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# Understanding and Benefiting from the Time Value of Money

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1. Understanding and Benefiting from the Time Value of Money

2. Contents • Introductory applied questions • Basic formulas and computations • Present Value and Future Value • Annuities • Practical applications of Time Value • Practice problems • Website links

3. Money is worth more as time progresses • An amount received today is worth more than that same amount received in the future.

4. The future value of a dollar and the present value of a dollar are based on the interest rate earned and how many years and compounding times per year are involved.

5. The time value of money is all about knowing which opportunity creates the maximum value over a given period of time.

6. Compounding • Simply refers to interest earning interest • Significantly influences how fast a dollar amount grows • May be computed annually or more frequently, such as quarterly, monthly, daily, or continually

7. If you invested \$1,000 for one year at 10%, how much would you have at the end of a year?

8. \$1,000 x 10% = \$100 interest earned • Add \$1000 principle to \$100 interest • Balance at end of first year is \$1,100 • Shortcut: \$1,000 x 1.10 = \$1,100 Thus, Present value (1 + interest rate) = Future Value

9. Shortcut • Check out a FVIF table and get factor for 1 year and 10% interest. Multiply this by the principle (PV) amount. Thus the FV = PV x FVIF

10. Future Value Interest Factors (1+I)n = FVIF

11. If you left that \$1,000 for two years, what would it be worth? \$1,100 x 1.10 = \$ 1,210 1,100 x 1.102 = \$ 1,210 or 1, 000 x 1.21 = \$ 1,210 • Thus, Future Value = PV x (1+ Interest rate)number years

12. Abbreviations • PV = Current/earliest lump sum amount • FV = Ending/latest lump sum amount • I = Interest rate earned or paid • N = number of years/compounding periods • FVIF = future value interest factor = (1+I)N

13. FV = PV (1+I)n Example: Invest \$500 for 4 years at 8 percent rate of return, how much will you have at the end of the fourth year?

14. Solution Using Formula & Table • FV = PV (1 + I )n • FV = PV (FVIF i, n) • FV = \$500 (1 + .08)4 = 500 (1.360) FV = \$ 680 With Calculator \$500 PV 4 n 8 I FV = \$680

15. If you are 20 years old and today invest \$1000 dollars at 7%, what will your \$1,000 be worth when you are 65 years old?

16. Solution • What is the future value of \$1000 today if invested for 45 years at 7% annual interest rate? • PV = \$1,000 • i = 7 (the I/YR key) • N = 45 • Solve for FV FV = \$ 21,002

18. Trouble Shooting • Only a few mistakes are common when computing time value of money problems on an HP 10B calculator. • Before starting a problem, make sure to check for the following: • How many payments is your calculator set on. To check, press shift (the yellow key), then C ALL (Input on the older HP calculator). • If a number other than what is needed appears, the number of payments may be changed by pressing the correct number of payments, then shift then PMT

19. Trouble Shooting (cont.) • Check to see whether your calculator is in the “begin” mode. This means that payments are assumed to be made at the beginning of the time period rather than at the end of the time period, which is typically when payments are considered. If begin appears on your screen when it is not needed, press shift then BEG/END to change back to the end of the time period.

20. Trouble Shooting (cont.) • ALWAYS clear the machine by pressing SHIFT then C ALL. Never presume the bottom left “C” button clears everything. It doesn’t. • When using more than one compounding period per year (as in monthly payments on a 5-year loan), press the number of years, then SHIFT then N. (eg., 5 SHIFT N after having the number of payments set on 12 per year - - 12 SHIFT PMT) • To change the number of decimal places shown to say 6 places, press SHIFT, DISP (=) 6

21. Now, let’s try another problem…

22. If you wait until you are 35 years old and invest \$1000 dollars at 7%, what will your \$1,000 be worth when you are 65 years old?

23. Solution • \$7,612 • 1000 PV • 7 I • 30 N • FV = 7,612.25

24. Hint: • Any of these variables may be computed just as easily as FV, such as a PV amount, an interest rate, or the number of years/payments, etc.

25. Present Value Factors • The reciprocal of the Future Value Factors • PVIF = present value interest factor = 1/(1+I)N

26. Present Value Interest Factors

27. Present Value Example You will need \$8,000 in 5 years to make a down payment on a house. How much would you need to put aside today to achieve your goal if you are earning 6% on your money?

