The Time Value

of Money
“The importance of money flows from it being
a link between the present and the future.”
—John Maynard Keynes

Get a Free $1,000 Bond with Every Car Bought

This Week!
There is a car dealer who appears on television regularly. He does his
own commercials. He is quite loud and is also very aggressive. According
to him you will pay way too much if you buy your car from anyone else in
town. You might have a car dealer like this in your hometown.
One of the authors of this book used to watch and listen to the television
commercials for a particular car dealer. One promotion struck him as
being particularly interesting. The automobile manufacturers had been
offering cash rebates to buyers of particular cars but this promotion had
recently ended. This local car dealer seemed to be picking up where the
manufacturers had left off. He was offering “a free $1,000 bond with every
new car purchased” during a particular week. This sounded pretty good.
The fine print of this deal was not revealed until you were signing the
final sales papers. Even then you had to look close since the print was
small. It turns out the “$1,000 bond” offered to each car buyer was what
is known as a “zero coupon bond.” This means that there are no periodic
interest payments. The buyer of the bond pays a price less than the face
value of $1,000 and then at maturity the issuer pays $1,000 to the holder of
the bond. The investor’s return is entirely equal to the difference between
the lower price paid for the bond and the $1,000 received at maturity.
How much less than $1,000 did the dealer have to pay to get this bond
he was giving away? The amount paid by the dealer is what the bond would
be worth (less after paying commissions) if the car buyer wanted to sell this
bond now. It turns out that this bond had a maturity of 30 years. This is how

194
long the car buyer would have to wait to receive the $1,000. The value of Learning Objectives
the bond at the time the car was purchased was about $57. This is what
the car dealer paid for each of these bonds and is the amount the car buyer After reading this chapter,
would get from selling the bond. It’s a pretty shrewd marketing gimmick you should be able to:
when a car dealer can buy something for $57 and get the customers to
believe they are receiving something worth $1,000. 1. Explain the time value of
money and its importance
In this chapter you will become armed against such deceptions. It’s all
in the business world.
in the time value of money.
2. Calculate the future value
Source: This is inspired by an actual marketing promotion. Some of the details have been changed, and all identities have and present value of a
been hidden, so the author does not get sued.
single amount.
3. Find the future and present
values of an annuity.
Chapter Overview 4. Solve time value of money
problems with uneven
A dollar in hand today is worth more than a promise of a dollar tomorrow. This cash flows.
is one of the basic principles of financial decision making. Time value analysis
is a crucial part of financial decisions. It helps answer questions about how much 5. Solve for the interest rate,
number or amount of
money an investment will make over time and how much a firm must invest now
payments, or the number
to earn an expected payoff later. of periods in a future or
In this chapter we will investigate why money has time value, as well as learn present value problem.
how to calculate the future value of cash invested today and the present value
of cash to be received in the future. We will also discuss the present and future
values of an annuity—a series of equal cash payments at regular time intervals.
Finally, we will examine special time value of money problems, such as how to
find the rate of return on an investment and how to deal with a series of uneven
cash payments.

195
196 Part II  Essential Concepts in Finance

Why Money Has Time Value

The time value of money means that money you hold in your hand today is worth more
than the same amount of money you expect to receive in the future. Similarly, a given
amount of money you must pay out today is a greater burden than the same amount
paid in the future.
In Chapter 2 we learned that interest rates are positive in part because people
prefer to consume now rather than later. Positive interest rates indicate, then, that
money has time value. When one person lets another borrow money, the first person
requires compensation in exchange for reducing current consumption. The person who
borrows the money is willing to pay to increase current consumption. The cost paid by
the borrower to the lender for reducing consumption, known as an opportunity cost, is
the real rate of interest.
The real rate of interest reflects compensation for the pure time value of money.
The real rate of interest does not include interest charged for expected inflation or the
risk factors discussed in Chapter 2. Recall from the interest rate discussion in Chapter 2
that many factors—including the pure time value of money, inflation, default risk,
illiquidity risk, and maturity risk—determine market interest rates.
The required rate of return on an investment reflects the pure time value of money,
an adjustment for expected inflation, and any risk premiums present.

Measuring the Time Value of Money

Financial managers adjust for the time value of money by calculating the future value
and the present value. Future value and present value are mirror images of each other.
Future value is the value of a starting amount at a future point in time, given the rate
of growth per period and the number of periods until that future time. How much will
$1,000 invested today at a 10 percent interest rate grow to in 15 years? Present value
is the value of a future amount today, assuming a specific required interest rate for a
number of periods until that future amount is realized. How much should we pay today
to obtain a promised payment of $1,000 in 15 years if investing money today would
yield a 10 percent rate of return per year?

The Future Value of a Single Amount

To calculate the future value of a single amount, we must first understand how money
grows over time. Once money is invested, it earns an interest rate that compensates for
the time value of money and, as we learned in Chapter 2, for default risk, inflation, and
other factors. Often, the interest earned on investments is compound interest—interest
earned on interest and on the original principal. In contrast, simple interest is interest
earned only on the original principal.
To illustrate compound interest, assume the financial manager of SaveCom decided
to invest $100 of the firm’s excess cash in an account that earns an annual interest rate
of 5 percent. In one year, SaveCom will earn $5 in interest, calculated as follows:
 Chapter 8  The Time Value of Money 197

Balance at the end of year 1 = Principal + Interest

= $100 + (100 × .05)

= $100 × (1 + .05)

= $100 × 1.05

= $105

The total amount in the account at the end of year 1, then, is $105.
But look what happens in years 2 and 3. In year 2, SaveCom will earn 5 percent
of 105. The $105 is the original principal of $100 plus the first year’s interest—so
the interest earned in year 2 is $5.25, rather than $5.00. The end of year 2 balance is
$110.25—$100 in original principal and $10.25 in interest. In year 3, SaveCom will
earn 5 percent of $110.25, or $5.51, for an ending balance of $115.76, shown as follows:

Beginning = Ending
Balance × (1 + Interest Rate) Balance Interest
Year 1 $100.00 × 1.05 = $105.00 $5.00
Year 2 $105.00 × 1.05 = $110.25 $5.25
Year 3 $110.25 × 1.05 = $115.76 $5.51

In our example, SaveCom earned $5 in interest in year 1, $5.25 in interest in

year 2 ($110.25 – $105.00), and $5.51 in year 3 ($115.76 – $110.25) because of the
compounding effect. If the SaveCom deposit earned interest only on the original
principal, rather than on the principal and interest, the balance in the account at the end
of year 3 would be $115 ($100 + ($5 × 3) = $115). In our case the compounding effect
accounts for an extra $.76 ($115.76 – $115.00 = .76).
The simplest way to find the balance at the end of year 3 is to multiply the original
principal by 1 plus the interest rate per period (expressed as a decimal), 1 + k, raised to
the power of the number of compounding periods, n.1 Here’s the formula for finding the
future value—or ending balance—given the original principal, interest rate per period,
and number of compounding periods:

Future Value for a Single Amount

Algebraic Method
FV = PV × (1 + k)n (8-1a)

where: FV = Future Value, the ending amount

PV = Present Value, the starting amount, or original principal
k = Rate of interest per period (expressed as a decimal)
n = Number of time periods

The compounding periods are usually years but not always. As you will see later in the chapter, compounding periods can be
1

months, weeks, days, or any specified period of time.

198 Part II  Essential Concepts in Finance

In our SaveCom example, PV is the original deposit of $100, k is 5 percent, and n

is 3. To solve for the ending balance, or FV, we apply Equation 8-1a as follows:
FV = PV × (1 + k)n
= $100 × (1.05)3
= $100 × 1.1576
= $115.76

We may also solve for future value using a financial table. Financial tables are a
compilation of values, known as interest factors, that represent a term, (1 + k)n in this
case, in time value of money formulas. Table I in the Appendix at the end of the book
is developed by calculating (1 + k)n for many combinations of k and n.
Table I in the Appendix at the end of the book is the future value interest factor
(FVIF) table. The formula for future value using the FVIF table follows:

Future Value for a Single Amount

Table Method

( )
FV = PV × FVIFk, n (8-1b)

where: FV = Future Value, the ending amount

PV = Present Value, the starting amount
FVIFk,n = Future Value Interest Factor given interest rate, k, and number of
periods, n, from Table I

In our SaveCom example, in which $100 is deposited in an account at 5 percent

interest for three years, the ending balance, or FV, according to Equation 8-1b, is as
follows:

Take Note
(
FV = PV × FVIF
k, n )
Because future value
interest factors are
= $100 × FVIF( 5%, 3 )
rounded to four decimal = $100 × 1.1576 (from the FVIF table)
places in Table I, you
= $115.76
may get a slightly
different solution
compared to a problem To solve for FV using a financial calculator, we enter the numbers for PV, n, and
solved by the algebraic k (k is depicted as I/Y on the TI BAII PLUS calculator; on other calculators it may be
method. symbolized by i or I), and ask the calculator to compute FV. The keystrokes follow.

TI BAII PLUS Financial Calculator Solution

Step 1: First press . This clears all the time value of money keys of all
previously calculated or entered values.
Step 2: Press 1 , . This sets the calculator to the
mode where one payment per year is expected, which is the assumption for
the problems in this chapter.

Older TI BAII PLUS financial calculators were set at the factory to a default
value of 12 for P/Y. Change this default value to 1 as shown here if you have
one of these older calculators. For the past several years TI has made the default
value 1 for P/Y. If after pressing you see the value 1 in the display
Step 2: Press 1 , . This sets the calculator to the
mode where one payment per year is expected, which
Chapter is the
8  The assumption
Time for
Value of Money 199
the problems in this chapter.

Older TI BAII PLUS financial calculators were set at the factory to a default
value of 12 for P/Y. Change this default value to 1 as shown here if you have
one of these older calculators. For the past several years TI has made the default
value 1 for P/Y. If after pressing you see the value 1 in the display
window you may press twice to back out of this. No changes to the P/Y
value are needed if your calculator is already set to the default value of 1.
You may similarly skip this adjustment throughout this chapter where the
financial calculator solutions are shown if your calculator already has its P/Y
value set to 1. If your calculator is an older one that requires you to change
the default value of P/Y from 12 to 1, you do not need to make that change
again. The default value of P/Y will stay set to 1 unless you specifically
change it.

