Compound Growth and Decay

Understanding Compound Growth

Compound growth refers to the process where the value of an investment or loan increases exponentially over time due to interest being calculated on the accrued interest in addition to the original amount.
The formula for compound growth can be given as A = P(1 + r/n)^(nt) where:
- A is the amount of money accumulated after n years, including interest.
- P is the principal or the initial amount of money.
- r is the annual interest rate (in decimal).
- n is the number of times that interest is compounded per unit t.

To apply this formula, you first convert the interest rate from a percentage to a decimal by dividing it by 100.
Note that the time should be in the same units as the interest rate. For instance, if the interest rate is per annum, t should be in years.
For example, if £500 is invested at an annual rate of 5% compounded yearly, the amount after two years will be A = 500(1 + 0.05/1)^(1*2) = £552.50.

Compound interest is calculated by adding the interest earned each period to the initial principal so that each subsequent interest calculation is based on a larger total amount.
Over multiple periods, the difference between the amount of simple and compound interest can become significant. With compound interest, the longer the time period, the larger the amount of money.

In addition to growth, the compound formula can also be applied to decay or depreciation. This is when the value of an item decreases over time, such as a car.
Here, the formula slightly changes to A = P(1 - r/n)^(nt). Note the minus sign.
For example, if a car depreciates at an annual rate of 5% and was bought for £5000, its value after 1 year will be A = 5000(1 - 0.05/1)^(1*1) = £4750.

Compound growth and decay operate on the same mathematical principles, but in reverse directions.
Compound growth is used when calculating how a sum of money or an investment might grow over time due to recurring increases like interest or dividends.
Compound decay, on the other hand, is used when calculating how a sum of money or an asset may decrease in value over time due to recurring decreases like depreciation or withdrawal.

Knowing how to calculate and understand compound growth and decay is crucial in real-life scenarios especially in financial planning, investing, retirement planning, and evaluating loan agreements.
It can also be applied in natural phenomena such as population growth and radioactivity decay in sciences.