Math 131
Spring 04
Savings
Savings:
Savings accounts are usually compounded monthly. This means that the annual interest rate (APR) is divided by twelve to form a monthly interest rate, and a new balance is computed at the end of each month.
For example, let's start with an "easy" number: Suppose we have an account which pays 12% annual interest; the monthly interest rate is then 12%/12=1%. Each month, we add 1% to the previous month’s balance. If we don’t add new money into the account, the next month’s balance is 101% of the previous month’s balance. Fill in the table for the first year:
Number of Months Passed Balance
0 $100
1 $101
2 $102.01
3
4
5
6
7
8
9
10
11
12
Compare your final result to a “simple” interest rate (i.e. compounded once) of 12%, where the final account balance would be $112.
Accounts can be compounded quarterly (divide the APR by 4), daily (divide by 365), or any number of times.
Important Note: Round off error makes a big difference here. After you divide the interest rate by 12, keep as many decimal places as your calculator will allow.
Example 1: Suppose the interest rate is 10% per month. This annual rate is $.10 for every $1. The monthly rate is .10/12 = $.0083333333 for every $1. Note that to compute the monthly interest we multiply by 1.0083333333 each month. The monthly interest rate is .83333333 %.
Example 2: Suppose the annual interest rate is 5.67%. This rate represents $.0567 for every $1. The monthly rate is (.0567)/12 = .004725, and we multiply by 1.004725 each month.
Investigating the effects of compounding:
1. Beginning with $100 and a 12% APR, use a spreadsheet to find the final balance after one year, when the account is compounded 2, 3, 4, 6, 12, 100, 365, and 1000 times per year. Be sure to include as many decimal places as possible for the interest rates you use.
Number of Times Compounded Balance After 1 Year
2. Describe your findings:
3.
Suppose the account were compounded even more often say every hour or every minute. What do you think would happen? Is there a limit on how high the interest can
go?
Investigating the effects of the interest rate:
For this set of problems, imagine that you are a grandparent, who has deposited $1000 in a college account for a new baby. The money will sit in the account for 18 years, collecting interest, compounded monthly.
4. Before doing any calculations, guess how much money will be in the account after 18 years at the following interest rates:
Interest Rate Guess for Balance after 18 Years
4%
6%
8%
10%
12%
14%
16%
18%
20%
5. Do your guesses follow any kind of pattern? Explain.
6. Now compute the actual balances. Also note the number of months it takes until you have $2000 and until you have $4000 in the account (if you get to these numbers).
Interest Rate Balance after 18 years Months to Reach $2000 Mths Rch $4000
4%
6%
8%
10%
12%
14%
16%
18%
20%
7. How do the actual balances after 18 years compare to your guesses? Explain. If your guesses were way off, what assumptions were you making that seem to not be true?
8. What patterns do you notice in the balances after 18 years at different interest rates? How does increasing the interest rate by 2 percentage points change the final balance? Are there different patterns with lower and higher rates?
9. What patterns do you notice in how long it takes for the balance to double (reach $2000)?
10. Convert the times for the balance to double into years (round to the nearest year). See if you can find an approximate relationship between the interest rate and the number of years to double the initial balance.
11. How does the time to reach $2000 compare with the time to reach $4000 (approximately)? Does this relationship continue with the time to reach $8000?
The Ancestor You Wish You Had.
For this set of problems, assume that you had an ancestor who put $1 in an account for you 200 years ago. For this problem, assume simple interest (compounded once per year).
12. Guess how much money would be in the account after 200 years at various interest rates (fill in the second column of the table below). See if you can use anything you’ve learned so far in this activity to help you estimate.
Interest Rate Guess for Balance after 200 Years Actual Balance
4%
6%
8%
10%
12%
14%
16%
18%
13. Now use Excel to compute numbers for the third column of the table.
14. How do your guesses compare to the actual numbers? Is there any pattern in your mistakes? Explain.
15. What patterns do you notice in the actual numbers? Can you relate them to patterns you’ve already found in this activity?
Saving for Something:
Think of something big you’d like to buy. Think of how much money you could save each month.
16. Set up a spreadsheet in Excel to figure out how long it would take you to save for your item at 6% APR, compounded monthly. This problem is like the amortization tables for credit cards, except that instead of subtracting a payment each month, you add one. Explain how you set up your spreadsheet and your results:
17. Now do the same problem, but at 12% APR.
18. In a systematic manner, vary the APR some more, and then vary the size of the payment you make each month. Make a table to record your results.
19. What patterns do you notice? Explain?
Copyright 2005, Debra K. Borkovitz. You may copy or edit this material for non-profit, educational use only.
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