Question

Explain and discuss how exponential functions effects Exponential growth or decline of money.

Answer 1

Solution:-

Exponential growth is a pattern of data that shows greater increases with passing time, creating the curve of an exponential function. For example, if a population of mice doubles every year starting with two in the first year, the population would be four in the second year, 16 in the third year, 256 in the fourth year, and so on. The population is growing to the power of 2 each year in this case

KEY TAKEAWAYS:

• Exponential growth is a pattern of data that shows sharper increases over time.
• In finance, compounding creates exponential returns.
• Savings accounts with a compounding interest rate can show exponential growth.

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 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 interest rate, you will earn $100 per year. The amount of interest paid will not change as long as no additional deposits are made.

If the account carries a compound interest rate, however, you will earn interest on the cumulative account total. Each year, the lender will apply the interest rate to the sum of the initial deposit, along with 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 chart, this curve starts slowly, remains nearly flat for a time before increasing swiftly to appear almost vertical. It follows the formula:

V = S * (1 + R) ^ T

The current value, V, of an initial starting point subject to exponential 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, stock market returns do not smoothly follow long-term averages each year.

Other methods of predicting long-term returns—such as the Monte Carlo simulation, which uses probability distributions to determine the likelihood of different potential outcomes—have seen increasing popularity. Exponential growth models are more useful to predict investment returns when the rate of growth is steady.

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