analytical techniques in financial management

ANALYTICAL TECHNIQUES IN

FINANCIAL MANAGEMENT

Riaz Hussain

Kania School of Management University of Scranton

Scranton, Pennsylvania

©

July 2012

ii

ANALYTICAL TECHNIQUES IN

FINANCIAL MANAGEMENT

Riaz Hussain

Kania School of Management University of Scranton

Scranton, Pennsylvania

©

July 2012

iii

This copy of

Analytical Techniques in Financial Management

by

Riaz Hussain

is for your personal use only.

You cannot give it, or sell it,

to anyone else in any form,

printed or electronic.

iv

PREFACE

Intended for MBA students taking the required course in managerial finance, this book

presents the fundamental principles of finance in a cohesive form. It emphasizes the need

for an analytical solution to a financial decision-making problem. It covers the main

topics for the long-term financial management of a firm. However, we can use the basic

ideas developed here to manage the current assets of a firm successfully.

Analytical Techniques is a workbook designed to help students understand the basic ideas

in finance, and to apply them in solving practical problems. The book develops and

applies different analytical techniques, such as discounting, net present value, and

probability models, in the financial decision-making. Students are encouraged to learn

Excel or Maple as an analytical tool and apply it in solving financial problems. They

should go through all the examples, and work out the problems.

Throughout the book, the emphasis is on the usefulness of the fundamental concept of net

present value. The capital asset pricing model analyzes the risk and return for a project.

The option pricing theory is now a part of many valuation models.

The students may use a standard textbook of finance, such as Corporate Finance, by

Ross, Westerfield, and Jaffe for supplementary reading.

I gratefully acknowledge the helpful suggestions of many students who worked their way

through the earlier versions of this text. Their critical feedback was essential in

completing this book.

Riaz Hussain

http://www.amazon.com/Corporate-Finance-McGraw-Hill-Insurance-Estate/dp/007733762X

v

contents

Chapter Topic Page

1. Finance and Analytical Tools 1

2. Time Value of Money 18

3. Valuation of Debt and Equity 40

4. Capital Budgeting under Certainty 62

5. Capital Budgeting under Uncertainty 83

6. Portfolio Theory 109

7. Capital Asset Pricing Model 120

8. Option Pricing Theory 137

9. Cost of Capital 156

10. Capital Structure Theory: Value Maximization 174

11. Capital Structure Theory: Minimizing WACC 198

12. Dividend Policy 208

13. Leasing 219

14. Investment Analysis 231

15. Review Problems 248

16. Formulas and Tables 260

1

1. Finance and Analytical Tools

Objectives: After reading this chapter, you will be able to

1. Get an overview of finance and basic algebra.

2. Use geometric series in financial calculations.

3. Understand the basic concepts of statistics.

4. Use Excel, Maple, or WolframAlpha to solve mathematical problems.

1.1 Field of Finance

When you look at the balance sheet of a company, you will see the assets and liabilities

are categorized as long-term, or short-term. In this course, we are dealing with the

management of long-term assets (machinery, equipment, buildings, land, etc.) and long-

term liabilities (bonds) and other long-term financing (stocks, preferred stock). Another

course, FIN 361, Working Capital Management, deals with the management of current

assets (cash, marketable securities, accounts receivable, and inventories) and current

liabilities (accounts payable, short-term financing, and accruals).

The following diagram outlines the relationship between the short-term and the long-term

assets and liabilities.

Assets Liabilities and Equity

FIN 361

Current Assets Cash

Marketable Securities

Accounts Receivable

Inventories

Current Liabilities Accounts Payable

Accruals

Notes Payable

Short

term

FIN 508

Long-term Assets

Plant and Equipment

Less Accumulated Depreciation

= Net Plant and Equipment

Long-term Liabilities Long-term bonds

Owners' Equity Common Stock

Preferred Stock

Retained Earnings

Long

term

Total Assets are equal to

Total Liabilities and Equity

First, one should learn the basic financial principles, such as time value of money, risk,

options, cash flows, and the measurable quantities such as the stock price, earnings per

share, debt ratio, etc. FIN 508, explains these basic ideas, and their application to long-

term financial management of a company.

You may say, “If I were a CEO, I would monitor spending daily.” You will be focusing

on the tan part of the diagram, which is a valid activity. You will monitor cash, accounts

receivables, inventory, and the daily cash flows. However, you have to know the whole

http://www.wolframalpha.com/

http://en.wikipedia.org/wiki/Balance_sheet

http://en.wikipedia.org/wiki/Bond_(finance)

http://en.wikipedia.org/wiki/Stock

http://en.wikipedia.org/wiki/Preferred_stock

http://134.198.33.115/hussain/fin361syl.htm

http://en.wikipedia.org/wiki/Marketable_securities

http://en.wikipedia.org/wiki/Accounts_receivable

http://en.wikipedia.org/wiki/Inventory

http://en.wikipedia.org/wiki/Accounts_payable

http://en.wikipedia.org/wiki/Accruals

Analytical Techniques 1. Finance and Analytical Tools _____________________________________________________________________________

2

picture. The most important decisions are made in the blue part of the diagram. Here you

are raising and investing millions, perhaps billions, of dollars. The recent IPO of

Facebook (May 18, 2012) is a good example of this activity. If you make an error, it may

turn out to be a big mistake.

When you are making a financial decision at a company, you cannot rely entirely on gut

feeling, just because you know your business. You have to do some homework first and

find the right course of action. If you are presenting your case to a board of directors, or

in a court of law, you have to convince the audience with facts and figures. We are

learning to do that here in this course. The relationship between different finance courses

is as follows.

→ FIN 582 Advanced Financial Management → FIN 583 Investment Analysis MBA 503C → FIN 508 → FIN 584 International Finance → FIN 585 Derivative Securities → FIN 586 Portfolio Theory

1.2 Problem Solving

One can learn finance efficiently by learning to solve financial problems analytically.

This textbook has plenty of problems, many of them are solved examples and the others

are the end-of-the-chapter exercises. There are two ways to look at any homework

problem. The first one is the quick one:

Homework Problem → Formula → Answer

Some students have the temptation to solve the problem quickly without understanding

the concept that the problem is supposed to develop. They miss the real purpose of the

exercise, which is to consolidate an idea and observe its application.

There is another way to look at the same problem.

Is there a similar problem,

worked out in the text,

which can be applied

here?

↓

Study the material in

the text to learn new

concepts

→

Study the worked out

examples to see how the

concepts can be applied to

practical problems

→

Homework Problem

→

Solution

↓ ↑ ↑

Concepts, such as

risk, time value of

money, required rate

of return, options,

CAPM

→

Can I write these concepts

in the form of

mathematical equations?

Is it possible to break this

problem into a set of

simpler problems, solve

them, and recombine

them?

http://en.wikipedia.org/wiki/Facebook

http://www.scranton.edu/faculty/hussain/teaching/mba503_online.shtml

http://www.scranton.edu/faculty/hussain/teaching/fin582.shtml





3

The second method is obviously more cumbersome, but it helps the student understand

the material.

Before we actually start studying finance and the financial management as a discipline, it

is worthwhile to review some of the fundamental concepts in mathematics first. This will

help us appreciate the usefulness of analytical techniques as powerful tools in financial

decision-making. We shall briefly review elementary algebra, basic concepts in statistics,

and finally learn Excel or Maple as a handy way to cut through the mathematical details.

Our approach toward learning finance is to translate a word problem into a mathematical

equation involving some unknown quantity, solve the equation and get the answer. This

will help us determine an exact answer, rather than just an approximation. This will lead

to a better decision.

1.3 Video 01A, Linear Equations

To review the basic concepts of algebra, we look at the simplest equations first, the linear

equations. These equations do not have any squares, square roots, or trigonometric or

other complicated mathematical functions.

Example

1.0. Suppose John buys 300 shares of AT&T stock at $26 a share and pays a commission

of $10. When he sells the stock, he will have to pay $10 in commission again. Find the

selling price of the stock, so that after paying all transaction costs, John’s profit is $200.

Let us define profit π as the difference between the final payoff F, after commissions, and

the initial investment I0, including commissions. We can write it as a linear equation as

follows

π = F − I0

We require a profit of $200, thus, π = 200. Suppose the final selling price of the stock per

share is x, the number we want to calculate. The initial investment in the stock, including

commission, is I0 = 300(26) + 10 = $7810. Selling 300 shares at x dollars each, and

paying a commission of $10, gives the final payoff as, F = 300x − 10. Make these

substitutions in the above equation to obtain

200 = 300x − 10 − 7810

Moving things around, we get 200 + 10 + 7810 = 300x

Or, 8020 = 300x

Or, x = 8020/300 = 26.73333333 $26.73 ♥

http://en.wikipedia.org/wiki/Linear_equations

http://en.wikipedia.org/wiki/Linear_equations

http://www.scranton.edu/faculty/hussain/camtasia/Module01A/Module01A.html


4

This means that the stock price should rise to $26.73 to get the desired profit. Note that

the answer has a dollar sign and it is truncated to a reasonable degree of accuracy,

namely, to the nearest penny.

Next, consider a somewhat more complicated problem involving dollars, doughnuts, and

coffee.

1.1. Jane works in a coffee shop. During the first half-hour, she sold 12 cups of coffee

and 6 doughnuts, and collected $33 in sales. In the next hour, she served 17 cups of

coffee and sold 8 doughnuts, for which she received $46. Find the price of a cup of coffee

and that of a doughnut.

This is an example where we have to find the value of two unknown quantities. The

general rule is that you need two equations to find two unknowns. We have to develop

two equations by looking at the sales in the first half-hour and in the second hour.

Suppose the price of a cup of coffee is x dollars, and that of a doughnut y dollars.

First half-hour, 12 cups and 6 doughnuts for $33, gives 12 x + 6 y = 33

Second hour, 17 cups and 8 doughnuts for $46, gives 17 x + 8 y = 46

Now we have to solve the above equations for x and y.

First, try to eliminate one of the variables, say y. You can do this by multiplying the first

equation by 8 and the second one by 6, and then subtracting the second equation from the

first. This gives

8*12 x + 8*6 y = 8*33

6*17 x + 6*8 y = 6*46

Subtracting second from first, (8*12 – 6*17) x = 8*33 – 6*46

Simplifying it, – 6 x = – 12, or x = 2 ♥

Substituting this value of x in the first equation, we have 12*2 + 6 y = 33

Or, 6y = 33 – 24 = 9

y = 9/6 = 3/2 ♥

The answer is that a cup of coffee sells for $2 and a doughnut for $1.50.

1.3 WolframAlpha

Mathematica is a useful analytical software, which has capabilities similar to Maple. It

can perform all the mathematical problems equally well. Mathematica has a website at

WolframAlpha, which is free to use. The instructions at WolframAlpha are almost

identical to those in Maple. You should explore this website and use it when you do not

http://www.wolfram.com/products/student/mathforstudents/index.html

http://www.wolfram.com/products/student/mathforstudents/index.html




5

have access to Maple. In this text, you can access WolframAlpha by clicking on this

button WRA.

For instance, to solve the equations in example 1.1

12 x + 6 y = 33

17 x + 8 y = 46

for x and y, enter the instructions as follows:

WRA 12x+6y=33,17x+8y=46

It provides the solution as x = 2, y = 3/2.

To see the sine wave of Figure 1.1, write

WRA Plot[Sin[x],{x,0,2Pi}]

1.2 Non-linear Equations

Non-linear equations contain higher powers of the unknown variable, or the variable

itself may show up in the power of a number. For instance, a quadratic equation is a non-

linear equation. The general form of a quadratic equation is

ax2 + bx + c = 0 (1.1)

The roots of this equation are

x = − b b

2 − 4ac

2a (1.2)

Consider the following examples of non-linear equations.

Examples

1.2. Solve for x: 1.113x = 2.678

First, we recall the basic property of logarithm functions, namely,

ln(ax) = x ln a

Taking the logarithm on both sides of the given equation, we obtain

x ln(1.113) = ln(2.678)

Or, x = ln(2.678)

ln(1.113) =

0.9850702

0.1070591 = 9.201 ♥






6

You can save some time by doing the calculation at WolframAlpha as follows:

WRA 1.113^x = 2.678

1.3. Solve for x, (2 + x)2.11

= 16.55

This gives 2 + x = (16.55)1/2.11

Or, x = (16.55)1/2.11

– 2 = 1.781 ♥

You can verify the answer at WolframAlpha as follows:

WRA (2+x)^2.11=16.55

1.4. Find the roots of 5x2 + 6x − 11 = 0

This is a typical quadratic equation. Use equation (1.2) and put a = 5, b = 6, and c = – 11.

This gives

x = –6 36 – 4(5)(–11)

10 =

–6 256

10 =

–6 16

10 = −

11

5 or 1 ♥

You can verify the answer at WolframAlpha as follows:

WRA 5*x^2+6*x-11=0

1.3 Geometric Series

In many financial management problems, we have to deal with a series of cash flows.

When we look at the present value, or the future value, of these cash flows, the resulting

series is a geometric series. Thus, geometric series will play an important role in

managing money. Let us consider a series of numbers represented by the following

sequence

a , ax , ax2 , ax

3 , ... , ax

n−1

The sequence has the property that each number is multiplied by x to generate the next

number in the list. There are altogether n terms in this series, the first one has no x, the

second one has an x, and the third one has x2. By this reasoning, we know that the nth

term must have xn1 in it. This type of series is called a geometric series. Our concern is

to find the sum of such a series having n terms with the general form

S = a + ax + ax2 + ax

3 + ... + ax

n−1 (1.3)

To evaluate the sum, proceed as follows. Multiply each term by x and write the terms on

the right side of the equation one-step to the right of their original position. We can set up

the original and the new series as follows:








7

S = a + ax + ax2 + ax

3 + ... + ax

n−1

xS = ax + ax2 + ax

3 + ... + ax

n−1 + ax

n

If we subtract the second equation from the first one, most of the terms will cancel out,

and we get

S − xS = a − axn

Or, S(1 − x) = a(1 − xn)

or, Sn = a (1 − x

n)

1 − x (1.4)

This is the general expression for the summation of a geometric series with n terms, the

first term being a, and the ratio between the terms being x. This is a useful formula,

which we can use for the summation of an annuity.

If the number of terms in an annuity is infinite, it becomes a perpetuity. To find the sum

of an infinite series, we note that when n approaches infinity, xn = 0 for x < 1. Thus, the

sum for an infinite geometric series becomes

S∞ = a

1 − x (1.5)

To obtain equations (1.4) and (1.5) at WolframAlpha, use the following instructions:

WRA Sum[a*xî,{i,0,n-1}]

WRA Sum[a*xî,{i,0,infinity}]

Examples

1.5. Find the sum of 3 + 6 + 12 + 24 ..., 13 terms

We identify a = 3, x = 2, and n = 13. Putting these in (1.4), we get

S = a (1 − x

n)

1 − x =

3 (1 – 213

)

1 – 2 = 3(2

13 – 1) = 24,573 ♥

WRA Sum[3*2î,{i,0,12}]

1.6. Find the sum of 1.7 + 2.21 + 2.873 ..., 11 terms

Here a = 1.7, and x = 2.21/1.7 = 1.3. Also, n = 11. This gives

S = a (1 − x

n)

1 − x =

1.7 (1 – 1.311

)

1 – 1.3 =

1.7(1.311

– 1)

1.3 − 1 = 95.89 ♥






8

At WolframAlpha, use the following instructions, and click on Approximate form.

WRA Sum[1.7*1.3^i,{i,0,10}]

1.7. Find the value of i=1

100

25

1.12i

The mathematical expression possibly means a sum of one hundred annual payments of

$25 each, discounted at the rate of 12% per annum. Write it as

i=1

100

25

1.12i =

25

1.12 +

25

1.122 +

25

1.123 +

25

1.124 + ... +

25

1.12100

This is a geometric series, with the initial term a = 25

1.12 , the multiplicative factor x =

1

1.12 , and the number of terms, n = 100. Use the equation

Sn = a (1 − x

n)

1 − x (1.4)

to get Sn = (25/1.12) (1 − 1/1.12

100)

1 − 1/1.12 = 208.33 ♥

The keystrokes needed to perform the calculation on a TI-30X calculator are as follows:

25 1.12 1 1 1.12 100 1 1 1.12

At WolframAlpha, use the following instruction, enter, and click on Approximate form,

WRA Sum[25/1.12^i,{i,1,100}]

1.4 Video 01B Elements of Statistics

Probability theory plays an important role in financial planning, forecasting, and control.

At this point, we shall briefly review some of the basic concepts of probability and

statistics. In many instances, we have to deal with quantities that are not known with

certainty. For example, what is the price of IBM stock next year or the temperature in

Scranton tomorrow? The future is unpredictable. The market may go up tomorrow, or

down. One way to get a handle on the unknown is to describe it in terms of probabilities.

For instance, there is a 30% chance that it may rain tomorrow. On the other hand, there is

an even chance that the market may go up or down on a given day. The sum of the

probabilities for all possible outcomes is, of course, one.





http://www.scranton.edu/faculty/hussain/camtasia/Module01B/Module01B.html


9

We may base the probabilities of different outcomes on the past observations of a certain

event. For instance, we look at the stock market for the last 300 trading days and we

notice that on 156 days it went up. Then it is fair to say that it has a 156/300 = 0.52 =

52% chance that it may go up tomorrow as well. A complete set of all probabilities is a

probability distribution. The probability distribution for the stock market may look like

this:

Outcome Probability

Market moves up 52%

Market moves down 48%

In the above case, we are assuming that the market does not end up exactly at the closing

level of the previous day.

The distribution in the previous example is a discrete probability distribution. Another

example of such a distribution is the set of probabilities for the outcomes of a roll of dice.

With a single die, the probability is 1/6 each of getting a 1, or 2, or 3, and so on.

A probability distribution may be continuous, such as the normal probability distribution.

The probability distribution describing the life expectancy of human beings, or machines,

is a continuous distribution. At present, we shall try to describe the uncertainty in terms

of discrete probability. We are going to use a subjective probability distribution to

describe the uncertain future.

We can find the expected value of a certain quantity by multiplying the probability of

each outcome by the value of that outcome.

Example

1.8. A project has the following expected cash flows

State of the Economy Probability Cash Flow X

Good 60% $10,000

Fair 30% $6,000

Poor 10% $2,000

To find the expected cash flow, we compute

E(X) = .6($10,000) + .3($6,000) + .1($2,000) = $8,000

Consider a random variable X. Its outcome is X

1 with a probability P

1, X

2 with a

probability P

2, and so on. In general, the outcome is X

i with a probability P

i. Then the

expected value of X is

E(X) = P1X1 + P2X2 + ... + PiXi

Write this as


10

Expected value of X, E(X) = i=1

n

PiXi = X —

(1.6)

Next, we would like to know how much scatter, or dispersion, is present in this expected

value of X. We may estimate this by the variance of X, or the standard deviation of X,

defining them as follows.

Variance of X, var(X) = i=1

n

Pi(Xi − X —

)2 (1.7)

Standard deviation of X, σ(X) = var(X) (1.8)

In the above example the standard deviation of the cash flow is

σ(X) = .6(10,000 − 8000)2 + .3(6000 − 8000)

2 + .1(2000 − 8000)

2 = $2683

This figure represents the uncertainty, or the margin of error, in the cash flow.

At times, it is necessary to find the mutual dependence of two different events. For

example, we start two separate projects X and Y. The following table shows their

expected cash flows. The first project X, is the same as the one discussed above on the

previous page.

State of the Economy Probability Cash Flow X Cash Flow Y

Good 60% $10,000 $12,000

Fair 30% $6,000 $8,000

Poor 10% $2,000 $6,000

The two projects seem to be in step, both making more money in good economy and less

in poor economy. They seem to be closely related. Is there a way to measure it

quantitatively? The answer is yes, by using a measure called the correlation coefficient.

First we define the covariance between two random variables X and Y as the

Covariance between X and Y, cov(X,Y) = i=1

n

Pi(Xi − X —

)(Yi − Y —

) (1.9)

To find the covariance between the two projects, we must first find the expected value of

Y. Do it as

Y —

= .6($12,000) + .3($8,000) + .1($6,000) = $10,200

Next, we find

cov(X,Y) = .6(10,000 8000)(12,000 10,200)

+ .3(6,000 8,000)(8,000 10,200) + .1(2,000 8,000)(6,000 10,200)

= 6,000,000


11

The six-million figure found above is not particularly meaningful. We next introduce a

more practical measure of interdependence of two projects, the correlation coefficient,

defined as

r(X,Y) = cov(X,Y)

σ(X)σ(Y) (1.10)

Write the above equation as

cov(X,Y) = r(X,Y)σ(X)σ(Y) (1.11)

We already know (X) to be $2683. We also evaluate (Y) to be

σ(Y) = .6(12,000 − 10,200)2 + .3(8000 − 10,200)

2 + .1(6000 − 10,200)

2 = $2272

The smaller value of (Y) indicates that the cash flows are more tightly bunched. Finally,

we find the correlation coefficient as

r(X,Y) = cov(X,Y)

σ(X)σ(Y) =

6‚000‚000

2683*2272 = .9843 ♥

Note that r(X,Y) is a pure number and its value always lies between +1 and −1. That is

−1 < r(X,Y) < 1 (1.11)

If the two projects are completely (meaning 100%), positively (meaning, moving in the

same direction) correlated, the correlation coefficient between them is +1. This will be

the case if one project is a carbon copy of the other one. If they are totally unrelated, the

coefficient should be 0. This will be the case if one project is completely independent of

the other one. If the two projects are such that whatever happens in one, the exact

opposite happens with the other, then their correlation coefficient is −1.

The high value of r(X,Y), .9843, in the above example is not particularly surprising

because the two projects go hand in hand, performing well in good times and poorly in

bad times. Some of these ideas are particularly helpful in understanding the risk and

return of different portfolios.

1.5 Excel

It is important that the students are able to set up finance problems using Excel, which is

now a standard of business and industry. A good working knowledge of this software

should be an integral part of every business student’s education. Almost all business

programs offer courses in the use of this software. If you want to brush up your skill in

the use of Excel, you may go the following Microsoft website for a variety of tutorials.

http://office.microsoft.com/en-us/training/CR100479681033.aspx

http://office.microsoft.com/en-us/training/CR100479681033.aspx


12

To get started on Excel, consider one of the previous problems that we solved by using

the logarithm function.

1.2. Solve for x: 1.113x = 2.678

Set up the table shown below. Adjust the number in the green cell B2 until the numbers

in cells B3 and B4 come very close together. B2 gives the answer.

A B

1 Base = 1.113

2 Unknown power = 9.201184226

3 Result (given) = 2.678

4 Result(calculated) = =B1^B2

It is possible to embed an Excel table within a Word document. To do that, go the Insert

tab in a Word document. When it opens, click on Table. In the Table menu, click on

Excel Spreadsheet near the bottom. An Excel sheet opens up, where you can do your

work. When you finish your Excel work, click anywhere on the Word document, and you

can leave Excel. To go back into the Excel spreadsheet, double-click on the table, which

will reveal all the calculations and formulas.

Next, consider example 1.8 on page 8 again. Set it up on Excel as follows. The numerical

results of the formulas in cells B5:B10 are given in green cells C5:C10. The principal

advantage of Excel is that it can handle large tables of numbers.

A B C D

1 State of the Economy Probability Cash Flow X Cash Flow Y

2 Good 60% 10000 12000

3 Fair 30% 6000 8000

4 Poor 10% 2000 6000

5 E(X) =B2*C2+B3*C3+B4*C4 8000

6 E(Y) =B2*D2+B3*D3+B4*D4 10200

7 Cov(X,Y) =B2*(C2-B5)*(D2-B6)+B3*(C3-B5)*(D3-B6)+B4*(C4-B5)*(D4-B6) 6000000

8 sigma(X) =SQRT(B2*(C2-B5)^2+B3*(C3-B5)^2+B4*(C4-B5)^2) 2683.28157

9 sigma(Y) =SQRT(B2*(D2-B6)^2+B3*(D3-B6)^2+B4*(D4-B6)^2) 2271.56334

10 r(X,Y) =B7/B8/B9 0.98437404

1.6 Video 01C Maple

Maple is an extremely powerful analytical software. Working with Maple is quite easy.

The help facility in Maple is very valuable and it can guide the user through various

steps, using plenty of examples. Maple has extensive application in science, mathematics,

engineering, and finance. Time spent in learning this program can pay rich dividends in

terms of greater accuracy and higher productivity. The following instructions will get you

started with Maple.

Since Maple interprets capital and lower case letters distinctly, we should use the

symbols carefully. Maple has many built in mathematical functions and constants, such

as

http://www.maplesoft.com/

http://www.scranton.edu/faculty/hussain/camtasia/Module01C/Module01C.html


13

ln, exp, Pi, sin, sqrt

Maple can do exact arithmetic calculations and displays the answer in its totality. For

example, we need the exact value of 264, or the factorial of 50, or the value of to 50

significant figures. We do this as follows: enter the commands at the > prompt, end each

line with a semicolon, and strike the return key.

2^64;

18446744073709551616 50!;

30414093201713378043612608166064768844377641568960512000000000000

evalf(Pi,50);

3.1415926535897932384626433832795028841971693993751

Here evalf

calculates the result in floating point with 50 significant figures. Maple can also do

algebraic calculations. For instance, to solve the equations

5x + 6y = 7

6x + 7y = 8

for x and y, enter the instructions as follows:

eq1:=5*x+6*y=7;

eq1 := 5 x + 6 y = 7

eq2:=6*x+7*y=8;

eq2 := 6 x + 7 y = 8

solve({eq1,eq2},{x,y});

{y = 2, x = -1}

The symbol := is used specifically to define objects in Maple. In other words, if we type

eq1;

then the computer will recall the equation defined as eq1 and display it as

5 x + 6 y = 7

Maple can also do differentiation and integration. Consider the function


14

x3 +

ln x

x

To differentiate this function with respect to x, we type in

diff(x^3+ln(x)/x,x);

3 x2 +

1

x2 −

ln(x)

x2

To integrate the result with respect to x, recreating the original function, we enter

int(%,x);

x3 +

ln x

x

Here we use % as a symbol to designate the previous expression.

We can also use Maple to plot functions. For instance, if we want to see the visual

representation of the well-known sine wave, as in Figure 1.1, we write.

plot(sin(x),x=0..2*Pi);

Fig. 1.1: Plot of sin x for 0 < x < 2

It is possible to add text in the plots, draw three-dimensional or animated plots, and draw

plots in color. All plots in this book are drawn with the help of Maple.


15

Problems

Solve the following equations:

1.9. 16x – 54 = 15x – 32 x = 22 ♥

1.10. (x +1) (x − 2) = (x – 1) (x + 2) x = 0 ♥

1.11. (10 x + 3) (3 x + 4) = (5 x + 6) (6 x + 7) x = −15/11 ♥

1.12. x – 2

x – 3 =

x – 7

x – 9 x = –3 ♥

1.13. x + 4

x + 5 =

x + 6

x + 8 x = –2 ♥

Solve the following equations for x and y:

1.14. 2x + 6y = 32

5x + 8y = 45 x = 1, y = 5 ♥

1.15. 3x + 4y = 15

5x + 8y = 45 x = –15, y = 15 ♥

1.16. At Wal-Mart, in the hardware department, a customer buys five gallons of paint

and six brushes and pays $97.52 for them, including 6% sales tax. Another person buys

eight gallons of paint and five brushes and pays $146.28, including the sales tax. Find the

price of a gallon of paint and that of a brush. Paint, $16 per gallon; brushes, $2 each ♥

Solve for x,

1.17. (1 + x)3.2

= 8.4 x = 0.9446 ♥

1.18. 1.767x = 3.876 x = 2.38 ♥

1.19. 3.909x = 15.99 x = 2.033 ♥

Find the roots of

1.20. 2x2 + 7x – 9 = 0 x = 1, –9/2 ♥

1.21. 3x2 + 4x – 7 = 0 x = 1, –7/3 ♥

Find the sum of the following series:

1.22. 2.5 + (2.5)(.3) + (2.5)(.3)(.3) ..., infinite terms 3.571 ♥


16

1.23. 1

1.1 +

1

1.12 + 1

1.13 + ... 9 terms 5.759 ♥

1.24. 30

1.12 +

30(1.05)

1.122 +

30(1.05)2

1.123 + ... 36 terms 386.60 ♥

1.25. i=1

10

500

1.12i 2825.11 ♥

1.26. i=1

100

25

1.12i 208.33 ♥

1.27. Write WolframAlpha instruction to find the sum, i=1

24

300

1.01i

Sum[300/1.01^i,{i,1,24}], 6373.02 ♥

1.28. The cash flows from two projects under different states of the economy are as

follows:

State of the economy Probability Project A Project B

Poor 20% $3000 $5000

Average 30% $4000 $7000

Good 50% $6000 $15,000

Find the coefficient of correlation between the two projects. .9922 ♥

1.29. The expected return from two stocks, Microsoft and Boeing, under different states

of the economy are as follows:

State of the economy Probability Microsoft Boeing

Poor 10% −5% −40%

Average 40% 10% −10%

Good 50% 20% 50%

(A) Find the expected return of Microsoft and of Boeing. 13.5%, 17% ♥

(B) Find the σ of Microsoft and of Boeing. 7.762%, 34.07% ♥

(C) Find the coefficient of correlation between the two stocks. .9471 ♥



17

Key Terms Accounts payable, 1

Accounts receivable, 1

Accruals, 1

annuity, 7

Cash, 1

Common stock, 1

correlation coefficient, 10, 11

covariance, 10

Excel, 1, 3, 11, 12

expected value, 9, 10

geometric series, 1, 6, 7

Inventories, 1

linear equation, 1

linear equations, 3, 5

Long-term bonds, 1

Maple, 1, 3, 12, 13, 14

Marketable securities, 1

normal probability

distribution, 9

Notes payable, 1

perpetuity, 7

Preferred stock, 1

probability distribution, 9

quadratic equation, 1, 5

Retained earnings, 1

standard deviation, 10

statistics, 1, 3, 8

18

2. TIME VALUE OF MONEY

Objectives: After reading this chapter, you should be able to

1. Understand the concept of compounding and discounting.

2. Calculate the present value, or the future value of a single payment, or a series of

payments.

3. See the effect of monthly, daily, or continuous compounding, or discounting.

2.1 Video 02A, Single Payment Problems

Suppose someone offers you to have $1000 today, or to receive it a year from now. You

would certainly opt to receive the money right away. A dollar in hand today is worth

more than a dollar you expect to receive a year from now. There are several reasons why

this is so.

First, you can invest the money and make it grow. For example, in July 2009, the rate of

interest paid by leading banks for a one-year certificate of deposit was 2%. This means

you can get the $1000 today, invest it, and earn $20 on it by next year. There is very little

risk in this investment.

Second, there is the risk of inflation that eats away the purchasing power of the dollar.

The rate of inflation in USA has been rather low recently, less than 1%, but in some other

countries, it has been much higher. Still you will need more than $1 next year to buy

what one dollar can buy today.

Third, there is an element of risk in this deal. If you receive the money today, you are

sure to have it in your pocket. On the other hand, the person who is promising you the

money may not be around next year, or he may change his mind.

Finally, you may take the money now and use it to buy some necessary things, such as

food and clothing. If you already have all the necessities, you may want to spend the

money on pleasurable pastimes, such as a vacation or a flat-screen television set. Human

beings prefer having pleasant things as soon as possible and postpone unpleasant

experiences.

The banks and thrift institutions realize this and in order to attract investors' savings they

offer to pay interest on deposits. Suppose you deposit $100 in a bank that offers 6%

annual interest. This amount will become $106 by next year, that is, it increases by a

factor of 1.06. After two years it will grow by a factor of 1.06 again and it becomes

100(1.06)(1.06) = $112.36. The additional $0.36 is the interest earned in the second year

on the $6 first year interest. In this way, compounding the interest annually, $100 will

grow to 100(1.06)3 = $119.10, after three years.

To get a general result, let assume that the interest rate is r; then the amount will increase

by a factor of (1 + r) every year. We define


Analytical Techniques 2. Time Value of Money _____________________________________________________________________________

19

PV = present value, or the initial deposit

FV = future value of this initial deposit

After n years,

FV = PV (1 + r)n (2.1)

This is one of the basic formulas in finance. It relates four quantities: FV the future value

of a sum of money, PV the present value of that money, r the rate of growth, or interest

rate per period, and n the number of periods. For instance, n could be 8 years, and r could

be 7% per year.

If the compounding is quarterly, then the rate of interest will be r/4 per quarter, but there

will be 4n periods for compounding. Thus (2.1) becomes

FV = PV (1 + r/4)4n

For monthly compounding, the rate of interest is r/12 and the number of periods 12n. In

general, if we carry out the compounding k times a year, then we may write (2.1) as

FV = PV (1 + r/k) kn

(2.2)

If k becomes very large, then the procedure will compound the money "continuously.”

Recall the definition of the exponential function

er =

limit

k → ∞

1 + r

k

k

(2.3)

which gives ern

= limit

k → ∞

1 + r

k

kn

In the case k approaches infinity, we see that

FV = PV ern

(2.4)

How is it possible to compare two cash payments that we will receive at different points

in time? For example, which is more desirable: $200 that we will get after 2 years, or

$250 available after 3 years? We find the answer, of course, by comparing their present

values. The present value of different future payments brings them to a common level,

namely, the present instant, and thus it is easy to compare them. To find the present value

we rewrite (2.1) as

PV = FV

(1 + r)n (2.5)

The above equation represents a very useful concept in finance, namely, discounting. If

we know the future value of a sum of money, we discount it to get its present value.


20

Examples

2.1. Wilkins Micawber has just received $20,000 and he is thinking of saving it for his

retirement that is 15 years away. First National Bank offers 7% interest on a 15-year

deposit, compounded annually. Second National Bank gives 6.9% annually, but

compounds it monthly. Third National Bank pays 6.5%, but compounds the interest

continuously. What should Micawber do?

Use (2.1) to find the final value of the deposit for the first bank as

FV(First) = 20,000(1.07)15

= $55,180.63

For the Second Bank, one has to use the monthly rate of interest, which is 6.9%/12 =

.069/12. The money grows at this rate for 12*15 = 180 months. The future value is

FV(Second) = 20,000(1 + .069/12)180

= $56,135.48

Many processes in nature show a continuous rate of growth. For example, the population

of a city grows continuously, but not uniformly. If the interest is added every instant in

time and added back to the principal, then the sum of money will grow continuously. In

mathematics, one can model continuous growth with the exponential function, hence the

term exponential growth. The exponential function y, has the form y = ex, where x is the

exponent and e has the approximate value 2.718281828. Use equation (2.4).

FV(Third) = 20,000e.065*15

= $53,023.34

To solve the problem in Excel, set up the following table

A B C D

1 Initial amount, $ = 20000 Time, years = 15

2 Rate Compounding Final value

3 First Nat Bank .07 Annually =B1*(1+B3)^D1

4 Second Nat Bank .069 Monthly =B1*(1+B4/12)^(D1*12)

5 Third Nat Bank .065 Continuously =B1*exp(B5*D1)

To verify the results at WolframAlpha, try these expressions.

WRA 20000*1.07^15

WRA 20000*(1+.069/12^180

WRA 20000*exp(.065*15)

He should put his money in Second National Bank, which gives the highest final value. ♥






21

2.2. Find the future value of the following investments:

(a) A deposit of $10,000 left in the bank for 10 years and accumulating interest at the rate

of 9% annually, compounded quarterly.

If the compounding is quarterly, then the rate of interest will be r/4 per quarter, but there

will be 4n periods for compounding. Thus (2.1) becomes

FV = PV (1 + r/4)4n

Or, FV = 10,000(1 + .09/4)40 = $24,351.89 ♥

(b) A $10,000 certificate of deposit maturing in 10 year, with 9% interest rate,

compounded continuously.

For continuous compounding, use

FV = PV ern

(2.4)

Put PV = $10,000, r = .09, and n = 10,

FV = 10,000 e(.09)(10) = $24,596.03 ♥

To verify the results at WolframAlpha, try these expressions.

WRA 10000*(1+.09/4)^40

WRA 10000*exp(.09*10)

2.3. You have a choice of receiving $10,000 now, or $11,100 a year from now, or

$12,500 two years from now. What is the best option? Assume that the discount rate is

10%.

By far the best way to compare the value of one payment and another one is to compare

their present values. This is a very important principle in finance and we will use it

frequently in the remaining chapters of the textbook. With a discount rate, r = 10%, and

using (2.5), the present values are:

PV(Option I) = $10,000

PV(Option II) = 11,100/1.1 = $10,090.91

PV(Option III) = 12,500/1.12 = $10,330.58

Comparing them, the third option is the best option. ♥





22

2.2 Video 02B, Multiple Payment Problems

There are many examples in finance where we have to deal with a series of payments.

For example, the paychecks that we receive over the course of a year, or the rent

payments from a rental property, or the payments we have to make to pay off an

installment loan. In a typical problem, we have to find the present value of a set of future

payments, or the final value of an account where we have made periodic deposits.

A series of payments constitute an annuity, even if the payments are not made annually.

Similarly, a perpetuity is a stream of payments that goes on forever. For a series of

payments, either we sum them one by one, or apply the result for the summation of a

geometric series. We define a geometric series as

S = a + ax + ax2 + ax

3 + ... + ax

n−1 (1.3)

where a is the initial term and x is the ratio between successive terms. The summation of

the series, Sn is

Sn = a (1 − x

n)

1 − x (1.4)

A very important problem in finance is that of finding the present value of a set of future

payments. Suppose we represent each payment, or cash flow, by C, and the discount rate

by r. Then the discounted value of n such cash flows is

PV = i=1

n

C

(1 + r)i = C

1 + r +

C

(1 + r)2 + C

(1 + r)3 + ... + C

(1 + r)n

To find the sum of this series, we notice that the first term is a = C

1 + r and the ratio

between the terms is x = 1

1 + r. By using equation (1.4) we get

PV =

C

1 + r

1 − 1

(1 + r)n

1 − 1

1 + r

After some simplification, it gives the result

i=1

n

C

(1 + r)i =

C[1 − (1 + r)−n

]

r (2.6)

To check equation (2.6), send the following instruction to WolframAlpha,

WRA Sum[C/(1+r)^i,{i,1,n}]

To find the value of an infinite geometric series, we need the relation





23

S∞ = a

1 − x (1.5)

The above equation is valid only if the ratio x < 1, then xn approaches zero when n

approaches infinity.

For instance, we want to find the PV of a perpetuity that pays C per year forever. With a

discount rate r, the result is

i=1

C

(1 + r)i =

C

r (2.7)

In equations (2.6) and (2.7), we assume that the first cash flow occurs after one period.

Now consider the case when the first cash flow is after k periods, and continues for the

next n periods. The present value of the cash flows is thus

C

(1 + r)k +

C

(1 + r)k+1 +

C

(1 + r)k+2 + ... +

C

(1 + r)k+n−1

Write it as

1

(1 + r)k–1 [C

1 + r +

C

(1 + r)2 + C

(1 + r)3 + ... + C

(1 + r)n] = 1

(1 + r)k–1 i=1

n

C

(1 + r)i

Thus the present value of n cash flows each one C, the first one available after k periods

is

PV = 1

(1 + r)k–1 i=1

n

C

(1 + r)i (2.8)

In many instances, we have to find the future value of a series of payments. An example

is the future value of a retirement account in which a person makes periodic payments

and the money is growing at a compound rate. Suppose, we make an initial deposit of C

right now, and then another similar one after one month, and so on for a total of n

deposits. Assume that the interest rate, or the rate of growth of money, is uniformly r per

month. The future value of the first deposit after n months will be C(1 + r)n. The future

value of the second deposit will be C(1 + r)n–1 because it has only n – 1 months to grow.

By extending the argument, we can find the future value of all n deposits as

C(1 + r)n + C(1 + r)n–1 + C(1 + r)n–2 + C(1 + r)n–3 + ... + C(1 + r)

This is a geometric series with a = C(1 + r)n, x = 1/(1 + r), and n = n. Substituting these

values in the general summation formula (1.4), we get

FV = C(1 + r)n[1 – 1/(1 + r)n]

1 – 1/(1 + r)

With some simplification, it becomes


24

FV = C(1 + r)[(1 + r)n – 1]

r (2.9)

Equation (2.9) gives the future value of n payments made at the beginning of every

period, each one C, which are accumulating interest at the periodic rate r. To verify (2.9),

use the following instruction

WRA Sum[C*(1+r)^i,{i,1,n}]

Examples

2.4. An investor deposits $100 at the beginning of each month in a savings account. The

bank pays 6% annual interest, but compounds it monthly. Find the total amount in this

account after 100 months.

The compounding rate is 0.5% monthly, or 0.005 per month. The first $100 are

compounded for 100 months, the second $100 for 99 months, and the last $100 for only

one month, the total amount is

FV = 100(1.005)100 + 100(1.005)99 + 100(1.005)98 + ... + 100(1.005)

This is a geometric series, with first term a = 100(1.005)100, ratio between the terms

x = 1/1.005, and n = 100 terms altogether. Using equation (1.4) we get

FV = 100 (1.005)100 [1 1.005–100]

1 1/1.005 = $12,998.04 ♥

Another way to look at the problem is to use (2.9)

FV = C(1 + r)[(1 + r)n – 1]

r (2.9)

Which gives the result as FV = 100(1.005)[1.005100 – 1]

.005 = $12,998.04 ♥

Look at the summation as

FV = i=1

100

100(1.005)i

To verify the result, use the Maple command as

sum(100*1.005^i,i=1..100);

The result comes out to be $12,998.04.

You can get the result on WolframAlpha by sending the following instruction.




25

WRA Sum[100*1.005î,{i,1,100}]

Video 02C 2.5. Vinson Massif is 24 years old. He has just started a savings program.

He would like to accumulate $2 million for his retirement at the age of 65. The savings

account pays interest at the annual rate of 9%, compounding it monthly. How much

money should Vinson put at the beginning of each month so that at the end of 41 years he

would attain his goal?

He will retire after 41 years and thus there are 12(41) = 492 monthly payments. Suppose

each payment is X. The first payment will accumulate interest for 492 months, the second

for 491 months, and the last one for one month. The monthly interest rate is 0.75% =

0.0075. Thus

2,000,000 = X(1.0075)492

+ X(1.0075)491

+ X(1.0075)490

+ ... + X(1.0075)

This is a geometric series with a = X(1.0075)492

, x = 1/1.0075, and n = 492. Thus by

equation (1.4),

2,000,000 = X (1.0075)

492 [1 − 1/1.0075

492]

1 − 1/1.0075

This gives us X = $386.74 ♥

To solve the problem in Maple, proceed as follows.

2000000=sum(X*1.0075î,i=1..492);

solve(%);

To solve the problem on WolframAlpha, copy and paste the following line:

WRA 2000000=sum(X*1.0075î,i=1..492)

One can do the problem on Excel by the following steps. Adjust the value in the green

cell B7 to get the final amount in cell B8 to be equal to the target amount in cell B4.

A B

1 Current age, years 24

2 Retirement age, years 65

3 Interest rate, per year 9%

4 Target amount, $ 2,000,000

5 Total months =12*(B2-B1)

6 Monthly rate =B3/12

7 Monthly deposit, $ 386.74

8 Final amount, $ =B7*((1+B6)^B5-1)/(1-1/(1+B6))

2.6. Axel Heiberg has just accepted a job with an annual salary of $24,000. He has

decided to put 10% of his gross monthly income into a retirement account at the

beginning of every month. The retirement account pays interest at the rate of 0.75% every






26

month. Axel also expects to receive an annual raise of 10% each year for the next several

years. How much money will he accumulate in his retirement fund at the end of two

years?

His pay is $2000 per month and 10% of that is $200. For the first year, the saving is $200

per month, but second year it goes up by 10% to $220 per month. The future value is thus

FV = 200(1.0075)

24 + 200(1.0075)

23 + ... 12 terms + 220(1.0075)

12 + 220(1.0075)

11 + ... 12 terms

The above expression is a sum of two geometric series. For the first series, a =

200(1.0075)24

, n = 12, x = 1/1.0075, and for the second series, a = 220(1.0075)12

, n = 12,

and x = 1/1.0075. Using (1.4), we get the answer as

FV = 200 (1.0075)

24 [1 − 1/1.0075

12]

1 − 1/1.0075 +

220 (1.0075)12

[1 − 1/1.007512

]

1 − 1/1.0075

= [200 (1.0075)

24 + 220 (1.0075)

12] [1 − 1/1.0075

12]

1 − 1/1.0075 = $5,529. ♥

To solve the problem on WolframAlpha, copy and paste the following line:

WRA Sum[200*1.0075^i,{i,13,24}]+ Sum[220*1.0075^i,{i,1,12}]

2.7. Suppose you deposit $200 at the beginning of every month in an account that pays

interest at the rate of 9% per year, compounding it monthly. How long will it take you to

accumulate $10,000 in this account?

The monthly interest rate is 0.75% or 0.0075, and the monthly growth factor is 1.0075.

Suppose it takes n months to accumulate the desired amount. Then

FV = 10,000 = 200(1.0075)n + 200(1.0075)

n−1 + ... + 200(1.0075)

We may sum up the series by using equation (1.4). In our case a = 200(1.0075)n, x =

1/1.0075, and n = n.

10,000 = 200 (1.0075)

n (1 − 1/1.0075

n)

1 − 1/1.0075

Or, 10,000 = 200 (1.0075

n − 1)

1 − 1/1.0075

Or, 10 000(1 – 1/1.0075)

200 + 1 = 1.0075

n

Or, 1.0075n = 1.372208435

Taking the natural logarithm of both sides, we get




27

n ln(1.0075) = ln(1.372208435)

which gives n = 42.347539.

At the end of 42 months, the amount in the account is

200 (1.0075) (1 − 1.0075

42)

1 − 1.0075 = $9904.39

Another $95.61 on the first day of the 43rd month will do it. ♥

To do the problem on Excel, use (2.9) to put the accumulated amount in cell B5 and set

the spreadsheet as follows. Since 10,000/200 = 50, the answer is 50 months if the bank

pays no interest. If the bank is paying interest, the answer should be somewhat less than

50 months; perhaps between 40 and 50 months. Start by putting 40 in cell B3.

A B C

1 Monthly payment, C 200

2 Monthly interest rate, r =.09/12

3 Number of months, n 42

4 Target amount 10000

5 Accumulated amount =B1*(1+B2)*((1+B2)^B3-1)/B2

6 Shortage =B4-B5 last month’s payment

7 Answer: =B3+1 months

Adjust the value in cell B4 until the shortage in cell B6 is less than 200. The answer is 43

months, with the last monthly payment being $95.61.

To solve the problem on WolframAlpha, copy and paste the following:

WRA 10000=Sum[200*1.0075^i,{i,1,n}]

2.8. Suppose you deposit $125 at the beginning of each month in an account that

compounds interest continuously at the annual rate of 8%. Find the total amount in this

account after 30 months.

By using equation (2.4), the FV of the first $125 after 30 months will be 125e.08(30/12)

.

Similarly, the FV of the second $125 after 29 months will be 125e.08(29/12)

, and so on. The

total amount will be

FV = 125e.08(30/12)

+ 125e.08(29/12)

+ 125e.08(28/12)

... 30 terms

This series can be summed by using (1.4), where n = 30, a = 125e.08(30/12)

= 152.6753448,

x = e−.08/12

= .9933555063 and n = 30. Thus




28

FV = 125e

.08(30/12) (1 − e

−.08(30/12))

1 − e−.08/12 =

125(e.08(30/12)

− 1)

1 − e−.08/12 = 4165.15

The account should have $4,165.15 at the end of 30 months. ♥

To solve the problem on Excel, set up a table like this one.

A B C

1 Amount of deposit = 125 dollars

2 Number of periods = 30 months

3 Continuously compounded interest rate = =.08/12 per month

4 Final amount = =B1*(exp(B3*B2)-1)/(1-exp(-B3))

To verify the answer at WolframAlpha, and click on Approximate form:

WRA Sum[125*exp(.08*i/12),{i,1,30}]

If you want to develop a general formula with continuous compounding, you may do it as

follows. Suppose you deposit A dollars per period, at the beginning of every period, for n

periods. The continuously compound rate of interest is r. Then the first deposit becomes

Aern

after n periods. The second deposit will become Aer(n−1)

after n−1 periods, the third

one Aer(n−2)

after n−2 periods, and so on. The total amount will be

S = Aern

+ Aer(n−1)

+ Aer(n−2)

+ n terms

This is a geometric series with first term a = Aern

, the ratio between the terms x = e−r

, and

the number of terms n = n. Use (1.4) to find the sum as

Sn = Ae

rn(1 − e

−rn)

1 − e−r =

A(ern

− 1)

1 − e−r

2.9. Clifford Montdale is going to put $100 at the beginning of each month in a savings

account that pays interest at the rate of 7% per year, compounding it monthly. This rate is

fixed for one year, and it will go down to 6% in the second year. Find the balance in this

account after two years.

Consider the future value of each deposit, and then sum them up. The first deposit will

remain in the bank earning interest at the rate of 7/12% per month for the first 12 months,

and then at a rate of 6/12 = 1/2% per month for the next 12 months. Its future value is

100(1 + .07/12)12(1 + .06/12)12.

The next deposit's future value will be 100(1 + .07/12)11(1 + .005)12, because it will earn

interest at the rate of 7/12% for the first 11 months, then at the rate of 6/12% = .5% =

.005 for the next 12 months. Continuing the procedure further, the future value of the first

12 payments will be

100(1 + .07/12)12(1.005)12 + 100(1 + .07/12)11(1.005)12 + ... + 100(1 + .07/12)1(1.005)12




29

During the second year, the interest rate is down to .5% per month. We can write the sum

of future values of the payments for second year as

100(1.005)12

+ 100(1.005)11

+ 100(1.005)10

+ ... + 100(1.005)1

This results in two geometric series. For the first series, the initial term

a = 100(1 + .07/12)24

(1.005)12

, The ratio between the terms x = 1/(1 + .07/12), and the

number of terms n = 12. For the second series, a = 100(1.005)12

, x = 1/(1.005), and the

number of terms n = 12. Applying (1.4), we get the total amount to be

S = 100(1 + .07/12)

12(1.005)

12[1 − 1/(1 + .07/12)

12]

1 − 1/(1 + 0.07/12) +

100(1.005)12

[1 − 1/1.00512

]

1 − 1/1.005

= $2563.09 ♥

The answer is quite reasonable because it consists of $2400 of actual deposits and $163

in interest for two years. To solve the problem on WolframAlpha, copy and paste the

following line:

Sum[100*1.005^12*(1+.07/12)^i,{i,1,12}]+Sum[100*1.005^i,{i,1,12}]

Video 02.10 2.10. Harold Brown is planning to put some money on the first of every

month in a savings account that pays 12% annual interest, compounded monthly. He will

start by putting $500 on February 1, 2008, but keep on increasing his deposits by 1%

every month. On what date will this account have more than $1 million for the first time?

This is a problem of future value and compounding. We find the final value as the sum of

all the deposits with proper compounding. Suppose Harold Brown reaches the million-

dollar mark after n months. The first $500 will earn interest at the rate of 1% per month,

compounding monthly, and after n months, its value will become 500(1.01)n.

Next month, the deposit will increase by 1% and it will be 500(1.01). This deposit will

grow for only n – 1 months, because one month has already elapsed. Its final value will

be 500(1.01)(1.01)n−1

. We can write it as 500(1.01)n. The final value of both deposits will

be identical because the second deposit starts out with a bigger amount, but has less time

to grow. The two factors cancel out precisely.

The deposit for the following month is 1% greater than the previous month’s deposit and

it equals 500(1.01)2. However, it can grow only for n − 2 months and it finally becomes

500(1.01)2(1.01)

n−2. Merging the powers, the result is 500(1.01)

n. This gives exactly the

same final value. We discover that the final value of each deposit is the same, namely,

500(1.01)n. Since there are n such deposits, their total final value should be n[500(1.01)

n].

We can summarize the previous discussion in the following equation.

1,000,000 = 500(1.01)n + 500(1.01)(1.01)

n−1 + 500(1.01)

2(1.01)

n−2 + ...


http://www.scranton.edu/faculty/hussain/camtasia/Prob0519/Prob0519.html


30

Or, 1,000,000 = 500n (1.01)n

We can solve the above equation for n by using Maple. We enter the instruction

fsolve(1000000=500*n*1.01^n,n);

and we get n = 221.2568054. Thus after 222 months, or, 18 years and 6 months, he

should have more than a million dollars. The exact amount is 500*222*1.01222

=

$1,010,806.30. ♥

To get the answer on WolframAlpha, enter the following

WRA 1000000 = 500*n*1.01^n

If Harold Brown decides to increase his monthly deposit by 2% every month, while the

bank keeps on paying 1% monthly interest, then the first equation will become

1,000,000 = 500(1.01)n + 500(1.02)(1.01)

n−1 + 500(1.02)

2(1.01)

n−2 + ...

In this case, the numbers do not cancel out neatly. Notice that the sum of the powers is n.

The factor (1.01) starts with power n and ends with power 1. The factor (1.02) starts with

power 0 and it ends with power n – 1. We can represent the above equation as follows

1,000,000 = i=1

n

500(1.02)i−1

(1.01)n+1−i

Note that the sum of the powers is i – 1 + n + 1 – i = n as required. The starting values for

the two powers are 0 and n, and the first term in the summation is 500(1.02)0(1.01)

n,

when i = 1. To solve the equation using Maple, enter

1000000=sum(500*1.02^(i-1)*1.01^(n+1-i),i=1..n);

solve(%);

The result is 162.1914189. This means Harold Brown can achieve his goal sooner, in 163

months. This is quite reasonable because he is increasing his initial deposit at a faster

rate.

For WolframAlpha, enter the following

1000000=Sum[500*1.02^(i-1)*1.01^(n+1-i),{i,1,n}]

It does not give an accurate answer. However, the two diagrams suggest a value of

around 162.

To do the problem on Excel, set up a table as follows.





31

A B

1 500 =A1*1.01

2 =A1*1.02 =(A2+B1)*1.01

3 =A2*1.02 =(A3+B2)*1.01

Column A will show the deposit for each month and column B the total accumulated in

the account up to that point. Next, highlight the cells (A2,B2,A3,B3) and drag down the

handle. When you reach row 163, the total accumulated in cell B163 will be

1,018,185.809, which is just over a million dollars.

2.11. The winner of the recent lottery received notice that he would get his money in 20

annual installments of $281,347 each. He will get the first installment right now. If the

discount rate is 12%, find the present value of his winnings.

PV = 281,347 + i=1

19

281‚347

1.12i

= 281,347 + 281347 (1 − 1.12

−19)

0.12 = $2,353,686 ♥

To get the answer on WolframAlpha, enter the following,

WRA Sum[281347/1.12^i,{i,0,19}]

2.12. A person has the following options in settlement of a life insurance policy:

(1) A cash settlement of $100,000, or

(2) A monthly payment of $1,000 available at the end of each month for the next 20

years.

If the proper discount rate in this case is 12%, which method is better for the recipient?

The present value of the monthly payments is the sum of discounted cash flows. For

monthly cash flows, the discount rate is 1% per month. There are 240 payments. For

summation of terms, use (2.6).

i=1

240

1000

1.01i = 1000(1 1.01240)

.01 = $90,819.42

Clearly, the cash settlement is better. ♥

2.13. You have just bought a car from your friend. He gives you two options: pay for the

car in 48 monthly installments of $109 each, or 60 monthly installments of $89 each. The

time value of money is 12% annually. Which is the cheaper method of payment?




32

Compare the present value of the two payment options by discounting them at the rate of

12% per year, or 1% per month.

PV of 48 monthly payments = i=1

48

109

1.01i = 109(1 − 1.01

−48)

.01 = $4,139.16

PV of 60 monthly payments = i=1

60

89

1.01i = 89(1 − 1.01

−60)

.01 = $4,001.00

It is better to take the 60-month option. ♥

2.14. You are interested in buying a new sports car costing $21,000. You can afford only

$5,000 as the down payment and the bank will finance the rest at 11.8%. What are the

level monthly payments that will pay off the loan in 48 months?

The basic financial principle in this problem is as follows:

The present value of the loan

= The present value of future payments

We can express it as

L = i=1

n

P

(1 + r)i (2.10)

The amount of loan is $16,000, the number of payments is 48, and the monthly interest

rate is 0.118/12. Thus if P is the monthly payment, then

i=1

48

P

(1 + .118/12)i =

P[1 − (1 + .118/12)−48

]

.118/12 = 16,000

Or, P = 16‚000 (0.118/12)

1 − (1 + 0.118/12)−48 = $419.77 ♥

To get the answer on WolframAlpha, enter the following expression

WRA Sum[P/(1+.118/12)^i,{i,1,48}]=16000

2.15. You have bought a piece of land in Wayne County for $12,000. You agreed to pay

the owner the price in five equal annual installments of $3,000 each, the first one right

now. What is the implicit rate of interest that the owner is charging?

Equating the cash price of the land to the discounted value of all the payments, we get




33

12,000 = 3000 + i=1

4

3000

(1 + r)i

Or, 9,000 = i=1

4

3000

(1 + r)i

To solve this equation using Maple, type the main instruction as

fsolve(9000=sum(3000/(1+r)^i,i=1..4),r,0..1);

The instruction "fsolve" requires the program to solve the following equation in

floating point and find the value of r within the range 0 to 1. The instruction "sum" takes

the summation of the following terms. After some computation the result .1258983250

shows up on the screen. This is the interest rate of 12.59%. ♥

A B C

1 Number of periods = 4

2 Each payment = -3000 dollars

3 Present value of loan = 9000 dollars

4 Implied interest rate = =RATE(B1,B2,B3) 13%

One can do the problem in Excel by using the RATE function. The inputs for the function

are =RATE(nper,pmt,pv,[fv],[type],[guess]), but the result is inaccurate.

At WolframAlpha, enter the following to get a positive real solution

WRA 12000=3000+Sum[3000/(1+r)^i,{i,1,4}]

2.16. You have borrowed $56,000 as a mortgage loan to buy a house. The bank will

charge interest at the rate of 9% annually and requires a minimum monthly payment of

$500. At the end of five years, you must pay off the entire mortgage by a “balloon

payment.” You plan to pay only the minimum amount each month and then pay off the

loan with the final payment. Find this balloon payment.

During uncertain economic times, when the interest rates are liable to fluctuate widely,

the lending institutions give out short-term loans that require a balloon payment to pay

off the loan. At the time of the balloon payment, the borrower can renegotiate the loan at

the current interest rates and it may include another balloon payment. Suppose the

balloon payment is B, then the equality of loan value to the present value of all payments

implies that

L = i=1

n

P

(1 + r)i +

B

(1 + r)n (2.11)

Put numerical values,




34

56,000 = i=1

60

500

1.0075i + B

1.007560

Or, 56,000 = 500(1 1.007560)

.0075 +

B

1.007560

Which gives B = [56,000 − 500(1 1.007560)

.0075](1.0075

60) = $49,966.07 ♥

You can check the answer with Maple by using the following steps.

L=sum(P/(1+r)^i,i=1..n)+B/(1+r)^n;

subs(L=56000,P=500,r=.09/12,n=5*12,%);

solve(%);

The following Excel table will solve the problem.

A B C

1 Interest rate = .0075 per month

2 Number of payments = 60

3 Payment per month = 500 dollars

4 Initial loan = -56000 dollars

5 Balloon payment = =FV(B1,B2,B3,B4) $49,966.07

To get an approximate answer on WolframAlpha, enter the following,

WRA 56000=Sum[500/1.0075^i,{i,1,60}]+B/1.0075^60

2.17. You have borrowed $500 from a friend with the understanding that you will pay

him back in three installments: $100 after one month, $200 after two months, and $220 at

the end of the third month. Find the implied interest rate in this arrangement.

Set the loan value to the PV of the three payments as

500 = 100

1 + r +

200

(1 + r)2 + 220

(1 + r)3

You cannot solve this problem by the Present Value tables but you can go to

WolframAlpha and paste the following instruction.

WRA 500=100/(1+r)+200/(1+r)^2+220/(1+r)^3

the answer comes out to be .017777 = 1.7777% monthly.

The nominal annual rate is 12 times the monthly rate, that is, 12(.017777) = .2133 =

21.33%. The effective annual rate is due to the monthly compounding and it comes out to






35

be 1.017777121 = 23.55% annually. The effective interest rate is higher than the

nominal rate. ♥

2.18. Sebastian Cabot has received $135,000 in compensatory damages. He has placed

the money in a trust account that pays 6% annual interest, compounded monthly. Cabot

will withdraw $1000 a month out of this account, starting one month after the initial

deposit. Calculate the money in the account after 25 withdrawals.

The monthly interest rate is .5% = .005. With the withdrawal rate of $1000 a month, the

present value of 25 withdrawals is i=1

25

1000

1.005i. This means the present value of amount of

money in the account after 25 months will be

135‚000 − i=1

25

1000

1.005i . This amount is

growing at the rate of .005 per month for 25 months. Thus, its future value is

FV =

135‚000 − i=1

25

1000

1.005i (1.005

25) = $126,368.29 ♥

In general, it becomes

FV =

A i=1

n

w

(1 + r)i (1 + r)

n (2.12)

Here FV is the future value of the account, which had an initial amount A. The account

pays interest at rate r. The owner of the account has made n withdrawals, each one equal

to w. This leads us to the question that if a person has a nest egg A from which he

regularly withdraws w per month, how long will it take him to exhaust his savings. To

answer that, we use Maple and type in the instructions:

0=(A-sum(w/(1+r)^i,i=1..n))*(1+r)^n;

solve(%,n);

After some simplification, we get the answer as

n =

ln

w

w – Ar

ln(1 + r) (2.13)

Consider a person with a total savings of $400,000, which is earning interest at the rate of

½% per month. He withdraws $3000 from it every month. He will exhaust his savings in

n =

ln

3000

3000 – 400‚000*.005

ln(1.005) = 220.3 months = 18.36 years ♥


36

2.19. Alabama Corporation has taken a loan of $60,000 with the understanding that it

will make the monthly payments of $600. The bank will charge the interest at the rate of

9% per year on the unpaid balance. After how many months will the balance become

$48,686.38?

The present value of the loan L in terms of the discounted future values is

L = i=1

n

P

(1 + r)i +

B

(1 + r)n (2.11)

where P is the regular monthly payment and B is the balloon payment, or balance after n

periods. We may solve it for n by using Maple as follows:

L=sum(P/(1+r)^i,i=1..n)+B/(1+r)^n;

solve(%,n);

The result is

n =

ln

rB – P

rL – P

ln(1 + r) (2.14)

Substituting r = 0.0075, B = $48,686.38, L = $60,000, P = $600, we get the value of n as

n =

ln

.0075*48‚686.38 – 600

.0075*60‚000 – 600

ln(1.0075) = 60

Thus after 60 months the balance will be $48,686.38. ♥

To do the problem on Excel, you can set up a table like this one. Adjust the number of

payments in cell B2, until the value in cell B5 becomes equal to required balance of

$48,686.38.

A B C

1 Interest rate = .0075 per month

2 Number of payments = 60

3 Payment per month = 600 dollars

4 Initial loan = -60000 dollars

5 Balance = =FV(B1,B2,B3,B4) $48,686.38

To check the answer at WolframAlpha, copy and paste the following instruction.

WRA 60000=Sum[600/1.0075^i,{i,1,n}]+48686.38/1.0075^n

2.20. A bank offers the following program to its customers. If you deposit $100 at the

beginning of every month for the next 7 years, then in return the bank will give you $100




37

a month forever, starting a month after your last monthly payment. If the time value of

money is 13.2% annually, would you join in this program?

First, find the present value of your payments. Suppose you make your first payment

today, the second payment at the end of the first month, and the 84th payment at the end

of the 83rd month. The monthly discount rate is 13.2/12 = 1.1% = .011. Because you

make the first payment now, the present value of 84 payments is

PV1 = 100 + i=1

83

100

1.011i = 100 +

100(1 − 1.011−83

)

.011 = $5524.33

Next, find the present value of bank’s payments. You made your last payment at the end

of the 83rd month and the bank will make its first payment at the end of the 84th month.

The present value of the bank's payments is

PV2 = 100

1.01184 +

100

1.01185 +

100

1.01186 + ... ∞

This is an infinite geometric series with a = 100/1.01184

and x = 1/1.011. Using (1.5),

PV2 = 100/1.011

84

1 − 1/1.011 = $3666.58

You are paying the bank an excess amount of 5,524.33 − 3,666.58 = $1,857.75,

calculated in present value dollars. The net present value of this transaction is

−$1,857.75. Because the net present value is negative, you should not join. ♥

You can find the net present value at WolframAlpha with the following instruction.

WRA -100-Sum[100/1.011^i,{i,1,83}]+Sum[100/1.011^i,{i,84,infinity}]

2.21. In the last problem, at what minimum rate of interest would the customers consider

making their deposits in this program?

The bank's payments have a smaller present value due to a higher discount rate. One has

to find a smaller discount rate that will equate these two values. At some equilibrium

point, the equation PV1 = PV2 will hold. This gives us

1 + i=1

83

1

(1 + r)i =

i=84

∞

1

(1 + r)i =

1

(1 + r)83

i=1

1

(1 + r)i

Or, 1 + 1 − (1 + r)

−83

r =

1

(1 + r)83

1

r

Multiplying by r throughout, we get




38

r + 1 − (1 + r)−83

= 1

(1 + r)83

Multiply by (1 + r)83

to get

(1 + r)84

= 2

Or, r = 21/84

− 1 = .008285892 monthly

The annual rate is 12(.008285892) = 0.099430704 = 9.943% annually. If the interest rates

in the market are less than 9.943%, then the depositors will come out ahead. In the time

of high inflation, when the interest rate is more than 10%, the bank will win this game. ♥

To perform the calculation on WolframAlpha, copy and paste the following:

WRA Sum[1/(1+r)^i,{i,0,83}] = Sum[1/(1+r)^i,{i,84,infinity}]

The result is .00828589.

