Compound Interest Formulas I

Example 1-2 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 real-world economic evaluations. First, take a look at Figure 1-2. It can help us to better understand the investment evaluation problems.

P		A	A	A		A	A	F

	0	1	2	3	...	n-1	n

Figure 1-2: Time diagram

The horizontal line represents the time. The left-hand end shows the present time and the right-hand 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 1-1 displays a method of notation that can help summarize the given information and avoid confusion.

Table 1-1: Variable relationship between P, F, and A and the Appropriate Factor
	To be Calculated Quantity	Given Quantity	Appropriate Factor (symbol)	Relationship
1	F	P	$F / P_{i, n}$	$\begin{array}{r} F = P * F / P_{i, n} \end{array}$
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:

Economic Evaluation

Click Here for Transcript of Economic Evaluation Video

PRESENTER: Hello. In this video, I'm going to summarize the basic economic evaluation problems, and I will explain how to approach each one. When facing a problem, we have to ask these five main questions. How much money is given? When is the money given, or where on the timeline? What is the time period, year, quarter, or month? What is the interest rate? What needs to be calculated?

The next step in approaching the problem is to draw the timeline. Here, as you can see, the horizontal line represents the time. The left hand end shows the present time and right hand end shows the future. Numbers below the line 0, 1, 2, 3, and n are time periods.

Now, let's add the variables. P on the left hand side is the present single sum of money at time zero. This is the amount of money that is received or paid at the present time, at time zero, at year zero or month zero. We could also write it-- write this P above the time zero. It would be the same.

The other variable is F, which is the future sum of money at the end of the period n. This is the amount of money that is received or paid in the future in the end of the end period, end year, end month. We could also write it above the end year, it's the same. The other parameter is A.

Above each time period starting from year one to year n, there is an A, which are called uniform series of equal payments at each compounding period. These A's show the money that has occurred, that is paid or received in those time periods. Here we assume all of them are equal payments.

When we face a problem, we just need to use the proper equation to solve it. And the next step is to figure out what type of problem we have. Based on given and unknown variables, there are six main categories of problems. In first category, P, money paid or received at the present time is given, and F, future value of that amount needs to be calculated.

Second category, F is given and P needs to be calculated. In third category, F needs to be calculated from given A, uniform and equal series of payments. In fourth category, A needs to be calculated from given f. Fifth category, P, present value, needs to be calculated from given A, and in sixth category, a needs to be calculated from given P. Note that in each type, we have only two money variables.

You can see these six categories in this table. The first column shows the unknown variable, the variable that needs to be calculated. The second column shows the given variable. And the third column shows the appropriate factor. Factor is just a notation, a symbol to summarize the problem.

The slash sign is not a division operator. The first letter on the left hand side of the slash sign shows the variable that needs to be calculated. And the second letter on the right hand side of the slash sign shows the given variable. The two subscripts on each factor are period interest rate, i, followed by the number of interest compounding period, n.

Credit: Farid Tayari

1. Single Payment Compound-Amount 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	...	n-1	n

Figure 1-3: Single Payment Compound-Amount 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 1-1: $F = P {(1 + i)}^{n}$ which can also be written regarding Table 1-1 notation as: $F = P * F / P_{i, n}$ . The mathematical expression ${(1 + i)}^{n}$ is called the “single payment compound-amount factor."

1. Single Payment Compound-Amount Factor

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