The Time Value of Money Learning objectives By

The Time Value of Money

Learning objectives By the end of this class you should be able to: Describe the concept of time value of money Calculate the future value of a single payment Calculate the present value of a single amount of money to be received in future Calculate the future value of annuities Calculate the present value of perpetuities 6 -2

What Companies Do Take the Money and … Park? Facing a huge projected 2009 budget deficit, Chicago Mayor Richard Daley struck a deal to lease the city’s 36, 000 parking meters to an investor group that included Morgan Stanley and its partners would pay Chicago $1. 2 billion up front; in return, they would collect revenue from parking meters for the next 75 years. Was it an unbelievable deal for the city of Chicago or for Morgan Stanley and its partners? To answer that question, you must know how to compare an up-front payment with a long-term stream of cash payments. This chapter will show you how to make that comparison. 6 -33

Time Value of Money Financial managers compare the marginal benefits and marginal cost of investment projects. Projects usually have a long-term horizon: timing of benefits and costs matters. Time-value of money: $1 received today worth > $1 received in the future. 6 -44

Future Value The value of an investment made today measured at a specific future date using compound interest. FVn = PV x (1+r)n Future Value depends on: Interest rate (r) Number of periods (n) Compounding interval (m) 6 -5

Simple interest calculation When you know the principal amount, the rate and the time, the amount of interest can be calculated by using the formula: Interest = P x r x t, where P = principal r = rate of interest t = time Example: Find the amount of interest if money is kept in the bank for 1 year and the bank pays an annual interest of 3%. Interest is only calculated on the principal 6 -6

Future Value of $200 for 4 years, 7% interest – compounding interest FV 4 = $262. 16 FV 3 = $245. 01 FV 2 = $228. 98 FV 1 = $214 PV = $200 0 1 2 3 End of Year 4 Compound interest: Interest earned both on the principal amount and on the interest earned in previous periods. 6 -7

Compounding Year 1: FV 1 = $214 Year 2: FV 2 = $228. 98 Year 3: FV 3 = $245. 01 Year 4: FV 4 = $262. 16 • Earns 7% interest on initial $200 • FV 1 = $200+$14 = $214 • Earn $14 interest again on $200 principal • Earns $0. 98 on previous year’s interest of $14: $14 x 7% = $0. 98 • FV 2 = $214+$0. 98 = $228. 98 • Earn $14 interest again on $200 principal • Earns $2. 03 on previous years’ interest of $28. 98: $28. 98 x 7% = $2. 03 • FV 3 = $228. 98+$14+$2. 03 = $245. 01 • Earn $14 interest again on $200 principal • Earns $3. 15 on previous years’ interest of $45. 01: $45. 01 x 7% = $3. 15 • FV 4 = $245. 01+$14+$3. 15 = $262. 16 6 -8

Future Value – Using Tables § § § FVn = PV (FVIFi, n) Where FVn = the future of the investment at the end of n year PV = the present value, or original amount invested at the beginning of the first year FVIF = Future value interest factor or the compound sum of $1 i = the interest rate n = number of compounding periods 6 -9

The Power of Compound Interest 20% 15% 10% 5% 0% Periods 6 - 10

Present Value The value today of a cash flow to be received at a specific date in the future, assuming an opportunity to earn interest at a specified rate. 6 - 11

Present Value of $200, 4 Years, 7% Interest Discounting 0 1 FV 1 = $200 PV = $186. 92 2 3 4 FV 2 = $200 FV 3 = $200 FV 4 = $200 End of Year PV = $174. 69 PV = $163. 26 PV = $152. 58 Discounting: The process of calculating present values. 6 - 12

Present Value – Using Tables PVn = FV (PVIFi, n) Where PVn = the present value of a future sum of money FV = the future value of an investment at the end of an investment period PVIF = Present Value interest factor of $1 i = the interest rate n = number of compounding periods 6 - 13

Present Value of One Dollar ($) The Power of Discounting 1. 00 0% 0. 75 0. 5 5% 0. 25 10% 15% 20% 0 2 4 6 8 10 12 14 16 18 20 22 24 Periods 6 - 14

Future Value of Annuities Annuity • A stream of equal periodic cash flows. 6 - 15

Future and Present Values of An Ordinary Annuity Compounding Future Value 0 $1, 000 1 2 3 $1, 000 4 $1, 000 5 End of Year Present Value Discounting 6 - 16

Future Value of an Ordinary Annuity 5 Years, 5. 5% Interest $1, 238. 82 $1, 174. 24 $1, 113. 02 $1, 055. 00 $1, 000. 00 0 $1, 000 1 2 3 4 $1, 000 5 End of Year Ordinary annuity: An annuity for which the payments occur at the end of each period. 6 - 17

FV Annuity example FVA = PMT (FVIFAi, n) What will $500 deposited in the bank every year for 5 years at 10% be worth? PMT = 500; i = 10%; n = 5 FVIFA = 6. 105 FVA = 500 X 6. 105 = 3, 052. 50 6 - 18

Present Value of Annuities 6 - 19

Present Value of An Ordinary Annuity 5 Years, 5. 5% Interest 0 1 2 3 4 $1, 000 5 $1, 000 End of Year $947. 87 $898. 45 $851. 61 $807. 22 $765. 13 6 - 20

PV of an Annuity – Using Table Calculate the present value of a $500 annuity received at the end of the year annually for 11 years when the discount rate is 9%. PVA = PMT (PVIFAi, n) = $500(6. 805) (From the table) = $3, 402. 5 6 - 21

Present Value of A Perpetuity • For a constant stream of cash flows that continues forever 6 - 22

Compounding More Frequently Than Annually • m compounding periods • continuous compounding • The more frequent the compound period, the larger the FV! 6 - 23

Compounding More Frequently Than Annually FV at end of 2 years of $125, 000 at 5% interest • Semiannual compounding: • Quarterly compounding: • Continuous compounding: 6 - 24

Stated Versus Effective Annual Interest Rates Stated annual rate • The contractual annual rate of interest charged by a lender or promised by a borrower. Effective annual rate • The annual rate of interest actually paid or earned, reflecting the impact of compounding frequency. 6 - 25

Stated rate vs. Effective rate Stated rate could be very different from the effective rate if compounding is not done annually. Example: $1 invested at 1% per month will grow to $1. 126825 ($1. 00 X 1. 0112) in one year. Thus even though the interest rate may be quoted or stated as 12% compounded monthly, the effective annual rate or APY is 12. 68%. 6 - 26

Stated rate versus Effective rate APY = (1 + quoted rate/m)m – 1 Where m = no. of compounding periods APY = (1 +. 12/12)12 – 1 = (1. 01)12 – 1 =. 126825 or 12. 6825% 6 - 27

Stated Versus Effective Annual Interest Rates Annual percentage rate (APR) • The stated annual rate calculated by multiplying the periodic rate by the number of periods in one year. Annual percentage yield (APY) • The annual rate of interest actually paid or earned, reflecting the impact of compounding frequency. • APY = EAR 6 - 28

Additional Applications of Time Value • Deposits needed to accumulate a future sum • Loan amortization • Implied interest or growth rates • Number of compounding periods 6 - 29

The Time Value of Money • Much of finance involves finding future and present values. • The time value of money is central to all financial valuation techniques. 6 - 30

End of Chapter