FV PV IF (Table 1)

where FV is the future value of the amount invested, PV is the present value (the amount

invested), and IF is the interest factor from Table 1.

The future value equation [FV PV(1 R)t] is

useful for determining the future value of an invest-

LEARNING NOTE

ment. In addition to this information, you might want

Future and present value calculations often contain rounding er-

to determine the amount earned on an investment

rors that depend on the number of decimal places included in

each period. For example, if you invested $500 for

computing interest factors. In this book, we round computations

three years at 8% interest, how much interest would

to the nearest cent or, for large amounts, to the nearest dollar.

you earn each year? A table like the one shown in Ex-

hibit 1 is useful for this purpose.

Exhibit 1 Interest Table for an Investment of $500 for Three Years at 8%

A B C D

Value Interest Earned Future Value at End of Year

Year at Beginning of Year (Column B Interest Rate) (Column B Column C)

1 500.00 40.00 540.00

2 540.00 43.20 583.20

3 583.20 46.66 629.86

Total 129.86

Column B shows the amount the investment is worth at the beginning of each year.

Column C shows the amount of interest earned each year, and column D reports the

F288 SECTION F2: Analysis and Interpretation of Financial Accounting Information

289

The Time Value of Money

amount the investment is worth at the end of each year. The total in column C is the

total interest earned for three years. This table illustrates how the value of an invest-

ment grows over time as interest is earned and reinvested. A bank could use the same

type of table to calculate the amount of interest expense incurred and the amount owed

the investor in a savings account each period.

Future Value of an Annuity

OBJECTIVE 3

The future value calculations so far have been limited to determining the future value

Determine the future

of a single investment, such as the future value of $1,000 invested on January 1, 2004.

value of an annuity.

Now, consider a situation in which a series of investments is made. For example, sup-

pose you invest $500 at the end of each year for three

LEARNING NOTE years. How much will your investments be worth at

the end of three years if you earn 8% interest each

It is important to know when amounts are paid or received when

year? This type of investment situation is known as

working with annuities. An annuity in which amounts are paid or

an annuity. An annuity is a series of equal amounts

received at the end of each fiscal period is known as an ordinary

received or paid over a specified number of equal

annuity. An annuity in which amounts are paid or received at the

time periods.

beginning of each fiscal period is known as an annuity due. We

limit our discussion to ordinary annuities, which are typical for Calculate the future value of these investments

most accounting transactions. by computing the future value of the amount invested

each year and adding all the amounts together.

End of Year 1 End of Year 2 End of Year 3 Future Value at End of Year 3

583.20 $500 1.082

Invested for 2 years $500 $

500 1.081

Invested for 1 year $500 540.00

500 1.080

Invested for 0 years 500.00

Future value of total investment $1,623.20

Total amount invested over 2 years* 1,500.00

Interest earned over 2 years $ 123.20

*Though three payments are made, the period covered is only two years because the first payment is made at

the end of year 1.

The $500 invested at the end of the first year is

LEARNING NOTE

worth $583.20 at the end of the third year. The $500

Any amount raised to the zero power is 1. Therefore, (1.08)0 invested at the end of the second year is worth $540.00

1. Also, (1.0)0 1, (1.1)0 1, and (200)0 1. Using an ex-

at the end of the third year, and the $500 invested at

0

pression such as $500 (1.08) is the same as saying that $500

the end of the third year is worth $500.00 at the end

invested at any point in time is worth $500 at that point in time

of the third year. Thus, the future value of the total

because no interest has been earned. Another way of saying the

investment is the sum of these amounts, $1,623.20.

same thing is to say that the future value of any amount at zero

This calculation is the same as:

periods in the future is that amount. The interest factor for any

interest rate at zero periods in the future is 1.0.

$500[(1.08)0 (1.08)1 (1.08)2]

$1,623.20

Therefore, we could use Table 1 to identify the interest factors for 8% and one and two

periods.

Excerpt from Table 1 Future Value of a Single Amount

Interest Rate

Period 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

1 1.01000 1.02000 1.03000 1.04000 1.05000 1.06000 1.07000 1.08000 1.09000

2 1.02010 1.04040 1.06090 1.08160 1.10250 1.12360 1.14490 1.16640 1.18810

3 1.03030 1.06121 1.09273 1.12486 1.15763 1.19102 1.22504 1.25971 1.29503

F289

CHAPTER F8: The Time Value of Money

290 The Time Value of Money

The interest factor for zero periods is 1. Thus, the interest factor for the annuity is

the sum of interest factors for zero, one, and two periods (3.2464 1.00 1.08

1.1664). Then, we multiply this interest factor times the amount invested each period

to compute the future value of the annuity:

$1,623.20 $500 3.2464

Alternatively, tables are available that contain the interest factors for computing the

future value of an annuity. Table 2 at the back of this book is this type of table. Using

the table simplifies the calculation by providing the interest factor.

Excerpt from Table 2 Future Value of an Annuity

Interest Rate

Period 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

1 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

2 2.01000 2.02000 2.03000 2.04000 2.05000 2.06000 2.07000 2.08000 2.09000

3 3.03010 3.06040 3.09090 3.12160 3.15250 3.18360 3.21490 3.24640 3.27810

USING EXCEL

We could use a spreadsheet and enter the amounts invested in a cell to calculate

their future value: (500*(1.08^2)) (500*1.08) 500. The first term in parentheses is

For Future Value of an

the future value of $500 at the end of two periods. The second term in parentheses

Annuity

is the future value of $500 at the end of one period, and the last term is the final in-

vestment of $500. The amount appearing in the cell is the future value of the annu-

ity ($1,623.20 in this example).

Spreadsheets also contain built-in functions for calculating the future value of an

annuity. Locate these functions by clicking on the Function ’x button.

From the pop-up menu, select the type of function you want. The future value

SPREADSHEET

of an annuity (FV) function is in the Financial category.

F290 SECTION F2: Analysis and Interpretation of Financial Accounting Information

291

The Time Value of Money

Click the OK button and the following box appears. Complete the box by enter-

ing the interest rate (Rate), number of periods for which investments will be made

(Nper), and the amount invested (Pmt). The Pv and Type boxes can be left blank.

Note that the amount invested is entered as a negative number because it is a cash

outflow to the investor.

The amount appearing in the cell used to reference the FV function is the future

value of the investment ($1,623.20 in this example). The function can be entered in the

worksheet™s cell directly by typing FV(.08,3, 500). It is important that the values (known

as arguments to a function) be entered in the cell in the correct order. The function for

the future value of an annuity is FV(Interest Rate, Number of Periods, Amount Invested).

Annuities are common in business activities. For example, suppose Mom™s Cookie

Company agrees to invest $5,000 for each of its employees in a retirement plan at the

end of each year. If the investment earns 7% and an employee works 20 years before

retiring, how much will be available when the employee retires? Using Table 2, we can

calculate the future value of an annuity of $5,000 per year for 20 years.

