1. $$ Entrepreneurial Finance, 5th Edition Adelman and Marks PRENTICE HALL ©2010 by Pearson Education, Inc. Upper Saddle River, NJ 07458 9-1 Chapter 9 Time Value of Money Part II: Annuities

2. $$ Entrepreneurial Finance, 5th Edition Adelman and Marks PRENTICE HALL ©2010 by Pearson Education, Inc. Upper Saddle River, NJ 07458 9-2 Learning Objectives  Explain the differences between an ordinary annuity and an annuity due.  Calculate the future value and present value annuity factors that are used to solve time value of money problems.  Integrate several of the methods provided in time value of money to solve real-life financial problems.

3. $$ Entrepreneurial Finance, 5th Edition Adelman and Marks PRENTICE HALL ©2010 by Pearson Education, Inc. Upper Saddle River, NJ 07458 9-3 Learning Objectives (continued)  Use a financial calculator to solve time value of money problems.  Present spreadsheet applications of the mathematics of finance.  Use financial tables to solve time value of money problems.  Calculate future and present value amounts by solving problems involving annuities and lump sums.

4. $$ Entrepreneurial Finance, 5th Edition Adelman and Marks PRENTICE HALL ©2010 by Pearson Education, Inc. Upper Saddle River, NJ 07458 9-4 Future Value of an Ordinary Annuity  What is the future value of a stream of payments for n periods at an i rate of return when the money is invested at the end of each compounding period?  Formula is found on the following page:

5. $$ Entrepreneurial Finance, 5th Edition Adelman and Marks PRENTICE HALL ©2010 by Pearson Education, Inc. Upper Saddle River, NJ 07458 9-5 Formula for Future Value of an Ordinary Annuity (continued)

6. $$ Entrepreneurial Finance, 5th Edition Adelman and Marks PRENTICE HALL ©2010 by Pearson Education, Inc. Upper Saddle River, NJ 07458 9-6 Future Value of an Ordinary Annuity (continued)  Example: What is the future value of $5,000 invested at the end of each year for 10 years if money earns 6% per annum?

7. $$ Entrepreneurial Finance, 5th Edition Adelman and Marks PRENTICE HALL ©2010 by Pearson Education, Inc. Upper Saddle River, NJ 07458 9-7 Future Value of an Ordinary Annuity (continued)  Example: What is the future value of $100 deposited at the end of each month at 12% annual interest compounded monthly for three years?

8. $$ Entrepreneurial Finance, 5th Edition Adelman and Marks PRENTICE HALL ©2010 by Pearson Education, Inc. Upper Saddle River, NJ 07458 9-8 Future Value of an Annuity Due  What is the future value of a stream of payments for n periods at an i rate of return when the money is invested at the beginning of each compounding period?  The formula is shown on the next slide:

9. $$ Entrepreneurial Finance, 5th Edition Adelman and Marks PRENTICE HALL ©2010 by Pearson Education, Inc. Upper Saddle River, NJ 07458 9-9 Formula for Future Value of an Annuity Due

10. $$ Entrepreneurial Finance, 5th Edition Adelman and Marks PRENTICE HALL ©2010 by Pearson Education, Inc. Upper Saddle River, NJ 07458 9-10 Future Value of an Annuity Due (Example)  What is the future value of $5,000 deposited at the beginning of each year for 10 years at 6% compounded annually?

11. $$ Entrepreneurial Finance, 5th Edition Adelman and Marks PRENTICE HALL ©2010 by Pearson Education, Inc. Upper Saddle River, NJ 07458 9-11 Future Value of an Annuity Due (Example)  What is the future value of $100 invested at the beginning of each month at 12% compounded monthly for 3 years?

12. $$ Entrepreneurial Finance, 5th Edition Adelman and Marks PRENTICE HALL ©2010 by Pearson Education, Inc. Upper Saddle River, NJ 07458 9-12 Future Value of an Annuity Due (Example)  What is the future value of $1,200 deposited at the beginning each year for 3 years at 12% if the money is compounded monthly?  Since mode of payment (annually) does not equal the mode of compounding (monthly) we have to find the effective rate.

13. $$ Entrepreneurial Finance, 5th Edition Adelman and Marks PRENTICE HALL ©2010 by Pearson Education, Inc. Upper Saddle River, NJ 07458 9-13 Future Value of an Annuity Due (Example) (continued)  We then use the effective rate as the actual interest rate to determine the future value of an annuity due factor and then solve for the future value as shown below.

14. $$ Entrepreneurial Finance, 5th Edition Adelman and Marks PRENTICE HALL ©2010 by Pearson Education, Inc. Upper Saddle River, NJ 07458 9-14 Comparing Ordinary Annuity and Annuity Due  Payment schedule differs Ordinary Annuity Annuity Due  So formulas differ

15. $$ Entrepreneurial Finance, 5th Edition Adelman and Marks PRENTICE HALL ©2010 by Pearson Education, Inc. Upper Saddle River, NJ 07458 9-15 Present Value of an Ordinary Annuity  What is the present value of a stream of payments for n periods at an i rate of return? The formula is shown below:

16. $$ Entrepreneurial Finance, 5th Edition Adelman and Marks PRENTICE HALL ©2010 by Pearson Education, Inc. Upper Saddle River, NJ 07458 9-16 Present Value of an Ordinary Annuity (Example)  What is the present value of a retirement account if you receive $60,000 per year at the end of each year for 30 years if money earns 8% compounded annually?

