The Time Value of Money

1. The Time Value of Money • Introduction to Time Value of Money, TVM • Future Value, FV • Lump-sum amount • Annuity • Uneven cash flow • Present Value, PV • Lump-sum amount • Annuity • Uneven cash flow • FV and PV Comparison • Solving for r and n • Intra-year Interest Compounding • Amortization

2. Time Value of Money • Why is it important to understand and apply time value of money concepts? • What is the difference between a present value amount and a future value amount? • What is an annuity? • What is the difference between the Annual Percentage Rate and the Effective Annual Rate? • What is an amortized loan? • How is the return on an investment determined?

3. The Time Value of Money • Time value of money is considered the most important concept in finance • Mathematics of finance • “Nuts & Bolts” of financial analysis—apply of TVM concepts to determine value • Interest = Rate of return = r = i = k = Y

4. The Time Value of Money • “$1 received today is more valuable than $1 received in one year.” Why? • Because if you have the opportunity to earn a positive return, investing the $1 today will cause it to grow to greater than $1 in one year. • For example, $1 invested at 5 percent will grow to $1.05 in one year because 5¢ of interest will be earned.

5. Future Value and Present Value • Future Value (FV)—determine to what amount an investment will grow over a particular time period • re-invested interest (earned in previous periods) earns interest • compounding—interest compounds or grows the investment • Present value (PV)—determine the current value of an amount that will be paid, or received, at some time in the future • PV is the future amount restated in current dollars; future interest has not been earned, thus it is not included in the PV • discounting—deflate, or discount, the future amount by future interest that can be earned (“deinterest” the FV)

6. Lump-Sum Amounts, Annuities, and Uneven Cash Flow Streams • Lump-sum amount—a single amount invested (received) today or in the future; growth in value is the result of interest only • Annuity—equal payments made (received) at equal intervals; growth in value is the result of additional payments as well as interest • ordinary annuity—end of period payments • annuity due—beginning of period payments • Uneven Cash Flows—payments that are not all equal that are generally made (received) at equal intervals; growth in value is the result of additional payments as well as interest

7. Cash Flow Time Lines • Helps you to visualize the timing of the cash flows associated with a particular situation • Constructing a cash flow time line is easy: Time 0 1 2 3 4 r = 10% Cash Flows FVn = ? -500

8. Approaches to TVM Solutions • Time line solution • Solve using a cash flow time line • Equation (numerical) solution • Use equations to solve the problem • Financial calculator solution • Financial calculators are programmed to solve time value of money problems using the numerical solution • Spreadsheet solution • Spreadsheets contain functions that can be used to solve time value of money problems using the numerical solution • Interest tables • Obsolete

9. Future Value • Determine to what amount an investment will grow over a particular time period if it is invested at a positive rate of return. • Compounding • Lump-sum amount • Annuity • Uneven cash flow stream

10. Time 0 1 2 3 4 r = 10% Cash Flows FVn = ? -500 Future Value, FV, of a Lump-Sum Amount Example: If you invest $500 today at 10%, what will the investment be worth in four years if interest is paid annually?

11. 2 0 3 1 4 10% Future Value Graphically, these computations are: × 1.10 × 1.10 × 1.10 -500 × 1.10 End of year amount = 732.05 = 665.50 = 550.00 = 605.00 FV4= 500(1.10 x 1.10 x 1.10 x 1.10) = 500(1.10)4 =732.05

12. Future Value The future value of an amount invested today for n years, FVn, can be found using the following equation: FVn = PV(1 + r)n = PV(interest multiple) FVn= future value in period n PV= present, or current, value r= interest rate per period n= number of periods interest is earned

13. Equation (Numerical) Solution Determined by applying the appropriate equation: FVn= PVx(1 + r)n FVn= 500x(1 + r)n FVn= 500x(1.10)n FV4= 500x(1.10)4 FV4= 500x(1.10)4 = 732.05 In our example: PV = $500, r = 10.0%, n = 4 In our example: PV = $500, r = 10.0%, n = 4 In our example: PV = $500, r = 10.0%, n = 4 In our example: PV = $500, r = 10.0%, n = 4

14. N I/Y PV PMT FV Financial Calculator Solution In our example: PV = $500, r = 10.0%, n = 4 In our example: PV = $500, r = 10.0%, n = 4 4 In our example: PV = $500, r = 10.0%, n = 4 410 In our example: PV = $500, r = 10.0%, n = 4 410-500 In our example: PV = $500, r = 10.0%, n = 4 410-5000 In our example: PV = $500, r = 10.0%, n = 4 410-5000? In our example: PV = $500, r = 10.0%, n = 4 410-5000? 732.05

15. Future Value of an Annuity • Annuity—a series of equal payments that are made at equal intervals • Ordinary annuity—end of the period • Annuity due—beginning of the period • The future value of an annuity, FVA, can be computed by solving for the future value of a lump-sum amount