Excerpt from Table 2 Future Value of an Annuity

Interest Rate

Period 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

1 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

2 2.01000 2.02000 2.03000 2.04000 2.05000 2.06000 2.07000 2.08000 2.09000

3 3.03010 3.06040 3.09090 3.12160 3.15250 3.18360 3.21490 3.24640 3.27810

• • • • • • • • • •

• • • • • • • • • •

• • • • • • • • • •

19 20.81090 22.84056 25.11687 27.67123 30.53900 33.75999 37.37896 41.44626 46.01846

20 22.01900 24.29737 26.87037 29.77808 33.06595 36.78559 40.99549 45.76196 51.16012

$204,977 $5,000 40.99549

where 40.99549 is the interest factor for 7% and 20 periods. The table is simply a labor-

saving device to reduce the number of calculations for this type of problem. Alterna-

F291

CHAPTER F8: The Time Value of Money

292 The Time Value of Money

tively, we could use a financial calculator or computer program that has future value

functions.

USING EXCEL

For Future Value The future value of the investment can be calculated in Excel by entering

FV(.07,20, 5000) in a cell.

SPREADSHEET

Using Table 2, we compute the future value of an annuity as follows:

FVA A IF (Table 2)

where FVA is the future value of the annuity, A is the amount invested at the end of

each period, and IF is the interest factor from Table 2.

In addition to determining the future value of an annuity, you might want to de-

termine the amount earned each period. For example, if you invested $500 at the end

of each year for three years at 8% interest, how much interest would you earn each

year? A table like the one shown in Exhibit 2 is useful for this purpose.

Exhibit 2 Interest Table for an Annuity of $500 at End of Each Year for Three Years at 8%

A B C D E

Amount

Value Interest Earned Invested Future Value at

at Beginning (Column B at End of End of Year

Year of Year Interest Rate) Year (Columns B C D)

1 0.00 0.00 500.00 500.00

2 500.00 40.00 500.00 1,040.00

3 1,040.00 83.20 500.00 1,623.20

Total 123.20 1,500.00

Column B shows the amount the investment is worth at the beginning of each year

before the contribution is made for that year. Column C contains the amount of in-

terest earned for the year, and column D contains the amount invested at the end of

each year. Column E reports the amount the investment is worth at the end of each

year. The total in column C is the total interest earned for three years. Interest earned

on the annuity is greater than that earned on a single investment of the same amount

because the investment is growing each period by the additional amount invested as

well as by the amount of interest earned.

Another question you might want to answer is, How much would you need to in-

vest each period to accumulate a certain amount? For example, suppose you want to

accumulate $1,000 over the next three years to take a trip to Mexico after you gradu-

ate from college. How much would you need to invest at the end of each year to accu-

mulate $1,000 at the end of three years, assuming that you invest the same amount each

year and can earn 6% on your investment?

F292 SECTION F2: Analysis and Interpretation of Financial Accounting Information

293

The Time Value of Money

We can answer this question using the future value of an annuity equation and

Table 2.

FVA A IF (Table 2)

$1,000 A 3.18360

A $1,000 3.18360

A $314.11

By investing $314.11 at the end of each year for three years, you can accumulate $1,000.

USING EXCEL

We can calculate the amount of the payment in Excel using the payment function

(PMT). To determine the amount, enter PMT(0.06,3,,1000) in a cell. The arguments

For Calculating a

of the function are PMT(Interest Rate, Number of Periods,, Future Value of the An-

Payment

nuity). Note that there are two commas following the number of periods because an

argument has been omitted. We will use that argument in a future computation. The

amount appearing in the cell where the function is entered is the amount of the an-

nuity payment. It appears as a negative amount because it is a cash outflow to the

investor.

SPREADSHEET

1 SELF-STUDY PROBLEM Harry Morgan recently graduated from college and started his

first full-time job. He wants to accumulate enough money in the

next five years to make a down payment on a house. He has $3,000 that he can invest

at the beginning of the five-year period. His investment will earn 8% interest.

Required

A. How much would Harry™s $3,000 investment be worth at the end of the five-year

period? How much interest would he earn for the five years?

B. Independent of part (A), suppose the amount Harry needs for a down payment at the

end of five years is $10,000. He wants to invest equal amounts at the end of each year

for the next five years to accumulate the $10,000 he needs. How much would he need

to invest each year, assuming that he earns 8% interest? How much would his invest-

ment be worth at the end of each year for the five-year period? How much would

Harry invest over the five years? How much interest would he earn over the five years?

The solution to Self-Study Problem 1 appears at the end of the chapter.

PRESENT VALUE

In many business activities, it is important to calculate the present value of an invest-

OBJECTIVE 4

ment rather than the future value. In this section, you will learn how to calculate the

Determine the present present value of an investment from information about the future value. To illustrate,

value of a single amount suppose a company offered to sell you an investment that pays $3,000 at the end of

to be received in the three years. You want to earn 8% return on your investment. How much would you

future. be willing to pay for the investment?

To solve this problem, you should recognize that the $3,000 to be received at the

end of three years is the future value of the investment. The amount you would pay for

the investment at the beginning of the three-year period is the present value of the in-

vestment. Use the future value equation described earlier to solve the problem.

R)t

FV PV(1

PV(1.08)3

$3,000

1

1.083

PV $3,000 $3,000

(1.08)3

PV $2,381.50

F293

CHAPTER F8: The Time Value of Money

294 The Time Value of Money

Thus, the present value of the investment”the amount you would be willing to pay at

the beginning of the three-year period”is $2,381.50.

We can also use Table 1 to solve the problem.

FV PV IF (Table 1)

$3,000 PV 1.25971

PV $3,000 1.25971

PV $2,381.50

We can rewrite the future value equation as a present value equation.

1

PV FV

R)t

(1

USING EXCEL

For the Present Value of The present value of the investment can be calculated in Excel by entering

a Single Amount 3000*(1/(1.08^3)) in a cell. The amount appearing in the cell ($2,381.50 in this ex-

ample) is the present value of the investment.

SPREADSHEET

Tables are available that provide interest factors for computing the present value

of an investment. Table 3 at the back of this book is an example of such a table. To

illustrate, the interest factor for computing the present value of an investment of three

periods at 8% is 0.79383 from Table 3.

Excerpt from Table 3 Present Value of a Single Amount

Interest Rate

Period 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

1 0.99010 0.98039 0.97087 0.96154 0.95238 0.94340 0.93458 0.92593 0.91743

2 0.98030 0.96117 0.94260 0.92456 0.90703 0.89000 0.87344 0.85734 0.84168

3 0.97059 0.94232 0.91514 0.88900 0.86384 0.83962 0.81630 0.79383 0.77218

The present value of an investment that pays $3,000 at the end of three years at 8%,

then, is as follows:

$2,381.49 $3,000 0.79383

This is the same amount computed above, except for the effect of a rounding error.

