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In Chapter 6 we used a financial calculator to solve time value of money problems.
In this Web Appendix, we discuss how we can use the interest factor tables, which are
given at the back of the text in Appendix A, to solve time value of money problems.
We should note that 20 years or so ago, before financial calculators and spreadsheets
were widely available, the tables were used to solve most time value problems. Today,
though, tables are rarely used in actual practice. Still, working through the tables can
provide useful insights into various time value issues.

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App6A_SW_Brigham_778312 1/23/03 5:15 AM Page 6A-1

**6A**

USING INTEREST FACTOR TABLES

In Chapter 6 we used a ?nancial calculator to solve time value of money problems.

In this Web Appendix, we discuss how we can use the interest factor tables, which are

given at the back of the text in Appendix A, to solve time value of money problems.

We should note that 20 years or so ago, before ?nancial calculators and spreadsheets

were widely available, the tables were used to solve most time value problems. Today,

though, tables are rarely used in actual practice. Still, working through the tables can

provide useful insights into various time value issues.

**S O LV I N G F O R F U T U R E VA L U E W I T H I N T E R E S T TA B L E S**

**Future Value Interest**

The**Future Value Interest Factor for i and n (FVIFi,n) **is de?ned as (1

i)n, and

**Factor for i and n**

these factors can be found by using a regular calculator as discussed in Chapter 6 and

**(FVIFi,n)**

then put into tables. Table 6A-1 is illustrative, while Table A-3 in Appendix A at the

The future value of $1 left

back of the book contains FVIFi,n values for a wide range of i and n values.

on deposit for n periods at a

Since (1

i)n

FVIFi,n, Equation 6-1, shown earlier in the text, can be rewritten

rate of i percent per period.

as follows:

FVn

PV(FVIFi,n).

To illustrate, the FVIF for our ?ve-year, 5 percent interest problem (discussed earlier

in this chapter) can be found in Table 6A-1 by looking down the ?rst column to

**T A B L E 6 A - 1**

**Future Value Interest Factors: FVIFi,n**

**(1 **

**i)n**

**PERIOD (n)**

**0%**

**5%**

**10%**

**15%**

1

1.0000

1.0500

1.1000

1.1500

2

1.0000

1.1025

1.2100

1.3225

3

1.0000

1.1576

1.3310

1.5209

4

1.0000

1.2155

1.4641

1.7490

5

1.0000

**1.2763**

1.6105

2.0114

6

1.0000

1.3401

1.7716

2.3131

7

1.0000

1.4071

1.9487

2.6600

8

1.0000

1.4775

2.1436

3.0590

9

1.0000

1.5513

2.3579

3.5179

10

1.0000

1.6289

2.5937

4.0456

**A P P E N D I X 6 A**

I

U S I N G I N T E R E S T F A C T O R TA B L E S

**6A-1**

App6A_SW_Brigham_778312 1/23/03 5:15 AM Page 6A-2

Period 5, and then looking across that row to the 5 percent column, where we see

that FVIF5%,5

1.2763. Then, the value of $100 after ?ve years is found as follows:

FVn

PV(FVIFi,n)

$100(1.2763)

$127.63.

Before ?nancial calculators became readily available (in the 1980s), such tables were

used extensively, but they are rarely used today in the real world.

**S O LV I N G F O R P R E S E N T VA L U E W I T H I N T E R E S T TA B L E S**

**Present Value Interest**

The term in parentheses in Equation 6-2, shown earlier in the text, is called the**Pres-**

**Factor for i and n**

**ent Value Interest Factor for i and n, **or **PVIFi,n, **and Table A-1 in Appendix A

**(PVIFi,n)**

contains present value interest factors for selected values of i and n. The value of

The present value of $1 due

PVIFi,n for i

5% and n

5 is 0.7835, so the present value of $127.63 to be received

n periods in the future

after ?ve years when the appropriate interest rate is 5 percent is $100:

discounted at i percent per

period.

PV

$127.63(PVIF5%,5) $127.63(0.7835) $100.

**F I N D I N G T H E I N T E R E S T R AT E W I T H I N T E R E S T TA B L E S**

To solve for the interest rate when n, FV, and PV are known, simply write out Equa-

tion 6-1 and susbtitute the known value into the equation as follows:

FVn

PV(1

i)n

PV(FVIFi,n)

$100

$78.35(FVIFi,5)

FVIFi,5

$100/$78.35

1.2763.

Find the value of the FVIF as shown above, and then look across the Period 5 row in

Table A-3 until you ?nd FVIF

1.2763. This value is in the 5% column, so the in-

terest rate at which $78.35 grows to $100 over ?ve years is 5 percent. (Note that

Equation 6-2 will work also. However, if Equation 6-2 is used, you would solve for

PVIF rather than FVIF.) This procedure can be used only if the interest rate is in the

table; therefore, it will not work for fractional interest rates or where n is not a whole

number. Approximation procedures can be used, but they are laborious and inexact.

**F I N D I N G T H E N U M B E R O F P E R I O D S W I T H I N T E R E S T TA B L E S**

To solve for the number of periods when i, FV, and PV are known, simply write out

Equation 6-1 and substitute the known values into the equation as follows:

FVn

PV(1

i)n

PV(FVIFi,n)

$100

$78.35(FVIF5%,n)

FVIF5%,n

$100/$78.35

1.2763.

**6A-2**

**A P P E N D I X 6 A**

I

U S I N G I N T E R E S T F A C T O R TA B L E S

App6A_SW_Brigham_778312 1/23/03 5:15 AM Page 6A-3

Now look down the 5% column in Table A-3 until you ?nd FVIF

1.2763. This

value is in Row 5, which indicates that it takes ?ve years for $78.35 to grow to $100

at a 5 percent interest rate.

**S O LV I N G F O R T H E F U T U R E VA L U E O F A N A N N U I T Y**

W I T H I N T E R E S T TA B L E S

The summation term in Equation 6-3, shown earlier in the text, is called the Future

Value Interest Factor for an Annuity (FVIFAi,n):1

n

FVIFAi,n

a (1

i)n t.

t

1

FVIFAs have been calculated for various combinations of i and n, and Table A-4 in

Appendix A contains a set of FVIFA factors. To ?nd the answer to the three-year,

$100 annuity problem (discussed earlier in the chapter), ?rst refer to Table A-4 and

look down the 5% column to the third period; the FVIFA is 3.1525. Thus, the future

value of the $100 annuity is $315.25:

FVAn

PMT(FVIFAi,n)

FVA3

$100(FVIFA5%,3) $100(3.1525) $315.25.

**S O LV I N G F O R T H E F U T U R E VA L U E O F A N A N N U I T Y**

**D U E W I T H I N T E R E S T TA B L E S**

In an annuity due, each payment is compounded for one additional period, so the fu-

ture value of the entire annuity is equal to the future value of an ordinary annuity

compounded for one additional period. Here is the solution for the annuity discussed

above, assuming that the annuity payments occur at the beginning of the year:

FVAn (Annuity due) PMT(FVIFAi,n)(1 i)

$100(3.1525)(1.05)

$331.01.

**S O LV I N G F O R T H E P R E S E N T VA L U E O F A N A N N U I T Y**

W I T H I N T E R E S T TA B L E S

**Present Value Interest**

The summation term in Equation 6-4, shown earlier in the text, is called the**Present**

**Factor for an Annuity**

**Value Interest Factor for an Annuity (PVIFAi,n), **and values for the term at dif-

**(PVIFAi,n)**

ferent values of i and n are shown in Table A-2 at the back of the book. Here is the

The present value interest

equation:

factor for an annuity of n

periods discounted at i

PVAn

PMT(PVIFAi,n).

percent.

1 Another form for this equation is as follows:

(1

i)n

1

FVIFAi,n

.

i

This form is found by applying the algebra of geometric progressions. This equation is useful in situations

when the required values of i and n are not in the tables and no ?nancial calculator or computer is available.