Problems

2.22. You have decided to deposit $100 in O'Neill National Bank at the beginning of

every month. The bank compounds interest on a monthly basis but at a variable rate

adjusting it annually. You know that during the first year, the interest will be 8%

annually, but during the second year, it may go up to 9%. Find the expected amount in

your account after two years. $2631.00 ♥

2.23. Conrad Aiken has a retirement account in which he has been adding a certain sum

of money on the first of every month. He started by depositing $250 in the first month but

kept on increasing his deposits by 1% every month. The bank was adding interest to his

account monthly at the rate of 0.8% per month during the first year and then 0.9% per

month during the second year. Find the total amount in this account at the end of two

years. $7,500.87 ♥

2.24. Homer Zeno has borrowed $50,000 from the bank with the understanding that he

will make a minimum payment of $500 per month. The bank will charge interest at the

rate of 1% per month on the unpaid balance. Homer plans to make $500 monthly

payments for the first 12 months, then $600 monthly payments for the next 48 months,

and then pay off the entire loan by making a balloon payment at the end of the 61st

month. Find this balloon payment. $44,316.52 ♥

2.25. You have taken a $100,000 mortgage loan at Albert Savings Association at the rate

of 9% annually. You are required to pay $1,000 monthly payment. Approximately, how

long will it take you to pay off the loan? 15 years 6 months ♥

2.26. Cooper Corporation has borrowed $120,000 from the bank at 8% annual interest

rate, compounded monthly. The company plans to pay $2,000 per month for the first 12




39

months, and then pay $2500 per month for the next 12 months. Find the remaining

balance of the loan after 24 months. $82,655.21 ♥

2.27. Atbara Corporation has taken a five-year loan from the local bank at the annual

interest rate of 9%. Atbara will pay back the $250,000 loan in monthly installments. Find

(a) the monthly payment, and (b) the balance of the loan after 24 payments.

(a) $5189.59, (b) $ 163195.99 ♥

2.28. Akron Corporation has borrowed $1.2 million from a bank with the understanding

that it will pay $50,000 a month, until the loan is paid off. The bank will charge 9% per

year interest on the unpaid balance, calculated monthly. Akron will make the payments at

the end of each month. Find the following:

(A) How long will it take Akron to pay off the loan? 27 months ♥

(B) What is the balance of the loan after 24 months? $126,272.71 ♥

2.29. You would like to accumulate a million dollars for your retirement. You have

another 35 years before you retire. The local bank, where you intend to keep the money,

will compound interest monthly at the annual rate of 6%. How much money should you

deposit at the beginning of each month to reach your goal? $698.41 ♥

2.30. Rutherford B. Hayes has borrowed $80,000 as a mortgage loan at 7.5% interest rate

and 30-year term. He has to pay the loan in monthly installments. After how many

payments will the unpaid balance become $40,000? 265 months ♥

Key Terms

annuity, 21

compounding, 17, 18, 20, 23,

24, 25, 27, 28, 33

discount rate, 20, 21, 22, 30,

36

discounting, 17, 18, 31

effective annual rate, 33

exponential function, 18

future value, 17, 18, 20, 22,

23, 25, 27, 28, 34

inflation, 17, 37

nominal annual rate, 33

perpetuity, 21, 22

present value, 17, 18, 20, 21,

22, 30, 31, 32, 34, 35, 36

risk, 17

40

3. VALUATION OF DEBT AND EQUITY

Objectives: After reading his chapter, you will

1. Understand the role of bonds in financial markets.

2. Distinguish between different types of bonds, such as zero-coupon, perpetual, discount,

convertible, and junk bonds and apply the bond pricing formulas to evaluate these

bonds.

3. Understand the concepts of equity capital, stock, and dividends.

4. Apply Gordon's growth model to evaluate the equity of a firm.

5. Find the value of a stock with supernormal growth for a few periods followed by

normal growth.

3.1 Video 03A Capital formation

Corporations need capital, meaning money, to run their business. They need the money

to make capital investments, which are investments in land, buildings, equipment, and

machinery. In order to acquire capital the firms turn to investors. Figure 3.1 represents

the relationship between the corporations and investors.

Investors

Capital

Return on investment

Corporation

Fig. 3.1: The relationship between the investors and a corporation.

Examining the long-term capital structure of a company, we find that the capital comes in

two forms: debt and equity. When a company acquires debt capital, it simply borrows

money on a long-term basis from the investors. A company can also borrow money from

a financial institution for the short-term. The firms issue financial instruments called

bonds and sell them to the investors for cash. Bonds are merely promissory notes that

promise to pay the investors the interest on the bonds regularly, and then pay the

principal when the bonds mature.

When a corporation wants to raise equity capital, it sells stock to the investors. The

stockholders then become part owners of the company. The ownership of stock gives

them an equity interest in the company. There are important differences between debt and

equity capital. For instance, the bonds mature after several years and the company must

redeem the bonds by paying the principal back to the investors. There is no maturity date

for the stock. The bondholders receive regular interest payments from the company. The

stockholders may or may not receive dividends from the company. The stockholders vote

for the election of board of directors, but the bondholders do not have nay voting rights.

The board of directors has the ability to make important decisions at the company, such

as hiring or firing of its president.


Analytical Techniques 3. Valuation of Bonds and Stocks _____________________________________________________________________________

41

3.2 Valuation of Bonds

The face amount of a typical bond is $1,000. The market value of the bond could be more

than $1,000, and then it is selling at a premium. A bond with a market value less than

$1,000 is selling at a discount, and a bond, which is priced at its face value, is selling at

par. Figure 3.2 shows an advertisement that appeared in the Wall Street Journal of July

23, 1997. Dynex Capital, Inc. issued bonds with a total face value of $100 million in July

1997. The bonds carried a coupon of 77/8%. This means that each bond pays $78.75 in

interest every year. Actually, half of this interest is paid every six months. The bonds will

mature after 5 years, which is relatively short time for bonds. They are senior notes in the

sense that the interest on these bonds will be paid ahead of some other junior notes. This

makes the bonds relatively safer.

$100,000,000

DYNEX

Dynex Capital, Inc.

77/8% Senior Notes Due July 15, 2002

Interest Payable January 15 and July 15

Price 99.900%

plus accrued interest from July 15, 1997

Paine Webber Incorporated Smith Barney Incorporated

Fig. 3.2: A bond advertisement in Wall Street Journal.

The price of these bonds is $999 for each $1,000 bond. Occasionally, the corporations

may reduce the price of a bond and sell them at a discount from their face value. This is

true if the coupon is less than the prevailing interest rates, or if the financial condition of

the company is not too strong. The buyer must also pay the accrued interest on the bond.

If an investor buys the bond on July 25, 1997, he must pay accrued interest for 10 days.

When the bonds are publicly traded, they will be listed as “Dynex 77/8s02.” The

information about the bonds is frequently displayed as: Madison Company 4.75s33. We

learn to interpret it as follows:

Madison Company: This is the name of the entity that issues the bonds

4.75: This is the coupon rate, or the annual rate of interest paid on the bonds, that is

4.75% per annum

s: This is just a separator between the two numbers

33: This is the year when the bond will mature, namely, 2033

The two companies listed at the bottom of the advertisement, Paine Webber Incorporated

and Smith Barney Incorporated, are the underwriters for this issue. Underwriters, or


42

investment banking firms, such as Merrill Lynch, will take a certain commission for

selling the entire issue to the public.

Since the appearance of this advertisement, several changes have occurred. On November

3, 2000, Paine Webber merged with UBS AG, a Swiss banking conglomerate. Smith

Barney is now owned by Citigroup. Corporations no longer use fractions in identifying

the coupon rates, instead decimals are used universally.

An important feature of every bond issue is the indenture. The indenture is a detailed

legal contract between the bondholders and the corporation that spells out the rights and

obligations of both parties. In particular, it gives the bondholders the right to sue the

company and force it into bankruptcy, if the company fails to pay the interest payments

on time. This provides safety to the bondholders, and puts serious responsibility on the

corporation.

The two factors that determine the interest rate of a bond are the creditworthiness of the

corporation and the prevailing interest rates in the market. A company that is doing well

financially, and has good prospects in the future, will have to pay a lower rate of interest

to sell bonds. A company that is close to bankruptcy will have a hard time selling its

bonds, and must attach a high coupon rate to attract the investors.

The term sinking fund describes the amount of money that a company puts aside to retire

its bonds. For example, a company issues bonds with face value $50 million, which will

mature in 20 years. During the last five years of their existence, the company may set

aside $10 million per year to buy back, or retire their bonds. This $10 million is the

sinking fund. This procedure spreads the loan repayment over a five-year period and is

easier for the company to manage.

To retire the bonds, the corporation may buy the bonds in open market if they are selling

below par. The corporation may also call the bonds, depending on the provisions of the

indenture, by paying the more than the face value of the bonds to the bondholders. Such

bonds are called callable bonds.

We can evaluate a bond by finding the present value of the interest payments and that of

the principal. The proper discount rate that calculates the present value depends on the

risk of the bonds. The risky bonds have a relatively higher discount rate. Further, the

discount rate is also the rate of return required by an investor buying that bond. The basic

financial principle is:

The present value of a bond is simply the present value of all future cash flows from the bond, properly discounted.

We may express the above statement as follows

B = i=1

n

C

(1 + r)i +

F

(1 + r)n (3.1)

http://en.wikipedia.org/wiki/Sinking_fund

http://en.wikipedia.org/wiki/Callable_bond


43

The first term on the right side is the present value of the coupon payments, or the interest

payments in dollars. The second term is the present value of the face amount of the bond

in dollars. This resembles equation (2.9).

Recently, some US corporations issued very long maturity bonds, 100 years to be exact.

A company in Luxembourg has issued bonds that will mature after 1,000 years. British

Government has issued perpetual bonds, called consols, which are still available today

and carry a coupon rate of 2½%. In principle, an American company can issue perpetual

bonds that will never mature but the Federal Government prohibits that.

Perpetual bonds have an infinite life span. In essence, they are perpetuities. The

bondholders continue to receive interest payments and if they want to, they can always

sell the bonds to other investors. Since the bond is never going to mature, the implicit

assumption is that the investors will never receive the face amount of such a bond. From

(2.7), when n approaches infinity, the summation becomes C/r. The second term for the

present value of the face amount also approaches zero. From (3.1) we get the simple

formula for perpetual bonds.

B = C

r (3.2)

Some companies try to conserve cash and they may sell zero-coupon bonds. These bonds

make no periodic interest payments and they pay the entire accumulated interest and the

principal at the maturity of the bond. Because of this feature, these bonds sell at a

substantial discount from their face value. For instance, General Motors issued zero

coupon bonds in 1996 due to mature in 2036. In January 2007, these bonds were selling

at 38.11, or $381.10 per $1000 bond. For zero-coupon bonds, the first term in (3.1) is

zero because C is zero. This leaves only the second term for the valuation of zero-coupon

bonds as follows:

B = F

(1 + r)n (3.3)

Occasionally, a company may issue convertible. A bondholder, at his discretion, can

exchange a convertible bond for a fixed number of shares of stock of the corporation. For

example, the bondholder may get 50 shares of stock by giving up the bond. If the price of

the stock is $10 a share, then the conversion value of the bond will be $500, that is, the

bond can be converted into $500 worth of stock. The market value of the bond will

always be more than the conversion value. If the price per share rises to $25, then the

price of the bond will be at least 50(25) = $1250. Thus, the convertible bonds are

occasionally trading above their face value.

At times, the financial health of a company deteriorates quite a bit. The company may

even stop paying interest on the bonds, and there is little hope of recovery of principal of

these bonds. Such bonds, with extremely high investment risk, are frequently labeled as

"junk" bonds.

http://en.wikipedia.org/wiki/Consols

http://en.wikipedia.org/wiki/Convertible_bond


44

An investor buys a bond for its rate of return, or its yield. We define the current yield, y,

of a bond as follows.

y = Annual interest payment of the bond

Current market value of the bond

The annual interest payment of the bond equals cF, where c is the coupon rate, and F is

the face value of the bond. With B being the market value of the bond, we may write

y = cF/B (3.4)

This represents the return on the investment provided the bond is held for a short period.

Holding a bond to maturity, one receives money in the form of interest payments, plus

there is a change in the value of the bond. The annual interest payment of the bond is cF,

as seen before. If you have bought the bond at a discount, it will rise in value reaching its

face value at maturity. Or, the bond may drop in price if it has been bought at a premium.

In any case, it should be selling for its face value at maturity. The total price change for

the bond is F−B, where F is the face value of a bond and B is its purchase price. This

change may be positive or negative depending upon whether F is more, or less, than B.

On the average, the price change per year is (F−B)/n, where n is the number of years

until maturity. On the average, the price of the bond for the holding period is (F + B)/2.

Thus the yield Y, of a bond is given, approximately, by dividing the annual return by the

average price. This is given by:

Y ≈ annual interest payment + annual price change

average price of the bond for the entire holding period

Or, Y cF + (F − B)/n

(F + B)/2 (3.5)

In equation (3.1), the discount rate r is also equal to the yield to maturity, Y. The reason

for the approximation in the equation (3.5) is that the value of a bond does not reach the

face amount linearly with time, as seen in Figure 3.3.

Consider a bond that has 8% coupon, pays interest semiannually, and will mature after 10

years. Assume that the investors require 10% return on these bonds. Then the current

value of the bond is

B = i=1

20

40

1.05i +

1000

1.0520 = $875.38

As the bond approaches maturity, its value reaches $1,000. This is shown in Fig. 3.3.

Notice that the curve is not a straight line. The bond value rises slowly at first and then

more rapidly when it is close to maturity.

Equation (3.5) calculates the yield to maturity of a bond only approximately. To find it

more accurately, we depend on the alternate definition of yield to maturity: The yield to

maturity of a bond is that particular discount rate, which makes the present value of the

http://en.wikipedia.org/wiki/Current_yield

http://en.wikipedia.org/wiki/Yield_to_maturity


45

cash flows to be equal to the market value of the bond. Thus, we go back to (3.1), put

known values of B, n, C, and F, and evaluate the unknown r. That is the yield to maturity

of the bond. We need a set of Maple or WolframAlpha instructions to get the final

answer.

Fig. 3.3: The value of a bond with respect to time to maturity. Face value $1000, coupon 8%, 10 years to

maturity, semiannual payments, yield to maturity 10%.

The US Government borrows heavily in the financial markets by issuing Treasury bonds.

They are issued with maturity date ranging from six months to thirty years. The yield of

these bonds fluctuates. The following table gives the yield of Treasury securities on

January 5, 2007.

US Treasury Bond Rates, January 5, 2007

Maturity Yield Yesterday Last Week Last Month

3 Month 4.88 4.87 4.84 4.83

6 Month 4.87 4.85 4.82 4.83

2 Year 4.72 4.67 4.78 4.56

3 Year 4.65 4.60 4.71 4.47

5 Year 4.62 4.57 4.65 4.43

10 Year 4.63 4.58 4.68 4.47

30 Year 4.72 4.69 4.79 4.58

Table 3.1: Source: http://finance.yahoo.com/bonds

One can plot the yield against the time to maturity to get the Treasury yield curve, shown

in Figure 3.4. The curve is plotted on a semilog scale to accommodate long maturity

dates. Normally, one expects that the longer maturity bonds have a higher yield, but this

is not the case in January 2007. Hence we see an inverted yield curve in the diagram.

http://finance.yahoo.com/bonds

http://en.wikipedia.org/wiki/Yield_curve


46

Figure 3.4. The inverted Treasury yield curve on January 5, 2007. On the x-axis, .1e2 means 10 years, and

.2e2 is 20 years.

Table 3.2 shows the yields of corporate bonds on January 5, 2007. The letters AAA, AA,

and A represent the quality of bonds, or bond rating, by Fitch. The least risky bonds are

designated by AAA, and so on. We notice two things. First, the longer maturity bonds of

the same quality rating have a higher yield. For instance, for bonds with A rating, the

yield for 2- year maturity is 5.13%; and for 20 years, it is 5.82%. Second, the yield is

higher for riskier bonds. Consider 5-year bonds. The yield rises from 5.06% to 5.20%

when the rating drops from AAA to A.

Corporate Bonds, January 5, 2007

Maturity Yield Yesterday Last Week Last Month 2yr AA 5.04 4.98 5.11 4.86

2yr A 5.13 5.08 5.20 4.92

5yr AAA 5.06 5.03 5.11 5.19

5yr AA 5.13 5.09 5.17 4.93

5yr A 5.20 5.16 5.23 4.99

10yr AAA 5.18 5.07 5.30 5.08

10yr AA 5.32 5.33 5.42 5.19

10yr A 5.43 5.37 5.47 5.26

20yr AAA 5.68 5.71 5.76 5.06

20yr AA 5.76 5.79 5.84 5.68

20yr A 5.82 5.85 5.90 5.71

Table 3.2: The yield of bonds as a function of quality and time to maturity. Source:

http://finance.yahoo.com/bonds January 5, 2007


47

Money Market Rates

Tuesday, July 31, 2007, Wall Street Journal

PRIME RATE: 8.25% (effective 6/29/06). This is a benchmark rate used by banks in

making loans to their commercial customers. The best customers pay close to the prime

rate, less creditworthy customers pay more.

DISCOUNT RATE: 6.25% (Primary) (effective 6/29/06). This is the rate charged by the

Federal Reserve for the loans made to the member banks.

FEDERAL FUNDS: 5.4375%, high, 4.500% low, 4.250% near closing bid, 5.00%

offered. Effective rate 5.32%. Reserves traded by the member banks for overnight use in

amounts of $1 million or more.

CALL MONEY: 7.00% (effective 6/29/06). This is the rate of interest used by

stockbrokers for making loans to their customers for the purchase of common stocks.

COMMERCIAL PAPER: placed directly by General Electric Capital Corporation,

5.24% 30 to 44 days, 5.25% 45 to 61 days, 5.28% 62 to 89 days, 5.29% 90 to 119 days,

5.30% 120 to 190 days, 5.29% 191 to 219 days, 5.28% 220 to 249 days, 5.27% 250 to

270 days.

CERTIFICATES OF DEPOSIT: 5.28% one month, 5.36% three months, 5.46% six

months.

LONDON INTERBANK OFFERED RATE (LIBOR): 5.3300% one month, 5.42625%

three months, 5.5150% six months, 5.5675% one year. The rate is set by the British

Banker's Association, and is used by one bank making loan to another bank. This is a key

rate used in international transactions, especially interest rate swaps.

FOREIGN PRIME RATES: Canada 6.25%, European Central Bank 4.00%, Japan

1.875%, Switzerland 4.42%, Britain 5.75%, Australia 6.25%, Hong Kong 8.00%.

TREASURY BILLS: Results of the Monday, August 31, 2007, auction of the short-term

T-bills sold at a discount in units of $1,000 to $1 million. 5.055% 4 weeks, 4.825% 13

weeks, 4.800% 26 weeks.

MERRILL LYNCH READY ASSETS TRUST: 4.70%, average rate of return, after

expenses, for the past 30 days; not a forecast of future returns.

CONSUMER PRICE INDEX: June 208.4, up 2.7% from a year ago. Bureau of Labor

Statistics.

Table 3.3: Money rates


48

Issue Price Coupon

%

Maturity

date

YTM

%

Current

Yield

Fitch

Ratings

Callable

Federal Home Ln Mtg 99.00 5.000 27-Jan-2017 5.128 5.051 AAA Yes

Goldman Sachs 104.40 5.750 1-Oct-2016 5.168 5.508 AA No

Emerson Electric 100.53 5.125 1-Dec-2016 5.056 5.098 A No

Clear Channel Comm. 90.90 7.250 15-Oct-2027 8.165 7.976 BBB No

Scotia Pacific 81.50 7.710 20-Jan-2014 11.634 9.460 BB No

Brookstone 99.88 12.000 15-Oct-2012 12.020 12.015 B Yes

Fedders No Am 72.50 9.875 1-Mar-2014 16.575 13.621 CCC Yes

Wise Metals 90.74 10.250 15-May-2012 12.678 11.296 CC Yes

Table 3.4: The yield of bonds as a function of quality and time to maturity. Source: Source:

http://finance.yahoo.com/bonds January 5, 2007

Table 3.4 shows a sampling of bonds available in the market in January 2007. They are

arranged in terms of their quality rating, the least risky bonds are the top and the riskiest

ones at the bottom.

Normally, when a buyer buys a bond he has to pay the accrued interest on the bond. This

is the interest earned by the bond since the last interest payment date. Occasionally some

bonds trade without the accrued interest and they are thus dealt in flat. Some corporations

gradually get deeper in financial trouble. As they come closer to bankruptcy, their bonds

lose their value drastically. Finally, they become junk bonds.

Video 03B Examples

3.1. An investor wants to buy a bond with face value $1,000 and coupon rate 12%. It

pays interest semiannually and it will mature after 5 years. If her required rate of return is

18%, how much should she pay for the bond?

The present value of a bond is the sum of the present value of its interest payments plus

the present value of its face value. The annual interest on the bonds is .12(1000) = $120,

and thus the semiannual interest payment is $60. The annual required rate is 18%, or 9%

semiannually. This is the discount rate. There are 10 semiannual periods in 5 years. Put n

= 10, r = .09, F = 1000 in (3.1), which gives

B = i=1

10

60

1.09i +

1000

1.0910 =

60(1 – 1.09–10

)

0.09 +

1000

1.0910 = $807.47

She should pay $807.47 for the bond. ♥

To verify the answer at WolframAlpha, use the following instruction.

WRA Sum[60/1.09^i,{i,1,10}]+1000/1.09^10

3.2. American Airlines bonds pay interest on January 15 and July 15, and they will

mature on July 15, 2017. Their coupon rate is 11%. Because of the risk characteristics of

http://finance.yahoo.com/bonds





49

American Airlines, you require a return of 15% annually on these bonds. How much

should you pay for a $1,000 bond on January 16, 2011?

The bonds will mature in 6.5 years and you will receive 13 interest payments of $55 each.

obtained by setting n = 13, r = .075, F = 1000 in (3.1), which gives the PV of these

interest payments, plus the discounted face value as

B = i=1

13

55

1.075i +

1000

1.07513 =

55[1 – 1.075–13

]

0.075 +

1000

1.07513 = $837.48 ♥

3.3. A zero coupon bond with face value $1,000 and 6.25 years until maturity is available

in the market. Because of its risk characteristics, you require a 11.5% return,

compounded annually, on this bond. How much should you pay for it?

For a zero-coupon bond, use B = F

(1 + r)n (3.3)

Put F = 1000, r = .115, n = 6.25,

to get B = 1000

1.1156.25 = $506.44 ♥

3.4. Canopus Corporation's 9% coupon bonds pay interest semiannually, and they will

mature in 10 years. You pay 30% tax on interest income, but only 15% on capital gains.

Your after-tax required rate of return is 12%. Assume that you pay taxes once a year.

What is the maximum price you are willing to pay for a $1,000 Canopus bond?

Suppose you pay x dollars for a $1,000 bond. The annual interest is $90; or $45 every six

months. For semiannual cash flows, the discount rate is 6%, which is one-half of the

annual required rate of return. In ten years, you will get 20 semiannual payments.

The annual tax on $90 interest income is .3(90) = $27.

After 10 years, you receive the face value of bond, $1,000, and you have a capital gain of

(1000–x). However, you have to pay tax on the capital gain, which comes to

(.15)(1000–x) = 150 – .15x. The after-tax amount is thus 1000–(150 – .15x) = 850 + .15x.

Apply the financial principle:

PV of the

bond

=

PV of 20 semiannual interest payments, discounted at 6%

−

PV of 30% of $90, paid in taxes

annually for 10 years

+

PV of the after-tax final payment, which is $1000 minus 15% of the difference between 1000 and x

Write it in symbols,

x = i=1

20

45

1.06i −

i=1

10

27

1.12i +

850 + .15x

1.1210


50

Or, x

1 – .15

1.1210 =

i=1

20

45

1.06i −

i=1

10

27

1.12i +

850

1.1210

Or, .9517040145 x = 516.1464548 − 152.5560218 + 273.6772511

x = 669.6070148 = $669.61 ♥

To verify the answer at WolframAlpha, use the following instruction.

WRA x=Sum[45/1.06î,{i,1,20}]-Sum[.3*90/1.12î,{i,1,10}]+(1000-(1000-x)*.15)/1.12^10

3.5. You have bought a $1,000 bond for $450, with a coupon of 5%, which has 10 years

until maturity. The interest is paid semiannually. Find the yield to maturity for this bond.

Here we use Y ≈ cF + (F − B)/n

(F + B)/2 (3.5)

Put c = .05, F = 1000, B = 450, n = 10, in (3.5), which gives

Y .05*1000 + (1000 − 450)/10

(1000 + 450)/2 = .1448

Or, about 14.5% per year. ♥

To find the yield accurately, we set the current price equal to the sum of discounted

future interest payments and the face value. Suppose the unknown yield to maturity is r,

which is also the proper discount rate to use in the bond valuation equation (3.1). Assume

the bond pays interest semiannually. Therefore, we should use r/2 as the discount rate for

$25 semiannual interest payments.

450 = i=1

20

25

(1 + r/2)i +

1000

(1 + r/2)20

We may solve this equation by using Maple. If we enter the command

fsolve(450=sum(25/(1+r)î,i=1..20)+1000/(1+r)^20,r);

the answer shows up as .1635007175

To solve the problem using WolframAlpha, write the above equation as

WRA 450=25*(1-1/(1+r/2)^20)/r*2+1000/(1+r/2)^20






51

Set up the spreadsheet in Excel as follows. Adjust the value in cell B5 until the value in

cell C5 becomes close to zero. The exact yield to maturity is in cell B5, with the quantity

in cell C5 equal to −.002, which is less than a penny.

A B C

1 Face amount of bond = 1000 Dollars

2 Market value of bond = 450 dollars

3 Coupon rate = 5%

4 Time to maturity = 10 years

5 Yield to maturity = 16.35% =B2-B3*B1/2*(1-1/(1+B5/2)^(2*B4))/B5*2-B1/(1+B5/2)^(2*B4)

This is the annual return, or 16.35%. This is the exact answer with four digit accuracy.

Video 03C 3.6. Bareilly Corporation bonds will mature after 3 years, and carry a

coupon rate of 12%. They pay interest semiannually. However, due to poor financial

condition of the company, you believe that there is a 30% probability Bareilly will go

bankrupt in any given year. In case of bankruptcy, you expect that the company will

make the interest payments for that year, and also pay only 20% of the principal at the

end of that year. If your required rate of return is 12%, find the value of this bond.

First, consider the probability of four possible outcomes:

Probability of going bankrupt during first year = .3

Probability of going bankrupt during second year = .7(.3) = .21

Probability of going bankrupt during third year = .7(.7)(.3) = .147

Probability of surviving three years = .7(.7)(.7) = 0.343

The total probability of all outcomes = .3 + .21 + .147 + .343 = 1

The present value of the bond is the present value of each cash flow multiplied by the

corresponding probability. The quantities in parentheses represent the PV of cash flows,

including the principal. Thus

PV = .3

i=1

2

60

1.06i +

200

1.062 + .21

i=1

4

60

1.06i +

200

1.064

+ .147

i=1

6

60

1.06i + 200

1.066 + .343

i=1

6

60

1.06i + 1000

1.066 = $570.43

The value of the bond is $570.43 ♥

You can also organize the calculation as follows.



52

Event Probability PV of cash flows Prob*PV

Bankrupt in first year .3 i=1

2

60

1.06i +

200

1.062 = 288.00

.3(288.00)

Bankrupt in second year .7(.3) = .21 i=1

4

60

1.06i +

200

1.064 = 366.33 .21(366.33)

Bankrupt in third year .7(.7)(.3) = .147 i=1

6

60

1.06i + 200

1.066 = 436.03 .147(436.03)

Survive all 3 years .7(.7)(.7) = .343 i=1

6

60

1.06i + 1000

1.066 = 1000 .343(1000)

Sum of the above 1 $570.43

3.7. Compton Company bonds pay interest semiannually, and they will mature after 10

years. Their current yield is 8%, whereas their yield to maturity is 10%. Find the coupon

rate and the market value of these bonds. Hint: use (3.1) and (3.4).

Since we have to find the value of two unknown quantities, the coupon rate, c and the

market value of the bond, B, we need to develop two equations. Recall that the yield to

maturity of a bond is the same as the required rate of return r. Assume semiannual

interest payments.

Put, semiannual required rate of return or the discount factor, r = .05,

The number of interest payments of the bond, n = 20,

The face value of the bond, F = 1000,

The dollar value of each interest payment, C = cF/2 = 500c, in (3.1)

B = i=1

20

500c

1.05i +

1000

1.0520 =

500c(1 − 1.05−20

)

.05 +

1000

1.0520

Or, with some simplification B = 6231.105171c + 376.8894829 (1)

This is the first equation. To get the second equation, put y = .08 in (3.4). This gives

.08 = c(1000)/B (2)

Solving (2) for c, we find c = (.08/1000)B

Substituting the above value of c in (1), we get

B = 6231.105171(.08/1000)B + 376.8894829

Or, B = .4984884137B + 376.8894829

Or, B(1 − .4984884137) = 376.8894829


53

Or, B = 376.8894829/(1 − .4984884137) = 751.5070303 $751.51

Going back to (2), c = (.08/1000) 751.5070303 = .06012056242 6.012%

This gives the coupon rate as 6.012%, and the market value of the bond to be $751.51. ♥

To solve the problem using WolframAlpha, write the two basic equations as

WRA B=Sum[500*c/1.05^i,{i,1,20}]+1000/1.05^20,.08=c*1000/B

3.3 Valuation of Stock

The two principal components of the capital structure of a company are its equity and

debt. A corporation sells its stock to the investors to raise equity capital. The financial

markets ultimately determine the value of a share of stock. If the company is in strong

financial condition and it has good earnings prospects, then the investors will

aggressively buy its stock and raise the price per share. The market value of a stock could

be quite different from its book value or its accounting value. The value of the stock

depends upon the expectations of the investors regarding the future earnings and growth

possibilities of the firm.

Table 3.5 gives the information about the stocks of some well-known companies. The

information is for close of business on January 5, 2007. The first column shows the range

of the stock price, in dollars, for the past 52 weeks. The next two columns give the name

of the company and its stock symbol. The fourth and fifth columns show the closing price

of the stock and its net change. General Electric, for instance, closed at $37.56 per share,

down 19¢. The next two columns show the annual dividend per share and the dividend

yield. For Boeing, the annual dividend is $1.40 per share and its dividend yield is

1.40/89.15 = .0157 = 1.57%.

52 Week

Range

Stock Sym-

bol

Close Net

Chg

Div Yld % PE Volume

1000s

Market

Cap

β

65.90-92.05 Boeing BA 89.15 -0.38 1.40 1.57 41.50 3,168 70.4B 0.62

44.81-57.00 Citigroup C 54.77 -0.29 1.96 3.60 11.79 13,130 269.1B 0.44

32.06-38.49 Gen Electric GE 37.56 -0.19 1.12 3.00 22.83 26,729 387.2B 0.51

32.85-43.95 Home Depot HD 37.79 -0.78 0.90 2.30 13.60 21,676 81.21B 1.28

21.46-30.26 Microsoft MSFT 29.64 -0.17 0.40 1.30 23.69 44,680 291.4B 0.71

27.83-37.34 PP&L PPL 35.55 -0.63 1.10 3.10 15.74 1,048 13.56B 0.24

Table 3.5: Stock prices. Source: Finance.Yahoo.com, January 7, 2007

The next column shows the P-E ratio, which is the ratio between the price of the stock per

share and the earnings per share. For Citigroup, it is 11.79. This gives the earnings per

share as 54.77/11.79 = $4.65 per share. Citigroup pays $1.96 as dividends out of this

money. Thus its dividend payout ratio is 1.96/4.65 = 42.15%. The next column shows the

trading volume. More than 44 million shares of Microsoft changed hands that day. The

next column shows the total market value of the company, in billions of dollars. The last

column shows the β of the stock. Beta is a measure of the risk of the stock. We shall look



http://finance.yahoo.com/


54

at it more closely in chapter 5. It is interesting to note that all stocks went down on this

trading day, but it is not surprising because all the major market indicators shown in

Table 3.6 also went down.

Symbol Last Change

Dow 12,398.01 ↓82.68 (0.66%)

NASDAQ 2,434.25 ↓19.18 (0.78%)

S&P 500 1,409.71 ↓8.63 (0.61%)

Table 3.6. The stock market indices on January 5, 2007, source, Finance.Yahoo.com

An investor buying the common stock of a corporation is looking at two possible returns:

the receipt of cash dividends and the growth of the company. Let us make a couple of

simplifying assumptions to develop a formula for stock valuation.

1. Assume that the firm is growing steadily, that is, its growth rate remains constant. This

also means that the dividends of the firm are also growing at a constant rate. In reality,

the firms grow in an uncertain way.

2. The growth is supposed to continue forever. This is also quite unrealistic because the

companies tend to grow rapidly at first, then the growth rate slows down, and some

mature firms actually decline in value.

Even though the assumptions are not very good, its gives a fairly accurate result. Let us

define:

P0 = price of the stock now

D1, D2, D3, ... = the stream of cash dividends received in year 1, 2, 3, ...

g = growth rate of the dividends

R = the required rate of return by the stockholders

Assuming that the company is going to grow forever, then the price of the stock now is

just the discounted value of all future dividends.

P0 = D1

1 + R +

D2

(1 + R)2 +

D3

(1 + R)3 + ... ∞

But D2 = D1(1 + g), D3 = D1(1 + g)2, D4 = D1(1 + g)

3, and so on. Thus

P0 = D1

1 + R +

D1(1 + g)

(1 + R)2 +

D1(1 + g)2

(1 + R)3 + ... ∞

This becomes an infinite geometric series, with the first term a = D1

1 + R and the ratio of

terms x = 1 + g

1 + R . Using equation (1.5),

http://finance.yahoo.com/


55

S∞ = a

1 − x (1.5)

we get

P0 = D1

(1 + R)

1 − 1 + g

1 + R

Simplifying it,

P0 = D1

R − g (3.6)

To solve the problem using WolframAlpha, write the above equations as

WRA Sum[D1*(1+g)^(i-1)/(1+R)^i,{i,1,infinity}]

Myron J. Gordon (1920-2010)

Equation (3.6) gives us the valuation of a common stock. It is a

well-known result called Gordon's growth model, named after

Myron Gordon. This result is valid only if R > g. Although it is

based on some unrealistic assumptions, it does provide fairly

accurate stock valuation. The following problems will illustrate

the general method of evaluation of equity when the growth rate

is not constant.

Examples

Video 03E 3.8. The common stock of Steerforth Inc has just paid a dividend of $2.00.

The dividends are expected to grow at the rate of 10% for the next three years, and then

at the rate of 5% forever. Find the price of this stock, assuming the required rate of return

is 20%.

The price of a stock is equal to the sum of discounted future dividends received by an

investor. The current dividend is $2, but it will grow at the rate of 10% for the next three

years. The dividend after one year is 2(1.1); after two years, it becomes 2(1.1)2; and after

three years, it is 2(1.1)3. After that, the growth slows down to 5%. Thereafter the

dividends are: 2(1.1)3(1.05) after 4 years, and 2(1.1)

3(1.05)

2 after 5 years, and so on. All

these numbers are discounted at the rate of 20%. If P0 is the current price of the stock,

then

P0 = 2 (1.1)

1.2 +

2 (1.1)2

1.22 +

2 (1.1)3

1.23 +

2 (1.1)3 (1.05)

1.24 +

2 (1.1)3

(1.05)2

1.25 + ...

Starting with the third term, 2 (1.1)

3

1.23 , it becomes an infinite series with a = 2(1.1/1.2)

3,

and x = 1.05/1.2. The sum of all terms is thus

P0 = 2 (1.1)

1.2 +

2 (1.1)2

1.22 +

2 (1.1/1.2)3

1 − 1.05/1.2 = $15.84 ♥



http://en.wikipedia.org/wiki/Myron_J._Gordon

http://www.scranton.edu/faculty/hussain/camtasia/Module03E/Module03E.html


56

3.9. Sirius Inc. common stock just paid a quarterly dividend of $1.00. Investors expect

this dividend to grow at annual rate of 4%, compounded quarterly, for the next 10

quarters. Then it will remain constant in future. The stockholders require a return of 12%

on their investment in Sirius. What is the current market price of Sirius common stock?

The growth rate of dividends is 4% per year, or 1% per quarter. The required rate of

return per year is 12%, or 3% per quarter. Adding together the discounted value of the

return from the first 10 quarters and the infinite many quarters thereafter will give the

value of the stock.

For the first 10 quarters, the dividends are growing at 1% per quarter and they are D1

=1.01, D2 = 1.012, D3 = 1.01

3, and so on. For the remaining quarters, starting with the

11th quarter, the dividend will remain constant at 1.0110

. The discount rate is 3%. Thus

P

0 = 1.01

1.03 +

1.012

1.032 +

1.013

1.033 + ... +

1.019

1.039 +

1.0110

1.0310 +

1.0110

1.0311 +

1.0110

1.0312 +

1.0110

1.0313 + ... ∞

Write the first ten terms as the summation of one geometric series and the remaining

terms as another infinite geometric series. For the first series, we let a = 1.01/1.03, x =

1.01/1.03, and n = 10 in equation (1.4). For the remaining terms, put a = 1.0110

/1.0311

,

and x = 1/1.03 in (1.5). This gives

P0 = (1.01/1.03)[1 − (1.01/1.03)

10]

1 − 1.01/1.03 +

1.0110

/1.0311

1 − 1/1.03

After simplification, the stock price comes out to be $36.39. ♥


WRA Sum[(1.01/1.03)^i,{i,1,10}]+ Sum[1.01^10/1.03^i,{i,11,infinity}]

3.10. Gabon Corporation is expected to have the following growth rates: 10% during the

first three years, 5% during the next three years, and then zero forever thereafter. Gabon

just paid its annual dividend of $5. What is the price of a share of Gabon stock if the

stockholders require a return of 10% on their investment?

We can find the stock price by the following summation. The PV of cash flows for the

first three years, next three years, and the remaining years, are shaded in different colors.

P0 = 5(1.1)

1.1 +

5(1.1)2

1.12 +

5(1.1)3

1.13 +

5(1.1)3(1.05)

1.14 +

5(1.1)3(1.05)

2

1.15 +

5(1.1)3(1.05)

3

1.16

+ 5(1.1)

3(1.05)

3

1.17 +

5(1.1)3(1.05)

3

1.18 +

5(1.1)3(1.05)

3

1.19 + ...




57

Canceling terms,

P0 = 5 + 5 + 5 + 5(1.05)

1.1 +

5(1.05)2

1.12 +

5(1.05)3

1.13 +

5(1.05)3

1.14 +

5(1.05)3

1.15 + ...

The shaded series is an infinite series. Add all items to get

P0 = 15 + 5(1.05/1.1) + 5(1.05/1.1)2 +

5(1.05/1.1)3

1 − 1/1.1 = $72.16 ♥

3.11. Troy Company has issued $5 cumulative preferred stock. The cumulative feature

means that if the company is unable to pay the dividends in any year, it must pay them

cumulatively next year. The probability that the company will actually pay the dividends

in any year is 70%. At the end of three years, Troy must pay all the dividends and buy

back the stock for $50 per share. If your required rate of return is 12%, how much should

you pay for a share of this stock?

The company will certainly make the final payment in year 3. Consider the three-year

period and the cash payments. There are four possible outcomes.

Year 1 Year 2 Year 3 Probability

$0 $0 $65 .3(.3) = 0.09

$0 $10 $55 .3(.7) = 0.21

$5 $0 $60 .7(.3) = 0.21

$5 $5 $55 .7(.7) = 0.49

The total probability of all outcomes is 1. The PV of the cash flows is thus

PV = 0.09

65

1.123 + 0.21

10

1.122 +

55

1.123 + 0.21

5

1.12 +

60

1.123 + 0.49

5

1.12 +

5

1.122 +

55

1.123

= $47.29

You should pay at most $47.29 for one share of stock. ♥

3.12. Mayfield Corporation stock is expected to pay a dividend of $2.00 one year from

now, $2.50 two years from now, $3.00 three years from now, and then $4.00 a year at the

end of the fourth and subsequent years. The stockholders of Mayfield require 15% return

on their investment. Find the price of a share of Mayfield stock now, and just after the

payment of the first dividend.

The value of the stock is the present value of all future dividends, discounted at the rate

of 15%. Display the cash flows and their present values in the following table.


58

Year 1 2 3 4 5 6 ... ∞

Dividend $2.00 $2.50 $3.00 $4.00 $4.00 $4.00 ... $4.00

Discounted

value

2

1.15

2.5

1.152

3

1.153

4

1.154

4

1.155

4

1.156

... 4

1.15∞

Discounting at the rate of 15%, write the present value of the cash flows as

P0 = 2

1.15 +

2.5

1.152 +

3

1.153 +

4

1.154 +

4

1.155 +

4

1.156 + … ∞

Look at the shaded terms. Note that starting with the fourth term, with $4 dividend, it

becomes an infinite geometric series. The first term in the series, a = 4

1.154 , and the ratio

between the terms, x = 1/1.15. Use equation (1.5) to find the sum.

S∞ = a

1 − x (1.5)

It gives,

P0 = 2

1.15 +

2.5

1.152 +

3

1.153 +

4/1.154

1 − 1/1.15 = $23.14 ♥

After the payment of the first $2 dividend, the next dividend will be $2.50 available one

year later, $3.00 two years later and so on. To visualize the cash flows, draw another

table as follows.

Year 1 2 3 4 5 6 ... ∞

Dividend $2.50 $3.00 $4.00 $4.00 $4.00 $4.00 ... $4.00

Proceeding on the same lines as before, we get the present value of the cash flows

P1 = 2.5

1.15 +

3

1.152 +

4

1.153 +

4

1.154 +

4

1.155 +

4

1.156 + … ∞

Starting with the third term, it becomes an infinite geometric series, with a = 4/1.153, and

x = 1/1.15. Use again to get

P1 = 2.5

1.15 +

3

1.152 +

4/1.153

1 − 1/1.15 + … ∞ = $24.61 ♥

Why does the value of the stock increase after a year? This is because the investors are

expecting to receive a higher set of dividends. This becomes clear when we compare the

two timelines and the cash flows.


59

Problems

3.13. Philadelphia Electric Co. (now Exelon) bonds were once selling at 120.25, with 27

years to maturity and semiannual interest payments. The coupon rate was 18%. If your

required rate of return were 16% at the time, would you have bought these bonds?

B = $1,123.04, no ♥

3.14. The Somerset Company bonds have a coupon of 9%, paying interest semiannually.

They will mature in 10 years. However, because of poor financial health of the company

you do not expect to receive more than 5 interest payments. Also, you do not expect to

receive more than 50% of the principal, after 4 years. Your required rate of return is 12%.

What is the maximum price you are willing to pay for these bonds? $503.26 ♥

3.15. Gambia Express bonds have a coupon of 8%, pay interest semiannually, have a

face value of $1,000 and will mature after 10 years. Your income tax rate for interest

income is 40%, but only 16% on capital gains. You pay the taxes once a year. How much

should you pay for a Gambia bond if your after-tax required rate of return is 10%?

$666.85 ♥

3.16. Leo Corporation bonds have a coupon of 9%; they pay interest semiannually; and

they will mature in 6 years. You pay 30% tax on ordinary income and 20% on capital

gains. What price should you pay for a Leo bond so that it gives you an after-tax return of

15%? $647.78 ♥

3.17. In 2008, Rumsfeld Co 13s2027 bonds paid interest annually, and their price was

quoted as 98. You had to pay 28% tax on interest income and capital gains, and your

required after-tax rate of return was 10%. Do you think you would have bought these

bonds as a long term investment? No, B = $943.90 ♥

3.18. Caruso Corporation 9% bonds will mature on January 15, 2019. They pay interest

semiannually. On July 16, 2007, these bonds are quoted as 87.375. If your required rate

of return is 11.5%, should you buy these bonds? No, B = $842.70, they sell at $873.75 ♥

3.19. Cincinnati Corporation 9% bonds pay interest semiannually, on April 15 and

October 15, and they will mature on April 15, 2019. They are selling at 89 on October 16,

2008. Considering its risk characteristics, your required rate of return for this bond is

10%.

(A) Do you think you should buy this bond? Yes, B = $935.89 ♥

(B) Suppose you buy the bond at the market price, what is its approximate yield to

maturity? 10.63% ♥

(C) Use Excel, Maple, or WolframAlpha to find its exact yield to maturity. 10.77% ♥

3.20. HAL Inc. stock is selling for $121 per share and it just paid an annual dividend of

$4.40. According to your careful analysis, you feel that HAL will continue to grow at the

rate of 25% per year for the next three years and then it will maintain a growth rate of

http://en.wikipedia.org/wiki/Exelon


60

10% per year forever. Its dividend payout ratio is expected to remain constant. Would

you invest your money in HAL, if your required rate of return is 16%?

S = $116.29, don't buy. ♥

3.21. White Rock Company stock just paid an annual dividend of $2.00. The dividends

are expected to grow at the rate of 3% annually for the next 10 years. After 10 years

White Rock will stop growing altogether, but will continue to pay dividends at a constant

rate. What is the price of this stock, assuming 12% discount rate. $20.20 ♥

3.22. Baffin Corporation stock has just paid the annual dividend of $4.00. The company

is expected to grow at the rate of 10% (along with its dividends) for the next three years,

then it is expected to grow at the rate of 3% forever. The investors require a return of

12% for their investment in the Baffin stock. What is the fair market price of a share of

the stock? $54.95 ♥

3.23. Timon Corporation stock has just paid the annual dividend of $2. This dividend is

expected to grow at the rate of 5% per year for the next ten years, and then it will remain

constant. If your required rate of return is 12%, how much should you pay for a share of

Timon stock? $23.01 ♥

3.24. McCormack Corporation just paid the annual dividend of $4.00. The dividends are

expected to have a growth rate as follows: g1 = 7%, g2 = 5%, g3 = 3%, g4 = g5 = ... = 0,

where the g's represent the growth in the first year, second year, etc. Your required rate of

return is 10%. How much should you pay for a share of McCormack stock? $45.86 ♥

3.25. Carpenter Corporation is expected to pay $2.00 dividend after one year, $3.00 after

2 years, $4.00 after 3 years, and then $5.00 a year uniformly after fourth and subsequent

years. If the stockholders of Carpenter require 12% return on their investment, find the

price of the stock now. What is its price just after the payment of the first $2.00 dividend?

$36.68, $39.08 ♥

3.26. Clifford Corporation stock is expected to pay a dividend on every January 25. In

2008, the dividend is $3.00, in 2009 $3.25, in 2010 $3.50, and in 2011 and all the

subsequent years it is expected to be $4.00. The shareholders of Clifford require a return

of 13% on their investment. Find the price of this stock on January 14, 2008, just before it

pays its dividend. What is its price on January 28, 2010, just after it has paid its dividend?

$32.71, $30.77 ♥

Key Terms accrued interest, 40, 47

beta, 52

board of directors, 39

bond rating, 45

bondholders, 39, 41, 42

bonds, 39, 40, 41, 42, 43, 44,

45, 47, 48, 50, 51, 57, 58

book value, 52

call money, 46

callable bonds, 41

capital, 39, 48, 52, 58

capital investments, 39

commercial paper, 46

Consumer Price Index, 46

convertible bond, 39, 42

coupon, 40, 41, 42, 43, 44,

47, 48, 49, 50, 51, 52, 57,

58

current yield, 43

debt capital, 39

discount rate, 41, 43, 48, 55,

58

Discount Rate, 46


61

dividends, 39, 52, 53, 54, 55,

56, 58, 59

equity, 39, 52, 54

equity capital, 39, 52

Federal Funds, 46

Gordon's growth model, 39,

54

indenture, 41

interest, 39, 40, 41, 42, 43,

46, 47, 48, 49, 50, 51, 57,

58

junior notes, 40

junk bonds, 42, 47

LIBOR, 46

P-E ratio, 52

perpetual bond, 39, 42

prime rate, 46

risk, 41, 42, 47, 48, 52, 58

senior notes, 40

sinking fund, 41

stock, 39, 42, 52, 53, 54, 55,

56, 58, 59

stockholders, 39, 53, 55, 56,

59

Treasury Bills, 46

underwriters, 40

yield, 43, 44, 45, 47, 49, 51,

52, 58

yield-to-maturity, 43

zero-coupon bond, 39, 42

y

62

4. CAPITAL BUDGETING UNDER CERTAINTY

Objectives: After reading this chapter, you will

1. Understand the concept of net present value.

2. Use NPV approach in decision-making.

3. Understand the meaning of internal rate of return. Appreciate the difficulty in applying

this concept, and its inability to give a unique, or optimal solution.

4. See the effect of depreciation and taxes on the investment decisions.

4.1 Video 04A Capital Budgeting

In the course of their business, firms have to make capital investment decisions. This

involves critical evaluation of long-term investments and their impact on the value of the

company. The corporations make large investments in buildings and land, and in plant

and equipment. There is a constant need for modernization of equipment due to changes

in technology. As the firm grows, they need larger facilities. A firm may embark on new

projects, which may entail large investment of time and capital. The firm considers all

these decisions in light of the long-term benefit of the corporation. From the financial

point of view, only those projects will be acceptable that add to the value of the firm, and

increase the wealth of the owners of the firm.

A company has to evaluate many projects. Some of these projects may be mutually

exclusive in the sense that you have to pick only one and exclude others. A company may

want to install gas heat, or oil heat, in a factory, but not both. The company may have to

evaluate several alternative projects and rank them according to their profitability.

Finally, they may have to pick only one or two projects that they can finance with the

available capital. Thus, capital budgeting becomes an important issue.

We will consider one of the most important concepts in finance, the net present value,

which is the optimal decision making model to screen out the profitable projects from the

unprofitable ones. The net present value, NPV, of a project is the discounted sum of all

cash flows of a project, negative and positive, present and future. We may treat the initial

investment as a negative cash flow at present. Discount the future cash flows at a rate that

depends on the cost of capital of the firm and the risk of the project. By definition,

NPV = − I0 + i=1

n

C

(1 + r)i (4.1)

In the above expression, we define the symbols as follows:

I0 = initial investment in the project,

C = after-tax annual earnings of the project,

n = life of the project in years, and

r = risk-adjusted discount rate for the cash flows.


Analytical Techniques 4. Capital Budgeting Under Certainty _____________________________________________________________________________

63

The decision rule in using NPV is that if the NPV is positive the project is acceptable,

otherwise not. This is also in concert with the fundamental aim of the corporation to

maximize its value. Alternative decision rules such as the internal rate of return, IRR or

payback period are inadequate in many situations and may give misleading results.

The payback period method is really quite easy to apply. For example if a firm spends

$5000 to start a project that generates an income of $1000 annually, then the payback

period is 5 years. On the other hand, a project that costs $8,000 and generates $2000

annually will have the payback period 4 years. Based on the criterion of payback period,

the second project with the shorter time is better.

There are three serious defects in the payback period method. The main problem with this

methodology is that this procedure ignores the time value of money. It does not discount

the future cash flows and treats them at par with the present investment. This violates a

very fundamental concept in finance.

The second problem is that we are not looking at the risk of the project. The riskier cash

flows should have less value than more secure cash flows. We should adjust the discount

rate according the risk involved.

The third drawback is that we are not looking at the entire set of cash flows, meaning, we

ignore the cash flows that occur after the time when we have recovered the initial

investment. Perhaps there are large negative cash flows that appear after the recovery of

the initial investment. This could change the calculation completely. We shall ignore this

method of project evaluation completely.

Closely related to the NPV method is the internal rate of return method. The internal rate

of return method has some merit. Actually, it is merely an extension of the NPV method

and we shall look at it in the next section.

We shall first consider simple problems in capital budgeting where the cash flows and

other outcomes are known with certainty. Later we shall include the complications due to

non-uniform cash flows, taxes and depreciation, and resale value. In the next chapter, we

shall continue the discussion of capital budgeting under uncertainty.

Examples

4.1. An investment requires the following cash outlays: $10,000 now and $5,000 a year

from now. The investment will give a cash return of $5,000 annually for 6 years, the first

payment coming in after 3 years. The risk-free rate is 6%. If the proper discount rate is

12%, would you accept this investment?

The firm should look at first two cash flows, $10,000 now, and $5000 a year from now,

as definite commitment to finance the project. The firm can certainly pay $5000 next year

by investing in a risk-free bond now, whose present value is 5000/1.06. Considering all

the cash flows,


64

NPV = –10,000 – 5000

1.06 +

5000

1.123 +

5000

1.124 +

5000

1.125 +

5000

1.126 +

5000

1.127 +

5000

1.128

We can either add them up separately, or combine them with the help of a formula. We

may write the above cash flows as

NPV = –10,000 – 5000

1.06 +

1

1.122 [

5000

1.12 +

5000

1.122 +

5000

1.123 +

5000

1.124 +

5000

1.125 +

5000

1.126]

NPV = –10,000 − 5000

1.06 +

1

1.122

i=1

6

5000

1.12i

Use, i=1

n

C

(1 + r)i =

C[1 − (1 + r)−n

]

r (2.6)

NPV = – 10,000 – 5000

1.06 +

1

1.122

5000(1 − 1.12

−6)

.12 = $1671

Because of the positive NPV, we should accept the investment. ♥


WRA -10000-5000/1.06+Sum[5000/1.12^i,{i,3,8}]

4.2. A young woman buys a life insurance policy on her 21st birthday. She has to pay an

annual premium of $147 through her 64th birthday. On her 65th birthday, she will receive

$10,000 as the surrender value of the policy. If she lives long enough to collect herself,

and assuming a discount rate of 12% in this case, find the NPV of this policy to the owner

of the policy.

Of course, the life insurance policy will also provide a $10,000 benefit to her heirs in case

she dies before reaching her 65th birthday. Here we are concerned only with the NPV of

her investment in case she lives to collect the benefits herself. She makes 44 payments of

$147 each, the first one right now. She also receives one payment of $10,000 after 44

years. Considering the present value of all the payments, we have

NPV = –147 – i=1

43

147

1.12i +

10‚000

1.1244 = –$1294.33

The negative NPV in this case does not mean that she should not buy the insurance. In

fact, it may be quite reasonable to provide $10,000 benefits to her children in case she

dies before she reaches the age of 65 by paying $1294 in current dollars. ♥

To solve the problem at WolframAlpha, use the following expression





65

WRA -Sum[147/1.12î,{i,0,43}]+10000/1.12^44

4.3. Devon Inc. wishes to invest $50,000 in a new project, which will give a return of

$10,000 annually for the first 5 years, and then an uncertain amount every year for the

next 5 years. The proper discount rate is 11% annually. Calculate the minimum value of

the uncertain return, which will make the project worthwhile for Devon.

Suppose the uncertain cash flow is x. To break even, the NPV of the project is zero. Thus,

we may write the problem as follows:

NPV = 0 = – 50,000 + i=1

5

10‚000

1.11i +

i=6

10

x

1.11i

The second summation on the right side is equivalent to 1

1.115

i=1

5

x

1.11i .

Using (2.6), we get

50,000 – 10‚000(1 – 1.11–5)

.11 =

1

1.115

x(1 – 1.11–5)

.11

Solving for x, we get

1.115

.11

1 − 1.11−5 [50,000 −

10 000(1 – 1.11–5)

.11] = x

This gives x = $5945.75 ♥

To solve the problem at WolframAlpha, use the following expression

WRA 0=-50000+Sum[10000/1.11î,{i,1,5}]+Sum[x/1.11î,{i,6,10}]

4.2 Video 04B Internal Rate of Return

The internal rate of return, or IRR, of a project is that particular discount r, which will

make the net present value of the project equal to zero. If we let

NPV = 0 = − I0 + i=1

n

C

(1 + r)i (4.2)

and solve the equation for r, then this particular discount rate is the internal rate of return.

Once we find the IRR, it is compared with the risk-adjusted discount rate for the given

project. If IRR is greater, the project is accepted.

This equation is difficult to solve in general. However, certain calculators, such as HP-

12C, have the capability of getting the answer. If the cash flows are uniform, using tables

and interpolating the value of the discount rate may solve the problem. The best way to

handle (4.2) is to use Maple.






66

Although many managers use the IRR as a decision-making tool to accept or reject a

project, it has some serious flaws.

First, a project may not have a unique IRR. This is because a quadratic or a higher degree

equation has multiple roots. Some of these values do not have any economic significance

whatsoever, and it is not always possible to identify the correct value. Second, one cannot

use the IRR method reliably to rank projects. This is again due to multiplicity of roots of

the equation.

All these problems are absent in the NPV method, which is the optimal method for

decision-making. The only advantage of using IRR is that one can compare it directly to a

hurdle rate, or a minimum acceptable rate of return set by the managers of a corporation.

Examples

4.4. Betsey Trotwood is planning to open a restaurant. Her initial investment will be

$50,000. She expects to receive $20,000 at the end of first, second, and third year. Find

the internal rate of return of her project.

In general, we can solve the internal rate of return problems using Maple. We equate the

present value of all cash flows to zero, and find the proper discount rate. In this case,

– 50,000 + i=1

3

20‚000

(1 + r)i = 0


WRA -50000+Sum[20000/(1+r)^i,{i,1,3}]=0

The real solution is r .0970103, which is about 9.7%.

To get the result using Excel, set up the spreadsheet as follows. Adjust the value of the

number in B1 until the result in cell B2 becomes very close to 0.

A B

1 IRR = .09701

2 NPV = 0 -50000+20000*(1-1/(1+B1)^3)/B1

4.5. An investment of $10,000 will return $3,000 at the end of each of the next five

years. Find the IRR of this investment.

To solve for IRR we set the NPV equal to zero. Thus

NPV = − 10,000 + i=1

5

3000

(1 + r)i = 0




67


WRA -10000+Sum[3000/(1+r)^i,{i,1,5}]=0

This gives the real solution as r .152382, which is 15.24%. ♥

4.6. An investment has an initial outlay of $1200; an income of $5,000 at the end of one

year; and an expense of $3,000 at the end of second year. Find the internal rate of return

of this investment.

By definition, the internal rate of return of an investment is the discount rate that will

make its net present value to be zero. Suppose the required discount rate is r. Then

NPV = −1200 + 5000

1 + r −

3000

(1 + r)2 = 0 (A)

Let 1 + r = x. Then

− 1200 + 5000

x −

3000

x2 = 0

or, multiplying by x2, – 1200 x

2 + 5000x – 3000 = 0

Dividing throughout by–200, we get

6x2 – 25x + 15 = 0

This is a quadratic equation and we may solve it by using the standard formula. The

solution of

ax2 + bx + c = 0

is

x = – b ± b

2 – 4ac

2a

In our case the solution is

x = 25 ± 625 – 4(6)(15)

12 =

25 ± 265

12 = 3.44 or 0.727

Since x = 1 + r, r = 2.44, – 0.273. This is a case of multiple internal rates of return. There

is not much economic sense in the two values of IRR calculated above. Therefore, we are

unable to decide the case on the basis of IRR. ♥

At WolframAlpha, write equation (A) as follows and click on Approximate forms.

WRA -1200+5000/(1+r)-3000/(1+r)^2=0






68

Fig. 4.1 shows the calculated value of NPV at different discount rates. When the discount

rate is zero, NPV = $800. Note that the NPV = 0 for r = .273 and 2.44. By differentiating

the function

−1200 + 5000

1 + r −

3000

(1 + r)2

with respect to r and setting the derivative equal to zero, we get the maximum value of

NPV as $883 when r = 20%. We can observe that in Fig. 4.1.

Fig. 4.1: The diagram shows the IRR for a project. The curve crosses the x-axis at r = −.273 and r = 2.44. 4.7. (A) Jefferson Corporation is considering a project that requires a cash outlay of

$4,000 now, and another $3,000 expense one year from now. The risk-free rate is 6%.

The project will terminate after two years, at which time it will generate a single positive

cash flow of $10,000. Calculate the internal rate of return of this project.

(B) If the cost of capital for Jefferson is 20%, should it undertake the above project?

(C) Verify your answer to (B) by calculating the NPV of the project.

(A) Setting the NPV equal to zero, we have

– 4000 − 3000

1.06 +

10‚000

(1 + r)2 = 0

Or, 10‚000

(1 + r)2 = 4000 +

3000

1.06 = 6830

Or, (1 + r)2 = 10,000/6830 = 1.464

Or, 1 + r = 1.209995206

Which gives r = .21 = 21% ♥


69

(B) With the cost of capital at 20%, which is less than the IRR, the project is acceptable.♥

(C) NPV = − 4,000 − 3000

1.06 +

10‚000

1.22 = $114.26

Since the NPV is positive, the project is indeed acceptable. ♥

4.3 Video 04D Taxes and Depreciation

At this point in our calculation of the net present value of a project, we must also include

two important considerations: depreciation and taxes. They have a strong impact on our

decision making process. We also have to contend with them in real life situations.

We know of the physical depreciation; that machinery and equipment wears down with

age. The value of old equipment decreases with time. The old equipment is subject to

frequent breakdowns and is not quite that productive as new equipment. Modern

technology tends to get obsolete rather quickly, that is, depreciates more rapidly. This

loss of value is the basis for depreciation as an accounting term.

Depreciation is a non-cash expense and companies use it to offset taxable income. One

can calculate the depreciation on a straight-line basis. A machine with a 5-year life will

have depreciation equal to 20% of its value in each year.

A faster method of depreciation is the sum-of-years-digits method. Since 1 + 2 + 3 + 4 +

5 = 15, the depreciation in the five years will be 5/15, 4/15, 3/15, 2/15 and 1/15,

respectively. A third method is the modified accelerated cost recovery system, or

MACRS. The following table gives a simplified version of MACRS for assets with a 3-

year or 5-year life.

Year MACRS, 3 year MACRS, 5 year

1 33.33% 20.00%

2 44.44 32.00

3 14.82 19.20

4 7.41 11.52

5 11.52

6 5.76

What is the optimal depreciation policy of a corporation? A company should use the

depreciation method, subject to IRS regulations, which gives it the maximum present

value of the tax benefits of depreciation.

Let us find the after-tax cash flows for a project as follows:

Pre-tax income per year = E

Depreciation per year = D

Taxable income per year = E − D

http://www.scranton.edu/faculty/hussain/camtasia/Module04D/Module04D.html


70

Income tax rate = t

Income tax due = t(E − D)

Earnings after taxes = E – t(E – D) = E – tE + tD = E(1 – t) + tD

This gives us the after-tax cash flow as

C = E(1 – t) + tD (4.3)

In the above equation, tD is called the tax benefit of depreciation. For a tax-exempt

entity, such as a university, t = 0. In that case (4.3) reduces to C = E, meaning after-tax

cash flow is the same s the pre-tax income. We should combine (4.1) and (4.3) to do the

NPV calculations involving taxes and depreciation.

As an example, consider an asset with initial value $50,000. The firm depreciates it with

MACRS, with three-year life, as shown in the previous table. Assume that the firm uses

11% as the discount rate and its tax rate is 32%. Then the present value of tax benefits of

depreciation is

= 50‚000(.3333)(.32)

1.11 +

50‚000(.4444)(.32)

1.112 +

50‚000(.1482)(.32)

1.113 +

50‚000(.0741)(.32)

1.114

= $13,090.08

In some cases, we have to include maintenance cost, or running cost, of a piece of

equipment. Maintenance expense is a tax-deductible item and its net cost is (1 – t)M. The

after tax cash flow in this situation is

C = E(1 – t) + tD – (1 – t)M

Or, C = (1 – t)(E – M) + tD (4.4)

4.4 Resale Value

A corporation may buy an asset, such as a car, or a machine, or a building, use it for a

number of years, and then sell it. We should, therefore, consider the additional factor, the

resale value of the machine. While using the asset, the corporation may depreciate the

asset and get the corresponding tax benefit, tD. When a company sells a piece of

equipment, it may, or may not, pay taxes on the sales price. It all depends upon the book

value of the equipment. The book value, B, of a capital asset is the original value of the

asset minus the depreciation already taken. For instance, if the initial value of a car is

$20,000 and it is depreciated at the rate of $5000 per year, then its book value after one

year is $15,000, after two years $10,000, after three years $5,000, and after four years,

when the car is fully depreciated, the book value is zero. By definition,

B = I0 – nD (4.5)


71

where I

0 is the price of the new equipment, n is the number of years it has been in service,

and D is the (uniform) annual depreciation. For non-uniform depreciation, nD represents

the total amount of depreciation. The book value of a brand new asset is I0, whereas the

book value of a fully depreciated asset is zero. Of course, the book value of an asset can

never be negative.

The tax, T due at the time of selling a piece of equipment is the income tax rate, t

multiplied by the difference between the sales price and the book value. Thus

T = t(S – B) (4.6)

where T = tax due, t = income tax rate, S = sale price of the equipment, and B = book

value of the equipment. If T is negative, the company gets a tax credit. This will happen

when the sale price is less than the book value of the asset. The after-tax value of the

sales price S becomes W, where

W = S – T = S – t(S – B)

Put B = I0 – nD from (4.5)

W = S − t[S − (I0 − nD)]

W = S – t(S − I0 + nD) (4.7)

Use equation (4.7) to find the after-tax resale value of an asset.

The following table gives the corporate income tax rate in USA in 2007.

Taxable income over Not over Tax rate

$0 $50,000 15%

50,000 75,000 25%

75,000 100,000 34%

100,000 335,000 39%

335,000 10,000,000 34%

10,000,000 15,000,000 35%

15,000,000 18,333,333 38%

18,333,333 .......... 35%

Examples

4.8. Dora Corporation is planning to buy a machine for $10,000, which will result in

$3,000 annual saving for the next five years. Dora will depreciate the machine in 5 years

using the straight-line method and then sell it for $1500. The tax rate of Dora is 30%, and

the proper discount rate is 15%. Should Dora make the investment?

