Chapter 5 5-1 Future Value Compute the future value in y.docx

Chapter 5
5-1 Future Value Compute the future value in year 9 of a
$2,000 deposit in year 1 and another $1,500 deposit at the end
of year 3 using a 10 percent interest rate. (LG5-1) LEARNING
GOALS
LG5-1 Compound multiple cash flows to the future.
Finding the Future Value of Several Cash Flows LG5-1
Consider the following contributions to a savings account over
time. You make a $100 deposit today, followed by a $125
deposit next year, and a $150 deposit at the end of the second
year. If interest rates are 7 percent, what’s the future value of
your deposits at the end of the third year? The time line for this
problem is illustrated as:
i
Note that the first deposit will compound for three years. That
is, the future value in year 3 of a cash flow in year 0 will
compound 3 (= 3 − 0) times. The deposit at the end of the first
year will compound twice (= 3 − 1). In general, a deposit in
year m will compound N− m times for a future value in year N.
We can find the total amount at the end of three years by
computing the future value of each deposit and then adding
them together. Using the future value equation from Chapter 4,
the future value of today’s deposit is $100 × (1 + 0.07)3 =
$122.50. Similarly, the future value of the next two deposits are
$125 × (1 + 0.07)2 = $143.11 and $150 × (1 + 0.07)1 = $160.50,

respectively.
Putting these three individual future value equations together
would yield:
FV3 = $100 × (1 + 0.07)3 + $125 × (1 + 0.07)2 + $150 × (1 +
0.07)1 = $426.11
The general equation for computing the future value of multiple
and varying cash flows (or payments) is:
In this equation, the letters m, n, and p denote when the cash
flows occur in time. Each deposit can be different from the
others.
5-3 Future Value of an Annuity what is the future value of a
$900 annuity payment over five years if interest rates are 8
percent? (LG5-2) Compute the future value of frequent, level
cash flows.
Future Value of Level Cash Flows LG5-2
Now suppose that each cash flow is the same and occurs every
year. Level sets of frequent cash flows are common in finance—
we call them annuities. The first cash flow of an annuity occurs
at the end of the first year (or other time period) and continues

every year to the last year. We derive the equation for the future
value of an annuity from the general equation for future value
of multiple cash flows, equation 5-1. Since each cash flow is the
same, and the cash flows are every period, the equation appears
as:
FVAN = Future value of first payment
× Future value of second payment + ··· + Last payment
= PMT × (1 + i)N−1 + PMT
× (1 + i)N−2 + PMT × (1 + i)N−3 + ··· + PMT(1 + i)0
The term FVA is used to denote that this is the future value of
an annuity. Factoring out the common level cash flow, PMT, we
can summarize and reduce the equation as:
annuity A stream of level and frequent cash flows paid at the
end of each time period—often referred to as an ordinary
annuity.
Suppose that $100 deposits are made at the end of each year for
five years. If interest rates are 8 percent per year, the future
value of this annuity stream is computed using equation 5-2 as:
We can show these deposits and future value on a time line as:
Five deposits of $100 each were made. So, the $586.66 future

value represents $86.66 of interest paid. As with almost any
TVM problem, the length of time of the annuity and the interest
rate for compounding are very important factors in
accumulating wealth within the annuity. Consider the examples
in Table 5.1. A $50 deposit made every year for 20 years will
grow to $1,839.28 with a 6 percent interest rate. Doubling the
annual deposits to $100 also doubles the future value to
$3,678.56. However, making $100 deposits for twice the amount
of time, 40 years, more than quadruples the future value to
$15,476.20! Longer time periods lead to more total
compounding and much more wealth. Interest rates also have
this effect. Doubling the interest rate from 6 to 12 percent on
the 40-year annuity results in nearly a five-fold increase in the
future value to $76,709.14.
Finding the Present Value of Several Cash Flows LG5-3
Consider the cash flows that we showed at the very beginning of
the chapter: You deposit $100 today, followed by a $125
deposit next year and a $150 deposit at the end of the second
year. In the previous situation, we sought the future value when
interest rates are 7 percent. Instead of future value, we compute
the present value of these three cash flows. The time line for
this problem appears as:
The first cash flow is already in year zero, so its value will not
change. We will discount the second cash flow one year and the
third cash flow two years. Using the present value equation
from the previous chapter, the present value of today’s payment
is simply $100 ÷ (1 + 0.07)0 = $100. Similarly, the present
value of the next two cash flows are $125 ÷ (1 + 0.07)1 =
$116.82 and $150 ÷ (1 + 0.07)2 = $131.02, respectively.
Therefore, the present value of these cash flows is $347.84 (=

$100 + $116.82 + $131.02).
Putting these three individual present value equations together
would yield:
PV = [$100 ÷ (1 + 0.07)0] + [$125 ÷ (1 + 0.07)1] + [$150 ÷ (1
+ 0.07)2] = $347.84
The general equation for discounting multiple and varying cash
flows is:
flows occur in time. Each deposit can differ from the others in
terms of size and timing.
The five TVM buttons/functions in financial calculators have
been fine, so far, for the types of TVM problems we’ve been
solving. Sometimes we had to use them two or three times for a
single problem, but that was usually because we needed an
intermediate calculation to input into another TVM equation.
Luckily, most financial calculators also have built-in
worksheets specifically designed for computing TVM in
problems with multiple no constant cash flows.
To make calculator worksheets as flexible as possible, they are
usually divided into two parts: one for input, which we’ll refer
to as the CF (cash flow) worksheet, and one or more for
showing the calculator solutions. We’ll go over the conventions
concerning the CF worksheet here, and we’ll discuss the output
solutions in Chapter 13.

The CF worksheet is usually designed to handle inputting sets
of multiple cash flows as quickly as possible. As a result, it
normally consists of two sets of variables or cells—one for the
cash flows and one to hold a set of frequency counts for the
cash flows, so that we can tell it we have seven $1,500 cash
flows in a row instead of having to enter $1,500 seven times.
Using the frequency counts to reduce the number of inputs is
handy, but you must take care. Frequency counts are only good
for embedded annuities of identical cash flows. You have to
ensure that you don’t mistake another kind of cash flow for an
annuity.
Also, using frequency counts will usually affect the way that the
calculator counts time periods. As an example, let’s talk about
how we would put the set of cash flows shown here into a CF
worksheet:
To designate which particular value we’ll place into each
particular cash flow cell in this worksheet, we’ll note the value
and the cell identifier, such as CF0, CF1, and so forth. We’ll do
the same for the frequency cells, using F1, F2, etc., to identify
which CF cell the frequency cell goes with. (Note that, in most
calculators, CF0 is treated as a unique value with an unalterable
frequency of 1; we’re going to make the same assumption here
so you’ll never see a listing for F0.) For this sample timeline,
our inputs would be:
To compute the present value of these cash flows, use the NPV
calculator function. The NPV function computes the present
value of all the future cash flows and then adds the year 0 cash
flow. Then, on the NPV worksheet, you would simply need to

enter the interest rate and solve for the NPV:
Note a few important things about this example:
1. We had to manually enter a value of $0 for CF3. If we hadn’t,
the calculator wouldn’t have known about it and would have
implicitly assumed that CF4 came one period after CF2.
2. Once we use a frequency cell for one cash flow, all
numbering on any subsequent cash flows that we enter into the
calculator is going to be messed up, at least from our point of
view. For instance, the first $75 isn’t what we would call
“CF5,” is it? We’d call it “CF7” because it comes at time period
7; but calculators usually treat CF5 as “the fifth set of cash
flows,” so we’ll just have to try to do the same to be consistent.
3. If we really don’t need to use frequency cells, we will usually
just leave them out of the guidance instructions in this chapter
to save space.
5-5 Present Value Compute the present value of a $2,000
deposit in year 1 and another $1,500 deposit at the end of year 3
if interest rates are 10 percent. (LG5-3) Discount multiple cash
flows to the present.
Finding the Present Value of Several Cash Flows LG5-3
Consider the cash flows that we showed at the very beginning of
the chapter: You deposit $100 today, followed by a $125
deposit next year, and a $150 deposit at the end of the second
year. In the previous situation, we sought the future value when
interest rates are 7 percent. Instead of future value, we compute
the present value of these three cash flows. The time line for

