Financial Management

Mr. Tahir Abbas University of South Asia (HIC) layyah Campus

Chapter#01 FINANCIAL MANAGEMENT

WHY STUDY FINANCE?

Accounting and Finance

Accountants are required to make financial decisions as well as understand the implications of new financial contracts

Financial analysts make extensive use of accounting information What is Business Finance?

In order to start any new business, the following issues become vital

What long-term investment should be taken on?

From where to get the long-term financing to pay for investment? Bring in other owners or borrow the money?

How to manage everyday financial activities? The Financial Manager

To create value, the financial manager should:

Try to make smart investment decisions.

Try to make smart financing decisions. Business Finance and Financial Manager

Financial Management Decisions

Capital Budgeting

Capital Structure

Working Capital Management Financial Management Decisions

Capital Budgeting

The process of planning and managing a firms long-term investments

Financial managers concern with how much, when and how likely is cash expected to receive

Evaluating the size, timing and risk of future cash flows is the essence of capital budgeting


The Corporate Firm

The corporate form of business is the standard method for solving the problems encountered in raising large amounts of cash.

However, businesses can take other forms. Forms of Business Organization There are three major forms

01. Sole proprietorship 02. Partnership

General

Limited 03. Corporation

Limited liability company

Sole Proprietorship A business which is run by one Person According to D.W.T. Stafford It is the simplest form of business organization, which is owned and controlled by one man. Sole proprietorship is the oldest form of business organization which is owned and controlled

by one person. In this business, one man invests his capital himself. He is all in all in doing

his business. He enjoys the whole of the profit. The features of sole proprietorship are:

Advantages Easiest to start Least regulated

Single owner keeps all the profits

Taxed once as personal income

Disadvantages

Limited to life of owner Equity capital limited to owners personal wealth

Unlimited liability

Difficult to sell ownership interest


Partnership According to Partnership Act 1932: Partnership is the relation between persons who have agreed to share the profits of a business carried on

by all or any of them acting for all.

Partnership means a lawful business owned by two or more persons. The profit of the business shared by the partners in agreed ratio. The liability of each partner is unlimited. Small and medium size business activities are performed under this organization

Advantages

Two or more owners

More capital available

Relatively easy to start Income taxed once as personal income

Disadvantages

Unlimited liability There are two types of partnership

General partnership Limited partnership

Partnership dissolves when one partner dies or wishes to sell

Difficult to transfer ownership

Corporation

A business created as a distinct legal entity owned by one or more individuals or entities. Advantages

Limited liability

Unlimited life

Separation of ownership and management

Transfer of ownership is easy

Easier to raise capital

Disadvantages

Separation of ownership and management Double taxation (income taxed at the corporate rate and then dividends taxed at personal rate)


Goal of the Corporate Firm

The traditional answer is that the managers of the corporation are obliged to make efforts to maximize shareholder wealth.

Alternatively, the goal of the financial manager is to maximize the current value per share of the existing stock

The Set-of-Contracts Perspective

The firm can be viewed as a set of contracts. One of these contracts is between shareholders and managers.

The managers will usually act in the shareholders interests. The shareholders can devise contracts that align the incentives of the managers with the goals of

the shareholders. The shareholders can monitor the managers behavior.

This contracting and monitoring is costly.

Managerial Goals Managerial goals may be different from shareholder goals Expensive perquisites

Survival

Independence

Increased growth and size are not necessarily the same thing as increased shareholder wealth.

Do Shareholders Control Managerial Behavior? Shareholders vote for the board of directors, who in turn hire the management team.

Contracts can be carefully constructed to be incentive compatible. There is a market for managerial talentthis may provide market discipline to the managersthey

can be replaced. If the managers fail to maximize share price, they may be replaced in a hostile takeover.

Managing Managers

Managerial compensation

Incentives can be used to align management and stockholder interests

The incentives need to be structured carefully to make sure that they achieve their goal

Corporate control

The threat of a takeover may result in better management

Other stakeholders


Financial Market

There are two type of Market

01. Primary Market When a corporation issues securities, cash flows from investors to the firm. Usually an underwriter is involved

02. Secondary Markets

Involve the sale of used securities from one investor to another. Securities may be exchange traded or trade over-the-counter in a dealer market.

Dealer Vs. Auction Markets

Auction markets are different from dealer markets in two ways: Trading in a given auction exchange takes place at a single site on the floor of the exchange.

Transaction prices of shares are communicated almost immediately to the public. Listing

The Balance Sheet

An accountants snapshot of the firms accounting value as of a particular date. The Balance Sheet Identity is:

a. Assets Liabilities + Stockholders Equity

When analyzing a balance sheet, the financial manager should be aware of three concerns: accounting liquidity, debt versus equity, and value versus cost.

The Balance-Sheet Model of the Firm

Total Value of Assets Total Firm Value to Investors

Current Assets Current Liabilities

Long-Term Debt Fixed Assets 1. Tangible

2. Intangible Shareholders Equity


Net Working Capital Net Working Capital Current Assets Current Liabilities NWC > 0 when Current Assets > Current Liabilities NWC < 0 when Current Assets < Current Liabilities NWC = 0 when Current Assets = Current Liabilities

NWC usually grows with the firm for the healthy firms.

The Balance-Sheet Model of the Firm

The Net Working Capital Investment Decision

Current Assets

Fixed Assets 1. Tangible 2. Intangible

Net Working Capital

How much short-

term cash flow does a

company need to pay

its bills?

Current Liabilities

Long-Term Debt

Shareholders Equity


The Financial Environment

In varying degrees, all businesses operate within the financial system, which consists of a

number of institutions and markets serving business firms, individuals, and governments. When

a firm invests temporarily idle funds in marketable securities, it has direct contact with financial

markets.

More important, most firms use financial markets to help finance their investment in assets. In

the final analysis, the market price of a companys securities is the test of whether the company

is a success or a failure. While business firms compete with each other in the product markets,

they must continually interact with the financial markets. Because of the importance of this

environment to the financial manager, as well as to the individual as a consumer of financial

services, this section is devoted to exploring the financial system and the ever-changing

environment in which capital is raised.

What is financial market?

Financial markets all institutions and procedures for bringing buyers and sellers of

financial instruments together

The Purpose of Financial Markets

The purpose of financial markets in an economy is to allocate savings efficiently to ulti-mate

users. If those economic units that saved were the same as those that engaged in capital

formation, an economy could prosper without financial markets. In modern economies,

however, most nonfinancial corporations use more than their total savings for investing in real

assets. Most households, on the other hand, have total savings in excess of total invest-ment.

Efficiency entails bringing the ultimate investor in real assets and the ultimate saver together at

the least possible cost and inconvenience.


What are the different types of financial market?

The types of financial market as under

Money market

The market for short-term (less than one year original maturity) government and

corporate debt securities It also includes government securities originally issued with

maturities of more than one year but that now have a year or less until maturity.

Capital market

The market for relatively long-term (greater than one year original maturity) financial

instruments (e.g., bonds and stocks)

Money and Capital Markets:

Financial markets can be broken into two classes the money market and the capital market.

The money market is concerned with the buying and selling of short-term (less than one year

original maturity) government and corporate debt securities. The capital market, on the other

hand, deals with relatively long-term (greater than one year original maturity) debt and equity

instruments (e.g., bonds and stocks). This section gives special attention to the market for long-

term securities the capital market

Primary market

A market where new securities are bought and sold for the first time (a new issues

market)

Secondary market

A market for existing (used) securities rather than new issues

Primary and Secondary Markets. Within money and capital markets there exist both primary

and secondary markets. A primary market is a new issues market. Here, funds raised through

the sale of new securities flow from the ultimate savers to the ultimate investors in real assets.

In a secondary market, existing securities are bought and sold. Transactions in these already

existing securities do notprovide additional funds to finance capital invest-ment. (Note: On

Figure 2.1 there is no line directly connecting the secondary market with the investment

sector.) An analogy can be made with the market for automobiles. The sale of new cars


provides cash to the auto manufacturers; the sale of used cars in the used-car market does not.

