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# Chapter 3

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1. Chapter 3 Time Value of Money

2. The Time Value of Money • Interest Rate • Simple Interest • Compound Interest • Future Value (Compounding) • Present Value (Discounting) • Annuities • Loan Amortization • Bond Valuation

3. Obviously, \$10,000 today. You already recognize that there is TIME VALUE TO MONEY!! Which would you prefer -- \$10,000 today or \$10,000 in 5 years? The Interest Rate

4. TIME allows you the opportunity to postpone consumption and earn INTEREST. Why is TIME such an important element in your decision? Why Time?

5. TIME VALUE OF MONEY • THE UNIVERSAL PREFERENCE FOR A DOLLAR TODAY OVER A DOLLAR AT SOME FUTURE TIME • UNCERTAINTY (RISK) • ALTERNATIVE USES • INFLATION

6. INTEREST RATES • THE PRICING MECHANISM FOR THE TIME VALUE OF MONEY • REFLECT INVESTORS’ TIME PREFERENCES FOR MONEY • MAY ALSO ACCOUNT FOR RISK AND INFLATION

7. INTEREST RATES REFLECT THE OPPORTUNITY COST OF NOT PUTTING MONEY TO ITS BEST USE.

8. SIMPLE INTEREST • MEANS THAT ONLY THE ORIGINAL PRINCIPAL EARNS INTEREST OVER THE LIFE OF THE TRANSACTION. • THE PRODUCT OF THE PRINCIPAL, THE TIME IN YEARS, AND THE ANNUAL INTEREST RATE.

9. Simple Interest Formula FormulaSI = P0(i)(n) SI: Simple Interest P0: Principal Deposited today (t=0) i: Interest Rate per Period n: Number of Time Periods

10. SI = P0(i)(n)= \$5,000(.08)(3) = \$1,200 Assume that you deposit \$5,000 in an account earning 8% simple interest for 3 years. What is the accumulated interest at the end of the 3rd year? Simple Interest Example

11. FV = P0 + SI = \$5,000+ \$1,200 =\$6,200 Future Valueis the value at some future time of a present amount of money, or a series of payments, evaluated at a given interest rate. What is the Future Value (FV) of the deposit? Simple Interest (FV)

12. The Present Value is simply the \$5,000 you originally deposited. That is the value today! Present Valueis the current value of a future amount of money, or a series of payments, evaluated at a given interest rate. What is the Present Value (PV) of the previous problem? Simple Interest (PV)

13. COMPOUND INTEREST • WHEN INTEREST IS EARNED AND CONVERTED TO PRINCIPAL MORE THAN ONCE DURING THE TIME OF THE INVESTMENT. • THE INTERVAL BETWEEN SUCCESSIVE CONVERSIONS IS CALLED THE CONVERSION PERIOD. • MONTHLY • QUARTERLY • DAILY • SEMI-ANNUALLY • ANNUALLY

14. COMPOUND AMOUNT - THE TOTAL AMOUNT AT THE END OF THE CONVERSION PERIOD. • COMPOUND INTEREST - IS THE DIFFERENCE BETWEEN THE COMPOUND AMOUNT AND THE BEGINNING PRINCIPAL.

15. COMPOUND INTEREST RATE (PERIODIC RATE) IS THE RATE PER CONVERSION PERIOD THAT IS CHARGED ON THE OUTSTANDING BALANCE AT THE BEGINNING OF THAT PERIOD. • Example: an annual rate of 12% is converted to a quarterly rate by dividing the annual rate by the number of conversions periods in a year; therefore, 12% ÷ 4 = 3% quarterly rate.

16. NOMINAL ANNUAL RATE - THE PERIODIC RATE CONVERTED TO AN ANNUAL BASIS. • Example: a monthly rate of 1.5%/month is converted to an annual rate by multiplying by the number of conversions periods in a year; therefore, the annual rate is 12 x 1.5 = 18% • EFFECTIVE ANNUAL RATE - THE RATE OF INTEREST ACTUALLY EARNED IN A YEAR. • With compounding the effective annual rate will be greater than the nominal annual rate.

17. Why Compound Interest? Future Value (U.S. Dollars)

18. COMPOUNDING • FUTURE VALUE OF A PRESENT SUM • FUTURE VALUE OF A SERIES OF PAYMENTS

19. Future Value of a Present Sum (graphic) Assume that you deposit \$5,000 at a compound interest rate of 8% for 2 years. 0 12 8% \$5,000 \$5,832 \$5,400 FV2

20. COMPOUNDING FUTURE VALUE OF A PRESENT SUM FV n = PVO (1+i)n OR FUTURE VALUE = PRESENT VALUE * (1 + COMPOUND RATE) CONVERSION PERIODS

21. Future Value of a Present Sum (formula) FV1 = P0 (1+i)1 = \$5,000(1.08) = \$5,400 Compound Interest You earned \$400 interest on your \$5,000 deposit over the first year. This is the same interest you would earn under simple interest.

22. Future Value of a Present Sum (formula) FV1 = P0(1+i)1 = \$5,000 (1.08) = \$5,400 FV2 = FV1 (1+i)1 = {P0 (1+i)}(1+i) = P0(1+i)2 =\$5,000(1.08)(1.08) = \$5,000(1.08)2 = \$5,832.00 You earned an EXTRA\$32.00 in Year 2 with compound over simple interest.

23. General Formula for Future Value FV1 = P0(1+i)1 FV2 = P0(1+i)2 General Future Value Formula: FVn = P0 (1+i)n or FVn = P0 (FVD in) -- See Table A1 etc.

