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# Compounding

## What is ‘Compounding’

Compounding is the process where the value of an investment increases because the earnings on an investment, both capital gains and interest, earn interest as time passes. This exponential growth occurs because the total growth of an investment along with its principal earn money in the next period. This differs from linear growth, where only the principal earns interest each period.

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## BREAKING DOWN ‘Compounding’

This phenomenon, which is a direct realization of the time value of money, is also known as compound interest. For example, suppose a $10,000 investment in Company X earns 20% the first year. The total investment is then worth $12,000. Next, assume that in the second year, the investment earns another 20%. In year two, the total balance of $12,000 would earn 20%, ending with a value of $14,400 instead of $14,000. The extra $400 of growth is due to the $2,000 earning of year one also growing at 20% in year two, along with the principal.

## The Effect of Compounding Periods On Future Value

When compound on an investment (or liability) occurs, the interest rate matters in determining the future value as well as the number of times that the compounding occurs per period. Assume a one-year time period. More compounding periods result in a higher ending future value of the investment. Two compounding periods per year are better than one. Four compounding periods per year are better than two. The generalized formula for compound interest is:

FV = PV x (1 + ( i / n)) ^ (n x t)

FV = future value

PV = present value

i = the annual interest rate

n = the number of compounding periods per year

t = the number of years

Given this formula, assume that an investment of $1 million earns 20% per year. The resulting future value, based on varying number of compounding periods is:

Annual compounding (n = 1): FV = $1,000,000 x (1 + (20%/1)) ^ (1 x 1) = $1,200,000

Semi-annual compounding (n = 2): FV = $1,000,000 x (1 + (20%/2)) ^ (2 x 1) = $1,210,000

Quarterly compounding (n = 4): FV = $1,000,000 x (1 + (20%/4)) ^ (4 x 1) = $1,215,506

Monthly compounding (n = 12): FV = $1,000,000 x (1 + (20%/12)) ^ (12 x 1) = $1,219,391

Weekly compounding (n = 52): FV = $1,000,000 x (1 + (20%/52)) ^ (52 x 1) = $1,220,934

Daily compounding (n = 365): FV = $1,000,000 x (1 + (20%/365)) ^ (365 x 1) = $1,221,336

The limit of this compounding process, based on calculus, is known as continuous compounding and is calculated using the formula:

FV = PV x e ^ (i x t), where e = the irrational number 2.7183. In the above example, the future value with continuous compounding equals:

FV = $1,000,000 x 2.7183 ^ (0.2 x 1) = $1,221,403