1. Lecture 13 Compound Interest Equations of Value Ana Nora Evans 403 Kerchof AnaNEvans@virginia.edu http://people.virginia.edu/~ans5k/ Math 1140 Financial Mathematics

2. Math 1140 - Financial Mathematics Homework 5 was A)Too hard B)Hard C)OK D)Easy E)Too easy 2

3. Math 1140 - Financial Mathematics On homework 5, I worked A)By myself B)I met with my friends C)I emailed with my friends D)I emailed the teacher E)I met with a tutor 3

4. Math 1140 - Financial Mathematics Homework 4 4

5. Math 1140 - Financial Mathematics Everybody is capable of doing well on the homework. Get help if you get stuck: ask questions in class call, skype a friend email your question to the class mailing list finmathfall2011@googlegroups.com email the class mailing list to find partners create a Facebook group, Twitter feed 5

6. Math 1140 - Financial Mathematics Use words to explain what the numbers mean. Don’t cram your answer on single line. Make it clear what you are doing. Don’t use part of the page for scrap paper. How to improve your homework 6

7. Math 1140 - Financial Mathematics I am happy a few of you tried the bonus questions. I hope more of you solved the bonus in homework 5. Good job on bonus questions 7

8. Math 1140 - Financial Mathematics Review Session Tonight at 7pm in Rice Hall, fifth floor (507). 8

9. Math 1140 - Financial Mathematics Questions About last class About homework 9

10. Math 1140 - Financial Mathematics Last time Discuss two homework problems Annual effective rate Section 3.4 10

11. Math 1140 - Financial Mathematics Today Annual effective rate of compound discount Compound rate formula Finding the time for an investment to grow Sections 3.5 and 3.6 11

12. Math 1140 - Financial Mathematics Annual Effective Rate Given a nominal interest rate i(m), the annual effective rate is the interest rate i such that if the same principal P is deposited in two accounts: one with nominal interest rate i(m) and one with yearly interest rate i, compounded yearly; at the end of one year the two accounts have the same balance. 12

13. Math 1140 - Financial Mathematics The compounded amount formula is S = P(1+i) n The balance in an account with nominal interest rate i(m) after one year is: S = P(1 + i(m)/m) m The balance in an account with interest rate i per year, compounded yearly, after one year is S = P(1 + i) 1 To calculate i P(1 + i) = P(1 + i(m)/m) m i = (1 + i(m)/m) m - 1 13

14. Math 1140 - Financial Mathematics Why does one calculate annual effective rate? 14 It allows us to compare different nominal interest rates. You are considering two different savings accounts. The first pays 4.7% compounded monthly and the second one pays 4.63% compounded daily. Which one is the better deal?

15. Math 1140 - Financial Mathematics Questions? 15

16. Math 1140 - Financial Mathematics A savings account pays a nominal rate of 7% convertible monthly. What is the equivalent annual effective rate? A)7% B)(7/12)% C)(1+0.07/12) 12 – 1 D)(1+0.07/12) -12 + 1 E)I don’t know Pledged Quiz 16

17. Math 1140 - Financial Mathematics Calculate the interest per conversion period using simple interest. After one conversion period add the interest to the principal. If the term is not a multiple of a period, use simple interest for the last part that does not fit in a conversion period. After one conversion period P(1+i) After two conversion periods P(1+i) 2 After n conversion periods P(1+i) n Compound interest 17

18. Math 1140 - Financial Mathematics d(m) is the nominal rate of compound discount payable m times a year. The discount rate per conversion period is d = d(m)/m For one conversion period P = S(1-dt) For two conversion periods P = S(1-dt) 2 …. For n conversion periods P = S(1-dt) n What if we use discount interest? 18

19. Math 1140 - Financial Mathematics Given a nominal discount rate d(m), the annual effective rate of compound discount is the discount rate d such that two accounts: one with nominal discount rate d(m) and one with yearly discount rate d, compounded yearly, with the same balance due after one year, pay the same proceeds. Proceeds of the account with nominal discount rate d(m) P = S( 1 - d(m)/m) m Proceeds of the account with yearly discount rate, compounded yearly P = S(1-d) 1-d = ( 1 - d(m)/m) m Annual Effective Rate of Compound Discount 19

20. Math 1140 - Financial Mathematics Find the nominal rate of discount compounded quarterly equivalent to an 8% nominal rate of interest compounded monthly. Application 20

21. Math 1140 - Financial Mathematics Questions? 21

22. Math 1140 - Financial Mathematics Can you write a simple computer program? A)Yes B)No C)Maybe 22

23. Math 1140 - Financial Mathematics Given: the amount S the principal P the number of conversion periods n Calculate the interest per period i S = P(1+i) n Compound Rate Formula 23

24. Math 1140 - Financial Mathematics A loan of $2,400 charges interest compounded monthly. Five years after the loan is made, the debt has grown to $4,600. What is the nominal rate if interest convertible monthly? 24

25. Math 1140 - Financial Mathematics Questions? 25

26. Math 1140 - Financial Mathematics Finding the time for an investment to grow You know how much money you have now, and you know how much money you want to have at some point in the future. The problem is to determine how long it will take to get the desired future value at a certain growing rate. 26

27. Math 1140 - Financial Mathematics Given: the principal P the amount S the nominal interest rate i(m) Want to calculate: the number of conversion periods n S = P(1+i) n (1+i) n = S/P n = ln(S/P)/ln(1+i) More precisely 27

28. Math 1140 - Financial Mathematics A)I need a quick review on logarithms B)Logarithm is my best buddy. Logarithms 28

29. Math 1140 - Financial Mathematics Tonight Review session at 7pm in Rice Hall, fifth floor(507) Friday class in Rice Hall, first floor First Exam (max 15 points): 26 September 2011 at 7pm in CLK 108 Charge 29