24. Valuation Using Table A1 FVD I,nis found in Table A1

25. Using Future Value Tables FV2 = \$5,000 (FVD 8%,2) = \$5,000 (1.166) = \$5,830 [ due to rounding]

26. PROBLEM: \$5000 @ 8% COMPOUNDED ANNUALLY FOR 3 YEARS FV n = 5000*(1.08)3 FV n =5000(1.259712) = 6,298.56

27. PROBLEM: \$5000 @ 8% COMPOUNDED QUARTERLY FOR 3 YEARS FV n = 5000*(1.02)12 FV n =5000(1.2682418) = 6,341.21

28. Example Problem Julie Miller wants to know how large her \$10,000 deposit will become at a compound interest rate of 10% for 5 years. 0 1 2 3 4 5 10% \$10,000 FV5

29. Problem Solution • Calculation based on general formula:FVn = P0 (1+i)nFV5= \$10,000 (1+ 0.10)5 = \$16,105.10 • Calculation based on Table A1: FV5= \$10,000(FVD 10%, 5)= \$10,000(1.6105) = \$16,105

30. DISCOUNTING • PROCEDURE WHEREBY THE PRESENT VALUE OF FUTURE INCOME IS DETERMINED. • PRESENT VALUE OF A FUTURE PAYMENT • PRESENT VALUE OF A SERIES OF PAYMENTS

31. PRESENT VALUE OF A FUTURE PAYMENT PVO = FVN /(1+i)n OR PRESENT VALUE = FUTURE VALUE / (1 + COMPOUND RATE) CONVERSION PERIODS

32. Present Value of a Single Deposit (graphic) Assume that you need \$5,000in 2 years. Let’s examine the process to determine how much you need to deposit today at a discount rate of 7%. 0 12 7% \$5,000 PV0 PV1

33. Present Value of a Single Deposit (formula) PV0 = FV2 / (1+i)2 = \$5,000/ (1.07)2 = FV2 / (1+i)2 = \$4367.19 0 12 7% \$5,000 PV0

34. General Formula for Present Value PV0= FV1 / (1+i)1 PV0 = FV2 / (1+i)2 General Present Value Formula: PV0 = FVn / (1+i)n or PV0 = FVn (PVD i,n) -- See Table A2 etc.

35. Valuation Using Table A2 PVD i,nis found on Table A2

36. Using Present Value Tables PV2 = \$5,000 (PVD 7%,2) = \$5,000 (.873) = \$4365.00

37. PROBLEM: \$6298.56 DISCOUNTED @ 8% FOR 3 YEARS PVO = 6298.56/(1.08)3 PVO = 6298.56/(1.259712) PVO = 5000

38. Example Problem Julie Miller wants to know how large of a deposit to make so that the money will grow to \$10,000in 5 years at a discount rate of 10%. 0 1 2 3 4 5 10% \$10,000 PV0

39. Problem Solution • Calculation based on general formula: PV0 = FVn / (1+i)nPV0= \$10,000/ (1+ 0.10)5 = \$6,209.21 • Calculation based on Table A2: PV0= \$10,000(PVD 10%, 5)= \$10,000(.6209) = \$6,209.00

40. CALCULATOR • PV = PRESENT VALUE • FV = FUTURE VALUE • I/YR (I/Y) = INTEREST RATE OR DISCOUNT RATE • N = PERIODS • PMT = PAYMENTS • P/YR = PAYMENTS PER YEAR

41. Types of Annuities • An Annuity represents a series of equal payments (or receipts) occurring over a finite period of time. • Ordinary Annuity: Payments or receipts occur at the end of each period. • Annuity Due: Payments or receipts occur at the beginning of each period.

42. FORMULA TO CALCULATE THE FUTURE VALUE OF AN ORDINARY ANNUITY • FV = Pmt * [{(1+i)n – 1}/i] • EXAMPLE: • Pmt = \$1000/year • 40 years • 8% annual compounding

43. FV = 1000 * [{(1.08)40 – 1}/0.08] • FV = 1000 * [{(21.72452) – 1}/0.08] • FV = 1000 * [20.72452/0.08] • FV = 1000 * 259.05652 = 259,056.52

44. WITH THE CALCULATOR • PV = 0 • PMT = -1000 • I/Y = 8 • N = 40 • P/Y = 1 • FV= ? 259,056.52

45. ANNUITY VS. A PERPETUITY • AN ANNUITY IS A CONSTANT INCOME STREAM THAT CONTINUES FOR A FINITE PERIOD. • A PERPETUITY IS A CONSTANT INCOME STREAM THAT CONTINUES FOR A INFINITE PERIOD.

46. Examples of Annuities • Student Loan Payments • Car Loan Payments • Insurance Premiums • Mortgage Payments • Retirement Savings

47. AMORTIZED LOAN • A LOAN THAT IS REPAID IN A SERIES OF PAYMENTS THAT COVER INTEREST AND PRINCIPAL. • IN OTHER WORDS, EACH PAYMENT INCLUDES BOTH PRINCIPAL AND INTEREST. • MAY BE LEVEL PAYMENTS OR DECREASING PAYMENTS

48. FULLY AMORTIZED LOAN • ONE WHERE THE PERIODIC LOAN PAYMENTS ARE SUFFICIENT TO PAY OFF THE ENTIRE PRINCIPAL AMOUNT OF THE LOAN OVER THE TERM OF THE LOAN

49. PARTIALLY AMORTIZED LOAN • THE PERIODIC LOAN PAYMENTS MAKE SOME REDUCTION IN THE PRINCIPAL BALANCE BUT DO NOT FULLY PAY OFF THE ENTIRE PRINCIPAL OVER THE TERM OF THE LOAN

50. BALLOON PAYMENT • A LUMP SUM PAYMENT OF PRINCIPAL DUE AT THE END OF THE TERM OF THE LOAN. • REPRESENTS THE REMAINING UNPAID PRINCIPAL BALANCE.