Introduction

All of a firm's major business decisions involve some form of comparing an amount of cash or earnings today or in the near future to an amount of cash
or earnings further into the future. A key factor affecting these decisions is the amount of time that is expected to pass between the 'now' and the
'future'. Why is this passage of time a key factor in business decision making? Because of the time value of money! Cash and earnings, in their form
as money, can gain, or lose, value over time. Your textbook or your professor can explain the theory behind this. The value that money has over time
is measured by the market interest rate. In this module, we show how to compute the amounts of money one can gain over some period of time
given some specified interest rate. This is the foundation of finance.

The basics of computing the time value of money center on what are called interest factors. There are two types
of interest factors, one for each time perspective. The first, more intuitive, time perspective is forward from the present. Starting from some
amount of money today, one looks forward to determine what amount of money one might have in the future. This is called finding the
future value and we use a future value interest factor to compute it.
Finding a future value is also known as compounding. The second, less intuitive, time perspective is backward
from the future. Starting from some amount of money at some point in the future, one looks backward to determine what amount of money might be needed
in the present to compound, or grow, into that future amount of money. This is called finding the present value
and we use a present value interest factor to compute it. Finding a present value is also known as
discounting.

There are two cases for finding future values and present values. One is for individual amounts, the other is for a series of equal amounts paid or
received at regular time intervals, called annuities.

This module will show the computation of the four pure scenarios; future value of an amount, future value of an annuity, present value of an amount,
and present value of an annuity. It will also show mixed scenarios when combinations of amount and annuity, or future value and present value are
required.

Your textbook should have a series of tables in its appendix that compute future value interest factors and present value interest factors for you to
use in your computations. Your professor may require you to use them. Most textbooks show how to solve future value and present value problems using
a financial calculator instead of the tables. Your professor may require you to purchase a financial calculator for your course. Most spreadsheet
programs have embedded financial tools that compute future values and present values.

We use neither the tables, nor hand-held calculators, nor spreadsheet calculators for this module, though you may if you prefer. We provide you with
a financial calculator written in JavaScript. Near the bottom of each section page and each question page, (though not this page), there is a link to
our financial calculator. It will open a new window or tab in which you can select any of the financial functions you need for this module.

Future Values of Amounts

Computing the future value of an amount means computing how much a given amount of money today will grow into, or compound to, over a given period
of time at a given interest rate. An example may be the purchase of a CD (certificate of deposit). A saver buys a CD for some amount with some
interest rate and some holding period. The saver would like to know - before the purchase - how much money he or she will have when the CD matures.

Say the CD amount is $10,000, the term is 1 year, and the interest rate is 10% with interest paid annually. The amount of money the saver has at the
end of that year is the CD amount of $10,000 times 1 plus the interest rate; written as Future Value = $10,000 * (1 + 0.10), where the interest rate is
written as a decimal. That yields a future value of $11,000.

Now say the CD has a term of 2 years with the same interest rate. How much money does the saver have at the end of 2 years? Well, at the end of 1
year, the saver has $11,000. The amount of money at the end of the second year is the $11,000 times (1 + 0.10) which yields $12,100. Notice that since
the first year's $11,000 can be written as $10,000 * (1 + 0.10), the future value of the 2-year CD can be expressed as Future Value = $10,000 * (1 + 0.10)^{2}.

The above expression shows the fundamental equation for computing the future values of amounts: FV = Present Amount * (1 + i)^{t}, where the
Present Amount is the money the saver has today, i is the interest rate per compounding period, and t is the number of compounding periods in the
holding period. The term (1 + i)^{t} is known as the Future Value Interest Factor (FVIF). This is the value you would look up in the tables
in your textbook. You might also compute it on your calculator. For a term of 2 years and an annual interest rate of 10% the FVIF is 1.1^{2} =
1.210. The future value is then $10,000 * 1.210 = $12,100. You won't need to compute these FVIFs as the calculator we provide computes the future value.
However, you should understand the computations the calculator is performing for you.

Example 1.

A person opens a savings account at a bank with a deposit of $8,000. The account has an annual interest rate of 4% paid annually. What will be the
balance in the account at the end of 5 years?

The present amount is $8,000. The FVIF = (1 + 0.04)^{5} = 1.217. The future value is $8,000 * 1.217 = $9,733.

Example 2.

A person opens a savings account at a bank with a deposit of $5,000. The account has an annual interest rate of 8% paid annually. What will be the
balance in the account at the end of 12 years?

The present amount is $5,000. The FVIF = (1 + 0.08)^{12} = 2.518. The future value is $5,000 * 2.518 = $12,591.

In the above examples, the interest was paid at the end of each year. That means the compounding period is one year. Interest can be paid every six
months yielding a semi-annual compounding period, or every 3 months yielding a quarterly compounding period. A compounding period is the time between
interest payments, how frequently interest is paid. When the compounding period is not one year, the annual interest rate and the holding period in
years must be adjusted to match the length of the compounding period. For example, semi-annual compounding must halve the annual interest rate to give
the interest per 6-month period and double the holding period to give the number of 6-month time periods in the total holding period.

Say the CD amount is $10,000, the term is 2 years, and the interest rate is 10% with interest paid semi-annually. The interest rate per 6 months is
10% / 2 = 5%. The number of 6-month periods in 2 years is 2 times 2 6-month periods per year = 4 compounding periods. The FVIF is then (1 + 0.05)^{4}
= 1.2155. The future value is $10,000 * 1.2155 = $12,155.

Example 3.

A person opens a savings account at a bank with a deposit of $8,000. The account has an annual interest rate of 4% paid semi-annually.
What will be the balance in the account at the end of 5 years?

