4.

8 Applications 139

4.8Applications
There are very abundant applications of simple annuities, certain, due, and immediate. Such
as seen in some of the examples in this chapter, a very common application is plans
purchase of all kinds of goods on credit (automobiles, real estate, household appliances,
etcetera.
In addition, there are applications that are very useful in corporate finance, especially
those used in the investment project evaluation topic. When a project is analyzed
In investment, market research, technical and risk studies are conducted, in addition to...
economic gods, which include financial analyses related to performance or
utility that is expected to be obtained from the project. The three methods of financial project evaluation
Types of investment that most commonly appear in texts that address this topic.1they are the ones with value
net present value (NPV), also known as net present value (NPV), the internal rate of return or
internal rate of return (IRR) and the payback period.
Generally, this financial evaluation of investment projects is based on the flows.
of cash associated with the project that can be grouped into four basic categories:
Net initial investment.
Future cash flows resulting from the project's operation.
Non-operating cash flows such as those required for a repair
important.
Net recovery value, which is the value at which they can be sold at the end of the project,
considerable value assets that may have needed to be acquired as part of the project.
The following explains several examples of the three methods of project evaluation.
investment.

4.8.1 Net present value

The net present value of an investment project is the present value of all cash flows related
associated with the project. In other words, it is the present value of all its costs (expenses) and its
income, from its beginning to its completion. This situation is illustrated in the following example.

Example 4.8.1

Suppose that there are plans to buy a building to renovate it and sell it for a profit.
dad. Its price is $4,200,000 and an additional investment of $3,000,000 would be required to renew it, during the
the following six months, at a rate of $500,000 per month. At the end of the six months, it is calculated that it will be
It could be sold for $9,100,000. Is this an attractive investment from a financial perspective?
It is illustrated below how to answer this question using the net present value criterion.
To calculate the NPV, a rate known as the cost of capital is used,2whose deter‑
Financing can be complicated, but if it is used as the cost of capital, simply the rate of

See, for example, Emery, Douglas R., Corporate Financial Management, Prentice Hall, Mexico, 2000; Gallagher, Timothy
1

J., Financial Administration, Colombia, 2001; or Ross, Stephen A., Westerfield, Randolph W., and Jaffe, Jeffrey F., Corporate Finance
Porativas, Irwin, Spain, 1995; Brealey, Scott and Brigham, Eugene F., Fundamentals of Financial Management, 12th ed.
McGraw-Hill/Interamericana de Mexico, 2001.
2The cost of capital corresponds to the compensation that investors receive for providing financial resources to the company.

that is, the payment received by both creditors (suppliers, banks, securities creditors, various creditors), as well as shareholders
In this way. Creditors receive interest in exchange for providing funds to the company in the form of debt; shareholders receive
dividends in exchange for the capital they contribute to their company.
Now, to assess the cost of capital, it is necessary to determine the price of the financial resources provided, the
which is measured in terms of rate. The cost of capital would then be the rate that is paid for the financial resources provided.
to the company. However, there are two types of resources (debt and equity), each with its rate. The cost of capital would be
therefore similar to the average costs of debt (interest) and equity (dividends), that is, similar to the pro‑
weighted average of both rates.
140 CHAPTER 4 Simple, Certain, Expired, and Immediate Annuities

interest that would have to be paid if money is borrowed from a bank, could be set
this cost of capital at, for example, 18% per year, compounded monthly. With those elements,
The net present value of this investment project is calculated as follows:

1 − (1.015)−6
VAN= −4,200,000− 500,000 + 9100 000(1.015)−6
0.015

Current value of Current value

Cost of
disbursements for sale of
purchase
monthly real estate

In the previous expression, the first term on the right side is the purchase cost of the edi-
The first is the fictitious value, the second is the present value of the six monthly disbursements to remodel it (under the
assuming that they are made at the end of each month) and, finally, the third term is the present value
from the proceeds obtained from the sale of the property. Note the negative signs of the unemployment
bags and the positive sign of the income and that all amounts are given at present value.
This net present value is:

1 − 0.914542
VAN= −4,200,000− 500,000 + 9100 000(0.9145422)=
0.015
= −4,200,000− 2,848,593.5+ 8,322,333.95= $1,273,740.37

Therefore, since the net present value is positive, the project is attractive in terms of
financial. In other words, if under the prevailing market conditions it can be considered
It is reasonable to have a capital cost of 18% per year convertible monthly; it is worth doing this.
investment project and, if there are no changes to the estimates, one could expect to obtain a uti-
net worth of $1,273,740.37, at current value. That is, if the investor were to borrow the entire amount.
Black that is required to acquire and remodel the property, upon selling it could liquidate the capital.
what was obtained as a loan and the corresponding interest, but it would also leave him with that gain
current value.
It is important to emphasize that the criterion for deciding whether to undertake the project or not must be based on
in the nature of the VAN, that is, whether it is positive or negative.
The criterion for deciding whether or not to carry out a project, according to the present value
neto is the following:
Net present value Decision
Positive Carry out the project
Negative Not to carry out the project

4.8.2 Internal Rate of Return (IRR)

The IRR is the rate at which the present value of the project's revenues is equal to the present value of the expenses.
SOS. The criterion for making decisions based on this method is to undertake the project when the
TIR is greater than the cost of capital, which is, expressed simply, a weighted average of
the costs of all the funds with which an organization operates, mainly equity and debt.

