Compound Interest Formulas II

3. Uniform Series Compound-Amount Factor

The third category of problems in Table 1-5 demonstrates the situation that equal amounts of money, A, are invested at each time period for n number of time periods at interest rate of i (given information are A, n, and i) and the future worth (value) of those amounts needs to be calculated. This set of problems can be noted as $F / A_{i, n}$ . The following graph shows the amount occurred. Think of it as this example: you are able to deposit A dollars every year (at the end of the year, starting from year 1) in an imaginary bank account that gives you i percent interest and you can repeat this for n years (depositing A dollars at the end of the year). You want to know how much you will have at the end of year n^th.


0		A	A		A	A	F=?

	0	1	2	...	n-1	n

Figure 1-4: Uniform Series Compound-Amount Factor, $F / A_{i, n}$

In this case, utilizing Equation 1-2 can help us calculate the future value of each single investment and then the cumulative future worth of these equal investments.

Future value of first investment occurred at time period 1 equals $A {(1 + i)}^{n - 1}$
Note that first investment occurred in time period 1 (one period after present time) so it is n-1 periods before the n^th period and then the power is n-1.

And similarly:
Future value of second investment occurred at time period 2: $A {(1 + i)}^{n - 2}$
Future value of third investment occurred at time period 3: $A {(1 + i)}^{n - 3}$
Future value of last investment occurred at time period n: $A {(1 + i)}^{n - n} = A$
Note that the last payment occurs at the same time as F.

So, the summation of all future values is

$F = A {(1 + i)}^{n - 1} + A {(1 + i)}^{n - 2} + A {(1 + i)}^{n - 3} + \dots + A$

By multiplying both sides by (1+i), we will have

$F (1 + i) = A {(1 + i)}^{n} + A {(1 + i)}^{n - 1} + A {(1 + i)}^{n - 2} + \dots + A (1 + i)$

By subtracting first equation from second one, we will have

$\begin{array}{l} F (1 + i) - F = A {(1 + i)}^{n} + A {(1 + i)}^{n - 1} + A {(1 + i)}^{n - 2} + \dots \\ + A (1 + i) - [A {(1 + i)}^{n - 1} + A {(1 + i)}^{n - 2} + A {(1 + i)}^{n - 3} + \dots + A] \\ F + F i - F = A {(1 + i)}^{n} + A {(1 + i)}^{n - 1} + A {(1 + i)}^{n - 2} + \dots \\ + A (1 + i) - A {(1 + i)}^{n - 1} - A {(1 + i)}^{n - 2} - A {(1 + i)}^{n - 3} - \dots - A \end{array}$

which becomes:

$F i = A {(1 + i)}^{n} - A$

then

$F = A [{(1 + i)}^{n} - 1] / i$

Equation 1-3

Therefore, Equation 1-3 can determine the future value of uniform series of equal investments as $F = A [{(1 + i)}^{n} - 1] / i$ . Which can also be written regarding Table 1-5 notation as: $F = A * F / A_{i, n}$ . Then $F / A_{i, n} = [{(1 + i)}^{n} - 1] / i$ .
The factor $[(1 + i) n - 1] / i$ is called “Uniform Series Compound-Amount Factor” and is designated by F/A_i,n. This factor is used to calculate a future single sum, “F”, that is equivalent to a uniform series of equal end of period payments, “A”.

Note that n is the number of time periods that equal series of payments occur.

Please review the following video, Uniform Series Compound-Amount Factor (3:42).

Uniform Series Compound Amount Factor

Click for the transcript of "Uniform Series Compound-Amount Factor" video.

PRESENTER: In the third category, equal amounts of money A are enlisted pay to receive at each time period for n number of time periods. n can be years or months, and interest rate is i. And the question asks you to calculate the future value of these payments, a single sum of money that is equivalent to all these series of payments A. Here, given information are A, n, and i. And F is the unknown parameter. These sets of problems can be displayed with the factor F slash A, or F/A. Again, the left side of this slash sign is the unknown parameter F, and the right side is the given variable, which is A.

