© Prentice Hall, Chapter 4 Foundations of Valuation: Time Value Shapiro and Balbirer: Modern Corporate Finance: A Multidisciplinary Approach to Value Creation Graphics by Peeradej Supmonchai

© Prentice Hall, Learning Objectives è Explain why money has time value and the importance of the interest rate in the valuation process. è Use the concepts of compound interest to determine the future value of both individual amounts as well as streams of payments. è Use discounting to determine the present value of both individual amounts as well as streams of payments.

© Prentice Hall, Learning Objectives (Cont.) è Explain how the concept of present value can be used to value assets ranging from plant and equipment to marketable securities. è Understand the difference between the stated and annual percentage rate (APR), and how this difference influences the present and future values of a stream of payments. è Understand the concept of an investment’s net present value (NPV) and how it relates to the building of shareholder value.

© Prentice Hall, Time Value of Money The time value of money is based on the simple idea that a dollar today is worth more than a dollar tomorrow. How much more depends on time preferences of individuals for consumption of goods and services, the rates of return that can be earned on available investments, and the expected rate of inflation.

© Prentice Hall, Future Value Formula FV = PV [(1+ k) n ] Where: k = the periodic interest rate n = the number of periods

© Prentice Hall, Future Value of $1,000 Investment ACCOUNT BALANCES FOR $1,000 INVESTMENT FIVE YEARS AT 6 PERCENT INTEREST BEGINNING INTEREST EARNED ENDING YEAR BALANCE DURING YEAR BALANCE 1 $1, $60.00 $1, , , , , , , , ,338.23

© Prentice Hall, Future Value Interest Factor Future Value Interest Factor= [(1+ k) n ] Where: k = the periodic interest rate n = the number of period

© Prentice Hall, Determinates of Future Value è Amount Invested è Interest Rate è Number of Compounding Periods

© Prentice Hall, Future Value of $1 r=10% r=5% r=3% r=1% Period Future Value

© Prentice Hall, Frequency of Compounding The future value in n years, when interest is paid m times a year is: F n,m = PV [ (1+k/m) nxm ] Where: k = the annual interest rate

© Prentice Hall, Frequency of Compounding - An Example Suppose you invested $1,000 for five years at a six percent interest rate. If interest were compounded semi- annually, the future value would be: F n,m = $1,000[1+(0.06/2)] 5x2 = $1,000[1.3439] = $1,343.90

© Prentice Hall, Annual Percentage Rate (APR) APR=FVIF k/m,m - 1 =[1+(k/m) m - 1]

© Prentice Hall, Annual Percentage Rate - An Example Suppose a U.S. corporate bond paying interest semiannually has a quoted rate of 9 percent. Its APR is: APR = [ (1.045) 2 ] -1 = 9.2 percent

© Prentice Hall, Financial Calculator Keystrokes N or n = the number of periods interest is compounded I or I/Y = the periodic interest rate FV = the future value of a current or present amount PV = the current or present value of a future amount PMT = the periodic payment or receipt. Used when dealing with a stream payments which are the same in each period. CPT = the “compute” button. Some calculators require that you hit this key prior to running a calculation.

© Prentice Hall, Financial Calculator Solutions - An Example Future value of $1,000 earning 6 percent for 5 years NIPVPMTFV Inputs 5 6 1,000 Answer:1,338.23

© Prentice Hall, Present Value Formula FV PV=  (1+ k) n Where: k = the discount rate n = the number of years

© Prentice Hall, Present Value - An Example Suppose you have the opportunity to buy a piece of land for $10,000 today, and sell it in eight years for $20,000. Is this a “good deal” if you can put your money in a risk-equivalent that is expected to earn 10 percent a year compounded annually?

© Prentice Hall, Present Value - An Example The present value of the $20,000 you expect to receive at the end of eight years is: PV = $20,000 [ 1/(1.10) 8 ] = $ This is a “bad deal” since the present value of return in eight years is less than the cost of the land.

© Prentice Hall, Calculator Solution NIFVPMTPV Inputs ,000 Answer:9,330.15

© Prentice Hall, Present Value Interest Factor (PVIF) 1 PVIF =  (1+ k) n Where: k = the discount rate n = the number of years

© Prentice Hall, Valuing a Zero-Coupon Bond Suppose that a zero-coupon bond matures in 20 years at a face value of $10,000. If an investor’s opportunity cost of money is 8 percent, the value of the bond would be: PV = FV(PVIF 8,20 ) = $10,000(0.2145) = $2,145.00

© Prentice Hall, Valuing a Zero-Coupon Bond - Calculator Solution NIFVPMTPV Inputs ,000 Answer:2,145.48

© Prentice Hall, Present Value of $1 Present Value Period

© Prentice Hall, The Discounting Period When interest is compounded more than once a year, the present value is: 1 PV=FV  (1+ k/m) nxm Where: k = the discount rate n = the number of years m = the number of times that interest is paid a year

© Prentice Hall, The Discount Period - An Example If you can earn 8 percent, compounded semiannually, the value of a zero- coupon bond maturing in 20 years at a face amount of $10,000 would be PV = FV(PVIF 4,40 ) = $10,000(0.2083) = $2,083.00

© Prentice Hall, Present Value of a Constant Perpetuity CF PV=  k Where: CF = Cash Flow per Period k = Opportunity Cost of Money

© Prentice Hall, Present Value of a Constant Perpetuity - An Example Suppose a console pays £50 a year, and the investor’s opportunity cost of money is 10 percent. The price of the console is: £50 Price=  0.10 = £ 500

© Prentice Hall, Present Value of a Growing Perpetuity CF PV=  (k - g) Where: CF = Cash Flow per Period k = Opportunity Cost of Money g=Growth Rate per Period

