1. Chapter 9 Time Value of Money © 2000 John Wiley & Sons, Inc.

2. 2 Chapter Outcomes n Explain what is meant by the time value of money n Describe the concept of simple interest n Describe the process of compounding n Describe discounting to determine present values

3. 3 Chapter Outcomes (Continued) n Find interest rates and time requirements for problems involving compounding and discounting n Describe the meaning of an ordinary annuity n Find interest rates and time requirements for problems involving annuities n Calculate annual annuity payments

4. 4 Chapter Outcomes (Concluded) n Make compounding and discounting calculations using time intervals that are less than one year n Describe the difference between the annual percentage rate and the effective annual rate n Describe the meaning of an annuity due (covered in Learning Extension)

5. 5 Time Value of Money Concepts n TIME VALUE OF MONEY: The mathematics of finance whereby interest is earned over time by saving or investing money n SIMPLE INTEREST: Interest earned only on the principal of the initial investment

6. 6 Time Value of Money Concepts (Continued) n PRESENT VALUE: Value of an investment or savings amount today or at the present time n FUTURE VALUE: Value of an investment or savings amount at a specified future time n BASIC EQUATION: Future value = Present value + (Present value x Interest rate)

7. 7 Time Value of Money Example: Simple Interest n BASIC INFORMATION: You have $1,000 to save or invest for one year and a bank will pay you 8% for use of your money. What will be the value of your savings after one year? n BASIC EQUATION: Future value = Present value + (Present value x Interest rate) Future value = $1,000 + ($1,000 x.08) = $1,000 + $80 = $1,080

8. 8 Time Value of Money Example: Simple Interest (Continued) n ALTERNATIVE BASIC EQUATION: Future value = Present value x (1 + Interest rate) Future value = $1,000 x (1 +.08) = $1,000 x 1.08 = $1,080

9. 9 Compounding to Determine Future Values n COMPOUNDING: An arithmetic process whereby an initial value increases at a compound interest rate over time to reach a future value n COMPOUND INTEREST: Interest earned on interest in addition to interest earned on the principal or investment

10. 10 Compounding to Determine Future Values: An Example n BASIC INFORMATION: You plan to invest $1,000 now for two years and a bank will pay you compound interest of 8% per year. What will be the value after two years? n BASIC EQUATION: Future value = Present value x [(1 + Interest rate) x (1 + Interest rate)] Future value = $1,000 x (1.08 x 1.08) = $1,000 x 1.166 = $1,166

11. 11 Using Interest Factor Tables to Solve Future Value Problems n FUTURE VALUE (FV n ) = PV(FVIF r,n ) Where: PV = present value amount FVIF r,n = pre-calculated future value interest factor for a specific interest rate (r) and specified time period (n) n EXAMPLE: What is the future value of $1,000 invested now at 8% interest for 10 years? FV 10 = $1,000(2.159) = $2,159

12. 12 Discounting to Determine Present Values n DISCOUNTING: n DISCOUNTING: An arithmetic process whereby a future value decreases at a compound interest rate over time to reach a present value n BASIC EQUATION: n BASIC EQUATION: Present value = Future value x {[1/(1 + Interest rate)] x [1/(1 + Interest rate)]}

13. 13 Discounting to Determine Present Values: An Example n BASIC INFORMATION: n BASIC INFORMATION: a bank agrees to pay you $1,000 after two years when interest rates are compounding at 8% per year. What is the present value of this payment? n BASIC EQUATION: n BASIC EQUATION: Present value = Future value x {[1/(1+ Interest rate)] x [1/(1 + Interest rate)]} Present value = $1,000 x (1/1.08 x 1/1.08) = $1,000 x 0.857 = $857

14. 14 Using Interest Factor Tables to Solve Present Value Problems n PRESENT VALUE (PV) = FV n (PVIF r,n ) Where: FV n = future value amount PVIF r,n = pre-calculated present value interest factor for a specific interest rate (r) and specified time period (n) n EXAMPLE: What is the present value of $1,000 to be received 10 years from now if the interest rate is 8%? PV = $1,000(0.463) = $463

15. 15 Interest Factor Tables: Finding Interest Rates or Time Periods n FOUR BASIC VARIABLES: FV = future value PV = present value r = interest rate n = number of periods n KEY CONCEPT: Knowing the values for any three of these variables allows solving for the fourth or unknown variable

