© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible Web site, in whole or in part. 1 Chapter 3: The Time Value of Money Corporate Finance, 3e Graham, Smart, and Megginson

© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible Web site, in whole or in part. 2 The Role of Financial Markets  Voluntary transfer of wealth  Between individuals  Financial intermediaries  Across time  Future value  Present value  The chance to earn a return on invested funds means a dollar today is worth more than a dollar in the future.

© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible Web site, in whole or in part. 3 Future Value The Value of a Lump Sum or Stream of Cash Payments at a Future Point in Time FV n = PV  (1+r) n Future Value depends on: – Interest Rate – Number of Periods – Compounding Interval

© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible Web site, in whole or in part. 4 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.

© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible Web site, in whole or in part. 5 Present Value  Compounding:  Finding the future value of present dollars invested at a given rate  Discounting:  Finding the present value of a future amount, assuming an opportunity to earn a given return ( r ), on the money

© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible Web site, in whole or in part. 6 Present Value Today's Value of a Lump Sum or Stream of Cash Payments Received at a Future Point in Time FV n = PV  (1+r) n PV = FV n (1+r) n

© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible Web site, in whole or in part. 7 FV of a Mixed Stream  The 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

© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible Web site, in whole or in part. 8 FV of a Mixed Stream Equation Future value of an n-year mixed stream of cash flows (FV) can be expressed as

© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible Web site, in whole or in part. 9 FV of Annuities: Formulas  FV of an ordinary annuity:  FV of an annuity due:

© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible Web site, in whole or in part. 10 Present Value Of Perpetuity Stream of Equal Annual Cash Flows that Lasts Forever or

© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible Web site, in whole or in part. 11 Present Value of a Growing Perpetuity The Gordon Growth model:

© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible Web site, in whole or in part. 12 Compounding More Frequently than Annually Semiannually Quarterly Monthly Continuous

© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible Web site, in whole or in part. 13 Compounding Intervals m compounding periods per year

© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible Web site, in whole or in part. 14 Compounding More Frequently Than Annually – For quarterly compounding, m = 4: – For semiannual compounding, m = 2: FV at End of 2 Years of $125,000 Deposited at 5.13% Interest

© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible Web site, in whole or in part. 15 Continuous Compounding  Interest Compounded Continuously FV n = PV  e r  n FV at End of 2 Years of $125,000 at 5.13% Annual Interest, Compounded Continuously

© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible Web site, in whole or in part. 16 The Stated Rate Versus the Effective Rate Effective Annual Rate (EAR) – The annual rate actually paid or earned Stated Rate – The contractual annual rate ( r ) charged by lender or promised by borrower

© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible Web site, in whole or in part. 17 Calculating Deposits Needed To Accumulate A Future Sum  Often need to find annual deposit required to accumulate a fixed sum of money in n years  Closely related to the process of finding the future value of an ordinary annuity

© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible Web site, in whole or in part. 18 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.

© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible Web site, in whole or in part. 19 Implied Interest or Growth Rates: Lump Sums  Lump sums: The interest or growth rate of a single cash flow over time can be found by solving for r in the following equation:  If we know the interest rate ( r ), we can calculate the number of periods ( n ) necessary for a present value to grow to a desired future value

© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible Web site, in whole or in part. 20 Implied Interest or Growth Rates: Lump Sums Annuities and mixed streams: Very difficult to solve for r using formulas Use an iterative trial-and-error approach. Spreadsheets and financial calculators can do this very quickly. Often referred to as finding the yield to maturity or internal rate of return (IRR)

© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible Web site, in whole or in part. 21 Number of Compounding Periods Lump Sums/Annuities: