A perpetuity is a series of cash flow payments occurring in equal amounts forever. The formula used to calculate the present value of a perpetuity is as follows:
present value of a perpetuity formula
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:
where; CF =$3.50, i = 0.07
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:
present value of annuity due formula
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:
where; PMT = $1,000, r = 0.07, n = 10
Using an HP12C calculator, we can solve the equation above using the following keystrokes:
The present value of an ordinary annuity formula can be used to calculate the present value of a stream of income payments, the formula is as follows:
present value of an ordinary annuity formula
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:
where; PMT = $1,000, r = 0.07, n = 10
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:
The concept of future value and present value interest factors is an important one to understand when you begin to calculate future and present values which take into account more complex forms of taxation.
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:
future value interest factor formula
Let’s construct a future value interest factor table using an assumed annual rate of 7%:
Rate
2.00%
3.00%
4.00%
5.00%
6.00%
7.00%
Year
1
1.0200
1.0300
1.0400
1.0500
1.0600
1.0700
2
1.0404
1.0609
1.0816
1.1025
1.1236
1.1449
3
1.0612
1.0927
1.1249
1.1576
1.1910
1.2250
4
1.0824
1.1255
1.1699
1.2155
1.2625
1.3108
5
1.1041
1.1593
1.2167
1.2763
1.3382
1.4026
6
1.1262
1.1941
1.2653
1.3401
1.4185
1.5007
7
1.1487
1.2299
1.3159
1.4071
1.5036
1.6058
8
1.1717
1.2668
1.3686
1.4775
1.5938
1.7182
9
1.1951
1.3048
1.4233
1.5513
1.6895
1.8385
10
1.2190
1.3439
1.4802
1.6289
1.7908
1.9672
future value interest factor table
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:
where; r = 0.07, n = 10
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:
present value interest factor formula
We can construct a PVIF table in the same manner:
Rate
2%
3%
4%
5%
6%
7%
Year
1
0.9804
0.9709
0.9615
0.9524
0.9434
0.9346
2
0.9612
0.9426
0.9246
0.9070
0.8900
0.8734
3
0.9423
0.9151
0.8890
0.8638
0.8396
0.8163
4
0.9238
0.8885
0.8548
0.8227
0.7921
0.7629
5
0.9057
0.8626
0.8219
0.7835
0.7473
0.7130
6
0.8880
0.8375
0.7903
0.7462
0.7050
0.6663
7
0.8706
0.8131
0.7599
0.7107
0.6651
0.6227
8
0.8535
0.7894
0.7307
0.6768
0.6274
0.5820
9
0.8368
0.7664
0.7026
0.6446
0.5919
0.5439
10
0.8203
0.7441
0.6756
0.6139
0.5584
0.5083
present value interest factor table
Notice, if you multiply any FVIF by its corresponding PVIF the product of the two numbers will equal one:
FVIF * PVIF
Using an HP12C calculator, we can solve for the FVIF in the example above using the following keystrokes:
hp12c
[1][.][0][7][ENTER] [10][y^x]
The Excel model for FVIF and PVIF table construction can be found here.
Recall, that the formula utilized to calculate the future value of a lump sum is as follows:
future value formula
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:
future value of a tax-deferred account
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:
where; PV = $1,000, rate = 0.07, n = 10, t = 30%
Using Excel, we can model what happens during each of the ten periods:
Year
PV
rate
FV
Tax (30%)
1
$ 1,000.00
7%
$ 1,070.00
2
$ 1,070.00
7%
$ 1,144.90
3
$ 1,144.90
7%
$ 1,225.04
4
$ 1,225.04
7%
$ 1,310.80
5
$ 1,310.80
7%
$ 1,402.55
6
$ 1,402.55
7%
$ 1,500.73
7
$ 1,500.73
7%
$ 1,605.78
8
$ 1,605.78
7%
$ 1,718.19
9
$ 1,718.19
7%
$ 1,838.46
10
$ 1,838.46
7%
$ 1,967.15
Tax
$ 590.15
Net ATFV
$ 1,377.01
future value of a tax-deferred account table
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:
future value of a tax-deferred account chart
Using an HP12C calculator, we can calculate the future value of a tax-deferred account with the following keystrokes:
hp12c
[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:
present value of a tax-deferred account formula
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.
The formula used to discount a future value to a present value today is as follows:
present value formula
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:
FV = $10,000; r = 0.07; t = 5
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:
Year
FV
rate
PV
5
$ 10,000.00
7.00%
$ 7,129.86
10
$ 10,000.00
7.00%
$ 5,083.49
15
$ 10,000.00
7.00%
$ 3,624.46
20
$ 10,000.00
7.00%
$ 2,584.19
25
$ 10,000.00
7.00%
$ 1,842.49
30
$ 10,000.00
7.00%
$ 1,313.67
present value table
The data can be represented visually as well:
present value chart
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:
