1. 6-1 Chapter 6 The Time Value of Money Future Value Present Value Rates of Return Amortization

2. 6-2 Cash Flow Time Lines CF 0 CF 1 CF 3 CF 2 0123 i% Tick marks Tick marks at ends of periods, so t=0 is today; t=1 is the end of Period 1; or the beginning of Period 2. Graphical representations used to show the timing of cash flows.

3. 6-3 Time line for a $100 lump sum due at the end of Year 2. 100 012 Year i%

4. 6-4 Time line for an ordinary annuity of $100 for 3 years. 100 0123 i%

5. 6-5 Time line for uneven CFs -$50 at t=0 and $100, $75, and $50 at the end of Years 1 through 3. 100 50 75 0123 i% -50

6. 6-6 Future Value The amount to which a cash flow or series of cash flows will grow over a period of time when compounded at a given interest rate. How much would you have at the end of one year if you deposited $100 in a bank account that pays 5 percent interest each year? FV n = FV 1 = PV + INT = PV + (PV x i) = PV (1 + i) = $100(1+0.05) = $100(1.05) = $105

7. 6-7 Finding FVs is Compounding. What’s the FV of an initial $100 after 3 years if i = 10%? FV = ? 0123 i=10% 100

8. 6-8 In general, FV n = PV (1 + i) n After 3 years: FV 3 =PV(1 + i) 3 =100 (1.10) 3 =$133.10. After 2 years: FV 2 =PV(1 + i) 2 =$100 (1.10) 2 =$121.00. After 1 year: FV 1 =PV + i = PV + PV (i) =PV(1 + i) =$100 (1.10) =$110.00. Future Value

9. 6-9 4 Solve the Equation with a Regular Calculator 4 Use Tables 4 Use a Financial Calculator Three ways to find Future Values:

10. 6-10 Financial Calculator Solution: Financial calculators solve this equation: There are 4 variables. If 3 are known, the calculator will solve for the 4 th.

11. 6-11 310-100 0 NI/YR PV PMTFV 133.10 INPUTS OUTPUT Here’s the setup to find FV: Clearing automatically sets everything to 0, but for safety enter PMT = 0. Set:P/YR= 1 Financial Calculator Solution:

12. 6-12 Present Value l Present value l Present value is the value today of a future cash flow or series of cash flows. l Discounting l Discounting is the process of finding the present value of a cash flow or series of cash flows, the reverse of compounding.

13. 6-13 What is the PV of $100 due in 3 years if i = 10%? 100 0123 10% PV = ?

14. 6-14 Solve FV n = PV (1 + i ) n for PV: What is the PV of $100 due in 3 years if i = 10%?

15. 6-15 Either PV or FV must be negative. Here PV = -75.13. Put in $75.13 today, take out $100 after 3 years. 310 0100 NI/YR PV PMTFV -75.13 INPUTS OUTPUT Financial Calculator Solution:

16. 6-16 If sales grow at 20% per year, how long before sales double? Solve for n: FV n = 1(1 + i) n ; 2 = 1(1.20) n.

17. 6-17 20 -1 0 2 NI/YR PV PMTFV 3.8 INPUTS OUTPUT 1234 Year 0 1 2 FV 3.8 Graphical Illustration: Financial Calculator Solution:

18. 6-18 Future Value of an Annuity èAnnuity: èAnnuity: A series of payments of an equal amount at fixed intervals for a specified number of periods. èOrdinary (deferred) Annuity: èOrdinary (deferred) Annuity: An annuity whose payments occur at the end of each period. èAnnuity Due: èAnnuity Due: An annuity whose payments occur at the beginning of each period.

19. 6-19 Ordinary Annuity Versus Annuity Due PMT 0123 i% PMT 0123 i% PMT

20. 6-20 What’s the FV of a 3-year Ordinary Annuity of $100 at 10%? 100 0123 10% 110 121 FV= 331

21. 6-21 310 0 -100 331.00 INPUTS OUTPUT NI/YRPVPMTFV Financial Calculator Solution:

22. 6-22 Present Value of an Annuity èPVA n = the present value of an annuity with n payments èEach payment is discounted, and the sum of the discounted payments is the present value of the annuity

23. 6-23 What is the PV of this Ordinary Annuity? 248.69 = PV 100 0123 10% 90.91 82.64 75.13

24. 6-24 310 100 0 -248.69 Have payments but no lump sum FV, so enter 0 for future value. Financial Calculator Solution: NI/YRPVPMTFV INPUTS OUTPUT

25. 6-25 Find the FV and PV if the Annuity were an Annuity Due. 100 0123 10% 100

26. 6-26 310 100 0 -273.55 INPUTS OUTPUT NI/YRPVPMTFV Switch from “End” to “Begin”. Then enter variables to find PVA 3 = $273.55. Then enter PV = 0 and press FV to find FV = $364.10. Financial Calculator Solution:

27. 6-27 Solving for Interest Rates with Annuities 250 0123 i = ? - 864.80 4 250 You pay $864.80 for an investment that promises to pay you $250 per year for the next four years, with payments made at the end of the year. What interest rate will you earn on this investment?