Example 4.8.2

With the same previous example, the approach would be as follows:

 4.8 Applications 141

1 − (1+ TIR)−6
4,200,000+ 500,000 = 9100 000(1+ TIR)−6
TIR

Cost Current value of Current value

of + disbursements = for sale of
purchase monthly property

Expenses Income

On the left side are the expenses and on the right side are the income. For its part, the
TIR is found when this equation is solved, starting by simplifying it as much as possible.
possible
1− (1+ TIR)−6
91(1+ TIR−6− 5 = 42
TIR
This equation can be solved by trial and error, with a calculator using in-
interpolation, as illustrated in section 4.7, to find its value of 0.05172837, which is the
monthly internal rate of return, since the cash flows are structured in months.
The annual IRR would be:
TIR= 1.05172812-1= 0.831, u 83.16%
Since this rate is (considerably) higher than the cost of capital, then, according to
According to the IRR criterion, the project should be carried out.
However, because the manual method of trial and error (with calculator) is very ...
borioso, the resolution is reserved for the next section, where the procedure is illustrated
to solve it using the Excel function that is precisely called IRR, which simplifies it
increases considerably.
At this point, it is worth noting that these first two methods that are exemplified
they can be equivalent in some cases, such as when it comes to independent projects,
in which the selection of a project does not depend on the selection of other projects and are the same
equivalent in cases of conventional projects where there are initial disbursements
in cash and a series of future cash flows (of income and expenses), also in cash, as in the
example that was presented earlier.
However, when the projects being evaluated are not independent, but rather
one depends on the other, these two evaluation criteria are not equivalent. They are also not
it can be applied interchangeably to cases where projects of different sizes are evaluated
or projects whose cash flows are significantly different, for example,
when one of them offers income flows only towards the end of the project and the other ones
promises throughout the entire life of the plan, because in these cases, relational issues also come into play.
related to the cash flow needs of the company, among others.
The details of these considerations are beyond the scope of this text, so to delve deeper...
to them, it is suggested to consult some of the works mentioned in the footnote
139 or in some other related to the topic.

4.8.3 Investment recovery period

The payback period of the investment can be simple or adjusted. The former is calculated simply.
add up all expected cash flows (without taking into account the time in which
are carried out or, in other words, without considering the differences in value at different times), pro‑
gradually, until the sum equals the initial disbursement. The difference between this moment and the
The moment the project starts is what is known as the payback period.
142 CHAPTER 4 Simple, certain, matured, and immediate annuities

This method can be useful as additional information to evaluate some projects.

specific terms but, as it does not take into account the value of money over time, lacks interest in
this text of financial mathematics, so its analysis is left here.
On the other hand, the adjusted payback period of the investment does take into account the value of the di-
black in time. To determine it, the present value of each of the income flows is calculated.
They are expected in the future and accumulate progressively until the total equals the disbursement.
initial. Given that it takes into account the loss of value that money suffers over the course of
time provides a more accurate measure of the risk involved in an investment project
Session. Its application is illustrated in the following example.

Example 4.8.3

Back to the investment project in a building for which $4,200,000 is paid, an investment is made
$500,000 at the end of each of six months to finally sell it for $9,100,000, also in six months.
Months later, the recovery period is clearly six months, which is when it is supposed to be.
that the sale is made and, with it, the payment is received.
On the other hand, it is worth emphasizing that sometimes cash flows can
not be made up of amounts that constitute annuities, such as in the $500,000
suales, for 6 months, which were handled in the two previous examples. Therefore, another is illustrated
common case in the following example.

Example 4.8.4

This example is adapted from a text about investment projects.3A project is being evaluated that
it involves a total initial investment of $360 million pesos, an amount that includes, among others,
incepts, land, construction, equipment, machinery, and furniture. Annual cash flows have been estimated.
income of $160, $143, $170, $162, $154, and $147 million pesos, which were determined
subtracting production costs, fixed expenses, and taxes from sales revenue, and to
those who, on the other hand, were also hit by depreciation. It is necessary to determine the decision that
It should be taken through the three methods of investment project evaluation that have been ...
positioned and using 20% annual cost of capital.

Solution:
I. According to the payback period method
Since $360 million was initially invested in the project, it is necessary to determine when.
this amount is recovered, at its current value, that is, on the same date (period) in which it was made
the disbursement. To carry out this operation step by step, at the end of the first year, there would be an in-
income of $160 (millions; henceforth, the mention of millions will be omitted for brevity)
position), which, at present value, would be:
VA1= 160(1.2)=-1 $133.33
With that amount, it is evident that the initial disbursement is not covered. Now, at the end of the...
in which year is an income of $143 obtained that, at current value, is:

VA2= 143(1.2)=-2 $99.31

with a total current value of income of $133.33+ 99.31= $232.64, Monday
still lower than the amount initially invested in the project. Next, with the
third year income, there is a new present value of:
VA3= 170(1.2)=-3 $98.38

3 Baca Urbina, Gabriel, Evaluation of Projects, 3rd ed., McGraw-Hill, Mexico, 1995, p. 198.