Step 3: Input values for principal (PV), interest rate (k or I/Y on the calculator), and
number of periods (n).

100 5 3 Answer: 115.76

In the SaveCom example, we input –100 for the present value (PV), 3 for number
of periods (N), and 5 for the interest rate per year (I/Y). Then we ask the calculator to
compute the future value, FV. The result is $115.76. Our TI BAII PLUS is set to display
two decimal places. You may choose a greater or lesser number if you wish.
We have learned three ways to calculate the future value of a single amount: the
algebraic method, the financial table method, and the financial calculator method. In
the next section, we see how future values are related to changes in the interest rate, k,
and the number of time periods, n.

The Sensitivity of Future Values to Changes in Interest Rates or the

Number of Compounding Periods
Future value has a positive relationship with the interest rate, k, and with the number of
periods, n. That is, as the interest rate increases, future value increases. Similarly, as the
number of periods increases, so does future value. In contrast, future value decreases
with decreases in k and n values.
It is important to understand the sensitivity of future value to k and n because
increases are exponential, as shown by the (1 + k)n term in the future value formula.
Consider this: A business that invests $10,000 in a savings account at 5 percent
for 20 years will have a future value of $26,532.98. If the interest rate is 8 percent
for the same 20 years, the future value of the investment is $46,609.57. We see
that the future value of the investment increases as k increases. Figure 8-1a shows
this graphically.
Now let’s say that the business deposits $10,000 for 10 years at a 5 percent annual
interest rate. The future value of that sum is $16,288.95. Another business deposits
$10,000 for 20 years at the same 5 percent annual interest rate. The future value of that
$10,000 investment is $26,532.98. Just as with the interest rate, the higher the number
of periods, the higher the future value. Figure 8-1b shows this graphically.
200 Part II  Essential Concepts in Finance

Future Value of $10,000 After 20 Years at 5% and 8%

$50,000
Future Value at 5%
$45,000
Future Value at 8%
$40,000

$35,000

Future Value, FV
$30,000

$25,000

$20,000

$15,000

$10,000
Figure 8-1a  Future Value
at Different Interest Rates $5,000
Figure 8-1a shows the
$0
future value of $10,000 after
20 years at interest rates of 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
5 percent and 8 percent. Number of Years, n

Future Value of $10,000 after 10 Years and 20 Years at 5%

$30,000
Future Value after 10 years
Future Value after 20 years
$25,000

$20,000
Future Value, FV

$15,000

$10,000

$5,000
Figure 8-1b  Future
Value at Different Times
Figure 8-1b shows the $0
future value of $10,000 after 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
10 years and 20 years at
an interest rate of 5 percent Number of Years, n
 Chapter 8  The Time Value of Money 201

The Present Value of a Single Amount

Present value is today’s dollar value of a specific future amount. With a bond, for
instance, the issuer promises the investor future cash payments at specified points
in time. With an investment in new plant or equipment, certain cash receipts are
expected. When we calculate the present value of a future promised or expected
cash payment, we discount it (mark it down in value) because it is worth less if
it is to be received later rather than now. Similarly, future cash outflows are less
burdensome than present cash outflows of the same amount. Future cash outflows
are similarly discounted (made less negative). In present value analysis, then, the
interest rate used in this discounting process is known as the discount rate. The
discount rate is the required rate of return on an investment. It reflects the lost
opportunity to spend or invest now (the opportunity cost) and the various risks
assumed because we must wait for the funds.
Discounting is the inverse of compounding. Compound interest causes the value
of a beginning amount to increase at an increasing rate. Discounting causes the present
value of a future amount to decrease at a decreasing rate.
To demonstrate, imagine the SaveCom financial manager needed to know how much
to invest now to generate $115.76 in three years, given an interest rate of 5 percent.
Given what we know so far, the calculation would look like this:

FV = PV × (1 + k) n

$115.76 = PV × 1.053

$115.76 = PV × 1.157625

PV = $100.00

To simplify solving present value problems, we modify the future value for a single
amount equation by multiplying both sides by 1/(1 + k)n to isolate PV on one side of
the equal sign. The present value formula for a single amount follows:

The Present Value of a Single Amount Formula

Algebraic Method
1
PV = FV ×
(1 + k) n (8-2a)

where: PV = Present Value, the starting amount

FV = Future Value, the ending amount
k = Discount rate of interest per period (expressed as a decimal)
n = Number of time periods

Applying this formula to the SaveCom example, in which its financial manager
wanted to know how much the firm should pay today to receive $115.76 at the end
of three years, assuming a 5 percent discount rate starting today, the following is the
present value of the investment:
202 Part II  Essential Concepts in Finance

1
PV = FV ×
(1 + k) n

1
= $115.76 ×
(1 + .05)3

= $115.76 × .86384

= $100.00

SaveCom should be willing to pay $100 today to receive $115.76 three years from
now at a 5 percent discount rate.
To solve for PV, we may also use the Present Value Interest Factor Table in Table
II in the Appendix at the end of the book. A present value interest factor, or PVIF, is
calculated and shown in Table II. It equals 1/(1 + k)n for given combinations of k and
n. The table method formula, Equation 8-2b, follows:

The Present Value of a Single Amount Formula

Table Method
PV = FV × (PVIFk, n) (8-2b)

where: PV = Present Value

FV = Future Value
PVIFk,n = Present Value Interest Factor given discount rate, k, and number
of periods, n, from Table II

In our example, SaveCom’s financial manager wanted to solve for the amount
that must be invested today at a 5 percent interest rate to accumulate $115.76
within three years. Applying the present value table formula, we find the following
solution:

PV = FV × ( PVIF )
k, n

= $115.76 × ( PVIF ) 5%, 3

= $115.76 × .8638 (from the PVIF table)

= $99.99 (slightly lower than $100 due to the rounding to four
places in the table)

The present value of $115.76, discounted back three years at a 5 percent discount
rate, is $100.
To solve for present value using a financial calculator, enter the numbers for future
value, FV, the number of periods, n, and the interest rate, k—symbolized as I/Y on the
calculator—then hit the CPT (compute) and PV (present value) keys. The sequence
follows:
 Chapter 8  The Time Value of Money 203

TI BAII PLUS Financial Calculator Solution

Step 1: Press to clear previous values.

Step 2: Press 1 , to ensure the calculator is in the
mode for annual interest payments.
Step 3: Input the values for future value, the interest rate, and number of periods,
and compute PV.
115.76 5 3 Answer: –100.00

The financial calculator result is displayed as a negative number to show that the
present value sum is a cash outflow—that is, that sum will have to be invested to earn
$115.76 in three years at a 5 percent annual interest rate.
We have examined how to find present value using the algebraic, table, and financial
calculator methods. Next, we see how present value analysis is affected by the discount
rate, k, and the number of compounding periods, n.

The Sensitivity of Present Values to Changes in the Interest Rate or

the Number of Compounding Periods
In contrast with future value, present value is inversely related to k and n values. In
other words, present value moves in the opposite direction of k and n. If k increases,
present value decreases; if k decreases, present value increases. If n increases, present
value decreases; if n decreases, present value increases.
Consider this: A business that expects a 5 percent annual return on its investment
(k = 5%) should be willing to pay $3,768.89 today (the present value) for $10,000 to be
received 20 years from now. If the expected annual return is 8 percent for the same 20
years, the present value of the investment is only $2,145.48. We see that the present value
of the investment decreases as k increases. The way the present value of the $10,000
varies with changes in the required rate of return is shown graphically in Figure 8-2a.
Now let’s say that a business expects to receive $10,000 ten years from now. If its
required rate of return for the investment is 5 percent annually, then it should be willing
to pay $6,139 for the investment today (the present value is $6,139). If another business
expects to receive $10,000 twenty years from now and it has the same 5 percent annual
required rate of return, then it should be willing to pay $3,769 for the investment (the
present value is $3,769). Just as with the interest rate, the greater the number of periods,
the lower the present value. Figure 8-2b shows how it works.
In this section we have learned how to find the future value and the present value
of a single amount. Next, we will examine how to find the future value and present
value of several amounts.

Working with Annuities

Financial managers often need to assess a series of cash flows rather than just one. One
common type of cash flow series is the annuity—a series of equal cash flows, spaced
evenly over time.
204 Part II  Essential Concepts in Finance

Present Value of $10,000 Expected in 20 Years at 5% and 8%

$12,000
Present Value at 5%
Present Value at 8%
$10,000

$8,000

Present Value, PV
$6,000

$4,000
Figure 8-2a  Present
Value at Different Interest
Rates $2,000

Figure 8-2a shows the

present value of $10,000 to
be received in 20 years at $0
interest rates of 5 percent and 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
8 percent. Number of Years, n

Present Value of $10,000 to Be Received

in 10 Years and 20 Years at 5%
$12,000
Present Value for 10-Year Deal
Present Value for 20-Year Deal
$10,000

$8,000
Present Value, PV

$6,000

$4,000

Figure 8-2b  Present

Value at Different Times $2,000

Figure 8-2b shows the

present value of $10,000 to
be received in 10 years and $0
20 years at interest rates of 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
5 percent and 8 percent. Number of Years, n
 Chapter 8  The Time Value of Money 205

Professional athletes often sign contracts that provide annuities for them after they retire,
in addition to the signing bonus and regular salary they may receive during their playing
years. Consumers can purchase annuities from insurance companies as a means of providing
retirement income. The investor pays the insurance company a lump sum now in order to
receive future payments of equal size at regularly spaced time intervals (usually monthly).
Another example of an annuity is the interest on a bond. The interest payments are usually
equal dollar amounts paid either annually or semiannually during the life of the bond.
Annuities are a significant part of many financial problems. You should learn to
recognize annuities and determine their value, future or present. In this section we will
explain how to calculate the future value and present value of annuities in which cash
flows occur at the end of the specified time periods. Annuities in which the cash flows
occur at the end of each of the specified time periods are known as ordinary annuities.
Annuities in which the cash flows occur at the beginning of each of the specified time
periods are known as annuities due.