In a real sense, a secondary market is a used-car lot for securities

Chapter#02

The Time Value of Money

Contents

01. The Interest Rate

02. Simple Interest

03. Compound Interest

Single Amounts Annuities Mixed Flows

04. Compounding More Than Once a Year

Semiannual and Other Compounding Periods

Continuous Compounding

Effective Annual

Interest Rate

05. Amortizing a Loan

06. Formulas

07. Questions

08. Problems

09. Solutions


The Interest Rate

Interest Money paid (earned) for the use of money

Which would you prefer $1,000 today or $1,000 ten years from today? Common sense tells us

to take the $1,000 today because we recognize that there is a time value to money. The

immediate receipt of $1,000 provides us with the opportunity to put our money to work and

earn interest. In a world in which all cash flows are certain, the rate of interest can be used to

express the time value of money. As we will soon discover, the rate of interest will allow us to

adjust the value of cash flows, whenever they occur, to a particular point in time. Given this

ability, we will be able to answer more difficult questions, such as: which should you prefer

$1,000 today or $2,000 ten years from today? To answer this question, it will be necessary to

position time-adjusted cash flows at a single point in time so that a fair comparison can be

made.

Simple Interest

Interest paid (earned) on only the original amount, or principal, borrowed

(lent)

Simple interest is interest that is paid (earned) on only the original amount, or principal,

borrowed (lent). The dollar amount of simple interest is a function of three variables: the

original amount borrowed (lent), or principal; the interest rate per time period; and the num-

ber of time periods for which the principal is borrowed (lent). The formula for calculating simple

interest is

SI=P0 (i)(n)

Where

SI=simple interest in dollars

P0=principal, or original amount borrowed (lent) at time period 0

i=interest rate per time period

n=number of time periods


For example, assume that you deposit $100 in a savings account paying 8 percent simple

interest and keep it there for 10 years. At the end of 10 years, the amount of interest

accumulated is determined as follows:

SI=$100(0.08)(10)

SI=$80

Future value

Future value (terminal value)The value at some future time of a present amount

of money, or a series of payments, evaluated at a given interest rate

To solve for the future value(also known as the terminal value) of the account at the end of 10

years (FV10 ), we add the interest earned on the principal only to the original amount invested.

Therefore

FV10=$100 +[$100(0.08)(10)] =$180

For any simple interest rate, the future value of an account at the end of nperiods is

FVn=P0+SI=P0+P0 (i)(n) Or

FVn=P0 [1 +(i)(n)]

Present value

The current value of a future amount of money, or a series of payments,

evaluated at a given interest rate

Sometimes we need to proceed in the opposite direction. That is, we know the future value of a

deposit at i percent for n years, but we dont know the principal originally invested the

accounts present value(PV0 =P0 ). A rearrangement of Eq. (3.2), however, is all that is needed.


PV0=P0=FVn /[1 +(i)(n)]

Now that you are familiar with the mechanics of simple interest, it is perhaps a bit cruel

to point out that most situations in finance involving the time value of money do not rely

on simple interest at all. Instead, compound interest is the norm; however, an understand-ing

of simple interest will help you appreciate (and understand) compound interest all the more.

Compound Interest

Compound interest interest paid (earned) on any previous interest earned, as well

as on the principal borrowed (lent).

The distinction between simple and compound interest can best be seen by example. illustrates

the rather dramatic effect that compound interest has on an investments value over time

when compared with the effect of simple interest. From the table it is clear to see why some

people have called compound interest the greatest of human inventions. The notion of

compound interest is crucial to understanding the mathematics of fin-ance. The term itself

merely implies that interest paid (earned) on a loan (an investment) is 3The Time Value of

Money

Future value of $1 invested for various time periods at an 8% annual interest rate

YEARS AT SIMPLE INTEREST AT COMPOUND INTEREST

2 $ 1.16 $ 1.17

20 2.60 4.66

200 17.00 4,838,949.59

Single Amounts

Future (or Compound) Value. To begin with, consider a person who deposits $100 into a savings

account. If the interest rate is 8 percent, compounded annually, how much will the $100 be

worth at the end of a year? Setting up the problem, we solve for the future value (which in this

case is also referred to as the compound value) of the account at the end of the year (FV1 )

FV1=P0 (1 +i) =$100(1.08) =$108


Interestingly, this first-year value is the same number that we would get if simple interest were

employed. But this is where the similarity ends. What if we leave $100 on deposit for two

years? The $100 initial deposit will have grown to $108 at the end of the first year at 8 percent

compound annual interest. Going to the end of the second year, $108 becomes $116.64, as $8

in interest is earned on the initial $100, and $0.64 is earned on the $8 in interest credited to our

account at the end of the first year. In other words, interest is earned on previously earned

interest hence the name compound interest. Therefore, the future value at the end of the

second year is

FV2=FV1 (1 +i)

=P0 (1 +i)(1 +i) =P0 (1 +i) 2

=$108(1.08) =$100(1.08)(1.08) =$100(1.08)2

=$116.64

At the end of three years, the account would be worth

FV3=FV2 (1 +i) =FV1 (1 +i)(1 +i) =P0 (1 +i)3

=$116.64(1.08) =$108(1.08)(1.08) =$100(1.08)3

=$125.97

In general, FVn , the future (compound) value of a deposit at the end of nperiods, is

FVn=P0 (1 +i)n or

FVn=P0 (FVIFi,n)

Where we let FVIFi,n (i.e., the future value interest factor at i% for n periods) equal (1 +i)n

Discount rate:

Discount rate (capitalization rate) Interest rate used to convert future values to present

values

Finding the present value (or discounting) is simply the reverse of compounding. Therefore,

lets first retrieve Eq.

FVn=P0 (1 +i)n


Rearranging terms, we solve for present value:

PV0=P0=FVn/(1 +i)n

=FVn [1/(1 +i)n]

Note that the term [1/(1 +i)n ] is simply the reciprocal of the future value interest factor at i%

for n periods (FVIFi,n ).This reciprocal has its own name the present value interest factor at i%

for n periods (PVIFi,n) and allows us to rewrite Eq as

PV0=FVn(PVIFi,n)

A present value table containing PVIFs for a wide range of interest rates and time periods

relieves us of making the calculations implied by Eq. as

PV0=P0=FVn/(1 +i)n

=FVn [1/(1 +i)n]

Every time we have a present value to solve

Annuities

Annuity a series of equal payments or over a specified number of periods in an ordinary

annuity, payments or receipts occur at the end of each period; in an annuity due,

payments or receipts occur at the beginning of each period

Ordinary Annuity An annuity is a series of equal payments or receipts occurring over a specified

number of periods In an ordinary annuity, payments or receipts occur at the end of each

period. Expressed algebraically, with FVAn defined as the future (compound) value of an

annuity, R the periodic receipt (or payment), and n the length of the annuity, the formula for

FVAn is

FVAn=R(1 +i)n1 +R(1 +i)n2 +... +R(1 +i)1 +R(1 +i)0

=R[FVIFi,n1+FVIFi,n2+... +FVIFi,1 +FVIFi,0]

As you can see, FVAn is simply equal to the periodic receipt (R) times the sum of the future

value interest factors at ipercent for time periods 0 to n1. Luckily, we have two shorthand

ways of stating this mathematically:


FVAn = R([(1+i)n -1] -1]/i) or

FVAn=R(FVIFAi,n)

where

FVIFAi,n stands for the future value interest factor of an annuity at i% for n periods.

we can write the general formula for the present value of an (ordinary) annuity for nperiods

(PVAn) as PVAn=R[1/(1 +i)1] +R[1/(1 +i)2] +... +R[1/(1 +i)n]

=R[PVIFi,1 +PVIFi,2 +... +PVIFi,n]

PVAn = R[(1-[1/(1+i)n ])/i]

and can be expressed even more simply as

PVAn=R(PVIFAi,n)

The present value of the $1,000 annuity for three years at 8 percent The PVIFA8%,3 is found

from the table to be 2.577. (Notice this figure is nothing more than the sum of the first three

numbers under the 8% column in Table of PVA) Employing Eq. (3.11), we get

PVA3=$1,000(PVIFA8%,3)

=$1,000(2.577) =$2,577

Unknown Interest (or Discount) Rate

A rearrangement of the basic future value (present value) of an annuity equation can be used

to solve for the compound interest (discount) rate implicit in an annuity if we know: (1) the

annuitys future (present) value, (2) the periodic payment or receipt, and (3) the number of

periods involved. Suppose that you need to have at least $9,500 at the end of 8 years in order

to send your parents on a luxury cruise. To accumulate this sum, you have decided to deposit

$1,000 at the end of each of the next 8 years in a bank savings account. If the bank compounds

interest annually, what minimum compound annual interest rate must the bank offer for your

savings plan to work?