The interest factor for computing the present value of an investment is 1 divided by the

interest factor for computing the future value of the same investment (same period and

interest rate). Thus, 0.79383 in Table 3 for three periods at 8% equals 1 1.25971 from

Table 1 for three periods at 8%.

Using Table 3, we can compute the present value of a single payment as follows:

PV FV IF (Table 3)

F294 SECTION F2: Analysis and Interpretation of Financial Accounting Information

295

The Time Value of Money

where PV is the present value, FV is the future value (the payment made in the future),

and IF is the interest factor from Table 3.

In addition to determining the present value of an investment, you might want to

determine the amount earned each period. For example, if your investment paid $3,000

at the end of three years and earned 8% interest, how much interest would you earn

each year? A table like the one shown in Exhibit 3 is helpful.

Exhibit 3 Interest Table for a Present Value of $2,381.49 for Three Years at 8%

A B C D

Present Value Interest Earned Value at End of Year

Year at Beginning of Year (Column B Interest Rate) (Column B Column C)

1 2,381.49 190.52 2,572.01

2 2,572.01 205.76 2,777.77

3 2,777.77 222.23 3,000.00

Total 618.51

Column B shows the amount the investment is worth at the beginning of each

year. Column C shows the amount of interest earned each year, and column D re-

ports the amount the investment is worth at the end of each year. The total in col-

umn C is the total interest earned for three years. Values in Exhibit 3 are calculated

the same way as in Exhibit 1. The present value of the investment must be calculated

before the table can be prepared. The amount computed for the future value at the

end of the investment period should be the future value of the investment (see $3,000

in Exhibit 3).

Present Value of an Annuity

The investment situation in the previous section assumed that a single amount was re-

OBJECTIVE 5

ceived at the end of an investment period. It is common for investments to pay an equal

Determine the present amount each period over an investment period. For example, assume that you could

value of an annuity. purchase an investment that would pay $1,000 at the end of each year for three years,

and that you expect to earn a return of 8%. How much would you be willing to pay

for the investment?

Calculate the present value of this annuity by calculating the present value of each

payment and adding them together.

Amount Received

Present Value

at Beginning of Year 1 End of Year 1 End of Year 2 End of Year 3

(1.08)1

$ 925.93 $1,000 $1,000

(1.08)2

857.34 $1,000 $1,000

(1.08)3

793.83 $1,000 $1,000

$2,577.10 Present value of total investment

$3,000.00 Total amount received over 3 years

2,577.10 Present value of total investment

$ 422.90 Interest earned over 3 years

The first row in this calculation is the present value of $1,000 received at the end

of the first year, the second row is the present value of $1,000 received at the end of the

second year, and the third row is the present value of $1,000 received at the end of the

third year.

F295

CHAPTER F8: The Time Value of Money

296 The Time Value of Money

Alternatively, we could use the interest factors from Table 3 for one, two, and three

periods at 8% and add them together to determine the interest factor.

Excerpt from Table 3 Present Value of a Single Amount

Interest Rate

Period 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

1 0.99010 0.98039 0.97087 0.96154 0.95238 0.94340 0.93458 0.92593 0.91743

2 0.98030 0.96117 0.94260 0.92456 0.90703 0.89000 0.87344 0.85734 0.84168

3 0.97059 0.94232 0.91514 0.88900 0.86384 0.83962 0.81630 0.79383 0.77218

$2,577.10 $1.000 (0.92593 0.85734 0.79383)

$2,577.10 $1.000 2.57710

USING EXCEL

Use the PV function in Excel to calculate the present value of an annuity. The func-

tion can be accessed by using the function button and completing the pop-up box

For the Present Value of

or by entering the function directly in a cell. The present value of an annuity function

an Annuity

is PV(Interest Rate, Number of Periods, Amount Invested each Period). Accordingly,

we can enter PV(0.08,3, 1000) in a cell. The amount appearing in the cell is the

present value of the annuity ($2,577.10 in this example). Remember that the amount

invested is entered as a negative value.

SPREADSHEET

To avoid the need to add interest factors together, tables are available that provide

this addition. Table 4 inside the back cover of this book is an example of this type of

table. Notice that the interest factor in this table for an annuity of three periods at 8%

is 2.57710, the sum of the interest factors for one, two, and three years from Table 3.

Using this table, we can calculate the present value of an annuity as follows:

PVA A IF (Table 4)

where PVA is the present value of an annuity, A is the amount of the periodic payment

of the annuity, and IF is the interest factor from Table 4.

Excerpt from Table 4 Present Value of an Annuity

Interest Rate

Period 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

1 0.99010 0.98039 0.97087 0.96154 0.95238 0.94340 0.93458 0.92593 0.91743

2 1.97040 1.94156 1.91347 1.88609 1.85941 1.83339 1.80802 1.78326 1.75911

3 2.94099 2.88388 2.82861 2.77509 2.72325 2.67301 2.62432 2.57710 2.53129

Again, in addition to determining the present value of an annuity, you might want

to determine the amount earned each period. For example, if you purchased an invest-

ment that paid $1,000 each year for three years at 8% interest, how much interest would

you earn each year? A table like the one shown in Exhibit 4 is useful for this purpose.

A B C D E F

F296 SECTION F2: Analysis and Interpretation of Financial Accounting Information

297

The Time Value of Money

Exhibit 4 Interest Table for an Annuity of $1,000 Each Year for Three Years at 8%

A B C D E F

Total Amount Amount Value at

Present Value Interest Earned Invested Paid at End of Year

at Beginning (Column B (Column B End of (Column D

Year of Year Interest Rate) Column C) Year Column E)

1 2,577.10 206.17 2,783.27 1,000.00 1,783.27

2 1,783.27 142.66 1,925.93 1,000.00 925.93

3 925.93 74.07 1,000.00 1,000.00 0.00

Total 422.90 3,000.00

This table is a bit more complicated than those presented earlier. The first step in

preparing this table is to calculate the present value of the annuity. This amount

($2,577.10) is the present value at the beginning of the first year. Interest for the first

year is earned on this amount, as shown in column C by multiplying the present value

in column B by the interest rate (8%). Column D shows the amount of the investment

after interest is earned for the period. It is the sum of columns B and C. However, an

amount ($1,000) is paid out of the investment each year. The amount left in the in-

vestment at the end of the year (column F) then is the amount in column D minus the

amount paid in column E. The amount in column F is what is available at the begin-

ning of the next year before interest is earned for the year (column B).

Observe that the value of the investment decreases over time because the amount

of the annuity ($1,000) is paid out each year. Once the final payment is made at the end

of the life of the investment, the value of the investment is zero (year 3, column F).

Exhibit 4 describes the amount of interest earned, the amount paid, and the value

for each period of an annuity. Note that the amount earned from the annuity over the

three years is equal to the difference between the amount received over the life of the

annuity and the present value of the annuity (the amount paid for the investment):

$422.90 $3,000 $2,577.10.

2 SELF-STUDY PROBLEM H. Greely has the option of buying either of two investments. One

investment pays $5,000 at the end of four years. The other in-

vestment pays $1,000 at the end of each year for four years. Both investments earn 8%

interest.