Future Value of an Ordinary Annuity

Financial managers often plan for the future. When they do, they often need to know
how much money to save on a regular basis to accumulate a given amount of cash at a
specified future time. The future value of an annuity is the amount that a given number
of annuity payments, n, will grow to at a future date, for a given periodic interest rate, k.
For instance, suppose the SaveCom Company plans to invest $500 in a money market
account at the end of each year for the next four years, beginning one year from today.
The business expects to earn a 5 percent annual rate of return on its investment. How
much money will SaveCom have in the account at the end of four years? The problem
is illustrated in the timeline in Figure 8-3. The t values in the timeline represent the end
of each time period. Thus, t1 is the end of the first year, t2 is the end of the second year,
and so on. The symbol t0 is now, the present point in time.
Because the $500 payments are each single amounts, we can solve this problem
one step at a time. Looking at Figure 8-4, we see that the first step is to calculate the
future value of the cash flows that occur at t1, t2, t3, and t4 using the future value formula
for a single amount. The next step is to add the four values together. The sum of those
values is the annuity’s future value.
As shown in Figure 8-4, the sum of the future values of the four single amounts is
the annuity’s future value, $2,155.05. However, the step-by-step process illustrated in

t0 t1 t2 t3 t4
(Today)

$500 $500 $500 $500

FV at FV at FV at
5% interest for 5% interest for 5% interest for
3 years 2 years 1 year

Figure 8-3  SaveCom

Σ = FVA = ? Annuity Timeline
206 Part II  Essential Concepts in Finance

Figure 8-4 is time-consuming even in this simple example. Calculating the future value
of a 20- or 30-year annuity, such as would be the case with many bonds, would take
an enormous amount of time. Instead, we can calculate the future value of an annuity
easily by using the following formula:

Future Value of an Annuity

Algebraic Method

 (1 + k) n − 1 
FVA = PMT ×  
 k  (8-3a)

where: FVA = Future Value of an Annuity

PMT = Amount of each annuity payment
k = Interest rate per time period expressed as a decimal
n = Number of annuity payments

Using Equation 8-3a in our SaveCom example, we solve for the future value of
the annuity at 5 percent interest (k = 5%) with four $500 end-of-year payments (n = 4
and PMT = $500), as follows:

 (1 + .05)4 − 1 
FVA = 500 ×  
 .05 

= 500 × 4.3101

= $2,155.05

For a $500 annuity with a 5 percent interest rate and four annuity payments, we see
that the future value of the SaveCom annuity is $2,155.05.

t0 t1 t2 t3 t4
(Today)

$500 $500 $500 $500

FVA
FV = 500 × (1 + .05) 3 FV = 500 × (1 + .05) 1
= 500 × 1.1576 = 500 × 1.05
FV = 500
= 578.80 = 525
FV = 500 × (1 + .05) 2 525
= 500 × 1.1025 551.25
= 551.25

578.80
Figure 8-4  Future Value
of the SaveCom Annuity Σ = FVA = $2,155.05
 Chapter 8  The Time Value of Money 207

To find the future value of an annuity with the table method, we must find the
future value interest factor for an annuity (FVIFA), found in Table III in the Appendix
at the end of the book. The FVIFAk, n is the value of [(1 + k)n – 1] ÷ k for different
combinations of k and n.

Future Value of an Annuity Formula

Table Method
FVA = PMT × FVIFAk, n (8-3b)

where: FVA = Future Value of an Annuity

PMT = Amount of each annuity payment
FVIFAk, n = Future Value Interest Factor for an Annuity from Table III
k = Interest rate per period
n = Number of annuity payments

In our SaveCom example, then, we need to find the FVIFA for a discount rate
of 5 percent with four annuity payments. Table III in the Appendix shows that the
FVIFAk = 5%, n = 4 is 4.3101. Using the table method, we find the following future value
of the SaveCom annuity:

FVA = 500 × FVIFA5%, 4

= 500 × 4.3101 (from the FVIFA table)
= $2,155.05

To find the future value of an annuity using a financial calculator, key in the
values for the annuity payment (PMT), n, and k (remember that the notation for
the interest rate on the TI BAII PLUS calculator is I/Y, not k). Then compute the
future value of the annuity (FV on the calculator). For a series of four $500 end-of- Take Note
year (ordinary annuity) payments where n = 4 and k = 5 percent, the computation Note that slight differences
is as follows: occur between the
table method, algebraic
method, and calculator
solution. This is because
TI BAII PLUS Financial Calculator Solution our financial tables round
interest factors to four
Step 1: Press to clear previous values. decimal places, whereas
the other methods
Step 2: Press 1 , . Repeat generally use many more
significant figures for
until END shows in the display to set the annual interest rate greater accuracy.
mode and to set the annuity payment to end of period mode.
Step 3: Input the values and compute.
0 5 4 500 Answer: 2,155.06

In the financial calculator inputs, note that the payment is keyed in as a negative
number to indicate that the payments are cash outflows—the payments flow out from
the company into an investment.
208 Part II  Essential Concepts in Finance

The Present Value of an Ordinary Annuity

Because annuity payments are often promised (as with interest on a bond investment)
or expected (as with cash inflows from an investment in new plant or equipment), it is
important to know how much these investments are worth to us today. For example,
assume that the financial manager of Buy4Later, Inc. learns of an annuity that promises
to make four annual payments of $500, beginning one year from today. How much
should the company be willing to pay to obtain that annuity? The answer is the present
value of the annuity.
Because an annuity is nothing more than a series of equal single amounts, we
could calculate the present value of an annuity with the present value formula for a
single amount and sum the totals, but that would be a cumbersome process. Imagine
calculating the present value of a 50-year annuity! We would have to find the present
value for each of the 50 annuity payments and total them.
Fortunately, we can calculate the present value of an annuity in one step with the
following formula:

The Present Value of an Annuity Formula

Algebraic Method
 1 
1 −
PVA = PMT ×  (1 + k) n

 
 k 
(8-4a)

where: PVA = Present Value of an Annuity

PMT = Amount of each annuity payment
k = Discount rate per period expressed as a decimal
n = Number of annuity payments

Using our example of a four-year ordinary annuity with payments of $500 per
year and a 5 percent discount rate, we solve for the present value of the annuity as
follows:

 1 
1 −
PVA = 500 ×  (1 + .05)4

 
 .05 

= 500 × 3.54595

= $1,772.97

The present value of the four-year ordinary annuity with equal yearly payments of
$500 at a 5 percent discount rate is $1,772.97.
We can also use the financial table for the present value interest factor for an
annuity (PVIFA) to solve present value of annuity problems. The PVIFA table is
found in Table IV in the Appendix at the end of the book. The formula for the table
method follows:
 Chapter 8  The Time Value of Money 209

The Present Value of an Annuity Formula

Table Method
PVA = PMT × PVIFAk, n (8-4b)

where: PVA = Present Value of an Annuity

PMT = Amount of each annuity payment
PVIFAk, n = Present Value Interest Factor for an Annuity from Table IV
k = Discount rate per period
n = Number of annuity payments

Applying Equation 8-4b, we find that the present value of the four-year annuity
with $500 equal payments and a 5 percent discount rate is as follows:2

PVA = 500 × PVIFA5%, 4

= 500 × 3.5460
= $1,773.00

We may also solve for the present value of an annuity with a financial calculator.
Simply key in the values for the payment, PMT, number of payment periods, n, and
the interest rate, k—symbolized by I/Y on the TI BAII PLUS calculator—and ask the
calculator to compute PVA (PV on the calculator). For the series of four $500 payments
where n = 4 and k = 5 percent, the computation follows:

TI BAII PLUS Financial Calculator Solution

Step 1: Press to clear previous values.

Step 2: Press 1 , . Repeat
until END shows in the display to set the annual interest rate
mode and to set the annuity payment to end of period mode.
Step 3: Input the values and compute.
5 4 500 Answer: –1,772.97

The financial calculator present value result is displayed as a negative number to

signal that the present value sum is a cash outflow—that is, $1,772.97 will have to
be invested to earn a 5 percent annual rate of return on the four future annual annuity
payments of $500 each to be received.

Future and Present Values of Annuities Due

Sometimes we must deal with annuities in which the annuity payments occur at the beginning
of each period. These are known as annuities due, in contrast to ordinary annuities in which
the payments occurred at the end of each period, as described in the preceding section.
Annuities due are more likely to occur when doing future value of annuity (FVA)
problems than when doing present value of annuity (PVA) problems. Today, for instance,
2
The $.03 difference between the algebraic result and the table formula solution is due to differences in rounding.
210 Part II  Essential Concepts in Finance

you may start a retirement program, investing regular equal amounts each month or year.
Calculating the amount you would accumulate when you reach retirement age would be
a future value of an annuity due problem. Evaluating the present value of a promised
or expected series of annuity payments that began today would be a present value of an
annuity due problem. This is less common because car and mortgage payments almost
always start at the end of the first period making them ordinary annuities.
Whenever you run into an FVA or a PVA of an annuity due problem, the adjustment
needed is the same in both cases. Use the FVA or PVA of an ordinary annuity formula shown
earlier, then multiply your answer by (1 + k). We multiply the FVA or PVA formula by (1
+ k) because annuities due have annuity payments earning interest one period sooner. So,
higher FVA and PVA values result with an annuity due. The first payment occurs sooner in
the case of a future value of an annuity due. In present value of annuity due problems, each
annuity payment occurs one period sooner, so the payments are discounted less severely.
In our SaveCom example, the future value of a $500 ordinary annuity, with k = 5%
and n = 4, was $2,155.06. If the $500 payments occurred at the beginning of each period
instead of at the end, we would multiply $2,155.06 by 1.05 (1 + k = 1 + .05). The product,
$2,262.81, is the future value of the annuity due. In our earlier Buy4Later, Inc., example,
we found that the present value of a $500 ordinary annuity, with k = 5% and n = 4, was
$1,772.97. If the $500 payments occurred at the beginning of each period instead of at
the end, we would multiply $1,772.97 by 1.05 (1 + k = 1 + .05) and find that the present
value of Buy4Later’s annuity due is $1,861.62.
The financial calculator solutions for these annuity due problems are shown next.