A savings of $3,000 is equivalent to a pre-tax additional income of $3,000. The

depreciation is 10,000/5 = $2,000 annually. The machine is fully depreciated after five


72

years and the company pays taxes on resale value. After taxes, it is 1500(1 − .3) = $1050.

Using (4.3), we get the net cash flow as

C = 3000(1 – .3) + .3(2000) = $2700

NPV = – 10,000 + i=1

5

2700

1.15i +

1050

1.155 = – 10,000 +

2700(1 – 1.15–5

)

.15 +

1050

1.155 = – $427.15

Based on the above considerations, Dora should reject the proposal. ♥

WRA -10000+sum[(3000*(1-.3)+.3*10000/5)/1.15î,{i,1,5}]+1500*(1-.3)/1.15^5

4.9. Tyree Corporation is considering the purchase of a machine, which will cost

$80,000. Tyree will depreciate it uniformly over 4 years although it will run for 5 years.

It will then sell the machine for $10,000. The tax rate of Tyree is 25%, and the proper

discount rate is 11%. Find the minimum earnings before taxes generated by this machine

that will make it profitable.

Suppose the minimum earnings before taxes is x. After taxes, it becomes

C = x(1 − t) + tD = x(1 − .25) + .25(80,000/4) = .75x + 5000

The tax benefit of depreciation, tD = $5000, will continue for 4 years, while the other

factor, .75x, will go on for 5 years. The company pays taxes on the resale value. After

taxes, it becomes 10,000(1 − .25) = $7500. Discounting all the cash flows and setting

NPV = 0, we get

NPV = 80,000 + i=1

5

.75x

1.11i +

i=1

4

5000

1.11i +

7500

1.115 = 0

Or, i=1

5

.75x

1.11i = 80,000 −

i=1

4

5000

1.11i −

7500

1.115

Or, 2.771922763x = 60,036.88659

Or, x = $21,658.93 ♥

The machine must generate at least $21,658.93, before taxes, annually to be profitable.

WRA -80000+Sum[.75*x/1.11î,{i,1,5}]+Sum[5000/1.11î,{i,1,4}]+7500/1.11^5=0

4.10. Tompkins Farms needs a harvesting machine that will need $4,000 in annual

maintenance costs. Tompkins will depreciate the machine fully over 10 years and then

sell it for 15% of its purchase price. It will save $18,000 in labor costs annually. The tax

rate of Tompkins is 30%, and the proper discount rate is 12%. How much should

Tompkins pay for the machine just to break even?




73

Suppose the initial investment in the machine is I0, which will result in the NPV to

become zero. The tax benefit of depreciation per year, tD = .3(I0/10) = .03 I0.

The annual cash flow C, including savings, maintenance expenses, depreciation and

taxes, is given by (4.4) as

C = (1 – .3)(18,000 – 4,000) + .03 I0 = 9800 + .03 I0

The resale value is .15I0. After taxes, it becomes .15(1 − .3) I0 = .105 I0. Using a discount

rate of 12%, (4.1) gives the NPV as

NPV = – I0 + i=1

10

9800 + .03 I0

1.12i +

.105 I0

1.1210 = 0

Isolating I0, I0 [− 1 + i=1

10

.03

1.12i +

.105

1.1210] = –

i=1

10

9800

1.12i

Or, − .7966861193 I0 = − 55,372.18568

Or, I0 = $69,503.14 ♥

Tompkins should not buy the machine for more than $69,503.14. ♥

WRA -x+Sum[(9800+.03*x)/1.12^i,{i,1,10}]+.105*x/1.12^10=0

4.5 Hurdle Rate

As the course progresses, you will develop a better understanding of the concepts. Take

for instance, the discount rate that a company uses in evaluating its projects. This depends

on two factors. The first factor is the cost of capital of a firm, which we will discuss in

Chapter 9. If the cost of capital for a firm is 12%, then the minimum acceptable rate of

return, and the discount rate, for a given project must be 12%.

The second factor is risk. The measurement of risk is more difficult. We will talk about it

in Chapter 6 and 7. Suppose a firm, with cost of capital 12%, is undertaking a rather risky

project. The risk-adjusted discount rate will be, perhaps, 14 or 15%. Additional source of

risk is due to leveraging, which will further increase the cost of capital. We will discuss

leveraged beta of a firm in chapter 11. Calculating the discount rate is a little complicated

but after completing this course, I hope you will have a better feel for this number.

In real life, firms use a “hurdle rate,” an arbitrarily chosen discount rate, which tends to

be too high, perhaps 25% or 30%. Otherwise, they will calculate the internal rate of

return of a project. If it is more than say 25%, they will accept the project. By using a

high discount rate, they are just playing it safe, but they are also rejecting many

reasonable projects with positive NPV.


http://en.wikipedia.org/wiki/Hurdle_rate


74

In Chapter 15, we will look at the analysis of investment opportunities. We have to look

at the proper discount rate again.

The US government occasionally provides investment tax credit to corporations to

encourage investment in plant and equipment. This gives a boost to the economic

development. For example, if 10% tax credit is available, the initial investment is 90% of

the price of the equipment and the remaining cost is deductible from the income tax of

the company. The total depreciation depends on the actual cost of the equipment, which

is, in this case 90% of the price of the equipment. Currently, however, this tax credit is

not available.

4.11. Grissom Corporation is planning to buy a new computer and is considering three

alternatives. The price of each computer, along with its annual revenues and maintenance

expenses, assumed to be at the end of each year, are shown below.

Brand Price Revenue Maintenance

IBM $80,000 $20,000 $4,000

Dell 70,000 18,000 3,000

Gateway 100,000 25,000 5,000

An investment tax credit of 10% is available and the company will depreciate the

computers over a 10-year period with no residual value. Grissom is in the 40% marginal

tax bracket and its hurdle rate is 15%. Which computer, if any, should Grissom acquire?

Consider IBM computer first. Its list price is $80,000, so Grissom is able to acquire it for

.9(80,000) = $72,000. Grissom will depreciate it over 10 years, thus the annual

depreciation is $7200.

Maintenance expense is a tax-deductible item and its after-tax cost is (1 – t)M. The after-

tax cash flow in this problem is

C = (1 – t)(E – M) + tD (4.4)

For IBM, C = (1 – .4)(20,000 – 4,000) + .4(72,000/10) = $12,480

NPV = – 72,000 + i=1

10

12‚480

1.15i = – 72,000 +

12‚480(1 – 1.15–10

)

.15 = – $9366

For Dell, C = (1 – .4)(18,000 – 3,000) + .4(63,000/10) = $11,520

NPV = – .9(70,000) + i=1

10

11‚520

1.15i = .9(70,000) +

11‚520(1 – 1.15–10)

.15 = $5184

For Gateway, C = (1 – .4)(25,000 – 5,000) + .4(90,000/10) = $15,600

NPV = –.9(100,000) + i=1

10

15‚600

1.15i = –.9(100,000) + 15‚600(1 – 1.15–10)

.15 =–$11,707

http://en.wikipedia.org/wiki/Investment_tax_credit

http://en.wikipedia.org/wiki/Investment_tax_credit

http://en.wikipedia.org/wiki/Hurdle_rate


75

Because all computers have a negative NPV, the firm should reject them all. However,

relative to one another, Dell computer is the best investment. ♥

4.12. The Cameron Company has these three options to buy a new machine:

Machine Cost Maintenance Cost of capital

Alpha $40,000 $5000 10%

Beta $45,000 $4500 9%

Gamma $50,000 $4000 8%

All the machines are identical in nature and each one will generate pre-tax revenue of

$18,000 annually. The company will depreciate any of the machines on a straight-line

basis over next 5 years with no residual value. The company is in 35% tax bracket.

Which machine should Cameron buy?

It is possible to have different discount rates. For instance, the manufacturer of the

machine may offer to loan the money to the firm to buy the machine at a lower rate to

compensate for the higher price of the equipment. The annual cash flow for each machine

and its NPV are as follows:

Alpha, C = (1 – .35)(18,000 – 5,000) + .35(8,000) = $11,250

NPV = –40,000 + i=1

5

11‚250

1.1i = –40,000 +

11‚250(1 – 1.1−5

)

.1 = $2646

Beta, C = (1 – .35)(18,000 – 4,500) + .35(9,000) = $11,925

NPV = –45,000 + i=1

5

11925

1.09i = –45,000 +

11925(1 – 1.09−5

)

.09 = $1384

Gamma, C = (1 – .35)(18,000 – 4,000) + .35(10,000) = $12,600

NPV = – 50,000 + i=1

5

12‚600

1.08i = – 50,000 +

12‚600(1 – 1.08−5

)

.08 = $308

Machine Alpha is the best buy on this basis. ♥

4.13. The Adams Company is planning to buy a new computer for $80,000 that will

increase the pre-tax earnings of the company by $30,000 annually. The company will

depreciate the computer fully on a straight-line basis over a 5-year period, and then sell it

for $10,000. The company has a tax rate of 40%. If the after-tax cost of capital of Adams

is 11%, should it purchase the computer?

This is a problem involving the resale value of the equipment. When the company sells

the fully-depreciated computer, its book value is zero. The tax applicable on the sale is


76

thus tS, by equation (4.4). The after-tax value of the sale is (1 – t)S = (1 – .4)*10,000 =

$6,000. This money is available after 5 years, and we must find its present value.

The cash flow is C = (1 – .4)(30,000) + .4(16,000) = $24,400

Including the sale of the equipment,

NPV = – 80,000 + i=1

5

24‚400

1.11i +

6000

1.115 (A)

This gives NPV = $13,741. It is a very good investment. Buy it. ♥

To solve the problem at WolframAlpha, write equation (A) as

WRA -80000+Sum[24400/1.11^i,{i,1,5}]+6000/1.11^5

4.14. Carver Corporation is planning to buy a machine costing $50,000, and it will

depreciate it fully along a straight line over 5 years. The machine will generate unknown

earnings before interest and taxes (EBIT), which will remain constant for the first 5 years

and then drop to half that value during the next five years. The tax rate of Carver is 30%,

and its discount rate is 10%. Calculate the EBIT for the machine to just break even, that

is, have zero NPV.

Suppose the unknown EBIT is E. The tax benefits of depreciation are available only

during the first five years. Then the cash flows are:

First five years, C = E(1 – .3) + .3(10,000) = .7 E + 3000

Next five years, C = (E/2)(1 – .3) = .35 E

To break even, set NPV = 0,

NPV = –50,000 + i=1

5

.7 E + 3000

1.1i + i=6

10

.35 E

1.1i = 0 (A)

Isolating the terms containing E, we get

– 50,000 + i=1

5

3000

1.1i + i=1

5

.7 E

1.1i + i=6

10

.35 E

1.1i = 0

Put i=6

10

.35 E

1.1i = 1

1.15

i=1

5

.35 E

1.1i in the above equation,

– 50,000 + 3000(1–1.1

−5)

0.1 = – E

0.7(1 – 1.1

−5)

0.1 +

1

1.15

0.35(1 –1.1−5

)

0.1

Or, 38,627.64 = 3.4773739 E




77

Or, E = $11,108 ♥

The machine must generate $11,108 per year for the first five years, and then half that

amount for the next five years just to break even.

To solve the problem using WolframAlpha, write equation (A) as

WRA -50000+Sum[(.7*x+3000)/1.1î,{i,1,5}]+Sum[.35*x/1.1î,{i,6,10}]=0

4.15. Gray Metals Company needs a new machine, which would save the company

$3,000 annually for the first five years and then $2,000 annually for another five years.

Gray will depreciate the machine on a straight-line basis for 10 years. Gray is in the 40%

tax bracket and its after-tax cost of capital is 8%. What is the break-even price of the

machine for Gray?

Suppose the break-even price is x, which should equal the discounted future cash flows,

including the tax benefits of depreciation. If the price of the machine is x, then

depreciation per year is x/10 or 0.1x and the corresponding tax advantage is 0.1(x)(0.4) =

0.04x. The quantity E(1 – t) for the first five years is 3000(1 – .4) = $1800, and for the

next five years, it is 2000(1 – .4) = $1200.

Consider the present value of three sets of cash flows:

(1) PV of E(1 – t) = $1800 annually for years 1-5

(2) PV of E(1 – t) = $1200 annually for years 6-10

(3) PV of tax benefits of depreciation, tD = 0.04x for years 1-10

The sum of these is the price of the machine x. Thus

x = i=1

5

1800

1.08i +

i=6

10

1200

1.08i +

i=1

10

.04x

1.08i (A)

Or, x

1 – i=1

10

.04

1.08i = i=1

5

1800

1.08i + 1

1.085

i=1

5

1200

1.08i

Or x

1 – 0.04(1 – 1.08

−10)

0.08 =

1800 (1 – 1.08−5

)

0.08 +

1

1.085

1200 (1 – 1.08−5

)

0.08

Or, x(.7315967) = 10,447.72

Or, x = 14,280.71

The break-even price of the machine is $14,280.71. ♥


WRA x=Sum[1800/1.08î,{i,1,5}]+Sum[1200/1.08î,{i,6,10}]+Sum[.04*x/1.08î,{i,1,10}]






78

4.16. Monroe Corporation needs a machine, which will cost $100,000. Monroe will

depreciate it on a straight-line basis over 5 years with no resale value. The tax rate of

Monroe is 28%, and its after-tax cost of capital is 11%. The machine will have an EBIT

of $18,000 a year for the first five years, and then an uncertain amount for the next five

years, years 6 through 10. Find the minimum amount of this uncertain EBIT, which will

make the purchase of this machine acceptable.

Suppose the unknown EBIT for the years 6 through 10 is E. The cash flows are:

For first five years, C = 18,000(1 – .28) + .28(20,000) = $18,560

For next five years, C = E(1 – .28) = .72 E

Setting NPV equal to zero, we have

NPV = –100,000 + i=1

5

18‚560

1.11i +

i=6

10

.72 E

1.11i = 0 (A)

Or, 100,000 – 18‚560(1 – 1.11

−5)

0.11 =

1

1.115

i=1

5

.72 E

1.11i =

1

1.115

0.72E(1 − 1.11

−5)

0.11

Or, E = [100,000 – 18‚560(1 – 1.11

−5)

0.11] (1.11

5)

.11

0.72(1 − 1.11−5

)

which gives E = $19,886 ♥


WRA -100000+Sum[18560/1.11^i,{i,1,5}]+Sum[.72*x/1.11^i,{i,6,10}]=0

4.6 Projects with Unequal Lives

All the problems that we have discussed so far involve either a single project, or two

projects with equal lives. In real life, we may have to compare the performance of two

machines with different lives. One machine may be cheaper, but it will not last as long as

a more expensive one. How can we possibly compare two machines with different lives?

One way to handle a problem like this one is to assume that you will continue to replace a

machine by a similar one forever. This means that the life of the project is infinity in each

case, but this life is spanned by one kind of machine or the other. Consider the following

problem.

4.17. You want to install a new heat pump in your house. Two different models are

available, Shinn and Gardner. They are both satisfactory in performance. Their

characteristics are as follows:




79

Heat Pump Cost Life Annual expenses

Shinn $3000 4 years $700

Gardner $4000 5 years $600

The proper discount rate is 12%. Which unit should you buy?

Assume that you cannot depreciate the equipment in your personal home. The problem

requires careful analysis since the machines have different lives. Assume that you will

replace the machines every 4 or 5 years forever. For the first unit, the NPV is

NPV = − 3000 − 700

1.12 −

700

1.122 −

700

1.123 −

700

1.124 −

3000

1.124 −

700

1.125 −

700

1.126 − ... (A)

The terms in the blue color represent the cost of new equipment and its replacement after

four years. The remaining terms are due to the annual expense of the heat pump. The

negative signs mean cash outflows. The entire series is consisting of two infinite series.

The ratio of the blue terms with numerator 3,000 is 1

1.124, and the ratio for the other terms

with numerator 700 is 1/1.12. Using (1.5), we get

NPV = − 3000

1 − 1

1.124

− 700

1.12 (1 − 1/1.12) = −$14,064.19

This figure, $14,064.19, represents the total cost (measured in present dollars) of using

the Shinn heat pump for an infinitely long period. The calculations for the Gardner unit

are done similarly. Change the cost of the equipment from $3000 to $4000; life of the

machine from 4 to 5 years, and annual cost from $700 to $600. Making these changes, for

the Gardner unit, we get

NPV = − 4000

1 − 1

1.125

− 600

1.12 (1 − 1/1.12) = −$14,246.99

Comparing the total costs, it is better to buy the Shinn Heat unit. ♥

To do the calculation using WolframAlpha, write equation (A) as

WRA -Sum[3000/1.12^(4*i),{i,0,infinity}]-Sum[700/1.12^i,{i,1,infinity}]

WRA -Sum[4000/1.12^(5*i),{i,0,infinity}]-Sum[600/1.12^i,{i,1,infinity}]

A second approach to do these problems is to find the equivalent annual cost. This cost is

the sum of the payments that will amortize the purchase price of the equipment, plus the

cost of running it. First, we find the amount needed to amortize the purchase cost of the

heat-pump. For Shinn, the amortization amount A is given by





80

3000 = i=1

4

A

1.12i

Solving it, we get A = $987.70. This means that either we can pay $3000 to buy the unit

now, or we may pay $987.70 every year for the next four years. Next, we add the annual

running cost, $700, to it. Then we get the total equivalent annual cost to be 987.70 + 700

= $1687.70.

Following the same procedure for the other unit, we get

4000 = i=1

5

A

1.12i

This gives A = $1109.64. The total equivalent annual cost is thus 1109.64 + 600 =

$1709.64. Comparing the two costs, we find that Shinn unit is cheaper.

If we find the PV of these costs in perpetuity, it comes out to be 1687.70/.12 =

$14,064.17 for Shinn, and 1709.64/.12 = $14,247.00 for Gardner. These numbers are the

same as found by the previous method. Thus, the two methods are equivalent.

4.18. Djibouti Corporation is considering the purchase of an air conditioning unit and it

has these two choices.

Unit Initial cost Annual cost Expected life

A $80,000 $10,000 5 years

B $70,000 $8,000 4 years

Djibouti is in the 40% tax bracket. Which unit should it buy? Assume that it will

depreciate each piece of equipment on a straight-line basis with no residual value during

its working life. The proper discount rate is 10%.

In this problem, we are concerned with three items: (1) The replacement cost of the unit

every four or five years (2) the annual after-tax cost of electricity and (3) the annual tax

benefits of depreciation of the unit.

For Unit A, the replacement cost is $80,000 every five years, the after-tax cost of

electricity is (1 .4)(10,000) = $6,000 annually, and the tax benefit from depreciation is

.4(80,000/5) = $6400. Combining the after-tax cost of electricity and the tax benefit of

depreciation, we have a net benefit of 6400 − 6000 = $400 annually. The NPV for an

infinite period is thus

NPV(A) = − 80,000 − 80‚000

1.15 − 80‚000

1.110 − ... +

400

1.1 +

400

1.12 +

400

1.13 + ... ∞


81

= − 80‚000

1 – 1/1.15 +

400

0.1 = − $207,038

For the second unit, the replacement cost is $70,000 every four years, the annual after-tax

cost of electricity is .6(8000) = $4800, and the tax benefit of depreciation is .4(70,000/4)

= $7000. Combining the last two we have a benefit of 7000 – 4800 = $2200. Using the

previous calculation as a guide, we have

NPV(B) = − 70 000

1 – 1/1.14 + 2200

0.1 = − $198,830

Based on these calculations, unit B is somewhat cheaper. ♥

Let us solve this problem using equivalent annual cost. The equivalent annual cost for

replacing the first unit is found from

80,000 = i=1

5

A

1.1i

This gives us 80,000 = A(1 – 1.1–5)

.1

Or, A = 80 000*.1

1 – 1.1–5 = $21,103.80

The tax benefit due to depreciation per year is .4*80,000/5 = $6400. The after-tax cost of

electricity is (1 – .4)10,000 = $6000. Thus the total cost for Unit A is 21,103.80 – 6400 +

6000 = $20,703.80.

The replacement cost for Unit B is found as B = 70‚000*.1

1 – 1.1–4 = $22,082.96. The tax benefit

due to depreciation per year is .4*70,000/4 = $7000. The after-tax cost of electricity is

(1 – .4)8,000 = $4800. Thus the total cost for Unit B is 22,082.96 – 7000 + 4800 =

$19,882.96.

Comparing their costs, $20,703.80 and $19,882.96, Unit B is cheaper,

To reconcile the answer, we also calculate the total cost for an infinite time horizon, we

get 20,703.80/.1 = $207,038 and 19,882.96/.1 = $198,830, as before. ♥

Problems

4.19. The Scranton Times is planning to buy a new press for $120,000 that will save the

company $30,000 annually. The press has a useful life of 10 years. The Times has a tax

rate of 40%, and it will depreciate the machine on a straight-line basis. The after-tax cost

of capital of the Times is 9.6%. Should it buy the new press? NPV = $22,536.20, yes ♥


82

4.20. Ellsmere Corporation plans to buy a new machine for $50,000, which will save the

company $12,000 annually. Ellsmere will depreciate the machine on the ACRS with

three-year life, the annual depreciation being 29%, 47%, and 24%. The company expects

that the machine will run for 5 years, and then it will sell it for $5,000. The after-tax cost

of capital to the company is 8%, and its tax rate is 40%. Should Ellsmere buy the

machine? NPV =–$1,970.99, no ♥

4.21. Cline Incorporated wants to buy a machine for $24,000, and depreciate it on

straight-line basis over 6 years. Cline has marginal tax rate of 35% and its after-tax cost

of capital is 7%. Calculate the minimum pre-tax annual earnings generated by this

machine to justify its purchase. $5592.46 ♥

4.22. Allen Corporation has to decide between the following two air conditioning units

for an office building. Both units are adequate in their performance.

Carrier Worthington

Initial cost $120,000 $80,000

Annual maintenance cost $10,000 $12,000

Annual electricity cost $20,000 $25,000

Expected life 6 years 5 years

The company will use straight-line depreciation, with no resale value. The tax rate of

Allen is 28%, and the proper discount rate is 10%. Which one of these units will prove to

be less costly in the long run?

NPV(Carrier) = –$435,529, NPV(Worthington) = –$432,638 (cheaper) ♥

4.23. Alcott Corp is interested in buying a machine for $40,000. It will depreciate the

machine uniformly to zero value over a 5-year period. During this period, the machine

will add $8,000 annually to the EBDIT of the company. Finally, Alcott will sell the

machine for $5,000. The tax rate of Alcott is 40% and the proper discount rate is 9%.

Should Alcott buy the machine? NPV = $6,933, no ♥

4.24. Darwin Corporation is going to buy a machine for $152,000 that will save the

company $20,000 annually. Darwin will depreciate the machine completely in five years

using straight-line method. The tax rate of company is 30%, and it uses a discount rate of

12%. Show that this machine will never be profitable.

Key Terms capital budgeting, 60, 61

capital investment, 60

cost of capital, 60, 66, 67, 70,

73, 74, 75, 79

depreciation, 60, 61, 67, 68,

69, 71, 73, 74, 78, 79

internal rate of return, 60, 61,

63, 64, 65, 66

mutually exclusive, 60

net present value, 60, 63, 65,

67

non-uniform cash flows, 61

payback period, 61

risk, 60, 61, 63, 66

taxes, 60, 61, 67, 68, 69, 73

83

5. CAPITAL BUDGETING UNDER UNCERTAINTY

Objectives: After reading this chapter, you should

1. Understand the basic ideas of discrete and continuous probability distributions.

2. Apply the concepts of probability to the problems of financial decision-making.

3. Be able to analyze problems involving inflation.

4. Understand the role of options in the capital budgeting decisions.

5.1 Probability and Capital Budgeting

We do not know the outcome of many future events with certainty. One way to handle

the problem is to use a probabilistic model that would describe the situation. This is

especially true of financial decisions where we do not know the future cash flows exactly.

One way to overcome this uncertainty is to develop a subjective probability distribution

about different possible outcomes. To find the expected value of the uncertain outcome,

we first multiply the probability of various possible outcomes with the value of each

outcome, and then sum them all. We express this in the form of an equation:

E(V) = i=1

n

PiVi (1.6)

Here E(V) is the expected value of an uncertain outcome, Pi is the probability that the

outcome Vi will occur, and n is the total number of possible outcomes.

If there are a large number of independent observations of a random event, then we may

approximate the result by the normal probability distribution. The two parameters

describing the distribution are the mean, or the expected value μ of a variable, and its

standard deviation σ. A smooth bell-shaped curve will represent the probabilities. The

area under the curve represents the cumulative probability of a certain outcome. These

values are available in a table set up so that the total area under the curve is unity, and the

area under half the curve is 0.5. The table provides the values for the area under the

curve, measured from the center, up to a point that is a certain number z of standard

deviations away from the mean value.

Examples

Video 05A 5.1. Mr. Barkis is considering an investment in Murdstone Inc stock. He

believes that the continuously compounded returns of the stock have a normal

distribution, with mean of 15%, and standard deviation 30%. What is the probability that

the continuously compounded return is more than 20%? What is the probability that his

loss is greater than 10%?

Using the concept of continuous compounding, we may relate the final stock price P1 and

the initial price P0 by the expression P1 = P0erT

. Here T is a certain time period, say one

http://en.wikipedia.org/wiki/Normal_probability_distribution


Analytical Techniques 5. Capital Budgeting Under Uncertainty _____________________________________________________________________________

84

year. We can find the value of the continuously compounded rate of return r from this

expression as

r = ln(P1/P0)

T (5.1)

We define z as the number of standard deviations that a given value is away from the

mean value. Here the mean value of returns is μ = 0.15, the required return is x = 0.2, and

the standard deviation is σ= 0.3. Since the required return is more than the expected

return, it is unlikely that the stock will accomplish that. The resulting probability is less

than 50%. Find z as

z = x − μ

σ =

0.2 – 0.15

0.3 =

0.05

0.3 = 0.1667

Now draw the normal probability distribution curve with z = 0 in the center and z =

0.1667 a little right of the center. The area to right of z = 0.1667, shaded yellow, under

the tail of the curve will represent the answer.

The probability table, in Chapter 16, shows the area under the curve from the mid point to

a given point z. This is the green area in the above diagram. In our case, we have to find

the probability of making more than 20% on our investment. This is equivalent to the

yellow area on the right side of z = .1667. In the table, the area corresponding to z = .16 is

.0636, and for z = .17, it is .0675. We have to find the area for z = .1667, which is 67% of

the way between z = .16 and z = .17. We have to interpolate the numbers between .0636

and .0675. The total area under the curve is 1, half of it is .5, and so we have

P(r > 0.2) = .5 – [.0636 + .67(.0675 – .0636)] = 0.4338, or 43.38%. ♥

You can check the naswer at Excel by copying the following instruction.

EXCEL =1-NORMDIST(.2,.15,.3,TRUE)

The answer is quite plausible. We expect to make 15%, and there is a good possibility

that we may make more than 20% in view the standard deviation of 30%.


85

For a loss of 10%, the return is – 0.1. For the loss to be greater than 10%, the return must

be less than –10%. We calculate z to be

z = x − μ

σ =

– 0.1 – 0.15

0.3 = −.8333

For a negative value of z, you just take the absolute value, because the probability table

gives the area under the curve on either side of the mean value. This time we have to look

for area on the left side of z = –.8333, under the tail of the curve, which corresponds to a

return of less than –10%.

From the table, we have first get the area between z = −.8333 and the center of the curve.

In the table, The area for z = .83 is .2967 and that for z = .84, it is .2995. Since we have to

go 33% of the way from .83 to .84 to reach .8333, we intertoplate the table as follows.

P(r < – 0.1) = .5 – [.2967 + .33(.2995 – .2967)] = .2024 or 20.24%. ♥

This result seems reasonable because there is a fair chance that the stock can indeed go

down by 10%.

A B

1 Expected return, μ .15

2 Standard deviation, σ .3

3 First required return, x .2

4 z = (x – μ)/σ =(B3-B1)/B2

5 Prob(R > .2) =1-NORMDIST(B4,0,1,true)

6 Second required return, x -.1

7 z = (x – μ)/σ =(B6-B1)/B2

8 Prob(R < -.1) =NORMDIST(B7,0,1,true)

In Excel, the function NORMDIST(x,mean,standard_dev,cumulative) serves the purpose of

the probability table in Chapter 16. For instance, NORMDIST(.2,.15,.3,true) will calculate

the cumulative area from the left end of the curve up to point x. If you need the area to the

right point x, you enter 1 − NORMDIST(.2,.15,.3,true). To do the problem, you can enter

the following instructions.



86

5.2. Black Ink is in financial distress. Its bonds have a 12% coupon rate and they pay the

interest semiannually. There is a 70% chance that it will go bankrupt after 1 year, and it

faces certain bankruptcy after 2 years. In case of bankruptcy, the company will pay

interest due on the bonds, but will pay only 30% of the principal, at the end of that year.

If your required rate of return is 12%, how much should you pay for a $1,000 Black Ink

bond?

In this problem, we look at the probability of an outcome and multiply it with the dollar

amount of that outcome, and then add all the products. There is a 70% chance that the

company will go bankrupt after one year, and 30% that it will be bankrupt after two

years. These are the only two possible outcomes because the company will not survive

after two years.

If the company goes bankrupt after one year, the bond should pay two interest payments

plus $300 at the end of one year. If it survives another year, it should pay four interest

payments, plus $300 at the end of the second year. The semiannual discount rate is 6%.

Multiplying the probability of an outcome with the present value of that outcome, and

adding the results, we have

P0 = 0.7

i=1

2

60

1.06i +

300

1.062 + 0.3

i=1

4

60

1.06i +

300

1.064 (A)

= 0.7

60(1 – 1.06

–2)

0.06 +

300

1.062 + 0.3

60(1 – 1.06

–4)

0.06 +

300

1.064 = $397.56 ♥

You may check the answer at WolframAlpha, by writing equation (A) as

WRA .7*(Sum[60/1.06^i,{i,1,2}]+300/1.06^2)+.3*(Sum[60/1.06^i,{i,1,4}]+300/1.06^4)

5.3. Mrs. Guinea just became a widow at the age of 65. She is eligible to receive

$100,000 in cash for the life insurance policy of her husband. She also has the alternate

choice of receiving $15,000 in annual installments for as long as she lives, with the first

payment available now. Mrs. Guinea can invest money with a return of 12%, and she has

chosen the second option. Find the minimum number of installments that she should

receive to come out ahead. Actuarial tables indicate that the expected life of a 65-year old

female is 16 years, with a standard deviation of 6 years. What is the probability that Mrs.

Guinea has made the right decision?

Suppose Mrs. Guinea lives for the next n years. The first payment is available

immediately. To break even,

100,000 = 15,000 + i=1

n

15000

1.12i (A)

Simplifying terms, 85 = 15 (1 – 1.12

–n)

0.12




87

Or, 1.12−n

= 0.32

Or, n = – ln(0.32)

ln(1.12) = 10.05


WRA 100000=15000+Sum[15000/1.12^i,{i,1,n}]

Thus, Mrs. Guinea must live another 11 years and collect 12 installments in all to come

out ahead. Since she expects to live another 16 years, there is more than 50% chance that

she will live another 11 years. We may find the probability that she may indeed live an

additional 11 years as follows. Find

z = x − μ

σ =

11 – 16

6 = −.8333

Draw a normal probability distribution curve, with z = 0 in the center and z = −.8333 to

the left of center. We need to find the area to the right of z = −.8333 to find the

probability that she will live more than 11 years. This area is well over 50%. Using the

table, we get the probability as

P(life > 11) = 0.5 + 0.2967 + .33(.2995 – .2967) = 0.7976 ≈ 80% ♥

EXCEL =1-NORMDIST(11,16,6,TRUE)

To do the problem on an Excel spreadsheet, enter the information as follows.

A B

1 Expected life, μ (years) = 16

2 Standard deviation, σ (years) = 6

3 Required life, x (years) = 11

4 Probability of attaining that = =1-NORMDIST(B3,B1,B2,true)




88

Video 05B, 5.4. Quincy Corporation is planning to buy a machine for $80,000. The

company will depreciate it over a 5-year period with no resale value. However, the

machine has an uncertain life as given in the following probability distribution table:

Probability Life (years)

40% 4

30% 5

30% 6

While the machine is running, it will produce an EBIT of $20,000 a year. The tax rate of

Quincy is 30%, and the proper discount rate is 12%. Should Quincy buy the machine?

The annual depreciation of the machine is $16,000 and its tax benefit is .3(16,000) =

$4800. The after-tax cash flow, C = 20,000(1 – .3) + .3(16,000) = $18,800.

If the machine breaks down at the end of the fourth year, the company can take the tax-

benefit of depreciation of the fifth year at that time. The cash flows are $18,800 for the

years 1–4, plus another $4800 for year 4.

If the machine runs for five years, the cash flow for each year is $18,800.

If the machine runs for six years, the cash flows for the first five years are $18,800 each.

For the sixth year, the tax benefit of depreciation is not available, and the cash flow is

only 20,000(1 – .3) = $14,000.

Including the probability of each outcome, we get

NPV = 80,000 + .4[i=1

4

18‚800

1.12i +

4800

1.124] + .3

i=1

5

18‚800

1.12i + .3[

i=1

5

18‚800

1.12i +

14‚000

1.126 ] (A)

= 80,000 + 24,061.06 + 20,330.94 + 22,458.79 = $13,149, reject ♥


.4*(Sum[18800/1.12î,{i,1,4}]+4800/1.12^4)+.3*Sum[18800/1.12î,

{i,1,5}]+.3*(Sum[18800/1.12î,{i,1,5}]+14000/1.12^6)-80000

5.5. Cavendish Company wants to acquire a new computer that will cost $150,000 and it

will save the company $33,000 annually. The following probability distribution table

represents the best estimate of its expected useful life and the corresponding salvage

value:

Probability Life Resale Value

20% 4 years $30,000

30% 5 years $20,000

50% 6 years $10,000

http://134.198.33.115/hussain/Camtasia/Module05B/Module05B.html



89

The company will depreciate the computer fully in 4 years. The tax rate of Cavendish is

30%, and the proper discount rate is 8%. Should Cavendish buy the computer?

The tax benefit of the computer for each year is tD = .3(150,000/4) = $11,250. The

present value of the tax benefits for all four years is thus

i=1

4

11‚250

1.08i = $37,261.43

Next, look at the probability of different lives, savings generated by the computer, and

the salvage value in each case. Since the company will depreciate the computer

completely when it sells it, the proceeds of the sale are fully taxable. After taxes, the

resale value after four years is 30,000( 1 − .3) = $21,000, after five years it is 20,000( 1 −

.3) = $14,000, and after six years it is 10,000( 1 − .3) = $7000.

The annual earnings after taxes, excluding the tax benefit of depreciation, are 33,000( 1 −

.3) = $23,100. Combining all the above numbers, we find the NPV as follows:

NPV = – 150,000 + 37,261.43 + 0.2

i=1

4

23‚100

1.08i +

21‚000

1.084 + 0.3

i=1

5

23‚100

1.08i + 14‚000

1.085

+ 0.5

i=1

6

23‚100

1.08i + 7000

1.086 (A)

= – $8,221.64, reject ♥


Sum[.3*150000/4/1.08î,{i,1,4}]+.2*(Sum[23100/1.08î,{i,1,4

}]+21000/1.08^4)+.3*(Sum[23100/1.08î,{i,1,5}]+14000/1.08^5

)+.5*(Sum[23100/1.08î,{i,1,6}]+7000/1.08^6)-150000

5.6. Jupiter Company needs a new computer costing $100,000. Jupiter will depreciate the

computer over 5 years with no resale value. There is, however, a 30% chance that it may

break down completely after 4 years. While the computer is running, it will add $40,000

annually to the pretax income of Jupiter, which has a tax rate of 40%. For a discount rate

of 8%, should Jupiter buy this computer?

Here C = 40,000(1 – .4) + .4(20,000) = $32,000

If the machine breaks down after 4 years, we have to take the fifth-year depreciation at

that time. The probability of breakdown after four years is 30% and there is a 70%

probability that it would run smoothly for 5 years. Thus

NPV = –100,000 + .3[i=1

4

32‚000

1.08i +

.4(20‚000)

1.084 ] + .7[

i=1

5

32‚000

1.08i ] (A)



90

= $22,997, buy it. ♥


.3*(Sum[32000/1.08^i,{i,1,4}]+.4*20000/1.08^4)+.7*Sum[32000

/1.08^i,{i,1,5}]-100000

Video 05.07, 5.7. Fisher Corporation does not expect to pay taxes in the near future. It

is planning to acquire a new machine, which will have a useful life of two years.

However, there is a 10% probability that the machine will break down after only one

year. There are two states of economy, good and bad, with the probability of occurrence

60% and 40% in any given year. If the economy is good, the after-tax cash flow from the

machine is $20,000 annually, and if the economy is bad, the cash flow is only $15,000.

The proper discount rate is 12%. What is the maximum amount that Fisher should pay for

this machine?

The company is not paying taxes, thus t = 0. Using (4.3),

C = E(1 – t) + tD (4.3)

we get C = E, meaning that the cash flows before and after taxes are identical. We find

the expected cash flow under two different states of the economy by multiplying the

probability by the corresponding cash flow and adding the results. This gives us

E(C) = .6(20,000) + .4(15,000) = $18,000

Next, consider two possible outcomes of the life of the project. To get the total NPV,

multiply each probability of life with the dollar outcome of that life, whether it is one

year or two years. To break even, we get

NPV = – I0 + .1

18‚000

1.12 + .9

18‚000

1.12 +

18‚000

1.122 = 0

Or, I0 = .1

18‚000

1.12 + .9

18‚000

1.12 +

18‚000

1.122 = $28,986 ♥

5.8. Quincy Corporation wants to buy a machine for $50,000, with a maximum life of 4

years. However, there is a 20% probability that the machine will break down after only 3

years. There is an investment tax credit of 6%. The company will depreciate the machine

on ACRS basis, with a life of 3 years. Assume that the depreciation in the first year is

25%, second year 38%, and 37% in the third year. The tax rate of the company is 34%

and its discount rate is 12%. What is the minimum pretax income of this machine to

make it profitable for Quincy?




91

Because of the 6% investment tax credit, the net cost of the equipment is 94% of its price,

namely, 0.94(50,000) = $47,000. The company can depreciate only this amount. The tax

benefit of depreciation is tD per year.

The PV of tax benefits of depreciation = .34(47,000)

.25

1.12 +

.38

1.122 +

.37

1.123 = $12,616.32

Suppose the minimum pre-tax earning is E, which is just enough to break even. Its after-

tax value is E(1 – .34) = .66E. There are two possible outcomes: the life of the machine is

either three years (probability 20%) or four years (probability 80%). Including the

probability of breakdown, we get

PV of earnings = .2i=1

3

.66E

1.12i + .8

i=1

4

.66E

1.12 i = [

.2(.66)(1 – 1.12–3

)

.12 +

.8(.66)(1 – 1.12–4

)

.12]E

= 1.920762 E

To break even, the NPV is equal to zero. Thus

– 47,000 + 12,616.32 + 1.920762 E = 0

This gives E = $17,901 ♥

You may check the answer at WolframAlpha, by using the following instruction.

47000=.34*47000*(.25/1.12+.38/1.12^2+.37/1.12^3)+.2*Sum[.66

*x/1.12î,{i,1,3}]+.8*Sum[.66*x/1.12î,{i,1,4}]

Video 05.09, 5.9. Galen Mining requires a digging machine that costs $50,000. It will

depreciate the machine uniformly over its life of 5 years. The tax rate of Galen is 35%

and the proper discount rate is 11%. Because of the uncertain price of the ore, the

expected pre-tax revenue from the machine is $15,000 annually, with a standard

deviation of $3,000. What is the probability that the machine will prove to be profitable?

If the after-tax cash flow is C, then to break even

NPV = 0 = – 50,000 + i=1

5

C

1.11i

WRA 0=-50000+Sum[x/1.11î,{i,1,5}]

This gives C = $13,528.52

The annual depreciation is $10,000. If E is the corresponding pre-tax earnings, then

13528.52 = E (1 – .35) + .35(10,000)





92

This gives E = $15,428.49.

The machine must generate $15,428.49 to break even. Since it is expected to make only

$15,000 annually, chances are less than 50% that it will be profitable. Further,

z = (15,428.49 − 15,000)/3000 = .1428

Draw a normal probability distribution curve, with z = 0 in the center and z = .1428

somewhat right of the center. The area further to the right of z = .1428, under the tail of

the curve, gives the answer. Checking the tables, we get

P(being profitable) = 0.5 − [.0557 + .28(.0596 − .0557)] = .4432 = 44.32% ♥

EXCEL =1-NORMDIST(15428.56,15000,3000,TRUE)

5.10. Adams Corporation is planning to buy a machine that will cost $40,000 and

depreciate it on a straight-line basis over a 5-year period with no residual value. The tax

rate of Adams is 30%, and the proper discount rate is 15%. The earnings before taxes

from the machine are uncertain, but their expected value is $15,000 a year, with a

standard deviation of $3,000. Calculate the probability that the machine will be a

profitable investment. Adams requires the probability of being profitable to be more than

60% to buy the machine. Based on your calculation, should Adams buy the machine?

First, find the break-even point, where the earnings before taxes E are just enough to

make the NPV of the investment equal to zero.

After-tax cash flow, C = E(1 – .3) + .3(8000) = .7E + 2400

At break even point, NPV = 0 = – 40,000 + i=1

5

.7E + 2400

1.15i

WRA 0=-40000+Sum[(.7*x+2400)/1.15^i,{i,1,5}]

This gives E = $13,618.



93

Since the breakeven point is $13,618 and the machine is expected to make $15,000,

chances are more than 50% that the machine will be profitable.

Further z = (15000 – 13,618)/3000 = 0.4607

From the tables, P(profitable) = 0.5 + .1772 + 0.07(0.1808 – 0.1772) = 67.75%

EXCEL =1-NORMDIST(13618,15000,3000,TRUE)

Since Adams requires the probability of profitability to be at least 60%, it should buy the

machine. ♥

Video 05.11, 5.11. Benton Corporation is planning to buy a machine that may add

$4000 to the pre-tax earnings of the company if the economy is good (probability 60%),

or only $3500 if the economy is bad (probability 40%). Benton will depreciate the

machine on the straight-line basis for four years, even though it has a 20% chance that it

may last for five years. The tax rate of Benton is 30%, and the proper discount rate is

11%. Find the maximum price that Benton should pay for this machine to make it a

profitable investment.

Suppose the break-even price of the machine is P and thus the depreciation per year is

P/4. The expected pretax earnings are 0.6(4000) + 0.4(3500) = $3800. The after-tax

earnings are

C = 3800(1 − .3) + .3(P/4) = 2660 + .075P, for the years 1-4

C = 3800(1 − .3) = $2660 for the fifth year.

To break even, the present value of the machine should be equal to the present value of

all cash flows. Including the 20% probability for the fifth-year cash flow, we get

P = i=1

4

2660 + .075P

1.11i +

.2(2660)

1.115

P

1 − i=1

4

.075

1.11i =

i=1

4

2660

1.11i +

.2(2660)

1.115

Solving for P, we get P = $11,166. If Benton buys the machine for less than $11,166, it

should be a profitable investment. ♥

WRA P=Sum[(2660+.075*P)/1.11^i,{i,1,4}]+.2*2660/1.11^5

Video 05.12, 5.12. Hawley Corporation needs a new computer that will produce annual

savings estimated at $5000 with a standard deviation of $2000. The company will buy the

computer for $20,000, depreciate if fully on a straight line over four years, and then sell it

for $3,000. The tax rate of Hawley is 25%, and it uses a discount rate of 11%. Find the

probability that the computer will have a positive NPV. Should Hawley buy the

computer?





94

In this problem, the additional factor to consider is the resale value of the machine. To

calculate the after-tax value of the resale price, we use

W = S(1 – t) + tB (4.7)

Since the machine is fully depreciated, its book value B is zero, Hawley has to pay taxes

on the sales price of the machine. After taxes, it becomes (1 − .25)(3000) = $2250.

To break even, suppose the earnings before taxes are E. With $5000 annual depreciation

and 25% tax rate, the after-tax cash flow is, by (4.3),

C = E( 1 − .25) + .25(5000) = .75E + 1250

This gives

NPV = 0 = – 20,000 + i=1

4

.75E + 1250

1.11i +

2250

1.114 (A)

20,000 – i=1

4

1250

1.11i –

2250

1.114 = E i=1

4

0.75

1.11i

Or, E = $6291.72

WRA 0=-20000+Sum[(.75*x+1250)/1.11^i,{i,1,4}]+2250/1.11^4

The computer is expected to generate only $5000 annually. Thus is it unlikely that it will

be profitable. Calculate the z-value as

z = 6291.72 – 5000

2000 = 0.6459

From the tables,

P(NPV > 0) = 0.5 – [0.2389 + 0.59(0.2422 – 0.2389)] = 25.92%


The probability that the computer is going to be profitable is only about 26%. Hawley

should not buy it. ♥

5.13. Columbus Corporation plans to acquire a computer that will last for 5 years, and

costs $100,000. The company will use straight-line method to fully depreciate the

computer in 4 years, and then sell it for $10,000 after 5 years. Columbus will use 9%

discount rate for this investment, and its income tax rate is 35%. The pretax saving from

this computer is uncertain, with expected value $24,000 per year, and standard deviation

$6,000. What is the probability that the computer will have a positive NPV? Should

Columbus buy this machine?



95

First we find the pretax earnings E that will make NPV = 0. The annual depreciation is

$25,000. The after-tax cash-flow is given by (4.3)

C = E(1 – .35) + .35(25,000) = .65E + 8750

Consider this as two cash flows: .65E, which will go on for five years and $8750 only for

the first four years. Finally, the after-tax value of the sales price is (1 – .35)10,000 =

$6500. Setting the NPV equal to zero, we get

0 = –100,000 + i=1

4

8750

1.09i +

i=1

5

.65E

1.09i +

6500

1.095 (A)

Or, 0 = –100,000 + 28,347.54 + 2.52827E + 4224.55

This gives E = $26,669.54

WRA 100000=Sum[8750/1.09^i,{i,1,4}]+ Sum[.65*x/1.09^i,{i,1,5}]+6500/1.09^5

Since the machine is expected to generate $24,000 annually in earnings, it is unlikely that

it will become profitable.

Now z = (26,669.54 – 24,000)/6,000 = .4449

P(profitable) = .5 – [.1700 + .49(.1736 – .1700)] = 32.82%


Considering the low probability of profitability, Columbus should not buy it. ♥

5.14. Lucas Corporation plans to buy a machine for $100,000, and depreciate it

uniformly over its useful life of 5 years. The machine will produce annual pretax revenue

of $27,000. The tax rate of Lucas is 35% and it will use 11% as the discount rate for this

investment. The resale value of the machine after five years has a mean of $20,000 with a

standard deviation of $5000. Calculate the probability that this machine will be

profitable. Should Lucas buy it?

Suppose the resale value of the machine is S to break even. Since the machine is fully

depreciated at the time of sale, the sale amount is taxable. Its after-tax value is (1 – .35)S

= .65S. The annual depreciation is $20,000, and thus the annual after-tax cash flow from

this machine is

C = (1 – .35)27,000 + .35(20,000) = $24,550

The gives

NPV = 0 = – 100,000 + i=1

5

24‚550

1.11i +

.65S

1.115 (A)



96

WRA 100000=Sum[24550/1.11^i,{i,1,5}]+.65*S/1.11^5

Solving for S, S = $24,020.45

Since the resale value is only $20,000, the probability is less than 50% that it will be

profitable. Further,

z = (24020 – 20000)/5000 = .8041

Using the tables,

P(profitable) = .50 – [.2881 + .41(.2910 – .2881)] = .2107


There is only a 21% probability that the machine will be profitable. The company should

not buy it. ♥

5.15. Delta Corporation is planning to buy a new machine at a cost of $30,000, which

will increase the pretax earning of the company by $10,000 annually. The maximum life

of the machine is 5 years, but there is also a 10% probability that it will break down after

3 years and a 20% probability that it will last only 4 years. Delta will depreciate the

machine on a straight-line basis for 5 years. In the case of a breakdown, Delta will

discard the machine, and lease a replacement machine for $8,000 annually, paying it in

advance every year. The after-tax cost of capital is 12%, and the company is in 30% tax

bracket. Should Delta proceed with the purchase? It cannot lease the machine for full five

years.

First, look at the earnings of the machine. The pretax earning from the machine, whether

it is the first one or the leased one, is $10,000 a year and its after-tax value is 10,000(1 −

.3) = $7,000. Take the PV for the full five years. This amounts to

PV of E(1 – t) = i=1

5

7000

1.12i = $25,233.43 (A)

WRA Sum[7000/1.12^i,{i,1,5}]

Next, consider the possibility that the machine breaks down after 3 years. The company

will write it off and lease a second one. At that time, the company does the following:

1. Takes the tax benefits of depreciation for the fourth and the fifth year. The depreciation

per year is $6,000 and its tax benefit, tD, is .3(6000) = $1800 per year. The total tax

benefit for the fourth and the fifth years is $3600.

2. Pays the lease payments for the two years in advance each year. The lease payments

are tax deductible, and so their after-tax value is 8000(1 – .3) = $5600 annually.

3. Considers the 10% probability that the machine breaks down after three years and

finds the PV of the cash flows.




97

PV of 3-year life = .1

i=1

3

1800

1.12i + 3600

1.123 – 5600

1.123 – 5600

1.124 = –$65.92 (B)

WRA .1*(Sum[1800/1.12î,{i,1,3}]+3600/1.12^3-5600/1.12^3-5600/1.12^4)

In the above expression, the summation is the PV of tax benefit of depreciation for the

first three years; $3600 is the tax benefit for years 4 and 5; $5600 is the after-tax value of

lease payment; and 0.1 is the probability factor.

Now consider the possibility that the machine runs for four years. At the end of the fourth

year, the company does the following:

1. Finds the PV of four years of tax benefits of depreciation, $1800 per year.

2. Takes the fifth year depreciation at the end of the fourth year. Its tax benefit is $1800.

3. Pays the lease payment for the fifth year, $5600 after taxes.

4. Incorporates the 20% probability of this event and calculates the PV of the cash flows.


i=1

4

1800

1.12i +

1800

1.124 −

5600

1.124 = $610.45 (C)

WRA .2*(Sum[1800/1.12î,{i,1,4}]+1800/1.12^4-5600/1.12^4)

The PV of a 5-year life for the first machine does not involve the second machine. The

probability of this event is 70%. Thus


i=1

5

1800

1.12i = $4542.02 (D)

WRA .7*Sum[1800/1.12î,{i,1,5}]

Combine (A), (B), (C), and (D) to get the NPV of all possibilities

NPV = – 30,000 + 25,233.43 – 65.92 + 610.45 + 4,542.02 = $319.98.

The machine is barely acceptable. ♥

5.2 Inflation

In financial modeling, one can also include the effects of inflation, which will influence

the future earnings or expenses of a corporation. Although inflation rates are difficult to

predict accurately, one can use the past as a proxy for the future and arrive at a reasonable

estimate of the rate of inflation. We should adjust the future cash flows accordingly.





98

Examples

5.16. A corporation has the following pension benefit for its retired employees. It pays

annual payments equal to 35% of an employee’s last salary at the time of retirement,

adjusting them upward according to the rate of inflation. Find the cost of these benefits to

the company for an employee who has just retired with an annual salary of $32,000. The

company expects to make 16 annual payments and expects the inflation rate to be 3.5%

during this period. The cost of capital to the company is 12%. The company will make

the first payment now.

First payment = 0.35(32,000) = $11,200

NPV = 11,200 + i=1

15

11‚200(1.035)

i

1.12i (A)

= 11200 + 11‚200 (1.035)

1.12 +

11‚200 (1.035)2

1.122 + ... + 11‚200 (1.035)15

1.1215

In this summation, a = 11‚200 (1.035)

1.12, x =

1.035

1.12 , and n = 15. Using (1.4), we get

NPV = 11,200 + 11‚200 (1.035)

1.12 [

1 − (1.035/1.12)15

1 − 1.035/1.12] = $105,834

The company may also offer to make a single payment of $105,834 as the settlement for

the pension benefits. ♥

WRA 11200+Sum[11200*1.035î/1.12î,{i,1,15}]

5.17. Benin Corporation has the following pension plan. It will give 35% of a person's

last annual salary as pension in annual installments, with the first payment at the time of

retirement. The pension has a cost of living adjustment, which is expected to be +3%

annually in the future. Benin uses a discount rate of 9%. Find the present value of the

pension cost for an employee who has just retired with an annual salary of $42,000. She

expects to receive 17 payments.

PV = 0.35(42,000) + 0.35(42,000)(1.03)

1.09 +

0.35(42,000)(1.03)2

1.092 + 17 terms (A)

= 0.35(42,000)[1 (1.03/1.09)17]

1 1.03/1.09 = $165,056 ♥

WRA Sum[.35*42000*(1.03/1.09)î,{i,0,16}]

5.18. California Gold Mining Company plans to buy a dredging machine, which costs

$40,000. The machine will produce an EBIT of $4,000 in the first year, but this will




99

increase at the rate of 10% annually due to the rising price of gold. The machine will last

for 10 years and the company will depreciate it on a straight-line basis. The tax rate of the

company is 30% and the proper discount rate is 12%. Should California buy the

machine?

Suppose the initial investment in the machine is I0. The depreciation per year is thus I0/n,

and the tax benefit of depreciation per year is t(I0/n). Suppose the EBIT from the machine

in the first year is E. Due to the inflation rate of f per year, it will become E(1 + f) in the

second year, E(1 + f)2 in the third year, and so on. To generalize, the earnings in the ith

year are E(1 + f)i−1

. Thus

NPV = − I0 + i=1

n

E(1 + f)

i−1(1 − t) + tI0/n

(1 + r)i

Substituting the numerical values, I0 = $40,000, E = $4,000, f = 0.1, n = 10, t = 0.3, r =

0.12, we get

NPV = − 40,000 + i=1

10

4000(1.1)

i−1(1 − .3) + .3(40,000)/10

1.12i

Use the Maple expression (red) or WolframAlpha (blue) to get the result.

-40000+sum((4000*.7*1.1^(i-1)+.3*40000/10)/1.12^i,i=1..10);

-40000+Sum[(4000*.7*1.1^(i-1)+.3*40000/10)/1.12^i,{i,1,10}]

The result is NPV = − $10,135.92, which is not acceptable. ♥

5.19. Carpenter Corporation is considering a project with an initial cost of $100,000. The

pre-tax cash inflows in current dollars, without adjustment for inflation, are as follows.

Year: 1 2 3 4 5

Amount: $40,000 $30,000 $30,000 $30,000 $40,000

The estimate for inflation is that it will be either 4%, with probability 30%; or 5%, with

probability 70%. Accordingly, the company should adjust the cash flow projections

upwards. The company is in 35% tax bracket and it uses a discount rate of 12%. Should it

accept the project?

Assuming that the inflation is 4% annually, the pretax cash flows will become

40,000(1.04), 30,000(1.04)2, 30,000(1.04)

3, etc. We multiply the cash flows by (1 .35)

to get their after-tax amount. Multiplying with the respective probabilities, and

discounting, we get the NPV as


100

NPV = – 100,000 + (1 – .35)

{0.3[40‚000(1.04)

1.12 +

30‚000(1.04)2

1.122 +

30‚000(1.04)3

1.123 +

30‚000(1.04)4

1.124 +

40‚000(1.04)5

1.125 ]

+ 0.7[40‚000(1.05)

1.12 +

30‚000(1.05)2

1.122 +

30‚000(1.05)3

1.123 +

30‚000(1.05)4

1.124 +

40‚000(1.05)5

1.125 ]}

= −$9264, reject the project. ♥

5.3 The Option to Postpone a Project In many situations, the investment decision of a corporation also has options embedded in

it. For example, a corporation can start a project now, or wait for a year and then start the

project. Therefore, the company has the option to delay the project. Later on, when the

project is becoming profitable, the company has the option to expand the project. If the

sales are not what they were originally anticipated, the firm may have the option to

contract the project. Finally, if the project is creating a big loss, then the firm should have

the option to abandon the project outright.

Each option listed above has a certain value, because it provides the company with

flexibility in its planning. For example, the company can build a factory now, or build it

next year. By building the factory now, the company exercises its option and thereby

loses the value of that option. It is also possible to find the value of this option.

Suppose a corporation can start a project right away, or wait for a while and then decide

whether to get on with the project. Consider the following simplified example.

Example

5.20. An oil company can invest $16,000 to drill a well and start producing oil. The

revenue from the well is uncertain, depending on the price of oil. At present, the price of

oil is $20 a barrel, but after a year, it could go up to $30 a barrel, or drop to $10 a barrel,

with equal probability. Let us assume that the new price of oil will stay constant forever.

The well will produce 100 barrels of oil a year forever with no cost. The oil revenues are

available at the end of each year. The cost of capital is 10%. Should the company drill the

well now, or wait for a year and then drill?

Suppose the company drills the well right now and starts producing the oil. The annual

revenue from the well is $3000, or $1000, depending on the price of oil. Both of these

outcomes are equally probable. The cash flows start a year from now and continue

forever. The following expression gives the NPV,

NPV = − 16,000 + .5i=1

3000

1.1i + .5

i=1

1000

1.1i = −16,000 + .5[3000/.1 + 1000/.1] = $4,000


101

Based on this incomplete analysis, the company should go ahead and drill the well. The

answer is, however, wrong because we have ignored the value of the option to delay the

decision for a year. The probability that the oil price will rise or drop is 50% each.

Suppose the risk-free rate is 6%, then

NPV(oil = $30) = .5

−16‚000

1.06 +

i=2

∞

3000

1.1i = $6089

NPV(oil = $10) = .5

−16‚000

1.06 +

i=2

∞

1000

1.1i = −$3002

If the company waits for a year and then drills the well, then its choice is clear – it will

drill only if the price of oil is $30 a barrel. For a price of $10, the NPV is negative and the

company will not get into oil production. The company must wait for a year and then

decide to drill.

Suppose the company did not have the option to wait. It is now or never. In that case, the

company should go ahead and drill now with an NPV of $4000. Since the second NPV is

$2089 more than the first NPV, this is the value of the option to wait. ♥

5.4 The Option to Abandon a Project

There may be several reasons for a company to stop a project: the market conditions have

changed, the plans have changed, or the project simply is not profitable any more. The

flexibility afforded by this option to continue or discontinue a project also adds value to

the company. It is possible to evaluate the optimal time to replace a car, or a machine, or

a whole factory. You may have to compare the NPV of abandoning versus the NPV of

continuing the project. The following example will illustrate the point.

Examples

5.21. A company has decided to buy a machine for $50,000. It will depreciate the

machine on a straight-line basis over its useful life of five years. The tax rate of the

company is 40% and the proper discount rate 10%. The following tables gives the resale

value, S of the machine and its pretax revenue, E.

Time, years Pretax revenue, E Resale value, S

0 $0 $50,000

1 $22,000 $38,000

2 $18,000 $27,000

3 $14,000 $17,000

4 $12,000 $8,000

5 $10,000 $0


102

The company has the option to keep or sell the machine at any time. What is the optimal

time to replace the machine?

Recall (4.3) and (4.7) to calculate the after-tax cash flows and the after-tax value of the

resale price of the machine. We augment the information in the above table as follows:

Time E S E(1 – t) + tD B S(1 – t) + tB

0 $0 $50,000 $0 $50,000 $50,000

1 $22,000 $38,000 $17,200 $40,000 $38,800

2 $18,000 $27,000 $14,800 $30,000 $28,200

3 $14,000 $17,000 $12,400 $20,000 $18,200

4 $12,000 $8,000 $11,200 $10,000 $8800

5 $10,000 $0 $10,000 $0 $0

To find the optimal replacement time, we find the NPV of the machine if it is used only

for one year, or two years, and so on. The NPV for all possibilities will be

NPV(1 year) = − 50,000 + 17‚200

1.1 +

38‚800

1.1 = $909

NPV(2 years) = − 50,000 + 17‚200

1.1 +

14‚800

1.12 +

28‚200

1.12 = $1174 (highest)

NPV(3 years) = − 50,000 + 17‚200

1.1 +

14‚800

1.12 +

12‚400

1.13 +

18‚200

1.13 = $858

NPV(4 years) = − 50,000 + 17‚200

1.1 +

14‚800

1.12 +

12‚400

1.13 +

11‚200

1.14 +

8‚800

1.14 = $844

NPV(5 years) = − 50,000 + 17‚200

1.1 +

14‚800

1.12 +

12‚400

1.13 +

11‚200

1.14 +

10‚000

1.15 = $1043

It is interesting to note that the NPV reaches a maximum after two years, equaling $1174.

Considering all these results, it is best to replace the machine after two years. ♥

The value of the option to stop at any time after the project has commenced is the

difference between the maximum NPV and the NPV for full 5 years. It comes out to be

1174 – 1043 = $131 ♥

5.22. A company can build a factory now at a cost of $900, or wait for a year and build it

at a cost of $1200, or $800. The probability of increased cost is 60%. The future cost

depends on the pending legislation. The output from the factory will have a net cash flow

of $100 per year forever. The proper discount rate for the cash flows is 10%, while the

risk free rate is 6%. Should the company build now or next year?

NPV(build now) = − 900 + i=1

∞

100

1.1i = $100


103

NPV(build next year) = .6

−1200

1.06 +

i=2

∞

100

1.1i + .4

−800

1.06 +

i=2

∞

100

1.1i = −$72.04

The company should build the factory now. ♥

5.23. A company can automate its payroll department by purchasing a computer. The

computer costs $25,000 now, but it may cost only $20,000 next year. The company

expects to save $6500 a year during its useful life of 5 years. The discount rate is 12%,

and the risk-free rate is 6%. The company uses straight-line depreciation and its tax rate

is 30%. Should the company install the computer this year, or next year?

The depreciation in the first case is $5000 annually, and in the second case $4000

annually.

NPV(install now) = −25,000 + i=1

5

6500(1 − .3) + .3(5000)

1.12i = −$3191.10

NPV(install next year) = − 20‚000

1.06 +

i=2

6

6500(1 − .3) + .3(4000)

1.12i = −$361.26

The company should not install the computer at present. It should re-evaluate the

situation after a year or two. Perhaps the price of the computer will drop futher, or the

savings will increase. ♥

5.24. A company has the option of buying a machine now, or waiting for a year. If the

company buys the machine now it will cost $12,000 while the current discount rate is

10%. If the company buys the machine next year, the machine will cost $13,000, but the

discount rate will be 9%. In any case, the machine has a useful life of 5 years with no

resale value. The machine will generate $4000 in annual pretax revenues. The tax rate of

the company is 30%. The risk-free rate is 6%. What is the better strategy?

Considering the after-tax cash flows in each case, find the NPV as

NPV(buy now) = − 12,000 + i=1

5

4000(1 − .3) + .3(2400)

1.1i = $1343.57

NPV(wait for a year) = − 13‚000

1.06 +

i=2

6

4000(1 − .3) + .3(2600)

1.09i = $511.034

The company should buy the machine now. The value of the option to wait is zero,

because the company will not wait. An option cannot have a negative value. ♥

Key Terms continuous, 81

discrete, 81

expected value, 81, 90, 92

inflation, 81, 95, 96, 97

mean, 81, 82, 83, 93, 102,

103, 105

normal probability

distribution, 81, 85

options, 81, 98


104

probability distributions, 81

random, 81

standard deviation, 81, 82,

84, 89, 90, 91, 92, 93, 102,

103, 104, 105

subjective probability

distribution, 81

Problems

5.26. Webster Company stock does not pay any dividends and it sells for $55 a share.

The continuously compounded expected return of the stock is .12, with standard

deviation of 0.22. Find the probability that the stock will be selling for more than $60

after one year. P(S > 60) = 55.96% ♥

5.27. Green Acres common stock pays no dividend, but its expected return is 13% with a

standard deviation of 30%. Currently the stock is selling for $65 a share. If you buy the

stock now, what is the probability that its price next year will be between $60 and $70?

P(60 < S < 70) = 18.46% ♥

5.28. You are looking for an investment opportunity that will provide a yield of at least

10%. You estimate that the returns from the Ford Motor Company stock have normal

distribution with mean 12.5% and standard deviation 8.5%. What is the probability of

achieving your objective if you buy Ford stock? What is the probability that your loss

will be limited to 5% of your investment? P(r > .1) = 61.57%, P(r > −.05) = 98.02% ♥

5.29. Marshall Company is in financial distress. Its bonds will mature on December 31,

2010, and they pay 8% interest annually on December 31. There is a 30% chance that the

company may become bankrupt in a given year. In case of bankruptcy, the company will

not pay interest on the bonds for that year and will settle the claims by paying 30% of the

principal at the end of the year. Suppose your required rate of return on these bonds is

14%, how much would you pay for a bond on January 1, 2008? $486.51 ♥

5.30. Benin Company 7.5s14 bonds are selling at 27. Your careful analysis reveals that

the company will survive the first year. It may go bankrupt during the second year

(probability 50%) or during the third year (probability 50%). Before bankruptcy, it will

continue to pay the semiannual interest payments. In case of bankruptcy, the company

will not pay any interest for that year, and you expect to get only 20% of the face amount

of the bond, which will be available one year after the bankruptcy. For instance, if Benin

goes bankrupt during the second year, you will receive the final payment at the end of the

third year. Your required rate of return is 12%. Do you think you should buy Benin

bonds? No, B = $232.58 ♥

5.31. Macmillan Corporation's cumulative preferred stock pays $5 annual dividend, the

first one will be received a year from now. The cumulative feature means that if the

company does not pay the dividends in a given year, it must pay the total dividends due

the following year. The payment of dividends is contingent upon the earnings after taxes,

and you feel that there is 90% chance that the dividends will indeed be paid in a given

year. In any case, at the end of 3 years, the company will buy back the stock by paying

$50 per share to the stockholders, plus any dividends due. The discount rate for this

investment is 14%. What price would you pay for a share of Macmillan stock? $45.25 ♥


105

5.32. The York Company wants to buy a new stamping machine that costs $100,000.

The machine has an expected life of 10 years with a standard deviation of 2 years. York

will depreciate the machine over a 10-year period. The cost of running the machine is

$10,000 annually, and it generates $30,000 annual revenue. The company has income tax

rate of 30%, and it uses 10% as the discount rate. What is the probability that the machine

will break down within 8 years? If it runs for only 8 years, what is its NPV?