16. 16 Future Value of an Annuity n ANNUITY: A series of equal payments that occur over a number of time periods n ORDINARY ANNUITY: Exists when the equal payments occur at the end of each time period (also referred to as a deferred annuity)

17. 17 Future Value of an Annuity (Continued) n BASIC EQUATION: FVA n = PMT{[(1 + r) n - 1]/r} n Where: FVA = future value of ordinary annuity PMT = periodic equal payment r = compound interest rate, and n = total number of periods

18. 18 Future Value of an Annuity: An Example n BASIC INFORMATION: You plan to invest $1,000 each year beginning next year for three years at an 8% compound interest rate. What will be the future value of the investment? n BASIC EQUATION: FVA n = PMT{[(1 + r) n - 1]/r} FVA 3 = $1,000{[(1 +0.08) 3 - 1]/0.08} = $1,000[(1.2597 - 1)/0.08] = $1,000(3.246) = $3,246

19. 19 Present Value of an Annuity n BASIC EQUATION: PVA n = PMT{[1 - (1/(1 +r) n )]/r} n Where: PVA = present value of ordinary annuity PMT = periodic equal payment r = compound interest rate, and n = total number of periods

20. 20 Present Value of an Annuity: An Example n BASIC INFORMATION: You will receive $1,000 each year beginning next year for three years at an 8% compound interest rate. What will be the present value of the investment? n BASIC EQUATION: PVA n = PMT{[1 - (1/(1 + r) n )]/r} PVA 3 = $1,000{[1 - (1/(1.08) 3 )]/0.08} = $1,000{[1 - 0.7938]/0.08} = $1,000(0.2062/0.08) = $2,577

21. 21 Annual Annuity Payments n AMORTIZED LOAN: A loan repaid in equal payments over a specified time period n PROCESS: Solve for the amount of the annual payment n LOAN AMORTIZATION SCHEDULE: A schedule of the breakdown of each payment between interest and principal, as well as the remaining balance after each payment

22. 22 Compounding or Discounting More Often than Once a Year n BASIC EQUATION: FV n = PV(1 + r/m) nxm Where: m = number of compounding periods per year and the other variables are as previously defined n EXAMPLE: What is the future value of a two-year, $1,000, 8% interest loan with semiannual compounding? FV n = $1,000( 1 + 0.08/2) 2x2 = $1,000(1.04) 4 = $1,000(1.170) =$1,170

23. 23 APR Versus EAR n ANNUAL PERCENTAGE RATE n ANNUAL PERCENTAGE RATE (APR): Determined by multiplying the interest rate charged (r) per period by the number of periods in a year (m) n APR EQUATION: n APR EQUATION: APR = r x m n EXAMPLE: n EXAMPLE: What is the APR on a car loan that charges 1% per month? APR = 1% x 12 months = 12%

24. 24 APR Versus EAR (Continued) n EFFECTIVE ANNUAL RATE (APR): true interest rate when compounding occurs more frequently than annually n EAR EQUATION: EAR = (1 + r) m - 1 n EXAMPLE: What is the EAR on a credit card loan with an 18% APR and with monthly payments? Rate per month = 18%/12 = 1.5% EAR = (1 + 0.015) 12 - 1 = 1.1956 -1 = 19.56%

25. 25 Learning Extension: Annuity Due Problems n ANNUITY DUE: Exists when the equal periodic payments occur at the beginning of each period n FUTURE VALUE OF AN ANNUITY DUE (FVAD n ) EQUATION: FVAD n = FVA n x (1 + r) FVAD n = PMT{[((1 + r) n - 1)/r] x (1 + r)}

26. 26 Learning Extension: Annuity Due Problems (Continued) n BASIC INFORMATION: You plan to invest $1,000 each year beginning now for three years at an 8% compound interest rate. What will be the future value of this investment? n EQUATION: FVAD n = FVA n x (1 + r) FVAD n = PMT{[((1 + r) n - 1)/r] x (1 + r)} FVAD 3 = $1,000{[((1.08) 3 - 1)/0.08] x (1.08)} = [$1,000(3.246)] x 1.08 = $1,000(3.506) = $3,506