this problem appears as:
The first cash flow is already in year zero, so its value will not
change. We will discount the second cash flow one year and the
third cash flow two years. Using the present value equation
from the previous chapter, the present value of today’s payment
is simply $100 ÷ (1 + 0.07)0 = $100. Similarly, the present
value of the next two cash flows are $125 ÷ (1 + 0.07)1 =
$116.82 and $150 ÷ (1 + 0.07)2 = $131.02, respectively.
Therefore, the present value of these cash flows is $347.84 (=
$100 + $116.82 + $131.02).
Putting these three individual present value equations together
would yield:
PV = [$100 ÷ (1 + 0.07)0] + [$125 ÷ (1 + 0.07)1] + [$150 ÷ (1
+ 0.07)2] = $347.84
The general equation for discounting multiple and varying cash
flows is:
flows occur in time. Each deposit can differ from the others in
terms of size and timing.
The five TVM buttons/functions in financial calculators have
been fine, so far, for the types of TVM problems we’ve been

solving. Sometimes we had to use them two or three times for a
single problem, but that was usually because we needed an
intermediate calculation to input into another TVM equation.
Luckily, most financial calculators also have built-in
worksheets specifically designed for computing TVM in
problems with multiple nonconstant cash flows.
To make calculator worksheets as flexible as possible, they are
usually divided into two parts: one for input, which we’ll refer
to as the CF (cash flow) worksheet, and one or more for
showing the calculator solutions. We’ll go over the conventions
concerning the CF worksheet here, and we’ll discuss the output
solutions in Chapter 13.
The CF worksheet is usually designed to handle inputting sets
of multiple cash flows as quickly as possible. As a result, it
normally consists of two sets of variables or cells—one for the
cash flows and one to hold a set of frequency counts for the
cash flows, so that we can tell it we have seven $1,500 cash
flows in a row instead of having to enter $1,500 seven times.
Using the frequency counts to reduce the number of inputs is
handy, but you must take care. Frequency counts are only good
for embedded annuities of identical cash flows. You have to
ensure that you don’t mistake another kind of cash flow for an
annuity.
Also, using frequency counts will usually affect the way that the
calculator counts time periods. As an example, let’s talk about
how we would put the set of cash flows shown here into a CF
worksheet:
To designate which particular value we’ll place into each

particular cash flow cell in this worksheet, we’ll note the value
and the cell identifier, such as CF0, CF1, and so forth. We’ll do
the same for the frequency cells, using F1, F2, etc., to identify
which CF cell the frequency cell goes with. (Note that, in most
calculators, CF0 is treated as a unique value with an unalterable
frequency of 1; we’re going to make the same assumption here
so you’ll never see a listing for F0.) For this sample timeline,
our inputs would be:
To compute the present value of these cash flows, use the NPV
calculator function. The NPV function computes the present
value of all the future cash flows and then adds the year 0 cash
flow. Then, on the NPV worksheet, you would simply need to
enter the interest rate and solve for the NPV:
Note a few important things about this example:
1. We had to manually enter a value of $0 for CF3. If we hadn’t,
the calculator wouldn’t have known about it and would have
implicitly assumed that CF4 came one period after CF2.
2. Once we use a frequency cell for one cash flow, all
numbering on any subsequent cash flows that we enter into the
calculator is going to be messed up, at least from our point of
view. For instance, the first $75 isn’t what we would call
“CF5,” is it? We’d call it “CF7” because it comes at time period
7; but calculators usually treat CF5 as “the fifth set of cash
flows,” so we’ll just have to try to do the same to be consistent.
3. If we really don’t need to use frequency cells, we will usually
just leave them out of the guidance instructions in this chapter