P(life < 8) = 15.87%, NPV = −$6507.21 ♥

5.33. Burundi Airlines is planning to acquire a Boeing 757 at a cost of $24 million. The

plane has an uncertain life span: it may last for 6 years (probability 50%), 7 years

(probability 30%), or 8 years (probability 20%). The airline will depreciate the plane on a

straight-line basis with a life of 6 years, with no residual value. While the plane is flying,

it will generate a pretax income of $6 million annually. The tax rate of Burundi is 40%

and its after-tax cost of capital is 9%. Should Burundi buy the new plane?

NPV = $672,782, buy it ♥

5.34. Usfan Company is interested in buying a computer with uncertain life. The

following table shows its expected life and resale value:

Expected life Probability Resale value

3 years 20% $20,000

4 years 30% $10,000

5 years 50% $5,000

The computer will save the company $40,000 annually while it is running. Usfan will

depreciate it fully on a straight-line basis in 3 years. The tax rate of Usfan is 30%, and the

proper discount rate in this case is 12%. The cost of the computer is $90,000. Should

Usfan buy it? Yes, NPV = $25,368 ♥

5.35. Pisces Corporation is planning to buy a machine for $20,000 that may run for 4

years (probability 60%), 5 years (probability 30%), or for 6 years (probability 10%).

Pisces will depreciate the machine on straight-line basis for 4 years, without any resale

value. The tax rate of Pisces is 30% and the proper discount rate is 8%. Calculate the

minimum annual earnings generated by this machine to be acceptable to Pisces.

$5887.52 ♥

5.36. Thales Corporation wants to buy a machine for $60,000. It will depreciate the

machine over a six-year period with no salvage value. The machine will generate pre-tax

revenue of $20,000 annually, the tax rate of Thales is 32%, and the proper discount rate is

11%. The machine has uncertain life: it may run for 5 years (probability 50%), 6 years

(probability 30%), or 7 years (probability 20%). Should Thales buy the machine?

NPV = $8841.69, yes. ♥

5.37. Cobb Company is planning to invest in a machine that will run for 5 years. Cobb

uses straight-line depreciation. The annual pretax revenue from the machine has normal

distribution, with mean $12,000 and standard deviation $4000. The tax rate of the


106

company is 30% and its after-tax cost of capital is 8%. How much should Cobb invest in

this machine so that there is greater than 80% probability that it will turn out to be

profitable. $31,731.46 ♥

5.38. Rayburn Corporation is considering the purchase of a new machine that has an

expected life of 5 years with a standard deviation of 2 years. The machine will cost

$40,000 and will generate a pre-tax income of $10,000 annually. The tax rate of Rayburn

is 40% and its cost of capital is 7%. Rayburn will depreciate the machine on a straight-

line basis over 5 years with no residual value. Calculate the probability that the machine

will have a life of between 4 and 7 years. Is the machine acceptable if it runs for only 4

years? 53.28%, NPV = $6396.48, no ♥

5.39. Acheson Corporation is planning to buy a machine that will cost $30,000. Its life is

uncertain: it may become obsolete after two years (probability 25%) and definitely after

three years. The economy in any given year may be good (probability 60%) or bad

(probability 40%). If the economy is good, the machine will generate pre-tax annual

revenue of $15,000 and if the economy is bad, the expected revenue will only be

$10,000. The tax rate of Acheson is 40% and it will depreciate the machine for 3 years on

a straight-line basis. The proper discount rate is 7%. Should Acheson buy the machine?

NPV = $567.71, no ♥

5.40. Martin Company is looking into a machine with a cost of $20,000 that will run for

5 years. Martin will depreciate the machine completely over this period. The tax rate of

the company is 30%, and its cost of capital is 8%. The expected pretax income from the

machine is $5,000 annually, with a standard deviation of $2,000. Calculate the

probability that the machine will turn out to be profitable. 41.26% ♥

5.41. Cooper Inc. plans to buy a new machine for $25,000 for a project that will last 5

years. The machine will generate $8,000 annually in pretax revenue. The tax rate of

Cooper is 30% and its after-tax cost of capital 8%. Assume straight-line depreciation for

five years. There is a 10% chance that the machine may break down after 3 years and a

20% probability of breakdown after 4 years. If the machine breaks down, Cooper will get

a used machine as a replacement for $12,000. The used machine has a two-year useful

life. At the end of the project, neither machine will have a resale value. Should Cooper

buy this machine? NPV = $1418.54, buy ♥

5.42. Salam Corporation would like to buy a new electric furnace for $42,000 that will

increase the EBIT of the company by $8,000 annually. The actual life of the furnace is

unpredictable, but for depreciation purposes, it is 7 years. The proper discount rate is 9%

and the tax rate is 30%. Find the minimum number of years that the furnace must run

before it would become a profitable investment. NPV(9 years) = $632.70 > 0 ♥

5.43. Richardson Company is considering the purchase of a machine that it expects will

run for 5 years, even though there is a 20% chance that it may break down completely

after 4 years. While the machine is running, it will generate $20,000 annually in pre-tax

income. Richardson will depreciate the machine on a straight-line basis over a five-year


107

period with no salvage value. The income tax rate of Richardson is 30% and its after-tax

cost of capital is 11%. The cost of the machine is $80,000. Should Richardson install the

machine? NPV = $12,116, no ♥

5.44. Laird Company is planning to invest in a project with expected after-tax cash flow

of $12,000 annually with a standard deviation $4000. The project requires an initial

investment of $75,000 and another expense of $25,000 at the end of the first year. The

first income will occur at the end of the second year. Calculate the probability that the

project will be profitable (that is, its NPV > 0) after 10 years. The cost of capital to Laird

is 9%. 7.33% ♥

5.45. Bush Inc. is planning to acquire a machine with expected life 5 years. Bush will

depreciate the machine on a straight-line basis without any salvage value. The following

table shows the pre-tax income generated by the machine depending on the state of the

economy.

State of the economy Probability Expected income

Good 30% $23,000

Fair 50% $18,000

Poor 20% $12,000

The economic conditions in any given year are independent of the conditions in the

previous year. Bush will not buy the machine unless it has NPV of at least $30,000. The

proper discount rate in this case is 12%, and the income tax rate for Bush is 35%.

Calculate the maximum price that Bush should pay for this machine. $17,225.36 ♥

5.46. Benue Company is interested in buying a machine that costs $80,000. Benue will

depreciate the machine over a 5-year period with no resale value. The actual life of the

machine is uncertain, but while it is operating, it will generate pretax revenue of $25,000

annually. The tax rate of Benue is 30%, and it uses 10% discount rate for such an

investment. Find the NPV of this investment if the machine lasts (a) 4 years, and (b) 7

years. (a) $6033.54, (b) $23,393.11 ♥

5.47. Bomu Corporation is planning to buy a $70,000 machine with a 5-year life. Bomu

will depreciate the machine fully over that period and then sell it for $10,000. The annual

pretax revenue from the machine is uncertain, with a mean of $25,000, and standard

deviation $10,000. The income tax rate of Bomu is 30% and the cost of capital 12%. Find

the probability that the machine will be profitable. 68.56% ♥

5.48. Syracuse Company is planning to buy a machine for $50,000 that will be

depreciated fully in five years on a straight-line basis. The machine is estimated to last 7

years, and then it will be sold for $5000. The before-tax earnings from the machine are

estimated to be $10,000 annually, with a standard deviation of $2000. The tax rate of

Syracuse is 30%, and its after-tax cost of capital 12%. Find the probability that this

machine will be profitable. 18.8% ♥


108

5.49. Cleveland Corporation needs a new machine that will cost $120,000 and it will

generate $25,000 annual pretax earnings. The company will depreciate the machine over

5 years, with no resale value. The discount rate for this investment is 8%, and the income

tax rate of Cleveland is 32%. The machine may actually run for 5 years (probability

30%), or 6 years (probability 70%). Should Cleveland buy this machine?

No, NPV = −$13,961 ♥

5.50. Chadwick Company has a pension plan that provides lifetime benefits to its retiring

employees, including cost of living adjustments. The first payment, paid a year after the

retirement, is equal to 60% of the last annual salary. However, the benefits are expected

to rise by 4% annually in the future. The expected number of annual payments to a 65-

year old person is 18. The cost of capital to Chadwick is 11%. Find the present value of

the benefits payable to a person whose last salary is $50,000 per year. $295,890.13 ♥

5.51. Lakeland Clinic, a tax-exempt entity, is interested in buying a MRI device now, or

postponing it for a year. The current price of the equipment is $3 million, but it is

expected to go up to $3.2 million next year. The hospital plans to use the machine for a

period of 6 years, and during this period the value of the machine is expected to drop at a

compound annual rate of 10%. The equipment will produce revenue of $500,000

annually. The cost of capital for hospital is 8%. What is the best course of action? What

is the value of the option to wait?

NPV(now) = $316,134, NPV(later) = $169,549, option value = 0 ♥

5.52. Western Airlines is planning to acquire a plane for $50 million. They will

depreciate it completely on a straight-line basis over 5 years. The tax rate of Western is

30% and its cost of capital 10%. The plane should generate $20 million annually in

pretax income, not including maintenance costs. The resale value and the annual

maintenance cost (in millions) of the plane are variable as shown in the following table:

Time Maintenance Resale value

0 year $0 $50 million

1 year $5 million $45 million





What is the optimal time to keep the airplane flying? What is the value of the option to

sell the plane ahead of time, rather than to keep it for full five years?

Optimal 2 years, NPV(2) = $2.273 million, value of the option = $2.289 million ♥

109

6. PORTFOLIO THEORY


1. Understand the basic reason for portfolio formation.

2. Calculate the risk and return characteristics of a portfolio.

6.1 Video 06A Portfolio Formation

A portfolio is a collection of projects, or securities, or investments, held together as a

bundle. For example, you may buy 100 shares of Boeing, 200 shares of Microsoft, and 5

PP&L bonds in an account. This is your portfolio of investments. If you own a home,

then you may include the value of the house in your portfolio. A portfolio may also

include less tangible items as your professional education, or even a license to practice

law. The total value of your portfolio may fluctuate with time.

Corporations also have a portfolio of different projects. They carefully select profitable

projects and invest in them. The banks loan money to individuals and corporations. They

have a loan portfolio, and they try to monitor the quality of their portfolios. The quality of

a loan portfolio can deteriorate if too many loans are non-performing.

Why do people, or corporations, form a portfolio? The simple answer is diversification.

You do not want to put all your eggs in one basket. It is a good idea to diversify your risk,

and if some of the investments do not pan out, the others will keep the value of the

portfolio intact.

6.2 Portfolio Theory

Harry Markowitz (1927- )

We can form a portfolio by carefully selecting a set of

securities. The two main features of a portfolio are its risk and

expected return. In 1952, Harry Markowitz first developed the

ideas of portfolio theory based on statistical reasoning. He

showed that one could reduce the risk for a given return by

putting together unrelated or negatively correlated securities.

We may define the realized return of a single security as the

sum of price appreciation and the dividends, divided by the

acquisition cost of the security. For example, if we buy a stock

at $50 a share, get a $2 dividend on it, and then sell it for $52,

then the return, in dollars, is $4. The return, as a percentage, is

4/50 = 8%. In general, we may define the return as

R = P1 − P0 + D1

P0 (6.1)

http://en.wikipedia.org/wiki/Harry_Markowitz


Analytical Techniques 6. Portfolio Theory _____________________________________________________________________________

110

where the purchase price of the stock is P0, its selling price is P1, and D1 is the dividend

paid, if any, at the end. The quantity P1 − P0 is the price appreciation of the stock, and

along with the dividend, is the total change in the value of the investment.

Modifying (6.1), we can represent the expected return of a security as

E(R) = E(P1) − P0 + E(D1)

P0 (6.2)

where E(•) is the expectations operator. Suppose we have a probability distribution pi,

with i = 1 ... n, describing n states of the economy and we also have the returns Ri under

each state, then

E(R) = i=1

n

piRi = R–

(6.3)

The second component of any investment is the amount of risk inherent in that

investment. We may use the standard deviation of return, σ(R), as a measure of risk. This

is because the standard deviation of a random outcome represents the uncertainty, or

spread, in that random variable. For a stock, it may represent the risk of that stock

investment. Using the notation of (1.6 − 1.8), we write

σ(R) = [i=1

n

pi(Ri − R–

)2]

1/2

(6.4)

To quantify the dependence of one stock on the other, recall equation (1.10) which

defines the correlation coefficient, rij. Mathemathically,

rij = cov(Ri,Rj)

σ(Ri)σ(Rj) (6.5)

For any two securities that are completely unrelated, the correlation coefficient between

them is zero, rij = 0. For perfectly positively correlated securities, rij = 1, and for those

that are perfectly negatively correlated, rij = −1. In real life, most of the securities are

partially positively correlated with one another.

6.3 Two-Security Portfolio

Let us first consider the simplest portfolio, the one that has only two securities in it, say

the stock of two major corporations, GM and IBM. If we add up all the weights of

securities in a given portfolio, the sum should be equal to one. For instance, if a portfolio

has 75% assets invested in GM and 25% in IBM, then 0.75 + 0.25 = 1. In general,

w1 + w2 = 1 (6.6)


111

The return defined by (6.1) is the realized return. This is in the past. What is the future

return of the stock? That is more difficult to predict. However, we can represent the

expected return by the symbol E(R). Similarly, we can represent the expected return of a

portfolio by E(Rp). For any two-security portfolio, the expected return of the portfolio,

E(Rp), will depend upon the return of the two securities, and their weights. Indeed, the

return is the weighted average of the returns of the individual securities. That is,

E(Rp) = w1 E(R1) + w2 E(R2) (6.7)

What is the risk of a two-security portfolio? The total risk will depend upon the weights

of the two securities. If more money is invested in GM, then the portfolio risk will tend to

be closer to the risk of GM.

Let us look at equation (6.4). This equation represents the risk of a portfolio. One way to

measure risk is to use the standard deviation of returns. σ(R). For more risky securities,

the standard deviation, or the spread of returns is higher. For less risky securities, the

spread is less, meaning we are more confident what the return will be. For risk-free

securities, the σ is zero.

For a two-security portfolio, the risk comes from both the securities. However, you

cannot add risk linearly. In general, two units of risk of one security plus two units of risk

from another security does not add up to four units of risk. In fact, the risk of one security

may partially cancel the risk of another security when you hold them together as a

portfolio. The greater is the diversification of the portfolio, the lesser is the risk of the

portfolio.

One could measure diversification in terms of correlation coefficient, r12 between two

stocks. You get greater diversification if you have stocks, which are unrelated to one

another. In other words, their correlation coefficient is smaller.

To begin with, σ is a non-linear quantity. As seen in (6.4), you have to find it by taking

the square root of a sum of squares. When you add the sigmas of two securities, you

cannot say that the sigma of the portfolio is the sum of their individual sigmas. You have

to add them by squaring them first, then adding them, and then you have to take the

square root. It also depends on the weights of the securities. Finally, risk of the portfolio,

σ(Rp) depends on their correlation coefficient, r12. It does become complicated. Using

statistical theory, one can prove that (6.8) represents the risk of a portfolio correctly.

Including the weights and risks of the two securities, and the correlation coefficient

between them, we can write the total risk of a portfolio as follows:

σ(Rp) = w12 σ1

2 + w2

2 σ2

2 + 2w1w2 σ1σ2r12 (6.8)

Going from a two- to a three-security portfolio complicates the problem further. In this

case, we have to consider the correlation coefficient between the first and the second

stock, between the first and the third stock, and between the second and the third stock.

That is, the formula should have r12, r13, and r23. This is what we find in equation (6.11).


112

It is better to use a computer program that can handle such equations to optimize the

formation or a real portfolio with perhaps thirty stocks.

6.4 Portfolios with Three or More Securities

We can readily extend the results for the two-security portfolio to the portfolios with

three or more securities. For a three-security portfolio, equations (6.6-8) become

w1 + w2 + w3 = 1 (6.9)

E(Rp) = w1 E(R1) + w2 E(R2) + w3 E(R3) (6.10)

σ(Rp) = w12σ1

2 + w2

2σ2

2 + w3

2σ3

2 + 2w1w2σ1σ2r12 + 2w1w3σ1σ3r13 + 2w2w3σ2σ3r23

(6.11)

We can extend our analysis for a portfolio with n securities. To summarize, let us define:

E(Rp) = expected return of the portfolio

E(Ri) = expected return of the security i

wi = weight of the security i, as a percentage of the total value of the portfolio

σ(Rp) = standard deviation of the returns of the portfolio

σi = standard deviation of the returns of the security i

cov(i,j) = covariance between the returns of securities i and j

rij = correlation coefficient between the securities i and j.

For a portfolio with n securities, (6.8) will become

w1 + w2 + w3 + ... + wn = 1

We may write it as

i=1

n

wi = 1 (6.12)

For an n-security portfolio, the expected return is

E(Rp) = w1 E(R1) + w2 E(R2) + w3 E(R3) + ... + wn E(Rn)

or,

E(Rp) = i=1

n

wi E(Ri) (6.13)

By definition, the covariance between the returns of the securities i and j is equal to the

product of the correlation coefficient between these securities and the standard deviations

of the returns of these two securities. Mathematically, we can write it as

cov(i,j) = σiσjrij (6.14)

For instance, we may write


113

σ12r11 = var(1) = cov(1,1) and σ1σ2r12 = cov(1,2)

Let us write (6.11) as follows:

(Rp) =

w1

21

2

w2w121r21

w3w131r31

+

+

+

w1w212r12

w222

2

w3w232r32

+

+

+

w1w313r13

w2w323r23

w323

2

1/2

The terms along the principal diagonal are the variance terms, and those off the diagonal

are the covariance ones. We can sum the terms in each row by using sigma notation with

j as an index, and then sum the rows using i as the index. Finally, we get for n securities,

σ(Rp) = [i=1

n

j=1

n

wiwjcov(i,j)]1/2

(6.15)

If a portfolio is composed of projects whose expected returns Ri and their standard

deviations i are all expressed in dollar amounts then we do not look at the weights of the

projects. That is, we drop the w's from the previous equations. We may write (6.6) and

(6.7) as

E(Rp) = E(R1) + E(R2) (6.16)

σ(Rp) = σ12 + σ2

2 + 2 σ1σ2r12 (6.17)

For an n-security portfolio, with dollar amounts of investments and returns, we rewrite

(6.12) and (6.14) as

E(Rp) = i=1

n

E(Ri) (6.18)

σ(Rp) = [i=1

n

j=1

n

cov(i,j)]1/2

(6.19)

Examples

Video 06.01 6.1. Cooper Corporation has the opportunity to invest in two of the

following three proposals. Which two projects should the company select, if the company

wants to maximize the ratio between expected NPV and the standard deviation?

Project A Project B Project C

Expected NPV $10,000 $11,000 $9,000

Standard deviation $2,000 $1,900 $1,500

Corr. coeff. between (A,B) = .4 (A,C) = .5 (B,C) = .8

In this problem, we regard the expected NPV as the expected return. The project costs

and returns are in dollars. For a two-security portfolio, with investments in projects 1 and

2, the sigma of the portfolio is given by



114

σ(Rp) = σ12 + σ2

2 + 2 σ1σ2r12 (6.17)

For the projects A + B, NPV = $21,000

σ= 20002 + 1900

2 + 2(0.4)(2000)(1900) = $3263

NPV/σ = 21,000/3263 = 6.4349

For A + C, NPV = $19,000

σ = 20002 + 1500

2 + 2(0.5)(2000)(1500) = $3041

NPV/σ= 19,000/3041 = 6.2472

For B + C, NPV = $20,000

σ= 19002 + 1500

2 + 2(0.8)(1900)(1500) = $3228

NPV/σ= 20,000/3228 = 6.1958

Cooper corporation should invest in projects A and B. ♥

6.2. Suppose you want to invest $40,000 in ExxonMobil, whose expected return is 15%

with standard deviation 20%, and $10,000 in Boeing whose expected return is 18% with

standard deviation 21%. The correlation coefficient between the securities is 0.75. Find

the expected total value of your portfolio after one year, and its standard deviation in

dollars.

The total value o0f the portfolio is 40,000 + 10,000 = $50,000. The weight of the first

security is 40,000/50,000 = .8. The weight of the second security is thus .2. Therefore,

w1 = 0.8, w2 = 0.2.

The expected return of the portfolio comes out to be

E(Rp) = 0.8(0.15) + 0.2(0.18) = 0.156

The standard deviation of the return of portfolio is

σ(Rp) = (.8)2(.2)

2 + (.2)

2(.21)

2 + 2(.8)(.2)(.21)(.2)(.75) = .1935

To find the expected value of the portfolio after one year, and its sigma, in dollars, we

calculate

E(V) = $50,000 (1.156) = $57,800 ♥

and σ(V) = $50,000 (0.1935) = $9,675 ♥


115

6.3. Suppose you have $40,000 that you would like to invest equally in four securities A,

B, C, and D. The expected returns from these securities are 10%, 11%, 12%, and 13%,

respectively. The standard deviations of these returns are 12%, 14%, 16%, and 18%,

respectively. The correlation coefficient between any two securities is 0.8. If the returns

are normally distributed, what is the probability that the portfolio will be worth more than

$50,000 after one year?

Since you want to invest your money equally among four securities, the weight of each

security is 25%. This means w1 = w2 = w3 = w4 = .25. Because the correlation coefficient

between any two securities is the same, 0.8, we have r12 = r13 = r14 = r23 = r24 = r34 = 0.8.

Find E(Rp) by multiplying the weight of each security by its expected return, and then

adding everything.

E(Rp) = 0.25(0.1) + 0.25(0.11) + 0.25(0.12) + 0.25(0.13) = 0.115 = 11.5%

Similarly, find the σ of the portfolio as follows,

σ(Rp) =

(.25)2(.12)

2 + (.25)

2(.14)

2 + (.25)

2(.16)

2 + (.25)

2(.18)

2

+ 2(.25)2(.12)(.14)(.8) + 2(.25)

2(.12)(.16)(.8)

+ 2(.25)2(.12)(.18)(.8) + 2(.25)

2(.14)(.16)(.8)

+ 2(.25)2(.14)(.18)(.8) + 2(.25)

2(.16)(.18)(.8)

1/2

= 0.1384

If we require the portfolio to be worth more than $50,000, then the required return is

(50,000 − 40,000)/40,000 = 0.25, or 25%. The expected return of the portfolio is 11.5%,

it quite unlikely that the return will exceed 25%. To calculate the probability, find

z = (0.25 − 0.115)/0.1384 = 0.9755

Figure 6.1. The shaded area to the right of z = .9755 represents the probability that the return of the

portfolio will be more than 25%.

Draw a normal probability distribution curve, with z = 0 in the center and z = .9755 to the

right of center. Since the expected portfolio return is 11.5%, and we require it to have a

return of more than 25%, it is unlikely that it will actually happen. The probability of it is

the area under the tail of the curve, on the right side of z = .9755. Using the tables,

calculate the probability as


116

P(R > 0.25) = .5 − [.3340 + .55(.3365 − .3340)] = 0.1643 = 16.43%

There is a 16.43% chance that the portfolio is worth more than $50,000 after one year. ♥

The following instructions will solve the problem in Excel.

A B C D E

1 E(R) 0.1 .11 .12 .13

2 σ(R) 0.12 .14 .16 .18

3 Corr Coeff 0.8

4 Weights 0.25

5 E(Rp) =B4*(B1+C1+D1+E1)

6 σ(Rp) =B4*SQRT(B2^2+C2^2+D2^2+E2^2+2*B3*(B2*C2+B2*D2+B2*E2+C2*D2+C2*E2+D2*E2))

7 Req retn =(50000-40000)/40000

8 z =(B7-B5)/B6

9 Probability =1-NORMDIST(B8,0,1,true)

6.4. Capella Corporation has an expected return of 12% and sigma .25, the expected

return of Rigel Corporation is 15% and its sigma 0.30. The coefficient of correlation

between the two companies is 0.25. Make a portfolio of these stocks so that the expected

return of the portfolio is 13%. What is the sigma of the portfolio?

We have to make a portfolio such that its expected return is 13%, but we do not know the

weights of the two securities. To find the weights, w1 and w2, you will need two

equations. The two equations are

w1 + w2 = 1 (6.6)

and

E(Rp) = w1 E(R1) + w2 E(R2) (6.7)

Substituting numbers in (6.6), we get

.13 = w1 (.12) + w2 (.15) (A)

To facilitate the calculation at WolframAlpha, subsitute w1 = x and w2 = y, then use this

expression,

WRA x+y=1,.13=.12*x+.15*y

The solution is x = .666667 and y = .333333, or w1 = 2/3, then w2 = 1/3. Using (6.8), we

get

σ(Rp) = w12σ1

2 + w2

2σ2

2 + 2w1w2σ1σ2r12

= (2/3)2(.25)

2 + (1/3)

2(.3)

2 + 2(2/3)(1/3)(.25)(.3)(.25) = 0.2147 ♥

We notice that the σ(Rp) = 0.2147 is less than σ1 = .25 and σ2 = .3. In other words, it is

quite possible to form a portfolio out of two securities so that the sigma of the portfolio is




117

less than the sigma of either of the two securities. This is because the risk of one security

can possibly offset the risk of the other one.

Video 06.06 6.6. Elizabeth Corporation is starting two new projects. Project A requires

an investment of $5,000, has expected return of 16% with standard deviation 14%.

Project B has initial investment of $15,000, expected return of 15% with standard

deviation 10%. The correlation coefficient between the projects is 0.75. Find the expected

return, in dollars, of the portfolio of these two projects. What is the probability that this

return is less than $4,000?

The total value of the portfolio is $20,000 and the weights are 0.25 and 0.75. We

calculate the expected return of the portfolio as

E(Rp) = 0.25(0.16) + 0.75(0.15) = 0.1525 = 15.25%

and in dollars, E(Rp) = 0.1525(20,000) = $3,050

σ(Rp) = (.25)2(.14)

2 + (.75)

2(.1)

2 + 2(.25)(.75)(.14)(.1)(.75) = 0.1038629

If the return is less than $4,000, it is less than 4,000/20,000 = 0.2 = 20%.

The expected return of the portfolio is 20%, but the required return is 15.25%, or less.

The portfolio can have a return of 15.25% quite easily and thus the required probability is

more than 50%. To find it, first calculate

z = (0.2 −0.1525)/0.1038629 = 0.4573.

Draw a normal probability distribution curve, with z = 0 in the center. z = .4573 will be

somewhat to the right of center. The area under the hump of the curve, to the left of z =

.4573, will give the required probability. From the tables,

P(R < $4,000) = 0.5 + 0.1736 + .73(.1772 − .1736) = 67.62% ♥

You can check the naswer at Excel by copying the following instruction.

EXCEL =NORMDIST(.2,.1525,.1038629,TRUE)

Key Terms correlation coefficient, 108,

109, 110, 112, 113, 114,

116

covariance, 110, 111

diversification, 107

expected return, 107, 108,

110, 111, 112, 114, 115

loan portfolio, 107

portfolio, 107, 108, 109, 110,

111, 112, 113, 114, 115,

116

portfolio formation, 107

portfolio theory, 107

probability distribution, 108,

114

realized return, 107, 108

risk, 107, 108, 109

standard deviation of return,

108

weights, 108, 109, 111, 115



118

Problems

6.7. Bankhead Corporation is considering the following projects, which are all

acceptable.


Expected return $4000 $5000 $6000

Standard deviation $2100 $2500 $2800

Correlation coefficients (A,B) = .8 (A,C) = .7 (B,C) = .6

Bankhead can take any one, any two, or all three projects. If the company wants to

maximize the ratio E(R)/σ(R), what is the best course of action?

Invest in B and C. Ratio = 2.3195 for B + C ♥

6.8. Suppose you have $50,000 that you would like to invest in two companies,

Bethlehem Books and Allentown Audio. Bethlehem has a return of 10% and standard

deviation 15%, while Allentown has return of 15% with a standard deviation of 20%.

The correlation coefficient between them is .5. Your portfolio should have a return of

12%. Find the standard deviation of this portfolio's returns. σ(Rp) = 14.73% ♥

6.9. Costello Corporation is undertaking these three projects:


Cost $235,000 $455,000 $310,000

Expected return 12% 11% 13%

Standard deviation 8% 9% 10%


Find the probability that the return on the portfolio is more than 15%. 33.87% ♥

6.10. The Lambda Corporation has the opportunity to invest in any of the following three

proposals:


Expected return $10,000 $11,000 $12,000

Standard deviation $2,000 $2,500 $2,800


If the company can invest in one, two, or three projects, what should it do to maximize

the ratio between expected return and standard deviation?

E(R)/ ratios: A = 5, B = 4.4, C = 4.28, (A + B) = 5.5630, (A + C) = 5.2680,

(B + C) = 4.5735, (A + B + C) = 5.2916. Take (A + B) ♥

6.11. Kagera Company has made a portfolio of these three securities:


119

Cost E(R) σ(R)

Treasury bond $100,000 6% 0

Kasai Corporation $80,000 15% 25%

Limpopo Company $70,000 16% 30%

The correlation coefficient between Kasai and Limpopo is 0.6. If the returns are normally

distributed, find the probability that the return of the portfolio is more than 15%.

41.04% ♥

6.12. Excel. The expected return from two stocks, Apple and Google, under different

states of the economy are as follows:

State of the economy Probability Apple Google

Poor 10% 0% −50%

Average 30% 20% 20%

Good 60% 20% 30%

You have invested $40,000 in Apple and $60,000 in Google to form a portfolio. Find the

following.

(A) Expected return of Apple and of Google. 18%, 19% ♥

(B) The σ of Apple and of Google. 6%, 23.43% ♥

(C) Coefficient of correlation between the two stocks. .9816 ♥

(D) Expected return and σ of the portfolio. 18.6%, 16.42% ♥

(E) Probability that the return of the portfolio will be more than 15%. 58.68% ♥

120

7. CAPITAL ASSET PRICING MODEL


1. Understand the concept of beta as a measure of systematic risk of a security.

2. Calculate the beta of a stock from its historical data.

3. Understand the Capital Asset Pricing Model.

4. Apply it to determine the risk, return, or the price of an investment opportunity.

7.1 Beta

In the section on capital budgeting, we saw the need for a risk-adjusted discount rate for

risky projects. The risk of an investment or a project is difficult to measure and to

quantify. This difficulty arises from the fact that different persons have different

perceptions of risk. What may be quite a risky project to one investor may appear to be

fairly safe to another person. After all, how can you quantify courage, or patience, or risk,

or beauty?

In the section on portfolio theory, we used σ as a measure of risk, which is really the

standard deviation of returns. Another useful measure of risk is the β of an investment.

Like σ, β is also a statistical measure of risk. We infer it from the observations of the past

performance of a stock. For example, we may want to find the risk of buying and holding

the stock of a particular corporation, such as IBM, and we are interested in finding the β

of IBM. We can start by looking at the historical value of three variables:

1. The returns of IBM stock, Rj. We define the return on a stock by the relation

Rj = P1 − P0 + D1

P0 (7.1)

In the above equation, P0 is the purchase price of the stock, P1 its price at the end of the

holding period, and D1 is the dividend paid, if any, at the end. The quantity P1 − P0 is the

price appreciation of the stock, and along with the dividend, is the total change in the

value of the investment. The return is equal to be the change in the value of the

investment divided by the original investment. For example to find the monthly rate of

return on the IBM stock, we may want to know the price of the stock at the beginning of

each month, the price at the end of the month, and the dividends paid during that

particular month. We have to develop a series of numbers representing the return for each

month for the last 24 months, say.

2. The returns of the market, Rm. A market index provides an overall measure of the

performance of the market. The oldest and the most popular market index is the Dow

Jones Industrial Average. The problem with this index is that it uses only 30 stocks in its

valuation. For a broader market index, we may have to look at S&P100, or S&P500

index. There is even an index for over-the-counter stocks called the NASDAQ Composite

Index. The value of these indexes is available daily.

Analytical Techniques 7. Capital Asset Pricing Model _____________________________________________________________________________

121

Let us track the market for the last 24 months. If we know the value of the index at the

start and finish of each month, we can find the return of the market for that month. The

dividend yield for the market is around 1.71% annually at present. Therefore, we define

the overall return on the market as

Rm = M1 − M0

M0 + d1 (7.2)

where M0 is the beginning value and M1 the ending value of the market index, and d1 is

the dividend yield as a percent for that period. With some effort, we may be able to

develop a set of market returns for each of the last 24 months.

3. The riskless rate of interest, r. The securities issued by the Federal government, such

as the Treasury bills, bonds, and notes, are, by definition, riskless. They are the safest

investments available, backed by the full faith and taxing power of the government. Their

rate of return depends on their time to maturity, and for longer maturity, the return is

generally higher. The Treasury yield curve is available on the Internet.

After some research, we may also get a series of riskless rates for each of the past 24

months.

Then we define two variables x and y as:

y = Rj −r

x = Rm −r

where Rj = return on the stock j each month for the last 24 months,

Rm = corresponding monthly returns on the market for the same period,

and r = riskless rate of interest per month, for the last 24 months.

By subtracting the riskless rate of interest, we are able to see the return due to the risk

inherent in the given stock, and the return from the risk in the market. Thus, we are

comparing the returns exclusively due to the risk in the investments.

A regression line drawn between the various observed values of x and y will show a

certain linear relationship between x and y. The slope of the line will give the rate of

change of y with respect to x. In other words, the slope will signify how much the return

on the stock will change corresponding to a given change in the return on the market. In

this diagram let us say that the slope of the line is β, and the y-intercept is α. The quantity

α is practically zero, and it is statistically insignificant. The quantity β, on the other hand,

represents an important concept.

This responsiveness of the stock return to the changing market conditions is called the

"beta" of the stock. Stocks with low betas will show very little movement due to the

fluctuations in the stock market. High beta stocks will tend to be jumpy showing a large

variation in response to small changes in the market.


122

Fig. 7.1: A regression line between y and x.

High β stocks, due to their large volatility, will be more unpredictable, and therefore,

more risky. Low beta stocks show relatively small volatility, and they are more

predictable and safe.

Beta is a statistical quantity, and it is a measure of the systematic risk, or the market

related risk of a stock. These results can also be expressed as a statistical formula,

βj = cov(Rj,Rm)

var(Rm) =

rjmσmσj

σm2 =

rjmσj

σm (7.3)

where cov(Rj,Rm) is the covariance between the returns on the stock j and the market, and

var(Rm) is the variance of the returns on the market. If we have collected sufficient

statistical data, we may find β by using

β = n(xy) − (x)(y)

nx2 − (x)

2 (7.4)

α = y − βx

n (7.5)

where n is the number of x and y values.

One can apply the concept of beta to a portfolio. The beta of a portfolio is simply the

weighted average of the betas of the securities in the portfolio,

Beta of a portfolio, βp = w1β1 + w2β2 + w3β3 + ... = i=1

n

wiβi (7.6)


123

The advantage of using β as a measure of risk is that it can combine linearly for different

securities in a portfolio, but the disadvantage is that it can measure only the market

related risk of a security. On the other hand, σ can measure the risk independent of the

market conditions, but its disadvantage is that it is non-linear in character and difficult to

apply in practice. Both β and σ are incomplete measures of risk; they change with time,

and are difficult to measure accurately.

By definition, the beta of a riskless investment is zero. Further, the beta of the market is

1. This is seen by setting j = m in (7.3) and noting that the covariance of a random

variable with itself is just its variance.

A security that has a high beta should show a large rise in price when there is an upward

movement in the market, and has a large drop in price in case of a downward movement.

These large price fluctuations can cause a considerable amount of uncertainty about the

return of this security, and greater risk associated with it. Therefore, a high beta security

is also a high-risk security. Thus, beta is frequently used as a measure of the risk of a

security. A low beta security is a defensive security and a high beta of a stock means a

more aggressive management stance.

The numerical value of β for different stocks is available from sources on the Internet,

such as www.etrade.com, and www.yahoo.com.

Examples

Video 07.01 7.1. Calculate the β of Hauck Corporation from the following data. The

prices are at the beginning and end of each year:

Year

Price of

Hauk stock

Dividend

per share

S&P 500

index

S&P 500

dividend

yield

Riskless

rate

beginning end beginning end

2005 $25 $27 $1.00 1000 1050 3.05% 6.00%

2006 $27 $29 $1.00 1050 1100 3.00% 6.00%

2007 $29 $32 $1.50 1100 1200 2.95% 5.95%

2008 $32 $33 $1.50 1200 1250 2.80% 5.90%

The return from the security in 2005 is capital gains ($2) plus dividends ($1) divided by

the initial price ($25), that is, 3/25 = 0.12. The riskless rate during 2005 was 0.06, thus

the excess return was 0.12 − 0.06 = 0.06. The return on the market for the same year was

5/100 + 0.0305 = 0.0805. The excess return was 0.0805 − 0.06 = 0.0205. Designating the

excess return for security as y and that for the market as x, we can tabulate the

calculations as:

Year Rj − r = y Rm − r = x

2005 3.00/25 − .06 = .06 5/100 + .0305 − .06 = 0.0205

2006 3.00/27 − .06 = .051111 5/105 + .03 − .06 = 0.017619

2007 4.50/29 − .0595 = .095672 10/110 + .0295 − .0595 = 0.060909

2008 2.50/32 − .059 = .019125 5/120 + .028 − .059 = 0.010667



124

Here n = 4, the number of periods, or x, y pairs

Σxy = (.0205)(.06) + (.017619)(.051111) + (.060909)(.095672) + (.010667)(.019125)

= 0.0081618

Σx = 0.0205 + 0.017619 + 0.060909 + 0.010667 = 0.109695

Σy = .06 + .051111 + .095672 + .019125 = 0.225908

Σx2 = (0.0205)

2 + (0.017619)

2 + (0.060909)

2 + (0.010667)

2 = 0.00455437

Using equation (7.4)

β = 4(0.0081618) − (0.109695)(0.225908)

4(0.00455437) − (0.109695)2 = 1.271927967 1.27

One can do the above problem with the help of Maple as follows:

#n is the number of periods, or returns

#n+1 is the number of price data points

n:=4;

#Price is an array to store price of stock

Price:=array(1..n+1,[25,27,29,32,33]);

#Div is an array to store dividends

Div:=array(1..n,[1,1,1.5,1.5]);

#Market is the array to store market index data

Market:=array(1..n+1,[100,105,110,120,125]);

#Markdiv is the array to store market dividends as percent

Markdiv:=array(1..n,[.0305,.03,.0295,.028]);

#RF is the array to store riskfree rate

RF:=array(1..n,[.06,.06,.0595,.059]);

#x, y are the arrays to store x, y values

x:=array(1..n); y:=array(1..n);

for i to n do

x[i]:=(Market[i+1]-Market[i])/Market[i]+Markdiv[i]-RF[i];

y[i]:=(Price[i+1]-Price[i]+Div[i])/Price[i]-RF[i] od;

unassign('i');

n*sum(x[i]*y[i],i=1..n)-sum(x[i],i=1..n)*sum(y[i],i=1..n);

n*sum(x[i]^2,i=1..n)-sum(x[i],i=1..n)^2;

beta=%%/%;

7.2. Calculate the β of Maine Corporation from the following data. The prices are at the

beginning and at the end of each year:


125

Year Price of

Maine

Dividend of

Maine

S&P 500

index

S&P 500

dividend

Riskless

rate

2000 25-27 $2.00 100-105 3.05% 8.0%

2001 27-29 $2.00 105-110 3.20% 8.5%

2002 29-32 $2.50 110-120 3.50% 7.5%

2003 32-33 $2.50 120-125 4.00% 7.0%

β = 0.89 ♥

7.2 Capital Asset Pricing Model

Beta is a measure of the market risk, or the systematic risk, of a security. A security with

a large beta will have large swings in its price in relation to the changes in the market

index. This will lead to a higher standard deviation in the returns of the security, which

will indicate a greater uncertainty about the future performance of the security.

Draw a diagram with the β of various securities along the X-axis and their expected

return along the Y-axis. We have already noticed that β is a linear measure of risk. If we

assume that a linear relationship exists between the risk and return, then only two points

are sufficient to draw a straight line in this diagram. The line, representing the

relationship between risk and expected return, is called the security market line. Under

equilibrium conditions, all other securities will also lie along this line. Higher β securities

will have a correspondingly higher expected return. Figure 7.2 shows this graphically.

Fig. 7.2: Security market line.

By definition, beta of the market is equal to 1. The securities with more than average risk

will have beta greater than 1, and less risky securities have beta less than 1. On this scale,

the beta of a riskless security is zero. Such securities will provide riskless rate of return, r,

to the investors. An example of such a security is the Treasury bill.


126

The security market line represents the risk-return characteristics of various securities,

assuming that there is linear relationship between risk and return. Point A represents a

riskless security with beta equal to zero and return r. Point B shows a market-indexed

security which could be a very large mutual fund portfolio, which is invested in a large

number of securities all weighted according to assets of the corporations whose securities

make up the portfolio. Point C shows an individual security whose beta is βi and whose

expected return is E(Ri). Since A, B, C all lie along the same straight line, then

Slope of segment AC = slope of segment AB

This gives, E(Ri) − r

βi =

E(Rm) − r

1

Or, E(Ri) = r + βi [E(Rm) − r] (7.7)

William Sharpe (1934- )

Equation (7.7) gives the expected return of a security i in terms of its

risk, expected return on the market, and the riskless rate. It is a

forward-looking model, and thus gives the expected values of the

returns. This equation represents what is known as the "Capital

Asset Pricing Model", CAPM for short, and was developed in the

1960s by William Sharpe, Jan Mossin, and John Lintner. The use of

this model is illustrated by the following examples.

Positive Alpha: Too Good to be True? New research from Robert Jarrow suggests that positive alpha is improbable.

During the past 25 years, an entire segment of the investment industry was constructed on the belief that

positive alphas exist and can be exploited by portfolio managers to yield greater profit at less risk. New

research by the Johnson School's Robert Jarrow strongly suggests that positive alphas are rare to

nonexistent.

"Every hedge fund in the world claims to have positive alpha, but I say it can't be," says Jarrow, Ronald P.

and Susan E. Lynch Professor of Investment Management at the Johnson School. "The claims for positive

alpha are too strong—professional investment managers are taking risks that are hidden."

Alpha, an estimate of an asset's future performance, after adjusting for risk, is a measure routinely

calculated by portfolio managers. Positive alpha suggests that an investor can realize higher returns at

lower risk than by holding an index. In other words, by investing in assets with positive alpha, one can

"beat the market," without exposure to the risk otherwise associated with the promised rate of return.

Jarrow used mathematical modeling to prove that positive alphas are equivalent to arbitrage opportunities.

And arbitrage opportunities—risk-free trading of an asset between two markets to take advantage of a price

differential—are rare in financial markets. According to Jarrow's research, positive alpha can exist only in

the presence of a true arbitrage opportunity. For this to occur, two stringent conditions must be met. First,

there must exist a market imperfection that enables the arbitrage opportunity to persist, even as arbitrageurs

capitalize upon it; second, there must be a source of financial wealth, on which the arbitrageurs draw, either

knowingly or unknowingly.

http://en.wikipedia.org/wiki/Capital_asset_pricing_model


http://en.wikipedia.org/wiki/William_Forsyth_Sharpe

http://en.wikipedia.org/wiki/Jan_Mossin

http://en.wikipedia.org/wiki/John_Lintner


127

"Academics have looked for arbitrage opportunities in financial markets, and haven't found many. So it

seems implausible to have so many positive alphas out there." Jarrow says. "To have positive alpha for any

length of time means that someone is consistently losing money to someone else, and that's hard to

believe."

In his paper "Active Portfolio Management and Positive Alphas: Fact or Fantasy?" forthcoming in the

Journal of Portfolio Management, Jarrow outlines his model and offers examples of both true and false

positive alphas, drawn from the pivotal events of the credit market crisis. His conclusions include a word of

caution to investors.

"The moral of this paper is simple," Jarrow writes. "Before one invests in an investment fund that claims to

have positive alphas, one should first understand the market imperfection that is causing the arbitrage

opportunity and the source of the lost wealth. If the investment fund cannot answer those two questions,

then the positive alpha is probably fantasy and not fact."

The Journal of Portfolio Management, Summer 2010, Vol. 36, No. 4: pp. 17-22

Examples

Video 07.03 7.3. Chicago Corp stock will pay a dividend of $1.32 next year. Its current

price is $24.625 per share. The beta for the stock is 1.35 and the expected return on the

market is 13.5%. If the riskless rate is 8.2%, what is the expected growth rate of Chicago?

Using the capital asset pricing model (CAPM),

E(Ri) = r + βi [E(Rm) − r] (7.7)

We first find the expected rate of return as

E(Ri) = 0.082 + 1.35 [0.135 − 0.082] = 0.15355 = R

The expected rate of return E(Ri), for a security is also its required rate of return R by the

investors. Using the growth model for a stock, equation (3.6),

P0 = D1

R − g

we get, R − g = D1/P0, or g = R − D1/P0,

which gives g = 0.15355 − 1.32/24.625 = 0.1. Thus the growth rate is 10%. ♥

Video 07.04 7.4. Peggotty Services common stock has a β = 1.15 and it expects to pay

a dividend of $1.00 after one year. Its expected dividend growth rate is 6%. The riskless

rate is currently 12%, and the expected return on the market is 18%. What should be a

fair price of this stock?

E(Ri) = r + βi [E(Rm) − r] (7.7)

we get E(Ri) = 0.12 + 1.15 [0.18 − 0.12] = 0.189




128

Thus, the expected return on the stock is 0.189, and the expected growth rate is 0.06.

Using (3.1) once again,

P0 = 1

0.189 − 0.06 = $7.75 ♥

Video 07.05 7.5. The beta of Vega Inc is 1.15, its rate of growth is 10%, it will give a

dividend of $3.00 next year, and its common stock sells for $50 a share. The riskless rate

is 8%. By careful planning and by selecting more secure projects, Vega has reduced its

risk. Its new beta is estimated to be 1, while everything else (income, dividends, growth

rate, capital structure, market return, etc.) is the same. What is its new share value?

The total return on a stock is the sum of its dividend return and the growth rate. If r is the

required rate of return, E(Ri) is the expected rate of return, g is the growth rate, D1 is the

dividend to be paid next year, and P0 is its price now, then

R = D1

P0 + g =

3

50 + 0.1 = 0.16 = E(Ri)

Use E(Ri) = r + βi [E(Rm) − r] (7.7)

Drop the subscript i, and solve for E(Rm), to get

Or, E(Rm) = r + E(R) − r

β

Or, E(Rm) = 0.08 + (0.16 − 0.08)/1.15 = 0.1496

The new β is 1, and since the β of the market is also 1, this implies that

E(R) = E(Rm) = 0.1496

Thus P0 = 3/(0.1496 − 0.1) = $60.53 ♥

7.6. Eastern Oil stock currently sells at $120 a share. The stockholders expect to get a

dividend of $6 next year, and they expect that the dividend will grow at the rate of 5%

per annum. The expected return on the market is 12% and the riskless rate is 6%. This

morning Eastern announced that it has won the multimillion dollar navy contract, and in

response to the news, the stock jumped to $125 a share. Find the beta of the stock before

and after the announcement.

Using Gordon's growth model, P0 = D1

R − g, we get R = D1/P0 + g, which is also the

expected return on the stock, E(R). But by CAPM,

E(Ri) = r + βi [E(Rm) − r]



129

we get β = E(Ri) − r

E(Rm) − r

Thus β = D1/P0 + g − r

E(Rm) − r =

6/120 + 0.05 − 0.06

0.12 − 0.06 = 0.667, before. ♥

And β = 6/125 + 0.05 − 0.06

0.12 − 0.06 = 0.633, after. ♥

7.7. Jupiter Gas Company is planning to acquire Saturn Water Company. The additional

pre-tax income from the acquisition will be $100,000 in the first year, but it will increase

by 2% in future years. Because of diversification, the beta of Jupiter will decrease from

1.00 to 0.9. Currently the return on the market is 12% and the riskless rate is 6%. What is

the maximum price that Jupiter should pay for Saturn? The tax rate of Jupiter is 35%.

The new beta for Jupiter is 0.9. Using CAPM, its expected return, and hence the cost of

capital will be

E(R) = 0.06 + 0.9(0.12 − 0.06) = 0.114

After tax income = 100,000 (1 − 0.35) = $65,000.

The total value of a firm is the present value of its future earnings, properly discounted.

Thus, the value added to Jupiter due to the acquisition of Saturn is the present value of

future after-tax earnings of Saturn, discounted at a rate equal to the cost of capital of

Jupiter, and summed up to infinity. Thus

PV = 65‚000

1.114 +

65‚000 (1.02)

1.1142 +

65‚000 (1.02)2

1.1143 ... ∞ = $691,489

WRA Sum[65000*1.02^(i-1)/1.114^i,{i,1,infinity}]

Jupiter should pay at most $691,489 for Saturn. ♥

7.8. Hamlin Dairies stock has a beta of 1.33. It has just paid its annual dividend of $1.20,

and it sells for $30 a share. Shareholders believe that Hamlin is growing at the rate of 7%

annually and will maintain a constant dividend payout ratio. Due to the unexpected death

of the chairperson, Hannibal Hamlin, the company is facing an uncertain future, and the

price per share dropped to $25. There is no other change in the company (dividends,

growth, sales, etc.) or in the market. The riskless rate is 6%. In light of the greater risk of

the company, find its new beta.

If the current dividend is $1.20, next year it will be 1.20(1.07) = $1.284. Apply Gordon’s

growth model, (3.6), to find the required rate of return for the stockholders. Before

Hamlin’s death, it is

R = D1/P0 + g = 1.20(1.07)/30 + 0.07 = 0.1128 (before)



130

After Hamlin’s death, the stock price drops suddenly, but the growth potential and the

current dividend remains intact. Thus the required rate of return after the death is

R = 1.20(1.07)/25 + 0.07 = 0.12136 (after)

The expected and the required rate of return for the stock are the same, meaning R =

E(R). We can use these numbers in CAPM, (7.7), to get two equations:

Before, 0.1128 = 0.06 + 1.33 [E(Rm) − 0.06]

After, 0.12136 = 0.06 + β[E(Rm) − 0.06]

Put β = x and E(Rm) = y temporarily. Copy and paste the following instruction at

WolframAlpha to solve the two equations simultaneously.

WRA .1128=.06+1.33*(y-.06),.12136=.06+x*(y-.06)

The approximate solution is x 1.54562 and y 0.0996992. Solving for beta, we get,

β = 1.55 ♥

The new β, 1.55, is higher than the previous β, 1.33, because of the uncertainty created by

the death of the chairperson. Greater uncertainty also means greater risk.

7.9. Epperly Fund invests in S&P500 companies and thereby simulates a market

portfolio. The expected return of Epperly is 13.5%, with a standard deviation of 10%.

Suppose you are able to borrow $10,000 at the riskless rate of 9%, and you already have

$10,000 of your own money. If you invest this $20,000 in Epperly Fund, what is the

probability that you will have a return greater than 25% on your own money?

The β of the market is 1, by definition. Epperly Fund mimics the market and therefore, its

β is also 1. When you borrow money to buy securities, the amount of borrowing is

equivalent to a negative cash position in your account. The β of cash is zero, because the

value of cash does not change due to fluctuations in the stock market. The total value of

the portfolio you own is $10,000, which equals your investment. Its composition is as

follows:

Value β Weight

Epperly Fund $20,000 1 2

Cash −10,000 0 −1

Portfolio 10,000 2 1

To find the β of the portfolio, use

βp = w1β1 + w2β2 = 2(1) + (−1)(0) = 2

This is highlighted in the previous table. With the help of CAPM, find




131

E(Rp) = .09 + 2(.135 − .09) = .18

The portfolio is consisting of two items with weights 2 and −1. The σ of the components

is, 10% for Epperly Fund and zero for cash. Using equation (6.4) for the σ of a two-

security portfolio, we have

σp = (2)2(.1)

2 + (−1)

2(0) + 2(2)(−1)(.1)(0)r12

Solving the above equation, we get σp = .2. We note that the expected return of the

portfolio is 18% with a standard deviation of 20%. The required return is more than 25%.

The probability of getting that return is less than 50%. To calculate it, first find

z = (25 − 18)/20 = 0.35

Draw a normal probability distribution curve, with z = 0 at the center and z = .35 to the

right of center. The probability of getting a return of greater than 25% is equal to the

shaded area to the right of z = .35. From the table, we get its value as,

P(R > 0.25) = 0.5 − .1368 = .3632 = 36.32%. ♥


7.10. Markham Co paid a dividend of $3.00 yesterday, but these dividends are expected

to grow at the rate of 5% in the long run. The beta of Markham is 0.95, the expected

return on the market is 15%, and the riskless rate is 10% at present. Find the price of one

share of Markham stock.

Using the CAPM, we have, E(R) = 0.10 + 0.95(0.15 − 0.1) = 0.1475

Using Gordon's growth model, we get the price of a share as

P0 = 3(1.05)

0.1475 − 0.05 = $32.31♥

7.11. You have developed the following information about two mutual funds:


132

Name of fund Beta Expected return

Dione Market Fund 1.00 14%

Rhea Energy Fund 0.80 13%

You have $5,000 to invest and you put $3,000 in Dione and $1,000 each in Rhea and

riskless bonds. Find the beta and expected return of your portfolio.

Let us first find the riskless rate. Dione has β of 1, the same as that of the market. Thus

the expected return of the market is also 0.14. Using CAPM, and using the information

about Rhea,

0.13 = r + 0.8(0.14 − r)

which gives the riskless rate, r = .09. The weights of securities are

w1 = 0.6, w2 = 0.2, and w3 = 0.2.

The beta of the portfolio is just the weighted average of the betas of the individual

securities. That is,

βp = 0.6(1.00) + 0.2(0.8) + 0.2(0) = 0.76 ♥

Similarly, the expected return on the portfolio is given by

E(Rp) = 0.6(0.14) + 0.2(0.13) + 0.2(0.09) = 0.128 ♥

You can also calculate the expected return of the portfolio by substituting β = .76, r = .09

and E(Rm) = .14 in CAPM. This gives

E(Rp) = .09 + .76[.14 − .05] = .128

7.12. Pindar Corporation stock is selling for $80 a share and its dividend next year is

expected to be $2. S&P500 index is 1437 at present, and it is expected to go up to 1550

after one year. The average dividend yield for the S&P500 is 1.52%, and the riskless rate

is 5.14%. If the beta of Pindar is 1.14, find the expected price of one share of Pindar after

one year.

Using the information about the market, find the expected percentage return on the

market as the sum of the dividend yield of the market (.0152) and its price appreciation

(1550 − 1437)/1437. This gives

E(Rm) = 0.0152 + (1550 − 1437)/1437 = .09384

Next, find the expected return of Pindar using CAPM. Put r = .0514, β = 1.14, and E(Rm)

= .09384.

E(Rj) = 0.0514 + 1.14(.09384 − 0.0514) = .09978

If x is the expected price of the stock next year, then the return of the stock is,


133

.09978 = (x − 80 + 2)/80

This gives x = 80(.09978) + 80 – 2 = $85.98 ♥

7.13. A portfolio is formed as follows:

Stock Amount invested β σ(R)

Childs Corporation $12,000 1.25 25%

Jermyn Company $13,000 1.20 22%

The riskless rate is 7%, and the expected return on the market is 14%. The covariance

between the two stocks is 0.0385. Find the expected return, and the standard deviation of

returns of the portfolio, in dollars, and as a percentage.

With the total value of the portfolio = 12,000 + 13,000 = $25,000, it is easy to see that the

weights are w1 = 12/25 = 0.48, and w2 = 0.52

Using CAPM, the expected returns are

E(R1) = 0.07 + 1.25(0.14 − 0.07) = 0.1575

and E(R2) = 0.07 + 1.2(0.14 − 0.07) = 0.1540

The portfolio return is E(Rp) = 0.48(0.1575) + 0.52(0.154) = 15.568% ♥

= 0.15568(25,000) = $3892 ♥

The correlation coefficient, r12 = cov(1,2)

σ1σ2 =

0.0385

(0.25)(0.22) = 0.7

The portfolio sigma, σ(Rp) = 0.482 0.25

2 + 0.52

2 0.22

2 + 2(0.48)(0.52)(0.25)(0.22)(0.7)

= 21.61% ♥

And in dollars, σ(Rp) = 0.2161(25,000) = $5403 ♥

7.14. The β of Blakely Company is 1.2. Blakely is planning to acquire Waymart

Corporation which will result in the combined company to have a β of 1.3. The riskless

rate is 6%, and the expected return on the market is 12%. Waymart Corporation is

expected to have first year earnings after taxes of $40,000, and these earnings are

expected to increase by 3% per annum in future. How much should Blakely pay for

Waymart?

With β = 1.3, use CAPM to find the risk adjusted discount rate = 0.06 + 1.3(0.12 − 0.06)

= 0.138


134

PV of earnings = 40‚000

1.138 +

40‚000(1.03)

1.1382 +

40‚000(1.03)2

1.1383 + ... ∞=

40‚000

1.138

1 − 1.03

1.138

= $370,370

The value of Waymart is thus $370,370 ♥

7.15. McNamara Fund has expected return of 10.5%, standard deviation of 17.5%, and

beta of 0.8. Schlesinger Fund has expected return of 12.5%, standard deviation of 21%

and beta of 1.1. The two mutual funds have correlation coefficient of 0.7. Find the

expected return and standard deviation of the market. What is the riskless rate of return?

First we use the CAPM, and put the numbers for the two funds, which gives us

McNamara: 0.105 = r + 0.8 [E(Rm) − r]

Schlesinger: 0.125 = r + 1.1 [E(Rm) − r]

To solve the equations, put r = x and E(Rm) = y temporarily. Then copy and paste the

following instruction at WolframAlpha.

WRA .105=x+.8*(y-x), .125=x+1.1*(y-x)

Solving the two equations, we get E(Rm) = 0.1183, and r = 0.05167 ♥

Next we construct the market portfolio out of these two funds. The β of the market, by

definition, is 1. The weights are w1 and w2, and they are combined to get β = 1 for the

market portfolio. Thus, we have

w1 + w2 = 1

0.8 w1 + 1.1 w2 = 1

To solve the equations, put w1 = x and w2 = y temporarily. Then copy and paste the

following instruction at WolframAlpha.

WRA x+y=1,8/10*x+11/10*y=1

Solving these two equations we get, w1 = 1/3 and w2 = 2/3. The same result can be

obtained by combining the expected returns of the two funds to get the expected return of

the market. Now we can find the sigma of the market as follows:

σ(Rm) = (1/3)2 0.175

2 + (2/3)

2 0.2

2 + 2(1/3)(2/3)(0.175)(0.21)(0.7) = 0.1856 ♥

7.16. Armstrong Corporation $6 preferred stock sells for $50 a share. The beta of this

stock is 1.25. The current riskless rate is 8%. Just yesterday, Louis Armstrong, the

founder and CEO, died and the stock dropped to $47 a share in response to the news.

Find the new beta of Armstrong preferred.






135

A preferred stock has fixed dividends, that is, there is no expectation of growth. This

means g = 0 in Gordon’s growth model, P0 = D1/(R – g), which becomes P0 = D1/R.

Rewrite it as R = D1/P0. This implies that the current return of the stock is 6/50 = .12.

This is quite reasonable. If you buy a stock for $50 a share and it pays a dividend of $6

annually, without any growth opportunity, the return is indeed 12%.

Using CAPM,

.12 = .08 + 1.25[E(Rm) − .08]

Solve this equation to get E(Rm) = (.12 − .08)/1.25 + .08 = .112

This is quite reasonable, because the stock, with its β1 = 1.25, has a return of .12; and the

market with its β = 1, should have a lower expected return, perhaps around 11%.

Let β1 = 1.25, beta of the stock before Armstrong’s death and β2 = beta of the stock after

his death. Now assume that Armstrong is just an insignificant player in the stock market,

and the market will ignore his demise. The expected return on the market will remain at

.112 and the risk-free rate at .08. The new return on the stock is 6/47. The CAPM gives

us

6/47 = .08 + β2[.112 − .08]

This gives β2 = (6/47 − .08)/(.112 − .08) = 1.49 ♥

The answer is quite reasonable, because Louis Armstrong was a very important

individual at Armstrong Company. His departure has introduced a substantial measure of

uncertainty, or risk, in the company, thereby increasing its β from 1.25 to 1.49.

Problems

7.17. The Washington Corp stock has a β of 1.15 and it will pay a dividend of $2.50 next

year. The expected rate of return of the market is 17% and the current riskless rate is 9%.

The expected rate of growth of Washington is 4%. Find the value of its common stock.

$17.61 per share ♥

7.18. Molopo Company has β = 1.2, whereas the return on the market is expected to be

12%, with a standard deviation of 8%. The riskless rate is 6% at present. The stock of

Molopo is selling at $100 a share, but it does not pay any dividends. Find the probability

that it will be selling for more than $120 by next year. Assume that the entire change in

the stock price is due to the change in the market. 23.94% ♥

7.19. Cheever Corp stock is selling at $40 a share. Its dividend next year will be $2 a

share and its beta is 1.25. Crane Company has the same growth rate as Cheever. The

current stock price of Crane is $55 a share, and its dividend this year is $3. The riskless

rate is 8% and the expected return on the market is 16%. Find the beta of Crane stock.

1.3955 ♥


136

7.20. Kingston Corporation has β = 1.2. It is interested in buying Plains Corporation

which also has β = 1.2. Kingston believes that after the acquisition, its β will be 1.1. The

expected after-tax earnings from Plains will be $50,000 for the first year, but this figure is

expected to increase by 3% per year in future. The expected return on the market is 12%,

and the riskless rate is 6%. Find the amount that Kingston should spend on this

acquisition. $520,833 ♥

7.21. Toledo Corporation estimates its β as 1.3, whereas the risk-free rate is 5% at

present. The expected return on the market is 11%, with a standard deviation of 7%.

Assume that the variation in the Toledo stock price is entirely due to the fluctuations of

the market. If you invest $10,000 in Toledo stock now, what is the probability that the

value of your investment will be more than $12,000 by next year? 21.45% ♥

7.22. Palmer Company stock has paid a dividend of $1.25 this year, which is in line with

its long-term growth rate of 5%. Its current β is 1.2 and the expected return of the market

is 12%. Today, after the company won the multimillion-dollar contract from the navy, the

stock jumped 3%, to $15.45 a share, in response to the good news. Find the risk-free rate

and the new β of the stock. r = 3.25%, new β = 1.171 ♥

7.23. Johnson Corporation preferred stock sells for $37 a share and pays an annual

dividend of $4. The β of this stock is 1.3. The current riskless rate is 3%. The common

stock of Johnson was upgraded by the analysts from ‘hold’ to ‘buy’ today. In response to

the news, the preferred stock jumped in price by $1. Find the new β of Johnson preferred.

1.253 ♥ Key Terms

beta, 116, 117, 118, 119, 120,

121, 123, 124, 125, 126,

127, 128, 129, 130, 131

Capital Asset Pricing Model,

116, 121, 122

capital budgeting, 116

CAPM, 122, 123, 124, 125,

127, 128, 129, 130

discount rate, 116, 129

dividend, 116, 117, 119, 121,

123, 124, 125, 127, 128,

130, 131

Dow Jones Industrial

Average, 116

linear relationship, 117, 121,

122

market index, 116, 117, 120,

121

regression line, 117, 118

return, 116, 117, 119, 121,

122, 123, 124, 125, 126,

127, 128, 129, 130, 131

risk, 116, 117, 118, 119, 121,

122, 124, 125, 130, 131

S&P500, 116, 126, 128

security market line, 121

standard deviation of returns,

116, 128

systematic risk, 116, 121

137

8. OPTION PRICING THEORY

Objectives: After reading this chapter, you will

1. Understand the role of options in financial markets and the terminology used.

2. Calculate the value of an option using the Black-Scholes model.

3. Apply the put-call parity theorem.

4. Use options in portfolio management and the valuation of risky securities.

8.1 Options

Suppose you believe that the price of gold is going to increase in the near future and you

want to buy some gold in anticipation of its price rise. However, you do not have enough

capital to finance your purchase and you do not want to take the risk of a major loss in

the event of a sharp drop in the price of gold. You can overcome both these problems by

buying a "call option" on gold. If the gold rises in price you can exercise your option to

buy gold at a preset price and resell it in the market for an immediate profit. If the price

drops, you have to do nothing, and your loss will be limited to the premium paid for the

call option. The call option gives you the right but not the obligation to buy an asset at a

previously agreed upon price.

There are several elements in a call option:

1. A call option is a contract between a buyer of the call option and a seller of the option.

The buyer and seller enter into the contract by mutual agreement.

2. The buyer of the call pays a certain amount of money to the seller of the call to initiate

this contract. This amount is non-refundable, and is called the call price or call premium.

3. This contract gives the buyer of the call the right but not the obligation to buy a certain

asset. The asset may be a piece of land, an ounce of gold, or 100 shares of Home Depot

stock. The buyer of the call exercises the call option if he buys the assets. Of course, he

may not exercise the option at all. If the option is exercised, the seller of the call is

obligated to sell the asset. It is an asymmetric contract. The buyer of the call must

compensate the seller of the call for this disadvantage by paying a premium for the call,

C.

4. There is a strict time limit for this contract, T. When this time has elapsed, the call

expires and the contract becomes void.

5. There is a certain exercise price, X, which is the purchase price of the asset. This is the

price that buyer of the call option must pay to the seller of the call if he (the buyer of the

call) decides to buy the asset by exercising the call during the life of the option.

The buyer of a call will exercise the call only if it gives him some financial advantage.

