Question

You deposit \(\$ 500\) in an account that pays \(4 \%\) interest compounded yearly. What is the balance after 5 years? after 10 years?

Short answer

The balance in the account after 5 years would be \$608.81 and after 10 years would be \$740.12.

Step-by-step solution

(Convert interest rate into decimal)

The interest rate is 4%, but for the formula, it needs to be in decimal form. So, divide the interest rate by 100 to convert it: \(r = 4/100 = 0.04\).

(Calculate the balance after 5 years)

Substitute P = \$500, r = 0.04, n = 1, and t = 5 into the formula. The formula is \(FV = P (1+ r/n)^{n*t}\). So, \(FV = 500 * (1+0.04/1)^{1*5} = \$608.81\).

(Calculate the balance after 10 years)

Substitute P = \$500, r = 0.04, n = 1, and t = 10 into the formula i.e., \(FV = P (1+ r/n)^{n*t} = 500 * (1+0.04/1)^{1*10} = \$740.12\).

Key concepts

Interest Rate Conversion

Understanding interest rate conversion is crucial when dealing with financial calculations like compound interest. In simple terms, converting an interest rate from a percentage to a decimal is a necessary step because it aligns with the mathematical conventions used in formulas.

For instance, an interest rate of 4% needs to be divided by 100, resulting in 0.04. This step might seem minor, but it's essential for accurate calculations. Always remember that an interest rate of 'x%' is equivalent to 'x/100' as a decimal, which is used in future value calculations.

Future Value Formula

The future value formula is a powerful tool used to calculate the amount of money an investment will grow to over a period at a given interest rate. The formula is expressed as
\[FV = P(1 + \frac{r}{n})^{n*t}\]
where

• \(FV\) is the future value of the investment,
• \(P\) is the initial principal balance,
• \(r\) is the interest rate in decimal form,
• \(n\) is the number of times that interest is compounded per unit t, and
• \(t\) is the time the money is invested for.

When interest is compounded yearly, as in our textbook example, \(n\) equals 1. By substituting the known values into the formula, you can determine the value of your investment after a certain number of years. This emphasizes the importance of understanding the interest rate conversion for proper use of the future value formula.

Exponential Growth

Exponential growth is a pattern of data that shows greater increases over time, creating a curve on a graph that resembles an increasingly steep slope. In financial contexts, it's the process by which an investment grows due to compound interest over time. This growth is not linear; it accelerates because the interest earned in each compounding period is added to the principal, resulting in interest being earned on interest.

Such growth is powerful due to its multiplying effect, as showcased in the balance of a savings account over time. The formula for future value reflects exponential growth,

The Magic of Compounding

as it includes an exponent, \((1 + \frac{r}{n})^{n*t}\), which shows how the initial investment will compound over the periods. Over long periods, this effect is particularly pronounced, demonstrating how potent exponential growth can be for financial investments.