**A P P E N D I X 6 A**

I

U S I N G I N T E R E S T F A C T O R TA B L E S

**6A-3**

App6A_SW_Brigham_778312 1/23/03 5:15 AM Page 6A-4

To ?nd the answer to the three-year, $100 annuity problem (discussed earlier in the

chapter), simply refer to Table A-2 and look down the 5% column to the third period.

The PVIFA is 2.7232, so the present value of the $100 annuity is $272.32:

PVAn

PMT(PVIFAi,n)

PVA3

$100(PVIFA5%,3) $100(2.7232) $272.32.

**S O LV I N G F O R T H E P R E S E N T VA L U E O F A N A N N U I T Y**

**D U E W I T H I N T E R E S T TA B L E S**

In an annuity due, each payment is discounted for one less period. Since its payments

come in faster, an annuity due is more valuable than an ordinary annuity. This higher

value is found by multiplying the PV of an ordinary annuity by (1

i). To ?nd the

present value of the annuity discussed above assuming that annuity payments occur

at the beginning of the year, we use the following equation:

PVAn (Annuity due) PMT(PVIFAi,n)(1 i)

$100(2.7232)(1.05)

$285.94.

**6A-4**

**A P P E N D I X 6 A**

I

U S I N G I N T E R E S T F A C T O R TA B L E S

USING INTEREST FACTOR TABLES

In Chapter 6 we used a ?nancial calculator to solve time value of money problems.

In this Web Appendix, we discuss how we can use the interest factor tables, which are

given at the back of the text in Appendix A, to solve time value of money problems.

We should note that 20 years or so ago, before ?nancial calculators and spreadsheets

were widely available, the tables were used to solve most time value problems. Today,

though, tables are rarely used in actual practice. Still, working through the tables can

provide useful insights into various time value issues.

The

i)n, and

these factors can be found by using a regular calculator as discussed in Chapter 6 and

then put into tables. Table 6A-1 is illustrative, while Table A-3 in Appendix A at the

The future value of $1 left

back of the book contains FVIFi,n values for a wide range of i and n values.

on deposit for n periods at a

Since (1

i)n

FVIFi,n, Equation 6-1, shown earlier in the text, can be rewritten

rate of i percent per period.

as follows:

FVn

PV(FVIFi,n).

To illustrate, the FVIF for our ?ve-year, 5 percent interest problem (discussed earlier

in this chapter) can be found in Table 6A-1 by looking down the ?rst column to

1

1.0000

1.0500

1.1000

1.1500

2

1.0000

1.1025

1.2100

1.3225

3

1.0000

1.1576

1.3310

1.5209

4

1.0000

1.2155

1.4641

1.7490

5

1.0000

1.6105

2.0114

6

1.0000

1.3401

1.7716

2.3131

7

1.0000

1.4071

1.9487

2.6600

8

1.0000

1.4775

2.1436

3.0590

9

1.0000

1.5513

2.3579

3.5179

10

1.0000

1.6289

2.5937

4.0456

I

U S I N G I N T E R E S T F A C T O R TA B L E S

App6A_SW_Brigham_778312 1/23/03 5:15 AM Page 6A-2

Period 5, and then looking across that row to the 5 percent column, where we see

that FVIF5%,5

1.2763. Then, the value of $100 after ?ve years is found as follows:

FVn

PV(FVIFi,n)

$100(1.2763)

$127.63.

Before ?nancial calculators became readily available (in the 1980s), such tables were

used extensively, but they are rarely used today in the real world.

The term in parentheses in Equation 6-2, shown earlier in the text, is called the

contains present value interest factors for selected values of i and n. The value of

The present value of $1 due

PVIFi,n for i

5% and n

5 is 0.7835, so the present value of $127.63 to be received

n periods in the future

after ?ve years when the appropriate interest rate is 5 percent is $100:

discounted at i percent per

period.

PV

$127.63(PVIF5%,5) $127.63(0.7835) $100.

To solve for the interest rate when n, FV, and PV are known, simply write out Equa-

tion 6-1 and susbtitute the known value into the equation as follows:

FVn

PV(1

i)n

PV(FVIFi,n)

$100

$78.35(FVIFi,5)

FVIFi,5

$100/$78.35

1.2763.