For instance, if the exercise price of a call is $40 and the stock is trading at $43 per share

Analytical Techniques 8. Option Pricing Theory _____________________________________________________________________________

138

just before the expiration of the call, then the owner of a call will exercise it and buy the

stock by paying only $40 per share for the stock. This gives him an advantage of $3 per

share.

The buyer of a call believes that the price of the asset will rise above the exercise price

during the life of the contract and that he will be able to buy the asset at less than its

market value. In case of a large drop in the value of the asset, his loss is limited to the

premium paid for the call.

The seller of the call believes that the price of the asset will remain the same, perhaps

drop a little. He expects that the call will not be exercised against him and that he will

keep the asset and pocket the premium. When the call expires at time T, which was not

exercised, he may want to sell another call.

If you own a call option, you may take any one of these actions:

1. Exercise your call and buy the asset, by paying the exercise price;

2. Sell the call to another investor before expiration, who may be interested in its profit

potential; or,

3. Do nothing, and let the option expire. After expiration, the value of a call is zero.

Another example of an option is the ticket to a sports event. If you buy a basketball ticket

for $5 from University of Scranton, you can do any of the three things: You can exercise

the option by watching the game, or, you can sell the ticket to a friend, or, you may let

the option expire by not attending the game. The University keeps the $5 in any case.

When you buy a put option, it gives you the right but not the obligation to sell an asset at

a certain exercise price within a given time. The buyer of a put believes that the value of

the underlying asset will fall in the near future and that he will be able to sell it at a fixed

price by exercising his put and thus make a profit. The seller of a put believes that the

value of the asset will actually rise and that he will keep the put premium.

The most important form of puts and calls are those on common stocks. For example you

can buy a call option on Boeing stock that will expire after 3 months. These options are

traded on well organized options exchanges. One can see real-time option prices on the

Internet. A good website for financial information is www.yahoo.com and its financial

section.

We make the following observations from the table.

(1) The call price decreases as the exercise price rises, for the same expiration time.

(2) For the same exercise price, the call price rises as the time to expiration increases.

(3) For the same time to maturity, the put price rises as the exercise price increases.

(4) For the same exercise price, the put price increases as the time to maturity increases.

http://www.yahoo.com/


139

Microsoft Corp. (MSFT) 30.45 0.64 (2.06%)

January 25, 2007, at 4:00 PM ET

CALL OPTIONS Expire at close Fri, Mar 16, 2007

Strike Last Chg Bid Ask Vol Open Int

25.00 5.80 0.44 5.50 5.70 85 290

27.50 3.30 0.56 3.20 3.40 1,428 601

30.00 1.40 0.30 1.35 1.40 6,127 1,414

32.50 0.40 0.04 0.40 0.45 7,046 5,515

35.00 0.15 0.05 0.10 0.15 761 53

PUT OPTIONS Expire at close Fri, Mar 16, 2007


27.50 0.20 0.10 0.15 0.20 1,069 642

30.00 0.80 0.30 0.75 0.80 7,987 3,294

32.50 2.27 0.45 2.25 2.35 722 127

35.00 4.18 0.12 4.50 4.60 5 780

37.50 6.64 0.00 6.90 7.10 10 10

CALL OPTIONS Expire at close Fri, Jan 18, 2008


20.00 11.30 0.51 11.10 11.30 732 68,052

22.50 8.90 0.60 8.80 9.10 97 56,618

25.00 6.90 0.40 6.70 6.90 369 186,080

27.50 4.80 0.40 4.80 5.00 127 113,036

30.00 3.30 0.20 3.20 3.30 1,420 365,211

32.50 2.00 0.11 1.95 2.00 1,358 483

35.00 1.05 0.10 1.00 1.10 2,726 105,582

40.00 0.25 0.05 0.25 0.30 450 59,188

PUT OPTIONS Expire at close Fri, Jan 18, 2008


20.00 0.15 0.00 0.10 0.15 4 227,277

22.50 0.23 0.03 0.20 0.30 478 86,706

25.00 0.45 0.05 0.45 0.50 467 145,824

27.50 0.90 0.20 0.85 1.00 743 112,323

30.00 1.65 0.25 1.65 1.70 774 84,062

32.50 2.70 0.16 2.85 3.00 93 1,768

35.00 4.70 0.50 4.60 4.80 261 8,781

40.00 9.08 0.00 9.40 9.60 2 155

Table 8.1: Option data for January 25, 2007

http://finance.yahoo.com/q/op?s=MSFT&k=25.000000



























140

When the call options expire, they are in the money if the stock price is higher than the

exercise price. They are at the money if the strike price is just equal to the stock price.

They are out of the money, hence worthless, if the stock price is less than the exercise

price. An option that is in the money has some value. You can unlock this value by

exercising it, and buying the stock at the exercise price, which is less than the current

stock price. For instance, at expiration, when the stock is selling at $50 and the exercise

price is $45, then the value of a call is just $5. We may generalize this result by the

equations

CT = ST − X, if ST > X (8.1a)

= 0, if ST ≤ X (8.1b)

where CT = call price at time T, that is, at expiration. Also, ST is the stock price at time T,

and X is the exercise price. The may write (8.1a) and (8.1b) as

CT = max(ST − X, 0) (8.2)

Here "max" means the greater of the two quantities in the parenthesis.

Before expiration, the value of a call option is the sum of its intrinsic value and its time

value.

Total value of an option

=

Intrinsic value

+

Time value

The intrinsic value is the value of the option if it is exercised immediately. If the stock

price is less than or equal to the exercise price then you do not want to exercise the

option. In that case the intrinsic value is zero.

Consider the Microsoft options of the Table 8.1. The stock is priced at $30.45. The

March30 is selling for $1.40. If we buy one of these calls and exercise it immediately, it

will give us a benefit of 45¢ per share, because we are able to buy the $30.45 stock for

only $30. The intrinsic value of this option is thus 45¢. Subtracting it from the total value

of the option, we find the time value of the call to be 1.40 − .45 = $0.95.

Next we consider the January35 call option that is selling for $1.05. Its entire value is its

time value, and it has no intrinsic value at all. The time value of an option is always

positive and it gradually becomes zero as the time to expiration dissipates.

8.2 Black-Scholes Option Pricing Model

We have already seen that the value of a call depends upon the stock price, the exercise

price, and the time to maturity. Its value at maturity is given by (8.2). Calculating its

value prior to maturity is a much more difficult problem. Further analysis reveals that it

depends upon two more factors, the riskless interest rate r and the volatility of the stock

measured by its σ.


141

Define the following:

S = Market or current price of the underlying asset. This asset could be an ounce of gold,

a share of IBM stock, a piece of land, or any other suitable asset. The price of this asset is

a stochastic variable: it may go up or down in price in a random manner.

X = Exercise price of the option. This is a fixed price, agreed upon by the buyer and the

seller, at which the option holder has a right to buy the asset. The exercise price of the

option can be above or below the market value of the asset.

T = The time period during which the option is viable. An "American" option can be

exercised at any time during this period whereas a "European" option can be exercised

only at the end of this period. The life of an option can be anywhere from one day to

several years.

σ = The standard deviation of the continuously compounded rate of return due to price

changes of the underlying asset. This is the volatility of the asset. As noted earlier, the

price S of the asset is a variable. If it is changing rapidly and by large amounts then σ is

large. If the price of an asset is not changing at all then its sigma is obviously zero.

r = Riskless rate of interest. One can determine this quantity by using the yield of

Treasury securities.

C = Price of a call option prior to maturity.

Fischer Black

1938-1995

Myron Scholes

1941-

Robert Merton

1944-

The relationship between the call price of an option and the other five parameters was

first discovered by Fischer Black and Myron Scholes in 1973, and independently by

Robert Merton. This remarkable result can be expressed as

C = S N(d1) − X e−rT

N(d2) (8.3)

http://en.wikipedia.org/wiki/Fischer_Black

http://en.wikipedia.org/wiki/Myron_Scholes

http://en.wikipedia.org/wiki/Robert_C._Merton


142

where d1 = ln(S/X) + (r + σ

2/2) T

σ T (8.4)

and d2 = ln(S/X) + (r − σ

2/2) T

σ T = d1 − σ T (8.5)

and N(d) is the cumulative normal density function, which is equal to the area under the

normal probability distribution curve from minus infinity to the point d. The table at the

end of this book give the numerical values to find out N(d). We may also express N(d) as

a definite integral as

N(d) = 1

2π −∞

d

ex2/2 dx (8.6)

Example (8.1) gives the Maple code to find the price of a call option using equations

(8.3) - (8.6).

It is possible to show that the call price is positively correlated with the asset price, time

to maturity, riskless rate, and variability of price returns, but it is negatively correlated to

the exercise price. We can express it as

C = f(S +, X

−, T

+, r

+, σ

+)

Black-Scholes formula gives the price of a European call option of a non-dividend paying

stock or some other asset. It also assumes that people are rational investors, that r and σ

remain constant, that there are no taxes or transaction costs, and that the capital markets

are efficient. Despite all these restrictions it is a remarkably accurate and practical

formula for options valuation.

Fig. 8.1: The value of a call, X = 100, T = .25, r = .05, σ = .3, for varying stock price.


143

Fig. 8.1 shows the value of a call with time to maturity T = 6 months, exercise price X =

$100, riskless rate r = 6%, volatility σ = .3 as the stock price S changes from $80 to $120.

Hans Stoll (1969) discovered a very important relationship between the value of a call

and the value of a put. We can write the relationship, known as the put-call parity

theorem, as

P + S = C + X e−rT

(8.7)

Substituting the value of C from (8.3), we get

P + S = S N(d1) − X e−rT

N(d2) + X e−rT

Or, P = S N(d1) − S − X e−rT

N(d2) + X e−rT

Or, P = S [N(d1) – 1] − X e−rT

[N(d2) −1] (8.8)

With the help of (8.), we can find the value of a European put on a stock.

By judicious use of put or call options, it is possible to manage the risk inherent in the

investment process. For example if you own the stock of a corporation, you may wish to

sell call options on your stock. In case of a drop in the price of the stock the options will

expire worthless. The premium you have already collected on the options will be yours to

keep and it will offset some of your loss in the value of the stock. What you have done is

to "hedge" your exposure to risk. In fact, it is possible to eliminate risk altogether by

setting up a riskless hedge. This can be done as follows.

Suppose you buy h shares of a stock and sell one call option. Here h is unknown but it is

the proper number of shares to set up the riskless hedge. The total money invested in the

hedge, or the value V of the hedge is

V = h S − C

where S is the price of the stock and C the price of the call option. The value of the hedge

should not vary as a result of variation in the price of the stock, and therefore the partial

derivative of V with respect to S should be zero.

∂V

∂S = 0

Or, ∂

∂S (h S − C) = 0

Or, h − ∂C

∂S = 0, or h =

∂C

∂S (8.9)

Since C = S N(d1) − X e−rT

N(d2) (8.3)


144

Differentiating the above expression with respect to S gives us, after considerable

algebra, ∂C

∂S = N(d1) (8.10)

Comparing (8.9) and (8.10) we note that

h = N(d1) (8.11)

The number of shares of stock, h, that one should buy for each option sold is called the

"hedge ratio" and it is just equal to N(d1). Hedging is also used to take advantage of any

temporary mispricing of the options. If the call options happen to be selling at a price

which is more than their theoretical value, one can sell them and buy an appropriate

number of shares. Likewise, if the options are relatively underpriced one can buy them

and sell the stock, using the same hedge ratio.

8.3 Options, Stockholders, and Bondholders

Consider a company that is financed partly by stockholders and partly by bondholders.

They are all stakeholders in the company. Because of the provisions of the indenture, the

bondholders have a stronger claim on the company. If the company is liquidated, the

bondholders will get their money first and then the stockholders. In other words, the

stockholders will get the leftover amount, after the bondholders are satisfied. This is also

the case if the bonds reach maturity and the bondholders are ready to receive the face

value of the bonds.

Consider a company with zero-coupon bonds with face value $25 million, which will

mature after 10 years. The bondholders are not getting any interest and they will have to

wait for 10 years before they receive their share. Suppose the value of the company is V

after 10 years, consider the following three possibilities for V and the division of that

amount. The amount received by each stakeholder will depend on the final value of the

firm.

Firm

value

Share of

Bondholders

Share of

Stockholders

Explanation

V > 25 25 V − 25 Suppose the total value of the firm is $35 million.

Bondholders will receive their share first, which

is $25 million, and the stockholders will get the

remaining value of the firm, which is $10 million

V = 25 25 0 If the value of the firm is $25 million, the

bondholders will liquidate the firm and get the

face value of the bonds, which is also $25 million.

The stockholders will get nothing.


145

V < 25 V 0 Suppose the value of the firm after 10 years is

only $20 million. The bondholders will not be

able to get the face value of their bonds. They will

just have to settle for $20 million. Each bond will

be worth only 20/25*1000 = $800. The

bondholders will get 80¢ on the dollar. The

stockholders will get nothing.

Now consider a call option and its payoff at maturity. Recall equation (8.2)

CT = max(ST − X, 0) (8.2)

This equation implies that the value of the call, at maturity, is the difference between the

stock price and the exercise price provided the stock price is higher than the exercise

price, otherwise it is zero.

Comparing the payoff of a call option and the relationship between bondholders and

stockholders, we reach a very important conclusion.

The stockholders of a corporation hold a call option on the assets of the firm,

with an exercise price equal to the face value of the bonds, and time to maturity equal to the maturity of the bonds held by the bondholders.

Examples 8.7-8.10 illustrate this relationship.

Examples

8.1. Anglia Corporation stock price is $40 a share. The riskfree rate is 6%, and the

volatility of the stock, σ is .4. Find the price of a call option that will expire after 6

months, with the exercise price $35.

First we write the information in symbolic form as follows: S = 40, X = 35, r = .06, T =

.5, and σ = .4. The Maple code for the problem is as follows:

1/sqrt(2*Pi)*int(exp(-x^2/2),x=-infinity..d);

simplify(%);

Nd:=%;

d1:=(ln(S/X)+(r+sigma^2/2)*T)/sigma/sqrt(T);

d2:=d1-sigma*sqrt(T);

Nd1:=subs(d=d1,Nd);

Nd2:=subs(d=d2,Nd);

C:=S*Nd1-X*exp(-r*T)*Nd2;

subs(S=40,X=35,r=.06,sigma=.4,T=.5,C);

evalf(%);

The result comes out to be $7.85. ♥


146

To do the problem in Excel, proceed as follows.

A B C

1 Stock price, S = 40 dollars

2 Exercise price, X = 35 dollars

3 Riskfree rate, r = 0.06 per year

4 Time to maturity, T = 0.5 year

5 Volatility, σ = 0.4 per √(year)

6 d1 = =(LN(B1/B2)+(B3+B5^2/2)*B4)/B5/SQRT(B4)

7 d2 = =B6-B5*SQRT(B4)

8 N(d1) = =NORMDIST(B6,0,1,true)

9 N(d2) = =NORMDIST(B7,0,1,true)

10 Call = =B1*B8-B2*EXP(-B3*B4)*B9 dollars

8.2. (A) Uriah Heep has just bought 100 oz of gold at $350 per ounce. He has calculated

that the standard deviation of returns in gold investment is 0.243, and that the riskless rate

is 11%. He would like to sell call options on gold at an exercise price of $375 per ounce,

with a maturity of three months. What is the correct value of these options?

(B) Suppose Uriah was able to sell options on 10 oz of gold. The price of gold at the end

of 3 months is $400 per ounce. Now he liquidates all his gold and settles the options,

what is his total profit?

(C) Using a discount rate of 15%, find the NPV of this investment.

(A) Here we are given that:

S = current price of the underlying asset = $350

X = exercise or the striking price of the option = $375

r = riskless rate of interest = 0.11 per year

T = time to maturity of the option = 0.25 years

σ = standard deviation of the continuously compounded rate of return from the

price fluctuations of the underlying asset = 0.243

The price of the option is calculated by using the Black-Scholes model as shown below:

d1 = ln(350/375) + [0.11 + 0.5 (0.243)

2] (0.25)

0.243 0.25 = − 0.2808

d2 = − 0.280755 − 0.243 0.25 = − 0.4023

Draw normal probability distribution curve, with 0 at the center and stretching up to ∞ on

both sides. First take d1 = − 0.2808, which will lie slightly left of center. N(d1) is defined

as the area under the curve from −∞ to d1, which will be somewhat less than .5. To find

its value, check the tables for .2808. This comes out as

N(d1) = .5 – [.1103 + .08(.1141 − .1103)] = 0.3894


147

Figure 8.2. N(d1) is defined as the area under the normal probability distribution fumction curve from −∞ to

d1. This is the shaded area in the diagram.

Similarly, N(d2) = .5 – [.1554 + .2256(.1591 − .1554)] = 0.3438

C = 350 (0.3894) − 375e−0.11(0.25)

(0.3438) = $10.86 ♥

(B) Since Heep sold options on only 10 oz of gold, he was able to sell 90 oz of gold at a

profit of $50 per oz. The options ended up in the money, that is, the final price of gold

was higher than the exercise price. As a result the buyers of the options exercised their

options by forcing Heep to sell the gold to them at the rate of $375 per oz. On these ten

ounces of gold he made $25 per ounce besides collecting the $10.86 option premium

calculated in part (A). Let us define profit as the difference between the final payoff and

the initial investment, without regard to the risk involved, and without the time value of

money. The total profit works out as follows:

Initial investment = cost of buying gold − cash received by selling the options

= 100(350) – 10(10.86) = $34,891.40

Final payoff = money received by selling 90 oz of gold in the open market at $400 an oz

+ money received by selling 10 oz of gold to the option holders at $375 an oz

= 90(400) + 10(375) = $39,750

Profit = 39750 − 34891.40 = $4858.60 ♥

(C) To calculate the NPV, we have to subtract the initial investment from the present

value of the future payoff, using a discount rate that includes the risk of the investment.

Using the continuously compounded discount rate, as we used it in the calculation of the

option price, NPV comes out as

NPV = − 34,891.40 + 39,750e−(.15)(.25)

= $3395.58 ♥

Note that NPV is less than the profit and it is a more conservative measure of the

profitability of an investment.


148

8.3. William Horner bought 100 oz of gold at $1663 an oz. Then he sold call options on

25 oz of gold, exercise price $1680, for $100 each; and options on 35 oz of gold, exercise

price $1700, for $80 each. The cost of capital for Horner is 9%. All the options will

expire after 6 months and then Horner will liquidate his position. Use continuous

discounting, to calculate the NPV of this hedge if the price of gold after 6 months is

expected to be $1700 an oz.

Initial investment = (value of 100 oz of gold at $1663 per oz)

– (value of 25 options sold, at $100 each, with X = 1680)

– (value of 35 options sold, at $80 each, with X = 1700)

= 100*1663 – 25*100 – 35*80 = $161,000

If the expected final price of gold is $1700, options with X = 1680 will be exercised, and

he will deliver 25 oz of gold and receive $1680 per oz. The options with X = 1700 will

expire worthless because when the stock price is exactly equal to the exercise price, at

expiration, then the value of the option is zero. Therefore, he will sell the remaining 75 oz

of gold in the market at $1700 per oz. Thus

Final payoff = money received because some of the options have been exercised + money

received by selling the rest of gold in open market = 25*1680 + 75*1700 = $169,500.

To summarize, his initial invest was $161,000 and the final payoff was $169,500. With

these two numbers, we can find the following on this investment.

Profit = 169,500 − 161,000 = $8,500

The above value of the profit is misleading because we did not consider the time value of

money and we did not take into account the risk involved. To compensate for these

factors, we should find the NPV of the investment. We can do it in two ways, in discrete

time and in continuous time. The results are as follows.

Discrete time, NPV = – 161,000 + 169,500(1.09)−.5

= $1351.56

With continuous discounting, NPV = – 161,000 + 169,500e−.09*.5

= $1041.57 ♥

8.4. Adam Diller bought 100 shares of Apple stock at $580.32 per share. He also sold 1

call option on the stock, at 41.66, with exercise price 590, and with 132 days till

expiration. Using a discount rate of 12%, continuously compounded, find the stock price

where Adam will just break even in this investment. Neglect transaction costs.

At the break-even point, the NPV of the investment will be zero. By selling the call, the

net cost of stock is reduced by $41.66 per share. The break-even price of the stock should

be around 580.32 − 41.66 = $538.66. The buyer of the option will not exercise the option

because the final stock price, $538.66 is much less than the exercise price of the call

option. Let us find the answer more accurately including the time value of money.

http://en.wikipedia.org/wiki/Apple_Inc.


149

The initial cost of stock = 100(580.22) = $58,022

Cash received by selling the call option = 100(41.66) = $4166

Net cost of this hedge = 58,022 − 4166 = $53,856

Suppose the final stock price is x. This is around $540, as seen by the approximate

calculation. The final payoff from selling the stock at x per share will be 100x. Its present

value, using 12% continuously compounded discount rate and 132 days to maturity, will

be 100xe−.12(132/365)

. Setting NPV = 0, we get

− 53,856 + 100xe−.12(132/365)

= 0

You may solve it at WolframAlpha by using the instruction

-53856+100*x*exp(-.12*132/365)=0

The result is x = $562.45. ♥

8.5. You own 1,000 shares of GM stock which is currently selling for $75 a share, and

your estimate of its sigma is 0.225. The riskless rate is 11.2%. What is the price of three

month call options at an exercise price of $80? How many call options should you sell in

order to set up a perfect hedge?

In this problem, S = 75, X = 80, T = .25, r = .112, σ = .225

d1 = ln(75/80) + (.112 + .225

2/2)(.25)

.225 .25 = −.2685368546

d2 = ln(75/80) + (.112 − .225

2/2)(.25)

.225 .25 = −.3810368545

N(d1) = =NORMDIST(-.2685368546,0,1,TRUE) = .3941430552

N(d2) = =NORMDIST(-.3810368545,0,1,TRUE) = .3515879509

C = 75*.3941430552 – 80*exp(−.112*.25)*(.3515879509) = 2.21032647

The call price is $2.21.

One can set up a hedge by buying the shares of GM and selling call options on them. In

order to set up a riskless hedge, one has to buy N(d1) shares of stock and sell one option.

With the proper hedge ratio as N(d1) = 0.3941, one should buy 0.3941 shares per call, or

1/0.3941 calls per share. But we already have 1,000 shares, therefore, we have to sell

1,000/0.3941 calls altogether. This works out to be 2537 calls.♥



150

As the time to maturity changes and the stock price fluctuates, the hedge ratio N(d1) also

changes. To keep the hedge riskless, we have to recalculate N(d1) periodically and adjust

the hedge accordingly.

8.6. (A) Stanley Corporation stock is currently selling for $76 a share, riskless rate is

12%, and the sigma for Stanley is 0.25. Find the price of a nine-month Stanley call option

with an exercise price of $70.

(B) Suppose the options in part (A) are selling for $10 each. Explain how you would set

up a hedge to take advantage of the mispricing.

(A) From the option pricing formula we get: d1 = 0.9038, d2 = 0.6873, N(d1) = 0.8169,

N(d2) = 0.7540, and call price = $13.85.

(B) Since options are selling for $10 apiece, which is substantially less than their

theoretical value of $13.85, you must buy them. To set up the hedge you buy the calls and

sell the stock short in the proper hedge ratio of 0.8169. The overall size of this hedging

operation depends upon the amount of available funds. For example, you may buy 1,000

options and sell 817 shares short, maintaining the hedge ratio of 0.817. When the price of

the options reach an equilibrium, you can take your guaranteed profits. The cost of

buying calls is 1,000(10) = $10,000. The proceeds from the short sale of stock is 817(76)

= $62,092. The net proceeds are 62,092 − 10,000 = $52,092. You should invest this

amount in riskless government bonds while waiting for the profits to occur.

Here we are assuming that our estimate of the σ of Stanley is absolutely correct and the

rest of the market does not know it yet. We also assume that there are no transaction

costs, that is, no brokerage commissions. We also assume that there are no restrictions

against short selling, and that we are continuously adjusting the hedge ratio while the

stock and option prices are changing. In practice this is very difficult to do. ♥

8.7. (A) Denver Corporation stock is currently selling for $100, riskless rate is 12%, and

the sigma for Denver is .25. Find the price of a nine month Denver call option with an

exercise price of $100.

(B) Suppose the options in the last problem are selling for $15 each. Explain how you

would set up a hedge to take advantage of the mispricing.

In part (A) we get: d1 = 0.5239, d2 = 0.3074, N(d1) = 0.6998, N(d2) = 0.6207, and call

price = $13.25.

In part (B) we notice that the calls are overpriced at $15 each compared to their

theoretical value of $13.25, and we should sell them. Because the hedge ratio N(d1) is

roughly 0.7, we should buy 0.7 shares of stock for each option sold. For example we can

sell 1,000 options but buy only 700 share of Denver. This will require a cash outlay of

700(100) − 1000(15) = $55,000. If we could borrow that money at a rate equal to the


151

riskless interest rate, and if we wait until the prices regain equilibrium then we should

make a profit of 1,000 (15 − 13.25) = $1,750. ♥

8.8. Glenn Corporation has an overall market value of $40 million. The firm has zero-

coupon bonds outstanding, maturing in 5 years, with the face value $25 million. The σ for

this company is 0.25, and the riskless rate is 8%. Glenn has one million shares of

common stock. What is the market value per share of its common stock?

The stockholders of a company hold a call option on the assets of the company after the

bondholders are satisfied. The bondholders have a senior claim on the assets of the firm

in the case of liquidation of the firm. The stockholders share whatever is left over, after

all the other claims are satisfied.

The value of the underlying asset is the current market value of the firm, namely, $40

million. The exercise price is the face value, not the market value, of the zero coupon

bonds, $25 million. The value of the option is the total market value of the common stock

of the firm. Substituting S = 40, X = 25, T = 5, r = .08, and σ = .25, in the Black-Scholes

formula, we find: d1 = 1.8358, d2 = 1.2768, N(d1) = 0.9668, N(d2) = 0.8992, and call price

= 23.60. This means that the value of 1 million shares of common stock is $23,600,000.

The price of the stock per share comes to $23.60. ♥

8.9. Fischer Black is the sole stockholder of Black Belt Co., which has an overall value

of $50,000. The company has borrowed some money from an investor, Myron Scholes,

and has promised to pay him back the entire amount as a lump sum of $30,000 after 5

years. The σ of Black Belt is 0.25, and the riskless rate is 6%. Find the market value of

the holdings of Black and Scholes individually.

Being a stockholder, Black holds an option on the assets of the firm after Scholes has

been satisfied. The value of the option can be found by the option pricing formula with S

= 50,000 X = 30,000 which gives C = $28,369. This represents Black's portion of the

assets. The total market value of a firm is equal to the market value of its common stock

plus the market value of its debt. The value of the debt in this case is thus 50,000 –

28,369 = $21,631. This is the value of Scholes' claims on the company.

If we evaluate his claim as if it were riskless, its value is 30,000 e−0.06(5)

= $22,225.

Because the debt is not riskless, its value is somewhat less. The difference between

$22,225 and $21,631 is $594, which is about 2.67% of $22,225. The debt is quite safe

because there is a good possibility that the $50,000 firm will have a terminal value of

$30,000 after 5 years. ♥

8.10. Carolina Inc has a total value of $5 million. It has zero coupon bonds maturing in

10 years with a face value of $4 million. The riskless rate is 10%, and σ of Carolina is

.25. Using Black-Scholes model, find the market value of a single $1,000 bond.

Use the following in the Black-Scholes formula: S = 5, X = 4, T = 10, r = .1, and σ = .25.

This gives us d1 = 1.9425, d2 = 1.1519, N(d1) = .9740, N(d2) = .8753, C = 3.582.


152

This means that the market value of the stock of Carolina is $3,582,000, and that of the

bonds 5,000,000 − 3,582,000 = $1,418,000. Since the face value of the bonds is

$4,000,000, each $1,000-bond is selling for 1,000*(1,418,000/4,000,000) = $354.50 ♥

If the bonds were riskless they would be selling for 1,000 e−0.1(10)

= $367.88 each. This

agrees with the price of the risky bonds found above, because the riskless bonds are

somewhat more valuable.

8.11. Calhoun's Saloon is run jointly by Calhoun and his brother-in-law Breckinridge.

Calhoun is the sole stockholder of the company, but the company owes Breckinridge

$10,000 which will be paid as a lump sum after 5 years. Considering the income

generated by the business, it is estimated that the value of the business is $20,000. The

risk of the business is measured by its σ which is estimated to be 0.5. The riskless rate is

10%. Calhoun wants to pay a fair price to Breckinridge for his loan and thus become the

sole proprietor of the business. How much should Calhoun pay Breckinridge now?

Here we have to use the Black-Scholes formula with the following values: S = 20,000, X

= 10,000, T = 5, σ = 0.5, r = 0.1. This gives: d1 = 1.626, d2 = 0.5082, N(d1) = 0.9481,

N(d2) = 0.6943, C = 14,750. In general, the stockholders of a firm hold a call option on

the assets of a firm, with an exercise price equal to the face value of the bonds of the

firm. Calhoun is the stockholder and he holds a call option on the assets of the firm after

the bondholder, Breckenridge, is satisfied. Thus the value of Calhoun's investment is

$14,750. The total value of a firm equals the value of stock plus the value of the debt.

The present value of Breckinridge's loan, the value of debt, is thus 20,000 − 14,750 =

$5,250. Therefore Calhoun should pay Breckinridge $5,250 to buy him out. ♥

8.12. Enceladus Corporation has a total value of $5 million. It has $2 million of zero-

coupon bonds maturing in 12 years. The sigma of Enceladus is .4 and riskless bonds with

12-year maturity have a yield of 9%. Find the market value of a $1,000 Enceladus bond.

Using Black-Scholes formula with S = 5, X = 2, r = 0.09, σ = 0.4, and T = 12, we get d1 =

2.1335, d2 = 0.74788, N(d1) = 0.98356, N(d2) = 0.77273, and C = 4.393. This means that

the value of equity is $4.393 million, and the value of debt is 5 − 4.393 = $0.607 million.

A thousand dollar bond sells for (1000)(0.607/2) = $303.50. ♥

Problems

8.13. The common stock of Zeta Corporation is currently selling for $56.375 per share

and the standard deviation of its continuously compounded rate of return is .256. The

riskless rate is 9.55%. Find the price of a call option at the exercise price of $60 with

maturity time of 9 months?

d1 = 0.1528, d2 = − 0.0689, N(d1) = 0.5607, N(d2) = 0.4725, C = $5.22 ♥

8.14. Arcturus Co common stock has market price $95 per share and its sigma is 0.2.

Find the value of a European call option with an exercise price of $90 and a term of 9

months on the Arcturus stock. The riskless rate is 12%. C = $14.48 ♥


153

8.15. Red Buttons has the option to buy a piece of land for $50,000 after one year. The

market value of the land is $40,000 at present, and the σ of returns for real estate

investments of this type is around 0.3. The riskless rate is 12%. What is the value of this

option? $3,149 ♥

8.16. Pluto Inc has $20 million face value zero-coupon bonds due in 5 years, and its σ is

0.45. The total market value of Pluto is $30 million and the riskless rate is 11%. The

company has 2 million shares outstanding. Find its price per share. $10.02 ♥

8.17. Rutherford Corporation has total assets of $40 million. The company has $20

million (face value) of zero-coupon bonds which will mature in 12 years. The riskless

rate is 9% and the σ of Rutherford Corporation is .45. Find the market value of

Rutherford bonds. What would be the value of these bonds if they were riskless?

Market value of bonds = $5,451,000. Their value, if riskless = $6,792,000. ♥

8.18. Capricornus Company has a total value of $40 million. Its debt is in the form of

zero coupon bonds which will mature in 10 years. The riskless rate is 6.5% at present.

The sigma of Capricornus is 0.45. Find the debt/assets ratio of Capricornus. The face

value of bonds is also $40 million. 32.99% ♥

8.19. Dulles Corporation has a total value of $80 million and it has $40 million (face

amount) of zero coupon bonds outstanding. The bonds will mature in 10 years. The

riskless rate is 8% and the sigma of Dulles is 0.25. Find the market value of the stock of

Dulles. $62.348 million ♥

8.20. Beaver Corporation is owned jointly by sisters Allison and Barbara, Allison's share

being 60%. The value of the corporation is $50,000 and its risk in terms of sigma is

estimated to be 0.5. Allison would like to buy Barbara out and offers her $20,000 cash, or

a note for $30,000 payable by Beaver Corporation after 5 years. The riskless rate is 9%.

Should Barbara take the cash or the note? PV of note = $15,724, take cash. ♥

8.21. The current price of gold is $457 an oz. The standard deviation of investment

returns in gold has been estimated to be 0.15, and the riskless rate is 6%. Calculate the

price of a six-month call option on gold with the exercise price of $475. C = $17.50. ♥

8.22. Miles Davis has bought 100 oz of gold at $460 an oz. He has sold call options on

30 oz of gold, with exercise price $480, for $10 each; and options on 40 oz of gold,

exercise price $470, for $20 each. All options will expire after 6 months and then Davis

will liquidate his position. Davis expects the price of gold after six months to be $475 an

oz. He uses 12%, continuously compounded, as the discount rate. Calculate the NPV of

this hedge. NPV = −$354.54 ♥

8.23. Charles Heston bought 100 shares of Priceline at $644.36 per share. He then sold 1

call at 27.36, with expiration time 41 days and exercise price 650. The risk-adjusted

discount rate for this hedge is 10%, continuously compounded. Suppose at expiration of

the option, the stock is selling at $660 per share. Find the NPV of this investment.

http://en.wikipedia.org/wiki/Priceline.com


154

$2573.95 ♥

8.24. Ryles is the only stockholder of Ryles & Hewish, but Hewish has lent a certain

sum of money to the business with the understanding that the company will pay him back

$100,000 after 6 years. The current value of the business is $150,000 and its σ is

estimated to be about 0.4. The riskless rate is 8%. If the company were to be liquidated

today, what would be a fair distribution of cash to Hewish and Ryles?

Ryles = $96,872, Hewish = $53,128. ♥

8.25. Dole Co stock is currently selling for $122 per share and its sigma is estimated to

be .234. The riskless rate is 6.55% at present. Find the price of call option on Dole with

an exercise price of $130 and expiring in 63 days. $2.27 ♥

8.26. Find the price of a call option on one share of Mackellar Corporation stock with

exercise price of $100 and a time to maturity of 9 months. The market price of Mackellar

stock is $106 per share and has a sigma of 0.45. The riskless rate is 7.5%.

d1 = 0.4887, d2 = 0.0990, N(d1) = 0.6875, N(d2) = 0.5394, C = $21.88 ♥

8.27. Rolls and Royce started a car company by investing £100,000 each. Rolls was a

stockholder, and thus the owner of the company. The corporation agreed to pay Royce

£300,000 after ten years for his share of the business. However, by mutual agreement the

company was sold after 5 years for £800,000 and the money was divided according to the

option pricing theory. The riskless rate at the time was 3%, and the sigma of Rolls-Royce

Company was estimated to be 0.4. Find the amount of money that went to Rolls and to

Royce.

S = 800, X = 300, σ = 0.4, T = 5 years, r = 0.03, d1 = 1.712, d2 = 0.8171, N(d1) = 0.9565,

N(d2) = 0.7931, Call price = 560.427, Rolls' share = £560,427, Royce's share = £239,573 ♥

8.28. Turabah Company has total value $50 million. It has zero-coupon bonds with face

value $40 million, maturing after 20 years. The riskfree rate is 6%. The volatility σ of

Turabah Company is .4. Using Black-Scholes model, find the value of its $1000 bond.

B = $195.63 ♥

8.29. Schnectedy Company has total value $200 million, and it has $100 million (face

value) of zero-coupon bonds maturing after 15 years. The σ of Schnectedy is estimated to

be .4 and the riskfree interest rate is 6%. Using Black-Scholes model estimate the

debt/assets ratio for the company. 15.77% ♥

Key Terms American option, 139

at the money, 138

Black-Scholes model, 135,

144, 148, 151

call option, 135, 149

call premium, 135

call price, 135, 136, 138, 139,

140, 146, 147

European option, 139

exercise, 135, 136, 138, 139,

140, 141, 143, 144, 145,

146, 147, 148, 149, 150

exercise price, 138, 150

expiration time, 136

hedge, 141, 142, 146, 147,

150

hedge ratio, 142, 146

in the money, 138, 145

intrinsic value, 138

option, 135, 136, 138, 139,

140, 141, 142, 143, 144,

145, 146, 147, 148, 149,

150, 151

out of the money, 138


155

put option, 136

put-call parity theorem, 135

riskless rate, 139

time value, 138, 145

transaction costs, 140, 147

volatility, 138, 139, 141, 143,

151

156

9. COST OF CAPITAL

Objectives: After studying this chapter, you will be able to

1. Calculate the cost of various forms of capital: stock, bonds, and preferred stock.

2. Apply the concept of original issue discount.

3. Find the weighted average cost of capital.

9.1 Cost of Capital

Capital is the lifeblood of any corporation. A company cannot invest in new machinery

and equipment without capital; it cannot embark on new projects without adequate

capitalization; it cannot even pay its current bills without sufficient working capital. Just

like any scarce resource, there is a cost associated with capital.

What is capital, and how do the corporations get it? Capital is the money, or cash, needed

by the firms to do their business. The corporations obtain capital from investors who have

saved some money and want to invest it. Of course, the investors require a certain return

on their investment depending on the amount of risk they are willing to take. The

following diagram illustrates the relationship between investors and corporations.

Corporations

Capital

Return on investment

Investors

Fig. 9.1. The relationship between investors and corporations.

Investors will not invest their money unless there is a reasonable return on their

investments. For instance, if the company offers bonds with coupon 8%, but the investors

require 10% return on such an investment, then the investors will not buy these bonds. If

the investors do not buy the bonds, their price will drop in the financial markets. The

price will drop to a point where the return on these bonds becomes 10%, and they reach

an equilibrium point. Thus we reach a very import conclusion:

The cost of capital to a corporation

= The required rate of

return for the investors

This cost of capital depends upon the supply and demand of capital in the capital markets.

During a recession, investors do not have enough savings to invest in the capital of

corporations. The Federal Reserve can lower the interest rates, or increase the money

supply, in an effort to facilitate the availability of capital.

The main components of the capital of a corporation are equity and debt. The firms

acquire equity capital by selling common stock. Similarly, they sell bonds to obtain debt

http://en.wikipedia.org/wiki/Working_capital

http://en.wikipedia.org/wiki/Federal_Reserve_System

http://en.wikipedia.org/wiki/Money_supply

http://en.wikipedia.org/wiki/Money_supply

http://en.wikipedia.org/wiki/Common_stock

http://en.wikipedia.org/wiki/Bond_(finance)

Analytical Techniques 9. Cost of Capital _____________________________________________________________________________

157

capital. Besides these two, the corporations may also sell preferred stock, convertible

bonds, and warrants as additional forms of capital.

The preferred stockholders get their dividends ahead of the common stockholders. This

gives them a more secure investment, relative to common stock, and thus the return is

lower. The cost of preferred stock to the firm is, therefore, less than the cost of common

equity.

Convertible bonds are hybrid securities that act like straight bonds in that they provide

regular coupon payments. They also act like stock because they can be converted into

common stock at the option of the bondholder. If the price of the stock rises, then the

price of the bond also rises. The cost of issuing convertible debt is difficult to ascertain

because of the complicated relationship between the stock value and the convertible bond

value. The cost of convertible bonds lies between the cost of debt and the cost of equity.

The bondholders of a company are in a secure financial position. This is due to the fact

that bond interest is paid before taxes or dividends. In case of liquidation of a

corporation, again the bondholders are the first ones to receive money from the sale of

assets. As a result of their greater security, the bondholders have a lower required rate of

return. The stockholders bear greater risk and expect to be compensated for it. Thus we

reach another important conclusion:

For a given firm,

Cost of debt, kd is always

less than the

Cost of equity, ke

Let us look at the cost of different components of capital.

9.2 Cost of Debt

We recall the principle that the cost of debt is equal to required rate of return by the

bondholders. Thus the cost of debt capital for a firm is equal to the yield to maturity for

its bonds for the bondholders. For most corporations, it is possible to find on the Internet

the bond prices, coupon rates, and the year of maturity. Using this information, we can

find the yield to maturity, and thus the cost of debt capital for that company.

The interest paid by a corporation is tax deductible, thereby lowering its cost of debt. For

example, a company has a debt of $1000, with an interest rate of 10%. It pays $100 in

interest annually. It can use the $100 as a tax deduction. Suppose the company has tax

rate of 30%, then its tax savings will be $30 because of this deduction. The net cost of

interest expense is $70, which means the after-tax cost of debt is 70/1000 = 7%. This 7%

is really (1 − .3)(.1) = 0.07. We may write it as (1 − t)kd, where kd is the cost of debt

before taxes, and t is the tax rate. Let us write this result as

If the cost of debt before taxes is kd

then the cost of debt after

taxes is (1 – t)kd.

http://en.wikipedia.org/wiki/Preferred_stock

http://en.wikipedia.org/wiki/Convertible_bonds

http://en.wikipedia.org/wiki/Convertible_bonds

http://en.wikipedia.org/wiki/Warrant_(finance)

http://en.wikipedia.org/wiki/Yield_to_maturity


158

When a company issues new bonds to raise additional capital, the company gets deeper

into debt. This reduces the security of bondholders. The company has to pay a higher

interest rate to sell the new bonds. The bonds of the companies that are in severe financial

distress are aptly called "junk bonds."

It is possible to get a more precise value of the after-tax cost of debt by including two

additional features of a bond. First, the corporation pays the interest semiannually; and

second, it gets the tax benfits of interest payments annually.

When a bond is selling at par, it is selling at its face value, namely $1000. Suppose a

company issues bonds at par with face value F. The coupon rate on these bonds is r and

they will mature after n years. The interest on the bonds is paid semiannually. Let us

assume that the company pays taxes once a year at a rate t. If the after-tax cost of debt is

k, then k is given by the following equation that says NPV = 0,

F

− i=1

2n

rF/2

(1 + k/2)i

+ i=1

n

rFt

(1 + k)i

− F

(1 + k)n

= 0

Face value

of the bond

PV of interest

payments

PV of tax benefits

of interest payments

PV of the

final payment

The above equation represents the common adage, “You get what you pay for.” In this

case, the corporation gets (1) the face value of the bonds when they are sold and (2) the

tax benefit of interest payments, in PV terms. The company pays for it (1) by paying

interest on the bonds, semiannually, and (2) paying the face value of the bonds at

maturity, measured in PV terms. Note that everything is reduced to its present value.

It is common to use Excel or WolframAlpha, to get a numerical solution to the problems.

For instance, if r = 10%, t = 30%, n = 10 years, then k 7.1188%. The result is somewhat

different from the simple answer of 7% from the expression (1 − .3)(.1). To verify the

result, use the following expression at WolframAlpha,

1000-Sum[.1*1000/2/(1+x/2)^i,{i,1,2*10}]+Sum[.1*1000*.3/(1+x)^i,{i,1,10}]-1000/(1+x)^10=0

Occasionally, a firm may issue bonds that sell at less than their face value. The difference

between the face value F and issue price G is called the original issue discount. For

instance, a corporation may issue a $1000 bond for only $900 and thus the original issue

discount is $100. Eventually, the firm must redeem the bond for its face value F. The

Internal Revenue Service allows the companies to treat the original issue discount as a

virtual form of interest, although they do not pay it to the bondholders. This is spread

over the life of the bond and its annual value is (F − G)/n. The annual tax benefit to the

corporation due to this item is t(F − G)/n.

Including the original issue discount, the after-tax cost of debt, k, for a bond issued at a

discount, G, where G < F, is given by the following expression. The NPV = 0 equation

includes the present value all cash flows, after taxes, as seen by the corporation.

http://en.wikipedia.org/wiki/Junk_bonds



http://en.wikipedia.org/wiki/Original_issue_discount

http://en.wikipedia.org/wiki/Internal_Revenue_Service


159

G

− i=1

2n

rF/2

(1 + k/2)i

+ i=1

n

[rF + (F − G)/n]t

(1 + k)i

− F

(1 + k)n

= 0

Original price of

the bond

PV of interest

payments

PV of tax benefits of interest

and original issue discount

PV of the final

payment

To understand the previous equation, consider a numerical example. A firm issues a bond

by selling it to the public at G = $700 with face value F = $1000. This is a discount bond

and the firm uses the original issue discount to find its after-tax cost of debt. Suppose the

income-tax rate of the company is 32%. The difference between the selling price and the

face value is $300, which is the original issue discount. The firm has to pay this amount

to the bondholders when the bonds mature. For tax purposes, the firm spreads it out over

ten years, claims a virtual interest payment of $30 per year, and gets a tax benefit .32(30)

= $9.60 every year. Suppose the coupon rate of the bond is r = 5%, and it will mature

after n = 10 years. The bond pays interest semiannually. Using the concept of NPV = 0

for all cash flows to the firm, one can write

700 − i=1

20

.05*1000/2

(1 + k/2)i +

i=1

10

[.05*1000 + (1000 − 700)/10](.32)

(1 + k)i −

1000

(1 + k)10

= 0

Simplifying it, we get

700 − i=1

20

25

(1 + k/2)i +

i=1

10

25.6

(1 + k)i −

1000

(1 + k)10 = 0

To solve it with Maple, enter the following instructions

700-sum(25/(1+k/2)î,i=1..20)+sum(25.6/(1+k)î,i=1..10)-1000/(1+k)^10=0;

solve(%);

The result is .06711987555, or about 6.712%. To do it with WolframAlpha, copy and

paste the following instruction

700-Sum[25/(1+x/2)î,{i,1,20}]+Sum[25.6/(1+x)î,{i,1,10}]-1000/(1+x)^10=0

It gives the answer, r .0671199. To verify if this is a valid answer, we find the

approximate yield to maturity for the bond by using equation (3.5) on page 36. This gives

Y [50 + (1000 – 700)/10]/850 = .09411764704

With tax rate 32%, the after-tax cost of debt is (1 − .32)(.09411764704) = .064 = 6.4%.

This is close to the previous value.

To do it on Excel, use the following table. Adjust the value of the unknown after-tax cost

of debt in cell B1, until the NPV of the bond in cell B9 becomes very close to zero. With

the after-tax cost of debt as 9.412%, the NPV in cell B9 is about $0.0007.



160

A B

1 After-tax cost of debt = 6.712%

2 Sale price of the bond = $ 700

3 Time to maturity = years 10

4 Coupon rate = % 5%

5 Income tax rate = % 32%

6 PV of interest payments = =-B4*1000*(1-1/(1+B1/2)^(2*B3))/B1

7 PV of tax benefits = =(B4*1000+(1000-B2)/B3)*B5*(1-1/(1+B1)^B3)/B1

8 PV of final payment = =-1000/(1+B1)^B3

9 NPV of bond = 0 =B2+B6+B7+B8

9.3 Cost of Equity

The cost of equity for a firm is more difficult to estimate primarily because of the greater

uncertainty in the cash flows per share of stock. We may use one of the following four

methods.

1. The first method is to find the cost of debt and then add a risk premium to it because

the stockholders bear greater risk than the bondholders do. This risk premium may vary

roughly from 3% to 8%, depending upon the financial health of the firm. If ke is the cost

of equity, and kd is the cost of debt, then an approximate relationship is:

ke = kd + risk premium (9.1)

2. The second method is to apply Gordon's growth model (3.6)

P0 = D1

R − g (3.6)

Here R is the required rate of return by the stockholders. But this is also the cost of equity

for the firm ke. This leads us to write the above equation as

ke = D1

P0 + g (9.2)

The right side of this equation has two terms. The first one, D1

P0 represents the dividend

yield of a stock, and the second one, g, the rate of growth of the company. The return of a

stock is indeed the sum of these two factors. High dividend stocks usually have a low

growth rate. For the stocks that pay no dividends at all, the investors are pinning their

hopes on the future growth potential of the company.

3. Another method of estimating the cost of equity is to apply the Capital Asset Pricing

Model (7.7)

E(Ri) = r + βi [E(Rm) − r] (7.7)

Here E(R) is the expected return of the stock, which is also the required rate of return for

an investor, and thus it is also the cost of equity capital for a firm. If we know the β of a

http://en.wikipedia.org/wiki/Dividend_discount_model




161

stock, the riskless rate of return r, and the expected return of the market, we can find the

cost of equity for that company as

ke = r + βi [E(Rm) − r] (9.3)

4. An approximate method for estimating the cost of equity is simply to look at the

previous performance of a stock. Using past as a proxy for the future, we assume that the

expected return on the stock is just equal to the previous return of that stock. This is then

the cost of equity for the firm,

ke = historical return of the stock

9.4. Cost of Preferred Stock

Note the following differences between a preferred stock and a common stock.

1. Preferred stock is like common stock; they both pay dividends.

2. Preferred stock is safer than the common, because the preferred dividends are paid out

before the common dividends.

3. Preferred stock has no growth potential and the dividends remain constant, whereas the

dividends of the common stock have the possibility of growing over time.

4. Preferred shares have a limited life in most cases and the company buys them back at

the issuing price after a few years.

5. A company that issues preferred stock does not get any tax benefit because the

dividends of preferred stock are not tax deductible. Thus, the pretax and after-tax cost of

capital for preferred stock is the same.

6. The preferred stockholders do not have voting rights and they cannot elect the board of

directors of a company.

Considering all that, one should use Gordon’s growth model, with g = 0, to find the cost

of capital for preferred stock.

9.5. Weighted Average Cost of Capital

Consider a corporation that uses only common stock and bonds in its capital structure.

Assume that both the stock and the bonds are publicly traded and they have a certain

market value. Then the total market value of the corporation is just equal to the market

value of the common stock plus the market value of the bonds. The real value of any

asset is determined by the market and it is not necessarily equal to its book value. The

market value of an asset can fluctuate for many reasons. For example when the interest

rates rise, the market price of bonds of a company will drop. If the company has poor


162

earnings projections, the price of its common stock will fall. The management, of course,

tries to maximize the value of the company, not just for the stockholders, but for the

bondholders as well.

The two major components of the capital of a company are the equity and the debt, or to

put it other words, common stock and bonds. We have already seen they have different

costs to the company, the debt being considerably cheaper. That is why almost all the

companies include debt in their capital structure. Let us define

S = market value of the entire common stock of the company

B = market value of the bonds of the company

V = market value of the company

then the total market value of the company is just equal to the sum of the market values

of its debt and equity. That is,

V = B + S (9.4)

The weight of debt, or the percentage of debt, in the capital structure is B/V. Similarly,

the weight of equity is S/V. The correct cost of debt is after taxes, namely, (1 − t)kd.

Combining these ideas, we find the weighted average cost of capital, WACC, to be

WACC = (1 − t) kd B

V + ke

S

V (9.5)

where t = income tax rate of the company

ke = cost of equity capital

kd = pre-tax cost of debt capital.

(1 − t)kd = after-tax cost of debt.

We may easily modify (9.5) to include preferred stock, or more than one bond issue.

The reason for calculating the WACC of a corporation is that it is the proper discount rate

that should be used in computing the NPV of the projects under consideration, provided

the projects are of average risk and the company is using the existing capital to finance

the new projects.

The capital cost of riskier projects is, of course, higher and the company should evaluate

it separately. If the corporation is raising new capital to finance a new project, it may

change the WACC and the company should make the acceptance decision under the new

WACC.

http://en.wikipedia.org/wiki/Weighted_average_cost_of_capital


163

Examples

9.1. Schirra Lumber Company is in 40% tax bracket. It wants to calculate the after-tax

cost of the following:

(A) A bond sold at par with a 13% coupon.

(B) A 10-year bond with 6% coupon and face value $1,000 sold for $600.

(C) A preferred stock sold for $25, with quarterly dividends of $0.50 each, if the

company plans to call the issue after 5 years at a price of $30 per share.

(D) A common stock at $15 a share, if the dividends are expected to grow at the rate of

3% annually, and the dividend next year is $2.00.

(A) If the bond has a coupon of 13% and it is sold at par, the pre-tax cost of capital is

13%. The interest is a tax deductible item, and so the cost of interest is reduced by the tax

rate. The after tax cost of capital is .13(1 − .4) = .078 = 7.8%. ♥

(B) Let us look at the problem from the point of view of an investor who buys this bond.

For him the approximate yield to maturity is, using (3.5),

Y ≈ 60 + (1000 – 600)/10

800 = .125

which is also the cost of debt to the company. After taxes, it should be (1 − .4)(.125) =

.075 = 7.5%. This is only an approximate answer.

Let us examine this problem a little more closely. First, the company receives $600 when

it sells a bond. Second, the company makes 20 semiannual interest payments of $30 each

which are discounted at the semiannual after-tax rate as r/2. The company makes a final

payment of $1000 for the bond. The difference between the initial and the final price of

the bond is thus $400. According to IRS regulations, the company is able to spread out

the difference, $400, over ten years as $40 per year. This is called the original issue

discount and it is a deductible expense. The total deduction per year is thus $100,

consisting of $40 in original issue discount and $60 in interest payments. This gives an

annual tax benefit of .4(100) = $40. Finally, the company has to pay $1000 to retire the

bond. Considering the PV of all the cash flows, the NPV of this operation is zero,

NPV = 600 − i=1

20

30

(1 + r/2)i +

i=1

10

100(.4)

(1 + r)i −

1000

(1 + r)10 = 0

PV of

sale

PV of interest

payments

PV of tax

benefits

PV of final

payment

Solve the above equation at WolframAlpha with this instruction:

600-Sum[30/(1+r/2)^i,{i,1,20}]+Sum[40/(1+r)^i,{i,1,10}]-1000/(1+r)^10=0

The result is r .0805132, or 8.051%. ♥





164

To do the problem on Excel, first simplify the above equation as follows.

NPV = 600 − 30[1 − (1 + r/2)

−20]

r/2 +

40[1 − (1 + r)−10

]

r −

1000

(1 + r)10 = 0

Since we already know that the answer is near 7.5%, we put .075 in cell B6 as an

approximate answer. Then fill the other cells with the above equation as follows. Adjust

the value in cell B6 until the result in cell B8 comes close to 0. Finally, when the number

in cell B6 is .08051, the value in cell B8 is -0.015649569. The difference is less than 2

cents. The after-tax cost of debt for the company is 8.051%.

A B

1 Face value of bond = 1000

2 Selling price of bond = 600

3 Coupon rate = .06

4 Number of years = 10

5 Income tax rate = .4

6 After-tax cost of debt = .08051

7 =B2-B3*B1*(1-1/(1+B6/2)^(2*B4))/B6

8 NPV = 0 =B7+B5*((B1-B2)/B4+B3*B1)*(1-1/(1+B6)^B4)/B6-B1/(1+B6)^B4

(C) The stock pays $2 annually, and is redeemed five years later at $30. Using (3.5),

Y 2 + (30 − 25)/5

27.50 = .1091

which gives the nominal annual cost of capital to be approximately 10.91%. ♥

There is no tax benefit of paying dividends. The following equation gives the exact result

0 = 25 − i=1

20

.5

(1 + r/4)i −

30

(1 + r)5

For WolframAlpha, use the following instruction

0=25-Sum[.5/(1+r/4)^i,{i,1,20}]-30/(1+r)^5

The answer is approximately r = 11.43%. ♥

(D) The cost of capital, k, to the company is the required rate of return, R, to an investor.

Thus R = k in Gordon's growth model for stock evaluation, namely, P0 = D1

R − g , where P0

is the price of the stock, D1 the dividend to be paid next year, k is the cost of equity, and g

is the growth rate. This gives

k = g + D1/P0 = 0.03 + 2/15 = .1633 = 16.33% ♥



165

Based on the above considerations it is cheapest to raise new capital by floating new

bonds at 13% coupon with an after-tax cost of capital of 7.8%. ♥

9.2. Verizon Corporation has decided to issue zero-coupon bonds to raise new capital.

The company will sell these bonds at 50% of their face value and their maturity date is

set at ten years. The tax rate of Verizon is 32%, and it gets tax benefits of the original

issue discount over the life of the bond. Find the after-tax cost of this bond.

To find the after-tax cost of capital for a zero-coupon bond, we equate the present value

of cash inflows to the present value of the cash outflows, discounted at the after-tax cost

of debt.

The first cash inflow is the money that the company receives by selling the bond, namely

$500. The second set of cash inflows are the tax benefits to the company during the life

of the bond. The final cash outflow is when the company pays $1000 at the maturity of

the bonds.

The original issue discount, 1000 – 500 = $500, is spread uniformly over ten years. The

discount per year is thus 500/10 = $50. Let us assume that the government allows this

$50 as the cost of borrowing the money for Verizon, and that it is tax deductible. The tax

benefit arising out of it is .32(50) = $16. We also have to find the present value of these

benefits for the next ten years, and add them to selling price of the bond, $500. This is

then equated to the present value of the face amount of the bond, $1000. This is best

explained with the help of the following equation,

B + i=1

n

t(F – B)

n(1 + r)i =

F

(1 + r)n

In this equation, we define:

B = the value of the bond when it first issued = $500,

F = the face amount of the bond = $1000,

t = income tax rate of the corporation = 32%,

n = the life of the bond = 10 years,

r = after-tax cost of debt capital, unknown.

This gives us 500 + i=1

10

.32(1000 – 500)

10(1 + r)i =

1000

(1 + r)10

To do it on WolframAlpha, copy and paste the following instruction.

500+Sum[.32*(1000-500)/10/(1+r)^i,{i,1,10}]=1000/(1+r)^10

It gives r .0481818 = 4.818% as the after-tax cost of debt. ♥



166

9.3. Colfax Markets has $200,000 in equity and $100,000 in long-term debt with interest

rate of 9%. The company needs $100,000 in additional capital, with $50,000 in equity

and $50,000 with bonds with a coupon of 9.5%. After the new financing, the company is

expected to have EBIT of $40,000, with a standard deviation of $10,000. Find the interest

coverage ratio of Colfax. Find the probability that this ratio is less than 1.

The current capital structure of the company consists of $200,000 in stock and $100,000

in bonds. The interest rate on the existing bonds is 9%. The company needs to raise

another $100,000. It will do that by selling $50,000 in stock and $50,000 in bonds, but

the new bonds will carry a coupon of 9.5%. We can find the total interest due by

multiplying the coupon rate on the bonds by their face value.

Total annual interest payment, after the new financing = 0.09(100,000) + 0.095(50,000)

= $13,750.

By definition, the interest coverage ratio is

Interest Coverage Ratio = EBIT

Total interest due =

40‚000

13‚750 = 2.909

The interest coverage ratio gives us the sense how comfortably can the company pay its

interest obligations. The interest coverage ratio can become less than 1, if the EBIT falls

below $13,750. This is quite unlikely to happen. To find the probability, we first calculate

the z-value as

z = (x − μ)/σ = (13,750 – 40,000))/10,000 = − 2.625.

Draw the normal probability table with z = 0 in the center and z = −2.625 far to the left.

The probability that EBIT < $13,750 is represented by the area to the left of z = −2.625.

This area is quite small. Next, we check the probability tables. The result is

P(EBIT < $13,750) = .5 − [.4956 + .5(.4957 − .4956)] = 0.435% ♥

EXCEL =NORMDIST(13750,40000,10000,true)


167

With the interest coverage ratio nearly 3, it is highly unlikely that the company will

default on its interest obligation. In reality, the companies keep some cash reserves too. If

the interest coverage ratio falls below 1, then they can still pay the interest due by dipping

into their cash reserves.

9.4. Western College, a tax exempt institution, plans to raise new capital by selling

bonds. The bonds will provide tax-free income to the investors. It may be able to sell 5-

year bonds with 8% coupon at 90. Or, it may issue zero coupon bonds, with 10 year term

to maturity, which will give the same return to the bondholders as the first issue.

Calculate the selling price of the zero coupon bonds.

The yield to maturity for the 5-year bonds is given approximately by

Y C + (F − P)/n

(F + P)/2 =

80 + (1000 − 900)/5

(1000 + 900)/2 = 0.1053

The yield for the zero coupon bonds is also 0.1053. Using FV = PV(1 + r)n we get

1000 = PV(1.1053)10

which gives PV = $367.45 ♥

To find a more precise answer, we use the equation

900 − i=1

10

40

(1 + r/2)i −

1000

(1 + r)5 = 0

For WolframAlpha, the instruction is

900-Sum[40/(1+r/2)^i,{i,1,10}]-1000/(1+r)^5=0

the answer comes out to be r = 10.85%. The value of the zero-coupon bonds is thus

PV = 1000(1.1085)–10

= $356.91 ♥

9.5. Walker Corporation has currently $80 million face value bonds with a coupon of

11%, and selling at par. It has 10 million shares of common stock outstanding, which is

expected to give a dividend of $4.00 next year. The stockholders require 18% return on

their investment, and they expect the dividends to grow at an annual rate of 5%. The

company also has 100,000 shares of preferred stock which has a dividend of $6.00. The

preferred shareholders require a return of 15%. The tax rate of the company is 40%. What

is its WACC?

The cost of debt, kd = 0.11, cost of equity, ke = .18, and the cost of preferred stock, kp =

0.15. Calculate the market values of the three components of the total capital of the

company.

Market value of bonds = B = $80 million



168

Using the equation P0 = D1

R − g , we get P0 =

4

0.18 − 0.05 = $30.769231 per share

Total value of common stock for 10 million shares = S = $307.69231 million

Price of preferred stock = 6/0.15 = $40 per share.

Total value of preferred stock for 100,000 shares = P = $4 million

Total value of the firm = S + B + P = V = 80 + 307.69231 + 4 = $391.69231 million

Modify equation (9.5) to include the cost of preferred stock as,

WACC = (1 − t)kd B

V + ke

S

V + kp

P

V (9.6)

WACC = 0.6 (0.11) (80) + 0.18 (307.69231) + 0.15 (4)

391.69231 = .1564 = 15.64% ♥

9.6. The beta of Kenner Corporation stock is 1.25, the market return is 12%, and the

riskless rate is 6%. The dividend on Kenner next year will be $2.50, and it is expected to

grow at 3.5% annually in the future. The company has 1 million shares of common

stock, 50,000 shares of $3 preferred stock whose holders require a return of 10% on their

investment, and $10 million face value of bonds with a coupon of 5%, and maturity of 20

years. The yield to maturity for the bonds is 8%. The income tax rate of the company is

35%. Find its WACC.

Here β = 1.25, E(Rm) = 0.12, r = 0.06. Using CAPM, find the cost of equity as

ke = r + βi [E(Rm) − r] (9.3)

we get

ke = 0.06 + 1.25(.12 − 0.06) = 0.135

The stock price per share is P0 = D1

R − g =

2.5

0.135 − 0.035 = $25

The total value of the stock for 1 million shares, S = $25 million.

The price of preferred stock = 3/0.1 = $30 per share

Total value of 50,000 preferred shares = 50,000 (30) = $1,500,000.

The yield to maturity of the bonds is not only the pre-tax cost of debt capital to the

company, but it is also the proper discount rate to evaluate the bonds. The annual interest

on the bonds is .05(10,000,000) = $500,000. The semiannual interest is half of that,

$250,000, with 40 payments. The market value of the bonds is thus


169

B = i=1

40

250‚000

1.04i +

10‚000‚000

1.0440 = $7,031,084

Total value of the firm = 25,000,000 + 1,500,000 + 7,031,084 = $33,531,084

WACC = (1 − .35) (0.08) (7,031,084) + 0.1 (1‚500,000) + 0.135 (25‚000‚000)

33‚531‚084

= 11.60% ♥

9.7. Shrike Company has 2 million shares of its common stock outstanding and they are

priced at $40 each. The current dividend is $3 per share, which is expected to grow at the

rate of 5% annually in the future. Shrike also has $40 million in long term bonds selling

at par with a coupon rate of 8%. It has 1 million shares of preferred stock with dividend

of $2 per share, and these shares yield 9% to the shareholders. The income tax rate of

Shrike is 36%. Find its WACC.

Using (9.2), find the cost of equity as

ke = D0(1 + g)

P0 + g = 3(1.05)/40 + 0.05 = 0.12875

Further, kd = 0.08 and kp = 0.09 are given.

The value of the stock and bonds are: S = $80 million, B = $40 million.

The total value of the preferred stock is P, where

P = 2‚000‚000

.09 = $22.222 million

The total value of firm, V = 80 + 40 + 22.222 = $142.222 million

Combining all these factors,

WACC = (1 − .36)(.08)(40) + .12875(80) + .09(22.222)

142.222 = 10.09% ♥

9.8. Southern Inns has 12% cost of debt and 18% cost of equity. At present it has $100

million (face amount), 8% coupon, 10-year bonds that pay interest semiannually. It also

has 5 million shares of common stock selling at $25 each. The tax rate of Southern Inns is

30%. Find its WACC.

First, find total the market value of the bonds. The discount rate is r = 12% per annum =

.06 semiannually. With 8% coupon, the annual interest is $8 million, and semiannual

interest $4 million. There are 10 years to maturity, or 20 semiannual payments. Thus,


170

B = i=1

n

C

(1 + r)i +

F

(1 + r)n =

i=1

20

4

1.06i +

100

1.0620 = $77.060 million.

The total market value of the stock, S and the market value of the company, V are:

S = 5(25) = $125 million, and V = 125 + 77.060 = $202.060 million.

With kd = .12, ke = .18, and t = .3, the WACC of the company is thus

WACC = (1 0.3)(0.12)(77.06/202.06) + 0.18(125/202.06) = 14.34% ♥

9.9. Vermont Corporation common stock sells for $40 a share. It will pay a dividend of

$4 next year, which is expected to grow at the rate of 5% annually. Vermont $5 preferred

stock is selling for $39 a share. The company also has perpetual bonds with coupon 3%,

but they sell at 35% of their face value. Vermont has a tax rate of 35%; it has 5 million

shares of common stock; 1 million shares of preferred stock; and $300 million (face

amount) of bonds. Using the existing capital, should Vermont undertake a project with a

return of 17%?

Find the cost for different components of the capital.

Using Gordon's growth model (9.2), ke = D1/P0 + g = 4/40 + .05 = 0.15.

