chapter 4

© Prentice Hall, 2000

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Chapter 4

Foundations of Valuation:

Time Value

Shapiro and Balbirer: Modern Corporate Finance:

A Multidisciplinary Approach to Value Creation

Graphics by Peeradej Supmonchai


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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.


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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.


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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.


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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


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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


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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


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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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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.


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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?


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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.


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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




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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


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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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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


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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


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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


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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


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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)


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Present Value Interest Factorof an Annuity (PVIFA)

(1+ k)n - 1PVIFA n,m =

k(1+ k)n




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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?


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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 )


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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


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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?


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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


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Future Value Interest Factor for an Annuity (FVIFA)

(1+k)n 1 FVIFAk,n =

k




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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.


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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


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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


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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


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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.


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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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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

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