Future Value of a Four-Year, $500 Annuity Due, k = 5%

TI BAII PLUS Financial Calculator Solution

Step 1: Press to clear previous values.

Step 2: Press 1 , . Repeat
command until the display shows BGN, to set to the annual
interest rate mode and to set the annuity payment to beginning of period mode.
Step 3: Input the values for the annuity due and compute.
5 4 500 Answer: 2,262.82

Present Value of a Four-Year, $500 Annuity Due, k = 5%

TI BAII PLUS Financial Calculator Solution

Step 1: Press to clear previous values.

Step 2: Press 1 , . Repeat
command until the display shows BGN, to set to the annual
interest rate mode and to set the annuity payment to beginning of period mode.
Step 3: Input the values for the annuity due and compute.
5 4 500 Answer: –1,861.62
 Chapter 8  The Time Value of Money 211

In this section we discussed ordinary annuities and annuities due and learned how
to compute the present and future values of the annuities. Next, we will learn what a
perpetuity is and how to solve for its present value.

Perpetuities
An annuity that goes on forever is called a perpetual annuity or a perpetuity. Perpetuities
contain an infinite number of annuity payments. An example of a perpetuity is the
dividends typically paid on a preferred stock issue.
The future value of perpetuities cannot be calculated, but the present value can be.
We start with the present value of an annuity formula, Equation 8-3a.

 1 
1 −
PVA = PMT ×  (1 + k) n

 
 k 

Now imagine what happens in the equation as the number of payments (n) gets
larger and larger. The (1 + k)n term will get larger and larger, and as it does, it will
cause the 1/(1 + k)n fraction to become smaller and smaller. As n approaches infinity,
the (1 + k)n term becomes infinitely large, and the 1/(1 + k)n term approaches zero. The
entire formula reduces to the following equation:

Present Value of Perpetuity

1 − 0 
PVP = PMT ×  
 k 

or

 1
PVP = PMT ×  
 k
(8-5)

where: PVP = Present Value of a Perpetuity

k = Discount rate expressed as a decimal

Neither the table method nor the financial calculator can solve for the present value
of a perpetuity. This is because the PVIFA table does not contain values for infinity and
the financial calculator does not have an infinity key.
Suppose you had the opportunity to buy a share of preferred stock that pays $70
per year forever. If your required rate of return is 8 percent, what is the present value
of the promised dividends to you? In other words, given your required rate of return,
how much should you be willing to pay for the preferred stock? The answer, found by
applying Equation 8-5, follows:
 1
PVP = PMT ×  
 k

 1 
= $70 × 
 .08 

= $875
212 Part II  Essential Concepts in Finance

The present value of the preferred stock dividends, with a k of 8 percent and a
payment of $70 per year forever, is $875.

Present Value of an Investment with Uneven Cash Flows

Unlike annuities that have equal payments over time, many investments have payments
that are unequal over time. That is, some investments have payments that vary over
time. When the periodic payments vary, we say that the cash flow streams are uneven.
For instance, a professional athlete may sign a contract that provides for an immediate
$7 million signing bonus, followed by a salary of $2 million in year 1, $4 million in
year 2, then $6 million in years 3 and 4. What is the present value of the promised
payments that total $25 million? Assume a discount rate of 8 percent. The present value
calculations are shown in Table 8-1.
As we see from Table 8-1, we calculate the present value of an uneven series of cash
flows by finding the present value of a single amount for each series and summing the totals.
We can also use a financial calculator to find the present value of this uneven series
of cash flows. The worksheet mode of the TI BAII PLUS calculator is especially helpful
in solving problems with uneven cash flows. The C display shows each cash payment
following CF0, the initial cash flow. The F display key indicates the frequency of that
payment. The keystrokes follow.

TI BAII PLUS Financial Calculator PV Solution

Uneven Series of Cash Flows
Keystrokes Display

CF0 = old contents

CF0 = 0.00
7000000 7,000,000.00
2000000 C01 = 2,000,000.00
F01 = 1.00
4000000 C02 = 4,000,000.00
F02 = 1.00
6000000 C03 = 6,000,000.00
Take Note
We used the NPV (net 2 F03 = 2.00
present value) key on
our calculator to solve I = 0.00
this problem. NPV will be
discussed in Chapter 10.
8 I = 8.00
NPV = 21,454,379.70

We see from the calculator keystrokes that we are solving for the present value of
a single amount for each payment in the series except for the last two payments, which
are the same. The value of F03, the frequency of the third cash flow after the initial
cash flow, was 2 instead of 1 because the $6 million payment occurred twice in the
series (in years 3 and 4).
 Chapter 8  The Time Value of Money 213

Table 8-1  The Present Value of an Uneven Stream of Cash Flows

Time Cash Flow PV of Cash Flow

1
t0 $7,000,000 $7,000,000 × = $7,000,000
1.080
1
t1 $2,000,000 $2,000,000 × = $1,851,852
1.081
1
t2 $4,000,000 $4,000,000 × = $3,429,355
1.08 2
1
t3 $6,000,000 $6,000,000 × = $4,762,993
1.083
1
t4 $6,000,000 $6,000,000 × = $4,410,179
1.08 4
Sum of the PVs = $21,454,380

We have seen how to calculate the future value and present value of annuities, the
present value of a perpetuity, and the present value of an investment with uneven cash
flows. Now we turn to time value of money problems in which we solve for k, n, or
the annuity payment.

Special Time Value of Money Problems

Financial managers often face time value of money problems even when they know
both the present value and the future value of an investment. In those cases, financial
managers may be asked to find out what return an investment made—that is, what
the interest rate is on the investment. Still other times financial managers must find
either the number of payment periods or the amount of an annuity payment. In the
next section, we will learn how to solve for k and n. We will also learn how to find
the annuity payment (PMT).

Finding the Interest Rate

Financial managers frequently have to solve for the interest rate, k, when firms make a
long-term investment. The method of solving for k depends on whether the investment
is a single amount or an annuity.

Finding k of a Single-Amount Investment  Financial managers may need to determine

how much periodic return an investment generated over time. For example, imagine
that you are head of the finance department of GrabLand, Inc. Say that GrabLand
purchased a house on prime land 20 years ago for $40,000. Recently, GrabLand sold
the property for $106,131. What average annual rate of return did the firm earn on its
20-year investment?
214 Part II  Essential Concepts in Finance

First, the future value—or ending amount—of the property is $106,131. The present
value—the starting amount—is $40,000. The number of periods, n, is 20. Armed with
those facts, you could solve this problem using the table version of the future value of
a single amount formula, Equation 8-1b, as follows:

FV = PV × ( FVIF )
k, n

$106,131 = $40,000 × ( FVIF k = ?, n = 20 )

$106,131 ÷ $40,000 = FVIFk = ?, n = 20

2.6533 = FVIFk = ?, n = 20

Now find the FVIF value in Table I, shown in part below. The whole table is in the
Appendix at the end of the book. You know n = 20, so find the n = 20 row on the left-
hand side of the table. You also know that the FVIF value is 2.6533, so move across
the n = 20 row until you find (or come close to) the value 2.6533. You find the 2.6533
value in the k = 5% column. You discover, then, that GrabLand’s property investment
had an interest rate of 5 percent.
Future Value Interest Factors, Compounded at k Percent for n Periods, Part of Table I
Interest Rate, k
Number
of
Periods,
n 0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%
18 1.0000 1.1961 1.4282 1.7024 2.0258 2.4066 2.8543 3.3799 3.9960 4.7171 5.5599
19 1.0000 1.2081 1.4568 1.7535 2.1068 2.5270 3.0256 3.6165 4.3157 5.1417 6.1159
20 1.0000 1.2202 1.4859 1.8061 2.1911 2.6533 3.2071 3.8697 4.6610 5.6044 6.7275
25 1.0000 1.2824 1.6406 2.0938 2.6658 3.3864 4.2919 5.4274 6.8485 8.6231 10.8347

Solving for k using the FVIF table works well when the interest rate is a whole
number, but it does not work well when the interest rate is not a whole number. To solve
for the interest rate, k, we rewrite the algebraic version of the future value of a single-
amount formula, Equation 8-1a, to solve for k:

The Rate of Return, k

1
 FV  n
k =  − 1 (8-6)
 PV 

where: k = Rate of return expressed as a decimal

FV = Future Value
PV = Present Value
n = Number of compounding periods
 Chapter 8  The Time Value of Money 215

Let’s use Equation 8-6 to find the average annual rate of return on GrabLand’s
house investment. Recall that the company bought it 20 years ago for $40,000 and sold
it recently for $106,131. We solve for k applying Equation 8-6 as follows:
1
 FV  n
k =  − 1
 PV 

1
 $106,131 20
=  − 1
 $40,000 

= 2.653275.05 − 1

= 1.05 − 1

= .05, or 5%

Equation 8-6 will find any rate of return or interest rate given a starting value, PV,
an ending value, FV, and a number of compounding periods, n.
To solve for k with a financial calculator, key in all the other variables and ask the
calculator to compute k (depicted as I/Y on your calculator). For GrabLand’s house-
buying example, the calculator solution follows:

TI BAII PLUS Financial Calculator Solution

Step 1: Press to clear previous values.

Step 2: Press 1 , .
Step 3: Input the values and compute.
40000 106131 20 Answer: 5.00

Remember when using the financial calculator to solve for the rate of return, you
must enter cash outflows as a negative number. In our example, the $40,000 PV is
entered as a negative number because GrabLand spent that amount to invest in the house.