From (3.2), for a perpetual bond, its market value B is the ratio between the annual

interest payment, C, and the required rate of return by the bond holders, r. That is, B =

C/r. In this case, B = $350 and C = $30 per year. Now r is also the pretax cost of debt.

Thus r = kd = 30/350 = 0.08571.

The preferred stock has no growth in dividends. Using (9.2), the cost of capital for

preferred stock is thus kp = 5/39 = 0.1282.

The market value of the stock, S = 5*40 = $200 million.

The market value of the bonds, B = .35*300 = $105 million.

The market value of the preferred stock, P = 39*1 = $39 million.

The total value of the company, V = 200 + 105 + 39 = $344 million.

WACC = (1 − .35)*.08571*105/344 + .15*200/344 + .1282*39/344 = 11.88%

With the cost of capital around 12%, a project with a return of 17% is quite attractive,

provided the risk of the new project is the same as the risk of the existing projects of the

company. ♥

9.10. Crookes Corporation has the following capital structure: $25 million (face amount)

of bonds with coupon 5%, which will mature in 10 years and sell at 70% of the face

value; and 5 million shares of stock priced at $10 each. It is estimated that the difference


171

between the cost of debt and equity is around 5%. The tax rate of Crookes is 30%. Find

its WACC.

Find the cost of debt, kd as the yield to maturity of the bonds, given by equation (3.5). Put

F = $1000, B = $700, n = 10 years, and cF = .05(1000) = $50

kd = Y = cF + (F − B)/n

(F + B)/2 =

50 + (1000 − 700)/10

(1000 + 700)/2 = 9.41%

Cost of equity is 5% higher than the cost of debt, ke = 14.41%,

Market value of the bonds, B = 0.7(25) = $17.5 million,

Market value of the stock, S = 5*10 = $50 million,

Total market value of the company, V = $67.5 million,

Income tax rate, t = 0.3

Thus WACC = 0.7(0.0941)(17.5/67.5) + 0.1441(50/67.5) = 12.38% ♥

Problems

9.11. Ellington Corporation has tax rate of 35%. It may raise new capital in one of the

following three ways. Find the after-tax cost of new capital.

(A) By selling common stock at $45 a share which will pay a dividend of $4 next year

which is expected to further grow at the rate of 5% per annum forever. 13.89% ♥

(B) By selling 8% bonds at 90 that will mature in 10 years. ~6.158%, 6.318% exactly ♥

(C) By selling $8 preferred stock at $75 a share, redeemable at par after 5 years.10.67% ♥

9.12. Aliquippa Company wants to issue discount bonds with a market value equal to

30% of their face value. The bonds will carry 6% coupon, paying interest semiannually,

and they will mature after 10 years. The income tax rate of Aliquippa is 40%.

(A) Calculate the approximate yield-to-maturity of the bonds and the after-tax cost of

debt for Aliquippa. YTM 20%, after-tax cost of debt 12% ♥

(B) Using the concept of original issue discount, write an equation that gives the after-tax

cost of debt for Aliquippa. Solve this equation by using Excel or Maple. 14.66%. ♥

9.13. Taïf Company has the following capital structure: 5 million shares of stock, selling

at $25 each, with = .9; zero-coupon bonds with face amount $50 million, maturing in

10 years, with yield to maturity 8%; and 1 million shares of preferred stock selling at $12

per share, paying a dividend of 30¢ per quarter. The income tax rate of Taïf is 40%. The

riskfree rate is 6%, and the expected return on the market 16%. Find the weighted

average cost of capital for Taïf. 13.15% ♥


172

9.14. Procyon Corporation has 55% debt and 45% equity (market values) in its capital

structure. The pretax cost of debt is 10%, and that of equity 15%. The total value of the

company is $15 million and its income tax rate is 35%. Procyon has to raise $2 million in

new capital, which will make the EBIT of the company to be $4 million, with a standard

deviation of $2 million. The company has decided to raise the new capital half with debt

and half with equity at the existing rates. Calculate Procyon's new WACC, and the

probability that its interest coverage ratio (ICR) will be less than one.

WACC = 10.38%, P(ICR < 1) = 6.21% ♥

9.15. Mercury Corporation stockholders expect a growth rate of 4% in the company, and

a dividend of $1.00 next year. The Mercury stock is currently selling for $10 a share.

There are 3 million shares of the common stock. The company also has $50 million face

value zero-coupon bonds which will be due after 10 years. The bondholders have a

required rate of return of 8%. Mercury has a tax rate of 35%. Find its WACC. 10.17% ♥

9.16. Blue Grass Co has the following capital structure. It has 2 million shares of

common stock selling for $20 each. The stock will pay a dividend of $2 next year and

this dividend is expected to grow at the rate of 5% annually. Blue Grass has just raised

$20 million by selling 10% coupon bonds at par. Blue Grass also has 1 million shares of

preferred stock which pays a dividend of $1.50 annually, and the preferred shareholders

have a required rate of return of 12%. Blue has a 35% income tax rate. Find the WACC of

Blue Grass. 12.14% ♥

9.17. Heisenberg Corporation has the following capital structure: $60 million (face

value) of 11% bonds selling at 95, maturing after 10 years; 10 million shares of common

stock selling at $10 each, with current dividend of $1.00 annually; and one million shares

of preferred stock selling at $40 each and paying an annual dividend of $5. The common

dividends are expected to grow at the rate of 3% annually, and the company's tax rate is

30%. Find the WACC of Heisenberg. 11.68% ♥

9.18. Libra Corporation has debt/assets ratio of .4, its cost of debt is 9% and that of

equity 13%. The tax rate of Libra is 30%. The company is not growing and its dividend

payout ratio is 100%. Libra has 2 million shares of common stock, with a dividend of $2

per share. Find the price per share of Libra, its total value, and its WACC.

$15.38, $51.28 million, 10.32% ♥

9.19. Haig Co has 4 million shares of common stock selling at $45 each. It has $70

million (face value) of bonds, with 6% coupon, maturing in 5 years, and selling at 90.

The difference between the cost of debt and the cost of equity for Haig is estimated to be

6%. The tax rate of Haig is 30%. The firm also has 2 million shares of preferred stock

that pay annual dividends of $5 each. The preferred shareholders get a return which is 2%

less than the return of the common shareholders. Find the WACC of Haig.

WACC = 12.26% ♥


173

Key Terms bonds, 152, 153, 154, 157,

160, 161, 162, 163, 164,

165, 166, 167, 168

capital, 152, 156, 157

convertible bonds, 153

cost of capital, 152, 158, 160,

165, 166

cost of debt, 153, 156, 158,

167

cost of equity, 156

debt, 152, 153, 154, 156, 157,

158, 159, 160, 161, 163,

165, 166, 167, 168

dividends, 153, 156, 158,

160, 163, 167, 168

equity, 152, 153, 156, 157,

158, 160, 161, 163, 164,

165, 166, 167, 168

Gordon's growth model, 156,

160, 165

junk bonds, 154

original issue discount, 152,

154, 159, 160, 167

preferred stock, 152, 153,

163, 164, 165, 167, 168

risk premium, 156

stock, 152, 153, 156, 157,

158, 160, 162, 163, 164,

165, 166, 167, 168

WACC, 158, 163, 164, 165,

166, 167, 168

warrants, 153

weighted average cost of

capital, 152, 158, 167

174

10. CAPITAL STRUCTURE THEORY: VALUE MAXIMIZATION


1. Maximize the EPS for a firm using either debt or equity.

2. Use the critical, or indifference EBIT, to decide the use of debt or equity.

3. Understand the concept of tax shield and bankruptcy costs.

4. Find the value of a leveraged firm.

10.1 Capital Structure of a Firm

Investors provide the capital to a corporation. They do so by buying the stock or the

bonds of that company. The company merges the money acquired from stockholders and

bondholders in a pool and does its business with that capital.

Consider a firm financed entirely by the capital provided by the stockholders. It is an all-

equity firm with no debt. The company is in a strong financial position because it does

not have to worry about interest payments. Is it a good idea to run a company that way?

No, because we have already seen that the cost of debt is less than the cost of equity. That

is why in real life the corporations carry fairly large amounts of debt. Fig. 10.1 shows the

capital structure of two companies with different percentages of debt and equity.

(A)

(B)

Fig. 10.1: Company with (A) 75% equity and 25% debt, and (B) 75% debt and 25% equity.

We know that debt is less costly than equity, why not finance a company entirely with

debt? Perhaps we should have 75% debt and 25% equity. The problem with this setup is

that in case of a lean year, or perhaps even a bad quarter, the company may not have

enough money to pay the interest due on the bonds. We also know that the bondholders

have the right to force the company into liquidation in case of default. To avoid this

unpleasant outcome, companies should avoid having too much debt.

Is there an optimal mix of debt and equity for a company? Yes. Ideally, the optimal mix

of debt and equity will maximize the value of a firm. It will possibly maximize its

Analytical Techniques 10. Capital Structure Theory: Value Maximization _____________________________________________________________________________

175

earnings per share. Perhaps the optimal capital structure will minimize the weighted

average cost of capital for the corporation. We shall explore all these possibilities.

The debt capacity of a company depends upon its line of business, the level, and

constancy of its earnings, and the need for new capital. The managers of a firm are

supposed to maximize the value of the firm. One quick and easy way to measure their

performance is to look at the value of the stock as it is published daily.

10.2 EBIT-EPS Analysis

One way to improve the value of a stock is to increase its earnings per share, as defined

by (10.1). It is easy to look up the price-earnings ratio of stocks, which gives the

investors a snapshot of the financial health of the company. The P-E ratio is the ratio

between price per share and earnings per share of a stock. A stock with a lower P-E ratio,

among similar stocks in the same industry, is more attractive to the investors because its

price is relatively less. As more and more investors buy that stock, they will bid up its

price and increase the P-E ratio.

Is it possible to enhance the expected earnings per share of a company by judicious use of

financing for new projects? The answer is yes.

The following diagram represents roughly the flow of funds in a corporation

Revenues

– fixed costs – variable costs – depreciation

=

Earnings before interest and taxes, EBIT

– interest, to bondholders (1)

=

Earnings before taxes, EBT

– income tax, to government (2) – sinking fund, to retire some debt – dividends, to preferred stockholders

=

Earnings after taxes, EAT

– dividends, to stockholders (3)

=

Retained earnings, RE

added back to the equity of the firm

Fig. 10.2: The diagram shows the flow of funds in a corporation.


176

The earnings-per-share is defined as

EPS = Total earnings after taxes

Number of shares of stock outstanding =

EAT

N (10.1)

Suppose a company is need of additional financing worth F dollars, which it can raise by

either selling new bonds or new stock. To find the method of raising new capital that will

maximize its EPS, let us define:

EBIT = expected earnings before interest and taxes after the new financing

EPS = earnings per share, after the new financing

I = interest that has to be paid on existing debt, if any

SF = sinking fund payments on existing debt, if any

PD = dividends on preferred stock, if any

t = corporate income tax rate

N = number of shares of stock outstanding

F = amount of new financing required

r = rate of interest on debt, if bonds are used for new financing

P = price per share of stock, if equity is used for new financing

The interest paid on the bonds is tax deductible. Thus the taxable income is (EBIT − I),

and the amount of tax paid is (EBIT − I)t. The amount left after paying taxes, or, earnings

after taxes, EAT is

EAT = (EBIT − I)(1 − t) (10.2)

After paying the sinking fund payments and the dividends to preferred stockholders, we

have (EBIT − I)(1 − t) − SF − PD. This amount is available to common stockholders

because we have satisfied all other claims. This the earnings after taxes, EAT. Thus, the

EPS comes out to be

EPS = (EBIT − I) (1 − t) − SF − PD

N (10.3)

Suppose the company goes ahead and sells bonds with face value F and coupon rate r.

The interest payable on the bonds is rF, which adds to the interest due. After this new

financing, the new EPS is given by

EPS(bonds) = (EBIT − I − r F) (1 − t) − SF − PD

N (10.4)

If the firm decides to use equity for new financing, it will sell new stock to raise capital.

The number of shares of new stock is F/P, where F is the total amount of new financing

and P is the price per share. This increases the total number of shares, making it N + F/P.

There is no change in the interest due; it is still I. The EPS in this case is

EPS(stock) = (EBIT − I) (1 − t) − SF − PD

N + F/P (10.5)


177

The financing that gives the greater EPS is clearly the better choice. It is also possible to

find the new EPS when a combination of debt and equity is used for the new capital.

Rewrite (10.3) as

EPS(bonds) = EBIT (1 − t)

N −

(I + r F) (1 − t) + SF + PD

N

Recall the equation of a straight line in the intercept form, namely,

y = m x + b

where m is the slope, and b the y-intercept. Note that the relationship between EPS and

EBIT is linear, and that the line representing them will have a positive slope 1 − t

N, and a

negative y-intercept, − (I + r F) (1 − t) + SF + PD

N

Similarly, write (10.4) as

EPS(stock) = EBIT (1 − t)

N + F/P −

I (1 − t) + SF + PD

N + F/P

The intercept for this line is also negative, − I (1 − t) + SF + PD

N + F/P , but it is smaller in

magnitude compared to the previous case. The slope is 1 t

N + F/P , which is also less than

the previous slope. Let us represent it on a diagram as shown below. The critical EBIT is

at the intersection of the two lines. If the expected EBIT is higher than the critical point

then bond financing is the better alternative.

In Fig. 10.3, the critical EBIT is at the intersection of the two lines. This is also the point

where the EPS for the bonds equals the EPS for the stock.

An alternate way to do the EBIT-EPS analysis is to find the critical EBIT where the EPS

for the two types of financing is equal. Use the right hand sides of equations (10.4) and

(10.5), and set them equal to one another. We designate the critical EBIT by E*.

(E* − I − r F) (1 − t) − SF − PD

N =

(E* − I) (1 − t) − SF − PD

N + F/P

We can solve the above equation by using Maple as follows

((e-i-r*F)*(1-t)-SF-PD)/N=((e-i)*(1-t)-SF-PD)/(N+F/P);

solve(%,e);

simplify(%);

http://www.purplemath.com/modules/strtlneq.htm


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Fig. 10.3. EPS for various EBIT, the intersection of lines is the critical EBIT.

The result comes out to be

E* = I + r(NP + F) + SF + PD

1 − t (10.6)

If the sinking fund term and the preferred dividends are missing, then the equation

becomes much simpler,

E* = I + r(NP + F) (10.7)

In (10.6), I = interest due on the current debt of the corporation

r = coupon rate on the new bonds, if the company uses debt for new financing

N = number of outstanding shares of the company

P = price per share for new shares, if the company uses equity financing

F = amount of new capital required by the company

Once E* is found by using the above expression, then we use the following decision rule,

If E(EBIT) > E*, use bonds If E(EBIT) < E*, use stock

Fig. 10.4. Decision rule for using stock or bonds for new financing.

If the expected EBIT after the financing is known along with its standard deviation, then

it is possible to calculate the probability of getting the interest coverage ratio to be less

than one, or the probability of having made the right decision. In Fig. 10.3, the two lines


179

intersect at the point where EBIT is $4.929 million. Since the actual EBIT is $6 million,

the bonds should give a higher EPS.

The previous discussion of EBIT-EPS analysis is myopic in nature. It looks at the EBIT

for the next year only. For example, a firm may opt for debt financing because its

expected EBIT is high, well over the critical E*. If the EBIT for subsequent years is

much lower, the company will not benefit from debt financing. In the next section, we

consider the long-term effects of debt financing, or leveraging.

10.3 Tax Shield, Bankruptcy Costs, and Optimal Capital Structure

As we have already seen, the cost of debt is lower than the cost of equity, it is desirable to

include debt financing in the capital structure. It is tempting to have a lot of debt and very

little equity. The drawback of this arrangement is that the company is overly exposed to

default risk. If the bondholders do not receive their interest payments on time, they can

force the company into bankruptcy. If the earnings of a company before payment of

interest or taxes are 5 or 6 times the amount of interest due, then the company is in a

rather safe position. If this interest coverage ratio is down to about 1.25, then there is

substantial probability of default.

Although debt is less expensive form of capital, too much of it can cause serious financial

problems for the firm. The lenders and stockholders foresee this and both expect higher

returns on their investment. Higher required rate of return will force the price of a

security downward. This will result in lower bond and stock prices and hence a lower

overall value of the firm. How can the management increase the value of a firm? It is

possible by having an optimal mix of debt and equity.

One cannot find the optimal blend of capital by an established formula. The company has

to base its decision on several variables, primarily on expected earnings and the stability

of earnings. If the earnings are high and quite stable, the company can afford to have

more debt. For a corporation with erratic earnings it is safer to finance the projects mostly

with equity.

If an all-equity firm wants to replace some equity with debt, it can do so by issuing bonds

and using the proceeds from the sale of bonds to repurchase its common stock. This

procedure will lower its WACC, which will increase its value. This increased value is

really due to the "tax shield" provided by the deductibility of interest from the income

before paying taxes. The present value of all future tax savings equals the tax shield, that

is, the increase in the value of the company.

Suppose a company is initially unleveraged, that is, it has no debt. Next, it issues a certain

amount of bonds, B, and uses the proceeds to buy back its own stock. The company is

just replacing one form of capital with another form of capital, without changing its

physical characteristics. If the bonds carry a coupon r, then the annual interest paid on

these bonds is rB. This interest is tax deductible. This means, the company is not paying

http://en.wikipedia.org/wiki/Tax_shield


180

interest on the amount rB. The resulting tax savings is trB, where t is the tax rate of the

company.

Once the company issues this debt, it will save an amount trB in taxes every year until the

debt is paid off. At that time, the company will issue new debt and thereby continue this

tax benefit forever. The company is thus increasing its value by an amount equal to the

present value of all future tax savings. This comes out to be

PV = i=1

∞

trB

(1 + r)i

We may solve this by using the formula for the present value of perpetuity,

i=1

C

(1 + r)i = C

r (2.7)

With C = trB in the above equation, we find

Tax shield = tB (10.8)

The above equation assumes that the discount rate of the tax savings equals the coupon

rate of the bonds. This assumption is not exactly right because the bonds introduce more

risk in the company. In other words, the tax shield is somewhat less than tB and (10.8) is

only an approximation.

This increase in value is linearly proportional to the amount of debt issued. However, the

increase cannot go on unchecked. After a while, the firm takes on too much debt and it

becomes too risky. This increases the probability of bankruptcy, and hence the

concomitant bankruptcy costs increase too. Higher bankruptcy costs then reduce the

value of the firm at an ever-increasing rate. The net result is that the firm falls in value

rapidly. There is a certain point where the firm reaches its maximum value due to the

right amount of equity and debt in its capital structure. This point represents the optimal

capital structure of the company.

Figure 10.5 illustrates the impact of debt on the value of a firm. Starting with the

unleveraged value VU, the value rises steadily with the increasing debt. However, the

bankruptcy costs become more and more important with growing debt and the value

reaches a peak when the amount of debt is at its optimal level. This represents the optimal

capital structure of the firm. We may represent these concepts by the equation

VL = VU + tB − b (10.9)

where VL = value of the leveraged firm

VU = value of the unleveraged firm

tB = tax shield provided by the debt B

b = bankruptcy costs

http://en.wikipedia.org/wiki/Bankruptcy_costs


181

The above equation provides a simple way of expressing the value of a leveraged firm

including the bankruptcy costs. The main difficulty is that no analytical formula exists,

which can calculate the bankruptcy costs.

To find the change in the value of a leveraged firm when it undergoes a change in its

capital structure, we may write equation (10.9) in its differential form as follows,

ΔVL = tΔB − Δb (10.10)

This equation says that the change in the total value of the corporation, ΔVL is the result

of two changes. First, a positive change due to tax shelter provided by additional debt,

tΔB, and second, a negative change due to higher bankruptcy costs, −Δb. Equation

(10.10) is useful in calculating the change in the value of a company when it undergoes a

change in its capital structure.

Consider the example of a corporation, which issues $10 million in new debt and buys

back its own stock from the proceeds. If its tax rate is 30%, it will increase its value by tB

= .3(10) = $3 million. If its bankruptcy costs increase by $1 million, then its value will

decrease by that amount. The net change in value will be $2 million, as predicted by

(10.10).

In a similar manner, an overleveraged company may reduce its debt by issuing additional

equity and buying back its own bonds. Since the company has reduced its debt, it is less

likely to go bankrupt and its bankruptcy costs will decrease. Reduction in bankruptcy

costs mean higher value of the company. At the same time, the lowering of debt reduces

its tax shield, and its value. The net change in the value of the company is again given by

(10.10).

When a company finds itself in a financial crisis, it tries to seek refuge in a bankruptcy

court. It presents its case in a federal court, asking for some time to find a solution to its

financial troubles, while the creditors do not demand any payments. If the federal judge

approves their financial plan, he appoints a trustee that oversees the financial dealings of

the corporation. The company must sell many of its assets to pay off the creditors. If it is

successful in raising enough money, it can get back on its feet. If the assets of the firm are

too small, then the judge has to decide how to distribute the money to various claimants.

The bankruptcy costs are either explicit or implicit. Some of the explicit costs are as

follows: legal bills, filing fees, trustee's fees, the losses incurred when the assets are sold

piecemeal. The value of the various components of the firm is much less than the viable,

moneymaking enterprise that it once was.

When the company is in financial difficulty, the suppliers become wary and they demand

the payment up front, in cash, which is already in short supply. The company is unable to

get trade credit. The banks also try to collect their loans from the firm. The customers

tend to desert a company, which is expected to go out to business soon. All these factors

add to the implicit cost of bankruptcy.

http://en.wikipedia.org/wiki/Bankruptcy_court

http://en.wikipedia.org/wiki/Bankruptcy_court


182

Fig. 10.5 shows a firm with unleveraged value $10 million and tax rate 30%. The straight

line represents the value of the firm if it could increase its value by tax shield alone. The

value will reach $12.4 million when the debt is $8 million. The curve represents the more

realistic situation including bankruptcy costs. In this case the firm reaches a peak value of

$11.017 million when it takes on the optimal debt B* = $4.613 million. When the debt is

$8 million, the value of the firm is reduced to $7.54 million.

Figure 10.5. The diagram shows the leveraged value of a firm. The upward sloping straight line represents

the value added to the firm due to tax shield tB. The increased bankruptcy costs force the value downward,

resulting in the curved line as the actual value of the firm. The peak of the curved line represents the

optimal capital structure of the firm.

Consider a simple example. A company has the following capital structure: $60 million

in equity and $40 million in debt. The total value of the company is $100 million. Its

debt/assets ratio is 40%.

The company keeps $10 million in a checking account and invests the remaining $90 in

other assets. One day the company decides to use all its cash to buy its own stock. Now

the company has no cash, $50 million in outstanding stock, and $40 million in debt. The

total value of the company is now 40 + 50 = $90 million. The debt/assets ratio is 40/90

44%.

Since the company has no cash, it is unable to pay its bills on time. It struggles to pay its

workers and suppliers with the new income that it generates. If the word gets around that

the company will not be able to pay the interest on its debt obligation, the threat of

bankruptcy rises. Suppose the bankruptcy costs, which are directly related to the

probability of going bankrupt, increase by $5 million. As a result, the value of the

company falls to $85 million. The debt is still $40 million and the debt/assets ratio of the

company will now be 40/85 47%.


183

Altogether, the debt ratio has increased from 40% to 47%. This is an extreme case and it

illustrates the relationship between available cash, debt, equity, bankruptcy costs, and

debt/assets ratio of a company.

What should a company do when it is facing a cash crunch? It must raise new capital by

selling stock or bonds, whichever is more advantageous, and keep enough cash on hand.

Note that not all stock buybacks are bad news. If a company is underleveraged, it can

increase its debt ratio by buying back its own stock. If the stock price is depressed in the

market, the managers of the company may buy their own stock at bargain prices.

The real world companies have to make decisions to reduce their debt (Ford), or buy their

own shares (Dollar General). Are they making the right decision? Companies have to

choose between debt or equity in raising new capital. They can decide by looking at the

resulting expected EPS. As the following problems illustrate, they can find the

probability of being right.

Examples

10.1. Florida Company, which is an all-equity firm, wants to raise $5 million in new

capital. After the new financing, its EBIT is expected to be $5 million, with a standard

deviation of $2 million. The company currently has 2 million common shares selling at

$10 each. The company can either sell stock at $9.75 a share, or sell bonds at par with a

coupon of 12%. What is the better method of financing, debt or equity? What is the

probability that you are right in your decision?

Use EBIT* = I + r(NP + F) (10.7)

Since the company is an all-equity firm, it has no existing debt and no interest to pay.

Thus I = 0. The interest rate on the new debt is 12%, which makes r = .12. The new

financing is F = $5,000,000. The existing number of shares, N = 2,000,000. The price per

share, P = $9.75. Substituting these values, we get

EBIT* = 0.12 [2,000,000 (9.75) + 5,000,000] = $2,940,000

Since the expected EBIT = $5 million, is greater than EBIT* = $2.94 million, it is

preferable to use bonds.

The probability of making the right decision is equal to the probability that E(EBIT) is

more than E*.To find the probability that the chosen method is the correct one, first find

the z-value,

z = (x − μ)/σ = (2.94 − 5)/2 = −1.03

http://online.wsj.com/article/SB10001424052748703465504575528071780343994.html?mod=WSJ_business_whatsNews

http://www.businessweek.com/ap/financialnews/D9IHIUIG0.htm


184

Draw a normal probability distribution curve with z = 0 in the center and z = −1.03 to the

left of center. The shaded area, to the right of z = −1.03 represents the probability of

EBIT to be more than $2.94 million. Since the probability is more than 50%, we find

from the tables

P(EBIT > 2.94) = .5 + .3485 = .8485

The probability that using bond financing is the right decision is about 85%. ♥

EXCEL =1-NORMDIST(2.94,5,2,true)

10.2. Fisher Corporation is an all-equity firm with 1 million shares outstanding, each

selling for $50. The company needs $10 million for expansion. It can raise the new

capital entirely by bonds with 8% coupon. Alternatively, it can raise $5 million in bonds

with coupon 7.5%, and $5 million in stock, at $47.50 a share. The EBIT for next year has

a normal distribution, with a mean of $4 million and standard deviation of $2 million.

Fisher’s marginal tax rate is 40%. Based on EBIT-EPS analysis, which method is better?

For the preferred method, what is the probability that the interest coverage ratio is less

than one?

First, calculate the earnings per share in each case using (10.4) and (10.5).

For $10 million in debt, we get

EPS(bonds) = (EBIT − I − r F) (1 − t)

N =

(4 − .08*10)(1 − .4)

1 = $1.92

For $5 million in debt and $5 million in equity, we find the interest as .075*5 ($ million)

and the number of additional shares as 5/47.5 (million).

EPS(bonds and stock) = (EBIT − r D) (1 − t)

N + n =

[4 − .075*5] (1 − 0.4)

1 + 5/47.5 = $1.97

Since the second method gives a higher EPS, it is better to use half debt and half equity to

acquire new capital. ♥


185

For the preferred method, the interest payments for the year will be 7.5% of $5 million,

or .075*5 = $0.375 million. The probability that EBIT is less than $0.375 million is quite

small, much less than 50%. Find it by calculating z,

z = (.375 − 4)/2 = −1.8125

Draw a normal probability distribution curve with z = 0 in the center and z = −1.8125

well to the left of center. The shaded area, under the tail of the curve, represents the

probability of EBIT to be less than $0.375 million. From the table, we get it as

P(EBIT < .375) = .5 − [.4649 + .25(.4656 − .4649)] = .0349 = 3.49% ♥

EXCEL =1-NORMDIST(.375,4,2,true)

10.3. Hamburg Company has the following capital structure: 10 million shares of

common stock selling at $25 a share; 7% bonds with face value $100 million, selling at

90; 1 million shares of $5 preferred stock, selling at $30 a share. Hamburg’s tax rate is

35%. The company has to raise $10 million in additional capital, either by selling bonds

with coupon 8%, or by selling common stock at $25 a share. The EBIT of the company,

after new financing, is expected to be $50 million, with a standard deviation of $20

million. The company must also pay $4 million of principal payments to the previous

bondholders. What is the better method of financing? For the method selected, what is the

probability that the company will be able to pay interest, sinking fund payments, and

preferred-stock dividends out of its current EBIT?

Use (10.4) and (10.5) to find the EPS for each type of financing.

For bonds, EPS = (EBIT − r1D − r2F) (1 − t) − SF − PD

N

= [50 − .07*100 − .08*10] (1 − .35) − 4 − 5

10 = $1.843

For stock, EPS = (EBIT − r1D) (1 − t) − SF − PD

N + F/P


186

= [50 − .07*100] (1 − .35) − 4 − 5

10 + 10/25 = $1.822

Based on the EBIT-EPS analysis, bond financing is slightly better. ♥

The company has to pay the following current obligations: (a) interest, 7% on $100

million on old debt plus 8% on $10 million in new debt, (b) taxes at the rate of 35%, (c)

sinking fund payments, $4 million, and (d) dividends for preferred stock, $5 million. The

minimum EBIT to cover all that is given by

(EBIT − .07*100 − .08*10) (1 − .35) − 4 − 5 = 0

EBIT = $21.646 million

Since the company expects to have $50 million in EBIT, the probability is well over 50%

that it will have $21.646 million. To find the probability, find z-value as

z = (21.646 − 50)/20 = −1.4177

Draw a normal probability curve, with z = 0 in the center and z = −1.4177 left of center.

The area to the right of z = −1.4177, which is more than 50%, gives the answer. From the

table, we get the result.

Prob(EBIT > 21.646) = .5 + .4207 + .77(.4222 − .4207) = .9219 = 92.19% ♥


10.4. Neptune Corporation has a dividend payout ratio of 40% and its tax rate is 40% as

well. Not expected to grow in the next several years, the annual dividend of Neptune is

$3.00. The stockholders have a required rate of return of 14%. The company has $100

million face value of bonds with a coupon of 9% selling at 93.75, and it has 20 million

shares of common stock. Find the EBIT and the total value of the company.

Since 40% of EPS is paid out as a $3.00 dividend,


187

0.4(EPS) = 3, or EPS = $7.50

Use EPS = (EBIT − I) (1 − t) − SF − PD

N (10.3)

Put EPS = $7.50, I = .09*100 = $9 million, t = .4, SF = PD = 0, and N = 20 million.

7.50 = (EBIT − 9) (1 − .4)

20

which gives EBIT = 20*7.50/.6 + 9 = $259 million ♥

From Gordon's growth model, P0 = D1

R − g (3.6)

one gets P0 = 3

.14 − 0 = $21.43 per share

Total value of stock = 21.43*20,000,000 = $ 428.571 million

Value of debt = $93,750,000

Total value of the firm = 428.571 + 93.75 = $ 522.321 million ♥

10.5. Janus Corporation has the following capital structure: 40 million common shares

each selling at $12; 7% bonds with face value $300 million priced at 65; and 1 million

shares of $6 preferred stock selling for $50 each. The tax rate of the company is 35%.

Janus needs $50 million in new capital, either debt or equity, whichever one gives higher

earnings per share. Janus can sell the new bonds at par with a coupon of 11%, and the

new stock at $11 a share, net. After the new financing, Janus expects its EBIT to be $70

million with a standard deviation of $20 million.

(A) Find the EBIT, which will make Janus indifferent toward debt or equity financing.

(B) Which form of financing is better?

(C) For the recommended financing, find the probability that Janus is unable to meet its

preferred dividends from the current EBIT.

Use E* = I + r(NP + F) + SF + PD

1 − t (10.6)

to find the critical EBIT, where Janus is indifferent between debt and equity. In (10.6), I,

the interest on the existing debt, is .07*300 = $21 million; r, the coupon rate on new debt

is 11%; N, the number of existing shares, is 40 million; P, the price per share of the new

stock is $11; F, the amount of new capital, is $50 million; SF, the sinking fund payment,

is zero; PD, the amount of preferred dividends, is $6 million; and t, the tax rate is 35%.

This gives us

E* = 21 + .11(40*11 + 50) + 0 + 6

1 − .35 = $84.131 million ♥


188

Since the expected EBIT, $70 million, is less than the critical value, it is better to use

equity financing. ♥

The minimum EBIT to cover interest, taxes, and $6 million in preferred dividends, in that

order, is given by

(EBIT − 0.07*300) (1 − .35) − 6 = 0

Solving for EBIT, we get the minimum EBIT as $30.23 million. Since the company

expects to have $70 million EBIT, the probability that EBIT will be less than $30.23

million is quite small. To find it, calculate z.

z = (30.23 − 70)/20 = −1.9885

Draw a normal probability curve, with z = 0 in the center and z = −1.9885 left of center.

The small area, to the left of z = −1.9885, under the tail of the curve gives the answer.

From tables,

P(EBIT < 30.23) = .5 − [.4761 + .85(.4767 − .4761)] = .0234 = 2.34% ♥

EXCEL =NORMDIST(30.23,70,20,true)

10.6. Robin Corporation has the following capital structure: $30 million in long-term

bonds selling at par with interest rate 9% and requiring a sinking fund payment of $3

million annually; 1 million shares of common stock selling at $50 each; and 1 million

shares of preferred stock selling at $10 each and paying a dividend of $1. The common

stockholders have a required rate of return of 15% on their investment. The income tax

rate for Robin is 35%. Robin needs $5 million in additional capital, which it can raise by

selling either common stock or bonds at the existing capital costs. The expected EBIT

after the new financing will be $18 million with the standard deviation $6 million. Based

on the EBIT-EPS analysis, find the preferred method of raising the new capital. Calculate

the probability that you have made the right choice.

First, find the critical EBIT from

E* = I + r(NP + F) + SF + PD

1 − t (10.6)

Which gives E* = .09*30 + .09(1*50 + 5) + 3 + 1

1 − .35

This gives E* as $13.804 million. Since the expected EBIT is $18 million, it is better to

use bond financing. ♥

In order to be right in this decision, the company should make at least $13.804 million in

EBIT. To find the probability of that happening, we first calculate the z-value as

z = (13.804 − 18)/6 = −0.6994


189

Draw a normal probability curve with z = 0 in the center and z = −0.6994 to the left of

center. The area to the right of z = −0.6994, under the hump of the curve, which is more

than 50%, gives the result. From the tables,

P(EBIT > 13.804) = .5 + .2549 + .94(.2580 − .2549) = .7578

The probability of being right is close to 76%. ♥


10.7. Defoe Company has $75 million in long-term bonds with 8% coupon; and 10

million shares of common stock priced at $25 each. Defoe needs another $10 million in

new capital, which it may raise by selling bonds with 8.5% coupon, or by selling stock at

$23 per share, net. The company has to pay $3 million in preferred dividends. Find the

EBIT of the company after the new financing which will make the EPS for bond and

stock financing to be equal. If the company expects to have EBIT of $17 million, what

type of financing will maximize the EPS. The income-tax rate of the company is 30%.

First, find the critical EBIT using

E* = I + r(NP + F) + SF + PD

1 − t (10.6)

E* = .08*75 + .085(10*23 + 10) + 0 + 3

1 − .3 = 30.686

For $30.686 million in EBIT, the company will be indifferent towards stock or bond

financing. Since the expected EBIT is $17 million, it is better to use stock financing. ♥

10.8. Cockcroft Corporation at present has $30 million in debt, and $60 million in equity.

It has decided to raise $10 million in additional debt at its current cost of debt of 10%.

The price per share of Cockcroft is $45. The expected EBIT after the new financing is

$20 million, with a standard deviation of $10 million. Calculate the probability that the

interest coverage ratio is less than one. Find the probability that it made the right choice

of financing with debt instead of equity.

Minimum EBIT to pay the interest = .1(30 + 10) = $4 million.

Since the company expects to have $20 million in EBIT, it should have no difficulty in

paying the $4 million interest payments. To find the probability of not being able to pay

it, find z as

z = (4 − 20)/10 = −1.6

Draw a normal probability curve with z = 0 in the center and z = −1.6 to the left of center.

The area under the tail of the curve, on the left of z = −1.6, gives the desired probability.

From the table,

P(default) = 0.5 − 0.4452 = 0.0548 = 5.48% ♥


190

EXCEL =NORMDIST(4,20,10,true)

Since the probability of default is quite small, it is better to finance with debt. Next, find

the critical EBIT from

E* = I + r(NP + F) (10.7)

Which gives E* = .1*30 + .1*(60 + 10) = $10 million

In order to be right, the E(EBIT) must be more than $10 million. The z-value is

z = (10 − 20)/10 = −1

Draw a normal probability curve with z = 0 in the center and z = −1 to the left of center.

The larger area on the right of z = −1, gives the desired probability. From the table,

P(being right) = 0.5 + 0.3413 = 84.13% ♥

This also makes sense because the expected EBIT, $20 million, is quite large compared to

the critical EBIT, $10 million.

EXCEL =1-NORMDIST(10,20,10,true)

Video 10.9 10.9. Delaware Corporation is an all equity firm with 30% tax rate. It has

1.5 million shares, each selling for $25, with a dividend of $3. The company plans to

issue $10 million (face amount) of bonds at 85 with a coupon of 8%. It will use the

proceeds of the bonds to repurchase the stock. Find the total value of the company before

and after the issuance of bonds.

Initially the company is an all equity firm and its total value is just the value of its stock,

S = 25 (1.5) = $37.5 million.

After the bond issue the firm gets a "tax shield" whose value = tB, where t is the

corporate income tax rate and B is the market value of the bonds. The amount of equity

replaced by debt equals the market value of the debt. The cost of capital cancels out.

Assume that there are no bankruptcy costs. Tax shield = 0.3 (0.85) (10) = $2.55 million

A more detailed look at the problem is as follows.

The discount factor r is the cost of debt = 80/850

Tax shield = PV = i=1

∞

tI

(1 + r)i =

tI

r =

.3*.08*10

80/850 = $2.55 million

The total value of the firm increases by an amount equal to the tax shield. Thus the new

value is 37.5 + 2.55 = $40.05 million. ♥

10.10. Bohr Corporation, an all-equity firm, has a total value of $30 million. The

company needs $5 million in new capital, which it can raise by selling common stock or



191

by selling zero-coupon bonds maturing after 10 years. The bonds will sell at 40. The

company is not paying any taxes at present. For equity financing, the company will sell

the common stock at $25 a share. The beta of stock is 0.78, the expected market return is

12%, and the risk-less rate is 6%. Find the preferred method of financing for Bohr.

Find the marginal cost of capital in both cases. For zero-coupon bonds, use

FV = PV (1 + r)n (2.1)

which gives

100 = 40(1 + r)10

Or, r =

100

40

1/10

– 1 = 0.09596 = 9.596%

Use CAPM to get ke,

ke = E(R) = r + β [E(Rm) − r] (7.7)

= 0.06 + 0.78[0.12 − 0.06] = 0.1068 = 10.68%

Since the company is not paying taxes, the after-tax cost of bonds is still 9.596%, and that

of stock 10.68%. Thus, it is better to use debt financing. To further check the result,

evaluate the WACC in both cases.

WACC(bonds) = (1 − 0)(0.09596)(5/35) + 0.1068(30/35) = .1053 = 10.53%

And WACC(stock) = 0.1068 = 10.68

This means that bond financing is slightly better. ♥

In this problem, the advantage of tax shield is zero, because the corporate tax rate is zero.

We cannot use the EBIT-EPS analysis here because the earning figures or number of

shares is unknown.

10.11. The Clinton Press has $45 million in long-term debt with interest rate of 12.5%,

and its income tax rate is 40%. Clinton has the optimal capital structure with debt/assets

ratio of 0.45; and it has to maintain this ratio. Clinton's dividend payout ratio is 0.35. The

company has EBIT of $15.625 million. Considering retained earnings to be part of equity,

how much new debt should the company acquire in order to maintain its present capital

structure?

The earnings after taxes EAT, earnings before interest and taxes EBIT, the interest due I,

and the income tax rate t are related by the equation

EAT = (EBIT − I)(1 − t)

This gives EAT = [15.625 − 0.125 (45)] (1 − 0.4) = $6 million.


192

Since the dividend payout ratio is 35%, the company retains 65% of the EAT. Retained

earnings = 6 (0.65) = $3.9 million.

The retained earnings become part of the capital in the form of equity. In other words, the

equity has increased by $3.9 million. To maintain the optimal capital structure, the

increase in capital should be 55% equity and 45% debt. Thus $3.9 million should be

matched by 3.9*45/ 55 = $3,190,909 of new debt. ♥

Video 10.12 10.12. Columbia Corporation has debt-to-assets ratio 50%, cost of equity

12%, and cost of debt 8%. Its tax rate is 40%. The total bankruptcy cost for Columbia is

estimated to be $10 million, but the probability of going bankrupt is 20%. If Columbia

reduces its leverage ratio to 40%, it will also reduce the probability of bankruptcy to

10%. The total value of Columbia is $40 million at present. If Columbia wants to

maximize its value, should it use the lower leverage ratio?

Because its debt/assets ratio is 50%, the current debt at Columbia is .5*40 = $20 million.

The probability of bankruptcy is 20%, and the total costs in case of actual bankruptcy are

$10 million. Thus the current bankruptcy costs are estimated to be 0.2*10 = $2 million.

The tax rate is 0.4, and the total value of Columbia is $40 million. Using

VL = VU + tB − b (10.9)

we get 40 = VU + 0.4(20) − 2

Or, VU = 40 − 0.4(20) + 2 = $34 million

This means that the value of the company will drop from $40 million to $34 million if it

eliminates its debt altogether. In other words, leveraging has added $6 million to the

value of Columbia.

Suppose the new value of the company is VL with bankruptcy costs down to $1 million

and debt/assets ratio 0.4. This gives us

VL = VU + 0.4(0.4)VL − 1

Substitute VU = $34 million in the above equation to get

VL = 34 + 0.4(0.4)VL – 1

WRA x=34+.4*.4*x-1

Solve it to get VL = $39.286 million

Since the new VL, $39.286 million, is less than the initial value of the company, $40

million, it is better to stay with the existing capital structure. ♥




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Video 10.13 10.13. Merle Company has debt/assets ratio of 0.4 at present. You believe

that it should increase to its optimal value 0.45. You believe that the bankruptcy costs

will not change appreciably due to this restructuring and the company will benefit from

the additional tax shield. The present value of Merle is $20 million and its tax rate is

30%. What will be the value of its debt and its equity after you implement your plan?

Present value, V1 = $20 million.

Since the debt/assets ratio is 40%, B1 = 0.4(20) = $8 million.

Suppose the debt increases from $8 million to B2.

Since there is no change in the bankruptcy costs,

The value added = additional tax shield = t B = 0.3 (B2 − 8).

The total value of Merle becomes V2 = 20 + 0.3 (B2 − 8).

But the new debt/assets ratio is 45%, thus

B2

20 + 0.3 (B2 − 8) = 0.45 (A)

Or, B2 = .45[20 + .3(B2 − 8)]

Or, B2 − .45(.3)B2 = .45(20) − .45(.3)(8)

Or, B2 = $9.156 million

WRA x/(20+.3*(x-8))=.45

Comparing B1 = $8 million and B2 = $9.156 million, Merle should issue $1.156 million in

bonds and buy back stock with that money.

The additional debt will add tΔB in tax shield, without any reduction in value due to

bankruptcy costs. The new value of Merle = 20 + 0.3(1.156) = $20.347 million.

Therefore, the new value of the stock is the difference between the new total value of the

company and its new total debt. The total new value of stock = 20.347 − 9.156 = $11.191

million. ♥

(B) If the initial stock price per share is $24, what is its stock price after the financial

restructuring?

Before: Total value of stock = $12 million

Number of shares = 12/24 = .5 million = 500,000

Face value of new bonds = $1,156,000

Number of shares repurchased = 1,156,000/24 = 48,167




194

After: Remaining shares = 500,000 − 48,167 = 451,833

Value of these shares, from part (A), $11,191,000

Price/share = 11,191,000/451,833 = $24.77 ♥

Comparing the initial price of the stock, $24, with its final price, $24.77, the company has

made the right move.

Video 10.14 10.14. Northern Electric has debt/assets ratio of 30% and tax rate 25%.

The total market value of the corporation is $100 million. The management has proposed

that the company increase its leverage ratio to 35% even though it would add $3 million

to the bankruptcy costs. If the firm wants to maximize its total value, should it move to

the higher debt level?

Use the equation

VL = VU + tB − b (10.9)

For existing setup, 100 = VU + 0.25(0.30)(100) − b,

For proposed debt level, VL = VU + 0.25(0.35) VL − b − 3,

Subtracting, 100 − VL = 7.5 − 0.0875 VL + 3,

WRA 100-x=7.5-.0875*x+3

which gives, VL = $98.082 million. Since the restructuring results in a lower overall value

of the firm, the firm should keep its present debt level. ♥

Problems

10.15. Aquarius Company has the following capital structure: 10 million shares of

common stock selling at $25 each, $150 million of long-term debt with coupon of 8%,

and 1 million shares of preferred stock with a dividend of $2.50. Aquarius would like to

raise $50 million in additional capital. It can do that by either selling bonds or equity at

the existing rates. The expected EBIT of Aquarius after the new financing is $30 million

with a standard deviation of $10 million. The tax rate of Aquarius is 32%. Which method

of financing will maximize its EPS? Using the preferred method, what is the probability

that Aquarius will be unable to pay its interest and preferred dividends out of its current

earnings? EPS(bonds) = $0.7020, EPS(stock) = $0.8117, P(default) = 7.60% ♥

10.16. Rusk Inc needs $50 million in new capital, which it may acquire by selling bonds

at par with coupon of 12% or by selling stock at $40 (net) per share. The current capital

structure of Rusk consists of $300 million (face value) of 10% coupon bonds selling at

90, and 10 million shares of stock selling at $43 apiece. After the new financing, the

EBIT of Rusk is expected to be $70 million with a standard deviation of $30 million.




195

Which method of financing do you recommend? What is the probability that you are

right? Sell stock. 67.96% ♥

10.17. Berks Corporation is expecting to have EBIT next year of $12 million, with a

standard deviation of $6 million. Berks has $30 million in bonds with coupon of 10%,

selling at par, which are being retired at the rate of $2 million annually. Further, Berks

has 100,000 shares of preferred stock, which pays annual dividend of $5 per share. The

tax rate of Berks is 40%. Calculate the probability that Berks will not be able to pay

interest, sinking fund, and preferred dividends, out of its current income, next year.

21.03% ♥

10.18. For Cambria Corporation debt-to-equity ratio, marginal tax rate, and dividend

payout ratio are all 40%. The cost of debt is 10%. Cambria has 1 million shares of

common stock, and $25 million in long-term bonds. Its dividend is $1 per share. Find the

EBIT and the price per share for Cambria. $6.667 million, $62.50 ♥

10.19. Holiday Inc has $50 million in long-term debt at 8% and one million shares of

common at $70 a share. It needs $10 million in new financing which it can raise by

selling new bonds with 8.5% coupon, or stock at $70 a share. The tax rate of Holiday is

35%. After the new financing, the expected EBIT is $10 million, with a standard

deviation of $2 million. If the objective is to maximize the EPS, what is the preferred

method of financing? What is the probability that you have made the right choice?

Sell stock, 65.54% ♥

10.20. Nunn Company has 3 million shares of common stock selling at $19 each. It also

has $25 million in bonds with coupon rate of 8%, selling at par. Nunn needs $10 million

in new capital, which it can raise by selling stock at $18, or bonds at 9% interest. The

expected EBIT after the new capitalization is $6 million, with a standard deviation of $3

million. What is the preferred method of raising new capital? What is the probability that

you are right? Sell stock, 72.13% ♥

10.21. Delta Corporation is an all equity firm with a total value of $20 million. It

requires an additional capital of $5 million, which may be either equity, or debt at the

interest rate of 10%. After the new capitalization, the expected EBIT is $5 million, with

standard deviation of $1.5 million. The company pays income tax at 30% rate, and it has

1 million shares outstanding. What is the preferred method of raising new capital, if the

objective is to maximize the EPS? What is the probability that you are right in your

decision? Debt financing is better. P(being right) = 95.2% ♥

10.22. Uranus Corporation currently has equity of $40 million and debt of $20 million,

in terms of market values. Its EBIT for next year is projected to be $12 million with a

standard deviation of $3 million. Uranus is in the 40% tax bracket. The bonds have a face

value of $25 million and they carry a coupon of 10%. Uranus has 1 million shares of

common stock outstanding. The company plans to raise an additional $6 million in

capital, half with equity, and half with debt. The new debt will have interest rate of 12%.


196

What will be EPS for next year? What is the probability that Uranus will not be able to

pay its interest next year? EPS = $5.10, P(EBIT < $2.86 m) = 0.1% ♥

10.23. Pauli Inc has 12 million shares of common stock selling at $12 each, and $45

million in long-term debt at a pre-tax cost of 12%. The tax rate of Pauli is 40%. The

company has to pay $1 million in preferred dividends and $4 million in sinking fund at

the end of this year. Pauli needs an additional capital of $25 million, which will then give

an expected EBIT next year of $35 million, with a standard deviation of $10 million. The

new financing can be with bonds at a coupon of 12.5%, or with common stock at $11.50

a share. Which type of financing will give the higher EPS for next year? In the method

selected, what is the probability that the company will be able to pay the interest, sinking

fund, and preferred dividends, out of its current income?

EPS(bonds) = $.9070, EPS(stock) = $.9002, bonds are slightly better. Prob = 96.52% ♥

10.24. Handy Inc has debt-to-assets ratio of 40%, tax rate of 35%, and total value of

$100 million. W. C. Handy, the CFO, would like to increase the leverage ratio to 42%,

and he believes that there will be no change in the bankruptcy cost of the company. How

many dollars worth of 12% coupon bonds should the company sell, and buy back its own

stock from the proceeds, to accomplish the financial restructuring? $2,344,666 ♥

10.25. Rochester Company plans to buy back 1 million shares of its own stock from its

cash reserves at $50 a share. This will increase the bankruptcy costs by $10 million, and

the debt/assets ratio from 35% to 40%. The income tax rate of the company is 30%. Find

the value of the stock per share after this buyback. Is the company making the right

move? $48.09, no ♥

10.26. Altoona Company has debt/assets ratio 50%, which is too high and it should be at

45% to be optimal. This debt reduction should also reduce the bankruptcy costs by $30

million. At present, Altoona has 5 million shares of common stock selling at $50 each.

The tax rate of Altoona is 30%.

(A) How many shares of stock should the company sell, and buy back bonds from the

proceeds, to attain its optimal capital structure? 265,896 shares ♥

(B) What is the total value of the company before and after the capital restructuring?

$500 million, $526.012 million ♥

10.27. Cypress Semiconductor Corp. has authorized a $600 million share buyback.

Cypress has 161 million shares outstanding and $424 million in cash. Cypress plans to

buy the stock at $15 a share. The company has no debt at present. To finance the stock

repurchase, Cypress will use up all its cash and borrow enough money to buy the stock,

which will increase its bankruptcy costs by $10 million. The tax rate of the company is

30%. Calculate the price per share after the stock buyback. $15.35 ♥

http://files.shareholder.com/downloads/ABEA-37EQD4/1080028340x0xS1193125-10-234634/791915/filing.pdf


197

Key Terms bankruptcy costs, 169, 175,

176, 177, 185, 187, 189,

191

critical, 172, 173, 182, 183,

184, 185

critical EBIT, 169

earnings per share, 170, 171,

179, 182

indifference EBIT, 169

leveraged firm, 169, 175

tax shield, 169, 174, 175,

176, 177, 185, 186, 187,

188

unleveraged firm, 174, 175,

176

198

11. CAPITAL STRUCTURE THEORY: MINIMIZING WACC

Objectives: After reading this chapter, you should be able to

1. Calculate the leveraged β and the unleveraged β of a corporation.

2. Find the cost of capital for a risky venture.

3. Understand how firms are able to minimize their WACC.

11.1 Leveraged Beta

Consider an all-equity firm, a company with no debt. By borrowing money, the firm can

leverage its own assets and it can generate a higher return on its equity capital. Thus, we

may think of a company that has no debt as unleveraged. Most of the companies in real

life are leveraged corporations, although some companies prefer to remain debt-free.

Some of the well-known examples of unleveraged firms are Microsoft, Apple, and

Google. We define the leverage ratio as the ratio between the total debt and the total

assets of a corporation.

By adding debt to its capital structure, a company becomes more risky. This is because of

the additional burden of paying interest when it is due. If the amount of debt is small, a

firm can pay the interest without much difficulty. As the debt increases, a firm gets more

risky due to higher probability of default. Highly leveraged companies are very risky

companies because they may not be able to pay their interest. This greater risk causes an

increase in their beta. The increased risk also leads to higher cost of capital for the firm.

We can develop a relationship between the beta of the stock of a firm when it is

unleveraged, βU, to the beta of the stock of the same firm when it is leveraged, βL. We

already know that the value of a leveraged firm increases due to the tax shield. Let us

assume that the value of a leveraged firm is VL, where VL = B + S, the sum of its debt and

equity. The beta of the entire firm βLF is the weighted average of the beta of its equity βL,

and the beta of its debt, βB. We may write it as

βLF = S βL

VL +

B βB

VL (11.1)

Assume that the bankruptcy costs are zero. Using (10.8), we get the value of the

leveraged firm to be

VL = VU + tB

Thus the beta of a leveraged firm VL, equals the weighted average of the beta of the

unleveraged firm VU, and the beta of the tax shield tB. Thus

βLF = VU βU

VL +

tB βB

VL (11.2)

Comparing (11.1) and (11.2), we get

Analytical Techniques 11. Capital Structure Theory: Minimizing WACC _____________________________________________________________________________

199

S βL

VL +

B βB

VL =

VU βU

VL +

tB βB

VL

Or, S βL + B βB = VU βU + tB βB

Substituting VU = VL − tB = B + S − tB = S + (1 − t)B in the above equation, we get

S βL + B βB = [S + (1 − t)B]βU + tB βB

Or, S βL = [S + (1 − t)B] βU − B βB + tB βB

Or, βL = [1 + (1 − t)B

S] βU − (1 − t)

B

S βB

If the risk of the bonds is negligible, then βB = 0, and the above equation becomes

βL = βU [1 + (1 − t)B

S] (11.3)

We define the symbols in this equation as

βL = beta of the stock of a leveraged firm

βU = beta of the stock of the same firm, if it were unleveraged

t = corporate income tax rate

B/S = debt-to-equity ratio in terms of market values

Robert S. Hamada (1937- )

Equation (11.3) is a well-known relationship first established by

Robert S. Hamada in 1969. This result is only an approximate one

and it is valid under these assumptions:

(1) The firm has only moderate leveraging.

(2) Its bonds are practically risk free.

(3) The bankruptcy costs are almost zero.

This is a very useful result. It enables us to separate the inherent business risk of firm

from its financial risk due to leveraging. We shall see several examples where we can use

this relationship.

11.2 Minimizing WACC

We have already seen that increasing the leverage of a company increases its value at

first. Then the value falls off with the greater possibility of bankruptcy. In a similar way

we can reason that the weighted average cost of capital will fall initially when a company

takes on debt, but then with increasing leverage, the WACC will actually rise.

http://en.wikipedia.org/wiki/Robert_Hamada_(professor)


200

In Figure 11.1, we examine the cost of capital of a firm. The firm has a value of $10

million when it is not leveraged. Then it gradually acquires debt from 0 to $8 million.

The increasing debt will increase the β of the firm, which will increase its cost of equity

capital. Assume that the cost of equity is 14% when there is no debt. With increasing

debt, the cost of equity rises to over 50%. The steep rise in the cost of equity is due to the

greater threat of bankruptcy.

It is possible to minimize the WACC of a firm by a suitable mix of debt and equity. In the

above example, assume that the firm is initially unleveraged. Further, assume that the

income tax rate of the company is 30%. The cost of capital, that is, the cost of equity is

14% initially. As the company acquires debt, the stock of the company becomes more

and more risky, and so the cost of equity rises. As the company gets deeper in debt, there

is a steep increase in the probability of bankruptcy.

Assume that the cost of debt is 6% initially, which is close to the riskless rate. The cost of

debt rises as the debt level increases, but at a slower rate compared to the increase in the

cost of equity. This is because, in case of bankruptcy, the bondholders have better

protection than the stockholders do.

The weighted average cost of capital starts out as the cost of equity because there is no

debt at that point. It tends to decrease as the company replaces equity with debt, the less

expensive form of capital. After reaching a minimum value when debt equals $2.46

million, in this example, the weighted average cost of capital rises again. At this

minimum value of WACC the company has an optimal mix of debt and equity.

Fig. 11.1. The cost of equity, cost of debt, and the weighted average cost of capital for a corporation as the

amount of debt increases in the capital structure of the firm.

Unfortunately, no analytical formula exists, which can calculate the optimal debt level

that will minimize WACC. The companies try to find the optimal debt level by

experimentation.


201

Examples

Video 11.1 11.1. Planck Company is an all equity firm with total value of its assets

equal to $20 million. Its β is 0.88 and the tax rate 30%. The company plans to issue $1

million of debt and simultaneously repurchase its stock worth $500,000. The company

will use the additional capital for expanding its own business. Assuming that the

bankruptcy costs are zero, what is the total value of the company after recapitalization?

Find the new β of Planck.

The total value of the firm increases due to (1) infusion of new capital of $500,000 and

(2) tax shield, tB. Note that the tax shield is due only to the equity that is replaced with

debt. This adds up to 500,000 + 0.3(500,000) = $650,000. The total value of the company

after recapitalization = 20 + 0.65 = $20.65 million. ♥

The new financing will change the company from an all-equity to a leveraged firm. Its

debt is $1 million, and equity $19.65 million. Assuming that the bankruptcy costs remain

negligible, this gives the new beta as

βL = βU [1 + (1 − t)B

S] = 0.88[1 + (1 − 0.3)(1/19.65)] = .9113 ♥

Video 11.2 11.2. Slayton Company has debt-to-assets ratio of 25%. Its cost of debt is

8% and the income tax rate is 30%. Using Shepard Video Games as a proxy, Slayton Co

wants to enter the video games business. Shepard has a debt-to-assets ratio of 20%, β of

1.10, and tax rate of 35%.

(A) Find the beta of this project for Slayton, using its current capital structure.

(B) If the riskless rate is 5%, expected return on the market is 12%, what return should

Slayton require on the new venture?

The β of a company measures its market-related risk. The risk is due to two factors: (1)

the nature of the business and (2) the financial leverage of the company. However, we

can separate the risk due to leveraging by calculating the unleveraged β of the firm. First,

look at Shepard Video Games. Find its debt/equity ratio, or B/S ratio, as follows:

For Shepard, B/V = .2, S/V = .8, B/S = (B/V)/(S/V) = .2/.8 = .25

Its unleveraged beta, using (11.3), is

βU(Shepard) = βL

1 + (1 − t) B/S =

1.1

1 + (1 − 0.35)(.25) = .9462

Thus, the risk inherent in the video games business is represented by the unleveraged β of

0.9462. Slayton has to take this risk for the new project, thus, for this project only,




202

βU(Shepard) = βU(Slayton). Using this as the unleveraged β for Slayton Company, find its

leveraged β as follows:

For Slayton, B/V = .25, S/V = .75, which gives B/S = .25/.75 = 1/3

βL(Slayton) = βU [1 + (1 − t)B

S] = .9462 [1 + (1 − 0.3)(1/3)] = 1.167

The β for this project for Slayton is 1.167. ♥

Find the cost of equity by using CAPM, (7.7),

ke = r + β [E(Rm) − r] = 0.05 + 1.167 [0.12 − 0.05] = 0.1317

The cost of debt for Slayton, kd = .08. Using (9.5), find the WACC as

WACC = (1 − 0.3)(0.08)(0.25) + (0. 1317)(.75) = 0.1128

The WACC for this project is 11.28% if Slayton uses its current capital to finance the

project. Thus, this is the required rate of return on the new project. ♥

11.3. Mimas Corporation has $175 million in equity and $30 million in long-term debt,

in terms of market values. Mimas has tax rate of 30% and its β is 1. Mimas plans to issue

$5 million of new bonds and buy back its own stock with the proceeds of the bond sale.

The bankruptcy costs are negligible. Find the new value, and the new β of the company.

The total value of the company is 175 + 30 = $205 million. The new debt will have a tax

shield of 0.3(5) = $1.5 million. Thus, the total value of the firm will rise to $206.5

million. This includes $35 million in debt. The new value of the equity will be 206.5 − 35

= $171.5 million. Initially B/S = 30/175. Using (11.3),

βU = βL

1 + (1 − t) B/S =

1

1 + (1 − .3)(30/175) = .8929

The new B/S = 35/171.5. Thus the new leveraged beta from (11.3) is

βL = βU [1 + (1 − t)B

S] = .8929 [1 + (1 − .3)(35/171.5)] = 1.02 ♥

11.4. Carter Corporation has debt/assets ratio 15%, tax rate 35% and (leveraged) beta

1.25. Carter is in the meat packing business. Sylvan Bus Lines has debt/assets ratio 25%,

tax rate 30%, and beta 1.6. The riskless rate is 5% and the expected rate of return from

the market is 11%. The cost of debt for Sylvan is 8%. Sylvan plans to start a meat

packing division with its existing capital. Calculate the minimum return on the new

venture acceptable to Sylvan.


203

First, find the unleveraged β of Carter, which represents the risk inherent in the meat

packing business. For Carter, B/V = .15, S/V = .85, thus B/S = .15/.85 = 15/85. Using

(11.3)

βU = βL

1 + (1 − t) B/S =

1.25

1 + (1 − .35) (15/85) = 1.25/(1+.65*15/85) = 1.121

Use this beta as a measure of the inherent risk in meat packing business to find the

leveraged beta for Sylvan as follows.

For Sylvan, B/V = .25, S/V = .75, thus B/S = .25/.75 = 1/3

βL = 1.121 [1 + (1 − 0.3)(1/3)] = 1.383

Using CAPM, (7.7), find the cost of equity as

ke = r + β[E(Rm) − r] = .05 + 1.383 (.11 − .05) = .1330

and finally the WACC,

WACC = (1 − 0.3)(0.08)(0.25) + .1330 (0.75) = .1137

which gives the minimum acceptable return on the new venture as 11.37%. ♥

The following table illustrates the way one can use Excel to solve the problem. The result

will show up in cell E7.

A B C D E F G

1 Carter B/V = 0.15 B/S = (B/V)/(1-B/V) =C1/(1-C1)

2 t = 0.35

3 betaL = 1.25 betaU = =C3/(1+(1-C2)*E1)

4 Sylvan B/V = 0.25 B/S = (B/V)/(1-B/V) =C4/(1-C4)

5 t = 0.3

6 betaL = 1.6 betaL = =E3*(1+(1-C5)*E4) ke = =C8+E6*(C9-C8)

7 kd = 0.08 WACC = =(1-C5)*C7*C4+G6*(1-C4)

8 Market r = 0.05

9 E(Rm) = 0.11

Video 11.5 11.5. Curie Corporation’s capital consists of $10 million in long-term debt,

and common stock with a market value of $70 million. Its β is 1.35 and the income tax

rate 30%. The cost of equity for Curie is 14% and the cost of debt 9%. The riskless rate is

6%. The company has just sold $10 million of new equity to finance new projects. Find

its (a) old WACC and (b) its new WACC.

(a) First, find the WACC of the company before the addition of new capital. Its debt is

$10 million and equity $70 million. Its total capital is $80 million. Thus B/V = 10/80 =

1/8, and therefore, S/V = 7/8. Using (9.5),

old WACC = (1 − .3)(.09)(1/8) + (.14)(7/8) = .130375 ♥



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(b) The ratio B/S = 1/7 for Curie. Find the unleveraged beta of the company,

βU = βL

1 + (1 − t) B/S =

1.35

1 + (1 − .3) (1/7) = 1.22727

With the new financing, the total equity becomes $80 million while the total debt remains

at $10 million. Thus the ratio B/S becomes 1/8. The new leveraged beta is

βL = 1.22727 [1 + (1 − 0.3)(1/8)] = 1.33466

The new beta, 1.335, is somewhat lower the old beta, 1.35. This is due to reduced debt-

to-equity ratio, from 1/7 to 1/8. Using CAPM, we have the old cost of equity, ke of the

company as

Old ke = r + β[E(Rm) − r]

.14 = .06 + 1.35 [E(Rm) − .06]

Solving for E(Rm), we get E(Rm) = (.14 − .06)/1.35 + .06 = .119259

Now find the new values, after refinancing,

New ke = 0.06 + 1.33466 (.119259 − 0.06) = .139091

With the addition of $10 million in new equity, the total stock of the company is now $80

million and its total value $90 million. This makes new B/V = 1/9 and new B/S = 8/9.

Thus

New WACC = (1 − 0.3) (0.09)(1/9) + .139091 (8/9) = .130636 ♥

As a result of additional equity capital, WACC increases from 13.0375% to 13.0636%, an

increase of three basis points. This is a very small increase, implying that the WACC

remains fairly constant due to a change in the capital structure. This small change is also

apparent in the flatness of the WACC curve in Figure 11.1.

11.6. ABC Corporation has $10 million in long-term bonds with 9% coupon selling at

par. It also has 3 million shares of stock selling at $30 each. Its β is 1.6. The income tax

rate of ABC is 32%, the riskless rate is 6%, and the expected return on the market 12%.

ABC is planning to borrow another $10 million at 9% interest to expand its business.

Find the WACC of the company before and after the new financing.

At present, β = 1.6, r = .06, E(Rm) = .12. Using CAPM, get the current cost of equity as

ke = r + β[E(Rm) − r] = .06 + 1.6(.12 − .06) = .156

Since B = $10 million and S = 3*30 = $90 million, V = $100 million. Further, t = .32, kd =

.09, B/V = 10/100 = .1 and S/V = .9. Next, calculate the present WACC as


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WACC = (1 − .32)(.09)(.1) + (.156)(.9) = .14652 ♥

The company is adding debt, thereby increasing its leverage ratio, its risk, and its beta.

However, it is not replacing any equity by debt and so its tax shield remains constant. To

find the new β, we have to unleverage the old β and then releverage it.

Using (11.3),

βU = βL

1 + (1 − t) B/S =

1.6

1 + (1 − .32) (1/9) = 1.4876

After adding debt, total debt becomes $20 million and the total value of the company

becomes 100 + 10 = $110 million. The value of the stock is still $90 million. Therefore,

B/S becomes 2/9. Thus

βL = βU [1 + (1 − t)B

S] = 1.4876[1 + (1 − .32)(2/9)] = 1.7124

The new cost of equity is

ke = r + β[E(Rm) − r] = .06 + 1.7124 (.12 − .06) = .16274

Combining all factors, we get the new WACC as

WACC = (1 − .32)(.09)(2/11) + .16274(9/11) = .14428 ♥

Comparing the initial value, 14.652% and the final value, 14.428%, note that the WACC

has gone down slightly due to addition of cheaper form of capital.

Key Terms leverage ratio, 192, 199, 201

leveraged β, 192, 196

unleveraged β, 192, 195 WACC, 192, 193, 194, 196,

197, 198, 199, 200, 201

Problems

11.7. Allen Corporation is planning to expand into the fast developing business of on-

demand movies. Allen has debt-to-equity ratio of 1, its pretax cost of debt is 15%, and its

marginal tax rate is 40%. The Gardner Corporation is already in the video business, has a

β of 1.5, debt-to-equity ratio of 0.75, and marginal tax rate of 25%. The riskless rate is

10% and the expected return on the market is 20%. What beta should Allen use in

evaluating this project? What is its required return on the project? 1.536, 17.18% ♥

11.8. Omega Corporation has debt-to-equity ratio of 0.7. Its cost of debt is 16%, and its

marginal tax rate is 40%. Omega is considering a project to go into fish-and-chips

business. Kappa Fish-and-Chips, which is already in the same business has a debt-to-


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equity ratio of 0.3, beta of 1.25, and a tax rate of 50%. If the riskless rate is 10% and the

return on the market is 18%, what is the required return for this project for Omega?

17.1% ♥

11.9. Venus Cosmetics Company is an all-equity firm producing only cosmetics. Its beta

is estimated to be 1.35. Mars Chemical Corporation is interested in starting a cosmetics

division. The total market value of Mars' stock is $25 million, and the bonds are worth

$20 million. The beta of Mars is 1.25 and its tax rate is 40%. Assuming that the new

division is financed by the existing capital of Mars, what is the required rate of return of

the new venture? The riskless rate is 10% and the market return is 15%. The cost of debt

for Mars is 12%. 14.31% ♥

11.10. Southampton Publishing Corporation has tax rate of 40%, debt-to-equity ratio of

40%, and has (leveraged) beta of 1.25. The riskless rate is 9% and the market return is

16%. Victoria Publishing Company is an all equity company and is in the same business.

What is the required rate of return by the Victoria stockholders? 16.06% ♥

11.11. Herter Company has debt/assets ratio of 0.3, but Mr. Herter feels that it should

increase to 0.4. The current tax rate of Herter Co is 35% and its beta is 1.25. The

increased leverage will add $10 million to the bankruptcy costs, and $35 million to the

tax shield of the company. By calculating the change in the total value of the firm, do you

think that the company should adopt the new plan? Calculate the new total value of the

company and its new beta.

Yes, current value = $900 million, new value = $925 million, new βL = 1.401 ♥

11.12. Dauphin Corporation has cost of equity 15%, tax rate 40%, and debt-to-equity

ratio of 30%. Fayette Corporation has tax rate 35% and debt-to-equity ratio of 50%. Both

Dauphin and Fayette are in the same business of selling automotive parts. If the riskless

rate is 8% and the expected return on the market is 13%, find the cost of equity for

Fayette. 15.86% ♥

11.13. Yukawa Corporation is an all-equity firm with 10% cost of capital. Its competitor

in the same line of business, Tomonaga, Inc. has WACC of 9.5%, but has debt-to-assets

ratio 30%. Both companies are in the 20% tax bracket. Find the cost of debt for

Tomonaga. The riskless rate is 6% and the expected return on the market is 10%.

kd = 6.42% ♥

11.14. The following table provides financial information of two soft-drink companies.

Firm Total equity Total debt Cost of debt Tax rate β

Troy Company $75 million $25 million 8% 30% 1.5

Utica Corporation $55 million $25 million 9% 35% ?

The riskless interest rate is 6% and the expected return on the market 14%. Which

company has lower WACC? WACC(Troy) = 14.90%, WACC(Utica) = 14.62% (lower) ♥


207

11.15. Ambler Company has $40 million in equity and $25 million in debt. Its tax rate is

35%; its cost of equity 15%; and its is 1.5. Arnold Corporation, its principal competitor

in the same line of business, has $60 million in equity and $30 million in debt. The tax

rate of Arnold is 35%, and its cost of debt is 10%. The riskless rate of interest is 6%. Find

the WACC of Arnold. 11.82% ♥

11.16. The following table provides the financial information of two companies, with the

dollar amounts in millions:

Company Debt Cost of debt Equity Cost of equity Tax rate Business

Lansing Co. $43 10% $77 18.6% 30% Publishing

Phoenix Co. $15 9% $66 17.0% 25% TV stations

At present, the risk-free rate is 6% and the expected return on the market 14%. If Lansing

Company wants to start a TV station as a side business, using its existing capital, what is

the minimum acceptable rate of return on the new venture? From your analysis, can you

deduce which is the riskier business, publishing or TV stations? 14.75%, TV ♥

11.17. Kemp Company has beta 1.5, debt/assets ratio 45%, and tax rate 30%. The cost of

debt for Kemp is 9%, and of equity 14%. The riskless rate is 8%. Find the WACC of

Kemp. If its debt/assets ratio is increased to 48% while its cost of debt remains

unchanged, what is the new WACC? Which leverage ratio is better?

WACC(1) = .10535, WACC(2) = .1045, second is better. ♥

11.18. Jeddah Restaurants has debt/equity ratio .5, and its leveraged beta is 1.6. Its tax

rate is 30%, and its cost of equity is 16%. The riskfree rate is 6%. Masturah Restaurants

has debt/equity ratio .4, and tax rate 35%. Find the cost of equity for Masturah. 15.33% ♥

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12. DIVIDEND POLICY


1. Understand the factors that lead to the dividend policy of a company.

2. See the effect of paying a cash dividend, giving a stock dividend, or declaring a stock

split, on the value of a firm.

3. Know the residual theory of dividends, and realize the possibility of dividend policy

being a residual decision, or an irrelevant issue.

12.1 Dividends and Stock Valuation

From the discussion on the flow of funds in a firm, we may recall that the corporations

pay out dividends to the stockholders after the bondholders have received interest

payments and the government has received its share of taxes. In other words, the

stockholders are the last ones in the line to receive the financial benefits from a company.

The dividend payments are made at the discretion of the board of directors. The directors

may choose to give little or no dividends to the stockholders if the company cannot afford

to pay them. This may happen if the company is suffering a temporary cash shortage, or it

has several profitable opportunities available that need to be funded first.

When a company gives out a dividend, say $2 per share, then the price of the stock, in

response to the payout, also drops by roughly $2 a share. This means that if a stockholder

owns 100 shares of a stock selling at $70 a share will receive a $200 dividend check, but

his stock will now be worth $6800. The income from the dividends is offset by the loss in

the value of the stock. It would be the same if an investor with a bank balance of $7,000

withdraws $200 from the bank.

The reason for the above phenomenon is quite simple. The $2 value of the dividend is

embedded in the total initial value of the stock, $70. The investors who buy the stock are

expecting this payment. If an investor buys the stock right after the payment of dividend,

he knows that he will not get the dividend and he will pay only $68 per share. We may

look at the financial structure of the entire company. Suppose the company has 1 million

shares. Then the company must have $2 million in cash in a checking account to pay the

dividends. Once the dividends are paid out, the cash is gone. The value of the company,

in particular, the value of its stock, decreases by $2 million. This amounts to $2 per share.

Occasionally a company may give a stock dividend, say 10%. This means that a

stockholder who already has 100 shares will now receive a stock certificate for 10

additional shares. He now owns 110 shares, but the market value of these shares is just

equal to the market value of the 100 shares before the declaration of the stock dividend. It

is the same pie but it is cut up in a larger number of relatively smaller pieces. The value

of the stock per share will also drop correspondingly. There is no change in the value of

your stock holdings when you receive a 10% stock dividend.

Analytical Techniques 12. Dividend Policy _____________________________________________________________________________

209

A stock split occurs when one share of stock is split into, say, three shares. The number

of shares is tripled, but the price of each share drops to one-third of its previous value.

Again, we notice that there is no change in the value of your stock investments whether

you own 100 shares at $30 apiece, or 300 shares at $10 each. The total value remains

constant at $3000.

In view of the above arguments, it seems that receiving dividends is quite immaterial for

the investors and the companies should not worry about their dividend policy. The

stockholders do not get richer by receiving the dividends; after all, it is their money

anyway. Yet, corporations pay a lot of attention to their dividend policy, as we shall see.

12.2 Mechanics of Dividend Payment

Most of the companies pay their dividends quarterly. A few weeks before the dividend

payment date, the board of directors would meet and declare a dividend. This is a formal

announcement by the company that it will indeed pay the dividend on time. The

announcement also fixes a record date: only those investors who are the owners of stock

at the end of business on the record date will be entitled to receive the dividends. If you

become the owner of a stock after the record date, then you will not get the dividend.

If you buy the stock, after the record date, it will be too late to receive the dividend. You

will end up buying the stock without the dividend, that is, ex dividend. The stocks that

will go ex dividend tomorrow are listed in today's newspaper. So today, you have the

opportunity to buy these stocks if you want to receive their dividends. The dividend

checks are mailed out a few weeks after the record date.

12.3 Dividend Policy

The payment of dividends is an important decision for any company. If the dividends are

too low, the stockholders who buy the stock for the dividend income would not be happy.

If the company has a generous dividend policy, it will not have money left for growth.

Besides, paying or not paying a dividend should not affect the shareholders wealth

anyway. This complicates the dividend puzzle further.

Let us consider three possible dividend policies for a corporation.

Dividend Policy #1: Suppose a company adopts the following dividend policy: it will

pay out a fixed ratio, say 50%, of its earnings after taxes, and keep the other 50% as

retained earnings. For instance, if earnings are $4 per share, the company will pay out $2

in dividend per share, and if the earnings are $5 a share, the dividend will be $2.50.

Another way of saying this is that the company maintains a constant dividend-payout-

ratio policy. The following diagram shows the EPS and DPS for the last several years.

This dividend policy is shown in Fig. 11.1.

Dividend Policy #2: This dividend policy says that the firm should maintain a constant

dollar dividend per share, and increase it whenever possible. For example, a company


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should pay out $2 per share annually, no matter what the annual EPS is. Further, it should

raise it to say $3, and then maintain it at that level.

Dividend Policy #3: This dividend policy follows these steps:

First, the firm should identify and finance the projects with positive NPV,

Second, it should maintain the optimal capital structure, before and after new financing,

Third, pay out any leftover earnings as dividends.

This is the essence of the residual theory of dividends.

Fig. 12.1: Constant dividend-payout-ratio policy.

Fig. 12.2: Constant dollar-amount dividend policy.


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The residual theory of dividends makes a lot of economic sense. When the investors buy

the stock of a corporation, it is with the understanding that the managers of the company

can make superior investment decisions; otherwise, the investors should invest the money

elsewhere. Now the managers have the money, they should search out and finance

profitable projects, those with positive NPV. The financing decision should be based on

the optimality of the capital structure. For instance, if the company already has too much

debt, it should finance the new projects with equity. The source of new equity capital is

the retained earnings. The residual amount of money, if there is any, should be paid out to

the stockholders. If the company keeps this money, they will end up investing it in

negative NPV projects.

According to the residual theory of dividends, the company should not follow Policy #1,

or Policy #2, because the dividend decision is merely a residual decision. They must

make the investment and financing decisions first.

The surprising fact is that most of the corporations follow Policy #2, where they try to

maintain a regular dollar amount of dividend per share, and increase it periodically. The

reasons for this anomaly are not known. However, several theories have been proposed to

explain this phenomenon.

Examples

12.1. Georgia Corporation has EBIT of $35 million this year. Next year the EBIT will

depend upon two factors: weather and labor problems. If the weather is good (probability

30%), the earnings will rise by 20%. If the weather is normal (probability 50%), the

earning will remain the same, but for poor weather (probability 20%), the earnings will

decline by 20%. In the case of a strike (probability 10%), the earnings will decline by

40%. The probability of the strike does not depend upon the weather. The company has

to pay $5 million interest next year and has a tax rate of 35%. If it intends to pay $10

million in dividends next year out of its earnings after taxes, what is its expected dividend

payout ratio?

Considering probabilities of different outcomes, find the expected EBIT:

Weather Labor Probability*outcome Expected EBIT

Good Strike .3 (.1) (1.2) (.6) (35) = 0.756

Good No strike + .3 (.9) (1.2) (1) (35) + 11.34

Normal Strike + .5 (.1) (1) (.6) (35) + 1.05

Normal No strike + .5 (.9) (1) (1) (35) + 15.75

Bad Strike + .2 (.1) (.8) (.6) (35) + 0.336

Bad No strike + .2 (.9) (.8) (1) (35) + 5.04

Adding all the numbers in the last column, gives the expected EBIT as $34.272 million.

An alternative method to find the expected EBIT is as follows:


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Labor effect = .1(.6) + .9(1) = .96 (EBIT will decrease by 4% due to labor problems)

Weather effect = .3(1.2) + .5(1) + .2(.8) = 1.02 (EBIT will increase 2% due to weather)

Combined effect = .96(1.02) = .9792 (EBIT will become 97.92% of this year)

Thus the expected EBIT is, E(EBIT) = .9792(35,000,000) = $34,272,000

Next, calculate EAT, which comes out to be

EAT = (EBIT − I)(1 − t) = (34.272 − 5)(1 − .35) = $19.03 million

The company wants to pay $10 million in dividends. Thus the expected dividend payout

ratio = 10/19.03 = .5256 = 52.56% ♥

12.2. Gamma Corporation common stock is selling for $50 per share. It has $10 million

in long-term bonds with coupon 8%. The income tax rate of Gamma is 30%. It has 1

million shares outstanding, and its current EBIT is $4 million. The company maintains

50% dividend payout ratio. The company has decided to repurchase its shares at $50 per

share, instead of giving a cash dividend. How many shares should the company buy, and

what would be the price of the shares after this acquisition?

The earnings after taxes for Gamma are

EAT = (EBIT − I)(1 − t) = (4 − .08*10)(1 − .3) = $2.24 million

From EAT, the company pays 50% in dividends, which is $1.12 million. If the company

pays the dividends in cash, the dividend per share will be $1.12. On the other hand, the

company can use this money to buy back the shares at $50 apiece. The number of shares

that the company should buy back = 1,120,000/50 = 22,400.

Let us calculate the value of the stock per share after the payment of dividends. The total

present value of the company is $60 million, which includes $1.12 million that the

company will pay out as dividends. After the payment of dividends, the value of the

company becomes 60 – 1.12 = $58.88 million. The value of the debt remains $10 million

and thus the value of the equity becomes $48.88 million. For 1 million shares, the price

per share becomes $48.88 per share. Every stockholder will get $1.12 per share in cash,

and the price of the stock will also go down by $1.12.

If the company buys 22,400 shares of its own stock, it must spend $1.12 million. This

reduces the value of total equity to $48.88 million as before. Some of the stockholders

will sell their stock (22,400 shares in all) and they will get $50 per share. This is the

market price of the stock. The company now has 1,000,000 – 22, 400 = 977,600 shares

outstanding. The total value of these shares is $48.88 million. The price per share is

48,880,000/977,6000 = $50. Thus, the stockholders who chose not to sell their shares

retain their stock value at $50 per share. ♥

12.3. Virginia Corporation has 5 million shares of common stock, each selling for

$14.69. The company has $10 million in long-term bonds with 7% coupon, selling at par.


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This year the company had EBIT of $12 million, which is expected to grow 5% next year.

The tax rate of the company is 35%. The P/E ratio of Virginia is expected to remain

constant. Next year the company wants to distribute 30% of its after-tax earnings to its

shareholders by the stock repurchase method. How many shares of stock should it buy

back?

For this year, the earnings after taxes, EAT are

EAT = (EBIT – I) (1 − t) = (12 − .07*10)(1 − .35) = $7.345 million

With 5 million shares, the earnings per share = 7.345/5 = $1.469

The P/E ratio of the company for this year is

P/E ratio = Price per share

Earnings per share = 14.69/1.469 = 10

Because the P/E ratio remains constant, the earnings after taxes and the stock price are

growing at the same rate.

For next year, the EBIT will grow by 5% and it will become 12*1.05 = $12.6 million.

The earnings after taxes, EAT next year will be

EAT = (EBIT – I) (1 − t) = (12.6 − .07*10)(1 − .35) = $7.735 million

With 5 million shares, the earnings per share = 7.735/5 = $1.547

Because P/E ratio remains constant at 10, the price per share will be ten times the

earnings, or 10(1.547) = $15.47. We need this share price to calculate the number of

shares to buy.

The company plans to distribute 30% of earnings after taxes as dividends next year. This

comes out to be .3(7.735) = $2.3205 million.

With the price per share at $15.47, the company should buy back x shares, where

Number of shares repurchased, x = 2,320,500/15.47 = 150,000 ♥

12.4. Burr Machine Co stock has price/earnings ratio of 12, annual dividend of $1.20 and

a dividend payout ratio of 30%. The company has 1 million shares outstanding. Burr has

$30 million in long-term debt carrying an interest rate of 11%, and it has a tax rate of

30%. Find (a) the EBIT and (b) the total value of Burr.

From (10.3), we get the dividend per share, DPS, as


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DPS = DPR [(EBIT − I) (1 − t) − SF − PD

N]

Substituting numbers,

1.20 = .3 [(EBIT − .11*30) (1 − .3) − 0 − 0

1] (A)

Solving for EBIT,

EBIT = 1.20/.3/.7 + .11*30 = $9.01429 million ♥

WRA 1.2=.3*(x-.11*30)*(1-.3)

Since P/E = 12, and EPS = $4.00, the price of stock is $48 per share. The value of equity

is $48 million, the value of debt is $30 million, and the value of the company is $78

million. ♥

12.5. Trenton Corporation expects to have EBIT next year of $120 million with a

standard deviation of $30 million. Trenton has tax rate of 35%. It also has $200 million

(face value) of 8.5% bonds and $300 million (face value) of 9% bonds. The company has

a dividend payout ratio of 45%. Trenton has 20 million shares outstanding. Find the

probability that the dividend next year will be more than $1.00 per share.

First, find the EBIT, which will give rise to a dividend of $1.00, by using (10.3),

[EBIT − 0.085(200) − 0.09(300)] (1 − 0.35)(0.45)

20 = 1 (A)

EBIT = 20

.45(1 − .35) + .085*200 + .09*300

This gives EBIT = $112.376 million

WRA (x-.085*200-.09*300)*(1-.35)*.45/20=1

Since the company expects to have $120 million in EBIT, and it needs $112.376 million,

the probability is more than 50% that it will be able to pay $1 dividend. Draw a normal

probability distribution curve, with μ = 120 in the center and x = 112.376 somewhat to

the left of center. The area to the right of 112.376 represents the probability of making

more than $112.376 million in EBIT. To calculate it, find

z = (120 − 112.376)/30 = .2541

From the tables,

P(DPS > $1) = 0.5 + .0987 + .41(.1026 − .0987) = .6003 = 60.03% ♥





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12.6. Phoebe Corporation stock has P/E ratio of 12, and it just paid a quarterly dividend

of $1.00. Phoebe has debt/equity ratio of 0.8, and has 2 million shares of common stock.

Phoebe is not growing and it pays all of its after-tax income as dividends. Find the total

value of Phoebe.

The annual dividend is $4. Since the company pays out all of its after-tax income as

dividends, the after-tax income per share must also be $4 annually. Since the P/E ratio is

12, price per share must be 12 times after-tax income. Thus, the price per share is 12*4 =

$48. Since there are 2 million shares outstanding, the total value of equity should be

2*48 = $96 million. Since the debt/equity ratio is 0.8, it means B/S = 0.8, or B = 0.8 (96)

= $76.8 million. Thus, the value of the debt is $76.8 million. Hence the total value of

Phoebe is 96 + 76.8 = $172.8 million. ♥

Video 12.7 12.7. The Zeta Co has EBIT of $10 million, $30 million of long-term debt at

11%, and tax rate of 30%. It has $5 million of profitable projects available. Zeta has 1

million shares outstanding, selling at $70 each. At present, the company has an optimal

mix of debt and equity. Using the residual theory of dividends, calculate its dividend per

share.

To follow the residual theory, the company must do two things: (a) finance its profitable

projects, and (b) maintain an optimal capital structure. The company has $30 million in

debt and $70 million in equity. Thus, its debt/assets ratio is 30%. The company should

finance the new projects in that ratio, namely, 0.3(5) = $1.5 million with debt, and 0.7(5)

= $3.5 million with equity.

The earnings after taxes = [10 − 0.11(30)](1 − .3) = $4.69 million. This represents new

equity. But the company needs only $3.5 million in new equity to finance new projects,

thus it should give out 4.69 − 3.5 = $1.19 million in dividends. This comes out to be

$1.19 per share. ♥

In order to maintain its optimal capital structure, Zeta should also borrow another $1.5

million in new debt. The retained earnings are $3.5 million, and along with the new $1.5

million in new debt, the company will be able to finance the new projects while

maintaining its optimal capital structure. ♥

12.8. Gill Corporation follows the residual theory of dividends. Its expected EBIT for

next year is $30 million with a standard deviation of $10 million. Gill has $60 million in

bonds with average coupon of 9%. It also pays $4 million in a sinking fund annually. Gill

will need $15 million to finance new projects while maintaining its current debt/assets

ratio at 30%. The tax rate of Gill is 35%, and it has 7 million shares of common stock.

Find the probability that it will be able to pay $1 dividend per share from its current

earnings next year.

The company needs $15 million to finance new projects, using .3(15) = $4.5 million in

new debt and .7(15) = $10.5 million in retained earnings. Since the company puts $4



216

million in the sinking fund to retire old debt, it must borrow an additional $4 million to

maintain its debt to assets ratio. Thus, the total new debt is $8.5 million.

The retained earnings, RE, and total common dividends, CD, are related by the equation

RE = (EBIT − I) (1 − t) − SF − PD − CD

Substituting numerical values, we get the required EBIT as

10.5 = (EBIT − .09*60) (1 − .35) − 4 − 0 − 1*7 (A)

EBIT = (10.5 + 4 + 7)/.65 + .09*60 = $38.477 million

WRA 10.5=(x-.09*60)*(1-.35)-4-0-1*7

Since the company expects to have only $30 million in EBIT, the probability is less than

50% that it will be able to pay $1 dividend per share. To find the probability, find z as

z = (38.477 − 30)/10 = .8477

Draw a normal probability curve with z = 0 in the center and z = .8477 to the right of

center. The area to the right of z = .8477, under the tail of the curve, represents the

probability that EBIT is more than $38.477 million. Checking the tables, we find the

probability as

P(divdend < $1) = .5 − [.2995 + .77(.3023 − .2995)] = .1983 = 19.83% ♥


Key Terms

dividend policy, 202, 203,

204, 211

ex dividend, 203

residual theory of dividends,

202, 204, 205, 209, 211,

212

stock dividend, 202

stock split, 202, 203



217

Problems

12.9. Taurus Corporation has the following dividend policy: if the earnings after taxes

are less than $1 million, the dividend payout ratio will be 35%, but if these earnings are

over $1 million, the dividend payout ratio will be 45%. Taurus expects that its EBIT for

next year will be $6 million with a standard deviation of $4 million. Taurus has $20

million in long-term bonds with coupon of 9%, and 1.5 million shares of common stock.

Calculate the probability that Taurus will give a dividend of more than $1 per share. The

tax rate of Taurus is 30%. 44.41% ♥

12.10. Rogers Corporation stock sells at $27 per share and its dividend per share is

$1.20. Rogers has price-earnings ratio 16. The company has $40 million worth of bonds,

selling at par, with 8.5% coupon. The EBIT of Rogers is $12 million and its tax rate is

30%. Calculate: (a) the dividend payout ratio, (b) the total number of shares, (c) the total

value of Rogers. (a) 71.11%, (b) 3.567 million, (c) $136.32 million ♥

12.11. The expected EBIT of Krupa Co next year is $20 million with standard deviation

of $4 million. The tax rate of Krupa is 30% and it has 10 million shares of common,

stock. Krupa has to pay $8 million in interest and $5 million in a sinking fund. What is

the probability that Krupa will be able to pay a dividend of $1 per share next year and

have no money left over as retained earnings? 0.92% ♥

12.12. Bethe Company stock sells at $45 a share and pays $2 dividends annually. The P-

E ratio of the stock is 5 and the tax rate of the company is 30%. Bethe has 2 million

shares outstanding. The expected EBIT next year is $30 million with a standard deviation

of $5 million. The interest payable next year is $5 million. Calculate the probability that

the company will be able to maintain its current dividend. Almost 100% ♥

12.13. Jefferson Company has $120 million of bonds outstanding, with coupon of

12.5%, selling at 95. It has 2 million shares of $4 preferred stock and 10 million shares of

common stock. Jefferson has EBIT of $76,958,042 this year, and it has income tax rate of

35%. Jefferson must also pay a principal payment of $5 million to the bondholders. The

company has decided to have a dividend payout ratio of 55%. What dividend should

Jefferson declare on the common stock per share? $1.50 ♥

12.14. Hofuf Company follows the residual theory of dividends. It has 4 million shares

of common stock, and it maintains its optimal debt/assets ratio at 30%. Its EBIT next year

is expected to be $12 million, with a standard deviation of $3 million. The income tax

rate of Hofuf is 35% and it has to pay $1 million in interest. It would like to finance $5

million in new projects from retained earnings and new borrowings. Find the probability

that it will be able to give a dividend of at least $1 next year. 42.88% ♥

12.15. Ithaca Company has 5 million shares of common stock selling at $50 each. It also

has $100 million in long-term bonds with coupon 8%, selling at 90. The tax rate of Ithaca

is 32%. Next year its EBIT is expected to be $20 million with a standard deviation of $8

million. The company plans to continue its $2 dividend per share. Ithaca also wants to


218

add $5 million to its total retained earnings next year. Find the probability that it will be

able to maintain its dividend. 10.43% ♥

12.16. Wilson Corporation follows the residual theory of dividends. It has 100 million

shares of common stock, and has EBIT of $120 million. Wilson has $200 million in long-

term bonds with coupon 10%. The tax rate of Wilson is 35%. The company has $40

million in new projects that are to be financed using current earnings and new bonds.

Wilson must also maintain its debt/assets ratio of 40%. Find (a) the amount of new bonds

that Wilson should sell, and (b) its dividend per share. $16 million, $0.41 ♥

219

13. LEASING


1. Make the optimal choice between leasing and buying an asset.

2. Calculate the lease payments, or the purchase price, which will make leasing

equivalent to buying.

3. Discern some of the advantages and disadvantages of leasing.

13.1 Leasing

A company may be in need of a piece of equipment. It can either lease the equipment or

buy it outright. In recent years, leasing has become a common source of financing for

corporations.

There are several reasons why companies want to lease an asset whether it is a computer,

an airplane, or a warehouse. First, it gives them some flexibility. If it is a startup

company, they may want to lease some office space and start their operations. They do

not want to invest in a building, for instance, where they have to tie up a large amount of

capital for a long of time. Second, a company may not have the cash needed to buy an

asset. They may be trying to conserve cash and invest in some profitable projects. Third,

they may simply need the asset for a short time and there is no point in buying it. Fourth,

it is perhaps cheaper for them to lease the equipment, as indicated by the NPV analysis of

the lease.

The companies may still want to buy certain assets, especially the ones that actually

increase in value, such as buildings and land. In many instances, they have to buy an

asset because it is not possible for them to lease it. A company may want to build a

factory to their own specifications and needs.

In some cases, a company may sell an asset and lease it back. For example, a bank may

sell the building that it owns, generate a substantial amount of cash, and also make a large

profit because it bought the building a long time ago. Then it simply leases the building

for its own use. It does not have to move out of its quarters. This is a sale-and-leaseback

arrangement.

IBM Corporation

(Lessor)

Computer (asset)

Lease payments

Mercy Hospital (Lessee)

Figure 13.1. In a direct lease, only two parties are involved

A lease may be direct lease, or a leveraged lease. In a direct lease, only two parties are

involved, the owner of the asset and the end user of the asset. The owner of the asset is

known as the lessor, and the end user, the lessee. An example may be IBM Corporation

leasing a computer to Mercy Hospital. The two parties negotiate various terms of the

http://en.wikipedia.org/wiki/Lease

http://en.wikipedia.org/wiki/Leasing

Analytical Techniques 13. Leasing _____________________________________________________________________________

220

lease, such as the amount of lease payments, when are they due, who will take care of the

maintenance costs, and so on. The lease arrangement is shown in Figure 13.1.

A leveraged lease involves three parties. The owner of the asset buys the asset by

borrowing the money from a bank, and then rents the asset out to the end user. The owner

of the asset is leveraging the lease and he does not want to tie up his personal capital in

the asset. For example, a property owner may buy an apartment building by borrowing

the money from the bank and then lease the apartments to individuals or families. The

arrangement for a leveraged lease is as shown below.

Bank

Mortgage loan

Loan payments

Landlord

Apartment house

Rent payments

Tenants

Fig. 13.2. In a leveraged lease, three or more parties may be involved.

In the above situation, it is possible to carry out the financial analysis from the point of

view of the bank, or the proprietor, or the tenant. The loan officer at the bank may want

to make sure that the loan is properly secured by examining the physical condition of the

building, the financial strength of the property owner, and assessing his managerial

ability. He should also look at the actual lease contracts, if available.

The property owner is an entrepreneur. He wants to analyze the lease and see if it is

profitable for him. He has to look at all the cash flows: mortgage loan payments, rents,

the maintenance expenses, the real estate taxes, his income tax rate, and the tax benefits

of depreciation of the building. The tenant wants to examine the advantages of renting the

apartment: convenience, predictable rent payments, and his personal preference. He may

also consider the advantages of borrowing the money and buying a house: the

deductibility of interest on the mortgage loan, the freedom to remodel the house, the

needs of his family, and so on. We are interested in the financial analysis only.

13.2 Capital Leases

Frequently corporations sign up long-term leases to make sure that they have access to a

particular asset without interruption. Wal-Mart may lease a store for several years, or a

corporation may lease a computer until it becomes obsolete. A long-term lease is called a

capital lease, or a financial lease, if it satisfies at least one of the following four

conditions:

1. The lessee will become the owner of the asset when the lease expires.

2. The lessee will have the right to purchase the asset at below its then market value at the

expiration of the lease.

3. The term of the lease exceeds 75% of the economic life of the asset.

4. The present value of the lease payments is more than 90% of the initial value of the

asset.

http://en.wikipedia.org/wiki/Leveraged_lease

http://en.wikipedia.org/wiki/Finance_lease

http://en.wikipedia.org/wiki/Finance_lease


221

If the lease does not satisfy any of the above terms, it is then an operating lease. A capital

lease is essentially similar to buying the asset. From the accounting point of view, capital

leases are shown as capital assets on the balance sheet of a corporation.

13.3 Lease Analysis

After calculating the NPV of all cash flows of buying or leasing, one can determine which

method is less expensive. The answer may very well depend upon the corporate tax rate,

the cost of capital, the depreciation schedule, whether or not outside financing is

involved. The lease payments are generally due at the beginning of each period and the

taxes are paid at the end of each period. The lease payments, the depreciation, and the

interest payments are all deductible business expenses for tax purposes.

It is generally accepted that the discount rate used to find the NPV is the after-tax cost of

debt, (1 − t)kd. There are several reasons for it. When a corporation is leasing an asset, it

has to sign a lease contract that requires regular cash payments, and if the corporation is

not making the lease payments on time, the owner has the right to repossess the asset. It

is quite similar to borrowing the money to buy the asset. If the borrower does not pay the

regular installment payments on time, the lender has a right to repossess the asset.

When a company is leasing an asset, it is simply borrowing it. The company has no

ownership right, or an equity interest in the asset. It has to return the asset to the owner at

the end of the lease in good condition. This is similar to borrowing money by selling a

bond where you have to pay the interest on the principal, and pay back the full amount

when the bond matures. The after-tax of cost of debt, the cost associated with a bond,

should also apply to a lease.

The corporations tend to use lease financing an alternative to debt financing. Although

they are not exact substitutes of one another, they play essentially similar roles.

Therefore, for lease analysis, we shall use the after-tax cost of debt, (1 − t)kd as the proper

discount rate.

Examples

Video 13.1 13.1. Alpha Power Corporation needs 25 new vans for their maintenance

crews. The vans cost $25,000 each, and an investment tax credit of 10% is available.

Alpha will depreciate the vans over a 4-year period on a straight-line basis and then sell

them for $5,000 each on the average. The company can borrow money at 9% interest by

making the loan payments at the end of each year for four years. The company expects to

pay income tax at 30% rate for the next 4 years. Beta Leasing Co leases the same vans at

$5,000 each annually for a 4-year period, with the payments made in advance for each

year. Alpha can take the tax benefits of lease payments at the end of each year. Using

NPV method, find the cheaper way to acquire the vehicles.

http://en.wikipedia.org/wiki/Operating_lease



222

The best way to do the problem is to calculate the NPV of leasing and the NPV of buying

a single van, compare the two, and select the cheaper one. The cheaper method is

applicable to one van or 25.

In all leasing problems, we will use the after-tax cost of debt, (1 – t)kd as the proper

discount rate. In this case, it is (1 – .3)(.09) = .063.

Because of the 10% investment credit, the vans will effectively cost 90% of their

purchase price, or, .9(25,000) = $22,500 each. Thus,

Initial investment = −$22,500

Since the company spends only $22,500 on a van, it is able to depreciate that amount

over 4 years. Thus, the depreciation per year is 22,500/4 = $5625. With the tax rate at

30%, the annual tax benefit of depreciation is .3(5625) = $1687.50. The present value of

the tax benefit of depreciation for four years is

PV(tax benefit of depreciation) = i=1

4

1687.50

1.063i =

1687.50(1 − 1.063−4

)

.063 = $5807.42

WRA Sum[.9*25000/4*.3/1.063î,{i,1,4}]

The company will sell the van for $5000 after 4 years. Since the asset is depreciated fully,

the company will pay taxes on the whole amount. The after-tax value is (1 – .3)(5000) =

$3500. Its present value of this amount is

PV(resale value of equipment) = 3500

1.0634 = $2741.16

WRA (1-.3)*5000/1.063^4

To find the NPV of buying a van, add the initial investment, the present value of the tax

benefits of depreciation, and the present value of after-tax final sales value. This gives

NPV(buying) = −22,500 + 5807.42 + 2741.16 = −$13,951.42

WRA -.9*25000+ Sum[.9*25000/4*.3/1.063î,{i,1,4}]+(1-.3)*5000/1.063^4

Now consider the cash flows due to leasing a van. The company has to make four

payments at the beginning of each of the four years. The present value of this is

PV(lease payments) = − 5000 − i=1

3

5000

1.063i = − 5000 –

5000(1 −1.063−3

)

.063 = −$18,291.22

WRA -Sum[5000/1.063î,{i,0,3}]






223

The tax benefit of a lease payment, tL, is .3(5000) = $1500 per year, at the end of each of

the next four years. Their present value is

PV(tax benefit of lease payments) = i=1

4

1500

1.063i =

1500(1 − 1.063−4

)

.063 = $5162.15

WRA Sum[.3*5000/1.063î,{i,1,4}]

Combining the present value of lease payments and their tax benefits, we get

NPV(leasing) = −18,291.22 + 5162.15 = −$13,129.07

WRA -Sum[5000/1.063î,{i,0,3}]+Sum[.3*5000/1.063î,{i,1,4}]

Comparing the NPV(buying) = −$13,951.42 with NPV(leasing) = −$13,129.07, we notice

that leasing is cheaper. ♥

To set it up in Excel, create the following spreadsheet.

A B

1 Purchase price = 25000

2 Investment tax credit = .1

3 Net cost = =B1*(1-B2)

4 Cost of debt = .09

5 Income tax rate = .3

6 Discount rate = =(1-B5)*B4

7 Sale price = 5000

8 Number of years = 4

9 PV of tax benefit of depreciation = =B5*B3/B8*(1-1/(1+B6)^B8)/B6

10 PV of after-tax sales price = =B7*(1-B5)/(1+B6)^B8

11 NPV(buy) = =-B3+B9+B10

12 Lease payment, in advance = 5000

13 PV of lease payments = =-B12-B12*(1-1/(1+B6)^(B8-1))/B6

14 Tax benefit of lease payments = =B5*B12*(1-1/(1+B6)^B8)/B6

15 NPV(lease) = =B13+B14

Video 13.2 13.2. Campbell Company has to decide between leasing a warehouse for

four years at the fixed annual rent of $24,098.04, payable in advance; and buying it

outright for $100,000. If Campbell buys the warehouse, it can borrow the money at 12%

interest for four years. The company will depreciate the warehouse on the straight-line

basis for 10 years, but actually sell it for $30,000 after 4 years. The marginal tax rate of

the company is 35%. Assume that the tax benefits of leasing are not available until the

end of that year. Should Campbell buy or lease the warehouse?

The after-tax cost of debt, the discount rate, is 0.12(1 − 0.35) = 0.078.

The depreciation of the warehouse is $10,000 annually, and therefore its tax benefit is

.35(10,000) = $3500. The book value of the warehouse after 4 years is $60,000. The

company plans to sell it for $30,000, which will create a loss of $30,000. But this loss





224

also has a tax benefit of 30,000(0.35) = $10,500. The final sale brings in $30,000 in cash

and $10,500 in tax benefits for a total of $40,500.

The NPV of the purchase decision is thus

NPV(buy) = −100,000 + i=1

4

3500

1.078i + 40‚500

1.0784 = − $58,365.52

Assume that the company makes the lease payments at the beginning of each year, but

their tax benefits are not available until the end of that year, the NPV of leasing is,

NPV(lease) = − 24,098.04 − i=1

3

24‚098.04

1.078i + i=1

4

24‚098.04(.35)

1.078i

= − 24,098.04 − 62,327.79 + 28,060.33 = − $58,365.50

Since the two costs are exactly alike, the company is indifferent between leasing and

buying the warehouse. ♥

You can use WolframAlpha to verify the answer as follows:

-100000+Sum[3500/1.078î,{i,1,4}]+40500/1.078^4

-Sum[24098.04/1.078î,{i,0,3}]+Sum[.35*24098.04/1.078î,{i,1,4}]

To do the problem in Excel, type in the following instructions:

A B

1 Buying

2 Initial cost 100000

3 Depreciation period 10

4 Holding period 4

5 Tax rate .35

6 Cost of debt .12

7 Discount rate =(1-B5)*B6

8 PV of tax benefit of depreciation =B2/B3*B5*(1-1/(1+B7)^B4)/B7

9 Book value =B2-B4*B2/B3

10 Sales price 30000

11 PV of sales price =(B10-(B10-B9)*B5)/(1+B7)^B4

12 NPV of buying =-B2+B8+B11

13 Leasing

14 Lease payments, given 24098.04

15 PV of lease payments =-B14-B14*(1-1/(1+B7)^(B4-1))/B7

16 PV of their tax benefits =B14*B5*(1-1/(1+B7)^B4)/B7

17 NPV of leasing =B15+B16

13.3. Altair Corporation is considering the alternatives of buying or leasing a new

machine. It may buy the machine for $1 million, use it for five years, and then sell it for

$100,000. Altair will depreciate the machine completely over this period. Altair's after-

tax cost of debt is 12% and its tax rate 30%. The lease will be for five years, with equal



225

payments made in advance each year, with the tax benefits available after one year. What

lease payment will make the cost of leasing equal to the cost of buying?

The after-tax cost of debt for the company is 12%, which is the proper discount rate.

Next, consider the net cost of buying the machine. The cost of the machine is $1 million

and the annual depreciation is $200,000. The annual tax benefit from the depreciation

will be .3(200,000) = $60,000. The final sale price is $100,000, but after taxes it becomes

100,000(1 − .3) = $70,000. Combining these numbers, we get

NPV(buy) = − 1,000,000 + i=1

5

60‚000

1.12i +

70‚000

1.125 = − $743,993.55

Suppose the lease payments are x dollars annually, payable in advance, and their tax

benefits, .3x, are available at the end of each year. Thus

NPV(lease) = − x − i=1

4

x

1.12i +

i=1

5

.3x

1.12i

= x[− 1 − 1 − 1.12

−4

.12 +

.3(1 − 1.12−5

)

.12] = −2.955916486x

Equate the two costs, as

− 743,993.55 = − 2.955916486x

This gives

x = $251,696.40 ♥

Thus, the annual lease payments are $251,696, in advance, which will equate the cost of

leasing to the cost of buying this machine.


-1000000+Sum[.3*1000000/5/1.12î,{i,1,5}]+100000*(1-

.3)/1.12^5=-Sum[x/1.12î,{i,0,4}]+Sum[.3*x/1.12î,{i,1,5}]

13.4. Dallas Corporation has to decide between buying and leasing a computer. If Dallas

buys the computer, it will cost $200,000, but an investment tax credit of 10% is also

available. Dallas will depreciate the computer using a 5 year ACRS, with depreciation of

18%, 33%, 25%, 16% and 8% for the five years respectively. After 5 years, Dallas

expects to sell the computer for $20,000. The tax rate of the company is 40% and its

after-tax cost of debt is 11%. King Leasing Co will lease the same computer to Dallas for

5 years, but will charge the annual lease payments in advance every year. Dallas will get

the tax benefits immediately. Find the lease payment, which will make Dallas indifferent

towards leasing or buying.



226

Let us first find the net cost of buying the computer. The purchase price is $200,000, the

investment tax credit is $20,000, and so the net initial investment is $180,000.

The PV of tax benefits due to depreciation

= 0.4 (180,000)

0.18

1.11 +

0.33

1.112 +

0.25

1.113 +

0.16

1.114 +

0.08

1.115 = + $55,128.13

The PV of after-tax proceeds of the sale = 20‚000 (1 − 0.4)

1.115 = + $7,121.42

Adding these, we get the NPV(buy) = −180,000 + 55,128.13 + 7,121.42 = − $117,750.45

Suppose the lease payments are X dollars, payable in advance, and the tax benefits are

also available at the same time. This gives

NPV(lease) = − (1 − .4)X − i=1

4

(1 − .4)X

1.11i = − .6X −

.6X(1 − 1.11−4

)

.11 = − 2.4614674 X

Equating the two costs, we get

− 2.4614674 X = − 117,750.45

X = the annual lease payment = $47,838 ♥


.4*180000*(.18/1.11+.33/1.11^2+.25/1.11^3+.16/1.11^4+.08/1.11^5)+

20000*(1-.4)/1.11^5-200000*(1-.1)=-Sum[(1-.4)*x/1.11^i,{i,0,4}]

13.5. Titan Trucking Company needs 10 new trucks. Each truck costs $45,000, lasts on

the average 6 years, with no residual value. Titan depreciates the trucks on a straight-line

basis. A 10% investment tax credit is also available. If it buys the trucks, Titan has to pay

$1,000 annually per truck for maintenance. Titan has tax rate of 40%. Hyperion Leasing

Company will lease these trucks to Titan at an annual cost of $10,000 per truck, payable

in advance. Titan can buy the trucks by borrowing money at the rate of 15% per annum.

What is your recommendation to Titan whether to buy or to lease the trucks?

Let us calculate the result for one truck. Because of the 10% investment tax credit, the net

cost of each truck is .9(45,000) = $40,500. The proper discount rate in this case is the

after-tax cost of debt = (1 − 0.4)(0.15) = 0.09. We find the NPV as follows:

NPV(buy) = − 40,500 − i=1

6

1000(1 − 0.4)

1.09i +

i=1

6

(40,500/6)(.4)

1.09i = − $31,079.57

Investment

in the truck

PV of after-tax cost of

maintenance

PV of tax benefits of

depreciation



227

Assuming that the lease payments are payable in advance each year and the tax benefits

are available at the end of the year, we get

NPV(lease) = − 10,000 − i=1

5

10‚000

1.09i +

i=1

6

.4(10,000)

1.09i = − $30.952.80

First lease

payment

PV of remaining

five payments

PV of tax benefits

of lease payments

Comparing the net cost of leasing and buying, we see that it is better to lease the trucks. ♥


-40500-Sum[1000*.6/1.09î,{i,1,6}]+Sum[40500/6*.4/1.09î,{i,1,6}]

-Sum[10000/1.09î,{i,0,5}]+Sum[.4*10000/1.09î,{i,1,6}]

13.6. Dalton Company needs a new computer, which it may buy for $140,000. It will

depreciate it on a straight-line basis over 7 years to zero value, and then sell for $10,000.

Alternately, Dalton may lease the same computer for 7 years with annual lease payments

of $25,000 payable in advance. Dalton can take the tax credit of lease payments

immediately. The tax rate of Dalton is 35%, and its pre-tax cost of debt 10%. Should it

buy or lease?

Using 6.5% as the after-tax cost of debt, set up the problem as follows:

NPV(buy) = − 140,000 + i=1

7

20‚000(.35)

1.065i +

10‚000 (1 − 0.35)

1.0657 = − $97,425.57

Initial

investment

PV of tax benefit of

depreciation

PV of after-tax

resale value

NPV(lease) = − 25,000(1 − .35) − i=1

6

25‚000(1 − .35)

1.065i = − $94,916.47

PV of first lease

payment, after tax

PV of remaining six

payments, after taxes

Comparing the cost of buying and leasing, we see that leasing is the better alternative. ♥

13.7. James Corporation needs a corporate jet for the next four years. It can buy the jet

for $15 million. The company will depreciate the jet on a straight-line basis over the next

15 years, but sell it for $11 million after four years. The cost of debt for the company is

12% and its income tax rate 25%. Calculate the annual lease payment, payable in advance

each year, which will make the cost of leasing equal to the cost of buying. Assume that

the tax benefits of the lease payments are available immediately.

The proper discount rate in this problem is (1 − t)kd = (1 − .25)(.12) = .09. The annual

depreciation is $1 million and its tax benefit tD is $.25 million. Finally, the jet is sold for

its book value, $11 million, and there is no payment of taxes. With amounts in $million,



228

NPV(buy) = − 15 + i=1

4

.25

1.09i +

11

1.094 = −$6.397392709 million

Suppose the lease payment is L and its tax benefit is available immediately, then after

taxes, it becomes (1 − .25)L = .75L. Thus

NPV(lease) = − .75L − i=1

3

.75L

1.09i = −$2.648470999L million

Equating the two costs,

−$2.648470999L = −$6.397392709

which gives L = 2.415504157 = $2.4155 million ♥


-15+Sum[.25/1.09^i,{i,1,4}]+11/1.09^4=-Sum[.75*L/1.09^i,{i,0,3}]

To simplify the leasing problems, we use the words immediately or right away to indicate

that if the annual lease payment L is made at time t = 0, its tax benefit tL is also

calculated at t = 0. This is the case in Examples 13.4, 13.6, and 13.7.

Similarly, the terms a year later, or at the end of the year signify that if the annual lease

payment L is made at time t = 0, its tax benefit tL is available at t = 1. This happens

frequently, as in Examples 13.1, 13.2, 13.3, and 13.5.

It is also possible to do the calculation twice, once with immediate benefits and once with

delayed benefits. The average of these values will give a more realistic result.

We do this to simply the calculations. In real life, the lease payments are on a monthly

basis and their cumulative tax benefit is available after several months. For example, a

company may lease a machine in November 2011, pay $2000 fee in November 2011, and

$1000 monthly lease payments in November and December 2011. The total lease-related

payments are $4000 in 2011. Suppose the income tax rate of the company is 30%. The

company can claim the tax benefit = .3(4000) = $1200 in April 2012 when it files its

income tax return.

Key Terms capital lease, 214, 215

discount rate, 215, 216, 217,

219, 220

financial lease, 214

leasing, 213, 215, 216, 217,

218, 219, 221, 222, 223

lessee, 213, 214

lessor, 213

leveraged lease, 213, 214

operating lease, 215



229

Problems

13.8. Aries Corporation needs a computer, which it can buy for $100,000. Aries will

depreciate the computer uniformly over its useful life of 4 years. An investment tax credit

of 10% is also available, and the computer will have no residual value. Aries plans to

borrow the money at an interest rate of 10% specifically to finance the purchase. The tax

rate of Aries is 40%. Gemini Leasing Corporation can also lease the same computer to

Aries for the same period. Calculate the annual lease payments, made in advance each

year, and their tax benefit taken right away, that will make Aries indifferent to leasing or

buying. $26,687 ♥

13.9. Muskie Corporation needs a new computer, which costs $120,000 depreciating it

on a straight-line basis over its economic life of 5 years. Muskie may be able to sell the

computer at that time for $20,000. The tax rate of Muskie is 35% and it is able to borrow

funds to acquire the computer at annual interest rate of 10%. Vance Corporation can lease

the same computer to Muskie at the annual fee of $25,000, payable in advance for the

next five years. Should Muskie buy or lease the computer, if the tax benefits of leasing

are delayed? NPV(buy) = −$75,604, NPV(lease) = −$74,283, lease. ♥

13.10. Bedford Corporation is planning to lease a machine for the next five years for an

annual lease payment of $2,000 paid in advance, plus a non-refundable initial fee of

$3,000. There is a one-year delay for the tax benefits of leasing. Bedford may buy the

machine, depreciate it fully over the next five years, and then sell it for 10% of the

purchase price. Bedford can borrow the money at 9% interest rate to finance the

purchase, and its tax rate is 40%. Calculate the price of the machine, which will make

purchasing or leasing to be equally costly. $12,207 ♥

13.11. Chandrasekhar Corporation plans to acquire a corporate jet, by either leasing it or

buying it. The five annual lease payments are $245,000 each, payable in advance. The

company can buy the jet by borrowing money at 9% interest. The tax rate of the company

is 30%, and it uses straight-line depreciation. Calculate the price of the jet, which will

equate the cost of buying to the cost of leasing. Assume that the tax benefits of lease

payments are available immediately. $1,016,546 ♥

13.12. Gore Inc is planning to lease a computer for $6,216 per annum, payable in

advance, for a period of 4 years. The lease will cover maintenance expenses. The

president of Gore feels that if he buys the same computer he should be able to sell it at

20% of the purchase price after 4 years. However, in case of purchase, the company must

pay annual maintenance expenses of $800 at the end of each year. The pretax cost of debt

of Gore is 10% and its income tax rate is 35%. If Gore buys the computer, it will

depreciate it fully in four years. What is the maximum price that Gore should pay for this

computer? Assume that Gore can take the tax credit for lease payments (a) immediately,

and (b) a year later. (a) Immediate: $21,629, (b) Delayed: $22,438 ♥

13.13. Violet Ray Inc needs a new computer. They may buy it for $50,000, depreciate it

completely on a straight-line basis for five years, and then sell it for a residual value of


230

$5,000. There will be an investment tax-credit of 10%. The cost of debt of the company

is 10%, and its marginal tax rate is 40%. Orange Computer will also lease the computer

to Violet for 5 years, charging the lease payments in advance each year. For what lease

payments will Violet be indifferent towards buying or leasing? $9925 ♥

13.14. Dammam Overnight Delivery Service would like to acquire 300 vans for its

business. It can buy each van for $30,000, depreciate it completely over 5 years, and then

sell it for $10,000. The tax rate of Dammam is 30%, and its cost of debt is 10%. Dhahran

Rental Company will lease these vans to Dammam for a period of 5 years at the annual

rate of $6,000, paid in advance. Dammam will get the tax benefits of the lease at the end

of each year. Should Dammam buy or lease these vans?

NPV(buy) = –$17,628.74, NPV(lease) = –$18,942.91, buy ♥

13.15. Albany Company needs a new computer that costs $50,000 with expected life of

5 years. Albany will depreciate it completely during this period on a straight-line basis

and then sell it for $10,000. Albany is also considering leasing the same computer for

five years, by paying an annual lease payment in advance, and taking the tax benefits

immediately. Find the amount of the lease payment, which will make the buying or

leasing to be equally costly for the company. The tax rate of Albany is 30%, and its cost

of debt 10%. L = $10,651 annually ♥

13.16. Liberia Company needs a car, which it may lease by paying an initial fee of

$2000, and lease payments for $400 a month in advance for 36 months. The cost of debt

for the company is 12% and its tax rate 25%. The company pays its income tax annually.

Liberia may also buy the car for $15,000, depreciate it fully over five years, but sell it at

some unknown price after three years. Find the selling price of the car that will make

buying and leasing to be equivalent. Should the company buy or lease the car?

$1323.45, buy the car ♥

231

14. INVESTMENT ANALYSIS

Objective: After reading this chapter, you will be able to analyze investment

opportunities, particularly the real-estate investments.

14.1 Real Estate Investments

In this chapter, we shall look at some of the investment opportunities that present

themselves to the corporations and individuals alike. We analyze the situations with the

help of a powerful tool, the NPV analysis. The desirability of an investment depends on

whether its NPV is positive or not. These examples provide an overview of the

investment process.

Investing in real estate is quite popular. Many people buy a house, live in it comfortably,

and then sell it at an appreciated price. We will look at real estate as an investment

opportunity. First, consider a simple example where an investor buys a house, rents it out

for a while, and then sells it at a profit.

To examine this situation analytically, assume that the purchase price of the house is H.

Its selling price after n years is Hn. The profit, Hn − H, is taxable income, and the tax due

at the time of sale is (Hn − H)t, where t is the income tax rate. The cash flow at the time

of sale is thus Hn − (Hn − H)t. = Hn(1 − t) + Ht. Suppose the risk-adjusted discount rate

for such an investment is r. Then the NPV of the investment is

NPV = − H + Hn(1 − t) + Ht

(1 + r)n (14.1)

Equation (14.1) is incomplete because it does not consider depreciation, rental income, or

maintenance expenses. We can make the problem more realistic by renting the house at

the annual rent R, and include the annual maintenance costs M. Assume that R and M are

calculated at the end of the year. The maintenance expenses may also cover the real estate

taxes. The net rental income R − M is taxable income, and its annual after-tax value is

(R − M)(1 − t). At the same time, the homeowner can use the depreciation of the house as

a tax deduction and create an annual benefit of tD, where D is the annual depreciation.

The annual cash flow, C is thus

C = (R − M)(1 − t) + tD.

When the investor sells the house, the capital gain on the sale is the selling price minus

the book value of the house. The book value of the house is given by H − nD. The after-

tax cash flow from the sale of the house is thus

Hn − (Hn − H + nD)t

Analytical Techniques 14. Investment Analysis _____________________________________________________________________________

232

Including rental income, maintenance, and depreciation, (14.1) becomes

NPV = − H + i=1

n

(R − M)(1 − t) + tD

(1 + r)i +


(1 + r)n (14.2)

We have assumed that the depreciation is on a straight-line basis and the income tax rate

for the ordinary income and the capital-gains income is the same. We can make the

model more complete by assuming a different tax rate t for ordinary income and tg for

capital gains. This will make (14.2) to be

NPV = − H + i=1

n

(R − M)(1 − t) + tD

(1 + r)i +

Hn − (Hn − H + nD)tg

(1 + r)n (14.3)

We should modify equation (14.2) if the annual rents are available in advance every year.

It will then become

NPV = − H + i=0

n−1

R

(1 + r)i +

i=1

n

(M + D − R)t − M

(1 + r)i +


(1 + r)n (14.4)

Next, we consider two other important costs: the closing costs when we buy the house

and the selling costs when we sell it. The closing costs include the transfer taxes,

attorney's fees, title insurance, loan origination fees, "points," document preparation fees,

and deed recording fees. The bank may charge a fee, called points, when it approves a

loan. It is usually between 0% and 3% of the amount of loan. In many places, the buyer

and the seller pay the transfer taxes to the local government. At present, the transfer tax

rate in Scranton, PA, is 2%, both for the buyer and the seller.

Let us consider a comprehensive problem about real estate investments. It is instructive to

follow the details of the problem and to modify it to solve simpler problems.

Purchase price of the house = $150,000 (land = $30,000, building = $120,000)

Depreciation schedule = 25 years, on a straight line

Initial rent = $1200 at the end of each month, expected to increase by 5% annually

Maintenance, including real estate taxes = $300 per month, expected to increase by 4%

per year

Expected rate of appreciation of the value of property = 5% per year

Plan to sell the property after 10 years

Borrow 80% of the value of the house and pay 2 points

Fixed closing costs at the beginning of the project = $335

Transfer tax rate = 1.85%

Fixed costs at the time of selling of the house = $100

Realtor's commission = 6%

Ordinary income tax rate = 28%

Capital gains tax rate = 14%

Risk-adjusted discount rate = 10%


233

Is it a worthwhile investment?

Divide the problem into the following simpler problems.

(1) What is the initial investment?

(2) What is the present value of the tax benefits of depreciation?

(3) What is the present value of rental income, including taxes and inflation?

(4) What is the present value of expenses, including taxes and inflation?

(5) What is the present value of final sales price, including taxes and inflation?

(1) To find the initial investment, consider the following expenses:

(a) Purchase price of the house, $150,000

(b) Payment of transfer taxes, .0185(150,000) = $2775

(c) “Points” paid to the bank, .02(.8)(150,000) = $2400

(d) Fixed cost at the time of purchase of the house (title insurance, lawyers fee, deed

recording fee, etc.) = $335

Adding these numbers, we get 150,000 + 2775 + 2400 + 335 = $155,510. The expenses

associated with buying the house, $5510 are tax deductible. Their tax benefit, available at

the end of the first year, is .28(5510) = $1542.80. Its present value = 1542.80/1.1 =

$1402.55. Subtracting it from the total expenses of buying the house, we get 155,510 −

1402.55 = $154,107.45, which is the initial investment. (Negative cash flow) ♦

Let us define: H = purchase price of the house

tt = transfer tax rate

p = points paid to the bank. For 2 points, p = .02

γ = loan-to-value ratio; the amount of mortgage loan divided by the property value

F1 = fixed closing costs at the time of buying the property

r = risk-adjusted discount rate used to discount all cash flows

t = income tax rate

Assume that the tax benefit of the expenses occurs at the end of the year. This gives the

first cash flow C1 as

C1 = − [H(1 + tt + pγ) + F1] + t[H(tt + pγ) + F1]

(1 + r) (14.5)

Using Maple, one can write it as

C1:=-(H*(1+tt+p*gamma)+F1)+t*(H*(tt+p*gamma)+F1)/(1+r);

subs(H=150000,tt=.0185,p=.02,gamma=.8,F1=335,r=.1,t=.28,C1);

(2) Only the building can be depreciated for tax purposes, not the land. The depreciation

per year is 120,000/25 = $4800. Its tax benefit is .28*4800 = $1344. The present value of

this amount over the 10-year holding period, discounted at the rate 10%, is


234

i=1

10

1344

1.10i = $8258.30 (Positive cash flow) ♦

Using the following symbols,

L = value of the land

N = number of years for depreciation

n = holding period of the property

one can write C2 as

C2 = i=1

n

t(H − L)/N

(1 + r)i =

t(H − L)[1 − (1 + r)−n

]

rN (14.6)

In Maple notation, it becomes,

C2:=t*(H-L)*(1-1/(1+r)^n)/r/N;

subs(t=.28,H=150000,L=30000,r=.1,n=10,N=25,C2);

(3) Consider only the first year’s rent. The monthly rent is $1200 and the monthly

discount rate is .1/12. The present value of the first-year rent is thus

i=1

12

1200

(1 + .1/12)i = $13,649.41

The total rent for the first year is 12(1200) = $14,400. Since the taxes are paid at the end

of the year, they are .28(14,400) = $4032. The present value of the taxes is 4032/1.1 =

$3665.45.

The present value of the first year rental income, after taxes, is 13,649.41 − 3665.45 =

$9,983.96.

The rental income will go up by 5% annually for several years, and it will be discounted

by 10% annually. The present value of 10-year rental income, including taxes, and

inflation, is thus

9,983.96 + 9,983.96(1.05/1.1) + 9,983.96(1.05/1.1)2 + ... 10 terms

This is a geometric series, with a = 9,983.96, x = 1.05/1.1, and n = 10. Do the summation

by using (1.4).

9‚983.96[1 − (1.05/1.1)

10]

1 − 1.05/1.1 = $81,706.63 (positive cash flow) ♦

Using the following symbols,


235

R = rent per month, collected at the end of the month

fr = annual rent inflation

we can write C3 as

C3 =

i=1

12

R

(1 + r/12)i −

12tR

1 + r

1 − [(1 + fr)/(1 + r)]

n

1 − (1 + fr)/(1 + r)

Or, C3 = 12R

1 − 1/(1 + r/12)

12

r −

t

1 + r

1 − [(1 + fr)/(1 + r)]

n

1 − (1 + fr)/(1 + r) (14.7)

In Maple notation, it becomes,

12*R*((1-1/(1+r/12)^12)/r-t/(1+r));

C3:=%*(1-((1+fr)/(1+r))^n)/(1-(1+fr)/(1+r));

subs(R=1200,r=.1,t=.28,fr=.05,n=10,C3);

(4) The calculation of the expenses proceeds in a similar way. The present value of the

first-year expenses is

i=1

12

300

(1 + .1/12)i = $3412.35

The annual tax benefit due to expenses is .28(3600) = $1008, and its present value is

1008/1.1 = $916.36. The present value of the first-year expenses, after taxes, is 3412.35 −

916.36 = $2495.99.

Use an inflation rate of 4%, and discount rate of 10% to find the present value of the

expenses for ten years as

2495.99 + 2495.99(1.04/1.1) + 2495.99(1.04/1.1)2 + ... 10 terms

Use (1.4) again to find the sum as

2495.99[1 − (1.04/1.1)10]/(1 − 1.04/1.1) = $19,644.76 (Negative cash flow) ♦

Defining the quantities,

M = monthly expenses, at the end of each month

fm = annual inflation rate for expenses

one can write C4 as,

C4 = −

i=1

12

M

(1 + r/12)i −

12tM

(1 + r)

1 − [(1 + fm)/(1 + r)]

n

1 − [(1 + fm)/(1 + r)]

Or, C4 = − 12M

1 − 1/(1 + r/12)

12

r −

t

(1 + r)

1 − [(1 + fm)/(1 + r)]

n

1 − (1 + fm)/(1 + r) (14.8)


236

In Maple notation, it is

-12*M*((1-1/(1+r/12)^12)/r-t/(1+r));

C4:=%*(1-((1+fm)/(1+r))^n)/(1-(1+fm)/(1+r));

subs(M=300,r=.1,t=.28,fm=.04,n=10,C4);

(5) Assume that the property values are increasing at the rate of 5% per year. After 10

years, the selling price of the house is 150,000(1.05)10

= $244,334.19. The realtor will

take 6% of this amount, namely, .06(244,334.19) = $14,660.05. Since the transfer taxes

are applied to both the buyer and the seller, you have to pay transfer tax again, which

amounts to .0185(244,334.19) = $4520.18. You pay another $100 as fixed costs at

selling.

After paying the realtor, the county transfer taxes, and fixed cost, you get 244,334.19 −

14660.05 − 4520.18 − 100 = $225,053.96. Defining,

fp = rate of appreciation of property values

one could write it as

H(1 + fp)n(1 − c − tt) − F2

In Maple notation, it looks like this

H*(1+fp)^n*(1-c-tt)-F2;

subs(H=150000,fp=.05,n=10,c=.06,tt=.0185,F2=100,%);

The total amount of depreciation for ten years is 10(4800) = $48,000. The book value of

the house is thus 150,000 – 48,000 = $102,000. The capital gain on the house is

225,053.96 – 102,000 = 123,053.96. The tax on the capital gain is .14(123,053.96) =

$17,227.55.

−{H(1 + fp)n(1 − c − tt) − F2 – [H – n(H – L)/N]}tg

-(H*(1+fp)^n*(1-c-tt)-F2-(H-n*(H-L)/N))*tg;

subs(H=150000,fp=.05,n=10,c=.06,tt=.0185,%);

subs(F2=100,L=30000,N=25,tg=.14,%);

The selling expenses on the house (realtor’s fee, transfer taxes, and fixed amount) add up

to 14660.05 + 4520.18 + 100 = $19,280.23. Using this expense as a deduction against

ordinary income, its tax benefit is .28(19,280.23) = $5398.47.

[H(1 + fp)n(c + tt) + F2]t

(H*(1+fp)^n*(c+tt)+F2)*t;

subs(H=150000,fp=.05,n=10,c=.06,tt=.0185,F2=100,t=.28,%);


237

After paying taxes, the cash from selling the house is 225,053.96 − 17,227.55 + 5398.47

= $213,224.87. The present value of this sum is 213,224.87/1.110

= $82,207.42 (positive

cash flow) ♦

C5 = [H(1 + fp)n(1 − c − tt) − F2 −{H(1 + fp)

n(1 − c − tt) − F2 – [H – n(H – L)/N]}tg

+ [H(1 + fp)n(c + tt) + F2]t]/(1 + r)

n (14.9)

In Maple notation, it becomes

H*(1+fp)^n*(1-c-tt)-F2;

%-(%-(H-n*(H-L)/N))*tg;

%+(H*(1+fp)^n*(c+tt)+F2)*t;

C5:=%/(1+r)^n;

subs(H=150000,fp=.05,c=.06,tt=.0185,F2=100,n=10,L=30000,%);

subs(N=25,tg=.14,r=.1,t=.28,%);

Adding all five cash flows, marked by a red diamond ♦, we get

NPV = −154,107.45 + 8258.30 + 81,706.63 – 19,644.76 + 82,207.42

NPV = −$1,579.87

In Maple notation, it is

NPV:=C1+C2+C3+C4+C5;

Considering the NPV, it is not a profitable project.