FVA8=R(FVIFAi,8)

$9,500 =$1,000(FVIFAi,8)

FVIFAi,8 =$9,500/$1,000 =9.5

Reading across the 8-period row in Table 3.5, we look for the future value interest factor of an

annuity (FVIFA) that comes closest to our calculated value of 9.5. In our table, that interest

factor is 9.549 and is found in the 5% column. Because 9.549 is slightly larger than 9.5, we

conclude that the interest rate implicit in the example situation is actually slightly less than 5

percent. (For a more accurate answer, you would need to rely on trial-and-error testing of

different interest rates, interpolation, or a financial calculator)

Unknown Periodic Payment (or Receipt)

When dealing with annuities, one frequently encounters situations in which either the future

(or present) value of the annuity, the interest rate, and the number of periodic payments (or

receipts) are known. What needs to be deter-mined, however, is the size of each equal

payment or receipt. In a business setting, we most frequently encounter the need to determine

periodic annuity payments in sinking fund (i.e., building up a fund through equal-dollar

payments) and loan amortization (i.e., extinguishing a loan through equal-dollar payments)

problems.

How much must one deposit each year end in a savings account earning 5 percent com-pound

annual interest to accumulate $10,000 at the end of 8 years? We compute the payment (R)

going into the savings account each year with the help of future value of an annuity In addition,

we use Table to find the value corresponding to FVIFA5%,8 and proceed as follows:

FVA8=R(FVIFA5%,8)

$10,000 =R(9.549)

R=$10,000/9.549 =$1,047.23

Therefore, by making eight year-end deposits of $1,047.23 each into a savings account earning

5 percent compound annual interest, we will build up a sum totaling $10,000 at the end of 8

years


Perpetuity

An ordinary annuity whose payments or receipts continue forever

Perpetuity is an ordinary annuity whose payments or receipts continue for-ever. The ability to

determine the present value of this special type of annuity will be required when we value

perpetual bonds and preferred stock in the next chapter.

PVA=R [(1 [1/ (1 +i) ])/i]

Because the bracketed term [1/(1 +i)] approaches zero, we can rewrite Eq as

PVA=R[(1 0)/i] =R(1/i)

Or simply

PVA=R/i

Thus the present value of a perpetuity is simply the periodic receipt (payment) divided by the

interest rate per period. For example, if $100 is received each year forever and the interest rate

is 8 percent, the present value of this perpetuity is $1,250 (that is, $100/0.08).

Annuity Due:

In contrast to an ordinary annuity, where payments or receipts occur at the

End of each period, an annuity due calls for a series of equal payments occurring at the

beginning of each period Luckily, only a slight modification to the procedures already outlined

for the treatment of ordinary annuities will allow us to solve annuity due problems.

FVADn=R(FVIFAi,n)(1 +i)

PVADn=(1 +i)(R)(PVIFAi,n)


Compounding More Than Once a Year

Nominal (stated) interest rate:

A rate of interest quoted for a year that has not been adjusted for frequency of

compounding. If interest is compounded more than once a year, the effective interest

rate will be higher than the nominal rate

Future (or Compound) Value

Up to now, we have assumed that interest is paid annually. It is easiest to get a basic

understanding of the time value of money with this assumption. Now, however, it is

time to consider the relationship between future value and interest rates for different

compounding periods. To begin, suppose that interest is paid semiannually. If you then

deposit $100 in a savings account at a nominal, or stated, 8 percent annual interest rate,

the future value at the end of six months would be

FV0.5=$100(1 +[0.08/2]) =$104

In other words, at the end of one half-year you would receive 4 percent in interest, not

8 per-cents. At the end of a year the future value of the deposit would be

FV1=$100(1 +[0.08/2])2

=$108.16

This amount compares with $108 if interest is paid only once a year. The $0.16

difference is caused by interest being earned in the second six months on the $4 in

interest paid at the end of the first six months. The more times during the year that

interest is paid, the greater the future value at the end of a given year. The general

formula for solving for the future value at the end of n years where interest is paid m

times a year is


FVn=PV0(1 +[i/m])mn

To illustrate, suppose that now interest is paid quarterly and that you wish to

know the future value of $100 at the end of one year where the stated annual

rate is 8 percent. The future value would be

FV1=$100(1 +[0.08/4])(4)(1)

=$100(1 +0.02)4

=$108.24

Which, of course, is higher than it would be with either semiannual or annual

compounding. The future value at the end of three years for the example with

quarterly compounding is

FV3=$100(1 +[0.08/4])(4)(3)

=$100(1 +0.02)12

=$126.82

Compared with a future value with semiannual compounding of

FV3=$100(1 +[0.08/2])(2)(3)

=$100(1 +0.04)6

=$126.53

And with annual compounding of

FV3=$100(1 +[0.08/1])(1)(3)

=$100(1 +0.08)3

=$125.97

Present (or Discounted) Value


When interest is compounded more than once a year, the formula for calculating

present value must be revised along the same lines as for the calculation of future

value Instead of dividing the future cash flow by (1 +i)n as we do when annual

compounding is involved, we determine the present value by

PV0=FVn/(1 +[i/m])mn

Where, as before, FVn is the future cash flow to be received at the end of year n,

m is the number of times a year interest is compounded, and iis the discount rate.

for example, to calculate the present value of $100 to be received at the end of

year 3 for a nominal discount rate of 8 percent compounded quarterly:

PV0=$100/(1 +[0.08/4])(4)(3)

=$100/(1 +0.02)12

=$78.85

If the discount rate is compounded only annually, we have

PV0=$100/(1 +0.08)3

=$79.38

Thus, the fewer times a year that the nominal discount rate is compounded, the

greater the present value this relationship is just the opposite of that for future

values.

Continuous Compounding

In practice, interest is sometimes compounded continuously. Therefore it is useful

to consider how this works. Recall that the general formula for solving for the

future value at the end of

year n,


FVn=PV0(1 +[i/m])mn

As m, the number of times a year that interest is compounded, approaches

infinity (), we get continuous compounding, and the term (1 +[i/m])mn

approaches e in, where e is approximately 2.71828. Therefore the future value at

the end of n years of an initial deposit of PV0 where interest is compounded

continuously at a rate of I percent is

FVn=PV0 (e) in

For our earlier example problem, the future value of a $100 deposit at the end of

three years with continuous compounding at 8 percent would be

FV3=$100(e)(0.08)(3)

=$100(2.71828)(0.24)

=$127.12

This compares with a future value with annual compounding of

FV3=$100(1 +0.08)3

=$125.97

Continuous compounding results in the maximum possible future value at the end

of n periods for a given nominal rate of interest. By the same token, when interest

is compounded continuously, the formula for the present value of a cash flow

received at the end of year n is

PV0=FVn/ (e)in

Thus the present value of $1,000 to be received at the end of 10 years with a

discount rate of 20 percent, compounded continuously, is

PV0=$1,000/(e)(0.20)(10)

=$1,000/(2.71828)2

=$135.34


We see then that present value calculations involving continuous compounding

are merely the reciprocals of future value calculations. Also, although continuous

compounding results in the maximum possible future value, it results in the

minimum possible present value.