Required Which investment is worth more at the beginning of the four-year period?

How much interest will Greely earn from each investment over the four-year period?

The solution to Self-Study Problem 2 appears at the end of the chapter.

LOAN PAYMENTS AMORTIZATION

AND

Economic decisions frequently require the use of present value calculations. These cal-

OBJECTIVE 6

culations are used in a variety of transactions recorded in accounting systems. For ex-

Determine investment ample, assume that you want to buy a used car. You negotiate with a dealer to purchase

values and interest a car for $5,000, which you arrange to borrow from a local bank. The bank charges 12%

expense or revenue for interest on the loan, which is to be repaid in two years in equal monthly payments.

various periods. How much will your payments be each month? How much interest will you pay over

the two years?

F297

CHAPTER F8: The Time Value of Money

298 The Time Value of Money

To answer these questions, you should recognize that this problem involves the

present value of an annuity. The amount you borrow ($5,000) is the present value of

the amount you will repay in equal installments. Because the repayment is in equal

monthly installments, this investment is an annuity. In effect, the bank is investing

$5,000 in you at the beginning of the two-year period in exchange for monthly pay-

ments that will earn 12% (annual) return for the bank.

When amounts are paid or received on less than an annual basis, the interest rate

and number of periods must be adjusted for the shorter period. An annual rate of 12%

is equivalent to a monthly rate of 1% (12% 12 months). Two years of monthly pay-

ments result in 24 monthly payments (12 2). Therefore, instead of an interest factor

of 12% for 2 years, we should use an interest factor of 1% for 24 months. Interest is

compounded monthly, and a portion of the loan principal is being repaid each month

along with the interest.

We can use Table 4 to solve this problem:

http://ingram.

PVA A IF (Table 4)

swlearning.com

$5,000 A 21.24339

Calculate a loan sched- A $5,000 21.24339

ule online. A $235.37

Excerpt from Table 4 Present Value of an Annuity

Interest Rate

Period 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

1 0.99010 0.98039 0.97087 0.96154 0.95238 0.94340 0.93458 0.92593 0.91743

2 1.97040 1.94156 1.91347 1.88609 1.85941 1.83339 1.80802 1.78326 1.75911

3 2.94099 2.88388 2.82861 2.77509 2.72325 2.67301 2.62432 2.57710 2.53129

23 20.45582 18.29220 16.44361 14.85684 13.48857 12.30338 11.27219 10.37106 9.58021

24 21.24339 18.91393 16.93554 15.24696 13.79864 12.55036 11.46933 10.52876 9.70661

Thus, the answer to the first question (How much would you pay each month?) is

$235.37. Remember that the interest rate used in this computation is 1% per month

and the number of periods is 24 months. The interest rate must be adjusted for the pe-

riod of the annuity. An annual rate of 12% is equivalent to a monthly rate of 1%.

USING EXCEL

Solve the problem using the payment function in Excel. Enter PMT (.01,24,5000).

The amount appearing in the cell is the amount of the annuity payment ($235.37 in

For Calculating a

this example). The arguments of the function are PMT(Interest Rate, Number of Pe-

Payment

riods, Present Value of the Annuity). Observe that this is the same function that we

used to calculate the amount of payments for the future value of an annuity. The pay-

ment function is PMT(Interest Rate, Number of Periods, Present Value of Annuity, Fu-

ture Value of Annuity). Either the third or fourth argument should be skipped depending

on which value is being calculated. The third argument (Present Value of Annuity) is

skipped if the last argument is included. In this case, an extra comma is needed af-

SPREADSHEET

ter the second argument to indicate that the third argument has been omitted.

F298 SECTION F2: Analysis and Interpretation of Financial Accounting Information

299

The Time Value of Money

To determine how much interest you would incur on the loan, we need to prepare

a table similar to the one in Exhibit 4. Exhibit 5 provides this information. This table

usually is referred to as a loan amortization (or loan payment) table. It is a little dif-

ferent from Exhibit 4. A column has been added in Exhibit 5 to calculate the principal

repaid each period, and the total investment column in Exhibit 4 has been omitted. Ex-

hibits 4 and 5 contain essentially the same information. Formats of these tables vary in

practice, but they all provide the same basic information.

Exhibit 5 Amortization Table for Automobile Loan of $5,000 for 24 Months at 1% per Month

A B C D E F

Principal Value at

Present Value Interest Incurred Paid End of Month

at Beginning (Column B Amount (Column D (Column B

Month of Month Interest Rate) Paid Column C) Column E)

1 5,000.00 50.00 235.37 185.37 4,814.63

2 4,814.63 48.15 235.37 187.22 4,627.41

3 4,627.41 46.27 235.37 189.10 4,438.31

23 463.70 4.64 235.37 230.73 232.97

24 232.97 2.33 235.30 232.97 0.00

Total 648.81 5,648.81

This exhibit provides useful information about the loan. The total interest incurred

over the life of the loan is $648.81. This amount equals the difference between the

amount paid over the life of the loan ($5,648.81) and the amount borrowed ($5,000).

The amount of interest incurred decreases each month (column C) because a portion

of the loan principal is repaid each month (column E). Notice that the payment made

each month ($235.37) repays a portion of the amount borrowed and pays the interest

expense for the month. The amount of the loan repaid each month (column E) is the

amount of the payment (column D) minus the interest expense for the month (col-

umn C). The amount owed decreases to zero over the 24 months. The final payment

in month 24 (column D) is adjusted slightly because of rounding error. These charac-

teristics are typical of many loan arrangements, especially consumer loans used to pur-

chase autos, appliances and similar goods, and homes.

Total Payment

Principal

Interest

$235.37

$185.37

$50.00

The information in Exhibit 5 can be used to determine the transactions that the bank

would record each month. It also can be used to determine transactions for the borrower

(for example, if the borrower were a company concerned with this information).

Consider first the transactions that would be recorded by the bank for the first

month of the loan, assuming the loan was made on April 1, 2004:

F299

CHAPTER F8: The Time Value of Money

300 The Time Value of Money

ASSETS LIABILITIES OWNERS™ EQUITY

Other Contributed Retained

Date Accounts Cash Assets Capital Earnings

Apr. 1, 2004 Notes Receivable 5,000.00

Cash 5,000.00

Apr. 30, 2004 Cash 235.37

Notes Receivable 185.37

Interest Revenue 50.00

The first transaction, at the beginning of the month (April 1), records the amount

of the loan as a decrease in Cash and an increase in Notes Receivable. The second trans-

action, at the end of the month (April 30), records the amount received from the cus-

tomer ($235.37) and the amount earned for the first month ($50.00). The balance in

Notes Receivable ($4,814.63 $5,000 $185.37) is the amount owed to the bank by

the customer at the end of the first month.