© Prentice Hall, Present Value of a Growing Perpetuity - An Example A firm’s cash flows are estimated to be $200,000 next year and are expected to grow at a five percent annual rate of return indefinitely. If the appropriate discount rate is 10 percent, the value of the firm is: $200,000 Value=  ( ) =$4,000,000

© Prentice Hall, Annuities An annuity is a series of equal cash flows per period for a specified number of periods. There are two basic kinds of annuities: è Annuity Due è Deferred Annuity

© Prentice Hall, Present Value of Annuity (PVA) Present Value Present Value Present Value PVA n = of Payment + of Payment +   + of Payment in Period 1 in Period 2 in Period n = PMT(PVIF k,1 ) + PMT(PVIF k,2 )    PMT(PVIF k,n )

© Prentice Hall, Present Value Interest Factor of an Annuity (PVIFA) (1+ k) n - 1 PVIFA n,m =  k(1+ k) n Where: k = the discount rate n = the number of years

© Prentice Hall, Present Value of an Annuity - An Example Suppose you are negotiating with a supplier to buy a piece of equipment that will reduce production costs. The after-tax savings are expected to be $50,000 a year for the next six years. How much is the equipment worth if your company’s opportunity cost of capital is 10 percent?

© Prentice Hall, Present Value of an Annuity - Solution PV = PMT (PVIFA 6,10 ) =$50,000 ( ) =$217,763

© Prentice Hall, Installment Payments on a Loan Suppose a small business borrows $200,000 from a bank at an interest rate of 12 percent compounded annually. The loan, including interest, is to be repaid in equal installments starting next year. The annual payments would be: $200,000 $200,000 PMT =  =  PVIFA 12, =$83,269.80

© Prentice Hall, LOAN AMORTIZATION SCHEDULE $200, PERCENT INTEREST Interest Principal Year-End Year Payment Portion Repayment Balance 1 $83, $24, $59, $140, , , , , , , , ( 0.04 )

© Prentice Hall, Future Value of an Annuity (FVA) Future Value Future Value Future Value Future Value FVA n = of Payment + of Payment +  + of Payment of Payment in Period 1 in Period 2 in Period n - 1 in Period n FVA n = PMT(1+k) n–1 + PMT(1+k) n–2 +   + PMT(1+k) 1 + PMT

© Prentice Hall, Future Value of an Annuity - An Example Suppose you were to receive $1,000 a year for three years, and then deposit each receipt in an account paying 8 percent interest, compounded annually. How much would you have at the end of three years?

© Prentice Hall, Future Value of an Annuity - Solution CALCULATING THE FUTURE VALUE OF A 3-YEAR ANNUITY Period Cash Flow Future Value 1 $1,000 x (1.08) 2 = $1, ,000 x (1.08) 1 = 1, ,000 x (1.08) = 1, $3,246.40

© Prentice Hall, Future Value Interest Factor for an Annuity (FVIFA) (1+k) n  1 FVIFA k,n =  k Where: k = the discount rate n = the number of years

© Prentice Hall, The Annuity Period FVA nm = PMT [FVIFA k/m, nm ] PVA nm = PMT [PVIFA k/m, nm ]

© Prentice Hall, Valuing Social Security Suppose you’re 25 years old and have just graduated with an engineering degree. You begin work for a company under a lifetime contract where your salary would remain unchanged at $30,000 a year until retirement in 40 years. Suppose that Social Security has been privatized, so that your 6.2 percent payment, plus the employers’ matching contribution can be put into a personal retirement account. With a salary of $30,000 a year, this means that $310 a month for 480 months will be put in an account earning 6 percent. You can also continue with the existing Social Security program, in which case $310/month would be sent to the government and credited to your account.

© Prentice Hall, Value of the Private Retirement Account FVA=PMT[FVIFA 0.50,480 ] = $310 [1,991.49] = $617,362.13

© Prentice Hall, VALUE OF $1,232 A MONTH SOCIAL SECURITY PAYMENT Life Expectancy Present Value of Beyond Age 65 Social Security Benefits Years (Months) Percent 5 (60) $ 63, (120) 110, (180) 145, (240) 171, (300) 191, (360) 205, (480) 223,913.02

© Prentice Hall, Present Value of Uneven Cash Flow Stream - Equipment Problem Revisited After-Tax Year Cash Flow X = Present Value 1 $50, $45, , , , , , , , , ,000* , Total Present Value = $190, * Includes an estimated $10,000 salvage value

© Prentice Hall, Present Value of Uneven Cash Flow Streams - Valuing John Smoltz’s Contract Year Payment X = Present Value 1997 $7,000, $6,481, ,750, ,644, ,750, ,152, ,500, ,247,754 Total Contract Value = $25,525,811

© Prentice Hall, Net Present Value (NPV) è The difference between the present value of an investment’s cash flows and its cost. è Measures how much better off we’ d be by taking on the investment. If the discount rate used in calculating present values represents the stockholders opportunity cost of money, taking on positive NPV projects will create shareholder value.

© Prentice Hall, Determinants of the Opportunity Cost of Money è Risk è Inflation è Taxes è Maturity

© Prentice Hall, Risk è Default Risk è Price, or Variability Risk è Type of Claim

© Prentice Hall, Expected Inflation - The Fisher Effect r = a + i + ai Where: r = the nominal interest rate a = the real or inflation-adjusted interest rate i = the expected rate of inflation

© Prentice Hall, Treasury Bill Rates versus Inflation Real Rate Inflation Interest rate % Year

© Prentice Hall, Generalized Fisher Effect 1 + r h 1 + i h  =  1 + r f 1 + i f Where: r h = the home country interest rates r f = the foreign currency interest rates i h = the home country inflation rates i f = the foreign country inflation rates