Summary Table of Key Compound Interest Formulas

Single Amounts:

FVn=P0(1 +i)n

=P0(FVIFi,n)

PV0=FVn[1/(1 +i)n]

=FVn(PVIFi,n)

Annuities:

01. FVAn=R([(1 +i)n1]/i)

=R(FVIFAi,n)

02. PVAn=R[(1 [1/(1 +i)n])/i] (3.10)

=R(PVIFAi,n)

03. FVADn=R(FVIFAi,n)(1 +i)

04. PVADn=R(PVIFAi,n1+1)

=(1 +i)(R)(PVIFAi,n)


Problems

01. The following are exercises in future (terminal) values:

a. At the end of three years, how much is an initial deposit of $100 worth, assuming a

compound annual interest rate of (i) 100 percent? (ii) 10 percent? (iii) 0 percent?

b. At the end of five years, how much is an initial $500 deposit followed by five year-end,

annual $100 payments worth, assuming a compound annual interest rate of (i) 10 per-cent? (ii)

5 percent? (iii) 0 percent?

c. At the end of six years, how much is an initial $500 deposit followed by five year-end, annual

$100 payments worth, assuming a compound annual interest rate of (i) 10 per-cent? (ii) 5

percent? (iii) 0 percent?

d. At the end of three years, how much is an initial $100 deposit worth, assuming a quar-terly

compounded annual interest rate of (i) 100 percent? (ii) 10 percent?

e. Why do your answers to Part (d) differ from those to Part (a)?

f. At the end of 10 years, how much is a $100 initial deposit worth, assuming an annual interest

rate of 10 percent compounded (i) annually? (ii) Semiannually? (iii) Quarterly? (iv)

Continuously?

02.The following are exercises in present values:

a. $100 at the end of three years is worth how much today, assuming a discount rate of

(i) 100 percent? (ii) 10 percent? (iii) 0 percent?

b. What is the aggregate present value of $500 received at the end of each of the next three

years, assuming a discount rate of (i) 4 percent? (ii) 25 percent?

c. $100 is received at the end of one year, $500 at the end of two years, and $1,000 at the end

of three years. What is the aggregate present value of these receipts, assuming a discount rate

of (i) 4 percent? (ii) 25 percent?

d. $1,000 is to be received at the end of one year, $500 at the end of two years, and $100


at the end of three years. What is the aggregate present value of these receipts assum-ing a

discount rate of (i) 4 percent? (ii) 25 percent?

e. Compare your solutions in Part (c) with those in Part (d) and explain the reason for the

differences.

03. Joe Hernandez has inherited $25,000 and wishes to purchase an annuity that will provide him with a steady income over the next 12 years. He has heard that the local savings and loan

association is currently paying 6 percent compound interest on an annual basis. If he were to

deposit his funds, what year-end equal-dollar amount (to the nearest dollar) would he be able

to withdraw annually such that he would have a zero balance after his last withdrawal 12 years

from now?

04. You need to have $50,000 at the end of 10 years. To accumulate this sum, you have decided to save a certain amount at the endof each of the next 10 years and deposit it in the

bank. The bank pays 8 percent interest compounded annually for long-term deposits. How

much will you have to save each year (to the nearest dollar)?

05. Same as Problem 4 above, except that you deposit a certain amount at the beginningof each of the next 10 years. Now, how much will you have to save each year (to the nearest

dollar)?

06. Vernal Equinox wishes to borrow $10,000 for three years. A group of individuals agrees to lend him this amount if he contracts to pay them $16,000 at the end of the three years. What is

the implicit compound annual interest rate implied by this contract (to the nearest whole

percent)?

07. You have been offered a note with four years to maturity, which will pay $3,000 at the end of each of the four years. The price of the note to you is $10,200. What is the implicit

compound annual interest rate you will receive (to the nearest whole percent)?

08. Sales of the P.J. Cramer Company were $500,000 this year, and they are expected to grow

at a compound rate of 20 percent for the next six years. What will be the sales figure at the end

of each of the next six years?

09. Suppose you were to receive $1,000 at the end of 10 years. If your opportunity rate is 10

percent, what is the present value of this amount if interest is compounded (a) annu-ally? (b)

Quarterly? (c) Continuously?


10. The Happy Hang Glide Company is purchasing a building and has obtained a $190,000

mortgage loan for 20 years. The loan bears a compound annual interest rate of 17 percent and

calls for equal annual installment payments at the end of each of the 20 years. What is the

amount of the annual payment?

11. You have borrowed $14,300 at a compound annual interest rate of 15 percent. You feel

that you will be able to make annual payments of $3,000 per year on your loan. (Payments

include both principal and interest.) How long will it be before the loan is entirely paid off (to

the nearest year)?

12. Lost Dutchman Mines, Inc., is considering investing in Peru. It makes a bid to the

government to participate in the development of a mine, the profits of which will be realized at

the end of five years. The mine is expected to produce $5 million in cash to Lost Dutchman

Mines at that time. Other than the bid at the outset, no other cash flows will occur, as the

government will reimburse the company for all costs. If Lost Dutchman requires a nominal

annual return of 20 percent (ignoring any tax consequences), what is the maximum bid it

should make for the participation right if interest is compounded (a) annually? (b)

Semiannually? (c) Quarterly? (d) Continuously?

13. Suppose that an investment promises to pay a nominal 9.6 percent annual rate of inter-est.

What is the effective annual interest rate on this investment assuming that interest is

compounded (a) annually? (b) Semiannually? (c) Quarterly? (d) Monthly? (e) Daily (365 days)?

(f) Continuously? (Note: Report your answers accurate to four decimal places e.g., 0.0987 or

(9.87 %.)


Solution

01.

a) FVn= P0(1 + i) n

(i) FV3= $100(2.0)3 = $100(8) = $800

(ii) FV3= $100(1.10)3 = $100(1.331) = $133.10

(iii) FV3= $100(1.0)3 = $100(1) = $100

b) FVn= P0(1 + i)n; FVAn= R[([1 + i]n- 1)/i]

(i) FV5= $500(1.10)5= $500(1.611) = $ 805.50

FVA5= $100[([1.10]5- 1)/(.10)] = $100(6.105) = 610.50

= $1,416.00

(ii) FV5= $500(1.05)5 = $500(1.276) = $ 638.00

FVA5= $100[([1.05]5- 1)/(.05)] = $100(5.526) = 552.60

=$1,190.60

(iii) FV5= $500(1.0)5= $500(1) = $ 500.00

FVA5= $100(5)* = 500.00

= $1,000.00

*[Note: We had to invoke l'Hospital's rule in the special case where i = 0; in short, FVIFAn= n

when i = 0.]

c) FVn= P0(1 + i)n ; FVADn= R[([1 + i]n - 1)/i][1 + i]

(i) FV6= $500(1.10)6= $500(1.772) = $ 886.00

FVAD5= $100[([1.10]5- 1)/(.10)] x [1.10] =

$100(6.105)(1.10) = 671.55


=$1,557.55

(ii) FV6= $500(1.05)6 = $500(1.340) = $ 670.00

FVAD5= $100[([1.05]5- 1)/(.05)] x [1.05] =

$100(5.526)(1.05) = 580.23

=$1,250.23

(iii) FV6= $500(1.0)6= $500(1) = $ 500.00

FVAD5= $100(5) = 500.00

$1,000.00

d) FVn= PV0(1 + [i/m])mn

(i) FV3= $100(1 + [1/4])12= $100(14.552) = $1,455.20

(ii) FV3= $100(1 + [.10/4])12 = $100(1.345) = $ 134.50

e) The more times a year interest is paid, the greater the future value. It is particularly

important when the interest rate is high, as evidenced by the difference in solutions between

Parts 1.a) (i) and 1.d) (i).

f) FVn= PV0(1 + [i/m])mn; FVn= PV0(e)in

(i) $100(1 + [.10/1])10= $100(2.594) = $259.40

(ii) $100(1 + [.10/2])20 = $100(2.653) = $265.30

(iii) $100(1 + [.10/4])40 = $100(2.685) = $268.50

(iv) $100(2.71828)1 = $271.83

2. a) P0= FVn[1/(1 + i)n]

(i) $100[1/(2)3] = $100(.125) = $12.50

(ii) $100[1/(1.10)3] = $100(.751) = $75.10

(iii) $100[1/(1.0)3] = $100(1) = $100


b) PVAn= R[(1 -[1/(1 + i)n])/i]