Finding k for an Annuity Investment  Financial managers may need to find the interest
rate for an annuity investment when they know the starting amount (PVA), n, and the
annuity payment (PMT), but they do not know the interest rate, k. For example, suppose
GrabLand wanted a 15-year, $100,000 amortized loan from a bank. An amortized loan
is a loan that is paid off in equal amounts that include principal as well as interest.3
According to the bank, GrabLand’s payments will be $12,405.89 per year for 15 years.
What interest rate is the bank charging on this loan?
To solve for k when the known values are PVA (the $100,000 loan proceeds), n
(15), and PMT (the loan payments $12,405.89), we start with the present value of an
annuity formula, Equation 8-3b, as follows:

3
Amortize comes from the Latin word mortalis, which means “death.” You will kill off the entire loan after making the
scheduled payments.
216 Part II  Essential Concepts in Finance

PVA = PMT × ( PVIFA k, n )

$100, 000 = $12, 405.89 × ( PVIFA k = ?, n = 15 )

8.0607 = PVIFA k = ?, n = 15

Now refer to the PVIFA values in Table IV, shown in part below. The whole table
is in the Appendix at the end of the book. You know n = 15, so find the n = 15 row on
the left-hand side of the table. You have also determined that the PVIFA value is 8.0607
($100,000/$12,405 = 8.0607), so move across the n = 15 row until you find (or come
close to) the value of 8.0607. In our example, the location on the table where n = 15
and the PVIFA is 8.0607 is in the k = 9% column, so the interest rate on GrabLand’s
loan is 9 percent.
Present Value Interest Factors for an Annuity, Discounted at k Percent for n Periods, Part of Table IV
Interest Rate, k
Number
of
Periods,
n 0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%
13 13.0000 12.1337 11.3484 10.6350 9.9856 9.3936 8.8527 8.3577 7.9038 7.4869 7.1034
14 14.0000 13.0037 12.1062 11.2961 10.5631 9.8986 9.2950 8.7455 8.2442 7.7862 7.3667
15 15.0000 13.8651 12.8493 11.9379 11.1184 10.3797 9.7122 9.1079 8.5595 8.0607 7.6061
16 16.0000 14.7179 13.5777 12.5611 11.6523 10.8378 10.1059 9.4466 8.8514 8.3126 7.8237

To solve this problem with a financial calculator, key in all the variables but k,
and ask the calculator to compute k (depicted as I/Y on the TI calculator) as follows:

TI BAII PLUS Financial Calculator Solution

Step 1: Press to clear previous values.

Step 2: Press 1 , . Repeat
until END shows in the display. Press after you see END in the display.
Step 3: Input the values and compute.
100000 15 12405.89 Answer: 9.00

In this example the PMT was entered as a negative number to indicate that the loan
payments are cash outflows, flowing away from the firm. The missing interest rate value
was 9 percent, the interest rate on the loan.

Finding the Number of Periods

Suppose you found an investment that offered you a return of 6 percent per year. How
long would it take you to double your money? In this problem you are looking for n,
the number of compounding periods it will take for a starting amount, PV, to double
in size (FV = 2 × PV).
 Chapter 8  The Time Value of Money 217

To find n in our example, start with the formula for the future value of a single
amount and solve for n as follows:

FV = PV × ( FVIF )k, n

(
2 × PV = PV × FVIFk = 6%, n = ? )
2.0 = FVIFk = 6%, n = ?

Now refer to the FVIF values, shown below in part of Table I. You know k = 6%,
so scan across the top row to find the k = 6% column. Knowing that the FVIF value
is 2.0, move down the k = 6% column until you find (or come close to) the value 2.0.
Note that it occurs in the row in which n = 12. Therefore, n in this problem, and the
number of periods it would take for the value of an investment to double at 6 percent
interest per period, is 12.
Future Value Interest Factors, Compounded at k Percent for n Periods, Part of Table I
Interest Rate, k
Number
of
Periods,
n 0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%
11 1.0000 1.1157 1.2434 1.3842 1.5395 1.7103 1.8983 2.1049 2.3316 2.5804 2.8531
12 1.0000 1.1268 1.2682 1.4258 1.6010 1.7959 2.0122 2.2522 2.5182 2.8127 3.1384
13 1.0000 1.1381 1.2936 1.4685 1.6651 1.8856 2.1329 2.4098 2.7196 3.0658 3.4523
14 1.0000 1.1495 1.3195 1.5126 1.7317 1.9799 2.2609 2.5785 2.9372 3.3417 3.7975

This problem can also be solved on a financial calculator quite quickly. Just key in
all the known variables (PV, FV, and I/Y) and ask the calculator to compute n.

TI BAII PLUS Financial Calculator Solution

Step 1: Press to clear previous values.

Step 2: Press 1 , .
Step 3: Input the values and compute.
1 2 6 Answer: 11.90

In our example n = 12 when $1 is paid out and $2 received with a rate of interest of
6 percent. That is, it takes approximately 12 years to double your money at a 6 percent
annual rate of interest.

Solving for the Payment

Lenders and financial managers frequently have to determine how much each payment—
or installment—will need to be to repay an amortized loan. For example, suppose you
are a business owner and you want to buy an office building for your company that
costs $200,000. You have $50,000 for a down payment and the bank will lend you the
218 Part II  Essential Concepts in Finance

$150,000 balance at a 6 percent annual interest rate. How much will the annual payments
be if you obtain a 10-year amortized loan?
As we saw earlier, an amortized loan is repaid in equal payments over time. The
period of time may vary. Let’s assume in our example that your payments will occur
annually, so that at the end of the 10-year period you will have paid off all interest and
principal on the loan (FV = 0).
Because the payments are regular and equal in amount, this is an annuity problem.
The present value of the annuity (PVA) is the $150,000 loan amount, the annual
interest rate (k) is 6 percent, and n is 10 years. The payment amount (PMT) is the
only unknown value.
Because all the variables but PMT are known, the problem can be solved by solving
for PMT in the present value of an annuity formula, equation 8-4a, as follows:

 1 
1 −
PVA = PMT ×  (1 + k) n

 
 k 

 1 
1 −
$150,000 = PMT ×  10 
(1.06) 

 .06 

$150,000 = PMT × 7.36009

$150,000
= PMT
7.36009

$20,380.19 = PMT

We see, then, that the payment for an annuity with a 6 percent interest rate, an n of
10, and a present value of $150,000 is $20,380.19.
We can also solve for PMT using the table formula, Equation 8-4b, as follows:

PVA = PMT × ( PVIFA ) k, n

$150,000 = PMT × ( PVIFA 6%, 10 years )

$150,000 = PMT × 7.3601 (look up PVIFA, Table IV)

$150,000
= PMT
7.3601

$20,380.16 = PMT (note the $0.03 rounding error)

The table formula shows that the payment for a loan with the present value of
$150,000 at an annual interest rate of 6 percent and an n of 10 is $20,380.16.
With the financial calculator, simply key in all the variables but PMT and have the
calculator compute PMT as follows:
The financial calculator will display the payment, $20,380.19, as a negative number
because it is a cash outflow.
 Chapter 8  The Time Value of Money 219

TI BAII PLUS Financial Calculator Solution

Step 1: Press to clear previous values.

Step 2: Press 1 , . Repeat
command until the display shows END, to set to the annual
interest rate mode and to set the annuity payment to end of period mode.
Step 3: Input the values for the annuity due and compute.
150,000 6 10 Answer: –20,380.19

Loan Amortization
As each payment is made on an amortized loan, the interest due for that period is paid,
along with a repayment of some of the principal that must also be “killed off.” After
the last payment is made, all the interest and principal on the loan have been paid. This
step-by-step payment of the interest and principal owed is often shown in an amortization
table. The amortization table for the ten-year 6 percent annual interest rate loan of
$150,000 that was discussed in the previous section is shown in Table 8-2. The annual
payment, calculated in the previous section, is $20,380.19.
We see from Table 8-2 how the balance on the $150,000 loan is killed off a little each
year until the balance at the end of year 10 is zero. The payments reflect an increasing
amount going toward principal and a decreasing amount going toward interest over time.

Compounding More Than Once per Year

So far in this chapter, we have assumed that interest is compounded annually. However,
there is nothing magical about annual compounding. Many investments pay interest
that is compounded semiannually, quarterly, or even daily. Most banks, savings and
loan associations, and credit unions, for example, compound interest on their deposits
more frequently than annually.
Suppose you deposited $100 in a savings account that paid 12 percent annual
interest, compounded annually. After one year you would have $112 in your account
($112 = $100 × 1.121).
Now, however, let’s assume the bank used semiannual compounding. With
semiannual compounding you would receive half a year’s interest (6 percent) after six
months. In the second half of the year, you would earn interest both on the interest earned
in the first six months and on the original principal. The total interest earned during the
year on a $100 investment at 12 percent annual interest would be:

$ 6.00 (interest for the first six months)

+ $ .36 (interest on the $6 interest during the second 6 months)4
+ $ 6.00 (interest on the principal during the second six months)
= $ 12.36 total interest in year 1

At the end of the year, you will have a balance of $112.36 if the interest is compounded
semiannually, compared with $112.00 with annual compounding—a difference of $.36.
4
The $.36 was calculated by multiplying $6 by half of 12%: $6.00 × .06 = $.36.
220 Part II  Essential Concepts in Finance

Table 8-2 Amortization Table for a $150,000 Loan, 6 Percent Annual Interest

Rate, 10-Year Term

Loan Amortization Schedule

Amount Borrowed: $150,000
Interest Rate: 6.0%
Term: 10 years
Required Payments: $20,380.19 (found using Equation 8-4a)
Col. 1 Col. 2 Col. 3 Col. 4 Col. 5
Col. 1 x .06 Col. 2 - Col. 3 Col. 1 - Col. 4
Beginning Total Payment Payment Ending
Year Balance Payment of Interest of Principal Balance
1 $ 150,000.00 $ 20,380.19 $ 9,000.00 $ 11,380.19 $ 138,619.81
2 $ 138,619.81 $ 20,380.19 $ 8,317.19 $ 12,063.01 $ 126,556.80
3 $ 126,556.80 $ 20,380.19 $ 7,593.41 $ 12,786.79 $ 113,770.02
4 $ 113,770.02 $ 20,380.19 $ 6,826.20 $ 13,553.99 $ 100,216.02
5 $ 100,216.02 $ 20,380.19 $ 6,012.96 $ 14,367.23 $ 85,848.79
6 $ 85,848.79 $ 20,380.19 $ 5,150.93 $ 15,229.27 $ 70,619.52
7 $ 70,619.52 $ 20,380.19 $ 4,237.17 $ 16,143.02 $ 54,476.50
8 $ 54,476.50 $ 20,380.19 $ 3,268.59 $ 17,111.60 $ 37,364.90
9 $ 37,364.90 $ 20,380.19 $ 2,241.89 $ 18,138.30 $ 19,226.60
10 $ 19,226.60 $ 20,380.19 $ 1,153.60 $ 19,226.60 $ 0.00