Similar transactions could be recorded by the customer:

ASSETS LIABILITIES OWNERS™ EQUITY

Other Contributed Retained

Date Accounts Cash Assets Capital Earnings

Apr. 1, 2004 Cash 5,000.00

Notes Payable 5,000.00

Apr. 30, 2004 Notes Payable 185.37

Interest Expense 50.00

Cash 235.37

The first transaction records the amount received from the bank and the liability

to the bank. The second transaction records the payment at the end of the first month.

The amount paid reduces the liability to the bank and pays interest expense for the first

month.

The bank records transactions for payments received each month over the life of

the loan. The amount of principal paid and the amount of interest earned change each

month. In the last month of the loan (March 2006), the bank would record the fol-

lowing:

ASSETS LIABILITIES OWNERS™ EQUITY

Other Contributed Retained

Date Accounts Cash Assets Capital Earnings

Mar. 31, 2006 Cash 235.30

Notes Receivable 232.97

Interest Revenue 2.33

After this transaction is recorded, the balance of Notes Receivable will be zero. The loan

will have been paid off.

F300 SECTION F2: Analysis and Interpretation of Financial Accounting Information

301

The Time Value of Money

The customer also could record transactions each month. The amount of princi-

pal repaid and the interest expense incurred would change each month. The final pay-

ment would reduce the Notes Payable balance to zero.

UNEQUAL PAYMENTS

Investments do not always involve single amounts or annuities. For example, suppose

Jill Johnson invested a portion of her salary at the end of each of four years. The amounts

she invested in those years were $700, $800, $900, and $1,000, respectively. How much

would her investments be worth at the end of the fourth year of investing if she earned

6% each year?

In this type of situation, each investment must be considered separately because

the amounts invested are not the same each period.

Amount Invested End End End Future Value

at End of Year 1 of Year 2 of Year 3 of Year 4 at End of Year 4

$700 $ 833.71

$800 898.88

$900 954.00

$1,000 1,000.00

Total $3,686.59

The future value of the amounts can be determined using interest factors from

Table 1.

FV PV IF (Table 1)

$833.71 $700 1.19102 (6%, 3 periods)

898.88 $800 1.12360 (6%, 2 periods)

954.00 $900 1.06000 (6%, 1 period)

1,000.00 $1,000 1.00000 (6%, 0 period)

$3,686.59

Observe that the amount invested at the end of the first year ($700) will be invested

for four years, the amount invested at the end of the second year will be invested for

three years, and so forth. Accordingly, the interest factor for the first investment is for

four years, and the number of periods decreases by one for each successive investment.

To continue the illustration, suppose you can

purchase an investment that is expected to pay $200,

LEARNING NOTE

$300, and $400 at the end of the next three years. You

A common mistake is to match investments with the incorrect

expect the investment to earn 7% interest. How much

period of investment. Consider how long an amount will be in-

should you pay for the investment?

vested until the end of the investment period when computing

You want to determine the present value of the

future value, or until the beginning of the investment period when

amounts you expect to receive. The relevant period

computing present value.

is from the time when the amount will be received to

the beginning of the investment period. For example,

the first amount ($200) will be received at the end of one year; therefore, the relevant

period is one year. The present value of the investment would be as follows:

Amount Received

Present Value at

Beginning of Year 1 End of Year 1 End of Year 2 End of Year 3

$186.92 $200

262.03 $300

326.52 $400

$775.47 Total

F301

CHAPTER F8: The Time Value of Money

302 The Time Value of Money

The present value of the investment would be as follows:

PV FV IF (Table 3)

$186.92 $200 0.93458 (7%, 1 period)

262.03 $300 0.87344 (7%, 2 periods)

326.52 $400 0.81630 (7%, 3 periods)

$775.47

COMBINING SINGLE AMOUNTS ANNUITIES

AND

In some cases an investment involves both a single amount and an annuity. For exam-

ple, suppose you could purchase an investment that offered to pay $100 at the end of

each year for 10 years and $1,000 at the end of the 10-year period. If you expect the in-

vestment to earn 8% interest, how much would you pay for the investment at the be-

ginning of the 10-year period? To answer this question, compute the present value of

the annuity and add the present value of the single amount.

PVA A IF (Table 4)

$671.01 $100 6.71008

PV FV IF (Table 3)

$463.19 $1,000 0.46319

Therefore, the amount you should pay is $1,134.20 $671.01 $463.19.

Any investment problem can be thought of as a single amount, a series of single

amounts, an annuity, or a combination of these arrangements.

SUMMARY FUTURE PRESENT VALUE CONCEPTS

OF AND

Future and present value consider timing differences between when cash is received or

paid and the present period. They are based on the simple concept that a dollar received

in the future is worth less than a dollar received at the present time. The difference be-

tween the two amounts depends on the rate of interest and the time period. Amounts

invested today must increase in value to compensate the investor for forgoing the use

of the amount invested. The higher the interest rate required from an investment, the

greater the future value must be relative to the present value. The longer an investor

must wait before receiving the future value, the larger the future value must be relative

to the present value. Exhibit 6 illustrates the basic concepts of future and present value.

Exhibit 6 Amount Invested Amount Received

at Present Time in Future

Future and Present

Value Concepts

Present Future

Value Value

Higher Interest Rate

Increases Difference

Longer Period

Increases Difference

The interest rate an investment is expected to earn depends on the risk associated

with the investment. The greater the uncertainty about the amount to be received from

F302 SECTION F2: Analysis and Interpretation of Financial Accounting Information

303

The Time Value of Money

an investment, the higher the interest rate investors require before they will invest.

Therefore, relatively safe investments, such as savings accounts, pay lower interest than

relatively risky investments, such as corporate debt, where the chance of bankruptcy af-

fects the amount an investor may receive. Similarly, the rate of interest a bank charges

a customer for a loan depends on the customer™s credit history as an indication of the

probability that the customer will repay the amount borrowed, plus interest, when due.

3 SELF-STUDY PROBLEM Required Calculate each of the following at the beginning of

year one.

A. The present value of $100 received at the end of two years at 10%.

B. The present value of $100 received at the end of three years at 10%.

C. The present value of $100 received at the end of three years at 8%.

D. The present value of $100 received at the end of each year for three years at 10%.

E. The present value of $50 received at the end of one year, $100 received at the end

of two years, and $150 received at the end of three years at 10%.

Use your answers to demonstrate the effect of time periods and interest rates on the

difference between the future and present values of investments by comparing A and

B, B and C, B and D, and D and E.

The solution to Self-Study Problem 3 appears at the end of the chapter.

REVIEW SUMMARY of IMPORTANT CONCEPTS

1. The future value of an investment is the amount the investment will be worth at some

particular time in the future.

a. The future value of an investment equals the present value times an interest factor

that depends on the rate of interest earned on the investment and the number of

periods it is invested.

b. The future value of an annuity is the future value of a series of equal amounts paid

at equal intervals.

2. The present value of an investment is the amount the investment is worth at the begin-

ning of an investment period.

a. The present value of an investment equals the future value times an interest factor.

