2. Exploration – Compound Interest Certain savings accounts compound interest periodically. That is, every given time period, say a month, the bank pays into the account a certain percentage. For example, suppose that you have $1 in a savings account that offers 100% interest that compounds every 6 months. Assuming that you leave the earned interest in the account, how much money will you have at the end of the year?

3. Exploration TimeMoney in the Account Interest EarnedMoney after compounding Now$1-----$1

4. Exploration TimeMoney in the Account Interest EarnedMoney after compounding Now$1-----$1 In 6 months$1$1(50%)=$0.50

5. Exploration TimeMoney in the Account Interest EarnedMoney after compounding Now$1-----$1 In 6 months$1$1(50%)=$0.50$1.50 In 12 months$1.50

6. Exploration TimeMoney in the Account Interest EarnedMoney after compounding Now$1-----$1 In 6 months$1$1(50%)=$0.50$1.50 In 12 months$1.50$1.50(50%)=$0.75

7. Exploration TimeMoney in the Account Interest EarnedMoney after compounding Now$1-----$1 In 6 months$1$1(50%)=$0.50$1.50 In 12 months$1.50$1.50(50%)=$0.75$2.25

8. Exploration TimeMoney in the Account Interest EarnedMoney after compounding Now$1-----$1 In 6 months$1$1(50%)=$0.50$1.50 In 12 months$1.50$1.50(50%)=$0.75$2.25 Notice that 100% interest is split evenly among the 2 compounding periods. Also, compounding twice gave you more interest ($1.25), than a simple interest payment of $1.00 at the end of the year.

9. Exploration – Quarterly Compounding TimeMoney in the Account Interest EarnedMoney after compounding Now$1-----$1 In 3 months In 6 months In 9 months In 12 months

10. Exploration – Quarterly Compounding TimeMoney in the Account Interest EarnedMoney after compounding Now$1-----$1 In 3 months$1$1(0.25)=$0.25$1.25 In 6 months In 9 months In 12 months

11. Exploration – Quarterly Compounding TimeMoney in the Account Interest EarnedMoney after compounding Now$1-----$1 In 3 months$1$1(0.25)=$0.25$1.25 In 6 months$1.25$1.25(0.25)=$0.3125$1.5625 In 9 months In 12 months

12. Exploration – Quarterly Compounding TimeMoney in the Account Interest EarnedMoney after compounding Now$1-----$1 In 3 months$1$1(0.25)=$0.25$1.25 In 6 months$1.25$1.25(0.25)=$0.3125$1.5625 In 9 months$1.5625$1.5625(0.25)=$0.3906$1.9531 In 12 months

13. Exploration – Quarterly Compounding TimeMoney in the Account Interest EarnedMoney after compounding Now$1-----$1 In 3 months$1$1(0.25)=$0.25$1.25 In 6 months$1.25$1.25(0.25)=$0.3125$1.5625 In 9 months$1.5625$1.5625(0.25)=$0.3906$1.9531 In 12 months$1.9531$1.9531(0.25)=$0.4882$2.44140625

14. Exploration – Quarterly Compounding TimeMoney in the Account Interest EarnedMoney after compounding Now$1-----$1 In 3 months$1$1(0.25)=$0.25$1.25 In 6 months$1.25$1.25(0.25)=$0.3125$1.5625 In 9 months$1.5625$1.5625(0.25)=$0.3906$1.9531 In 12 months$1.9531$1.9531(0.25)=$0.4882$2.44140625 Notice that you receive about $0.19 more if you compound quarterly as compared to semi-annually. If we increase the number of compounding periods, is there a limit to the interest that may be earned?

15. Exploration

17. 1 (yearly) 4 (quarterly) 12 (monthly) 365 (daily) 8760 (hourly) 17520 (every 30 minutes) 525600(every minute) 31,536,000 (every second)

18. Exploration 1 (yearly) 4 (quarterly) 12 (monthly) 365 (daily) 8760 (hourly) 17520 (every 30 minutes) 525600(every minute) 31,536,000 (every second)

19. Exploration 1 (yearly) 4 (quarterly) 12 (monthly) 365 (daily) 8760 (hourly) 17520 (every 30 minutes) 525600(every minute) 31,536,000 (every second)

20. Exploration From the table above, it becomes evident that as the number of compounding periods increases (approaches infinity), the money at the end of the year appears to be converging (approaching) a value:

21. Exploration

23. Examples

26. Examples - LN

29. Assignment – p.557