# Capitalization

Capitalization is the process of projecting initial capital at a later period of time, based on an interest rate .

Capitalization (simple or compound) is the process by which a certain amount of capital increases in value. In fact, it is a mathematical expression of a real phenomenon. For example, they give us 2% of income on our initial capital annually for 3 years. At the end of the three years we will have 6%.

From the above, we can see that it is an expression that calculates the evolution of said capital. The opposite of capitalizing is updating or discounting. That is, the opposite of capitalization is the discount or update.

The capitalization process implicitly carries an interest rate . So the projected capital in the future depends on what kind of interest we project the initial capital. Therefore, the final capital is a function of the initial and the interest rate.

Imagine the following situation:

• We invest \$ 1,000 in a financial assetfor a term of five years.
• This product gives an annual interest rate of 1%.

The value of our initial investment after five years depends on the initial capital and those interest generated. It will also depend on the type of capitalization that is applied in the operation. Since this will condition how interest rates are applied to the initial capital. And, therefore, the final value will vary based on this.

## Components of capitalization

To understand the mathematical formulas that regulate the relationship between capitals and the interests they generate, it is necessary to know that the nomenclature used is as follows:

C 0 : Initial capital or capital in year 0.

n :  Capital in year “n”.

i:    Interest rate of the operation.

n:    Number of years.

The nomenclature may vary depending on the bibliographic reference. For example, instead of 0 we can have CI (initial capital acronym). Also, instead of n we could simplify and refer to the final capital with the acronym CF.

## Types of capitalization

There are two main types, depending on whether the interest earned is incorporated or not into the initial capital.

• Simple capitalization:The interest generated in any period is proportional to the duration of the period and the initial capital. This type of capitalization is usually used for periods of less than one year. Because of this, because this capitalization system does not capitalize the interest generated. And, in addition, the reinvestment of those interests does not include the final capital.
• Compound capitalization:Interest generated in one period is accumulated at the initial capital for the following period. In this case the interest if they are capitalized, just the opposite of simple capitalization. Therefore, this type of capitalization is usually used for periods longer than one year. Therefore, here interests generate more interests. In the case of operations greater than one year, this type of capitalization will generate a greater final amount than the simple one.
• Continuous capitalization:Interest is generated infinitely times a year. That is, they accumulate continuously in every second. This type of capitalization implies the continuous reinvestment of those interests. Therefore, compared to compound capitalization, this will generate a higher final capital value.

Interest is generated infinitely times a year. That is, they accumulate continuously in every second. This type of capitalization implies the continuous reinvestment of those interests. Therefore, compared to compound capitalization, this will generate a higher final capital value. In the following graph we can see the difference between them:

The red line refers to simple capitalization, the orange line to compound capitalization and the green line to continuous capitalization.

## Capitalization Example

To better understand the concept of capitalization, let’s solve two examples of capitalization. One of them will be of simple capitalization and another of compound capitalization.

In both cases we will start from the same example. Suppose we have an initial capital of \$ 20,000 and the return on an investment is 3%. annual. The investment will last three years.

### Example of simple capitalization

In the example of simple capitalization we do not accumulate interest. That is, if they are going to be 3 years and the interest is 3%, we do the following operation: 3 x 3 = 9%. This is similar to withdrawing interest each year and starting from scratch.

Final capital = 20,000 x (1 + 0.09) = \$ 21,800

In the same way, we could also calculate the interest paid each year and add it to the initial capital:

Interest paid each year = 0.03 x 20,000 = \$ 600

Being three years, we multiply the 600 dollars they pay us each year for the three years and add them to the initial capital:

Final capital = 20,000 + (600 x 3) = 21,800

### Example of compound capitalization

In the case of compound capitalization, we accumulate interest. That is, every year instead of starting from scratch, we add the interest generated. Therefore, each year we have a higher initial capital. The formula allows us to calculate the interest of a large number of periods when the interest generated remains constant.

That is, instead of multiplying 1 + r to the result of each year, we directly apply the following formula:

Final capital = 20,000 x (1 + 0.03) 3

We perform the calculation and we have to:

Final capital = 20,000 x 1.092727 = 21.854.54

This is the same result as if we do the following:

Year 1: 20,000 x 1.03 = 20,600

Year 2: 20.600 x 1.03 = 21.218

Year 3: 21,218 x 1.03 = 21,854.54

Obviously, it is faster to use the formula. Especially when it comes to large periods.