The interest factor for the present value is the reciprocal of the interest factor for

the future value: interest factor for PV 1 interest factor for FV.

b. The present value of an annuity is the present value of a series of equal amounts

paid at equal intervals.

3. Loan payments are determined from the present value of the loan (the amount bor-

rowed) and the interest factor (interest rate and time period).

a. A loan amortization table is useful for determining the amount of interest incurred,

the amount of principal repaid each period on the loan, and the amount owed at

the end of each period.

b. A loan amortization table provides a basis for transactions recorded by the bor-

rower and lender.

4. Future and present value calculations may involve a series of unequal payments or a

combination of an annuity and a single amount.

5. Three factors important for calculating any future or present value are the amount of

the payments, the interest rate, and the time periods when payments are made.

F303

CHAPTER F8: The Time Value of Money

304 The Time Value of Money

DEFINE TERMS and CONCEPTS DEFINED in this CHAPTER

annuity (F288) present value (F285)

future value (F285)

SELF-STUDY PROBLEM SOLUTIONS

SSP8-1 A. FV PV IF (Table 1)

FV $3,000 1.46933

FV $4,407.99

The amount of interest earned would be $1,407.99 $4,407.99 $3,000.

B. FVA A IF (Table 2)

$10,000 A 5.86660

A $1,704.56

Column E in the following interest table identifies the amount the investment is worth at the

end of each year. The total of column D is the amount Harry invested over the five years. The

total of column C is the amount of interest earned over the five years.

A B C D E

Amount

Value Interest Earned Invested Future Value at

at Beginning (Column B at End of End of Year

Year of Year Interest Rate) Year (Columns B C D)

1 0.00 0.00 1,704.56 1,704.56

2 1,704.56 136.36 1,704.56 3,545.48

3 3,545.48 283.64 1,704.56 5,533.68

4 5,533.68 442.69 1,704.56 7,680.93

5 7,680.93 614.48 1,704.56 9,999.97

Total 1,477.17 8,522.80

Option 1:

SSP8-2

PV FV IF (Table 3)

$3,675.15 $5,000 0.73503

Interest earned $1,324.85 $5,000 $3,675.15

Option 2:

PVA A IF (Table 4)

$3,312.13 $1,000 3.31213

Interest earned $687.87 $4,000 ($1,000 per year 4 years) $3,312.13

Option 1 is worth more, even though he will receive payments sooner from option 2.

PV FV IF

SSP8-3

A. $82.65 $100 0.82645 (Table 3)

B. $75.13 $100 0.75131 (Table 3)

C. $79.38 $100 0.79383 (Table 3)

D. $248.69 $100 2.48685 (Table 4)

E. $45.45 $50 0.90909

$82.65 $100 0.82645

$112.70 $150 0.75131

$240.80

F304 SECTION F2: Analysis and Interpretation of Financial Accounting Information

305

The Time Value of Money

Comparison of A and B: The present value of an investment decreases relative to the fu-

ture value as the time until the investment is received increases.

Comparison of B and C: The present value of an investment decreases relative to the

future value as the interest rate increases. A higher interest rate results in a higher amount

of interest being earned for investment B ($24.87 $100 $75.13) than for investment C

($20.62 $100 $79.38).

Comparison of B and D: The present value of an investment increases as the number of

payments received increases. Thus, an annuity is more valuable than a single payment when

each annuity payment is as large as the single payment.

Comparison of D and E: Both D and E pay $300 over three years. Investment D is worth

more than investment E, however, because a larger amount is received sooner from D than

from E.

Thinking Beyond the Question

How much will it cost to borrow money?

Most debt requires periodic payments of principal and interest. Consequently,

debt often involves computations of annuities, particularly the present value of

annuities. Determining the payment amount and the total cost of borrowing de-

pends on the interest rate and the number of periods over which the debt is re-

paid.

As a borrower, you may be able to negotiate the number of periods over

which debt is repaid. For example, you may be able to repay a loan over five

or ten years. Often, agreeing to a shorter borrowing period means getting a loan

with a lower interest rate. Why would that be true? What factors would encour-

age a lender to require a higher or lower rate of interest from a borrower? What

kinds of information would a lender look for in the financial statements of a bor-

rower like Mom™s Cookie Company? Why? What financial information could help

a borrower bargain for a lower rate or more money?

QUESTIONS

A friend remarks, “I just got out of an accounting lecture about future value and present value.

Q8-1

Frankly, I don™t have a clue what the professor was talking about. And we have a quiz on

Obj. 1

Wednesday. Help!” Come to your friend™s rescue. Clearly and concisely explain what is meant

by the terms future value and present value.

What does this statement mean? “The future value of $1,000 is an amount greater than $1,000.”

Q8-2

Obj. 1

Why is the present value of $10,000 less than $10,000?

Q8-3

Obj. 1

You are inspecting Table 1 at the back of this book. At the intersection of the 10% column

Q8-4

and the 18-period row you find the following number: 5.55992. Interpret that number. What

Obj. 2

does it mean?

Freida invested $3,000 in an investment plan that guarantees 7% compound interest annu-

Q8-5

ally. The interest is deposited into the account at the end of each year. The account has now

Obj. 2

been open 30 years. As the years went by, were the earnings from interest in any given year

larger than the year before, smaller than the year before, or the same as the year before? Ex-

plain the reason for your answer.

F305

CHAPTER F8: The Time Value of Money

306 The Time Value of Money

Q8-6 Why do the interest factors in Table 1 (at the back of this book) get larger and larger as you

move from the upper left corner of the table to the lower right corner?

Obj. 2

Your boss has asked you what the ending balance will be if he puts $8,000 into an investment

Q8-7

earning 9% interest compounded annually. He plans to leave the money untouched for 29

Obj. 2

years. Unfortunately, you only have your accounting textbook available to help you and you

find that Table 1 (inside the front cover) only goes up through 25 periods. How can you solve

this problem using only Table 1? What will be the ending amount in the account?

Kelly Walker places 5% of her salary each year into a company-sponsored 401k retirement

Q8-8

plan. Assume her annual salary is $100,000 and deposits are made to the retirement plan at

Obj. 3

the end of each year. What will be the balance in her 401k account at the end of 25 years if

she never receives a pay raise and the plan earns 11% per year?

You are inspecting Table 2 at the back of this book. At the intersection of the 6% column and

Q8-9

the 13-period row you find the following number: 18.88214. Interpret that number. What

Obj. 3

does it mean?

Why do the interest factors in Table 3 (inside the back cover of this book) get smaller and

Q8-10

smaller as you move from the upper left corner of the table to the lower right corner?

Obj. 4

Your boss has asked you what amount she must invest today at 8% interest so that she will

Q8-11

have $350,000 available to pay off a lump-sum debt that comes due in 32 years. In other words,

Obj. 4

what is the present value of $350,000 that must be paid in 32 years assuming an 8% rate? Un-

fortunately, you only have your accounting textbook available to help you and you find that

Table 3 (inside the back cover) only goes up through 25 periods. How can you solve this prob-

lem using only Table 3? What is the present value?

