Basic Financial Tools: A Review

T he building blocks of finance include the time value of money, risk and its

relationship with rates of return, and stock and bond valuation models.

These topics are covered in introductory finance courses, but because of

their fundamental importance, we review them in this chapter.1

Time Value of Money

Time value concepts, or discounted cash ¬‚ow analysis, underlie virtually all

the important topics in ¬nancial management, including stock and bond

valuation, capital budgeting, cost of capital, and the analysis of ¬nancing

vehicles such as convertibles and leasing. Therefore, an understanding of

time value concepts is essential to anyone studying ¬nancial management.

Future Values

An investment of PV dollars today at an interest rate of i percent for n peri-

ods will grow over time to some future value (FV). The following time line

shows how this growth occurs:

2 n

1 n 1

3

0 i% ...

PV FV1 FV2 FV3 FVn FV n

1

PV(1 i) FV1 (1 i) FV2 (1 i) FVn 2 (1 i) FVn 1 (1 i)

This process is called compounding, and it can be expressed with the fol-

lowing equation:

i)n (28-1)

FVn PV(1 PV(FVIFi,n).

The term FVIFi,n is called the future value interest factor.

The future value of a series of cash ¬‚ows is the sum of the future values

of the individual cash ¬‚ows. An ordinary annuity has equal payments, with

1

This review is limited to material that is necessary to understand the chapters in the main text. For a more detailed

treatment of risk, return, and valuation models, see Chapters 2 through 5. For a more detailed review of time value of

money concepts, see Eugene F. Brigham and Michael C. Ehrhardt, Financial Management, 10th edition (Cincinnati,

OH: South-Western, 2001), or some other introductory ¬nance text.

28-2 Basic Financial Tools: A Review

Chapter 28

symbol PMT, that occur at the end of each period, and its future value

(FVAn) is found as follows:

0 1 2 3 n 1 n

...

PMT PMT PMT PMT PMT

...

PMT(1 i)n 1 PMT(1 i)n 2

Future value FVAn

PMT(1 i) PMT

(1 i)n 1

PMT c d

(28-2)

i

PMT(FVIFAi,n).

The second and third forms of Equation 28-2 represent more convenient

ways to solve the equation set forth on the ¬rst row.2 The term FVIFAi,n is

called the future value interest factor of an annuity at i percent for n periods.3

Here are some applications of these concepts. First, consider a single pay-

ment, or lump sum, of $500 made today. It will earn 7 percent per year for

25 years. This $500 present value will grow to $2,713.72 after 25 years:

0.07)25

Future value $500(1 $500(5.42743) $2,713.72. (28-1a)

Now suppose we have an annuity with 25 annual payments of $500 each,

starting a year from now, and the interest rate is 7 percent per year. The

future value of the annuity is $31,624.52:

0.07)25

$500 c d

(1 1

Future value $500(63.24904) $31,624.52. (28-2a)

0.07

A ¬nancial calculator could be used to solve this problem. On most cal-

culators, the N button is for the number of periods. We recommend setting

the calculator to one period per year, with payments occurring at the end of

the year. The I (or I/Y) button is for the interest rate as a percentage, not as

a decimal. The PV button is for the value today of the future cash ¬‚ows, the

FV button is for a lump sum cash ¬‚ow at the end of N periods, and the PMT

button is used if we have a series of equal payments that occur at the end of

each period. On some calculators the CPT button is used to compute pres-

ent and future values, interest rates, and payments. Other calculators have

different ways to enter data and ¬nd solutions, so be sure to check your spe-

ci¬c manual.4 To ¬nd the future value of the single payment in the example

above, input N 25, I 7, PV 500 (negative because it is a cash out¬‚ow),

and PMT 0 (because we have no recurring payments). Press CPT and then

the FV key to ¬nd FV $2,713.72. To calculate the future value of our

annuity, input N 25, I 7, PV 0, PMT 500, and then press CPT

2

See the Extension to Chapter 5 for a derivation of the sum of a geometric series.

3

An annuity in which payments are made at the start of the period is called an annuity due. The future value interest

factor of an annuity due is (1 i)FVIFAi,n.

4

Our Technology Supplement contains tutorials for the most commonly used ¬nancial calculators (about 12 typewrit-

ten pages versus much more for the calculator manuals). Our tutorials explain how to do everything needed in this book.

See our Preface for information on how to obtain the Technology Supplement.

28-3

Time Value of Money

and then the FV key to ¬nd FV $31,624.52. Some ¬nancial calcu-

lators will display the negative of this number.

Present Values

The value today of a future cash ¬‚ow or series of cash ¬‚ows is called the

present value (PV). The present value of a lump sum future payment, FVn,

to be received in n years and discounted at the interest rate i, is

1

PV FVn

(1 i)n (28-3)

FVn(PVIFi,n).

PVIFi,n is the present value interest factor at i percent due in n periods.

The present value of an annuity is the sum of the present values of the indi-

vidual payments. Here is the time line and formula for an ordinary annuity:

0 1 2 3 n 1 n

...

