Chapter 3 Time Value of Money © 2007 Thomson South-Western Professor XXX Course name/number

2 The Role of Financial Markets Financial intermediates Competition and financial costs Corporate perspective –Required return (cost of capital, opportunity cost) The chance to earn a return on invested funds means a dollar today is worth more than a dollar (or euro, pound, franc, or yen) in the future

3 Future Value The Value of a Lump Sum or Stream of Cash Payments at a Future Point in Time FV n = PV x (1+r) n Future Value depends on: – Interest Rate – Number of Periods – Compounding Interval

4 Future Value of $100 (5 Years, 6% Interest )

5 Periods 0% Future Value of One Dollar ($) % 5% 15% 20% The Power Of Compound Interest 40.00

6 Future Value Two key points: –(1) the higher the interest rate, the higher the future value –(2) the longer the period of time, the higher the future value

7 Present Value Compounding: –find the future value of present dollars invested at a given rate Discounting –find the present value of a future amount, assuming an opportunity to earn a given return (r), on the money

8 Present Value Today's Value of a Lump Sum or Stream of Cash Payments Received at a Future Point in Time

9 Present Value of $1,700 (8 Years, 8% Interest )

10 The Power Of High Discount Rates – Present Value of $1 Periods Present Value of One Dollar ($) % 5% 15% 20% 0%

11 FV of a Mixed Stream Future value of any stream of cash flows measured at the end of a specified year is the sum of the future values of the individual cash flows at that year’s end. Sometimes called the terminal value

12 FV of a Mixed Stream Future value of an n-year mixed stream of cash flows (FV) can be expressed as or

13 FV of a Mixed Cash Flow Stream Invested at 9% Interest

14 FV of Annuities - Formulas FV of an ordinary annuity: FV of an annuity due:

15 FV of an Ordinary Annuity of $1,000 Per Year Invested at 7% Interest

16 FV of an Annuity Due of $1,000 Per Year Invested at 7% Interest

17 $1,000 $1,000 $1,000 $1,000 $1,000 End of Year $ $ $ $ $ Present Value of Ordinary Annuity (5 Years, 5.5% Interest Rate)

18 $1,000 $1,000 $1,000 $1,000 $1,000 End of Year $ $ $ $ Present Value of Annuity Due (5 Years, 5.5% Interest Rate)

19 FV and PV of Mixed Stream (5 Years, 4% Interest Rate) PV $5, $10,000 $3,000 $5,000 $4,000 $3,000 $2,000.0 Discounting End of Year FV $6,413.8 Compounding - $12,166.5 $3,509.6 $5,624.3 $4,326.4 $3,120.0 $4,622.8 $3,556.0 $2,564.4 $1,643.9 $2,884.6

20 Present Value Of Perpetuity Stream of Equal Annual Cash Flows That Lasts “Forever” or

21 Present Value Of a Growing Perpetuity The Gordon Growth model:

22 Present Value Of a Growing Perpetuity Growing Perpetuity CF 1 = $1,000 r = 7% per year g = 2% per year $1,000 $1,020 $1,040.4 $1,061.2 $1,082.4 …

23 Compounding More Often than Annually Semiannually Quarterly Continuous The more frequently interest compounds, the greater the amount of money that accumulates.

24 Compounding Intervals m compounding periods – For Semiannual Compounding, m = 2 – For Quarterly Compounding, m = 2

25 Compounding More Frequently Than Annually – For Quarterly Compounding, m Equals 4: – For Semiannual Compounding, m Equals 2: FV at End of 2 Years of $125,000 Deposited at 5.13% Interest

26 Continuous Compounding Interest Compounded Continuously FV n = PV x (e r x n ) FV at End of 2 Years of $125,000 at 5.13 % Annual Interest, Compounded Continuously FV n = $138,506.01

27 The Stated Rate Versus the Effective Rate Effective Annual Rate (EAR) – The Annual Rate Actually Paid or Earned Stated Rate – The Contractual Annual Rate Charged by Lender or Promised by Borrower

28 FV of $100 at End of 1 Year, Invested at 5% Stated Annual Interest, Compounded: –Annually: FV = $100 (1.05) 1 = $105 –Semiannually: FV = $100 (1.025) 2 = $ –Quarterly: FV = $100 (1.0125) 4 = $ The Stated Rate Versus The Effective Rate Stated Rate of 5% Does Not Change. What About the Effective Rate?

29 Continuous Compounding The stated and effective rates are equivalent for annual compounding. The effective annual rate increases with increasing compounding frequency.

30 Deposits Needed To Accumulate A Future Sum Often need to find annual deposit needed to accumulate a fixed sum of money in n years Closely related to the process of finding the future value of an ordinary annuity

31 Loan Amortization Common application of TV is finding loan payment amounts Amortized loans are loans repaid in equal periodic (annual, monthly) payments or

32 Loan Amortization Generalize the formula to more frequent compounding periods by dividing the interest rate by m and multiplying the number of compounding periods by m.

33 Implied Interest or Growth Rates Lump sums: the interest or growth rate of a single cash flow over time Annuities: interest rate associated with an annuity Mixed streams: very difficult using formulas or PV tables –Can be accomplished by an iterative trial-and-error approach –Often referred to as finding the yield-to-maturity or internal rate of return (IRR)

34 Number of Compounding Periods Calculate the unknown number of time periods necessary to achieve a given cash flow goal: –Lump Sums: If we know the PV, FV, and the interest rate (r), we can calculate the number of periods (n) necessary for a present amount to grow to equal the future amount. –Annuities: D etermine the unknown life of an annuity intended to achieve a specified objective, such as to repay a loan of a given amount with a stated interest rate and equal annual end-of-year payments.