Determine the Present Value of the Following Single Amounts

Present Value Formulas, Tables and Calculators

The easiest and about authentic fashion to summate the present value of whatever future amounts (single amount, varying amounts, annuities) is to use an electronic financial calculator or computer software. Some electronic financial calculators are now available for less than $35.

Important!
Empathize that each fiscal calculator operates differently and may operate in a manner that is different from our Explanation. Be sure to read and thoroughly empathize the directions earlier operating any calculator or computer software. 1 incorrect assumption, i wrong input, or the utilize of a button that works differently from what you assumed, will mean an wrong answer—and that could accept a significant financial consequence!

If yous don’t have admission to an electronic financial estimator or software, an easy way to summate present value amounts is to apply
nowadays value tables
(PV tables). PV tables cannot provide the same level of accuracy as financial calculators or computer software because the factors used in the tables are rounded off to fewer decimal places. In addition, they unremarkably contain a express number of choices for interest rates and fourth dimension periods. Despite this, nowadays value tables remain popular in bookish settings because they are piece of cake to contain into a textbook. Because of their widespread use, we will use present value tables for solving our examples.

Behind every table, figurer, and piece of software, are the
mathematical formulas
needed to compute present value amounts, interest rates, the number of periods, and the futurity value amounts. Nosotros will, at the get-go, testify you several examples of how to use the nowadays value formula in add-on to using the PV tables.

Except for small-scale differences due to rounding, answers to equations below will be the same whether they are computed using a financial computer, calculator software, PV tables, or the formulas.

Calculating the Present Value of a Single Amount (PV)

In this section nosotros will demonstrate how to observe the present value of a single future cash amount, such as a receipt or a payment. Nosotros’ll refer to the present value of a single corporeality as
PV.

Do #1

Let’s assume nosotros are to receive $100 at the end of two years. How do we calculate the nowadays value of the corporeality, assuming the interest charge per unit is 8% per year compounded annually?

The post-obit timeline depicts the information nosotros know, forth with the unknown component (PV):

Adding Using the PV Formula

The nowadays value formula for a single corporeality is:

Using the second version of the formula, the solution is:

The answer,
$85.73, tells us that receiving $100 in ii years is the aforementioned equally receiving $85.73 today, if the fourth dimension value of money is eight% per twelvemonth compounded annually. (“Today” is the aforementioned concept as “time flow 0.”)

Calculation Using a PV of i Table

It’south common for accounting and finance textbooks to provide nowadays value tables to apply in calculating nowadays value amounts. In a PV of 1 table, each
column
heading displays an interest rate (i), and the
row
indicates the number of periods into the future before an amount will occur (n). At the intersection of each column and row is the correlating nowadays value of 1 (PV of 1) factor. The PV of i factor tells us what the present value will exist, at time period 0, for a single corporeality of $1 at the end of fourth dimension period (n). Click the following to come across a nowadays value of 1 table:
PV of 1 Table.

The PV of i tabular array has two limitations: (1) values are rounded (ours has the rounding to 3 decimal places for ease of utilize) and thus the table sacrifices a bit of accuracy, and (2) the table displays only a limited number of choices for rates and years.

Once you determine the PV of 1 gene from the tabular array, simply use it to substitute for the post-obit term in the PV formula:


[ 1 ÷ (i + i)^{due north}
]

Using the information presented in Do #1, we can solve for the nowadays value of receiving $100 at the end of two years, when discounted by an interest rate of 8% compounded annually:

Because the PV of 1 table had the factors rounded to iii decimal places, the answer ($85.lxx) differs slightly from the amount calculated using the PV formula ($85.73). In either example, what the reply tells us is that $100 at the cease of ii years is the equivalent of receiving approximately $85.70 today (at fourth dimension menstruum 0) if the time value of coin is viii% per year compounded annually.

Practise #2

We need to calculate the nowadays value (the value at time period 0) of receiving a single corporeality of $ane,000 in 20 years. The interest rate for discounting the future amount is estimated at 10% per twelvemonth compounded annually.

The following timeline depicts the information we know, along with the unknown component (PV):

Adding Using the PV of 1 Formula

Using the formula to make up one’s mind the present value, we have:

The answer tells the states that receiving $1,000 in 20 years is the equivalent of receiving $148.64 today, if the time value of coin is 10% per year compounded annually.

Adding Using a PV of one Table

Use the PV of 1 table to find the (rounded) present value factor at the intersection of n = 20 and i = 10%. To calculate the nowadays value of receiving $one,000 at the end of 20 years with a 10% interest charge per unit, insert the gene into the formula:

Nosotros see that the present value of receiving $one,000 in 20 years is the equivalent of receiving approximately $149.00 today, if the fourth dimension value of coin is 10% per year compounded annually.

Practice #iii

What is the present value of receiving a unmarried amount of $5,000 at the cease of three years, if the time value of coin is 8% per year, compounded
quarterly?

The following timeline depicts the data nosotros know, along with the unknown component (PV):

Notice that the timeline shows due north = 12, considering there are 12 quarters in the 3-twelvemonth menstruation. Considering the time periods are three months long, the charge per unit for discounting is i = ii% (the
quarterly
charge per unit that results from the annual rate of 8% divided past the four quarters in each twelvemonth).

Calculation Using the PV Formula

Using the formula to make up one’s mind the present value, we accept:

The reply tells us that receiving $5,000 iii years from today is the equivalent of receiving $3,942.45 today, if the time value of money has an almanac rate of eight% that is compounded
quarterly.

Calculation Using a PV of one Table

The present value of receiving $5,000 at the end of three years when the interest rate is compounded
quarterly, requires that (n) and (i) be stated in quarters. Utilise the
PV of ane Table
to find the (rounded) present value figure at the intersection of
n = 12
(3 years x four quarters) and
i = ii%
(eight% per year ÷ 4 quarters). Insert the factor into the formula:

Nosotros encounter that the present value of receiving $five,000 three years from today is approximately $iii,940.00 if the fourth dimension value of money is 8% per year, compounded quarterly.

Exercise #iv

What is the present value of receiving a single amount of $10,000 at the cease of v years, if the time value of money is half dozen% per yr, compounded
semiannually?

The following timeline depicts the information nosotros know, along with the unknown component (PV):

Notice that the timeline shows northward = 10, because there are 10 half dozen-month (or semiannual) periods in five-years time. Considering the compounding occurs semiannually, the charge per unit for discounting is i = 3% per six-month menses (the annual charge per unit of 6% divided by the two semiannual periods in each twelvemonth).

Calculation Using the PV Formula

Using the formula to determine the present value, we have:

The answer tells usa that receiving $10,000 five years from today is the equivalent of receiving $7,440.ninety today, if the time value of money has an annual rate of half dozen% compounded
semiannually.

Adding Using a PV of 1 Table

The nowadays value of receiving $10,000 at the terminate of five years when the compounding is semiannual, requires that
n = 10
(5 years X 2 semiannual periods per twelvemonth) and that
i = 3%
(6% per year ÷ two semiannual periods in each year). Use the
PV of 1 Table
to find the (rounded) present value factor at the intersection of n = 10 and i = three%. Insert the cistron into the formula:

Nosotros encounter that the present value of receiving $10,000 v years from today is the equivalent of receiving approximately $7,440.00 today, if the time value of money has an almanac charge per unit of 6% compounded
semiannually.