3. Uniform Series CompoundAmount Factor
The third category of problems in Table 15 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  ...  n1  n 
Figure 14: Uniform Series CompoundAmount Factor, $F/{A}_{i,n}$
In this case, utilizing Equation 12 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 $A{\left(1+i\right)}^{n1}$
Note that first investment occurred in time period 1 (one period after present time) so it is n1 periods before the n^{th} period and then the power is n1.
And similarly:
Future value of second investment occurred at time period 2: $A{\left(1+i\right)}^{n2}$
Future value of third investment occurred at time period 3: $A{\left(1+i\right)}^{n3}$
Future value of last investment occurred at time period n: $A{\left(1+i\right)}^{nn}=A$
Note that the last payment occurs at the same time as F.
So, the summation of all future values is
$$F=A{\left(1+i\right)}^{n1}+A{\left(1+i\right)}^{n2}+A{\left(1+i\right)}^{n3}+\dots +A$$
By multiplying both sides by (1+i), we will have
$$F\left(1+i\right)=A{\left(1+i\right)}^{n}+\text{}A{\left(1+i\right)}^{n1}+\text{}A{\left(1+i\right)}^{n2}+\dots +\text{}A\left(1+i\right)$$
By subtracting first equation from second one, we will have
$$\begin{array}{l}F\left(1+i\right)\u2013F=A{\left(1+i\right)}^{n}+\text{}A{\left(1+i\right)}^{n1}+\text{}A{\left(1+i\right)}^{n2}+\dots \text{}\\ +A\left(1+i\right)\u2013\left[A{\left(1+i\right)}^{n1}+A{\left(1+i\right)}^{n2}+A{\left(1+i\right)}^{n3}+\dots +A\right]\\ F+Fi\u2013F=A{\left(1+i\right)}^{n}+A{\left(1+i\right)}^{n1}+\text{}A{\left(1+i\right)}^{n2}+\dots \text{}\\ +\text{}A\left(1+i\right)\u2013\text{}A{\left(1+i\right)}^{n1}\text{}A{\left(1+i\right)}^{n2}\text{}A{\left(1+i\right)}^{n3}\dots A\end{array}$$
which becomes:
$$Fi=A{\left(1+i\right)}^{n}\u2013A$$
then
Therefore, Equation 13 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 15 notation as: $F=A*F/{A}_{i,n}$. Then $F/{A}_{i,n}=\left[{\left(1+i\right)}^{n}1\right]/i$.
The factor $\left[\left(1+i\right)n1\right]/i$ is called “Uniform Series CompoundAmount 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 CompoundAmount Factor (3:42).
Example 13:
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 13, we will have
$$\begin{array}{l}F=A*F/{A}_{i,n}=A\left[{\left(1+i\right)}^{n}1\right]/i\\ F=A*F/{A}_{6\%,20}=4000\text{}*\text{}\left[{\left(1+0.06\right)}^{20}1\right]/0.06\\ F=4000*36.78559\text{}=147142.4\end{array}$$
So, you will have 147,142.4 dollars at 20th year.
Factor  Name  Formula  Requested variable  Given variables 

F/A_{i,n}  Uniform Series CompoundAmount 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. SinkingFund Deposit Factor
The fourth group in Table 15 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  ...  n1  n 
Figure 15: SinkingFund Deposit Factor, $A/{F}_{i,n}$
Equation 13 can be rewritten for A (as unknown) to solve these problems:
Equation 14 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 15 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 “sinkingfund deposit factor”, and is designated by $A/{F}_{i,n}$ . The factor is used to calculate a uniform series of equal endofperiod 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 14:
Referring to Example 13, 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 14, 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%.
Factor  Name  Formula  Requested variable  Given variables 

$A/{F}_{i,n}$  SinkingFund Deposit Factor  $i/\left[{\left(1+i\right)}^{n}1\right]$  A: Uniform series of equal endofperiod payments  F: cumulated future value of investments n: number of time periods i: interest rate 
Note that $i/\left[{\left(1+i\right)}^{n}1\right]$