + APR ) (n Y ) n n = number of compounding periods per year, and Y = number of years. Note that Y is not necessarily an integer. For example, a calculation for three and a half years would have Y = 3.5. However, n Y, the total number of compoundings, must be an integer. Example. Answer the following questions using the above compound interest formula. A bank account has a 6% APR which is compounded monthly. If $1200 is invested for 5.5 years, what amount will be in the account afterward? A bank account has a 10% APR compounded semi-annually. After investing an initial principal, the amount in the account after 7 years was $2500. What was the initial investment (the principal)? An initial investment of $6500 was made in a bank account with an APR which is compounded quarterly. After 10 years of investing, the amount in the account is $15000. What is the APR for this account? 5
Definition The annual percentage yield (or APY) is the annual percentage by which a balance increases in one year. It is equal to the APR if interest is compounded annually. It is greater than the APR if interest is compounded more than once a year. The APY does not depend on the starting principal. The APY is sometimes also called the effective yield or simply the yield. One method for computing APY: Invent an amount of principle. Find the amount which would be in an account with that principle after one year, then Find the relative change from the initial principle to the amount after one year. This relative change is the annual percentage yield. Example. Find the annual percentage yield for the following bank accounts: an APR of 8% compounded yearly an APR of 8% compounded quarterly an APR of 8% compounded monthly an APR of 8% compounded daily 6
Example. A bank account at Bank X has an APR of 6% compounded annually. A bank account at Bank Y has an APR of 5.9% compounded quarterly. A bank account at Bank Z has an APR of 5.85% compounded daily. Which bank account will have the highest yield? If your bank compounds interest more often per year, the return is greater. So what if you could find a bank that would compound your interest infinitely many times in a year? A Fun Fact from Calculus As n gets really large, we have that P ( 1 + AP R ) n Y P e (AP R Y ), n where e = 2.71828..., P is the principal, and AP R is the annual percentage rate. So if you compound more often, there is a cap on how much your account will accrue. Epic Fail. Compound Interest Formula for Continuous Compounding (AP R Y ) A = P e where A = accumulated balance after Y years, P = starting principal (initial deposit), AP R = annual percentage rate (as a decimal), and Y = number of years. 7
Example. Answer the following questions. If you invest $150 at an APR of 6% compounded continuously, how much money would be in your account after 4 years? If you invest at 3% compounded continuously and amass a total of $500 after 4 years, what was the initial investment? Which account has a higher APY, an account with an APR of 7% compounded monthly or an account with an APR of 6.9% compounded continuously? 8