17. $$ Entrepreneurial Finance, 5th Edition Adelman and Marks PRENTICE HALL ©2010 by Pearson Education, Inc. Upper Saddle River, NJ 07458 9-17 Present Value of an Ordinary Annuity (Example)  What is the present value of $100 received at the end of each month for 3 years if money earns 12% compounded monthly?

18. $$ Entrepreneurial Finance, 5th Edition Adelman and Marks PRENTICE HALL ©2010 by Pearson Education, Inc. Upper Saddle River, NJ 07458 9-18 Present Value of an Annuity Due  What is the present value of a stream of payments that are received at the beginning of each time period? The formula is shown below.

19. $$ Entrepreneurial Finance, 5th Edition Adelman and Marks PRENTICE HALL ©2010 by Pearson Education, Inc. Upper Saddle River, NJ 07458 9-19 Present Value of an Annuity Due (Example)  What is the present value of $50,000 paid at the beginning of each year for the next 20 years if money can earn 7% per year?

20. $$ Entrepreneurial Finance, 5th Edition Adelman and Marks PRENTICE HALL ©2010 by Pearson Education, Inc. Upper Saddle River, NJ 07458 9-20 Present Value of an Annuity Due (Example)  How much do we have to place in an account today to receive $100 payment at the beginning of each month if money earns 12% annual interest compounded monthly?

21. $$ Entrepreneurial Finance, 5th Edition Adelman and Marks PRENTICE HALL ©2010 by Pearson Education, Inc. Upper Saddle River, NJ 07458 9-21 Present Value of an Annuity Due (Example) (continued) Payments Differ Present Value of an Annuity Due So formulas differ Present Value of an Ordinary Annutiy

22. $$ Entrepreneurial Finance, 5th Edition Adelman and Marks PRENTICE HALL ©2010 by Pearson Education, Inc. Upper Saddle River, NJ 07458 9-22 Present Value of an Annuity Due (Example)  Many banks actually calculate daily interest on accounts. For example: What is the present value of $1,200 a year, received at the beginning of each year, at 12% annual interest compounded daily?  First, we have to calculate the effective annual rate.

23. $$ Entrepreneurial Finance, 5th Edition Adelman and Marks PRENTICE HALL ©2010 by Pearson Education, Inc. Upper Saddle River, NJ 07458 9-23 Present Value of an Annuity Due (Example) (continued)  We then calculate the present value using the effective annual rate of 12.75% as follows:

24. $$ Entrepreneurial Finance, 5th Edition Adelman and Marks PRENTICE HALL ©2010 by Pearson Education, Inc. Upper Saddle River, NJ 07458 9-24 Present Value and Amortization  All bank loans are based on the present value of an ordinary annuity. Example: using a 30-year mortgage:

25. $$ Entrepreneurial Finance, 5th Edition Adelman and Marks PRENTICE HALL ©2010 by Pearson Education, Inc. Upper Saddle River, NJ 07458 9-25 Present Value and Amortization (continued)  Example: Using a 15-year mortgage

26. $$ Entrepreneurial Finance, 5th Edition Adelman and Marks PRENTICE HALL ©2010 by Pearson Education, Inc. Upper Saddle River, NJ 07458 9-26 Amortization  The reduction of a loan balance by applying each month’s principal payment.  An amortization table (Table 9-3) is based on the reduction of the loan over time and provides the lender and borrower with the following components:

27. $$ Entrepreneurial Finance, 5th Edition Adelman and Marks PRENTICE HALL ©2010 by Pearson Education, Inc. Upper Saddle River, NJ 07458 9-27  Components of an amortization table: › Payment number › Date of payment › Payment amount › Interest paid › Principal paid › Remaining loan balance Amortization (continued)

28. $$ Entrepreneurial Finance, 5th Edition Adelman and Marks PRENTICE HALL ©2010 by Pearson Education, Inc. Upper Saddle River, NJ 07458 9-28

29. $$ Entrepreneurial Finance, 5th Edition Adelman and Marks PRENTICE HALL ©2010 by Pearson Education, Inc. Upper Saddle River, NJ 07458 9-29 Types of Mortgages  Types are based on scores determined by Fair Isaac Corporation (FICO) score. FICO Range is 500 to 850. › 500 extremely poor credit › 850 for outstanding credit.  Three basic types of mortgage loans granted by lenders. 1. Prime Mortgages ›outstanding credit scores ›have sufficient capital to make substantial down payments. 2. Prime Mortgagee (Alt A) ›aimed primarily at individuals who are improving their scores ›have had a credit problem in the past. 3. Subprime loans ›poor credit ratings ›considered to be high risk.

30. $$ Entrepreneurial Finance, 5th Edition Adelman and Marks PRENTICE HALL ©2010 by Pearson Education, Inc. Upper Saddle River, NJ 07458 9-30 Combining Lump Sums and Annuities  You open a 401k with $300 per month and increase the monthly payments by $200 each year until you have the maximum $1,250 monthly payment ($15,000 per year). The 401k is in mutual funds that average 11%. You are currently 26 and plan to retire at 67.  We therefore have to construct a table that incorporates both the lump sum contributions and the annuities as illustrated on the following page:

31. $$ Entrepreneurial Finance, 5th Edition Adelman and Marks PRENTICE HALL ©2010 by Pearson Education, Inc. Upper Saddle River, NJ 07458 9-31 Combining Lump Sums and Annuities (continued)