Here, you can see the equation to calculate F from A, i and n. The mathematical proof of this equation is straightforward, and they explain it in Lesson One. We can write this equation, regarding the factor notation, F equals A multiply the factor. This factor is called uniform series compound-amount factor. And it is used to calculate the future single sum F that is equivalent to uniform series of equal ends of period payments A.

Let's work on an example to see how this factor can be used. Assume you save $4,000 per year and deposit it, in the end of the year, in an imaginary saving account or some other investment that gives you 6% interest rate per year, compounded annually, for 20 years, starting from year 1 to year 20th. And you want to know how much money will you have in the end of the 20th year.

First, we draw the time line. Left-hand side is the present time. We don't have anything there. Note that your investment, it starts from year 1 to year 20th. If there is no extra information in the question, and question says you invest for 20 years, you need to assume your investment, it starts from year 1. So there is no payment at present time, or year zero.

Right-hand side is the future time, which is a single amount future value, and it is unknown. Your investment takes 20 years, so n equals 20. And above each year, you have to write $4,000, because you have a payment of $4,000 in the end of each year. So A equals $4,000, n number of years is 20, i interest rate 6%, and F needs to be calculated.

And F equals A times the factor F/A. In this factor, i is 6% and 20. And we use the equation to calculate the F. And we find the answer. So if you invest $4,000 per year for 20 years, with 6% interest rate, you will have about $147,000 at the end of the 20th year.

Credit: Farid Tayari

Example 1-3:

Assume you save 4000 dollars per year and deposit it at the end of the year in an imaginary saving account (or some other investment) that gives you 6% interest rate (per year compounded annually), for 20 years. How much money will you have at the end of the 20th year?


0		$4000	$4000		$4000	$4000	F=?

	0	1	2	...	19	20

So
A =$4000
n =20
i =6%
F=?

Please note that n is the number of equal payments.

Using Equation 1-3, we will have

$\begin{array}{l} F = A * F / A_{i, n} = A [{(1 + i)}^{n} - 1] / i \\ F = A * F / A_{6 %, 20} = 4000 * [{(1 + 0.06)}^{20} - 1] / 0.06 \\ F = 4000 * 36.78559 = 147142.4 \end{array}$

So, you will have 147,142.4 dollars at 20th year.

**Table 1-8**: Uniform Series Compound-Amount Factor
Factor	Name	Formula	Requested variable	Given variables
F/A_i,n	Uniform Series Compound-Amount Factor	$[{(1 + i)}^{n} - 1] / i$	F: Future value of uniform series of equal investments	A: uniform series of equal investments n: number of time periods i: interest rate

4. Sinking-Fund Deposit Factor

The fourth group in Table 1-5 is similar to the third group but instead of A as given and F as unknown parameters, F is given and A needs to be calculated. This group illustrates the set of problems that ask you to calculate uniform series of equal payments (or investment), A, to be invested for n number of time periods at interest rate of i and accumulated future value of all payments equal to F. Such problems can be noted as $A / F_{i, n}$ and are displayed in the following graph. Think of it as this example: you are planning to have F dollars in n years and there is a saving account that can give you i percent interest. You want to know how much you have to deposit every year (at the end of the year, starting from year 1) to be able to have F dollars after n years.

0		A=?	A=?		A=?	A=?	F

	0	1	2	...	n-1	n

Figure 1-5: Sinking-Fund Deposit Factor, $A / F_{i, n}$

Equation 1-3 can be rewritten for A (as unknown) to solve these problems:

$A = F {i / [{(1 + i)}^{n} - 1]}$

Equation 1-4

Equation 1-4 can determine uniform series of equal investments, A, given the cumulated future value, F, the number of the investment period, n, and interest rate i. Table 1-5 notes these problems as: $A = F * A / F_{i, n}$ . Then $A / F_{i, n} = i / [{(1 + i)}^{n} - 1]$ . The factor $i / [{(1 + i)}^{n} - 1]$ is called the “sinking-fund deposit factor”, and is designated by $A / F_{i, n}$ . The factor is used to calculate a uniform series of equal end-of-period payments, A, that are equivalent to a future sum F.