Find the value of the FVIF as shown above, and then look across the Period 5 row in

Table A-3 until you ?nd FVIF

1.2763. This value is in the 5% column, so the in-

terest rate at which $78.35 grows to $100 over ?ve years is 5 percent. (Note that

Equation 6-2 will work also. However, if Equation 6-2 is used, you would solve for

PVIF rather than FVIF.) This procedure can be used only if the interest rate is in the

table; therefore, it will not work for fractional interest rates or where n is not a whole

number. Approximation procedures can be used, but they are laborious and inexact.

To solve for the number of periods when i, FV, and PV are known, simply write out

Equation 6-1 and substitute the known values into the equation as follows:

FVn

PV(1

i)n

PV(FVIFi,n)

$100

$78.35(FVIF5%,n)

FVIF5%,n

$100/$78.35

1.2763.

I

U S I N G I N T E R E S T F A C T O R TA B L E S

App6A_SW_Brigham_778312 1/23/03 5:15 AM Page 6A-3

Now look down the 5% column in Table A-3 until you ?nd FVIF

1.2763. This

value is in Row 5, which indicates that it takes ?ve years for $78.35 to grow to $100

at a 5 percent interest rate.

W I T H I N T E R E S T TA B L E S

The summation term in Equation 6-3, shown earlier in the text, is called the Future

Value Interest Factor for an Annuity (FVIFAi,n):1

n

FVIFAi,n

a (1

i)n t.

t

1

FVIFAs have been calculated for various combinations of i and n, and Table A-4 in

Appendix A contains a set of FVIFA factors. To ?nd the answer to the three-year,

$100 annuity problem (discussed earlier in the chapter), ?rst refer to Table A-4 and

look down the 5% column to the third period; the FVIFA is 3.1525. Thus, the future

value of the $100 annuity is $315.25:

FVAn

PMT(FVIFAi,n)

FVA3

$100(FVIFA5%,3) $100(3.1525) $315.25.

In an annuity due, each payment is compounded for one additional period, so the fu-

ture value of the entire annuity is equal to the future value of an ordinary annuity

compounded for one additional period. Here is the solution for the annuity discussed

above, assuming that the annuity payments occur at the beginning of the year:

FVAn (Annuity due) PMT(FVIFAi,n)(1 i)

$100(3.1525)(1.05)

$331.01.

W I T H I N T E R E S T TA B L E S

The summation term in Equation 6-4, shown earlier in the text, is called the

ferent values of i and n are shown in Table A-2 at the back of the book. Here is the

The present value interest

equation:

factor for an annuity of n

periods discounted at i

PVAn

PMT(PVIFAi,n).

percent.

1 Another form for this equation is as follows:

(1

i)n

1

FVIFAi,n

.

i

This form is found by applying the algebra of geometric progressions. This equation is useful in situations

when the required values of i and n are not in the tables and no ?nancial calculator or computer is available.

I

U S I N G I N T E R E S T F A C T O R TA B L E S

App6A_SW_Brigham_778312 1/23/03 5:15 AM Page 6A-4

To ?nd the answer to the three-year, $100 annuity problem (discussed earlier in the

chapter), simply refer to Table A-2 and look down the 5% column to the third period.

The PVIFA is 2.7232, so the present value of the $100 annuity is $272.32:

PVAn

PMT(PVIFAi,n)

PVA3

$100(PVIFA5%,3) $100(2.7232) $272.32.

In an annuity due, each payment is discounted for one less period. Since its payments

come in faster, an annuity due is more valuable than an ordinary annuity. This higher

value is found by multiplying the PV of an ordinary annuity by (1

i). To ?nd the

present value of the annuity discussed above assuming that annuity payments occur

at the beginning of the year, we use the following equation:

PVAn (Annuity due) PMT(PVIFAi,n)(1 i)

$100(2.7232)(1.05)

$285.94.

I

U S I N G I N T E R E S T F A C T O R TA B L E S

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