Examples

14.1. Suppose your marginal income-tax rate is 25%. Assume that 60% of the capital

gains and the first $200 in dividends are tax exempt. The interest paid on borrowed funds

and the transaction costs are tax deductible. On November 1, 2007, you borrow $8,000

from the bank at 7.5% interest rate. You already have $10,000 of your own funds. With

this $18,000, you buy 100 shares of IBM at 90, with a dividend of $1.20 per share. You

invest the remaining funds in PP&L 8s2018 bonds at 90. On November 1, 2008, you

liquidate your portfolio, IBM at 100, and PP&L bonds at 96, and pay off the bank loan.

The total transaction costs are $20 for stock and $50 for bonds. Calculate the after-tax

rate of return on your own funds.

Consider the stock investment first. Money invested in IBM stock = $9000.

Capital gain on IBM, including transaction costs = 10,000 − 9000 − 20 = $980.

Since 60% of the capital gain is tax-exempt, the tax is due on 40% of the gain.

Thus tax due = .4(980)(.25) = $98

Dividends received on IBM stock = $120 (all tax exempt)

After-tax capital gain on stock, plus dividends = 980 + 120 − 98 = $1002 (A)


238

You bought the bonds at 90, that is, 90% of their face value. Thus, you buy $10,000 face

value of bonds for $9000. You sell the bonds at 96, meaning, 96% of their face value.

Capital gain on bonds, including transaction cost = 10,000(0.96 − 0.90) − 50 = $550.

The tax due = .4(550)(.25) = $55.

The after-tax capital gain on bonds = 550 – 55 = $495 (B)

Interest received on bonds = 0.08 (10,000) = $800

Interest paid to the bank = 0.075 (8,000) = $600

Net interest income = 800 − 600 = $200

Tax due on interest income = 200(0.25) = $50

After-tax interest income = 200 − 50 = $150 (C)

Adding (A), (B), and (C), the total dollar return = 1002 + 495 + 150 = $1647

Overall percentage rate of return on your own investment = 1647/10,000 = 16.47% ♥

14.2. You are looking into the possibility of operating a Laundromat. You will rent a

storefront and buy the washing machines and dryers. You expect to generate $20,000

annually in revenue. The rent of the building is $3,000 annually, payable in advance for

the next five years. You will depreciate the machines on a straight-line basis over 5 years,

with no resale value. You are in a 30% tax bracket, and your after-tax required rate of

return is 12%. Assuming that the cash inflows occur at the end of the year, how much

should you pay for the equipment?

To simplify the problem, assume that other expenses such as electricity and water are

zero. To break even, the NPV of this investment is zero, arranged as follows:

Action Present value Equals

1 Buy the Laundromat −x −x

2 PV of after-tax cash from machines i=1

5

20‚000(1 − .3)

1.12i

50,466.86683

3 PV of rent payments, in advance − 3000 −

i=1

4

3000

1.12i

−12,112.04804

4 PV of tax benefit of rent i=1

5

3000(.3)

1.12i

3,244.298582

5 PV of tax benefit of depreciation i=1

5

(x/5)(.3)

1.12i

.2162865721 x

6 NPV Sum of the above = 0

Write the numbers in the last column as an equation,

−x + 50466.86683 − 12112.04804 + 3244.298582 + .2162865721 x = 0

Simplify it to get

.7837134279 x = 41599.11737


239

This gives x = 53,080. You should pay $53,080 for the equipment. ♥

You may do the problem on WolframAlpha as follows.

-x+Sum[20000*(1-.3)/1.12î,{i,1,5}]-

Sum[3000/1.12î,{i,0,4}]+Sum[3000*.3/1.12î,{i,1,5}]+Sum[x/

5*.3/1.12î,{i,1,5}]=0

To do the problem using Maple, enter the following instructions:

-I0;

%+sum(20000*(1-.3)/1.12î,i=1..5);

%-sum(3000/1.12î,i=0..4);

%+sum(3000*.3/1.12î,i=1..5);

%+sum(I0/5*.3/1.12î,i=1..5)=0;

solve(%);

To do the problem using Excel, set up the spreadsheet as follows. Adjust the value of the

initial investment in cell B2 until the NPV in cell B15 becomes close to zero.

A B C D E F G

1 Time, years 0 1 2 3 4 5

2 Initial investment, $ 53080

3 Project life, years 5

4 Cost of capital 0.12

5 Income tax rate, t 0.3 1 − t =1-B5

6 Cash from machines, $

20000 =C6 =C6 =C6 =C6

7 After-tax cash from machines, $

=C6*D5 =D6*D5 =E6*D5 =F6*D5 =G6*D5

8 Annual rent, $ 3000 =B8 =B8 =B8 =B8

9 Depreciation for year =B2/B3 =B2/B3 =B2/B3 =B2/B3 =B2/B3

10 Tax benefit of depreciation

=B5*C9 =B5*C9 =B5*C9 =B5*C9 =B5*C9

11 Tax benefit of rent =B5*B8 =B5*B8 =B5*B8 =B5*B8 =B5*B8

12 Total cash flow =-B2-B8 =C7-C8+C10+C11

=D7-D8+D10+D11

=E7-E8+E10+E11

=F7-F8+F10+F11

=G7+G10+G11

13 Discount factor 1 =1/(1+B4) =1/(1+B4)^2 =1/(1+B4)^3 =1/(1+B4)^4 =1/(1+B4)^5

14 PV of cash flows =B12*B13 =C12*C13 =D12*D13 =E12*E13 =F12*F13 =G12*G13

15 NPV =SUM(B14:G14)

Video 14.3 14.3. Elbridge Gerry is planning to buy a house and rent it out for the next 5

years, collecting an annual rent of $6,000 in advance each year. He is in the 22% tax

bracket. He will depreciate the house on a straight-line basis for 25 years. Gerry plans to

sell the house after 5 years at a price that will be 20% higher than the purchase price,

taking the profit as a long-term capital gain. Assume 60% of such capital gains are tax

exempt. The after tax cost of capital for Gerry is 8%. Ignore the maintenance expenses of

the house and real estate taxes. How much should Gerry pay for the house to break even?

Suppose the purchase price of the house is H. We can find the NPV by considering these

factors and their respective present values:

(1) The initial investment = − H

http://134.198.33.115/hussain/camtasia/Prob1402/Prob1402.html


240

(2) PV of rents, in advance annually = 6000 + i=1

4

6000

1.08i = $25,872.76

(3) Assume that the taxes are due at the end of each year at the rate of 22% of rents.

PV of taxes paid on the rental income = − i=1

5

6000(.22)

1.08i = − $5270.38

(4) The annual depreciation is H/25.

The PV of tax benefits of depreciation = i=1

5

(H/25)(.22)

1.08i = .03514 H

(5) The sale price of the house is 1.2H. Gerry has already taken 20% of the depreciation

of the house leaving its book value to be .8H. The profit on the sale is (1.2H − .8H). Only

40% of it is taxable at the rate of 22%.

PV of after-tax sales price = 1.2H − (1.2H − .8H)(.4)(.22)

1.085 = .7927433078 H

NPV is the sum of all these figures, and to break even, it should be zero. Thus

− H + 25872.76104 − 5270.377249 + .03513584833 H + .7927433078 H = 0

Solving for H, we have H = $119697.2041, or approximately, $119,700 ♥

We can also do the problem by using (14.4)

NPV = − H + i=0

n−1

R

(1 + r)i +

i=1

n

(M + D − R)t − M

(1 + r)i +


(1 + r)n (14.4)

We have to modify the equation slightly to accommodate the following changes:

(1) There are no maintenance expenses, which makes M = 0.

(2) The tax benefit of depreciation is available at the end of year 1-5. He pays the taxes

on rental income at the end of the year.

(3) Since 60% of capital gain is tax exempt, we should take 40% of the capital gain, and

pay taxes on it at the rate of 22%.

Incorporating these changes, equation (14.3) becomes

NPV = − H + i=0

n−1

R

(1 + r)i +

i=1

n

(D − R)t

(1 + r)i +

Hn − (Hn − H + nD)(.4)t

(1 + r)n

Now, put n = 5 years, R = $6000, t = .22, r = .08, Hn = 1.2H, D = H/25, and NPV = 0.

This gives


241

− H + i=0

4

6000

1.08i +

i=1

5

.22(H/25 − 6000)

1.08i +

1.2H − (1.2H − H + 5H/25)(.4)(.22)

1.085 = 0

The WolframAlpha instruction to solve the equation is as follows.

-H+Sum[6000/1.08î,{i,0,4}]+Sum[.22*(H/25-

6000)/1.08î,{i,1,5}]+(1.2*H-(.2H+H/5)*.4*.22)/1.08^5=0

The result is $119,697.

To do the problem on Excel, set up the spreadsheet as follows. Adjust the value of the

house in cell B2 until the NPV in cell B18 is almost zero.

A B C D E F G

1 Time, years 0 1 2 3 4 5

2 Purchase price of house, $

119700 Depreciable life, years

25 Price appreciation

0.2

3 Project life, years 5

4 Cost of capital 0.08

5 Income tax rate, t 0.22 1 − t =1-B5

6 Rent, $ 6000 =B6 =B6 =B6 =B6

7 Tax on rent, $ =B5*B6 =B5*B6 =B5*B6 =B5*B6 =B5*B6

8 After-tax rent, $ =B6 =C6-C7 =C6-C7 =C6-C7 =C6-C7 =-C7

9 Depreciation for year =B2/D2 =B2/D2 =B2/D2 =B2/D2 =B2/D2

10 Tax benefit of depreciation

=B5*C9 =B5*C9 =B5*C9 =B5*C9 =B5*C9

11 Sale price of house, $ =(1+F2)*B2

12 Book value of house =B2-B3*B2/25

13 Capital gain =G11-G12

14 Tax on capital gain =0.4*B5*G13

15 Total cash flow =-B2+B8 =C8+C10 =C8+C10 =C8+C10 =C8+C10 =G8+G10+G11-G14

16 Discount factor 1 =1/(1+B4) =1/(1+B4)^2 =1/(1+B4)^3 =1/(1+B4)^4 =1/(1+B4)^5

17 PV of cash flows =B15*B16 =C15*C16 =D15*D16 =E15*E16 =F15*F16 =G15*G16

18 NPV =SUM(B17:G17)

To do the problem using Maple, we let

-H+sum(6000/1.08î,i=0..4);

%-sum(.22*6000/1.08î,i=1..5);

%+sum(.22*H/25/1.08î,i=1..5);

%+(1.2*H-(1.2*H-.8*H)*.4*.22)/1.08^5=0;

solve(%);

14.4. Ralph Boston plans to buy a house and rent it out for a period of 5 years collecting

an annual rent of $6,000 in advance each year. He will sell the house after five years and

he believes that the price of the house will appreciate at the compound rate of 6% per

annum. The maintenance and real estate taxes, paid at the end of each year, will be

$1,000 annually. Boston's income tax rate is 30%, and his after-tax cost of capital is 12%.

He will depreciate the house on a straight-line basis over a 20-year period. How much

should he pay for the house so that the NPV of this project is $5,000?

Suppose the purchase price of the house is H.



242

Increasing at the rate of 6% annually, its value after 5 years will become 1.065H =

1.338225578H.

The accumulated depreciation for 5 years will be 5(H/20) = .25H.

The book value of the house will be H − .25H = .75H.

The capital gain on the house will be (1.338 − .75)H = .5882H.

The tax on this capital gain will be .3(.5882H) = .1765H.

The after-tax proceeds from the sale will be (1.338 − .1765)H = 1.162H, with present

value = (1.162/1.125)H = .6592H. Write the cash flows in a table:

Action Present value of cash flow Equal to

1 Buy the house −H −H

2 Tax benefit of depreciation i=1

5

.3H/20

1.12i

.05407H

3 Rent for 5 years, in advance 6000 +

i=1

4

6000

1.12i

24,224

4 Tax on rental income −

i=1

5

.3(6000)

1.12i

-6489

5 Maintenance expenses, after taxes −

i=1

5

(1000)(1 − .3)

1.12i

-2523

6 Sell the house 1.065H − [1.06

5H − (H − .25H)](.3)

1.125

.6592H

7 Required NPV Sum of the above 5000

Adding (1) through (7) and equating it to (8), we get

− H + .05407H + 6000 + 18224 − 6489 − 2523 + .6592H = 5000

Solve for H, H = $35,617.70 ♥


-H+Sum[6000/1.12î,{i,0,4}]+ Sum[(.3H/20-.3*6000-1000*(1-

.3))/1.12î,{i,1,5}]+(1.06^5-(1.06^5-(1-

.25))*.3)*H/1.12^5=5000

The Maple code to solve the problem is as follows.

-H+sum(.3*H/20/1.12î,i=1..5)+6000+sum(6000/1.12î,i=1..4);

%-sum(.3*6000/1.12î,i=1..5)-sum(1000*(1-.3)/1.12î,i=1..5);

%+(H*1.06^5-.3*(H*1.06^5-.75*H))/1.12^5=5000;

solve(%);



243

14.5. Hart Co intends to buy an office building for $2 million and depreciate it on a

straight-line basis over a 20-year period. However, Hart plans to sell the building after 5

years for $2.5 million. The tax rate of the company is 30% and its WACC is 8%. What is

the minimum acceptable rental income (after paying expenses) calculated at the end of

each year? The capital gains are fully taxable.

Write the cash flows in $million. Include the initial investment, PV of tax benefit of

depreciation, PV of after-tax rental income, PV of after-tax resale value of building. To

break even, set the NPV equal to zero. Arrange the items in a table.

Action Present value of cash flow Equal to

1 Buy the building −2 −2

2 Tax benefit of depreciation i=1

5

.3(2/20)

1.08i

.1197813011

3 After-tax rental income i=1

5

R(1 − .3)

1.08i

2.794897026 R

4 Sell the building 2.5 − [2.5 − (2 − 5*2/20)](.3)

1.085

1.497283033

5 NPV Sum of the above 0

− 2 + .1197813011 + 2.794897026 R + 1.497283033 = 0

This gives R = .1370124418 = $137,012 annually ♥


-2+Sum[(.3*2/20+R*(1-.3))/1.08î,{i,1,5}]+(2.5-(2.5-(2-

5*2/20))*.3)/1.08^5=0

The Maple code for this problem is as follows:

-2+sum(2/20*.3/1.08î,i=1..5);

%+sum(R*(1-.3)/1.08î,i=1..5);

%+(2.5-(2.5-2+5*2/20)*.3)/1.08^5=0;

solve(%);

14.6. Binghamton Company is planning to buy ATM machines and install them

nationwide inside supermarkets. Each machine costs $6,000 and it will be depreciated on

a straight-line basis over 5 years, although the useful life of each machine is expected to

be 10 years. Binghamton will charge the users $1 per withdrawal. The company expects

the expenses to be as follows: annual rent to the supermarket $3,000, payable in advance

each year; maintenance, insurance, and service $2000 per year; and 25 cents per

transaction to the banks whose cards are used at the machine. The income tax rate of

Binghamton is 30% and its cost of capital 12%. Find the minimum transactions per year

to break even. Realistically, is it a good project?



244

Suppose x is the number of annual withdrawals per machine to break even. Write an

equation with the following items in it:

(1) Initial investment

(2) PV of tax benefit of depreciation

(3) PV of after-tax transaction fees collected

(4) PV of after-tax rent paid, and

(5) PV of after-tax maintenance costs

Set the NPV equal to zero. This gives us

NPV = − 6000 + i=1

5

.3(6000/5)

1.12i +

i=1

10

(1 − .25)(1 − .3)x

1.12i

– 3000(1 − .3) − i=1

9

3000(1 − .3)

1.12i −

i=1

10

2000(1 − .3)

1.12i = 0

NPV = − 6000 + 1297.719433 + 2.966367090 x – 13,289.32456 − 7910.312240 = 0

It gives x = 8732 withdrawals/year ♥


-6000+Sum[.3*6000/5/1.12î,{i,1,5}]- Sum[3000*(1-

.3)/1.12î,{i,0,9}]+ Sum[((1-.25)(1-.3)*x-2000*(1-

.3))/1.12î,{i,1,10}]=0

The Maple solution is as follows:

-6000;

%+sum(.3*6000/5/1.12î,i=1..5);

%+sum((1-.25)*(1-.3)*x/1.12î,i=1..10);

%-3000*(1-.3)-sum(3000*(1-.3)/1.12î,i=1..9);

%-sum(2000*(1-.3)/1.12î,i=1..10);

solve(%=0);

8732 withdrawals per year are equal to 8732/365 = 24 transactions per day. It may work

out as a reasonable project. ♥

14.7. Ardmore Corporation wants to set up a car wash. It will buy the land for $50,000,

build a building for $150,000, and buy the car-wash equipment for $50,000. The machine

will last for 5 years with no resale value. The company uses straight-line depreciation.

Ardmore will depreciate the building over a 20-year period, but will sell the building and

land for $200,000 after 5 years. The WACC for Ardmore is 12% and its income tax rate

30%. The company plans to charge $4.00 per car. Ardmore estimates per car expenses as,

electricity 25¢, water 25¢, detergent 25¢, labor 50¢. Assume that all revenues and



245

expenses are available at the end of each year. In order to break even, how many cars

should Ardmore wash per year?

Including machinery, land, and building, the initial investment in the car-wash business is

$250,000. (1)

The depreciation for the building per year is 150,000/20 = $7500, and for equipment

50,000/5 = $10,000, with a total of $17,500 per year. You cannot depreciate land. The

present value of the tax benefit of depreciation for 5 years is

= i=1

5

.3(17,500)

1.12i = 18,925.07506 (2)

Suppose the company washes x cars per year to break even. The profit per car is (4.00 −

.25 − .25 − .25 − .50) = $2.75. After taxes it becomes 2.75(1 − .3) = $1.925. The present

value of after-tax income for 5 years = i=1

5

1.925x

1.12i = 6.939194190x (3)

The original value of building and land was $200,000. The total amount of depreciation

taken on building and land is 7500*5 = $37,500. Its book value is thus 200,000 − 37,500

= $162,500. The taxable profit on building and land is 200,000 − 162,500 = $37,500. The

taxes due are .3(37,500) = $11,250. The company sells the building and land for

$200,000.

PV of after-tax sales = 200‚000 − 11‚250

1.125 = 107,101.8190 (4)

Combine the cash flows (1) through (4), and set the total equal to zero to break even.

Thus

−250,000 + 18,925.07506 + 6.939194190 x + 107,101.8190 = 0

Solve for x, x = 17,866 ♥

The car-wash must wash 17,866 cars every year to break even. If it is open 300 days a

year, it should have 17,866/300 = 60 customers every day. It is unlikely to have that kind

of traffic.

To verify the answer at WolframAlpha, try this:

-250000+Sum[.3*(150000/20+50000/5)/1.12^i,{i,1,5}]+Sum[2.75*(1-

.3)*x/1.12^i,{i,1,5}]+(200000-.3*(200000-(200000-

5*7500)))/1.12^5=0



246

Problems

14.8. John Fulton is planning to buy a house for $50,000 by borrowing money at the rate

of 9%. He expects to rent the house for 5 years, collecting $4,000 annual rent in advance

each year. He thinks that he can sell the house for $55,000 after five years. Fulton has

income tax rate of 40%. He will have to pay $2,000 annually in maintenance and real

estate taxes, and he will depreciate the house on a straight-line basis for 20 years. The

risk-adjusted discount rate in this project is 10%. If all the expenses are fully deductible,

and all gains are taxable, should he undertake this project? No, NPV = −$10,340 ♥

14.9. Texas Company is interested in buying a house and renting it out for $6000 a year,

collecting the rent in advance each year. It will depreciate the house over 25 years, but

sell it after 15 years at twice its purchase price. The maintenance expenses and real estate

taxes, at the end of each year, are $1000 annually. The after tax cost of capital for Texas

is 10% and its income tax rate 25%. Find the price of the house that Texas should pay so

that it can make $5000, in current dollars, from this project. $51,924 ♥

14.10. John Lewis is looking into the possibility of buying several coin-operated vending

machines and placing them in the local hospitals. Each machine costs $2000, which he

will depreciate on a straight-line basis over 8 years. The machine will dispense Coke cans

at 75 cents each and Coca Cola Company will replenish them at 40 cents each. Each

machine is expected to sell 1500 cans a month. The hospitals will provide the space and

electricity for the machines for $200 a month at the end of every month. The tax rate of

John Lewis is 25% and the after tax cost of capital 12%. Assume that the income and

bills occur at the end of each month, but the taxes are paid annually. Should John Lewis

get into this venture? NPV = $13,464, yes ♥

14.11. Bahrah Corporation is interested in buying a 50-unit apartment complex, and

renting out the individual apartments at $600 per month. The tenants pay their own heat

and utilities. The company estimates the land value as 20% of the total value of the

complex. Bahrah expects to sell the property after 10 years at a price that is 50% higher

than its current value. Bahrah will depreciate the buildings on a straight-line basis over 25

years. Bahrah estimates maintenance expenses, including real estate taxes, to be 20% of

the rental income. Assume that all cash flows occur at the end of each year. The discount

rate for this investment is 12%, and the tax rate of Bahrah is 30%. Calculate the price that

Bahrah should pay for the complex just to break even. $2,101,620 ♥

14.12. Ace Car Rental plans to start its business by buying 10 cars at the average price of

$18,000 each, depreciating them completely over 5 years using the straight-line method.

It will rent space in a parking lot for $300 a month, paying the rent in advance each

month. Ace expects that it will rent five cars on an average day, charging $40 per day per

car. The maintenance expense for each car is $60 a month. After 5 years, Ace will sell the

cars at 40% of the original value. Ace receives all the income and pays all the bills,

(except rent) at the end of each month, but it pays the taxes once a year. Its income tax

rate is 25% and it will use 12% as the discount rate. Assume that there are 30 days in a

month. Is it a worthwhile project for Ace? Yes, NPV = $57,067 ♥


247

14.13. Barquisimeto Corporation is interested in buying a 50-room motel that costs $1.5

million. The value of the building is $1 million and that of the land $500,000. It will

depreciate the building over 20 years using the straight-line method. However,

Barquisimeto expects to sell the property after 5 years for $2 million. The average

occupancy rate in the motel is 80%, that is, on the average only 40 rooms are rented out

daily. The cost of cleaning a room after the use by a guest is $12 (maid-service, laundry,

etc.) The total bill for major expenses (maintenance, heating, electricity, real estate taxes,

etc.) for the motel is estimated to be $60,000 annually. The salary of the manager is

$40,000 annually. The tax rate of Barquisimeto is 30%, and the proper discount rate for

this project is 12%. Find the daily room rent to break even. Assume that there are 365

days in a year. $30.76 ♥

14.14. Judy Garland is planning to open a stall at the local mall, paying $2500 rent, in

advance each month. She will buy $25,000 in costume jewelry as the initial inventory,

and buy the display cases for $4000. She expects that the average sales per month will be

$15,000, of which she will use $5000 to replenish her inventory. She will hire an assistant

for $2000 a month. At the end of five years, she plans to sell the entire business,

including inventory and display cases, for $30,000. Her income tax rate is 25% and she

will use a discount rate of 12%. Assume that all the cash flows occur at the end of the

month, except rent, and the taxes are due at the end of the year. Ignore depreciation and

capital gains. Is it a worthwhile project? Yes, NPV = $174,673 ♥

Key Terms

appreciation, 226, 230, 235

closing costs, 226, 227

deed, 226, 227

discount rate, 225, 226, 227,

228, 229, 240, 241

investment, 225, 226, 227,

231, 232, 233, 235, 237,

238, 239, 240

points, 226, 227

real estate, 225, 226, 233,

235, 240, 241

title insurance, 226, 227

transfer taxes, 226, 227, 230

248

15. REVIEW PROBLEMS

15.1. Alger Corp wants to buy a bulldozer for $50,000, which has a useful life of 4 years

with no salvage value. Alger uses straight-line depreciation, has a tax rate of 30%, and an

after-tax cost of capital of 10%. The machine will generate a pretax income of $16,000

for the first year, but this figure will decline by 5% annually for the remaining three

years. Should Alger buy this machine? NPV = −$4,985, don't buy ♥

15.2. Bierce Company stock is not paying any dividend at present. However, you feel

that it may start paying an annual dividend of $1 after one year (probability is 30%). If it

does not start the dividend after one year then it may start the dividend after two years,

with a probability of 50%. It will certainly start paying dividends after three years. Once

the dividend start, they will continue uniformly in the future. If your required rate of

return is 15%, how much should you pay for a share of Bierce stock? $5.79 per share ♥

15.3. Togo Company follows the residual theory of dividends. It expects to have $45

million in earnings after taxes next year, and $45 million in profitable projects at that

time. Togo must maintain its optimal capital structure with debt-to-assets ratio at 45%.

The company has 4.5 million shares of common stock. Find the expected dividend per

share next year. $4.50 ♥

15.4. You have developed the following information about two stocks, whose correlation

coefficient is 0.5:

Name E(R) σ(R)

Burroughs Corp 14% 20% 0.85

Chandler Corp 16% 23% 1.10

(A) Make a portfolio that simulates the market. Find the expected return on the market,

E(Rm), and the standard deviation of this return, σ(Rm). E(Rm) = .152, σ(Rm) = .191 ♥

(B) Find the probability that the actual return on the market is more than 20%. 40.08%. ♥

15.5. Annapolis Corporation needs a computer, which costs $100,000. Annapolis will

depreciate it completely on a straight-line basis over 5 years, and then sell it for $5,000.

Alternatively, it may lease the same computer for 5 years, paying the lease payments in

advance each year and taking their tax benefits at the end of each year. The cost of debt

to Annapolis is 15%, and its income tax rate is 30%. Find the amount of annual lease

payments that will make the cost of buying equal to that of leasing the computer.

L = $25,031 ♥

15.6. Columbus Corporation needs $12 million in new capital, which it may acquire by

selling 12% coupon bonds at par, or by selling stock at $14 net per share after paying the

flotation costs. At present Columbus has 2 million shares of common and $10 million

face amount of bonds with 11% coupon. After the new financing, Columbus expects to

Analytical Techniques 15. Review Problems _____________________________________________________________________________

249

have an EBIT of $4 million, with a standard deviation of $1.5 million. Which method of

financing is better, if the objective is to maximize the EPS of the company? What is the

probability that you have made the right choice? What is the probability that the company

may default on its interest payment, if it uses the method decided above?

Use stock, P(being right) = 89.74%, P(default) = 2.66% ♥

15.7. Austin Co has the following capital structure: $56 million (face value) of bonds

with coupon of 9%, maturing after 11 years, selling at 79; $40 million, 12% coupon

bonds, selling at par; 36 million shares of common selling at $6 each; and 1 million

shares of preferred stock which pays a dividend of $2.50 and sells at $17 each. The beta

of the common stock is 1.24, the expected return of the market is 16%, and the riskless

rate is 9%. The tax rate of Austin is 25%. Find the WACC of Austin. WACC = 15.24% ♥

15.8. Albany Corp has zero coupon bonds maturing after 15 years. The overall value of

the firm is 3 times the face value of the bonds. The company has sigma of .25 and the

riskless rate is 8%. Using the option pricing theory, find the market value of a $1000

bond. B = $298.74 ♥

15.9. Richmond Corporation needs a new machine, which is expected to add $15,000

annually to the EBIT of the company, with a standard deviation of $3,000. The machine

will run for 5 years. Richmond will depreciate it on a straight line with no salvage value.

The tax rate of Richmond is 30% and its after-tax cost of capital is 12%. If the cost of

machine is $50,000, find the probability that it will be a profitable investment.

NPV = $1335.52, P(NPV>0) = 43%, reject ♥

15.10. Aquascutum Company is planning to acquire Austin Reed using its existing

capital. Both companies sell men's clothing. Aquascutum currently has $40 million in

debt at an average rate of 8%, and $60 million in equity. The beta of Aquascutum is 1.35,

its tax rate is 35%, the expected return on the market is 13%, and the riskless rate is 7%.

The additional earnings (before taxes) due to this acquisition will be $1 million for the

first year, and they will rise 3% annually thereafter. Find the price of Austin Reed that

Aquascutum should pay. $7,985,285 ♥

15.11. Dickins & Jones, Inc is a company formed by two partners. Dickins owns the

entire stock of the company, but Jones has lent some money to the company with the

understanding that the company will pay him $100,000 after 5 years. The σ of the

company is 0.4, and the riskless rate is 7%. Dickins and Jones are interested in selling the

company to a buyer for $200,000. Based on option pricing theory, what is an equitable

distribution of the proceeds of the sale between Dickins and Jones?

Dickins $135,500, Jones $64,500 ♥

15.12. Comcast Corporation has the following capital structure: $40 million face amount

of 4% bonds due in 10 years, selling at 60; 4 million shares of common stock selling at

$16 each; and 1 million shares of preferred stock selling at $15 a share. The β of Comcast

is 1.35, the riskless rate is 6%, and the expected return on the market is 12%. The tax rate


250

of Comcast is 30%, and the cost of capital for the preferred stock is exactly halfway

between that of debt and equity. Find the WACC of Comcast. 12.15% ♥

15.13. Christies has debt/assets ratio of 30%, but the management believes that it should

be raised to 40%. However, the additional debt will add $5 million to the bankruptcy

costs of the company. The current value of Christies is $77 million, and its tax rate is

35%. Should the company move to higher debt level?

New V = $74.32 million, stay at 30% level ♥

15.14. Harrod's is a department store with $50 million in debt and $60 million in equity.

Its tax rate is 40%, cost of debt 8%, and beta 1.35. The riskless rate is 6% and the

expected return on the market 12%. Harrod's would like to start a limousine service using

its existing capital. Lillywhite provides only limousine service. Lillywhite has $1 million

in debt and $4 million in equity, with tax rate of 30% and beta 1.2. Find the required rate

of return for Harrod's in the new venture? 10.46% ♥

15.15. Selfridges Corporation is in need of a corporate jet, which costs $5 million. The

jet has a useful life of 10 years. Selfridges will depreciate it on a straight line. The tax rate

of Selfridges is 30%, and its cost of debt is 10%. It may also lease the jet by making a

certain lease payment in advance each year and take the tax credit of the lease payments

immediately. Find the annual lease payments which will make leasing or buying to be

equal in cost. $750,184 ♥

15.16. You have assembled the following information about two stocks:

Stock E(R) σ(R)

Harrison Co. 0.8 12.4% 0.15

Johnson Co. 1.6 18.8% 0.3

The correlation coefficient between the companies is 0.6. Find riskless rate and the

expected return on the market. Construct a portfolio with β = 1 and find its σ(Rp).

r = .06, E(Rm) = .14 and σ(Rp) = .1685 ♥

15.17. Marks & Spencer is planning to get a new machine for $18,000. The expected

income from the machine is $4,000 annually, with the standard deviation of $2,000. The

machine will run for 6 years and M&S will depreciate it on a straight line. The tax rate of

M&S is 30%, and the after tax cost of capital 9%. Find the probability that this machine

will be profitable. 41.16% ♥

15.18. Sixty-five years old Ashley Taylor has received $300,000 as a lump sum pension

settlement. She has invested the money in an account that pays 6% interest per annum,

compounded monthly. Ashley plans to withdraw $2,000 per month from this account.

The first withdrawal will be after one month. How long will it be before this money is

exhausted? If Ashley expects to live another 17 years, with a standard deviation of 7

years, what is the probability that the money will be used up during her life? 18.93% ♥


251

15.19. You are considering the following two bonds in 2011: Moosic Corporation 8.5s30

selling at 95 and Duryea Company 8.75s33 selling at 99. Both the bonds are rated by

S&P as BBB. Your required rate of return is 9%. Which bond(s) will you buy? First ♥

15.20. Wyoming Coal Company is interested in buying a machine for $40,000, which it

will depreciate uniformly over a four-year period. An analysis of the life expectancy of

such machines reveals that 30% break down after 3 years, 60% run for 4 years, and 10%

last for 5 years. The tax rate of Wyoming is 35% and its cost of capital is 9%. If the

machine can generate $10,000 per year in pretax earnings, should Wyoming buy it?

No, NPV = $8,494.83 ♥

15.21. Pittston Corporation is considering the following three projects:

Project Investment E(R) σ(R) Correlation coefficient

A $10,000 0.15 0.20 A, B = 0.3

B $20,000 0.14 0.18 A, C = 0.5

C $50,000 0.12 0.16 B, C = 0.7

Pittston may take one, or two, or all three projects. The company wants to maximize the

ratio E(R)/σ(R). What do you recommend? Take A and B together. ♥

15.22. Ghana Company expects to have $40 million in EBIT next year, with standard

deviation $10 million. The company has $100 million in long-term bonds with coupon

9%, and it has to pay $6 million in preferred dividends. Ghana has dividend payout ratio

40%, and 10 million shares of common stock. The income tax rate of the company is

35%. Find the probability that the dividend next year is more than 60¢ per share.

44.80% ♥

15.23. Mountaintop Corporation expects to pay a dividend of $3.00 next year, $3.25

after two years, $3.50 after three years, and $3.75 after the fourth and subsequent years.

The required rate of return for the stockholders is 16%. Find the price of the stock now,

and just after the payment of the first $3.00 dividend. $22.26, $22.82 ♥

15.24. Churchill Corporation has total value of $100 million. It has $50 million (face

amount) of zero-coupon bonds outstanding in 2008, which will mature in 2030. The

riskless rate is 6% at present. The risk of Churchill measured in terms of its σ is .30.

Using option pricing theory, find the debt/assets ratio of Churchill. B/V = .1188 ♥

15.25. Attlee Company stock has 25 million shares of stock selling at $15 each. The β of

stock is estimated to be 1.25. Attlee also has $625 million (face amount) of zero-coupon

bonds maturing after 25 years. These bonds are selling at $150 each. The riskless rate is

6%. The expected return on the market is 12%. The income tax rate of Attlee is 30%.

Find the weighted average cost of capital of Attlee. 11.90% ♥

15.26. Heath, Inc. has $45 million in long-term bonds, selling at par. It also has 3 million

shares of common stock priced at $25 each. The bankruptcy costs of the company are


252

estimated to be $10 million. The tax rate of the company is 30%. Mr. Heath has proposed

that the company should reduce its debt/assets ratio to 30% by selling some stock and

buying back bonds from the proceeds. The new capital structure will also reduce the

bankruptcy costs to $5 million. Is it a good proposal? What is the new debt and equity of

the company after implementing this proposal?

Yes. New debt = $36.758 million, new equity = $85.769 million ♥

15.27. Thatcher Airlines is estimated to be 1.5. Thatcher has $700 million in equity

and $300 million in debt and its tax rate is 30%. Major Airlines has $800 million in

equity and $400 million in debt and has income tax rate 25%. The riskless rate is 7% and

the expected return on the market is 12%. Find the cost of equity capital for Major.

14.93% ♥

15.28. Chamberlain Leasing Corporation would like to buy a computer for $125,000,

depreciate it fully over a five year period, and then sell it for $10,000. Chamberlain can

lease the computer to Macmillan Company for 5 years, charging them $30,000 annually,

in advance each year. The maintenance expenses, $3,000 annually, will be paid by

Chamberlain. The tax rate of Chamberlain is 30%, and the cost of debt 10%. Will this

lease be profitable to Chamberlain? NPV = − $3,153.47, no ♥

15.29. Baldwin Company is interested in buying a new corporate jet for $6 million. It

will depreciate the jet fully in 5 years and then sell it for $5 million. The jet will use

$60,000 in fuel annually, and its maintenance will be $40,000 annually. The tax rate of

Baldwin is 35% and its WACC 10%. Find the minimum annual savings generated by the

jet to justify its purchase. $1.167 million ♥

15.30. Callaghan Company stock is selling at $45 a share and its price/earnings ratio is

15. The P/E ratio is based on current price and the earnings for the last 12 months. The

earnings for next year is uncertain. It is expected to be $4 a share, with a standard

deviation of $1. Assuming that the stock maintains its current P/E ratio, find the

probability that Callaghan stock will be selling for more than $55 a share next year.

63.06 % ♥

15.31. Turin Corporation is borrowing $200,000 from the bank with the understanding

that the loan will be paid off in 12 monthly installments, and the interest will be

calculated at the rate of 12% per annum on the unpaid balance. Turin has decided to

structure the payments so that they will increase at the rate of 4% every month, in line

with the increasing cash flow at the company. Find the first month's payment.

$14,257.38 ♥

15.32. Milan Corporation is interested in buying a machine that will cost $50,000, and it

will be completely depreciated on the straight line basis over a 5 year period. The

machine is expected to last for 7 years and then it would be sold for $5,000. The expected

earnings before taxes from the machine is $15,000 with a standard deviation of $5,000.

The income tax rate of Milan is 35%, and its after-tax cost of capital 10%. Find the

probability that this machine will be profitable. 78.39% ♥


253

15.33. Rome Corporation is planning to acquire a machine for $20,000. The life of the

machine is uncertain: it may last for 4 years (probability 30%), 5 years (probability 50%),

or 6 years (probability 20%). The machine will be fully depreciated in five years on a

straight line basis with no residual value. The after-tax cost of capital for Rome is 12%

and its tax rate 30%. Rome will not buy the machine unless it has a NPV of $3000. Find

the minimum earnings before taxes that this machine must generate to justify its

purchase. $7,534.91 ♥

15.34. A portfolio is made of 400 shares of Naples Corporation, selling at $20 each, and

1700 shares of Palermo Corporation that sell at $10 each. The returns of these securities

under different conditions are presented below:

State of Economy Probability Return of Naples Return of Palermo

Good 25% 22% 25%

Fair 50% 15% 15%

Poor 25% 5% 0%

Find the expected return and standard deviation of return of the portfolio.

13.91%, 8.008% ♥

15.35. Abington Corporation is valued at $50,000, and is owned equally by two brothers

John and Bill. John would like to sell his share of business to Bill. Bill accepts that, and

offers to pay him $25,000 in cash. Or, John can take a note guaranteed by the

corporation, payable after 5 years, with a face value of $35,000. The riskless rate of

interest is 6%, and the σ of the company is .3. Using option pricing theory, what is your

suggestion for John?

Value of the note = $23,910, take cash. ♥

15.36. Pfizer Incorporated has 2 million shares of common stock, selling at $18 each.

The β of the stock is 1.5, T-bill rate is 6%, and the expected return on the market is 12%.

Pfizer also has $20 million (face amount) of bonds, with coupon 6%, which will mature

after 8 years. The required rate of return for the bondholders is 10%. Use equation (3.1)

to find the market value of the bonds. The income tax rate of Pfizer is 40%. Find the

WACC of Pfizer. 12.27% ♥

15.37. Ambridge Company has 6 million shares of common stock selling at $26 a share,

and $50 million in bonds, selling at par, with coupon 7%. The company needs $50

million in new capital which can be raised by selling stock at $25 a share, or by selling

bonds with 8% coupon. After the new financing, the expected EBIT of the company is

$60 million, with a standard deviation of $20 million. Ambridge has to pay $6 million in

preferred dividends and $10 million in sinking fund payments. The income tax rate of

Ambler is 30%. By calculating the EBIT that will make it indifferent to debt or equity

financing for the new capital, find the preferred method of financing. What is the

probability that you are right? E* = 42.357 million, P(being right) = 81.12% ♥


254

15.38. Archbald Corporation can buy a computer of $70,000, depreciate it fully over 7

years, and then sell it for $5,000. There is a 6% investment tax credit available. The

income tax rate of Archbald is 40%, and its cost of debt is 12%. Archbald can also lease

the computer for 7 years, paying a certain lease payment in advance each year whose tax

benefit is available at the end of the year. Find the annual lease payment that will make

leasing or buying to be equally costly. $12,187.68 ♥

15.39. Arnold Corporation wants to buy a machine that costs $50,000, which is expected

to run for 5 years. It will be depreciated on a straight line basis over that period with no

resale value. However, there is a 20% chance that the machine may break down after just

4 years. In that case a used machine will be purchased for $15,000 which will be used for

only one year without any resale value. The tax rate of Ardmore is 30%, and its after-tax

cost of capital is 12%. The machine will generate an income of $13,000 annually. Should

Arnold buy the machine? No, NPV = −$7,737.22 ♥

15.40. Dayton Corporation is considering these three projects whose returns are

normally distributed:

Project Investment E(R) σ(R) Correlation coefficient

A $20,000 12% 20% A, B = 0.4

B $30,000 14% 20% A, C = 0.5

C $50,000 16% 30% B, C = 0.6

Find the probability that the return on the portfolio is more than 20%. 40.11% ♥

15.41. Garfield Corporation bonds will mature after 3 years and they pay interest

annually at the rate of 8%. The first interest payment is due a year from now. The bonds

are regarded as junk because there is a 25% probability that the company may go

bankrupt in a given year. In case of bankruptcy, the company will not pay any interest on

the bonds and it is expected to pay only 20% of the principal to bondholders one year

after the bankruptcy. If your required rate of return is 15%, how much should you pay for

a Garfield bond? $464.33 ♥

15.42. Arthur Company is planning to acquire a machine for $90,000 which has an

uncertain life. The machine may break down after 4 years (probability 10%), 5 years

(probability 20%), or 6 years (probability 70%). The machine will be depreciated on a

straight line basis for 5 years with no resale value. The income tax rate of Arthur is 30%,

and its after-tax cost of capital 8%. Find the annual earnings generated by this machine so

that its NPV is $10,000. $25,657 ♥

15.43. Grant Corporation is interested in buying a machine which costs $60,000, and

which will be depreciated linearly to zero value in 5 years, but then it will be sold for

$5,000. The earnings from this machine are expected to be $15,000 per year, with

standard deviation $5,000. The income tax of Grant is 35%, and its after-tax cost of

capital 9%. Find the probability that this machine will turn out to be profitable. 38.71% ♥


255

15.44. Cleveland Company has made a portfolio of three projects as follows:

Project Cost E(R) σ(R) Correlation coefficient

A $25,000 10% 20% (A,B) = .3

B $35,000 12% 25% (A,C) = .4

C $40,000 15% 30% (B,C) = .5

Find the probability that the return of the portfolio is more than 15%. 45.56% ♥

15.45. Cumaná Company is interested in acquiring a computer. It can buy the computer

for $150,000, depreciate it fully over 4 years using straight line depreciation, use it for 5

years, and then sell it for $30,000. Or, it can lease the computer for 5 years by paying

$25,000 annual lease payments in advance. The tax benefits of lease payments are

available immediately. In case of leasing, the company also has to pay a non-refundable

fee of $10,000 in advance. The cost of debt for Cumaná is 10%, whereas its income tax

rate is 35%. Which method of acquisition is better for Cumaná, buying or leasing?

NPV(buy) = –$90,803.72, NPV(lease) = –$78,419.23, leasing ♥

15.46. The following information is available for two corporations. The debt and equity

are in millions.

Equity Debt Cost of debt Tax rate β Business

Cabimas $500 $300 10% 35% 1.4 Retail

Caracas $100 $20 9% 30% 1.5 Fast food

The riskfree rate is 5%, and the expected return on the market is 12%. Cabimas Company

would like to start fast food restaurants within their stores, using their existing capital.

Find the required rate of return for the new venture. 13.56% ♥

15.47. Maturín Corporation is owned by two brothers José and Miguel. José is a

stockholder in the company, but Miguel holds a promissory note that entitles him to

receive $3 million from the corporation after 5 years. They have decided to sell the

company for $5 million and split the proceeds according to the Black-Scholes model. The

σ of Maturín is estimated to be .3 and the riskfree rate is 6%. Find the amount of money

that each brother should receive. José’s $2,896,191, Miguel $2,103,809 ♥

15.48. Merck & Co has the following capital structure. It has 5 million shares of

common stock that sell at $26 each. The stock just paid its annual dividend of $1.75, and

it is expected to grow at annual rate of 7% in the foreseeable future. The company also

has $150 million (face amount) in zero-coupon bonds that will mature after 8 years, and

they sell at $500 for each $1,000 bond. The income tax rate of Merck is 30%. Find its

weighted average cost of capital. 11.32% ♥

15.49. Maracaibo Company is interested in buying a machine that costs $50,000. The

machine will be depreciated on a straight line basis over 4 years. The machine, however,

is expected to run for 5 years (probability 30%) or 6 years (probability 70%). While the


256

machine is running it will generate a pretax revenue of $10,000 annually. The income tax

rate of Maracaibo is 35%, and the proper discount rate for this machine is 10%. Should

Maracaibo buy this machine? NPV = –$8,923.37 ♥

15.50. Maracay Corporation is interested in a project that will cost $100,000. The cost of

capital for Maracay is 10%. The annual after-tax income from this project is uncertain.

However, its expected value is $20,000, with a standard deviation of $10,000. This

project will run for 6 years.

(a) Find the probability that the project will turn out to be profitable. 38.36% ♥

(b) How much should Maracay spend on this project so that the probability of it being

profitable is 80%? $50,443 ♥

15.51. Niger Company is interested in buying a house and renting it out for $6000 a

year, collecting the rent in advance each year. The house will be depreciated over 20

years, but sold after 5 years at a price which is 20% more than its purchase price. The

maintenance expenses and real estate taxes, paid at the end of each year, are $2000

annually. The after tax cost of capital for Niger is 10%, and its income tax rate 25%. Find

the price of the house that Niger should pay so that it can make $10,000 in current dollars

from this project. $13,148.22 ♥

15.52. Zaraza Company is in financial distress. Its bonds are not paying interest at

present, and are trading at 23% of their face value. You have estimated that there is a

25% probability that Zaraza will go bankrupt after 1 year. If it does not become bankrupt,

then there is a 50% probability that it will become bankrupt after 2 years. It will certainly

go bankrupt after 3 years. You believe that one year after the bankruptcy, the company

will pay $400 per bond to settle with the bondholders. If your required rate of return is

11%, should you buy these bonds? Yes, B = $289.65 ♥

15.53. Barranquilla Corporation has borrowed $200,000 from Bogotá bank with the

following terms:

(a) Bogotá Bank will charge interest at 12% per annum, with monthly compounding.

(b) Barranquilla will make $10,000 monthly installments to pay off the loan.

Find the balance of the loan after 1 year. $98,539.98 ♥

15.54. Bucaramanga Company is in financial distress. The probability of it becoming

bankrupt in a given year is 20%. The bonds of the company have a 9% coupon, but they

pay interest only once a year, on October 15, and they will mature on October 15, 2012.

In case of bankruptcy, you do not expect to get interest for that year, and only 30% of the

face amount of the bond. You are thinking of buying one of these bonds on October 16,

2009. How much should you pay for a bond, if your required rate of return is 12%.

$626.60 ♥


257

15.55. Buenaventura Corporation stock is selling at $55 a share. The company will pay a

dividend of $3 at the end of one year, $4 at the end of two years, and then $5 at the end of

three years. However this last dividend is expected to grow at the rate of 8% forever. If

your required rate of return is 16%, do you think you should buy this stock? No, $52.01 ♥

15.56. Cartagena Company is planning to get a machine that will save the company

$50,000 annually, with a standard deviation of $10,000. The company uses straight line

depreciation. The tax rate of Cartagena is 30%, and the proper discount rate in this case is

10%. The machine will cost $300,000 and is expected to last for 8 years. Calculate the

probability that the machine will turn out to be profitable. 7.69% ♥

15.57. Cali Company is planning to buy a computer for $80,000 that is expected to save

the company $20,000 annually. The tax rate of Cali is 30%, and the proper discount rate

in this case is 12%. The computer will be depreciated on a straight line basis over the

next 4 years. The useful life of the computer is uncertain. The probability that the

computer will become obsolete after a certain number of years, and the resale value of the

equipment at that time is given in the following table. Should Cali buy this computer?

Probability Expected life Resale value

30% 4 years $12,000

30% 5 years $8,000

40% 6 years $3,000

No, NPV = –$7,874.74 ♥

15.58. Ibagué Company stock sells at $35 a share. It has = 1.54 and = .4. The riskfree

rate is 6%, and the expected return on the market is 12%. You have formed a portfolio

with these two items in it:

(1) 1000 shares of Ibagué stock.

(2) A zero-coupon riskfree bond with face value $50,000, maturing after 1 year.

Calculate the following:

(a) Initial value of the portfolio. $82,169.81 ♥

(b) Expected value of the portfolio after one year. $90,334 ♥

(c) The σ of the portfolio. 17.04% ♥

15.59. Manizales Corporation stock sells at $73 a share. The σ of the stock is .4, and the

riskfree rate is 6% at present. Find the value of a call option on this stock with exercise

price $75, and expiring after 73 days. $4.71 ♥

15.60. The value of Senegal Corporation is $200 million. It has $100 million (face

amount) in zero-coupon bonds that will mature in 10 years. The σ of Senegal is estimated

to be 0.4, and the riskfree rate is 6%. Using the option pricing theory, calculate the debt-

to-assets ratio of the company. 22.78% ♥


258

15.61. You have assembled the following information about two companies, with dollar

figures in millions:

β Equity Debt Tax rate Cost of debt

Algerian Airlines 1.6 $80 $20 30% 9%

Libyan Car Rental 1.5 $30 $20 $35 10%

The riskfree rate is 6%, and the expected return on the market 14%. If Algeria wants to

start a car rental subsidiary, with its existing capital, find the minimum acceptable rate of

return on the new project. 13.93% ♥

15.62. Monsanto Company has 3 million shares, selling at $25 each. The company has

just paid a dividend of $1.35 and the next year's dividend is expected to be $1.50, which

is in line with its long-term growth rate. Monsanto has $50 million (face amount) of

bonds, maturing in 10 years, with coupon 8%. The yield to maturity for the bonds is 10%.

(Use the definition of yield to maturity on page 36 of the textbook.) The income tax rate

of the company is 30%. Find its WACC. 13.38% ♥

15.63. The debt-to-assets ratio of Mali Corporation is 45%. The chief financial officer at

the company feels that it should be reduced to 40%. This should result in lowering the

bankruptcy costs by $20 million. The total value of Mali is $200 million, and its tax rate

35%. Should the change be implemented? If so, what is the value of the stock that should

be sold to buy back the bonds? Yes, V

2 = $219.2 million, sell stock for $2.326 million ♥

15.64. (Advanced) Consider example 2.5. Axel Heiberg has just accepted a job with an

annual salary S. He has decided to put a fraction a of his gross monthly income into a

retirement account at the beginning of every month. The retirement account pays interest

at the rate of r% every month. Axel also expects to receive an annual raise of g% each

year for the next n years. Show that the amount in his retirement fund at the end of n

years will be

j=0

n−1

i=1

12

aS(1 + r)

i+12(n−j−1)(1+g)

j

12 ♥

15.65. (Advanced) A bank offers the following program to its customers. If you deposit

C at the beginning of every month for the next n years, then in return the bank will give

you C a month forever, starting a month after your last monthly payment. Find the

minimum value of r, the interest rate per year, which will make depositors join this

program. r = 12[21/(12n)

– 1] ♥

15.66. (Advanced) A loan L is payable in n months, with a monthly interest rate r. The

unpaid balance after k months is B. Show that

k =

ln

B + (L − B)(1 + r)

n

L

ln(1 + r)


259

15.67. (Advanced) A bond with face value F will mature after n years. Its coupon rate is

c and it pays interest semiannually. An investor who buys this bond pays tax annually, at

the rate T for interest income and t for the capital gain. The investor uses r as the annual

discount rate. Show that the value of this bond for the investor is

B = F[cT + r(1 − t) − 2

2nc(1 + r)

n(2 + r)

−2n + c(1 − T)(1 + r)

n]

r[(1 + r)n − t]

260

16. FORMULAS AND TABLES

Geometric series, S = a + ax + ax2 + ax

3 + ... + ax

n−1 (1.3)

Sum of geometric series, n terms Sn = a (1 − x

n)

1 − x (1.4)

Sum of infinite geometric series, S∞ = a

1 − x (1.5)

Expected value of X, E(X) = i=1

n

PiXi = X —

(1.6)

Variance of X, var(X) = i=1

n

Pi(Xi − X —

)2 (1.7)

Standard deviation of X, σ(X) = var(X) (1.8)

Covariance between X and Y, cov(X,Y) = i=1

n

Pi(Xi − X —

)(Yi − Y —

) (1.9)

Correlation coefficient between X and Y, r(X,Y) = cov(X,Y)

σ(X)σ(Y) (1.10)

Range of the correlation coefficient, −1 < r(X,Y) < 1 (1.11)

Future value of a sum of money, FV = PV (1 + r)n (2.1)

FV with continuous compounding, FV = PV ern (2.4)

Present value of a future payment, PV = FV

(1 + r)n (2.5)

Present value of an annuity, i=1

n

C

(1 + r)i =

C[1 − (1 + r)−n

]

r (2.6)

Present value of a perpetuity, i=1

C

(1 + r)i =

C

r (2.7)

PV of n cash flows, starting after k periods, PV = 1

(1 + r)k–1 i=1

n

C

(1 + r)i (2.8)

Future value of n cash flows, starting now, FV = C(1 + r)[(1 + r)n – 1]

r (2.9)

Loan amortization, L = i=1

n

P

(1 + r)i (2.10)

Loan amortization with balloon payment B, L = i=1

n

P

(1 + r)i + B

(1 + r)n (2.11)

Analytical Techniques 16. Formulas and Tables _____________________________________________________________________________

261

Future value, including withdrawals, FV =

A − i=1

n

w

(1 + r)i (1 + r)

n (2.12)

Time to exhaust savings, n =

ln

w

w – Ar

ln(1 + r) (2.13)

Present value of a coupon bond, B = i=1

n

C

(1 + r)i +

F

(1 + r)n (3.1)

Present value of a perpetual bond, B = C

r (3.2)

Present value of a zero-coupon bond, B = F

(1 + r)n (3.3)

Current yield of a bond, y = cF/B (3.4)

Yield-to-maturity of a bond, Y ≈ cF + (F − B)/n

(F + B)/2 (3.5)

Gordon's growth model, P0 = D1

R − g (3.6)

Net present value, NPV = − I0 + i=1

n

C

(1 + r)i (4.1)

Internal rate of return, NPV = 0 = I

0 + i=1

n

C

(1 + IRR)i (4.2)

After-tax cash flow, C = E(1 − t) + tD (4.3)

With maintenance cost M, C = (1t)(E M) + tD (4.4)

Book value of an asset, B = I0 – nD (4.5)

Tax due on the sale of an asset, T = t(S B) (4.6)

After-tax value of resale price, W = S(1 – t) + tB (4.7)

For a two-security portfolio, w1 + w2 = 1 (6.6)

E(Rp) = w1 E(R1) + w2 E(R2) (6.7)

σ(Rp) = w12 σ1

2 + w2

2 σ2

2 + 2w1w2 σ1σ2r12 (6.8)

Covariance between i and j, cov(i,j) = σiσjrij (6.5)

For an n security portfolio, i=1

n

wi = 1 (6.12)

Expected return of portfolio, E(Rp) = i=1

n

wi E(Ri) (6.13)


262

Its standard deviation, σ(Rp) = [i=1

n

j=1

n

wiwjcov(i,j)]1/2

(6.5)

For dollar amounts, E(Rp) = i=1

n

E(Ri) (6.18)

σ(Rp) = [i=1

n

j=1

n

cov(i,j)]1/2

(6.19)

Return on a stock, Rj = P1 − P0 + D1

P0 (7.1)

Return on the market, Rm = M1 − M0

M0 + d1 (7.2)

Definition of β, βj = cov(Rj,Rm)

var(Rm) =

rjmσmσj

σm2 =

rjmσj

σm (7.3)

Calculation of β, β = n(xy) − (x)(y)

nx2 − (x)

2 (7.4)

Calculation of α, α = y − βx

n (7.5)

Beta of a portfolio, βp = w1β1 + w2β2 + w3β3 + ... = i=1

n

wiβi (7.6)

Capital Asset Pricing Model, E(Ri) = r + βi [E(Rm) − r] (7.7)

Black-Scholes model, C = S N(d1) − X e−rT

N(d2) (8.3)

where d1 = ln(S/X) + (r + σ

2/2)T

σ T (8.4)

and d2 = ln(S/X) + (r − σ

2/2)T

σ T = d1 − σ T (8.5)

Put-Call Parity Theorem, P + S = C + X e−rT

(8.7)

Value of a European put, P = S [N(d1) – 1] − X e−rT

[N(d2) −1] (8.8)

Hedge ratio, h = ∂C

∂S = N(d1) (8.11)

Total value of a firm, V = B + S (9.4)

WACC, WACC = (1 − t) kd B

V + ke

S

V (9.5)

Earnings per share, EPS = EAT

N (10.1)


263

Earnings after taxes, EAT = (EBIT − I)(1 − t) (10.2)

Earnings per share, EPS = (EBIT − I) (1 − t) − SF − PD

N (10.3)

Bond financing, EPS(bonds) = (EBIT − I − r F) (1 − t) − SF − PD

N (10.4)

Stock financing, EPS(stock) = (EBIT − I) (1 − t) − SF − PD

N + F/P (10.5)

Critical EBIT, E* = I + r(NP + F) + SF + PD

1 − t (10.6)

For SF = PD = 0, E* = I + r (N P + F) (10.7)

Tax shield = tB (10.8)

Value of a leveraged firm, VL = VU + tB − b (10.9)

Change in the value of a firm ΔVL = tΔB − Δb (10.10)

Leveraged beta, βL = βU [1 + (1 − t)B

S] (11.3)

House, H NPV = − H + i=1

n

12(R − M)(1 − t) + tD

(1 + r)i +


(1 + r)n (14.2)


264

16.1 How to Use the Probability Tables

In some finance problems, you need to estimate the probability of something happening.

For example, a typical question is to find the probability that a given machine may run

for more than 5 years. If we assume that the life of the machine is normally distributed,

and we know the two parameters that describe the distribution, namely the expected value

μ and the standard deviation, σ, then we can answer the question. Further, you need to

learn the use of the probability tables or the NORMDIST function in Excel. The tables

are a little easier to use, although Excel does the calculations automatically.

Suppose you have a portfolio whose expected return is 11%, with normal distribution and

a standard deviation of 7%. You want to find the probability that, by chance, it will

provide a return of 15% or higher. By reading the problem, you know that it is unlikely to

happen. You expect to have a return of 11%, but you require a return of 15%. The result

should be less than 50%.

Start by calculating the parameter z = |μ − x|/σ. It comes out as z = |.11 − .15|/.07 = 4/7 =

.5714285716. Let us truncate it to four figures because that is the accuracy of the tables.

Thus z = .5714.

Next, draw a bell-shaped probability distribution curve. Put z = 0 at the center and mark

off points at −3, −2, −1, 0, 1, 2, and 3. These points are σ, 2σ, and 3σ from the center. The

required return is .15, which is equivalent to z = .5714. It is about ½σ away from the

center. Since we need the probability of return to be higher than .15, we need the area to

the right of z = .5714, which is under the tail of the curve.

The tables are set up to provide the area from the center to the point z. In our case, z =

.5714. To read the tables, first find the area for the first two digits, .57 and then add to it,

14% of the difference between the areas corresponding to .58 and .57. The number 14%

comes from the last two digits of z = .5714. This gives the required area for .5714.

Whatever area we get, we have to subtract it out .5 to find the area under the tail of the

curve. Put it all together as follows:

P(R > .15) = .5 – [.2157 + .14(.2190 − .2157)] = .2838 = 28.38% ♥


265

The portfolio with expected return 11%, and standard deviation 7%, has 28.38%

probability that it may actually attain a return of 15% or higher. This is a reasonable

answer.

You will need the table to find the values of N(d1) and N(d2) for use in the Black-Scholes

model. In this case N(d1) and N(d2) represent the total area under the normal probability

curve, measured from −∞ up to the point d1 or d2. Suppose d1 = .5462, which is positive.

You can place it a to the right of zero, or the midpoint of the curve. In this case the area

N(d1) will be somewhat more than .5, as shown in the following diagram.

By the use of the tables, we get

N(d1) = .5 + .2054 + .62(.2084 − .2054) = .7073

If d2 is negative, say, −.4142, then it will lie on the left side of the center of the curve.

The shaded area will be less than .5, as seen the next diagram.

The result in this case will be

N(d2) = .5 – [.1591 + .42(.1628 − .1591)] = .3393 ♥


266

Area under the normal probability curve, from the center to a point z, where z = x − μσ

z Area z Area z Area z Area z Area z Area

0.01 .0040 0.51 .1950 1.01 .3438 1.51 .4345 2.01 .4778 2.51 .4940

0.02 .0080 0.52 .1985 1.02 .3461 1.52 .4357 2.02 .4783 2.52 .4941

0.03 .0120 0.53 .2019 1.03 .3485 1.53 .4370 2.03 .4788 2.53 .4943

0.04 .0160 0.54 .2054 1.04 .3508 1.54 .4382 2.04 .4793 2.54 .4945

0.05 .0199 0.55 .2088 1.05 .3531 1.55 .4394 2.05 .4798 2.55 .4946

0.06 .0239 0.56 .2123 1.06 .3554 1.56 .4406 2.06 .4803 2.56 .4948

0.07 .0279 0.57 .2157 1.07 .3577 1.57 .4418 2.07 .4808 2.57 .4949

0.08 .0319 0.58 .2190 1.08 .3599 1.58 .4429 2.08 .4812 2.58 .4951

0.09 .0359 0.59 .2224 1.09 .3621 1.59 .4441 2.09 .4817 2.59 .4952

0.10 .0398 0.60 .2257 1.10 .3643 1.60 .4452 2.10 .4821 2.60 .4953

0.11 .0438 0.61 .2291 1.11 .3665 1.61 .4463 2.11 .4826 2.61 .4955

0.12 .0478 0.62 .2324 1.12 .3686 1.62 .4474 2.12 .4830 2.62 .4956

0.13 .0517 0.63 .2357 1.13 .3708 1.63 .4484 2.13 .4834 2.63 .4957

0.14 .0557 0.64 .2389 1.14 .3729 1.64 .4495 2.14 .4838 2.64 .4959

0.15 .0596 0.65 .2422 1.15 .3749 1.65 .4505 2.15 .4842 2.65 .4960

0.16 .0636 0.66 .2454 1.16 .3770 1.66 .4515 2.16 .4846 2.66 .4961

0.17 .0675 0.67 .2486 1.17 .3790 1.67 .4525 2.17 .4850 2.67 .4962

0.18 .0714 0.68 .2517 1.18 .3810 1.68 .4535 2.18 .4854 2.68 .4963

0.19 .0753 0.69 .2549 1.19 .3830 1.69 .4545 2.19 .4857 2.69 .4964

0.20 .0793 0.70 .2580 1.20 .3849 1.70 .4554 2.20 .4861 2.70 .4965

0.21 .0832 0.71 .2611 1.21 .3869 1.71 .4564 2.21 .4864 2.71 .4966

0.22 .0871 0.72 .2642 1.22 .3888 1.72 .4573 2.22 .4868 2.72 .4967

0.23 .0910 0.73 .2673 1.23 .3907 1.73 .4582 2.23 .4871 2.73 .4968

0.24 .0948 0.74 .2704 1.24 .3925 1.74 .4591 2.24 .4875 2.74 .4969

0.25 .0987 0.75 .2734 1.25 .3944 1.75 .4599 2.25 .4878 2.75 .4970

0.26 .1026 0.76 .2764 1.26 .3962 1.76 .4608 2.26 .4881 2.76 .4971

0.27 .1064 0.77 .2794 1.27 .3980 1.77 .4616 2.27 .4884 2.77 .4972

0.28 .1103 0.78 .2823 1.28 .3997 1.78 .4625 2.28 .4887 2.78 .4973

0.29 .1141 0.79 .2852 1.29 .4015 1.79 .4633 2.29 .4890 2.79 .4974

0.30 .1179 0.80 .2881 1.30 .4032 1.80 .4641 2.30 .4893 2.80 .4974

0.31 .1217 0.81 .2910 1.31 .4049 1.81 .4649 2.31 .4896 2.81 .4975

0.32 .1255 0.82 .2939 1.32 .4066 1.82 .4656 2.32 .4898 2.82 .4976

0.33 .1293 0.83 .2967 1.33 .4082 1.83 .4664 2.33 .4901 2.83 .4977

0.34 .1331 0.84 .2995 1.34 .4099 1.84 .4671 2.34 .4904 2.84 .4977

0.35 .1368 0.85 .3023 1.35 .4115 1.85 .4678 2.35 .4906 2.85 .4978

0.36 .1406 0.86 .3051 1.36 .4131 1.86 .4686 2.36 .4909 2.86 .4979

0.37 .1443 0.87 .3078 1.37 .4147 1.87 .4693 2.37 .4911 2.87 .4979

0.38 .1480 0.88 .3106 1.38 .4162 1.88 .4699 2.38 .4913 2.88 .4980

0.39 .1517 0.89 .3133 1.39 .4177 1.89 .4706 2.39 .4916 2.89 .4981

0.40 .1554 0.90 .3159 1.40 .4192 1.90 .4713 2.40 .4918 2.90 .4981

0.41 .1591 0.91 .3186 1.41 .4207 1.91 .4719 2.41 .4920 2.91 .4982

0.42 .1628 0.92 .3212 1.42 .4222 1.92 .4726 2.42 .4922 2.92 .4982

0.43 .1664 0.93 .3238 1.43 .4236 1.93 .4732 2.43 .4925 2.93 .4983

0.44 .1700 0.94 .3264 1.44 .4251 1.94 .4738 2.44 .4927 2.94 .4984

0.45 .1736 0.95 .3289 1.45 .4265 1.95 .4744 2.45 .4929 2.95 .4984

0.46 .1772 0.96 .3315 1.46 .4279 1.96 .4750 2.46 .4931 2.96 .4985

0.47 .1808 0.97 .3340 1.47 .4292 1.97 .4756 2.47 .4932 2.97 .4985

0.48 .1844 0.98 .3365 1.48 .4306 1.98 .4761 2.48 .4934 2.98 .4986

0.49 .1879 0.99 .3389 1.49 .4319 1.99 .4767 2.49 .4936 2.99 .4986

0.50 .1915 1.00 .3413 1.50 .4332 2.00 .4772 2.50 .4938 3.00 .4987

analytical techniques in financial management

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