Problem 33 - Applied Calculus

Question

Find the future value in 15 years of a $$\$ 20,000$$ payment today, if the interest rate is \(3.8 \%\) per year compounded continuously.

Step-by-Step Solution

Answer

The future value is approximately $35,392.60.

Step 1: Understand Continuous Compounding Formula

First, we need to recognize that when interest is compounded continuously, we use the formula: \( FV = PV \times e^{rt} \), where \( FV \) is the future value, \( PV \) is the present value, \( r \) is the interest rate (expressed as a decimal), \( t \) is the time in years, and \( e \) is the base of the natural logarithm (approximately 2.71828).

Step 2: Identify and Assign Values

Given values are: \( PV = 20,000 \), \( r = 3.8 \% = 0.038 \), and \( t = 15 \) years. These values will be substituted into the continuous compounding formula.

Step 3: Substitute the Values into the Formula

Substitute the given values into the formula: \( FV = 20,000 \times e^{0.038 \times 15} \).

Step 4: Calculate the Exponent

Compute the exponent: \( 0.038 \times 15 = 0.57 \). So, the expression becomes: \( FV = 20,000 \times e^{0.57} \).

Step 5: Calculate \( e^{0.57} \)

Find the value of \( e^{0.57} \). Using a calculator, \( e^{0.57} \approx 1.76963 \).

Step 6: Compute the Future Value

Multiply the present value by \( e^{0.57} \): \( FV = 20,000 \times 1.76963 = 35,392.60 \).

Step 7: State the Final Result

The future value of the $20,000 after 15 years with continuous compounding at an interest rate of 3.8% is approximately $35,392.60.

Key Concepts

Future Value Calculation

Calculating the future value is an essential concept in finance. It helps us understand how much an investment will grow over time. The future value (FV) represents the amount an investment will be worth at a specified time in the future. To find this, we consider the amount initially invested, known as the present value (PV), the interest rate, and the duration of time the money is invested. In exercises involving continuous compounding, the future value is computed using the formula:\[ FV = PV \times e^{rt} \]where:

• \(FV\) is the future value we want to calculate,
• \(PV\) is the present value or initial investment, which is \(20,000 in the exercise,
• \(r\) is the annual interest rate expressed as a decimal, which is 0.038 in this case,
• \(t\) is the time in years that the money is invested, which is 15 years here,
• \(e\) is the base of the natural logarithm, approximately 2.71828.

By substituting the given values into this formula, we can calculate how much the initial \)20,000 will grow over a period of 15 years.

Exponential Growth

Exponential growth is a fundamental concept in understanding how investments can grow over time. This type of growth occurs when the value of an investment increases at a rate proportional to its current value, leading to growth that becomes faster as time goes on. In the context of continuous compounding, this is seen in the formula \[ FV = PV \times e^{rt} \]Here, the term \(e^{rt}\) represents the exponential growth factor. As time passes, the money invested keeps growing at a rate determined by both the interest rate and the period of investment. This form of growth is potent because not only does the original investment earn interest, but the interest itself also earns interest as the timeline extends.For example, in our exercise, the investment at 3.8% interest starts to grow quicker over the 15 years due to the compound effect. The calculation of \(e^{0.57}\), turning out to be approximately 1.76963, plays a critical role in illustrating the exponential growth over this timeframe.

Compounding Interest

Compounding interest is the process whereby interest is added to the initial principal, and then new interest is calculated based on this new amount. Continuous compounding is a special case where interest is compounded at every conceivable instant. This leads to a slightly more robust future value compared to other compounding frequencies (such as annually or semi-annually).When interest is compounded continuously, it means that the compounding effect happens without any breaks. This approach utilizes the mathematical constant \(e\), allowing even small rates to accumulate into significant growth over time. This is evident in the scenario we evaluated, where a \(20,000 investment grows to \)35,392.60 over 15 years with a continuous 3.8% rate.Some benefits of continuous compounding include:

• Maximizing growth: The constant compounding results in the highest possible growth for an investment of given rate and time.
• Understanding potential growth: It's a valuable model for showing the potential of investment growth when looking at long-term strategies.

In real world applications, while true continuous compounding isn't commonly practiced directly, it helps finance professionals and students alike to appreciate the impact of continuous growth calculations.