Example 12 was about one single sum; what if you want to add some savings to your bank account each year? So, we need to learn some more techniques to be prepared for realworld economic evaluations. First, take a look at Figure 12. It can help us to better understand the investment evaluation problems.
P  A  A  A  A  A  F  


0  1  2  3  ...  n1  n 
Figure 12: Time diagram
The horizontal line represents the time. The lefthand end shows the present time and the righthand end shows the future. The numbers below the line (1, 2, 3, …, n) are time periods. Above each time period, there is a sum A, which shows the money that occurs in that time period; here, we assume all of them are equal payments, so:
A is a uniform series of equal payments at each compounding period;
P is a present single sum of money at the time zero;
F is a future sum of money at the end of period n. And i is the compound interest rate.
In order to understand an economic evaluation problem we have to determine:
 How much money is given?
 When is the money given (where on the timeline)?
 What is the time period (year, quarter, or month)?
 What is the interest rate?
 What needs to be calculated?
Following these steps, we just need to use the proper equation to solve the problem. Based on the unknown (asked) variable, there are six basic categories of problems here:
 F (future value) needs to be calculated from given P
 F (future value) needs to be calculated from given A
 P (present value) needs to be calculated from given F
 P (present value) needs to be calculated from given A
 A (uniform and equal period values) needs to be calculated from given F
 A (uniform and equal period values) needs to be calculated from given P
Table 11 displays a method of notation that can help summarize the given information and avoid confusion.
To be Calculated Quantity  Given Quantity  Appropriate Factor (symbol)  Relationship  

1  F  P  $F/{P}_{i,n}$  $\begin{array}{r}\hfill F=P*F/{P}_{i,n}\end{array}$ 
2  P  F  $P/{F}_{i,n}$  $P=F*P/{F}_{i,n}$ 
3  F  A  $F/{A}_{i,n}$  $F=A*F/{A}_{i,n}$ 
4  A  F  $A/{F}_{i,n}$  $A=F*A/{F}_{i,n}$ 
5  P  A  $P/{A}_{i,n}$  $P=A*P/A{}_{i,n}$ 
6  A  P  $A/{P}_{i,n}$  $A=P*A/{P}_{i,n}$ 
Note: “/” in the Appropriate Factor (symbol) column is not a division operator, the entire $F/{P}_{i,n}$or $F/{A}_{i,n}$, … is a factor (symbol). The first letter shows the variable that needs to be calculated and the second letter shows the given variable. The two subscripts on each factor are the given period interest rate, i, followed by the number of interest compounding periods, n.
The new notation helps us summarize the problem. The factor actually give a gives us a coefficient that when multiplied by given parameter, gives the unknown parameter.
All time value of money calculations involves writing an equation or equations to calculate F, P, or A. Each of terms in the column “Appropriate Factor (symbol)” has a name that you will learn later in this course.
Please watch the following (4:32) video:
1. Single Payment CompoundAmount Factor
The first category of six categories that were introduced explains the situation that the present value of money is given and asks you to calculate the future value according to the given interest rate of i per period and n period from now. This problem can be summarized with the factor (symbol) of $F/{P}_{i,n}$and can be shown as:
P  _  _  _  _  _  F=?  


0  1  2  3  ...  n1  n 
Figure 13: Single Payment CompoundAmount Factor, F/P_{i,n}
As explained earlier, the future value of money after n period with an interest rate of i can be calculated using the Equation 11: $F=P{\left(1+i\right)}^{n}$ which can also be written regarding Table 11 notation as: $F=P*F/{P}_{i,n}$. The mathematical expression ${\left(1+i\right)}^{n}$ is called the “single payment compoundamount factor."