(i) $500[(1 -[1/(1 + .04)3])/.04] = $500(2.775) = $1,387.50

(ii) $500[(1 -[1/(1 + .25)3])/.25] = $500(1.952) = $ 976.00

c) P0= FVn[1/(1 + i)n]

(i) $ 100[1/(1.04)1] = $ 100(.962) = $ 96.20

500[1/(1.04)2] = 500(.925) = 462.50

1,000[1/(1.04)3 ] = 1,000(.889) = 889.00

$1,447.70

(ii) $ 100[1/(1.25)1] = $ 100(.800) = $ 80.00

500[1/(1.25)2 ] = 500(.640) = 320.00

1,000[1/(1.25)3] = 1,000(.512) = 512.00

$ 912.00

d) (i) $1,000[1/(1.04)1] = $1,000(.962) = $ 962.00

500[1/(1.04)2] = 500(.925) = 462.50

100[1/(1.04)3] = 100(.889) = 88.90

$1,513.40

(ii) $1,000[1/(1.25)1] = $1,000(.800) = $ 800.00

500[1/(1.25)2] = 500(.640) = 320.00

100[1/(1.25)3] = 100(.512) = 51.20

$1,171.20


e) The fact that the cash flows are larger in the first period for the sequence in

Part (d) results in their having a higher present value. The comparison illustrates

the desirability of early cash flows.

3. $25,000 = R(PVIFA6%,12) = R(8.384)

R = $25,000/8.384 = $2,982

4. $50,000 = R(FVIFA8%,10) = R(14.486)

R = $50,000/14.486 = $3,452

5. $50,000 = R(FVIFA8%,10)(1 + .08) = R(15.645)

R = $50,000/15.645 = $3,196

6. $10,000 = $16,000(PVIFx%,3)

(PVIFx%,3) = $10,000/$16,000 = 0.625

Going to the PVIF table at the back of the book and looking across the row for n =

3, we find that the discount factor for 17 percent is 0.624 and that is closest to

the number above.

7. $10,000 = $3,000(PVIFAx%, 4) (PVIFAx%,4) = $10,200/$3,000 = 3.4

Going to the PVIFA table at the back of the book and looking across the row for

n = 4, we find that the discount factor for 6 percent is 3.465, while for 7 percent it

is 3.387. Therefore, the note has an implied interest rate of almost 7 percent.


8. Year Sales

1 $ 600,000 = $ 500,000(1.2)

2 720,000 = 600,000(1.2)

3 864,000 = 720,000(1.2)

4 1,036,800 = 864,000(1.2)

5 1,244,160 = 1,036,800(1.2)

6 1,492,992 = 1,244,160(1.2)

09. Amount Present Value Interest Factor Present Value

$1,000 1/(1 + .10)10 = .386 $386

1,000 1/(1 + .025)40 = .372 372

1,000 1/e(.10)(10) = .368 368

10. $190,000 = R (PVIFA17%,20) = R(5.628)

R = $190,000/5.628 = $33,760

11. $14,300 = $3,000(PVIFA15%,n) (PVIFA15%,n) = $14,300/$3,000 = 4.767

Going to the PVIFA table at the back of the book and looking down the column

for i = 15%, we find that the discount factor for 8 years is 4.487, while the

discount factor for 9 years is 4.772. Thus, it will take approximately 9 years of

payments before the loan is retired.


12. A) $5,000,000 = R[1 + (.20/1)]5= R(2.488)

R = $5,000,000/2.488 = $2,009,646

b) $5,000,000 = R[1 + (.20/2)]10= R(2.594)

R = $5,000,000/2.594 = $1,927,525

c) $5,000,000 = R[1 + (.20/4)]20= R(2.653)

R = $5,000,000/2.653 = $1,884,659

d) $5,000,000 = R(e)(.20)(5) = R(2.71828)(1)

R = $5,000,000/2.71828 = $1,839,398

13. Effective annual interest rate = (1 + [i/m])m- 1

a. (annually) = (1 + [.096/1])1- 1 = .0960

b. (semiannually) = (1 + [.096/2])2- 1 = .0983

c. (quarterly) = (1 + [.096/4])4- 1 = .0995

d. (monthly) = (1 + [.096/12])12- 1 = .1003

e. (daily) = (1 + [.096/365])365- 1 = .1007

Effective annual interest rate with continuous compounding = (e)i- 1

f. (continuous) = (2.71828).096- 1 = .1008


Chapter#03

The Valuation of Long-Term Securities

Distinctions among Valuation Concepts

Liquidation value:

The amount of money that could be realized if an asset or a group of assets (e.g.,

a firm) is sold separately from its operating organization

Going-concern value

The amount a firm could be sold for as a continuing operating business

Book value

(1) An asset: the accounting value of an asset the assets cost minus its

accumulated depreciation; (2) a firm: total assets minus liabilities and preferred

stock as listed on the balance sheet

Market value

The market price at which an asset trades

Intrinsic value

The price a security ought to have based on all factors bearing on valuation

Bond Valuation

Bond

A long-term debt instrument issued by a corporation or government

Face value

The stated value of an asset In the case of a bond, the face value is usually $1,000.

Coupon rate


The stated rate of interest on a bond; the annual interest payment divided by the

bonds face value

Perpetual Bonds

The first (and easiest) place to start determining the value of bonds is with a

unique class of bonds that never matures. These are indeed rare, but they help

illustrate the valuation technique in its simplest form. Originally issued by Great

Britain after the Napoleonic Wars to consolidate debt issues, the British consol

(short for consolidated annuities) is one such example. This bond carries the

obligation of the British government to pay a fixed interest payment in perpetuity.

The present value of a perpetual bond would simply be equal to the capitalized

value of an infinite stream of interest payments. If a bond promises a fixed annual

payment of I forever, its present (intrinsic) value, V, at the investors required rate

of return for this debt issue, kd , is

V= i/(1+kd)1 + i/(1+kd)2++ i/(1+kd)

V= (i/1+kd) t

V=I(PVIFAkd,)

we know should reduce to

V=I/kd

Consol

A bond that never matures; perpetuity in the form of a bond

Thus the present value of a perpetual bond is simply the periodic interest

payment divided by the appropriate discount rate per period. Suppose you could

buy a bond that paid $50 a year forever. Assuming that your required rate of


return for this type of bond is 12 percent, the present value of this security would

be

V=$50/0.12 =$416.67

This is the maximum amount that you would be willing to pay for this bond. If the

market price is greater than this amount, however, you would not want to buy it

Bonds with a Finite Maturity

Non zero Coupon Bonds

If a bond has a finite maturity, then we must consider not only the interest stream

but also the terminal or maturity value (face value) in valuing the bond. The

valuation equation for such a bond that pays interest at the end of each year is

V= i/(1+kd)1 + i/(1+kd)2++ i/(1+kd) n + MV/(1+kd)n

V= (i/1+kd)t + MV/(1+kd)n

=I(PVIFAkd,n) +MV(PVIFkd,n)

Where n is the number of years until final maturity and MVis the maturity value

of the bond. We might wish to determine the value of a $1,000-par-value bond

with a 10 percent coupon and nine years to maturity. The coupon rate

corresponds to interest payments of $100 a year. If our required rate of return on

the bond is 12 percent, then

V= $100/(1+.12)1 +$100/(1+.12)1 +..+$100/(1+.12)9 + $1000/(1+.12)9

V=$100(5.328) +$1,000(0.361)

=$532.80 +$361.00 =$893.80


Zero-coupon bond

A bond that pays no interest but sells at a deep discount from its face

value; it provides compensation to investors in the form of price

appreciation.

A zero-coupon bond makes no periodic interest payments but instead is sold

at a deep discount from its face value. Why buy a bond that pays no

interest? The answer lies in the fact that the buyer of such a bond does

receive a return. This return consists of the gradual increase (or

appreciation) in the value of the security from its original, below-face-value

purchase price until it is redeemed at face value on its maturity date.