Note that n is the number of time periods that equal series of payments occur.

Please watch the following video, Sinking Fund Deposit Factor (4:42).

Sinking Fund Deposit Factor

Click for the transcript of "Sinking Fund Deposit Factor" video.

PRESENTER: The fourth group is similar to the third one. But A is the unknown and F is the given variable. This set of problems asks you to calculate uniform series of equal payments, A, to be invested for n number of time periods at interest rate of i. And the accumulated future value of all payments or equivalent future value is F.

This set of problems can be summarized with the factor A over F or A slash F. The left side of this last sign is the unknown parameter. Here it is A. And the right side is the given variable, which is F.

Equation 1-3 for uniform series compound amount factor can be rewritten for A as unknown to solve these problems, which gives the Equation 1-4. Equation 1-4 can determine uniform series of equal investments, A, for accumulated future value, F, number of investment period n and interest rate i.

We can write this equation according to the factor notation, A equals F times the factor A over F. This factor is called the Sinking-Fund Deposit Factor. And it is displayed by A slash F. The factor is used to calculate the uniform series of equal end of the period payments, A, that are equivalent to a future sum, F.

For example, referring to example 1-7 in previous video, let's say you plan to have $200,000 after 20 years. And you are offered an investment, which can be the imaginary savings account, that gives you 6% per year compound interest rates. And you want to know how much money, equal payments, you need to save each year, or invest-- deposit in your account in the end of each year.

So in summary, you want to have $200,000 after 20 years. And you can invest your money with 6% interest rate. The question is, how much you need to invest per year?

Again, the first step is drawing the time line. Left-hand side is the present time. We won't have any payment. So there is no payment at present time or time zero. Right-hand side is the future. And you want to have a single amount of $200,000. So you write $200,000 in the 20th year, or in the end of right-hand side of the time line.

Note that $200,000 has the same time dimension as the last payment, A. Both are in the year 20th. Your investment takes 20 years, so n equals 20. And above each year, you have to write A, which is unknown and needs to be calculated.

So F equals $200,000. n number of years is 20. i, interest rate, 6%. And A needs to be calculated.

We can use the factor notation to summarize the equation. In this factor, i is 6%, n is 20, and F is given, and A needs to be calculated. And we calculate the result. So if you want to have $200,000 in 20 years from now with 6% interest rate, you will need to invest equal amounts of $5,437 per year at the end of each year for 20 years, starting from year one.

Credit: Farid Tayari

Example 1-4:

Referring to Example 1-3, assume you plan to have 200,000 dollars after 20 years, and you are offered an investment (imaginary saving account) that gives you 6% per year compound interest rate. How much money (equal payments) do you need to save each year and invest (deposit it to your account) in the end of each year?

0		A=?	A=?		A=?	A=?	F=200,000

	0	1	2	...	19	20

So
F=$200,000
n=20
i=6%
A=?

Using Equation 1-4, we will have

$\begin{array}{l} A = F * A / F_{i, n} = F {i / [{(1 + i)}^{n} - 1]} \\ A = F * A / F_{6 %, 20} = 200, 000 * 0.06 / [{(1 + 0.06)}^{20} - 1] \\ A = 200, 000 * 0.027185 = 5436.912 \end{array}$

So, in order to have 200,000 dollars at 20th year, you have to invest 5,436.9 dollars in the end of each year for 20 years at annual compound interest rate of 6%.

**Table 1-9**: Sinking-Fund Deposit Factor
Factor	Name	Formula	Requested variable	Given variables
$A / F_{i, n}$	Sinking-Fund Deposit Factor	$i / [{(1 + i)}^{n} - 1]$	A: Uniform series of equal end-of-period payments	F: cumulated future value of investments n: number of time periods i: interest rate

Note that $i / [{(1 + i)}^{n} - 1]$

3. Uniform Series Compound-Amount Factor

Example 1-3:

4. Sinking-Fund Deposit Factor

Example 1-4:

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