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Understanding and Benefiting from the Time Value of Money 1

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

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

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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. 4

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

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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 6

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

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$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 8

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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 9

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Future Value Interest Factors (1+I) n = FVIF Periods1%3%6%8%10%

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

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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 12

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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? 13

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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 $500PV 4n 8I FV = $680 14

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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? 15

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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 FVFV = $ 21,002 16

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If you didnt get the correct answer, perhaps the next three slides will help you 17

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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 18

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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. 19

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Trouble Shooting (cont.) ALWAYS clear the machine by pressing SHIFT then C ALL. Never presume the bottom left C button clears everything. It doesnt. 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 SHIFT PMT) To change the number of decimal places shown to say 6 places, press SHIFT, DISP (=) 6 20

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Now, lets try another problem… 21

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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? 22

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Solution $7, PV 7I 30N FV= 7,

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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. 24

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

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Present Value Interest Factors Periods1%3%6%8%10%

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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? 27

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

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

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Solution PV$10,000 FV 20,000 I6 N12 years (11.9 years) 30

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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 = 723 x 24 = 72 4 x 18 = 726 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 31

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Annuities Now, lets 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 32

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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? 33

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

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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) 35

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

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

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

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Mortgage Solutions 15-year note 100,000PV 7.5I 15shift N pmts per year) PMT year note $100,000 PV 8.0%I 30shift pmts per year PMT$

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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? 40

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

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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 42

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Amortization Example On the 15 and 30-year mortgage examples, what would the ending balances be at the end of the 1 st 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) 43

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Amortization Answer 15- year loan 1st years payments: ($11,124.15) Interest $ Principle $3, Balance $ 96, year loan 1st years payments: ($8,805.17) Interest $ 7, Principle $ Balance $ 99,

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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. 45

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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: 46

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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) 47

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

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Practice Problems 49

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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? 50

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What equal annual payments are needed to repay $35,000 of student loans over eight years, if the interest rate being charged is 9%? (omit any retroactive interest charges) 51

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Maura has just graduated from the university and will start earning $24,000 per year. In 10 years, what will she need to earn to maintain the same purchasing power if inflation is 3% per year? 52

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Would you lend anyone money for 30 years expecting to receive no interest? If so, what would you pay today for a 30-year zero coupon $100,000 bond? Hint, pick an interest rate that you need to receive over the years, say 8%. 53

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If a credit company charges 12% per year on monthly accounts of its customer loans, what is the effective annual interest rate? 54

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At a 15% cost of money, which is more valuable, a) an annuity of $2000 per year for 10 years, b) a lump sum of $15,000 now, or c) a lump sum of $35,000 at the end of 10 years? 55

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Louisa and Michael just got married and received $11,000 cash in wedding gifts. If they place this money in a mutual fund earning 11%, how much will they have on their first anniversary? Their 10 th anniversary? Their 25 th anniversary? Their 50 th anniversary? 56

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It is your 18 th birthday and you have just won a lottery worth $250,000. However, you may not receive any funds until you are 21. If your windfall earns 8% between now and your 21 st birthday, how much will you receive on you 21 st birthday? 57

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If you are 21 and start depositing $2,000 each year into an IRA which earns 9% interest, how much will you have accumulated at age 65? 58

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What return are you receiving on a Series EE bond that you bought in 1999 and that will mature in 2016? Hint: a Series EE sells for half of par and pays no annual interest 59

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Daniel Paul, age 1, just inherited $50,000 from his great grandmother. If this amount is invested for him at 13%, how long will it take the little tot to become a millionaire? 60

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Charles Casco has just retired at age 65. He has $700,000 accumulated in his pension plan and would like to withdraw it in equal annual dollar amounts so that he has withdrawn it all by age 85. How much should he withdraw each year until age 85 assuming 7% interest? 61

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. You plan to make the following deposits to save money to buy a car in five years. At 9% interest, find the value of your deposits at the end of the 5th years: You deposit $100 at the end of the 1 st year You deposit $200 at the end of the next 2 years You deposit $500 at the end of the 4 th year You deposit $1000 at the end of the 5 th year 62

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Capital Budgeting Problem A new robot for your factory will cost you $45,000 but will save you variable/manual labor costs of $14,000 per year for 10 years. What is the present value of the investment if you must pay 8.5% for money to buy the robot? 63

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Suppose Olivia deposits $10,500 into a 403B plan at the end of each of the next 25 years, what will her investment be worth in 25 years if her deposits earn 9% each year? What would this investment be worth at the end of the 25 th year if she makes the deposits at the beginning of the year instead? 64

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What is the most that you should pay for a 30-year zero coupon bond if you are earning 12% on the money you would use to buy the bond? 65

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If you buy an annuity today for $15,000, it will pay you $500 per month for 10 years, what rate of return are you earning on your money? 66

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Applicable Links On-line calculators Mutual Funds and Misc. Investments Stock Market Money rates Mortgage and Real Estate Economic Data Insurance Quotes Tax Services 67

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On-line Calculators

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Mutual Funds and Miscellaneous Investments

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Stock Market

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Money Rates

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Mortgage and Real Estate

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Economic Data

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Insurance Quotes

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Tax Services

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