28. 6-28 Financial Calculator Solution: PVA n = PMT(PVIFA i,n ) $846.80= $250(PVIFA i = ?,4 ) 4 ? - 846.80 250 0 =7.0 INPUTS OUTPUT NI/YRPVPMTFV INPUTS OUTPUT NI/YRPVPMTFV

29. 6-29 Uneven Cash Flow Streams èA series of cash flows in which the amount varies from one period to the next. Ù Payment (PMT) designates constant cash flows Ù Cash flow (CF) designates cash flows in general, including uneven cash flows

30. 6-30 What is the PV of this Uneven Cash Flow Stream? 0 100 1 300 2 3 10% -50 4 90.91 247.93 225.39 -34.15 530.08 = PV

31. 6-31 4 Input in “Cash Flow” register: CF 0 = 0 CF 1 =100 CF 2 =300 CF 3 =300 CF 4 = -50 4 Enter I = 10%, then press NPV button to get NPV = 530.09. (Here NPV = PV.) Financial Calculator Solution:

32. 6-32 $100 (1 + i ) 3 = $125.97. What interest rate would cause $100 to grow to $125.97 in 3 years? 3-100 0 125.97 INPUTS OUTPUT NI/YRPVFV PMT 8%

33. 6-33 Semiannual and Other Compounding Periods èAnnual compounding èAnnual compounding is the arithmetic process of determining the final value of a cash flow or series of cash flows when interest is added once a year. èSemiannual compounding èSemiannual compounding is the arithmetic process of determining the final value of a cash flow or series of cash flows when interest is added twice a year.

34. 6-34 LARGER! If compounding is more frequent than once a year--for example, semi-annually, quarterly, or daily-- interest is earned on interest more often. Will the FV of a lump sum be larger or smaller if we compound more often, holding the state i% constant? Why?

35. 6-35 0123 10% 100 133.10 0123 5% 456 134.01 123 0 100 Annually: FV 3 = 100(1.10) 3 = 133.10. Semi-annually: FV 6/2 = 100(1.05) 6 = 134.01.

36. 6-36 Distinguishing Between Different Interest Rates Simple (Quoted) Rate i SIMPLE = Simple (Quoted) Rate used to compute the interest paid per period Effective Annual Rate EAR= Effective Annual Rate the annual rate of interest actually being earned Annual Percentage Rate APR = Annual Percentage Rate periodic rate X the number of periods per year

37. 6-37 How do we find EAR for a simple rate of 10%, compounded semi-annually? m EAR= 1+ i SIMPLE     m - 1   = 1+ 0.10 2 - 1.0 = 1.05 - 1.0 = 0.1025= 10.25%.       2 2

38. 6-38 FV of $100 after 3 years under 10% semi-annual compounding? Quarterly? = $100(1.05) 6 = $134.01 FV 3Q = $100(1.025) 12 = $134.49

39. 6-39 0.75 10 - 100 0 ? =107.41 INPUTS OUTPUT NI/YRPVPMTFV Fractional Time Periods 00.250.500.75 8% - 100 1.00 FV = ? Example: $100 deposited in a bank at 10% interest for 0.75 of the year

40. 6-40 Amortized Loans l Amortized Loan: l Amortized Loan: A loan that is repaid in equal payments over its life. l Amortization tables are widely used-- for home mortgages, auto loans, business loans, retirement plans, etc. lThey are very important! l Financial calculators (and spreadsheets) are great for setting up amortization tables.

41. 6-41 Construct an amortization schedule for a $1,000, 10% annual rate loan with 3 equal payments. Step 1: Find the required payments PMT 0123 10% -1000 3 10 -1000 0 INPUTS OUTPUT NI/YRPVFV PMT 402.11

42. 6-42 Step 2: Find interest charge for Year 1 Step 2: Find interest charge for Year 1 INT t = Beg bal t (i) INT 1 = 1000(0.10) = $100. Step 3: Find repayment of principal in Year 1 Step 3: Find repayment of principal in Year 1 Repmt.= PMT - INT = 402.11 - 100 = $302.11. Step 4: Find ending balance after Year 1 Step 4: Find ending balance after Year 1 End bal=Beg bal - Repmt =1000 - 302.11 = $697.89. Repeat these steps for Years 2 and 3 to complete the amortization table.

43. 6-43 Interest declines. Tax Implications. Loan Amortization Table 10 Percent Interest Rate

44. 6-44 $ 0123 402.11 Interest Principal Payments 302.11 Level payments. Interest declines because outstanding balance declines. Lender earns 10% on loan outstanding, which is falling.

45. 6-45 4 i SIMPLE : Written into contracts, quoted by banks and brokers. Not used in calculations or shown on time lines. 4 i per : Used in calculations, shown on time lines. If i SIMPLE has annual compounding, then i per = i SIMPLE /1 = i SIMPLE. 4 EAR : Used to compare returns on investments with different payments per year. (Used for calculations if and only if dealing with annuities where payments don’t match interest compounding periods.) Comparison of Different Types of Interest Rates

46. 6-46 4 i SIMPLE is stated in contracts. Periods per year (m) must also be given. 4 Examples: 8%; Quarterly 8%, Daily interest (365 days) Simple (Quoted) Rate

47. 6-47 4 Periodic rate = i per = i SIMPLE /m, where m is periods per year. m = 4 for quarterly, 12 for monthly, and 360 or 365 for daily compounding. 4 Examples: 8% quarterly: i per = 8/4 = 2% 8% daily (365): i per = 8/365 = 0.021918% Periodic Rate

48. 6-48 4 Effective Annual Rate: The annual rate which cause PV to grow to the same FV as under multiperiod compounding. 4 Example: EAR for 10%, semiannual: FV=(1 + i SIMPLE /m) m =(1.05) 2 = 1.1025. EAR=10.25% because (1.1025) 1 = 1.1025. Any PV would grow to same FV at 10.25% annually or 10% semiannually. Effective Annual Rate