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© Prentice Hall, 2000
1
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, 2000
2
Learning Objectives
Explain why money has time value and the importance of the interest rate in the valuation process.
Use the concepts of compound interest to determine the future value of both individual amounts as well as streams of payments.
Use discounting to determine the present value of both individual amounts as well as streams of payments.
© Prentice Hall, 2000
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Learning Objectives (Cont.) Explain how the concept of present value can
be used to value assets ranging from plant and equipment to marketable securities.
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.
Understand the concept of an investment’s net present value (NPV) and how it relates to the building of shareholder value.
© Prentice Hall, 2000
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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, 2000
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Future Value Formula
FV = PV [(1+ k)n]
Where:
k = the periodic interest rate
n = the number of periods
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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 $1,000.00 $60.00 $1,060.00
2 1,060.00 63.60 1,123.60
3 1,123.60 67.42 1,191.02
4 1,191.02 71.46 1,262.48
5 1,262.48 75.75 1,338.23
© Prentice Hall, 2000
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Future Value Interest Factor
Future Value Interest Factor = [(1+ k)n]
Where:
k = the periodic interest rate
n = the number of period
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Determinates of Future Value
Amount Invested
Interest Rate
Number of Compounding Periods
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Future Value of $1
1
2
3
4
5
6
7
0 5 10 15 20
r=10%
r=5%
r=3%r=1%
Period
Fut
ure
Val
ue
© Prentice Hall, 2000
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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, 2000
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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:
Fn,m = $1,000[1+(0.06/2)]5x2
= $1,000[1.3439] = $1,343.90
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12
Annual Percentage Rate (APR)
APR = FVIF k/m,m - 1
= [1+(k/m)m - 1]
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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
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Financial Calculator KeystrokesN or n = the number of periods interest is
compoundedI or I/Y = the periodic interest rateFV = the future value of a current or
present amountPV = the current or present value of a
future amountPMT = 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, 2000
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Financial Calculator Solutions -An Example
Future value of $1,000 earning 6 percent for 5 years
N I PV PMT FV
Inputs 5 6 1,000
Answer: 1,338.23
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Present Value Formula
FVPV =
(1+ k)n
Where: k = the discount rate
n = the number of years
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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, 2000
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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] = $9330.15
This is a “bad deal” since the present value of return in eight years is less than the cost of the land.
© Prentice Hall, 2000
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Calculator Solution
N I FV PMT PV
Inputs 8 10 20,000
Answer: 9,330.15
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Present Value Interest Factor (PVIF)
1PVIF =
(1+ k)n
Where: k = the discount rate
n = the number of years
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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(PVIF8,20) = $10,000(0.2145) = $2,145.00
© Prentice Hall, 2000
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Valuing a Zero-Coupon Bond - Calculator Solution
N I FV PMT PV
Inputs 20 8 10,000
Answer: 2,145.48
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23
Present Value of $1
0.00.10.20.30.40.50.60.70.80.91.0
0 10 20 30 40 50
Pre
sent
Val
ue
Period
© Prentice Hall, 2000
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The Discounting Period
When interest is compounded more than once a year, the present value is:
1PV = FV
(1+ k/m)nxm
Where: k = the discount raten = the number of yearsm = the number of times that
interest is paid a year
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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(PVIF4,40) = $10,000(0.2083) = $2,083.00
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Present Value of a Constant Perpetuity
CFPV =
k
Where: CF = Cash Flow per Period k = Opportunity Cost
of Money
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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:
£50Price=
0.10
= £ 500
© Prentice Hall, 2000
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Present Value of a Growing Perpetuity
