Problem 33 - Applied Calculus

Question

Find the future value in 8 years of a \(\$ 10,000\) payment today, if the interest rate is \(3 \%\) per year compounded continuously.

Step-by-Step Solution

Answer

The future value is approximately $12,712.

Step 1: Understanding Continuous Compounding

In continuous compounding, the future value of an investment is calculated using the formula: \[ A = P e^{rt} \] where \( A \) is the future value, \( P \) is the principal amount (initial investment), \( r \) is the annual interest rate, and \( t \) is the time in years.

Step 2: Identifying Given Values

From the problem statement, we know \( P = 10,000 \), \( r = 0.03 \), and \( t = 8 \) years. These values will be substituted into the continuous compounding formula to find the future value.

Step 3: Substitute Values into Formula

Substitute the known values into the formula: \[ A = 10,000 \cdot e^{0.03 imes 8} \] This sets up the equation to find the future value \( A \).

Step 4: Calculate the Exponent

Calculate \( 0.03 \times 8 = 0.24 \). This is the exponent that will be used in the formula for continuous compounding.

Step 5: Calculate the Future Value

Now calculate \( A = 10,000 \cdot e^{0.24} \). Using a calculator, you find \( e^{0.24} \approx 1.2712 \). So, \( A = 10,000 \times 1.2712 = 12,712 \).

Step 6: Final Answer

The future value of a $10,000 payment in 8 years with continuous compounding at an interest rate of 3% is approximately $12,712.

Key Concepts

Future Value

Future value is a financial concept that describes the value of an investment at a specific point in the future. It accounts for various factors like the time period, interest rate, and initial investment amount. Essentially, it helps in predicting how much an investment made today will be worth after a certain duration. When dealing with continuous compounding, the calculation considers a scenario where interest is added to the investment an infinite number of times over the duration.

• Future value helps in understanding the growth potential of savings or investments.
• It's a crucial concept in financial planning and assessing investment opportunities.
• This calculation helps determine how much you need to invest now to achieve a particular financial goal.

By grasping the future value concept, one can make informed financial decisions, as it provides a snapshot of what your current investments will yield over time.

Exponential Growth

Exponential growth is a process that occurs when the growth rate of a value is proportional to its current value. In the context of continuous compounding, it demonstrates how investments grow at a rate that continuously builds upon itself.

• This type of growth is characterized by the rapid increase in value over time, particularly noticeable in higher interest rates or longer time periods.
• The mathematical function used to describe this type of growth is the exponential function, often expressed as \( e^{rt} \), where \( e \) is the base of the natural logarithm.
• It reflects the potential compounding effect of interest over time, leading to significant increases in value.

Understanding exponential growth is fundamental in finance, as it helps visualize how powerful compounding can be as a tool for wealth accumulation.

Compound Interest Formula

The compound interest formula is a mathematical expression used to calculate the interest accrued on an initial investment. In cases of continuous compounding, the formula becomes \[A = P e^{rt}\]This formula captures how continuously compounding interest affects the future value of an investment. Here's a breakdown of the key components:

• **\( A \)** is the future value, or the total value of the investment after time \( t \).
• **\( P \)** represents the principal amount, or initial investment.
• **\( r \)** is the annual interest rate expressed as a decimal.
• **\( t \)** is the time the interest is applied, in years.
• **\( e \)** is the mathematical constant approximately equal to 2.71828.

This formula is especially useful when interest is compounded continuously, allowing investors to see how small changes in interest rate or time can lead to substantial differences in future value.