to save space.
5-7 Present Value of an Annuity what’s the present value of a
$900 annuity payment over five years if interest rates are 8
percent? (LG5-4) Compute the present value of an annuity.
Present Value of Level Cash Flows LG5-4
Page 109You will find that this present value of an annuity
concept will have many business and personal applications
throughout your life. Most loans are set up so that the amount
borrowed (the present value) is repaid through level payments
made every period (the annuity). Lenders will examine
borrowers’ budgets and determine how much each borrower can
afford as a payment. The maximum loan offered will be the
present value of that annuity payment. The equation for the
present value of an annuity can be derived from the general
equation for the present value of multiple cash flows, equation
5-3. Since each cash flow is the same, and the borrower pays
the cash flows every period, the present value of an annuity,
PVA, can be written as:
Suppose that someone makes $100 payments at the end of each
year for five years. If interest rates are 8 percent per year, the
present value of this annuity stream is computed using equation
5-4 as:
TABLE 5.2 Magnitude of the Annuity, Number of Years
Invested, and Interest Rate on PV

The time line for these payments and present value appears as:
Notice that although five payments of $100 each were made,
$500 total, the present value is only $399.27. As we’ve noted
previously, the span of time over which the borrower pays the
annuity and the interest rate for discounting strongly affect
present value computations. When you borrow money from the
bank, the bank views the amount it lends as the present value of
the annuity it receives over time from the borrower. Consider
the examples in Table 5.2.
A $50 deposit made every year for 20 years is discounted to
$573.50 with a 6 percent discount rate. Doubling the annual
cash flow to $100 also doubles the present value to $1,146.99.
But extending the time period does not impact the present value
as much as you might expect. Making $100 payments for twice
the amount of time—40 years—does not double the present
value. As you can see in Table 5.2, the present value increases
less than 50 percent to only $1,504.63! If the discount rate
increases from 6 percent to 12 percent on the 40-year annuity,
the present value will shrink to $824.38.
Your firm needs to buy additional physical therapy equipment
that costs $20,000. The equipment manufacturer will give you
the equipment now if you will pay $6,000 per year for the next
four years. If your firm can borrow money at a 9 percent
interest rate, should you pay the manufacturer the $20,000 now
or accept the 4-year annuity offer of $6,000?
SOLUTION:

We can find the cost of the 4-year, $6,000 annuity in present
value terms using equation 5-4:
The cost of paying for the equipment over time is $19,438.32.
This is less, in present value terms, than paying $20,000 cash.
The firm should take the annuity payment plan.
Similar to Problems 5-7, 5-8, self-test problem 2
Page 110The present value of a cash flow made far into the
future is not very valuable today, as Figure 5.2 illustrates.
That’s why doubling the number of years in the table from 20 to
40 only increased the present value by approximately 30
percent. Notice how the present value of $100 annuity payments
declines for the cash flows made later in time, especially at
higher discount rates. The $100 cash flow in year 20 is worth
less than $15 today if we use a 10 percent discount rate; they’re
worth more than double, at nearly $38 today, if we use a
discount rate of 5 percent. The figure also shows how quickly
present value declines with a higher discount rate relative to a
lower rate. As we showed above, the present values of the
annuities in the figure are the sums of the present values shown.
Since the present values for the 10 percent discount rate are
smaller, the present value of an annuity is smaller as interest
rates rise.
Present Value of Multiple Annuities
Just as we can combine annuities to solve various future value
problems, we can also combine annuities to solve some present
value problems with changing cash flows. Consider Alex
Rodriguez’s (A-Rod’s) baseball contract in 2000 with the Texas
Rangers. This contract made A-Rod into the “$252 million
man.” The contract was structured so that the Rangers paid A-
Rod a $10 million signing bonus, $21 million per year in 2001