The valuation equation for a zero-coupon bond is a truncated version of that

used for a normal interest-paying bond. The present value of interest

payments component is lopped off, and we are left with value being

determined solely by the present value of principal payment at maturity,

or

V= MV/(1+kd)n

=MV(PVIFkd,n)

Suppose that Espinosa Enterprises issues a zero-coupon bond having a 10-year

maturity and a $1,000 face value. If youre required return is 12 percent, then

V=$1,000/(1.12)10

V= $322


Semiannual Compounding of Interest

Although some bonds (typically those issued in European markets) make interest

payments once a year, most bonds issued in the United States pay interest twice a

year. As a result, it is necessary to modify our bond valuation equations to

account for compounding twice a year.

V= (i/2/1+kd/2)t + MV/(1+kd/2)2n

=I/2(PVIFAkd/2, 2n) +MV(PVIFkd/2, 2n)

Where kd is the nominal annual required rate of interest, I/2 is the semiannual

coupon payment, and 2n is the number of semiannual periods until maturity.

Preferred Stock Valuation

Preferred stock:

A type of stock that promises a (usually) fixed dividend, but at the discretion of

the board of directors It has preference over common stock in the payment of

dividends and claims on assets

Most preferred stock pays a fixed dividend at regular intervals. The features of

this financial instrument are discussed in Chapter 20. Preferred stock has no

stated maturity date and, given the fixed nature of its payments, is similar to a

perpetual bond. It is not surprising, then, that we use the same general approach

as applied to valuing a perpetual bond to the valuation of preferred stock.

Thus the present value of preferred stock is

V=Dp/kp

Where Dp is the stated annual dividend per share of preferred stock and kpis the

appropriate discount rate. If Margana Cipher Corporation had a 9 percent, $100-

par-value preferred stock issue outstanding and your required return was 14

percent on this investment, its value per share to you would be


V=$9/0.14 =$64.29

Common Stock Valuation

Common stock:

Securities that represent the ultimate ownership (and risk) position in a

corporation

Are Dividends the Foundation?

When valuing bonds and preferred stock, we determined the discounted value of

all the cash distributions made by the firm to the investor. In a similar fashion, the

value of a share of common stock can be viewed as the discounted value of all

expected cash dividends provided by the issuing firm until the end of time

V= D1/(1+ke)1 + D2/(1+ke)2 +..+ D/(1+ke)

V= Dt/(1+ke)t

Where Dt is the cash dividend at the end of time period t and ke is the investors

required return, or capitalization rate, for this equity investment. This seems

consistent with what we have been doing so far. But what if we plan to own the

stock for only two years? In this case, our model becomes

V= D1/(1+ke)1 + D2/(1+ke)2 + P2/(1+ke)2

Where P2 is the expected sales price of our stock at the end of two years this

assumes that investors will be willing to buy our stock two years from now. In

turn, these future investors will base their judgments of what the stock is worth

on expectations of future dividends and a future selling price (or terminal value).

And so the process goes through successive investors


Dividend Discount Models

Dividend discount models are designed to compute the intrinsic value of a share

of common stock under specific assumptions as to the expected growth pattern

of future dividends and the appropriate discount rate to employ. Merrill Lynch, CS

First Boston, and a number of other investment banks routinely make such

calculations based on their own particular models and estimates. What follows is

an examination of such models, beginning with the simplest one.

Constant Growth

Future dividends of a company could jump all over the place; but, if dividends are

expected to grow at a constant rate, what implication does this hold for our basic

stock valuation approach? If this constant rate is g, then

V= D0(1+g)/(1+ke)1 + D0(1+g)/(1+ke)2 +..+ D0(1+g)/(1+ke)

Where D0 is the present dividend per share Thus the dividend expected at the end

of period n is equal to the most recent dividend times the compound growth

factor, (1 +g)n This may not look like much of an improvement over. However,

assuming that ke is greater than g (a reasonable assumption because a dividend

growth rate that is always greater than the capitalization rate would imply an

infinite stock value), can be reduced to

V=D1/(keg)

Rearranging, the investors required return can be expressed as

ke=(D1/V) +g

Suppose that LKN, Inc.s dividend per share at t=1 is expected to be $4, that it is

expected to grow at a 6 percent rate forever, and that the appropriate discount

rate is 14 percent. The value of one share of LKN stock would be


V=$4 / (0.14 0.06) =$50

If we multiply both sides of Eq. (4.13) by (1 +ke)/(1 +g) and subtract Eq.

D0(1+g)/(1+ke)1 + D0(1+g)/(1+ke)2 +..+ D0(1+g)/(1+ke) from

the product, we get

[V (1+ke)/ (1+g)] V = Do [(1+g)/(1+ke)]

Because we assume that ke is greater than g, the second term on the right-hand

side approaches zero. Consequently,

V([(1+ke)/ (1+g)]-1)=Do

V([(1+ke)-(1+g)/ (1+g)])=Do V (ke g) =D0(1 +g) =D1

V=D1/(ke g)

This model is sometimes called the Gordon Dividend Valuation Model

after Myron J. Gordon, who developed it from the pioneering work done by

John Williams. See Myron J. Gordon, the Investment, Financing, and

Valuation of the Corporation (Homewood, IL: Richard D. Irwin, 1962).

No Growth

A special case of the constant growth dividend model calls for an expected

Dividend growth rate, g, of zero Here the assumption is that dividends will be

maintained at their current level forever. In this case, Eq. (4.14) reduces to

V=D1/ke


Not many stocks can be expected simply to maintain a constant dividend forever.

However, when a stable dividend is expected to be maintained for a long period

of time, Eq. V=D1/ke can provide a good approximation of value.

Growth Phases

When the pattern of expected dividend growth is such that a constant growth

model is not appropriate, modifications A number of valuation models are based

on the premise that firms may exhibit above-normal growth for a number of years

(g may even be larger than ke during this phase), but eventually the growth rate

will taper off. Thus the transition might well be from a currently above-normal

growth rate to one that is considered normal. If dividends per share are expected

to grow at a 10 percent compound rate for five years and thereafter at a 6

percent rate, Eq.

D0(1+g)/(1+ke)1 + D0(1+g)/(1+ke)2 +..+ D0(1+g)/(1+ke)

becomes

V=[ Do(1.10)t/(1+ke)t] + [D5(1.06)t-5/(1+ke)t] Note that the growth in dividends in the second phase uses the expected dividend

in period 5 as its foundation. Therefore the growth-term exponent is t5, which

means that the exponent in period 6 equals 1, in period 7 it equals 2, and so forth.

This second phase is nothing more than a constant-growth model following a

period of above-normal growth. We can make use of that fact to rewrite Eq.

[Do(1.10)t/(1+ke)t] + [D5(1.06)t-5/(1+ke)t] as follows:

V= [Do(1.10)t/(1+ke)t]+ [1/(1+ke)5][(D6/ke-0.06)]


If the current dividend, D0 , is $2 per share and the required rate of return, ke , is

14 percent, we could solve for V.

V= [$2(1.10)t/(1.14)t]+ [1/(1.14)5][($3.41/0.14-0.06)] =$8.99 +$22.13 =$31.12

PHASE 1: PRESENT VALUE OF DIVIDENDS

TO BE RECEIVED OVER FIRST 5 YEARS

END OF YEAR PRESENT VALUE CALCULATION PRESENT VALUE

(DIVIDEND PVIF14%,t ) OF DIVIDEND

1 $2(1.10)1 = $2.20 0.877 = $1.93

2 2(1.10)2= 2.42 0.769 = 1.86

3 2(1.10)3= 2.66 0.675 = 1.80

4 2(1.10)4= 2.93 0.592 = 1.73

5 2(1.10)5= 3.22 0.519 = 1.6

Or

[ $2(1.10)t/(1.14)t] = $8.99


PHASE 2: PRESENT VALUE OF CONSTANT GROWTH COMPONENT

Dividend at the end of year 6 =$3.22(1.06) =$3.41

Value of stock at the end of year 5 =D6/(ke g) =$3.41/(0.14 0.06) =$42.63

Present value of $42.63 at end of year 5 =($42.63)(PVIF14%,5)

=($42.63)(0.519) =$22.13

PRESENT VALUE OF STOCK

V=$8.99 +$22.13 =$31.12

The transition from an above-normal rate of dividend growth could be specified as more gradual than the two-phase approach just illustrated. We might expect dividends to grow at a

10 percent rate for five years, followed by an 8 percent rate for the next five years and a 6 per-cent growth rate thereafter. The more growth segments that are added, the more closely the growth in dividends will approach a curvilinear function. But no firm can grow at an above-normal rate forever. Typically, companies tend to grow at a very high rate initially, after which their growth opportunities slow down to a rate that is normal for companies in general. If maturity is reached, the growth rate may stop altogether.