The present amount is $8,000. The 6-month interest rate is 4% / 2 = 2%. The number of periods is 5 years * 2 6-month periods per year = 10 periods.
The FVIF = (1 + 0.02)^{10} = 1.219. The future value is $8,000 * 1.219 = $9,752.

Example 4.

A person opens a savings account at a bank with a deposit of $5,000. The account has an annual interest rate of 8% paid quarterly. What will be the
balance in the account at the end of 12 years?

The present amount is $5,000. The 3-month interest rate is 8% / 4 = 2%. The number of periods is 12 years * 4 quarters per year = 48 periods.
The FVIF = (1 + 0.02)^{48} = 2.587. The future value is $5,000 * 2.587 = $12,935.

You should notice that the future values in examples 3 and 4 are larger than those in examples 1 and 2. That is
because the future value increases when the length of the compounding period decreases. We prefer not to discuss theoretical issues but this
concept is extremely important in finance. The impact of a shorter compounding period, such as from annual to semi-annual, is that interest is being
paid more frequently, every 6 months instead of every 12 months. For the semi-annual compounding scenario, interest paid after the first 6 months of
a year will earn interest in the second 6 months of the year. Thus, compared to the annual compounding scenario, semi-annual compounding will yield
higher future values because interest is being earned on interest at a faster rate. Quarterly compounding yields a higher future value than
semi-annual compounding for the same reason.

The phenomenon of earning interest on interest is the major reason why amounts of money grow over time.

Sample Questions about Future Values of Amounts.

Present Values of Amounts

Computing the present value of an amount means computing how much money is needed today for it grow into, or compound to, a given amount of money in
the future over a given period of time at a given interest rate. An example may be the promise of some money. A person promises to give a relative
a gift of a certain amount of money at sometime in the future with some interest rate. The person receiving the gift may like to know how much money is
needed today to grow into the promised amount during the given time period and at the given interest rate. Another way to conceptualize present value
is to ask what is the gift to be received in the future worth today to the person receiving it.

Say the gift amount is $10,000, the gift will be received in one year, and the interest rate is 10% with interest paid annually. The amount of money
the gift is worth today is the amount of $10,000 divided by 1 plus the interest rate; written as Present Value = $10,000 / (1 + 0.10), where the
interest rate is again written as a decimal. That yields a present value of $9,091.

Now say the gift is to be received in 2 years with the same interest rate. How much is the gift worth today? Well, at the end of 1 year, the gift
was worth $9,091. To account for the second year, the preent value for the first year is divided by (1 + 0.10) a second time which yields $8,264.
Notice again that since the first year's $9,091 can be written as $10,000 / (1 + 0.10), the present value of the gift in 2 years can be expressed as
Present Value = $10,000 / (1 + 0.10)^{2} or Present Value = $10,000 * (1 + 0.10)^{-2}.

The above expression shows the fundamental equation for computing the present values of amounts: PV = Future Amount * (1 + i)^{-t}, where the
Future Amount is the money the person will receive in the future, i is the interest rate per compounding period, and t is the number of compounding
periods in the holding period. The term (1 + i)^{-t} is known as the Present Value Interest Factor (PVIF). This is the value you would look
up in the tables in your textbook the same as FVIF. Again, you might also compute it on your calculator. For a term of 2 years and an annual interest
rate of 10% the PVIF is 1.1^{-2} = 0.8264. The present value is then $10,000 * 0.8264 = $8,264. Again, you won't need to compute these PVIFs
as the calculator we provide computes the present value. However, you should understand the computations the calculator is performing for you.

Example 1.

A person enters into a 5 year contract which pays $8,000 at the end of the 5 years. The annual market interest rate of 4%. What is the future amount
of $8,000 worth today?

The future amount is $8,000. The PVIF = (1 + 0.04)^{-5} = 0.8219. The present value is $8,000 * 0.8219 = $6,575.

Example 2.

A person enters into a 12 year contract which pays $5,000 at the end of the 12 years. The annual market interest rate of 8%. What is the future amount
of $12,000 worth today?

The future amount is $5,000. The PVIF = (1 + 0.08)^{-12} = 0.3971. The present value is $5,000 * 0.3971 = $1,986.

The compounding period, called the discounting period for present value, can again be different then 1 year. The same adjustment that was made for
future value computation is also made for present value computation.

Say the gift amount is $10,000 to be received in 2 years, and the market interest rate is 10% with interest paid semi-annually. The interest rate per
6 months is 10% / 2 = 5%. The number of 6-month periods in 2 years is 4 discounting periods. The PVIF is then (1 + 0.05)^{-4} = 0.8227.
The present value is $10,000 * 0.8227 = $8,227.

Example 3.

A person enters into a 5 year contract which pays $8,000 at the end of the 5 years. The annual market interest rate of 4% paid semi-annually.
What is the future amount of $8,000 worth today?

The future amount is $8,000. The 6-month interest rate is 4% / 2 = 2%. The number of periods is 5 years * 2 6-month periods per year = 10 periods.
The PVIF = (1 + 0.02)^{-10} = 0.8203. The present value is $8,000 * 0.82103 = $6,563.

Example 4.

A person enters into a 12 year contract which pays $5,000 at the end of the 12 years. The annual market interest rate of 8% paid quarterly.
What is the future amount of $12,000 worth today?

The future amount is $5,000. The 3-month interest rate is 8% / 4 = 2%. The number of periods is 12 years * 4 quarters per year = 48 periods.
The PVIF = (1 + 0.02)^{-48} = 0.3865. The present value is $5,000 * 0.3865 = $1,933.

You should notice again that the present values in examples 3 and 4 are smaller than those in examples 1 and 2. That
is because the present value decreases when the length of the discounting period decreases. It is due to the same phenomenon that makes
future values increase as the length of the compounding periods decreases. More frequent discounting makes present values smaller.

Sample Questions about Present Values of Amounts.

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