Why do the interest factors in Table 4 (inside the back cover of this book) get smaller as you

Q8-12

move from left to right, but larger as you move from top to bottom?

Obj. 5

Q8-13 Jeraldo invested $4,100 into a financial instrument that promised to pay him $1,000 at the

end of each year for the next five years. The salesperson explained that this would earn for

Obj. 5

him a 5% rate of return. At the end of five years, Jeraldo noticed that the dollar amount of

his interest earnings had been smaller and smaller as the years went by. Explain why this hap-

pened.

Determine whether the rows in Tables 1 through 4 (at the back of your textbook) are labeled

Q8-14

in years or in periods. Does it make a difference whether they are labeled in years or periods?

Obj. 6

Why?

Imelda is making equal-sized monthly payments of $288 on her car loan. She has only 18 pay-

Q8-15

ments left. Each month, the portion of her payment that goes to pay interest and the portion

Obj. 6

that goes to repay principal is different. Why? Is there any pattern to this change in portions?

Explain why or why not.

If your instructor is using Personal Trainer in this course, you may complete online the assign-

EXERCISES ments identified by .

Write a short definition for each of the terms listed in the Terms and Concepts Defined in this

E8-1

Chapter section.

Assume that you borrow $25,000 on April 1, 2004, at an annual rate of 7%. How much will

E8-2

you owe on March 31, 2005 if you make no payments until that date? How much will you

Obj. 2

owe on March 31, 2006 if you make no payments until that date? If you pay the interest in-

curred for the first year on March 31, 2005, how much will you owe on March 31, 2006 if you

make no other payments until that date?

Today is Dave™s 40th birthday. He is experiencing a midlife crisis and is thinking about re-

E8-3

tirement for the first time. To supplement his expected retirement pension, he deposits $8,000

Obj. 2

in an investment account guaranteed to return him 6% interest annually.

a. What balance will he have in this account on his 65th birthday?

b. How much interest will he earn between now and then?

F306 SECTION F2: Analysis and Interpretation of Financial Accounting Information

307

The Time Value of Money

E8-4 You just won last night™s lotto drawing for the $1,000,000 prize. It will be paid to you in 20

installments of $50,000 each. You will receive the first payment today and receive an addi-

Obj. 2

tional $50,000 payment at the end of each of the next 19 years. What is the present value of

your winnings if 7 1/2% is the appropriate rate? Use an Excel spreadsheet and the PV func-

tion to determine the solution. (Hint: What is the present value of the amount you received

SPREADSHEET

today? How many periods long is the remaining annuity?)

Renalda is saving for a once-in-a-lifetime trip, to begin seven years from today. She plans to

E8-5

visit the South Pacific and Far East. Today she starts her savings plan; she will deposit $1,500

Obj. 3

at the end of each of the next seven years into a 5% savings account.

a. What amount will she have in her account when she begins her trip?

b. What amount of the total will be from her own deposits and what amount will she

have earned in interest?

Optimism, Inc. anticipates the need for factory expansion four years from today. The firm has

E8-6

determined that it will have the necessary funds for expansion if it puts $400,000 per year into

Obj. 3

a stock portfolio expected to earn 9% per year. Deposits will be made at the end of each year.

a. How much is the company planning to raise toward factory expansion with this

plan?

b. What amount would the company expect to raise if it could invest $400,000 per

year for seven years?

c. Why is the answer to part b more than twice as large as the answer to part a even

though the length of the annuity is less than twice as long?

Use an Excel spreadsheet and the FV function to determine the following:

E8-7

Obj. 3

a. The future value of a $2,000 annuity for 30 years at 8% compounded annually.

b. The future value of a $2,000 annuity for 30 years at 8% compounded semiannually.

c. The future value of a $2,000 annuity for 30 years at 8% compounded quarterly.

d. How much extra interest is earned on the annuity above simply by changing the com-

pounding period from annually to quarterly?

e. How much extra interest would be earned by changing from annual compounding to

daily compounding?

SPREADSHEET

f. Louisa set up an individual retirement account (IRA account) on her 35th birthday.

She contributed $2,000 to the account on each subsequent birthday through her 65th.

The account earned 8% compounded annually. What amount of interest (in dollars)

will she have earned on this investment?

g. Suppose she had started the IRA account 10 years earlier when she was 25 years old.

How much additional interest would she have earned on this investment by age 65?

(Hint: Her first payment into the account was on which birthday?)

Assume you will receive $1,000 at the end of year 1. What is its present value at the beginning

E8-8

of year 1 if you expect an 8% rate of return? What is the present value if you expect a 9% re-

Obj. 4

turn? 10%? What can you conclude about the effect of the rate of return on the present value

of cash to be received in the future?

What is the present value of $800 to be received at the end of one year if it must provide a re-

E8-9

turn of 8%? What is the present value of $900 to be received at the end of one year? $1,500?

Obj. 4

What can you conclude about the effect of the amount expected to be received on its present

value?

What is the present value of $1,000 to be received at the end of one year if it must provide a

E8-10

return of 5%? What is the present value of $1,000 to be received at the end of two years? Three

Obj. 4

years? What can you conclude about the effect of time until receipt on the present value of

future cash inflows?

E8-11 Assume that you received a loan on July 1, 2004. The lender charges annual interest at 5%.

On June 30, 2009, you owe the lender $510.52. Assuming that you made no payments for

Obj. 4

principal or interest on the loan during the five years, how much did you borrow?

What is the present value of an annuity of $200 per year for five years if the required rate of

E8-12

return is 8%? What is the present value of the annuity if the required rate of return is 10%?

Obj. 5

F307

CHAPTER F8: The Time Value of Money

308 The Time Value of Money

E8-13 A wealthy uncle has offered to give you either of two assets: (a) an asset that pays $500 at the

end of three years or (b) an asset that pays $100 at the end of each year for five years. Assume

Obj. 5

that both assets earn a 7% annual rate of return. Which asset should you choose?

Lincoln Corporation expanded recently by investing $100,000 in new business assets. This has in-

E8-14

creased annual operating cash inflows by $50,500 and annual operating cash outflows by $30,200.

Obj. 5

These increases are expected for a total of six years. At that time, these new assets will be obso-

lete and worthless. (Assume that operating cash inflows and outflows occur at the end of the year.)

a. If the corporation requires a return of 8% on its investments, was this a wise invest-

ment decision? Show calculations to prove your answer.

b. If you decide that the company has not met its investment goals, what minimum an-

nual net cash inflow over the six-year period would give the desired rate of return?

c. Assuming that operating cash outflows do not change, what is the necessary increase in

operating cash inflows that is needed to earn an 8% return?

Katina Washington is currently employed as a computer programmer by Megatel Company.