Here’s how to find answers to problems in which the compounding period is less
than a year: Apply the relevant present value or future value equation, but adjust k and
n so they reflect the actual compounding periods.
To demonstrate, let’s look at our example of a $100 deposit in a savings
account at 12 percent for one year with semiannual compounded interest. Because
we want to find out what the future value of a single amount will be, we use that
formula to solve for the future value of the account after one year. Next, we divide
the annual interest rate, 12 percent, by two because interest will be compounded
twice each year. Then, we multiply the number of years n (one in our case) by two
because with semiannual interest there are two compounding periods in a year.
The calculation follows:

FV = PV × (1 + k/2)n × 2

= $100 × (1 + .12/2)1 × 2

= $100 × (1 + .06)2

= $100 × 1.1236

= $112.36
 Chapter 8  The Time Value of Money 221

The future value of $100 after one year, earning 12 percent annual interest
compounded semiannually, is $112.36.
To use the table method for finding the future value of a single amount, find the
FVIFk, n in Table I in the Appendix at the end of the book. Then, divide the k by two
and multiply the n by two as follows:

FV = PV × ( FVIFk / 2, n × 2 )
= $100 × ( FVIF 12% / 2, 1 × 2 periods )
= $100 × ( FVIF 6%, 2 periods )
= $100 × 1.1236

= $112.36

To solve the problem using a financial calculator, divide the k (represented as I/Y
on the TI BAII PLUS calculator) by two and multiply the n by two. Next, key in the
variables as follows:

TI BAII PLUS Financial Calculator Solution

Step 1: Press to clear previous values.

Step 2: Press 1 , .
Step 3: Input the values and compute.
100 6 2 Answer: 112.36

The future value of $100 invested for two periods at 6 percent per period is $112.36.5
Other compounding rates, such as quarterly or monthly rates, can be found by
modifying the applicable formula to adjust for the compounding periods. With a quarterly
compounding period, then, annual k should be divided by four and annual n multiplied
by four. For monthly compounding, annual k should be divided by twelve and annual n
multiplied by twelve. Similar adjustments could be made for other compounding periods.

Annuity Compounding Periods

Many annuity problems also involve compounding or discounting periods of less than
a year. For instance, suppose you want to buy a car that costs $20,000. You have $5,000
for a down payment and plan to finance the remaining $15,000 at 6 percent annual
interest for four years. What would your monthly loan payments be?
First, change the stated annual rate of interest, 6 percent, to a monthly rate by
dividing by 12 (6%/12 = 1/2% or .005). Second, multiply the four-year period by 12 to
obtain the number of months involved (4 × 12 = 48 months). Now solve for the annuity
payment size using the annuity formula.

5
Note that we “lied” to our TI BAII PLUS calculator. It asks us for the interest rate per year (I/Y). We gave it the semiannual
interest rate of 6 percent, not the annual interest rate of 12 percent. Similarly, n was expressed as the number of semiannual
periods, two in one year. As long as we are consistent in expressing the k and n values according to the number of compounding
or discounting periods per year, the calculator will give us the correct answer.
222 Part II  Essential Concepts in Finance

In our case, we apply the present value of an annuity formula, equation 8-4a, as follows:

 1 
 1 − 
PVA = PMT × (1 + k) n
 
 k 

 1 
1 −
$15,000 = PMT ×  48 
(1.005) 

 .005 

$15,000 = PMT × 42.5803

$150,000
= PMT
42.5803

$352.28 = PMT

The monthly payment on a $15,000 car loan with a 6 percent annual interest rate
(.5 percent per month) for four years (48 months) is $352.28.
Solving this problem with the PVIFA table in Table IV in the Appendix at the end
of the book would be difficult because the .5 percent interest rate is not listed in the
PVIFA table. If the PVIFA were listed, we would apply the table formula, make the
adjustments to reflect the monthly interest rate and the number of periods, and solve
for the present value of the annuity.
On a financial calculator, we would first adjust the k and n to reflect the same
time period—monthly, in our case—and then input the adjusted variables to solve the
problem as follows:

TI BAII PLUS Financial Calculator Solution

Step 1: Press to clear previous values.

Step 2: Press 1 , . Repeat
command until the display shows END, to set to the annual
interest rate mode and to set the annuity payment to end of period mode.
Step 3: Input the values for the annuity due and compute.
15,000 .5 48 Answer: –352.28

Note that once again we have lied to our TI BAII PLUS financial calculator. The
interest rate we entered was not the 6 percent rate per year but rather the .5 percent rate
per month. We entered not the number of years, four, but rather the number of months, 48.
Because we were consistent in entering the k and n values in monthly terms, the calculator
gave us the correct monthly payment of –352.28 (an outflow of $352.28 per month).

Continuous Compounding
The effect of increasing the number of compounding periods per year is to increase the
future value of the investment. The more frequently interest is compounded, the greater
 Chapter 8  The Time Value of Money 223

the future value. The smallest compounding period is used when we do continuous
compounding—compounding that occurs every tiny unit of time (the smallest unit of
time imaginable).
Recall our $100 deposit in an account at 12 percent for one year with annual
compounding. At the end of year 1, our balance was $112. With semiannual
compounding, the amount increased to $112.36.
When continuous compounding is involved, we cannot divide k by infinity and
multiply n by infinity. Instead, we use the term e, which you may remember from your
math class. We define e as follows:

h
 1
e = lim 1 +  = 2.71828
 h

h →∞

The value of e is the natural antilog of 1 and is approximately equal to 2.71828. This
number is one of those like pi (approximately equal to 3.14159), which we can never
express exactly but can approximate. Using e, the formula for finding the future value
of a given amount of money, PV, invested at annual rate, k, for n years, with continuous
compounding, is as follows:

Future Value with Continuous Compounding

FV = PV × e(k x n) (8-7)

where k (expressed as a decimal) and n are expressed in annual terms

Applying Equation 8-7 to our example of a $100 deposit at 12 percent annual interest
with continuous compounding, at the end of one year we would have the following
balance:

FV = $100 × 2.71828(.12 × 1)
= $112.75

The future value of $100, earning 12 percent annual interest compounded

continuously, is $112.75.
As this section demonstrates, the compounding frequency can impact the
value of an investment. Investors, then, should look carefully at the frequency
of compounding. Is it annual, semiannual, quarterly, daily, or continuous? Other
things being equal, the more frequently interest is compounded, the more interest
the investment will earn.

What’s Next
In this chapter we investigated the importance of the time value of money in
financial decisions and learned how to calculate present and future values for a
single amount, for ordinary annuities, and for annuities due. We also learned how
to solve special time value of money problems, such as finding the interest rate
or the number of periods.
224 Part II  Essential Concepts in Finance

The skills acquired in this chapter will be applied in later chapters, as we evaluate
proposed projects, bonds, and preferred and common stock. They will also be used when
we estimate the rate of return expected by suppliers of capital. In the next chapter, we
will turn to the cost of capital.

Summary
1. Explain the time value of money and its importance in the business world.
Money grows over time when it earns interest. Money expected or promised in the
future is worth less than the same amount of money in hand today. This is because we
lose the opportunity to earn interest when we have to wait to receive money. Similarly,
money we owe is less burdensome if it is to be paid in the future rather than now. These
concepts are at the heart of investment and valuation decisions of a firm.

2. Calculate the future value and present value of a single amount.

To calculate the future value and the present value of a single dollar amount, we may
use the algebraic, table, or calculator methods. Future value and present value are mirror
images of each other. They are compounding and discounting, respectively. With future
value, increases in k and n result in an exponential increase in future value. Increases
in k and n result in an exponential decrease in present value.

3. Find the future and present values of an annuity.

Annuities are a series of equal cash flows. An annuity that has payments that occur at
the end of each period is an ordinary annuity. An annuity that has payments that occur
at the beginning of each period is an annuity due. A perpetuity is a perpetual annuity.
To find the future and present values of an ordinary annuity, we may use the
algebraic, table, or financial calculator method. To find the future and present values of
an annuity due, multiply the applicable formula by (1 + k) to reflect the earlier payment.

4. Solve time value of money problems with uneven cash flows.

To solve time value of money problems with uneven cash flows, we find the value of
each payment (each single amount) in the cash flow series and total each single amount.
Sometimes the series has several cash flows of the same amount. If so, calculate the
present value of those cash flows as an annuity and add the total to the sum of the present
values of the single amounts to find the total present value of the uneven cash flow series.