28. Solution • \$8,000 FV • 5 N • 6 I • PV = \$5,978 , i.e., is needed today to have the \$8,000 in five years.

29. Problem • If you invest \$10,000 at 6% annual interest, how long will it take your money to double?

30. Solution • PV \$10,000 • FV 20,000 • I 6 • N 12 years (11.9 years)

31. Rule of 72 • Any two numbers when multiplied together give you 72, will indicate the annual interest rate or the length of time until a sum doubles. For example, • 2 x 36 = 72 3 x 24 = 72 • 4 x 18 = 72 6 x 12 = 72….. • Thus, at 2% it takes 36 years for a sum to double. Likewise, at 36%, it takes 2 years for a sum to double. • Eg., in the previous example, 6 x 12 = 72

32. Annuities • Now, let’s consider making an investment or payment more than just one time. • Multiple payment situations are generically referred to as annuities. • Payments may be made annually (once a year) or more frequently • If payments are made more than once a year, be sure the calculator knows the payment frequency by pressing the number of payments per year, then SHIFT then PMT

33. Practice Problem If you are 25 years old and invest \$1000 dollars each year at 7%, what will your investments be worth when you are 65 years old? If you waited until you were 35 to start making those payments, how much would you have at age 65?

34. Solutions • Starting at age 25 \$1000 PMT 7 I 40 N FV \$199,635 • Starting at age 35 \$1000 PMT 7 I 30 N FV \$94,461

35. Car Payments • You want to borrow \$15,000 for a new car and make monthly payments for 2 years. If the bank will lend you the money for 8%, how much will your monthly payments be? • Hint: Solve for PMT rather than PV • Must set calculator on 12 payments per year (press 12 shift pmt)

36. Solution to Car Payments • \$15,000 PV • 8 I • 2 shift N (will assume 24 pmts) PMT ?

37. Car Loan Alternatives • What would your monthly payments be if you were charged 10% interest? • \$692.17 • What would your monthly payments pay if you borrowed the money for 3 years at 8%? • \$470.04

38. Mortgages • If you were to borrow \$100,000 for a house, consider the following alternatives: • 15 years at 7.5 % • 30 years at 8%

39. Mortgage Solutions • 15-year note • 100,000 PV • 7.5 I • 15 shift N (@12 pmts per year) • PMT 927.01 • 30-year note • \$100,000 PV • 8.0% I • 30 shift N • @12 pmts per year • PMT \$733.76

40. How much more would you pay for the house over the life of the loan if you chose the 30-year note rather than the 15-year note?

41. Solution: 15 vs. 30-year notes 15-year note PMT \$ 927.01 TOTAL PAID: (PMT x 12 x 15) = \$ 166,862 30-year note PMT \$733.76 TOTAL PAID: = \$ 264,154 Difference: \$97,688

42. Amortization • The depleting or repaying of borrowed funds • Term also used to mean the “using up of an asset” • May be used to determine the balance of a loan at any time

43. Amortization Example • On the 15 and 30-year mortgage examples, what would the ending balances be at the end of the 1st year of each loan? • HINT: Solve for the payment, then press shift AMORT (FV) then =. You will then see the time period for which the amortization will be given (eg., period 1-12 for the first year when there are monthly payments). Then press = to get the amount of interest paid during the first year; press = again to get the amount of principal paid, then = again to get the ending balance after the first year)

44. Amortization Answer 15- year loan 1st year’s payments: (\$11,124.15) Interest \$7372.79 Principle \$3,751.36 Balance \$ 96,248.64 30-year loan 1st year’s payments: (\$8,805.17) Interest \$ 7,969.80 Principle \$ 835.37 Balance \$ 99,164.63

45. Ordinary Annuity vs. Annuity Due • When payments are made on a regular (say, annual) basis into an investment, the timing of the payments is important. • If an amount is paid each year into a fund earning interest, the present or future value of that investment fund is determined by whether the payment is made at the beginning or the end of each year.

46. Ordinary Annuity vs. Annuity Due (cont.) • In a typical situation (ordinary annuity), the PV or FV is computed assuming the payments are all made at the end of each period. • 3 Questions:

47. Ordinary vs. Annuity Due Examples • If payments of \$1000 are made each year for 10 years at 12%, how much will be accumulated at the end of the 10th year? • How much would be accumulated if the payments were made at the beginning of each year? • How much would you pay today for an annuity that paid you \$1000 a year for 10 years if interest rates were 12%? (assume payments at end of period)

48. Annuity Solutions Ordinary annuity \$1000 pmt (1/year) 10 n 12 I FV \$17,548.74 PV 5,650.22 Annuity Due Turn on Begin (shift BEG/END) \$1000 PMT 10 n 12 I FV \$19,654.58 PV 6,328.25

49. Practice Problems

50. Oliver will need \$15,000 for a down payment on a house, and has \$10,000 in savings (in a certificate of deposit - CD) earning 5.5%. How long will it take Oliver to buy a house?