# What Is Compounding?

## What is compounding?

Capitalization is the process by which the income from an asset, either capital gains Where interest, are reinvested to generate additional income over time. This growth, calculated using exponential functions, occurs because the investment will generate income both on its initial capital and on the cumulative income of previous periods.

Compounding therefore differs from linear growth, where only the principal earns interest each period.

### Key points to remember

• Compounding is the process by which interest is credited on an existing principal amount as well as interest already paid.
• Compounding can therefore be interpreted as interest on interest, the effect of which is to amplify interest returns over time, the so-called “miracle of compounding”.
• When banks or financial institutions credit compound interest, they use a compounding period such as annual, monthly, or daily.
• Compounding can occur on an investment where savings grow faster or on debt where the amount owed may increase even though payments are being made.
• Compounding occurs naturally in savings accounts; certain investments that pay dividends may also benefit from compounding.

## Understand composition

Compounding generally refers to the increase in value of an asset due to interest earned on both a principal and accrued interest. This phenomenon, which is a direct realization of the time value of money (TMV) concept, is also known as compound interest.

Capitalization is crucial in finance, and the gains attributable to its effects are the motivation behind many investment strategies. For example, many companies offer dividend reinvestment plans (DRIPs) that allow investors to reinvest their cash dividends buy additional shares of Stock. Reinvesting in more of these dividend-paying stocks worsens investors’ returns because increasing the number of stocks will consistently increase future income from dividend payments, assuming stable dividends.

Investing in dividend growth stocks in addition to dividend reinvestment adds another layer of compounding to this strategy which some investors call dual compounding. In this case, not only are the dividends reinvested to buy more shares, but these dividend growth stocks also increase their payouts per share.

## Compound interest formula

The formula for the future value (FV) of a current asset is based on the concept of compound interest. It takes into account the present value of an asset, the annual interest rate, the frequency of capitalization (or the number of capitalization periods) per year and the total number of years. The general formula for compound interest is:



F

V

=

P

V

×

(

1

+

I

)

not

where:

F

V

=

Future value

P

V

=

Current value

I

=

Annual interest rate

not

=

Number of capitalization periods per year

\begin{aligned}&FV=PV\times(1+i)^n\\&\textbf{where:}\\&FV=\text{Future Value}\\&PV=\text{Current Value}\\&i= \text{Annual interest rate}\\&n=\text{Number of compounding periods per year}\end{aligned}

FV=PV×(1+I)notwhere:FV=Future valuePV=Current valueI=Annual interest ratenot=Number of capitalization periods per year

This formula assumes that no additional changes other than interest are made to the initial principal balance.

### 536 870 912

Curious what the 100% daily composition looks like? One Grain of Rice, Demi’s folk tale, centers around a reward where a single grain of rice is awarded on the first day and the number of grains of rice awarded each day are doubled over 30 days. By the end of the month, over 536 million grains of rice would be awarded on the last day.

## Increase in capitalization periods

The effects of compounding become stronger as the frequency of compounding increases. Assume a period of one year. The more compounding periods throughout that year, the greater the future value of the investment, so naturally, two compounding periods per year is better than one, and four compounding periods per year is better only two.

To illustrate this effect, consider the following example given the formula above. Suppose an investment of $1 million returns 20% per year. The resulting future value, based on a variable number of compounding periods, is: • Annual composition (n = 1): FV =$1,000,000 × [1 + (20%/1)] (1×1) = $1,200,000 • Semester composition (n = 2): FV =$1,000,000 × [1 + (20%/2)] (2×1) = $1,210,000 • Quarterly composition (n = 4): FV =$1,000,000 × [1 + (20%/4)] (4×1) = $1,215,506 • Monthly composition (n = 12): FV =$1,000,000 × [1 + (20%/12)] (12×1) = $1,219,391 • Weekly composition (n = 52): FV =$1,000,000 × [1 + (20%/52)] (52×1) = $1,220,934 • Daily composition (n = 365): FV =$1,000,000 × [1 + (20%/365)] (365×1) = 1,221,336 It is obvious that the future value increases by a smaller margin even if the number of compounding periods per year increases significantly. The frequency of compounding over a fixed term has a limited effect on the growth of an investment. This limit, based on calculation, is known as continuous composition and can be calculated using the formula:  F V = P × e r you where: e = Irrational number 2.7183 r = Interest rate you = Time \begin{aligned}&FV=P\times e^{rt}\\&\textbf{where:}\\&e=\text{Irrational number 2.7183}\\&r=\text{Interest rate}\ \&t=\text{Time}\end{aligned} FV=P×eryouwhere:e=Irrational number 2.7183r=Interest rateyou=Time In the example above, the future value with continuous compounding is equal to: FV =1,000,000 × 2.7183 (0.2×1) = $1,221,403. Capitalization is an example of the “snowball effect” where a low-level situation turns into a larger and more serious condition. ## Capitalization of investments and debt Compound interest works on both assets and passive. Although compounding increases the value of an asset more quickly, it can also increase the amount owed on a loan because interest accrues on unpaid principal and past interest charges. Even if you repay your loan, compound interest may cause the amount you owe to increase in future periods. The concept of compounding is particularly problematic for credit card balances. Not only is the interest rate on credit card debt high, but interest charges can be added to the principal balance and result in interest assessments on itself in the future. For this reason, the concept of capitalization is not necessarily “good” or “bad”. Compounding effects can work for or against an investor depending on their specific financial situation. ## Example of composition To illustrate how compounding works, suppose$10,000 is held in an account that earns 5% interest per year. After the first year or compounding period, the account total grew to $10,500, with a mere reflection of$500 interest being added to the $10,000 director. In the second year, the account achieves 5% growth on the initial principal and$500 of interest in the first year, resulting in a gain of $525 in the second year and a balance of 11$025.

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