5. S
 olve special time value of money problems, such as finding the interest rate, number
or amount of payments, or number of periods in a future or present value problem.
To solve special time value of money problems, we use the present value and future
value equations and solve for the missing variable, such as the loan payment, k, or n.
We may also solve for the present and future values of single amounts or annuities in
which the interest rate, payments, and number of time periods are expressed in terms
other than a year. The more often interest is compounded, the larger the future value.
 Chapter 8  The Time Value of Money 225

Equations Introduced in This Chapter

Equation 8-1a. Future Value of a Single Amount—Algebraic Method:

FV = PV × (1 + k)n

where: FV = Future Value, the ending amount

PV = Present Value, the starting amount, or original principal
k = Rate of interest per period (expressed as a decimal)
n = Number of time periods

Equation 8-1b. Future Value of a Single Amount—Table Method:

(
FV = PV × FVIFk, n )
where: FV = Future Value, the ending amount
PV = Present Value, the starting amount
FVIFk,n = Future Value Interest Factor given interest rate, k, and number of
periods, n, from Table I

Equation 8-2a. Present Value of a Single Amount—Algebraic Method:

1
PV = FV ×
(1 + k) n

where: PV = Present Value, the starting amount

FV = Future Value, the ending amount
k = Discount rate of interest per period (expressed as a decimal)
n = Number of time periods

Equation 8-2b. Present Value of a Single Amount—Table Method:

(
PV = FV × PVIFk, n )
where: PV = Present Value
FV = Future Value
PVIFk, n = Present Value Interest Factor given discount rate, k, and number
of periods, n, from Table II
226 Part II  Essential Concepts in Finance

Equation 8-3a. Future Value of an Annuity—Algebraic Method:

 (1 + k) n − 1 
FVA = PMT ×  
 k 

where: FVA = Future Value of an Annuity

PMT = Amount of each annuity payment
k = Interest rate per time period expressed as a decimal
n = Number of annuity payments

Equation 8-3b. Future Value of an Annuity—Table Method:

FVA = PMT × FVIFAk, n

where: FVA = Future Value of an Annuity

PMT = Amount of each annuity payment
FVIFAk, n = Future Value Interest Factor for an Annuity from Table III
k = Interest rate per period
n = Number of annuity payments

Equation 8-4a. Present Value of an Annuity—Algebraic Method:

 1 
 1 − 
PVA = PMT × (1 + k) n
 
 k 

where: PVA = Present Value of an Annuity

PMT = Amount of each annuity payment
k = Discount rate per period expressed as a decimal
n = Number of annuity payments

Equation 8-4b. Present Value of an Annuity—Table Method:

PVA = PMT × PVIFAk, n

where: PVA = Present Value of an Annuity

PMT = Amount of each annuity payment
PVIFAk, n = Present Value Interest Factor for an Annuity from Table IV
k = Discount rate per period
n = Number of annuity payments
 Chapter 8  The Time Value of Money 227

Equation 8-5. Present Value of a Perpetuity:

1
PVP = PMT
k

where: PVP = Present Value of a Perpetuity

k = Discount rate expressed as a decimal

Equation 8-6. Rate of Return:

1
 FV  n
k =  − 1
 PV 

where: k = Rate of return expressed as a decimal

FV = Future Value
PV = Present Value
n = Number of compounding periods

Equation 8-7. Future Value with Continuous Compounding:

FV = PV × e(k x n)

where: FV = Future Value

PV = Present Value
e = Natural antilog of 1
k = Stated annual interest rate expressed as a decimal
n = Number of years

Self-Test
ST-1. J ed is investing $5,000 into an eight-year ST-3. H
 eidi’s grandmother died and provided in her
certificate of deposit (CD) that pays 6 percent will that Heidi will receive $100,000 from
annual interest with annual compounding. How a trust when Heidi turns 21 years of age, 10
much will he have when the CD matures? years from now. If the appropriate discount
rate is 8 percent, what is the present value of
ST-2. T
 im has found a 2010 Toyota 4-Runner on sale this $100,000 to Heidi?
for $19,999. The dealership says that it will
finance the entire amount with a one-year loan, ST-4. Z
 ack wants to buy a new Ford Mustang
and the monthly payments will be $1,776.98. automobile. He will need to borrow $20,000
What is the annualized interest rate on the loan to go with his down payment in order to afford
(the monthly rate times 12)? this car. If car loans are available at a 6 percent
annual interest rate, what will Zack’s monthly
payment be on a four-year loan?
228 Part II  Essential Concepts in Finance

ST-5. B
 ridget invested $5,000 in a growth mutual ST-6. I f Tom invests $1,000 a year beginning today
fund, and in 10 years her investment had grown into a portfolio that earns a 10 percent return
to $15,529.24. What annual rate of return did per year, how much will he have at the end
Bridget earn over this 10-year period? of 10 years? (Hint: Recognize that this is an
annuity due problem.)

Review Questions
1. What is the time value of money? 7. H
 ow does continuous compounding benefit an
investor?
2. Why does money have time value?
8. I f you are doing PVA and FVA problems, what
3. W
 hat is compound interest? Compare
difference does it make if the annuities are ordinary
compound interest to discounting.
annuities or annuities due?
4. H
 ow is present value affected by a change in
9. W
 hich formula would you use to solve for the payment
the discount rate?
required for a car loan if you know the interest rate,
5. What is an annuity? length of the loan, and the borrowed amount? Explain.

6. S
 uppose you are planning to make regular
contributions in equal payments to an investment
fund for your retirement. Which formula would
you use to figure out how much your investments
will be worth at retirement time, given an
assumed rate of return on your investments?

Build Your Communication Skills

CS-1. O
 btain information from four different CS-2. I nterview a mortgage lender in your
financial institutions about the terms of community. Write a brief report about the
their basic savings accounts. Compare the mortgage terms. Include in your discussion
interest rates paid and the frequency of the comments about what interest rates are
compounding. For each account, how much charged on different types of loans, why rates
money would you have in 10 years if you differ, and what fees are charged in addition to
deposited $100 today? Write a one- to two- the interest and principal that mortgagees must
page report of your findings. pay to this lender. Describe the advantages and
disadvantages of some of the loans offered.
 Chapter 8  The Time Value of Money 229

Problems
All these problems may be solved using algebra, tables, financial calculator, or Excel.
The algebra, table, and financial calculator approaches are presented in the body of this
chapter. Those who wish to solve these problems using Excel may learn the skills to do
so for each type of problem presented here by going to www.textbookmedia.com and
going to the area of the website for this textbook.
8-1. What is the future value of $1,000 invested today if you earn 4 percent Future Value
annual interest for five years?
8-2. Calculate the future value of $20,000 ten years from now if the annual Future Value
interest rate is
a. 0 percent
b. 5 percent
c. 10 percent
d. 20 percent
8-3. How much will you have in 10 years if you deposit $5,000 today and earn Future Value
6 percent annual interest?
8-4. Calculate the future value of $50,000 fifteen years from now based on the Future Value
following annual interest rates:
a. 3 percent
b. 6 percent
c. 9 percent
d. 12 percent
8-5. Calculate the future values of the following amounts at 4 percent for twenty- Future Value
five years:
a. 50,000
b. 75,000
c. 100,000
d. 125,000
8-6. Calculate the future value of $60,000 at 5 percent for the following years: Future Value
a. 5 years
b. 10 years
c. 15 years
d. 20 years
8-7. What is the present value of $20,000 to be received ten years from now Present Value
using a 7 percent annual discount rate?
8-8. Calculate the present value of $20,000 to be received twenty years from now Present Value
at an annual discount rate of:
a. 0 percent
b. 5 percent
c. 10 percent
d. 20 percent
230 Part II  Essential Concepts in Finance

Present Value 8-9. Norton is going to receive a graduation present of $9,000 from his
grandparents in four years. If the annual discount rate is 6 percent, what is
this gift worth today?

Present Value 8-10. Calculate the present values of $15,000 to be received in ten years using the
following annual discount rates:
a. 3 percent
b. 6 percent
c. 9 percent
d. 12 percent

Present Value 8-11. Calculate the present values of the following using a 4 percent annual
discount rate at the end of 15 years:
a. $50,000
b. $75,000
c. $100,000
d. $125,000

Present Value 8-12. Calculate the present value of $80,000 at a 5 percent annual discount rate to
be received in:
a. 5 years
b. 10 years
c. 15 years
d. 20 years

Present Value 8-13. What is the present value of a $500 ten-year annual ordinary annuity at a
of an Ordinary 4 percent annual discount rate?
Annuity
Present Value 8-14. Calculate the present value of a $15,000 thirty-year annual ordinary annuity
of an Ordinary at an annual discount rate of:
Annuity
a. 0 percent
b. 10 percent
c. 20 percent
d. 50 percent

Present Value 8-15. What is the present value of a ten-year ordinary annuity of $20,000, using
of an Ordinary a 4 percent annual discount rate?
Annuity
Present Value 8-16. Find the present value of a four-year ordinary annuity of $10,000, using the
of an Ordinary following annual discount rates:
Annuity
a. 9 percent
b. 13 percent
c. 15 percent
d. 21 percent
 Chapter 8  The Time Value of Money 231

8-17. What is the future value of a five-year annual ordinary annuity of $300, Future Value
using a 9 percent annual interest rate? of an Ordinary
Annuity
8-18. Calculate the future value of a twelve-year, $8,000 annual ordinary annuity,
using an annual interest rate of: Future Value
of an Ordinary
a. 0 percent Annuity
b. 2 percent
Future Value
c. 10 percent of an Ordinary
d. 20 percent Annuity

8-19. What is the future value of a ten-year ordinary annuity of $5,000, using an Future Value
annual interest rate of 4 percent? of an Ordinary
Annuity

8-20. What is the future value of an eight-year ordinary annuity of $5,000, using Future Value
an 7 percent annual interest rate? of an Ordinary
Annuity
8-21. Find the future value of the following five-year ordinary annuities, using a 5 Future Value
percent annual interest rate. of an Ordinary
Annuity
a. $1,000
b. $10,000
c. $75,000
d. $125,000

8-22. Starting today, you invest $5,000 a year into your individual retirement Future Value of
account (IRA). If your IRA earns 12 percent a year, how much will be an Annuity Due
available at the end of 40 years?

8-23. John will deposit $500 at the beginning of each year for five years into an Future Value of
account that has an interest rate of 4 percent. How much will John have to an Annuity Due
withdraw in five years?

8-24. An account manager has found that the future value of $10,000, deposited Future Value of
at the end of each year, for five years at an annual interest rate of 4 percent an Annuity Due
will amount to $54,163.23. What is the future value of this scenario if the
account manager deposits the money at the beginning of each year?

8-25. If your required annual rate of return is 6 percent, how much will an Present Value of
investment that pays $80 a year at the beginning of each of the next 20 years an Annuity Due
be worth to you today?

8-26. Sue has won the lottery and is going to receive $30,000 per year for Present Value of
25 years; she received her first check today. The current annual discount rate an Annuity Due
is 7 percent. Find the present value of her winnings.

8-27. Kelly pays a debt service of $1,300 a month and will continue to make this payment Present Value of
for 15 years. What is the present value of these payments discounted at 6 percent if an Annuity Due
she mails her first payment in today? Be sure to do monthly discounting.

