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Week 5 - Part 1 - Compound Interest - Chapter 4: Sec. 4.8

Almost every day you are bombarded with opportunities to borrow money. Credit card applications are everywhere. Many of these offers talk about the APR or Annual Percentage Rate that is associated with the money or credit offer. The APR is also called the "nominal rate" (meaning in name only). In this section we will address compound interest and look at interest that is compounded in discrete periods such as daily, weekly, monthly and yearly and we will look at interst that is compounded continuously. When you are done with this section you should be able to:


Nominal and effective rates

YearAnnualMonthlyDaily
0$1,000.00$1,000.00$1,000.00
1$1,050.00$1,051.16$1,051.27
2$1,102.50$1,104.94$1,105.16
3$1,157.63$1,161.47$1,161.82
Let's look at a $1,000 deposit made to an account that pays 5% interest per year. The table shows the balance in the account if we assume annual compounding (once a year), monthly compounding and daily compounding. As you can see from the table the accounts grow at different rates. The more frequent the compounding period the faster the account balance grows. This means that the effective yearly interest rates of the accounts must be different (the amount of interest actually paid during a year). They differ because your account earns interest on interest when compounding takes place. The effective interest rates can be calculated as follows.

YearContinous
0$1,000.00
1$1,051.27
2$1,105.17
3$1,161.83
It we have $1,000 and can invest is at 5% compounded continuously we can expect the following balances shown in the table on the right. The effective annual interest rate for this account will be e.05(1)=1.0513 or 5.13%. The nominal rate would be 5%. It is not by accident that the effective continuous compounding rate and the effective daily compounding rate are close, if you kept increasing the number of compounding events during the year you are approaching continuous compounding and the effective rates approach one another.

How long will it take our money to grow to a specified amount?

You and a friend each have $1,000. You put yours in an account that pays 5% per year compounded semi-annually (twice a year). Your friends money goes into an account that pays 4.9 % compounded continuously. How long will it you each of the accounts to grow to $2,500?