CFPV =
(k - g)
Where:
CF = Cash Flow per Periodk = Opportunity Cost of Moneyg = Growth Rate per Period
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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:
$200,000Value =
(0.10 - 0.05)
= $4,000,000
© Prentice Hall, 2000
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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:
Annuity Due
Deferred Annuity
© Prentice Hall, 2000
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Present Value of Annuity (PVA)
Present Value Present Value Present Value
PVAn = of Payment + of Payment + + of Payment
in Period 1 in Period 2 in Period n
= PMT(PVIFk,1) + PMT(PVIFk,2) PMT(PVIFk,n)
© Prentice Hall, 2000
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Present Value Interest Factorof an Annuity (PVIFA)
(1+ k)n - 1PVIFA n,m =
k(1+ k)n
Where: k = the discount rate
n = the number of years
© Prentice Hall, 2000
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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, 2000
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Present Value of an Annuity - Solution
PV = PMT (PVIFA 6,10)
= $50,000 (4.35526)
= $217,763
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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 $200,000PMT = =
PVIFA 12,3 2.4018
= $83,269.80
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LOAN AMORTIZATION SCHEDULE
$200,000 LOAN @ 12 PERCENT INTEREST
Interest Principal Year-End
Year Payment Portion Repayment Balance
1 $83, 269.80 $24,000.00 $59,269.80 $140,730.20
2 83,269.80 16,887.62 66,383.20 74,348.00
3 83,269.80 8,921.76 74,348.04 ( 0.04 )
© Prentice Hall, 2000
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Future Value of an Annuity (FVA)
Future Value Future Value Future Value Future Value
FVAn = of Payment + of Payment + + of Payment of Payment
in Period 1 in Period 2 in Period n - 1 in Period n
FVAn = PMT(1+k)n–1 + PMT(1+k)n–2 + + PMT(1+k)1 + PMT
© Prentice Hall, 2000
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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, 2000
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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 = $1,166.40
2 1,000 x (1.08)1 = 1,080.00
3 1,000 x (1.08) = 1,000.00
$3,246.40
© Prentice Hall, 2000
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Future Value Interest Factor for an Annuity (FVIFA)
(1+k)n 1 FVIFAk,n =
k
Where: k = the discount rate
n = the number of years
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The Annuity Period
FVAnm = PMT [FVIFAk/m, nm]
PVAnm = PMT [PVIFAk/m, nm]
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Valuing Social SecuritySuppose 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, 2000
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Value of the Private Retirement Account
FVA = PMT[FVIFA0.50,480 ]
= $310 [1,991.49]
= $617,362.13
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VALUE OF $1,232 A MONTH
SOCIAL SECURITY PAYMENT
Life Expectancy Present Value of
Beyond Age 65 Social Security Benefits
Years (Months) Discounted @0.5 Percent
5 (60) $ 63,725.89
10 (120) 110,970.50
15 (180) 145,996.33
20 (240) 171,963.51
25 (300) 191,214.85
30 (360) 205,487.27
40 (480) 223,913.02
© Prentice Hall, 2000
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Present Value of Uneven Cash Flow Stream - Equipment Problem Revisited
After-Tax
Year Cash Flow X PVIF@10% = Present Value
1 $50,000 0.9091 $45,455.00
2 48,000 0.8264 39,667.20
3 45,000 0.7513 33,808.50
4 40,000 0.6830 27,320.00
5 35,000 0.6209 21,731.50
6 40,000* 0.5645 22,580.00
Total Present Value = $190,562.20
* Includes an estimated $10,000 salvage value
© Prentice Hall, 2000
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Present Value of Uneven Cash Flow Streams -
Valuing John Smoltz’s Contract
Year Payment X PVIF@8% = Present Value
1997 $7,000,000 0.9259 $6,481,481
1998 7,750,000 0.8573 6,644,376
1999 7,750,000 0.7938 6,152,200
2000 8,500,000 0.7350 6,247,754
Total Contract Value = $25,525,811
© Prentice Hall, 2000
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Net Present Value (NPV)
The difference between the present value of an investment’s cash flows and its cost.
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, 2000
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Determinants of the Opportunity Cost of Money
Risk
Inflation
Taxes
Maturity
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Risk
Default Risk
Price, or Variability Risk
Type of Claim
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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
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51
Treasury Bill Rates versus Inflation
-5.00
0.00
5.00
10.00
15.00
1955 1965 1975 1985 1995Real Rate
Inflation
Interest rate
%
Year
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Generalized Fisher Effect
1 + rh 1 + ih
= 1 + rf 1 + if
Where: rh = the home country interest rates
rf = the foreign currency interest rates
ih = the home country inflation rates
if = the foreign country inflation rates