What Is Exponential Growth?

Exponential growth is a pattern of data that shows greater increases with passing time, creating the curve of an exponential function. The formula for exponential growth is V = S x (1+R)^T, where S is the starting value, R is the interest rate, T is the number of periods that have elapsed, and V is the current value.

To demonstrate exponential growth, suppose a population of mice rises exponentially by a factor of two every year starting with two in the first year, then four i🍌n the second year, eight in the third year, 16 in the🍌 fourth year, and so on. In this case the population is growing by a factor of two each year. If the mice instead gave birth to four pups in the first year, you would have four, then 16, then 64, then 256.

Exponential growth (which is multiplicative) can be contrasted with linear growth (which is additive) and geometric growth (which is raised to a p🍷ower).

Understanding Exponential Growth

In finance, compound returns cause exponential growth. The power of compounding is one of the most powerful forces in finance. This concept allows investors to create large sums with little initial capital. Savings accounts that carry a 澳洲幸运5官方开奖结果体彩网:compound interest rate are common examples of exponential growth.

Applications of Exponential Growth

Assume you deposit $1,000 in an account that earns a guaranteed 10% rate of interest. If the account carries a simple 澳洲幸运5官方开奖结果体彩网:interest rate, you will earn $100 per year. The amount of interest paid w⭕ill not change as long as no additional deposits are made. You will always earn $100 each year.

If the account carries a compoun🐼d interest rate, however, you will earn interest on the cumulative account total. Each year the lender will app🌃ly the interest rate not to the initial deposit but to the sum of the initial deposit and any interest previously paid.

In the first year the interest earned is still 10% or $100. In the second year, however, the 10% rate is applied to the new total of $1,100, yielding $110. With each subsequent year, the amount of interest paid grows, creating rapidly accelerating, or exponential, growth. After 30 years, with no other deposits required, your account would be worth $17,449.40.

The Formula for Exponential Growth

On a cha🧔rt this curve starts slowly and remains nearly fl🦹at for a time before increasing swiftly to appear almost vertical. It follows the formula mentioned above:

$V=S\times(1+R)^T$V=S×(1+R)T

The current value, V, of an initial starting point subject to exponen🌠tial growth can be determined by multiplying the starting value, S, by the sum of one plus the rate of interest, R, raised to the power of T, or the number of periods that have elapsed.

Special Considerations

While exponential growth is often used in financial modeling, the reality is often more complicated. The application of exponential growth works well in the example of a savings account, because the rate of interest is guaranteed and does not change over time. In most investments, this is not the case. For instance, 澳洲幸运5官方开奖结果体彩网:stock market returns do not smoothly follow🍃 lon꧋g-term averages each year.

Other methods of predicting long-term returns—such as the 澳洲幸运5官方开奖结果体彩网:Monte Carlo simulation, which uses 澳洲幸运5官方开奖结果体彩网:probability distributions to determine the likelihood of different potential outcomesꦛ—have seen increasing popularity. Exponential growth models are more useful to predict investment returns when the rate of growth is steady.

The Bottom Line

Compounding is considered one of the great miracles of investing, because money grows faster over longer periods♏ due to the returns being calculated not only on the principal but also on the previous returns. This type of exponential growth underscores the idea that starting to invest early in your life has clear ben🌱efits.