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

So, the summation of all future values is
$$F=A{\left(1+i\right)}^{n-1}+A{\left(1+i\right)}^{n-2}+A{\left(1+i\right)}^{n-3}+\dots +A$$

By multiplying both sides by (1+i), we will have
$$F\left(1+i\right)=A{\left(1+i\right)}^n+A{\left(1+i\right)}^{n-1}+A{\left(1+i\right)}^{n-2}+\dots +A\left(1+i\right)$$

By subtracting first equation from second one, we will have
$$\begin{array}{l}F\left(1+i\right)–F=A{\left(1+i\right)}^n+A{\left(1+i\right)}^{n-1}+A{\left(1+i\right)}^{n-2}+\dots \\ +A\left(1+i\right)–\left[A{\left(1+i\right)}^{n-1}+A{\left(1+i\right)}^{n-2}+A{\left(1+i\right)}^{n-3}+\dots +A\right]\\ F+Fi–F=A{\left(1+i\right)}^n+A{\left(1+i\right)}^{n-1}+A{\left(1+i\right)}^{n-2}+\dots \\ +A\left(1+i\right)–A{\left(1+i\right)}^{n-1}-A{\left(1+i\right)}^{n-2}-A{\left(1+i\right)}^{n-3}-\dots -A\end{array}$$

which becomes:
$$Fi=A{\left(1+i\right)}^n–A$$

then

Therefore, Equation 1-3 can determine the future value of uniform series of equal investments as $F=A\left[{\left(1+i\right)}^n-1\right]/i$ . Which can also be written regarding Table 1-5 notation as: $F=A*F/A_{i,n}$. Then $F/A_{i,n }=\left[{\left(1+i\right)}^n-1\right]/i$.
The factor $\left[\left(1+i\right)n-1\right]/i$ 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).

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

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/Ai,n	Uniform Series Compound-Amount Factor	$\left[{\left(1+i\right)}^n-1\right]/i$	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:

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 $A/F_{i,n}=i/[{\left(1+i\right)}^n-1]$. The factor $i/[{\left(1+i\right)}^n-1]$ 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).

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

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	$i/\left[{\left(1+i\right)}^n-1\right]$	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 $i/\left[{\left(1+i\right)}^n-1\right]$