16. Time 0 1 2 3 Cash Flows Future Value of an Annuity, FVA 7% -100 -100 -100 x (1.07)0 100.00 x (1.07)1 107.00 x (1.07)2 FVA = 321.49 114.49 FVA = 100(1.07)2 + 100(1.07)1 + 100(1.07)0 = 321.49 FVA = 100(1.07)2 + 100(1.07)1 + 100(1.07)0 = 321.49 = 100[(1.07)2 + (1.07)1 + (1.07)2] = 100(3.2149) = 321.49

17. FVA—Equation (Numerical) Solution In our example: PMT = $100, r = 7%, n = 3

18. FVA—Annuity Due • Annuity due is an annuity with cash flows that occur at the beginning of the period. • When compared to an ordinary annuity, which has end-of-period cash flows, the cash flows of an annuity due earn one additional period of interest.

19. FVA—Annuity Due 0 1 2 3 Time 7% -100-100-100 Cash Flows -100 -100 -100 x (1.07) x (1.07)0 107.00 100.00 x (1.07) x (1.07)1 107.00 114.49 x (1.07)2 x (1.07) 114.49 114.49 FVA = 321.49 122.50 343.99 FVA(DUE) =

20. FVA(DUE)n = PMT FVA= PMT FVA(DUE)—Equation (Numerical) Solution In our example: PMT = $100, r = 7%, n = 3

21. N N I/Y I/Y PV PV PMT PMT FV FV 3 7.0 0 -100? FVA—Financial Calculator Solution In our example: n = 3, r = 7%, PMT = $100 37.00-100 In our example: n = 3, r = 7%, PMT = $100 37.00 In our example: n = 3, r = 7%, PMT = $100 37.0 In our example: n = 3, r = 7%, PMT = $100 3 In our example: n = 3, r = 7%, PMT = $100 In our example: n = 3, r = 7%, PMT = $100 37.00-100? FVA = 321.49 BEGIN FVA(DUE) = 343.99

22. Solutions: Future value computations FV of $25,000 lump-sum amount: N = 5, I/Y = 5, PV = -25,000, PMT = 0, FV = ? FV of $25,000 lump-sum amount: N = 5, I/Y = 5, PV = -25,000, PMT = 0, FV = 31,907 FV of $5,499.40 annuity: N = 5, I/Y = 5, PV = 0, PMT = -5,499.40, FV = ? FV of $5,499.40 annuity: N = 5, I/Y = 5, PV = 0, PMT = -5,499.40, FV = 31,907

23. Uneven Cash Flow Streams • Uneven cash flow stream—cash flows that are not all the same (equal) • Simplifying techniques (that is, using a single equation) used to compute FVA cannot be used

24. 0 1 2 3 FV—Uneven Cash Flow Streams 4% -600 -400 -200 200.00 416.00 648.96 _______ 1,264.96

25. FV of Uneven Cash Flow Streams—Equation (Numerical) Solution

26. FV of Uneven CF Streams—Calculator Solution • Input the cash flows, find the present value, PV, and then compute the future value, FV, of PV. • Discussed in the next section.

27. Present Value • Determine the current value of an amount that will be paid, or received, at a particular time in the future. • Finding the present value (PV), or discounting, an amount to be received (paid) in the future is the reverse of compounding, or determining the future value of an amount invested today. • We find the PV by “de-interesting” the FV.

28. Present Value—Lump-Sum Amount What is the PV of $800 to be received in four years if your opportunity cost is 8 percent? Stated differently: How much would you be willing to pay today for an investment that will pay $800 in four years if you have the opportunity to invest at 8 percent per year?

29. 2 0 3 Time 1 4 Present Value—Lump-Sum Amount 8% 800 Cash Flows PV = ?

30. Present Value—Equation (Numerical) Solution FV= PVx(1 + r)n In our example, FV4 = 800, n = 4, r = 8.0% Remember that FV is computed as follows: FV= PVx(1 + r)n Remember that FV is computed as follows: 800=PVx(1.08)4 800= PVx1.36049 800= PVx(1 + r)n In our example, FV4 = 800, n = 4, r = 8.0% 800= PVx(1.08)n In our example, FV4 = 800, n = 4, r = 8.0% 800= PVx(1.08)4 In our example, FV4 = 800, n = 4, r = 8.0% PV=800/1.36049 PV=800/1.36049 = 588.02

31. é ù 1 = 800 ê ú 4 ê ú (1.08) ë û = = 800(0.7350 3) 588.02 Present Value—Equation (Numerical) Solution PV Equation: In our example: FV = $800, r = 8.0%, and N = 4

32. 2 0 3 1 4 8% Present Value—Time Line Solution Graphically, this computation is: 588.02 635.07 685.87 740.74 800.00 End of year amount

33. N I/Y PV PMT FV PV Lump-Sum Amount—Financial Calculator Solution In our example: FV = $800, r = 8.0%, n = 4 In our example: FV = $800, r = 8.0%, n = 4 4 In our example: FV = $800, r = 8.0%, n = 4 48.0 In our example: FV = $800, r = 8.0%, n = 4 48.00800 In our example: FV = $800, r = 8.0%, n = 4 48.00 In our example: FV = $800, r = 8.0%, n = 4 48.0?0800 -588.02

34. 0 1 2 3 Present Value of an Annuity, PVA 7% 100 100 100 93.46 87.34 81.63 262.43 81.63 = PVA 262.43

35. PVA—Equation (Numerical) Solution In our example: PMT = $100, r = 7%, n = 3

36. PVA—Annuity Due Annuity due is an annuity with cash flows that occur at the beginning of the period.

37. 0 0 0 7% 7% 7% 1 1 1 2 2 2 3 3 3 100 100 100 100 100 100 100 100 100 100 100 93.46 87.34 100.00 81.63 262.43 = PVA 93.46 87.34 280.80 PVA—Annuity Due (1.07) (1.07) (1.07) (DUE)

38. PVA = PMT PVA(DUE)n = PMT x (1 + r) PVA(DUE)—Equation (Numerical) Solution In our example: PMT = $100, r = 7%, n = 3

39. N N I/Y I/Y PV PV PMT PMT FV FV 37.01000 PVA—Financial Calculator Solution In our example: n = 3, r = 7%, PMT = $100 In our example: n = 3, r = 7%, PMT = $100 37.0?1000 In our example: n = 3, r = 7%, PMT = $100 37.01000 = PVA -262.43 BEGIN 37.0?1000 BEGIN 37.01000 -280.80 = PVA(DUE)

40. Calculator Solution: Calculator Solution: N = 5, I/Y = 5, PMT = -5,499.40, FV = 0, PV = 25,000 Calculator Solution: N = 5, I/Y = 5, PMT = -5,499.40, FV = 0, PV = ? Numerical Solution:

41. Uneven Cash Flow Streams • Uneven cash flow stream—cash flows that are not all the same (equal) • Simplifying techniques (that is, using a single equation) used to compute PVA cannot be used

42. 0 1 2 3 Present Value of an Uneven Cash Flow Stream 4% 600 400 200 576.92 369.82 177.80 1,124.54 177.80

43. PV of Uneven Cash Flows— Equation (Numerical) Solution

44. PV of Uneven Cash Flows—Financial Calculator Solution • Use the cash flow (CF) register (see calculator instructions) • Input CFs in the order they occur—that is, first input CF1, then input CF2, and so on • CF0—most calculators require you to input a value for before entering any other cash flows • Enter the value for I • NPV = PV of uneven cash flows

45. PV of Uneven Cash Flows—Financial Calculator Solution CF0= 0 CF1=600 CF2=400 CF3=200 r=4% Compute NPV = 1,124.54

46. FV of Uneven CF Streams—Calculator Solution • Input the cash flows, find the present value, PV, and then compute the future value, FV, of PV • In our example, PV = $1,124.54, so the future value is: FV = $1,124.54(1.04)3 = $1,264.95

47. Numerical solution: PV = ($2 million)/(1.06)1 + ($4 million)/(1.06)2 + ($5 million)/(1.06)3 = ($2 million)(0.943396) + ($4 million)(0.889996) +($5 million)(0.839619) = 1.8868 + 3.5600 + 4.1981 = 9.6449 Calculator solution: CF0 = 0 CF1 = 2 million CF2 = 4 million CF3 = 5 million I = 6 NPV = 9.6448

48. 0 1 2 3 7% 100 100 100 PVA = 262.43 FVA = 321.49 Comparison of PVA, FVA, and Lump-Sum Amount • PMT= $100; r = 7%; n = 3 • FVA= $321.49 • PVA= $262.43 C A B FV = 262.43 x (1.07)3 = 321.49 PV = 321.49/(1.07)3 = 262.43

49. PVA, FVA, and Lump-Sum Amount PMT = $100; r = 7%; n = 3; PVA = $262.43 PMT = $100; r = 7%; n = 3; PVA = $262.43 BeginningInterest Ending Payment/ Year Balance @ 7%BalanceWithdrawal 1$262.43$18.37$280.80$100.00 2180.8012.66193.46100.00 393.466.54100.00100.00 FVA = 321.49

50. Solving for Time (n) and Interest Rates (r)—Lump Sums • The computations for lump-sum amounts included four variables: n, r, PV, and FV. • If three of the four variables are known, then we can solve for the unknown variable—e.g., if n, PV, and FV are known, we can solve for r.