through 2004, $25 million per year in 2005 and 2006, and $27
million per year in 2007 through 2010.1 Notethatadding the
signing bonus to the annual salary equals the $252 million
figure. However, Rodriguez would receive the salary in the
future. Using an 8 percent discount rate, what is the present
value of A-Rod’s contract?
The reported values for many sports contracts may be
misleading in present value terms.
FIGURE 5.2 Present Value of Each Annuity Cash Flow
We begin by showing the salary cash flows with the time line:
First create a $27 million, 10-year annuity. Here are the
associated cash flows:
Now create a –$2 million, six-year annuity:
Notice that creating the −$2 million annuity also resulted in the
third annuity of −$4 million for four years. This time line shows
three annuities. If you add the cash flows in any year, the sum
is A-Rod’s salary for that year. Now we can find the present
value of each annuity using equation 5-4 three times.
Adding the value of the three annuities reveals that the present

value of A-Rod’s salary was $158.67 million (= $181.17m −
$9.25m − $13.25m). Adding in the $10 million signing bonus
produces a contract value of $168.67 million. So, the present
value of A-Rod’s contract turns out to be quite considerable, but
you might not call him the $252 million man!
5-39 Loan Payments You wish to buy a $25,000 car. The
dealer offers you a 4-year loan with a 9 percent APR. What are
the monthly payments? How would the payment differ if you
paid interest only? What would the consequences of such a
decision be? (LG5-9) Compute payments and amortization
schedules for car and mortgage loans.
Say that you have your heart set on purchasing a beautiful, old
Tudor-style house for $250,000. A mortgage broker says that
you can qualify for a mortgage for 80 percent (or $200,000) of
the price. If you get a 15-year mortgage, the interest rate will be
6.1 percent APR. A 30-year mortgage costs 6.4 percent. One of
the factors that will help you decide which mortgage to take is
the magnitude of the monthly payments. What will they be? 5
SOLUTION:
To pay off the mortgage in only 15 years, the payments would
have to be larger than for the 30-year mortgage. The higher
payment will be eased somewhat because the interest rate is
lower on the 15-year mortgage. The payment for the 15-year
mortgage is:
The payment for the 30-year mortgage would be:

So, the payments on the 15-year mortgage are nearly $450 more
each month than the 30-year mortgage payments. You must
decide whether the cost of paying the extra $450 each month is
worth it to own the house with no debt 15 years sooner. The
decision would depend on your financial budget and the
strength of your desire to be debt free.
Similar to Problems 5-39, 5-40, self-test problems 1 and 4
time out!
the Math Coach on...
Using the AMORT Function in TVM Calculators
Page 121imagesTVM calculators have preprogrammed functions
to compute the amount of principal paid part way through a
mortgage. For example, if you took out a 30-year, $200,000
mortgage at a 6 percent APR, how much principal have you paid
after five years? How much do you still owe? How much
interest have you paid?
To answer these questions, first enter the mortgage information
to compute the monthly payments. Then use the AMORT
function. This example uses the Texas Instruments BA II + as
an example. The Hewlett-Packard and other TVM calculators
have similar functions. The AMORT function allows you to
compute the loan balance at any time during the mortgage
period. It also computes the amount of principal and interest
that has been paid during any time period. To answer the
questions above:

1. Press 2nd AMORT and P1 = 1 appears.s (This refers to the
first payment of the mortgage.)
2. Press the down arrow ↓, P2 = appears. (This refers to the last
payment made.)
3. The question refers to 5 years of payments, which is 60
months. Enter 60 and press ENTER.
The calculator has now computed the loan balance after the 60th
payment and the amount of principal and interest that have been
paid between the first and the 60th payment.
4. Press the down arrow ↓, displayed is BAL = 186,108.71,
which is the loan balance.
5. Press the down arrow ↓, displayed is PRN = 13,891.29, which
is the principal paid in the first 5 years.
6. Press the down arrow ↓, displayed is INT = −58,054.78,
which is the interest paid in the first 5 years.
Note that in the beginning of a mortgage, far more interest is
paid than principal.

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