E8-15

Her dream, however, is to start her own computer software firm. To provide cash to start her

Objs. 2, 5

own business in six years she will invest $10,000 today. She thinks the investment will earn a

12% annual return.

a. How much would Katina have in her account at the end of six years if she earns 12%

on the investment? How much of this would be interest earned during the six years?

b. Assume, instead, that Katina has decided she needs $20,000 to begin business. She

wants to invest equal amounts at the end of each year for the next six years to accumu-

SPREADSHEET

late the $20,000 needed at that time.

i. How much must be invested each year, assuming that it earns 12% interest?

ii. How much will the investment be worth at the end of each of the next six years?

iii. How much will Katina have put into the account over the six years?

iv. How much interest will be earned over the six years?

I. M. Cansado is about to retire. He has a retirement account that allows two payment op-

E8-16

tions. Under Option 1, he can choose to receive $140,000 at the end of six years. Under Op-

Objs. 4, 5

tion 2, he can choose to receive $20,000 at the end of each year for six years. An interest rate

of 10% is applicable to both plans. (a) Which retirement plan has the highest present value at

the beginning of the six-year period? (b) Which option would you recommend?

Complete the tables and answer the questions.

E8-17

Objs. 2, 3, 4, 5

a.

Single Compounding Interest Future

Sum Rate Time Frequency Factor Value

$1,000 12% 2 years Annual ________ ________

$1,000 12% 2 years Semiannual ________ ________

$1,000 12% 2 years Quarterly ________ ________

$1,000 12% 2 years Monthly ________ ________

b. Summarize the effect of changing the compounding period on the future value of a

single sum. Explain why this effect appears reasonable.

c. What effect do you think that changing the compounding period of an annuity would

have on its future value? Explain why you think this.

d.

Single Compounding Interest Present

Sum Rate Time Frequency Factor Value

$1,000 12% 2 years Annual ________ ________

$1,000 12% 2 years Semiannual ________ ________

$1,000 12% 2 years Quarterly ________ ________

$1,000 12% 2 years Monthly ________ ________

e. Summarize the effect of changing the compounding period on the present value of a

single sum. Explain why this effect appears reasonable.

f. What effect do you think that changing the compounding period of an annuity would

have on its present value? Explain why you think this.

F308 SECTION F2: Analysis and Interpretation of Financial Accounting Information

309

The Time Value of Money

E8-18 An investment is expected to pay a return of $100 per year. The interest rate for the invest-

ment is 6%. What will the price of the investment be if it has a life of 5 years? 10 years? 20

Obj. 6

years?

What is the present value of an investment that pays $80 at the end of each year for 10 years

E8-19

and pays an additional $1,000 at the end of the tenth year if the required rate of return is 7%?

Obj. 6

8%? 9%?

E8-20 An investment has a life of 10 years. The rate of return for the investment is 6%. What will

the price of the investment be if it is expected to pay a return of $10 per year? $100 per year?

Obj. 6

What is the maximum amount a company should pay for equipment that it expects will in-

E8-21

crease its net income and cash flow by $250,000 per year for five years? The company requires

Obj. 6

a 12% return on its investment.

Old Money Company borrowed $1 million from a bank on January 1, 2004. The loan is to be

E8-22

repaid in annual installments over a three-year period. The bank requires a 9% return.

Obj. 6

a. What is the amount of Old Money™s required payment to the bank each year?

b. How much interest expense will Old Money incur each year?

c. Show how this loan would be entered into Old Money™s books on January 1, 2004.

d. Show how Old Money™s first annual installment payment would be entered into its

books. Use the format below for parts C and D.

ASSETS LIABILITIES OWNERS™ EQUITY

Other Contributed Retained

Date Accounts Cash Assets Capital Earnings

Lily Pewshun negotiated a three-year, 9%, $46,000 loan from her bank. It called for three

E8-23

equal-sized year-end payments.

Obj. 6

a. Determine the amount of her payment each year (round to the nearest dollar).

b. Show how the loan, and each of the three payments, would be entered into her account-

ing system (round each amount to the nearest dollar). Use the format shown below.

ASSETS LIABILITIES OWNERS™ EQUITY

Other Contributed Retained

Date Accounts Cash Assets Capital Earnings

E8-24 a. Calculate each of the following:

i. The present value of $300 to be received at the end of three years if invested at 6%.

Objs. 4, 5, 6

ii. The present value of $300 to be received at the end of four years if invested at 6%.

iii. The present value of $300 to be received at the end of four years if invested at 5%.

iv. The present value of $300 to be received at the end of each year for four years if in-

vested at 6%.

F309

CHAPTER F8: The Time Value of Money

310 The Time Value of Money

v. The present value of $100 to be received at the end of one year, $200 to be received

at the end of two years, $300 to be received at the end of three years, and $600 to be

received at the end of four years at 6%.

b. Inspect your results. What do they suggest to you about the effect of time periods and

interest rates on the present value of amounts to be received in the future?

Use an Excel spreadsheet and the FV, PV, and PMT functions to determine the amount of

E8-25

each of the following. R the annual interest rate and t number of years. When there are

Objs. 3, 5, 6

multiple cash flows per year, the amount of the annuity shown below is the amount of each

individual cash flow (not the total cash flow for the year). Round all answers to the nearest

dollar.

a. Present value of a $500 annuity when R 11% compounded annually and t 18

b. Future value of a $2,400 annuity when R 5% compounded annually and t 25

c. Future value of a $950 annuity when R 12.8% compounded semiannually and t 15

SPREADSHEET

d. The annual annuity payment that will provide $13,400 in eight years when R 9%

compounded annually

e. Present value of a $10,000 annuity when R 8% compounded quarterly and t 10

f. Future value of a $238 annuity when R 7% compounded annually and t 16

g. Present value of a $1,000 annuity when R 6 % compounded annually and t 3

h. Present value of a $700 annuity when R 10% compounded semiannually and t 11

i. The semiannual annuity payment that will pay off, over six years, a $9,860 debt owed

today if R 13%

j. Future value of a $1 annuity when R 8% compounded annually and t 200

If your instructor is using Personal Trainer in this course, you may complete online the assign-

PROBLEMS ments identified by .

Computing Future Value

P8-1

Objs. 2, 3 Arthur has just graduated from college and has his first job. His salary is that of an entry-level

employee, so he has to budget his money carefully. However, he does understand the need to

save money for the future.

Required

A. Assume that he deposits $500 at the end of each year for 10 years into an investment

account earning 7%. He then stops making deposits and uses the money instead for

house and car payments. How much will be in the investment account at the end of

the 10-year period?

B. Assume Arthur decides to keep the investment but does not make any additional con-

tributions. How much will be in the account when he retires, after working for another

25 years?

C. Assume that Arthur does not begin saving until he has worked for 20 years. If he plans

to retire in 15 years from that time, how much would he have to invest at the end of

each year, in an account earning 7%, to equal the balance in the account in part B?

D. Calculate the total amount of cash that Arthur would pay in under parts A and B com-

bined and the amount he would pay in under part C. Why is there a difference?

Computing Future Value