Rates of Return (or Yields)

Yield to maturity:


(YTM)The expected rate of return on a bond if bought at its current market price

and held to maturity

Yield to Maturity (YTM) on Bonds

The market required rate of return on a bond (kd) is more commonly referred to

as the bonds yield to maturity. Yield to maturity (YTM) is the expected rate of

return on a bond if bought at its current market price and held to maturity; it is

also known as the bonds internal rate of return (IRR).Mathematically, it is the

discount rate that equates the present value of all expected interest payments

and the payment of principal (face value) at maturity with the bonds current

market price. For an example, lets return to Eq. (V= (i/1+kd)t + MV/(1+kd)n ),

the valuation equation for an interest-bearing bond with a finite maturity.

Replacing intrinsic value (V) with current market price (P0 ) gives us

Po= i/(1+kd)t + MV/(1+kd)n

If we now substitute actual values for I, MV, and P0 , we can solve for kd,

which in this case would be the bonds yield to maturity. However, the

precise calculation for yield to maturity is rather complex and requires bond

value tables, or a sophisticated handheld calculator, or a computer

Interpolation

Interpolate Estimate an unknown number that lies somewhere between two

known numbers

If all we have to work with are present value tables, we can still determine

an approximation of the yield to maturity by making use of a trial-and-error

procedure. To illustrate, consider a $1,000-par-value bond with the

following characteristics: a current market price of $761, 12 years until

maturity, and an 8 percent coupon rate (with interest paid annually). We

want to determine the discount rate that sets the present value of the

bonds expected future cash-flow stream equal to the bonds current market


price. Suppose that we start with a 10 percent discount rate and calculate

the present value of the bonds expected future cash flows. For the

appropriate present value interest factors, we make use of Tables II and IV in

the Appendix at the end of the book.

V=$80(PVIFA10%,12) +$1,000(PVIF10%,12)

=$80(6.814) +$1,000(0.319) =$864.12

A 10 percent discount rate produces a resulting present value for the bond

that is greater than the current market price of $761. Therefore we need to

try a higher discount rate to handicap the future cash flows further and

drive their present value down to $761. Lets try a 15 percent discount rate:

V=$80(PVIFA15%,12) +$1,000(PVIF15%,12)

=$80(5.421) +$1,000(0.187) =$620.68

This time the chosen discount rate was too large. The resulting present

value is less than the current market price of $761. The rate necessary to

discount the bonds expected cash flows to $761 must fall somewhere

between 10 and 15 percent

This time the chosen discount rate was too large. The resulting present

value is less than the current market price of $761. The rate necessary to

discount the bonds expected cash flows to $761 must fall somewhere

between 10 and 15 percent. To approximate this discount rate, we

interpolate between 10 and 15 percent as follows


0.10 $864.12

0.05 X YTM $761.00 $103.12 $243.44

0.15 $620.68

X/0.05=$103.12/$243.44 Therefore, (0.05)X($103.12)/$243.44 =0.0212

In this example X=YTM0.10. Therefore, YTM =0.10 +X=0.10 +0.0212 =0.1212, or 12.12 percent.

The use of a computer provides a precise yield to maturity of 11.82 percent. It is important to

keep in mind that interpolation gives only an approximation of the exact percentage; the

relationship between the two discount rates is not linear with respect to present value.

However, the tighter the range of discount rates that we use in interpolation, the closer the

resulting answer will be to the mathematically correct one. For example, had we used 11 and

12 percent, we would have come even closer to the true yield to maturity.

Behavior of Bond Prices

1. When the market required rate of return is more than the stated coupon rate, the price of

the bond will be less than its face value. Such a bond is said to be selling at a discount from face

value. The amount by which the face value exceeds the current price is the

Bond discount

Bond discount the amount by which the face value of a bond exceeds its current price

2. When the market required rate of return is less than the stated coupon rate, the price of the

bond will be more than its face value. Such a bond is said to be selling at a premium over face

value. The amount by which the current price exceeds the face value is the

Bond premium

Bond premium the amount by which the current price of a bond exceeds its face value

3. When the market required rate of return equals the stated coupon rate, the price of the

bond will equal its face value. Such a bond is said to be selling at par

Key Point:

a. If a bond sells at a discount, then P0coupon rate

b. If a bond sells at par, then P0=par and YTM =coupon rate.

c. If a bond sells at a premium, then P0>par and YTM


3. If interest rates rise so that the market required rate of return increases, the bonds

price will fall. If interest rates fall, the bonds price will increase. In short, interest rates

and bond prices move in opposite directions just like two ends of a childs seesaw.

From the last observation, it is clear that variability in interest rates should lead to

variability in bond prices. This variation in the market price of a security caused by

changes in interest rates is referred to as interest-rate (or yield) risk. It is important to

note that an investor incurs a loss due to interest-rate (or yield) risk only if a security is

sold prior to maturity and the level of interest rates has increased since time of

purchase. A further relationship, not as apparent as the previous four observations,

needs to be illustrated separately.

Interest-rate (or yield) risks the variation in the market price of a security caused by

changes in interest rates.

5. For a given change in market required return, the price of a bond will change by a greater

amount, the longer its maturity.

In general, the longer the maturity, the greater the price fluctuation associated with a given

change in market required return. The closer in time that you are to this relatively large

maturity value being realized, the less important are interest payments in determining the

market price, and the less important is a change in market required return on the market price

of the security. In general, then, the longer the maturity of a bond, the greater the risk of price

changes to the investor when changes occur in the overall level of interest rates.

6. For a given change in market required rate of return, the price of a bond will change by

proportionally more, the lower the coupon rate. In other words, bond price volatility is

inversely related to coupon rate.

YTM and Semiannual Compounding

As previously mentioned, most domestic bonds pay interest twice a year, not once. This real-

world complication is often ignored in an attempt to simplify discussion. We can take

semiannual interest payments into account, however, when determining yield to maturity by

replacing intrinsic value (V) with current market price (P0) in bond valuation

V= (i/2/1+kd/2)t + MV/(1+kd/2)2n


Solving for kd /2 in this equation would give us the semiannual yield to maturity.

The practice of doubling the semiannual YTM has been adopted by convention in bond circles

to provide the annualized (nominal annual) YTM or what bond traders would call the bond-

equivalent yield.The appropriate procedure, however, would be to square 1 plus the

semiannual YTM and then subtract 1: that is,

(1 +semiannual YTM)2 1 =(effective annual) YTM

Yield on Preferred Stock

Substituting current market price (P0 ) for intrinsic value (V) in preferred stock valuation

P0= Dp/kp

To illustrate, assume that the current market price per share of Acme Zarf Companys 10

percent, $100-par-value preferred stock is $91.25. Acmes preferred stock is therefore priced to

provide a yield of

kp=$10/$91.25 =10.96%

Yield on Common Stock

The rate of return that sets the discounted value of the expected cash dividends from a share of

common stock equal to the shares current market price is the yield on that common stock. If,

for example, the constant dividend growth model was appropriate to apply to the common

stock of a particular company, the current market price (P0) could be said to be

P0=D1 /(keg)

Solving for ke, which in this case is the market-determined yield on a companys common stock,

we get

ke=D1 /P0+g

From this last expression, it becomes clear that the yield on common stock comes from two

sources. The first source is the expected dividend yield, D1/P0 ; whereas the second source, g,

is the expected capital gains yield. Yes, g wears a number of hats. It is the expected com-pound


annual growth rate in dividends. But, given this model, it is also the expected annual percent

change in stock price (that is, P1/P01 =g) and, as such, is referred to as the capital gains yield.

