where:

CF = the periodic cash flow of the perpetuity
i = the discount rate

Assume an investor wanted to purchase a preferred stock that paid an annual dividend of $3.50, using a discount rate of 7%, what is the value of the preferred stock?

We can calculate the present value using the formula above as follows:

In this particular scenario, the present value of the stream of dividend payments for this particular security would be $50.00.

Compared to an ordinary annuity, the present value of an annuity due can be calculate by modifying the formula above with the addition of the quantity
(1 + r) as follows:

Where:
PMT = payment
r = rate
n = periods

Assume an individual won the lottery and the prize was to be a series of $1,000 payments received at the beginning of each year, over a ten year period. The winner has the option of choosing between the stream of payments or a lump sum discounted at a required rate of 7%, we can calculate what the present value of the stream of payments is as follows:

Using an HP12C calculator, we can solve the equation above using the following keystrokes:

[g][BEG]
[1000][PMT]
[7][i]
[10][n][PV]

Where:
PMT = payment
r = rate
n = periods

Assume you won the lottery and the prize is a $1,000 series of payments to be received over the next ten years, at the end of each year. As the winner you could choose either the $1,000 stream of payments or a lump sum discounted at a required rate of 7%. We can calculate the lump sum as follows:

Give the result above, as long as the lump sum is exactly $7,023.58, and assuming you could realize a 7% return over ten years, you should be indifferent to receiving a lump sum or the stream of payments. If the lump sum offered is less than $7,023.58, you should chose the income; however, if the lump sum offered is greater than $7,023.58 you should choose the lump sum over the payment stream.

Using an HP12C calculator, you can calculate the present value of an ordinary annuity with the variables above using the following keystrokes:

[1000][PMT]
[7][i]
[10][n][PV]

For instance, to determine the future value of an account that taxes interest or dividends on an annual basis at some specified rate would require you to calculate the future value interest factor using a formula that solves for such method of taxation.

Before the age of calculators and computers, solving future value and present value equations required the use of interest factor tables. Fortunately, solving for the factors is easier than in sounds.

The future value interest factor (FVIF) is simply the quantity that the present value is compounded by:

Let’s construct a future value interest factor table using an assumed annual rate of 7%:

Let’s assume you wanted to calculate the future value interest factor for an investment that will grow at 7% for ten years, plugging those values into the future value interest factor equation will yield the following:

Using the table above and finding the area where n = 10 and r = 7% intersect indicates that the FVIF is 1.9672. Any dollar amount multiplied by the FVIF of 1.9672 will yield the future value of an investment that grew at 7% for ten years.

Present value interest factors (PVIF) are simply the inverse of FVIFs:

We can construct a PVIF table in the same manner:

Notice, if you multiply any FVIF by its corresponding PVIF the product of the two numbers will equal one:

Using an HP12C calculator, we can solve for the FVIF in the example above using the following keystrokes:

[1][.][0][7][ENTER]
[10][y^x]

The Excel model for FVIF and PVIF table construction can be found here.

Where:
FV = Future Value
PV = Present Value
r = rate
n = periods

Calculating the future value of a tax-deferred account incorporates the tax paid on the money when it is withdrawn during the final period. We can account for the taxes paid by adjusting the present value after it has been compounded by the specified rate and number of periods:

The addition of the quantity (1 – t) adjusts the future value in the final period by the tax that is owed.

Assume you have a present value of $1,000, the will grow at a rate of 7% for ten years, with an assumed tax rate of 30%. Plugging those values into the formula will yield the following:

Using Excel, we can model what happens during each of the ten periods:

Notice how the tax is paid during the final period. In the United States, this is how the future value of a Traditional IRA would be calculated. We can represent the table above visually with the following chart:

Using an HP12C calculator, we can calculate the future value of a tax-deferred account with the following keystrokes:

[1000][PV]
[7][i]
[10][n][FV]
[.][7][*]

The formula can be rearranged as follows to find the present value of a tax-deferred account:

The present value of a tax-deferred account formula is usually only seen on tests which require you to calculate the present value of a tax-deferred account based on an initial investment an investor made in the past, given some current value in the future.

A copy of the Excel model can be found here.

Where:

PV = Present Value
FV = Future Value
r = rate
t = time

Assume you would like to have a future lump sum of $10,000. How much would you have to invest today, if the initial contribution grew at required rate of 7.00% for five years? Plugging those values into the formula would yield the following:

The amount that is required today, in order to have $10,000 in the future will decrease as a function of either a longer time-frame, or a higher discount rate. Using Excel, we can model the amounts required given a specific time-frame or rate:

The data can be represented visually as well:

Viewing the chart above, you can see that the initial investment required today, decreases exponentially as a function of time.

Using an HP12C calculator, the present value can be calculated using the following keystrokes:

[10,000][FV]
[7][i]
[5][n]
[PV]

A copy of the Excel model can be found here