8-28. You invested $50,000, and 10 years later the value of your investment has grown Solving for k
to $118,368. What is your compounded annual rate of return over this period?
 232 Part II  Essential Concepts in Finance

Solving for k 8-29. You invested $1,000 five years ago, and the value of your investment has
fallen to $815.37. What is your compounded annual rate of return over
this period?

Solving for k 8-30. What is the rate of return on an investment that grows from $50,000 to
$118,368 in 10 years?

Present Value 8-31. What is the present value of a $50 annual perpetual annuity using a discount
of a Perpetuity rate of 4 percent?

Present Value 8-32. A payment of $70 per year forever is made with an annual discount rate of
of a Perpetuity 9 percent. What is the present value of these payments?

Present Value 8-33. You are valuing a preferred stock that makes a dividend payment of $65 per
of a Perpetuity year, and the current discount rate is 8.5 percent. What is the value of this
(Preferred Stock) share of preferred stock?

Future Value 8-34. Joe’s Dockyard is financing a new boat with an amortizing loan of $24,000,
which is to be repaid in 10 annual installments of $3,576.71 each. What
interest rate is Joe paying on the loan?

Solving for the 8-35. On June 1, 2012, Sue purchased a home for $220,000. She put $20,000
Monthly Interest down on the house and obtained a 30-year fixed-rate mortgage for the
Rate on a remaining $200,000. Under the terms of the mortgage, Sue must make
Mortgage
payments of $898.09 a month for the next 30 years starting June 30. What is
the effective annual interest rate on Sue’s mortgage?

Present Value 8-36. What is the amount you have to invest today at 3 percent annual interest to
be able to receive $10,000 after
a. 5 years?
b. 10 years?
c. 20 years?

Future Value 8-37. How much money would Ruby Carter need to deposit in her savings account
at Great Western Bank today in order to have $16,850.58 in her account after
five years? Assume she makes no other deposits or withdrawals and the bank
guarantees an 6 percent annual interest rate, compounded annually.

Future Value 8-38. If you invest $35,000 today, how much will you receive after
a. 7 years at a 5 percent annual interest rate?
b. 10 years at a 7 percent annual interest rate?

Solving for k 8-39. The Apple stock you purchased twelve years ago for $55 a share is now
worth $490.39. What is the compounded annual rate of return you have
earned on this investment?

Solving for n 8-40. Amy Jolly deposited $1,000 in a savings account. The annual interest rate is
10 percent, compounded semiannually. How many years will it take for her
money to grow to $4,321.94?
 Chapter 8  The Time Value of Money 233

8-41. Beginning a year from now, Bernardo O’Reilly will receive $50,000 a year Present Value of
from his pension fund. There will be fifteen of these annual payments. What an Ordinary
is the present value of these payments if a 6 percent annual interest rate is Annuity
applied as the discount rate?
8-42. If you invest $4,000 per year into your pension fund, earning 7 percent Future Value
annually, how much will you have at the end of twenty years? You make of an Annuity Due
your first payment of $4,000 today.
8-43. What would you accumulate if you were to invest $250 every quarter for Future Value of an
five years into an account that returned 8 percent annually? Your first deposit Ordinary Annuity
would be made one quarter from today. Interest is compounded quarterly.
8-44. If you invest $2,000 per year for the next ten years at an 6 percent annual Future Value of an
interest rate, beginning one year from today compounded annually, how Ordinary Annuity
much are you going to have at the end of the tenth year?

8-45. It is the beginning of the quarter and you intend to invest $450 into your Future Value of an
retirement fund at the end of every quarter for the next thirty years. You are Ordinary Annuity
promised an annual interest rate of 8 percent, compounded quarterly. (Challenge Problem)

a. How much will you have after thirty years upon your retirement?
b. How long will your money last if you start withdrawing $6,000 at the end
of every quarter after you retire?
8-46. A $30,000 loan obtained today is to be repaid in equal annual installments Solving for a
over the next seven years starting at the end of this year. If the annual interest Loan Payment
rate is 4 percent, compounded annually, how much is to be paid each year?
8-47. Dannifer invested $1,000 in a certificate of deposit (CD) that pays 4% Future Value
annual interest, compounded continuously, for fifteen years. How much
money will she have when this CD matures?
8-48. Allie Fox is moving to Central America. Before he packs up his wife and Present Value
son he purchases an annuity that guarantees payments of $10,000 a year in of a Perpetuity
perpetuity. How much did he pay if the annual interest rate is 5 percent?
8-49. Matt and Christina Drayton deposited $1,000 into a savings account the day Future Value
their daughter, Joey, was born. Their intention was to use this money to help
pay for Joey’s wedding expenses when and if she decided to get married.
The account pays 5 percent annual interest with continuous compounding.
Upon her return from a graduation trip to Hawaii, Joey surprises her parents
with the sudden announcement of her planned marriage to John Prentice.
The couple set the wedding date to coincide with Joey’s twenty-third
birthday. How much money will be in Joey’s account on her wedding day?
8-50. You deposit $1,000 in an account that pays 8 percent annual interest, Time to Double
compounded annually. How long will it take to triple your money? Your Money

8-51. Upon reading your most recent credit card statement, you are shocked to learn Time to Pay Off
that the balance owed on your purchases is $4,000. Resolving to get out of debt a Credit Card
once and for all, you decide not to charge any more purchases and to make Balance
regular monthly payments until the balance is zero. Assuming that the bank’s
credit card annual interest rate is 19.5 percent and the most you can afford to pay
each month is $250, how long will it take you to pay off your debt?
 234 Part II  Essential Concepts in Finance

Solving for k 8-52. Joanne and Walter borrow $16,936.06 for a new car before they move to
Stepford, Connecticut. They are required to repay the amortized loan with four
annual payments of $5,000 each. What is the annual interest rate on their loan?

Future Value of 8-53. Norman Bates is planning for his eventual retirement from the motel
an Annuity business. He plans to make quarterly deposits of $1,000 into an IRA starting
(Challenge Problem) three months from today. The guaranteed annual interest rate is 4 percent,
compounded quarterly. He plans to retire in 15 years.
a. How much money will be in his retirement account when he retires?
b. Norman also supports his mother. At Norman’s retirement party, Mother tells
him they will need $2,000 each month in order to pay for their living expenses.
Using the preceding interest rate and the total account balance from part a, for
how many years will Norman keep Mother happy by withdrawing $6,000 at the
end of each quarter? It is very important that Norman keep Mother happy.
Challenge Problem 8-54. Jack Torrance comes to you for financial advice. He hired the Redrum
Weed-N-Whack Lawn Service to trim the hedges in his garden. Because of
the large size of the project (the shrubs were really out of control), Redrum
has given Jack a choice of four different payment options. Which of the
following four options would you recommend that Jack choose? Why?
• Option 1. Pay $5,650 cash immediately.
• Option 2. Pay $6,750 cash in one lump sum two years from now.
• Option 3. Pay $800 at the end of each quarter for two years.
• Option 4. P ay $1,000 immediately plus $5,250 in one lump sum two years
from now.
Jack tells you he can earn 6 percent annual interest, compounded quarterly, on
his money. You have no reason to question his assumption.
Solving for a 8-55. Sarah wants to borrow $250,000 from a mortgage banker to purchase a
Mortgage Loan $300,000 house. The mortgage loan is to be repaid in monthly installments
Payment over a thirty-year period. The annual interest rate is 6 percent. How much
will Sarah’s monthly mortgage payments be?
Solving for a 8-56. The Robinsons have found the house of their dreams. They have $50,000 to use
Mortgage Loan as a down payment and they want to borrow $250,000 from the bank. The current
Payment mortgage annual interest rate is 6 percent. If they make equal monthly payments
for twenty years, how much will their monthly mortgage payment be?
Solving for Auto 8-57. Slick has his heart set on a new Miata sportscar. He will need to borrow
Loan Payments $28,000 to get the car he wants. The bank will loan Slick the $28,000 at an
annual interest rate of 6 percent.
a. How much would Slick’s monthly car payments be for a four-year loan?
b. How much would Slick’s monthly car payments be if he obtains a six-
year loan at the same interest rate?
 Chapter 8  The Time Value of Money 235

8-58. Assume the following set of cash flows: Challenge Problem

(Value of Missing
End of End of End of End of Cash Flow)
Year 1 Year 2 Year 3 Year 4
$100 $150 ? $100

At an annual discount rate of 10 percent, the total present value of all the
cash flows above, including the missing cash flow, is $471.01. Given these
conditions, what is the value of the missing cash flow?

For more, see

6e Spreadsheet Templates
for Microsoft Excel

8-59. You are considering financing the purchase of an automobile that costs Loan
$22,000. You intend to finance the entire purchase price with a five-year Amortization Table
amortized loan with an 6 percent annual interest rate.
a. Calculate the amount of the monthly payments for this loan.
b. Construct an amortization table for this loan using the format shown in
Table 8-2. Use monthly payments.
c. If you elected to pay off the balance of the loan at the end of the thirty-
sixth month how much would you have to pay?
For more, see
6e Spreadsheet Templates
for Microsoft Excel
236 Part II  Essential Concepts in Finance

Answers to Self-Test
ST-1. F
 V = $5,000 × 1.068 = $7,969.24 = Jed’s balance when the eight-year
CD matures.

ST-2. (
$19,999 = $1,1776.89 × PVIFA k = ?, n = 12

)
11.2551 = PVIFAk = ?, n = 12
k = 1% monthly (from the PVIFA Table at the end of the n = 12 row,
k = 1%)
Annual Rate = 1% × 12 = 12%

ST-3. PV = $100,000 × (1/1.0810) = $46,319.35

= the present value of Heidi’s $100,000

ST-4. $20,000 = PMT × [1 – (1/1.00548)] / .005

$20,000 = PMT × 42.5803
PMT = $469.70 = Zack’s car loan payment

ST-5. $5,000 × FVIFk = ?, n = 10 = $15,529.24

FVIFk = ?, n = 10 = 3.1058, therefore k = 12% = Bridget’s annual rate of

return on the mutual fund.

ST-6. ( )
F V = PMT × FVIFA k%, n × (1 + k )
(
= $1,000 × FVIFA10%, 10 ) × (1 + .10)
= $1,000 × 15.9374 × 1.10
= $17,531.14