Question:

What market yield is implied by a share of common stock currently selling for $40 whose

dividends are expected to grow at a rate of 9 percent per year and whose dividend next year is

expected to be $2.40?

Answer:

The market yield, ke , is equal to the dividend yield, D1 /P0 , plus the capital gains yield, g, as

follows:

ke=$2.40/$40 +0.09 =0.06 +0.09 =15%

Problems

1. Gonzalez Electric Company has outstanding a 10 percent bond issue with a face value of $1,000 per bond and three years to maturity. Interest is payable annually. The bonds are

privately held by sure safe Fire Insurance Company. Sure safe wishes to sell the bonds,

and is negotiating with another party. It estimates that, in current market conditions,

the bonds should provide a (nominal annual) return of 14 percent. What price per bond

should sure safe be able to realize on the sale?

2. 3. Superior Cement Company has an 8 percent preferred stock issue outstanding, with each share having a $100 face value. Currently, the yield is 10 percent. What is the

market price per share? If interest rates in general should rise so that the required

return becomes 12 percent, what will happen to the market price per share?

3. The stock of the Health Corporation is currently selling for $20 a share and is expected to pay a $1 dividend at the end of the year. If you bought the stock now and sold it for

$23 after receiving the dividend, what rate of return would you earns?


4. Delphi Products Corporation currently pays a dividend of $2 per share, and this dividend is expected to grow at a 15 percent annual rate for three years, and then at a 10 percent

rate for the next three years, after which it is expected to grow at a 5 percent rate

forever. What value would you place on the stock if an 18 percent rate of return was

required?

5. North Great Timber Company will pay a dividend of $1.50 a share next year. After this, earnings and dividends are expected to grow at a 9 percent annual rate indefinitely.

Investors currently require a rate of return of 13 percent. The company is considering

several business strategies and wishes to determine the effect of these strategies on the

market price per share of its stock.

a. Continuing the present strategy will result in the expected growth rate and required rate

of return stated above.

b. Expanding timber holdings and sales will increase the expected dividend growth rate to

11 percent but will increase the risk of the company. As a result, the rate of return required

by investors will increase to 16 percent.

c. Integrating into retail stores will increase the dividend growth rate to 10 percent and

increase the required rate of return to 14 percent.

6. Waynes Steaks, Inc., has a 9 percent, non callable, $100-par-value preferred stock issue

outstanding. On January 1 the market price per share is $73. Dividends are paid annually on

December 31. If you require a 12 percent annual return on this investment, what is this stocks

intrinsic value to you (on a per share basis) on January 1?

7. The 9-percent-coupon-rate bonds of the Melbourne Mining Company have exactly 15 years

remaining to maturity. The current market value of one of these $1,000-par-value bonds is

$700. Interest is paid semiannually. Melanie Gibson places a nominal annual required rate of

return of 14 percent on these bonds. What dollar intrinsic value should Melanie place on one of

these bonds (assuming semiannual discounting)?

8. Assume that everything stated in Problem 11 remains the same except that the bonds are

not perpetual. Instead, they have a $1,000 par value and mature in 10 years.

a. Determine the implied semiannual yield to maturity (YTM) on these bonds. (Tip: If all you

have to work with are present value tables, you can still determine an approx-imation of the


semiannual YTM by making use of a trial-and-error procedure coupled with interpolation. In

fact, the answer to Problem 11, Part (a) rounded to the near-est percent gives you a good

starting point for a trial-and-error approach.)

b. Using your answer to Part (a), what is the (nominal annual) YTM on these bonds? The

(effective annual) YTM on these bonds?

9. Burp-Cola Company just finished making an annual dividend payment of $2 per share on its

common stock. Its common stock dividend has been growing at an annual rate of 10 percent.

Kelly Scott requires a 16 percent annual return on this stock. What intrinsic value should Kelly

place on one share of Burp-Cola common stock under the following three situations?

a. Dividends are expected to continue growing at a constant 10 percent annual

rate.

b. The annual dividend growth rate is expected to decrease to 9 percent and to

remain constant at that level.

c. The annual dividend growth rate is expected to increase to 11 percent and to

remain constant at the level.

SOLUTIONS TO PROBLEMS

1. End of Discount

Year Payment Factor (14%) Present Value

1 $ 100 .877 $ 87.70

2 100 .769 76.90

3 1,100 .675 742.50

Price per bond $ 907.10

2. Current price: P0= Dp/kp = (.08)($100)/(.10) = $80.00

Later price: P0= Dp/kp = ($8)/(.12) = $66.67


$13.33

3. Rate of return = $1 dividend + ($23 - $20) capital gain /$20 original price

= $4/$20 = 20%

5. a) P0= D1/(ke- g): ($1.50)/(.13 - .09) = $37.50

b) P0= D1/(ke- g): ($1.50)/(.16 - .11) = $30.00

c) P0= D1/(ke- g): ($1.50)/(.14 - .10) = $37.50

Either the present strategy (a) or strategy (c) Both result in the same market price per share

6. V = Dp/kp

= [(.09)($100)]/(.12) = $9/ (.12) = $75

7. V = (I/2)(PVIFA7%,30) + $1,000(PVIF7%,30)

= $45(12.409) + $1,000(.131)

= $558.41 + $131 = $689.41

8. Trying a 4 percent semiannual YTM as a starting point for a trial-and-error approach, we get

P0= $45(PVIFA4%,20) + $1,000(PVIF4%,20)

= $45(13.590) + $1,000(.456)

= $611.55 + $456 = $1,067.55

Since $1,067.55 is less than $1,120, we need to try a lower discount rate, say 3 percent


P0= $45(PVIFA3%,20) + $1,000(PVIF3%,20)

= $45(14.877) + $1,000(.554)

= $669.47 + $554 = $1,223.47

To approximate the actual discount rate, we interpolate between 3 and 4 percent as

follows:

0.03 $1,223.47

0.01 X YTM $1,120.00 $103.47 $155.92

0.15 $1,067.55

X/0.01 = $103.47/ $155.92 Therefore, (.01) x ($103.47)/ $155.92 = .0066 percent. (The use of a computer provides a precise semiannual YTM bfigure of 3.64 percent.)

b) (Semiannual YTM) x (2) = (nominal annual) YTM

(.0366) x (2) = .0732

c) (1 + semiannual YTM)2- 1 = (effective annual) YTM (1 + .0366)2- 1 = .0754

9. D0(1 + g)/(ke- g) = V

a. $2(1 + .10)/(.16 - .10) = $2.20/.06 = $36.67

b. $2(1 + .09)/(.16 - .09) = $2.18/.07 = $31.14

c. $2(1 + .11)/(.16 - .11) = $2.22/.05 = $44.40


Chapter#04

Financial Statement Analysis

Financial Statements

Financial (statement) analysis:

The art of transforming data from financial statements into information that is

useful for informed decision making.

Balance sheet:

A summary of a firms financial position on a given date that shows total assets

=total liabilities +owners equity.

Income statement

A summary of a firms revenues and expenses over a specified period, ending with

net income or loss for the period

Balance Sheet Information

Cash equivalents:

Highly liquid, short-term marketable securities that are readily convertible to

known amounts of cash and generally have remaining maturities of three months

or less at the time of acquisition.

Shareholders equity

Total assets minus total liabilities Alternatively, the book value of a companys

common stock (at par) plus additional paid-in capital and retained earnings


MARCH 31

Aldine Manufacturing

Company balance sheets (in thousands) 1

ASSETS2 20X2 20X1

Cash $ 178 $ 175

Accounts receivable3 678 740

Inventories, at lower of cost or market4 1,329 1,235

Prepaid expenses5 21 17

Accumulated tax prepayments 35 29

Current assets6 $2,241 $2,196

Fixed assets at cost7 1,596 1,538

Less: Accumulated depreciation8 (857) (791)

Net fixed assets $ 739 $ 747

Investment, long term 65

Other assets, long term 205 205

Total assets9 $3,250 $3,148

LIABILITIES AND MARCH 31

SHAREHOLDERS EQUITY10,11

Documents

Financial Management