PMT PMT PMT PMT PMT

PMT PMT PMT PMT

...

Present value PVAn

(1 i)n

2

(1 i)n 1

(1 i) (1 i)

1

£

1 (28-4)

(1 i)n §

PMT

i

PMT(PVIFAi,n).

PVIFAi,n is the present value interest factor for an annuity of i percent for

n periods.5

The present value of a $500 lump sum to be received in 25 years when the

interest rate is 7 percent is $500/(1.07)25 $92.12. The present value of a

series of 25 payments of $500 each discounted at 7 percent is $5,826.79:

1

1

(1.07)25 ¢

° (28-4a)

PV $500 500(11.65358)

0.07

$5,826.79.

Present values can be calculated using a ¬nancial calculator. For the lump

sum payment, enter N 25, I 7, PMT 0, and FV 500, and then

press CPT and then the PV key to ¬nd PV $92.12. For the annuity, enter

N 25, I 7, PMT 500, and FV 0, and then press CPT and then

the PV key to ¬nd PV $5,826.79.

Present values and future values are directly related to one another. We

saw just above that the present value of $500 to be received in 25 years is

5

The present value interest factor of an annuity due is (1 i)PVIFAi,n.

28-4 Basic Financial Tools: A Review

Chapter 28

$92.12. This means that if we had $92.12 now and invested it at a 7 percent

interest rate, it would grow to $500 in 25 years. For the annuity example, if

you put $5,826.79 in an account earning 7 percent, then you could with-

draw $500 at the end of each year for 25 years, and have a balance of zero

at the end of 25 years.

An important application of the annuity formula is ¬nding the set of equal

payments necessary to amortize a loan. In an amortized loan (such as a

mortgage or an auto loan) the payment is set so that the present value of the

series of payments, when discounted at the loan rate, is equal to the amount

of the loan:

Loan amount PMT(PVIFAi,n), or

PMT (Loan amount)/PVIFAi,n.

Each payment consists of two elements: (1) interest on the outstanding

balance (which changes over time) and (2) a repayment of principal (which

reduces the loan balance). For example, consider a 30-year, $200,000 home

mortgage with monthly payments and a nominal rate of 9 percent per year.

There are 30(12) 360 monthly payments, and the monthly interest rate is

9%/12 0.75%. We could use Equation 28-4 to calculate PVIFA0.75%,360,

and then ¬nd the monthly payment, which would be

PMT $200,000/PVIFA0.75%,360 $1,609.25.

It would be easier to use a ¬nancial calculator, entering N 360, I 0.75,

PV 200000, and FV 0, and then press CPT and then the PMT key to

¬nd PMT $1,609.25.

To see how the loan is paid off, note that the interest due in the ¬rst

month is 0.75 percent of the initial outstanding balance, or 0.0075

($200,000) $1,500.00. Since the total payment was $1,609.25, then

$1,609.25 1,500.00 $109.25 is applied to reduce the principal balance.

At the start of the second month the outstanding balance would be

$200,000 $109.25 $199,890.75. The interest on this balance would be

0.75 percent of the new balance or 0.0075($199,890.75) $1,499.18, and

the amount applied to reduce the principal would be $1,609.25 1,499.18

$110.07. This process would be repeated each month, and the resulting amor-

tization schedule would show, for each month, the amount of the payment

that is interest and the amount applied to reduce principal. With a spread-

sheet program such as Excel, we can easily calculate amortization schedules.

Nonannual Compounding

Not all cash ¬‚ows occur once a year. The periods could be years, quarters,

months, days, hours, minutes, seconds, or even instantaneous periods.

Discrete Compounding The procedure used when the period is less than

a year is to take the annual interest rate, called the nominal, or quoted rate,

and divide it by the number of periods in a year. The result is called the

periodic rate. In the case of a monthly annuity with a nominal annual rate

of 7 percent, the monthly interest rate would be 7%/12 0.5833%. As a

decimal, this is 0.005833. Note that interest rates are sometimes stated as

decimals and sometimes as percentages. You must be careful to determine

which form is being used.

28-5

Time Value of Money

When interest is compounded more frequently than once a year, interest

will be earned on interest more frequently. Consequently, the effective rate

will exceed the quoted rate. For example, a dollar invested at a quoted

(nominal) annual rate of 7 percent but compounded monthly will earn

7/12 0.5833 percent per month for 12 months. Thus, $1 will grow to

$1(1.005833)12 1.0723 over one year, or by 7.23%. Therefore, the effec-

tive, or equivalent, annual rate (EAR) on a 7 percent nominal rate com-

pounded monthly is 7.23 percent.

If there are m compounding intervals per year and the nominal rate is

inom, then the effective annual rate will be

inom m

a1 b (28-5)

EAR (or EFF%) 1.0.

m

The larger the value of m, the greater the difference between the nominal

and effective rates. Note that if m 1, which means annual compounding,

then the nominal and effective rates are equal.6

Solving for the Interest Rate

The general formula for the present value of a series of cash ¬‚ows, CFt, dis-

counted at some rate i, is as follows:

n

CFt

CF1 CF2 CFn

...