Savings Accounts
Savings accounts are an example of exponential growth with a fixed percent increase. The money grows exponentially by a fixed interest rate which is called the annual percentage rate (APR).
Accounts that compound annually calculate and add interest into the account once a year.
Example: deposit $500 compounded annually with an annual percentage rate (APR) of 3%
Using the formula P*(1+r), the formula in B3 is =B2*(1+.03)
|
year |
money |
|
0 |
$ 500.00 |
|
1 |
$ 515.00 |
|
2 |
$ 530.45 |
|
3 |
$ 546.36 |
|
4 |
$ 562.75 |
|
5 |
$ 579.64 |
|
6 |
$ 597.03 |
|
7 |
$ 614.94 |
|
8 |
$ 633.39 |
|
9 |
$ 652.39 |
|
10 |
$ 671.96 |
To find the value of the account after 10 years without using excel, use the “by hand” formula Y = P * (1+r)x : 500*(1+.03)^10=$671.96
However, most savings accounts compound more often than once a year; most compound monthly. For accounts that compound monthly, the balance increases by 1/12 of the APR each month. For compounded monthly divide the APR by 12 and for the “by hand” formula multiply the years by 12. Remember each period in the table is one month NOT one year.
The formula in B3 is =B2*(1+.03/12)
|
month |
money |
|
0 |
$ 500.00 |
|
1 |
$ 501.25 |
|
2 |
$ 502.50 |
|
3 |
$ 503.76 |
|
4 |
$ 505.02 |
|
5 |
$ 506.28 |
|
6 |
$ 507.55 |
|
7 |
$ 508.82 |
|
8 |
$ 510.09 |
|
9 |
$ 511.36 |
|
10 |
$ 512.64 |
|
11 |
$ 513.92 |
|
12 |
$ 515.21 |
|
13 |
$ 516.50 |
|
14 |
$ 517.79 |
|
15 |
$ 519.08 |
|
16 |
$ 520.38 |
|
17 |
$ 521.68 |
|
18 |
$ 522.98 |
|
19 |
$ 524.29 |
|
20 |
$ 525.60 |
|
21 |
$ 526.92 |
|
22 |
$ 528.23 |
|
23 |
$ 529.55 |
|
24 |
$ 530.88 |
So to calculate the amount in an account compounded monthly after 2 years:
500*(1+.03/12)^24=$530.88
If you compare the value after 2 years in both accounts, you will find that the monthly account has more than the yearly account ($530.88 vs $530.45). Assuming you start with the same amount and the same APR, the more often the interest is compounded, the more money you earn.
What we have noticed is that with an account that compounds other than annually, the account increases by more than the annual percentage rate in one year. The actual rate earned by the account is called the Annual Percentage Yield (APY).
The APY is the percent change in value of the account after ONE year.
APY = (new-old)/old = (amount after one year - original amount) / original amount
For our monthly example, (515.21-500)/500 = 3.042%
Express the APY in percent form with at least 3 decimal places. This shows that the account is earning an APY of 3.034%
The annual percentage yield will allow you to compare two different accounts to determine which will earn more money. For example, if you were offered an account that compounded annually at a 6.2% APR and another account that compounded monthly at a 6.15% APR, you don’t know which will earn you more money without calculating the APY. To do so, assume a starting value for each account ($100) and figure out how much is in each account after one year. To do this you can either set up a table in Excel or use the “by hand” formula:
| year | balance |
| 0 | $ 100.00 |
| 1 | $ 106.20 |
Or 100*(1+.062)^1 = 106.20
Then calculate the APY =(106.20-100)/100 = 6.2% (Note: for annual compounding, the APY = APR)
| month | balance |
| 0 | $ 100.00 |
| 1 | $ 100.51 |
| 2 | $ 101.03 |
| 3 | $ 101.55 |
| 4 | $ 102.07 |
| 5 | $ 102.59 |
| 6 | $ 103.11 |
| 7 | $ 103.64 |
| 8 | $ 104.17 |
| 9 | $ 104.71 |
| 10 | $ 105.24 |
| 11 | $ 105.78 |
| 12 | $ 106.33 |
Or 100*(1+.0615/12)^12 = 106.33
APY = (106.33-100)/100 = 6.33%
This shows us that the monthly account has a higher APY and therefore earns more money.