Problem 34 - 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 of the investment is $35,380.

Step 1: Understand the Problem

To find the future value of a continuous compounding investment, we utilize the formula for continuous compounding: \( FV = PV \times e^{rt} \) where \( FV \) is the future value, \( PV \) is the present value, \( r \) is the annual interest rate as a decimal, \( t \) is the time in years, and \( e \) is the base of the natural logarithm.

Step 2: Identify Given Values

From the problem, we know: \( PV = 20,000 \), \( r = 0.038 \) (convert the percentage to a decimal), and \( t = 15 \) years. We aim to find the future value \( FV \).

Step 3: Apply the Formula

Substitute the known values into the continuous compounding formula: \[ FV = 20,000 \times e^{0.038 \times 15} \] Simplify the expression \( 0.038 \times 15 = 0.57 \).

Step 4: Calculate Exponential

Calculate \( e^{0.57} \). Using a calculator, \( e^{0.57} \approx 1.7690 \).

Step 5: Final Calculation

Now, multiply the result by the present value: \[ FV = 20,000 \times 1.7690 = 35,380 \] Therefore, the future value in 15 years is \( 35,380 \).

Key Concepts

Future Value

The future value is a crucial concept in finance and investment, as it indicates how much money will grow over a certain period, given a specific interest rate. In the context of continuous compounding, it represents the total amount of money an investment will yield after it has accrued interest continuously over time. Understanding future value helps investors make informed decisions about the potential returns of various financial options.

In continuous compounding, the future value (\( FV \)) is calculated using the formula:

• \( FV = PV \times e^{rt} \)

This formula shows that to find the future value, you need to know three things:

• Present Value (\( PV \)): The initial amount of money invested.
• Annual Interest Rate (\( r \)): Presented as a decimal.
• Time (\( t \)): The period of time the money is invested for, in years.

The exponential function (\( e \)) is a constant approximately equal to 2.71828, and it represents the base of the natural logarithms. By understanding and applying this formula, one can calculate the future value of investments under continuous compounding.

Present Value

Present Value is the starting point of any investment calculation. It refers to the current value of a sum of money or asset as it stands today, before any interest is applied or earned over time. The present value is fundamental when planning investments as it allows investors to determine how much they need to invest now to achieve a desired future sum given a specific interest rate.

Calculating present value under continuous compounding involves considering the present amount of money, and how this money will grow over time due to interest. Essentially, it provides the initial value input in the future value equation. For instance, in our exercise, the present value is given as $20,000, and this is the seed amount from which future computations begin.

Investors can use present value to decide the feasibility of investments or compare different investment opportunities. The larger the present value, or the smaller the time to the future value, the better the investment potentially is.

Exponential Functions

Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent. These functions are essential in finance, especially in calculating compound interest and future value. Continuous compounding specifically relies on exponential functions due to its nature of applying interest at every possible instant.

The exponential function in the formula for continuous compounding is represented by the symbol \( e \). It is a mathematical constant approximately equal to 2.71828, known as Euler's number. The continuous growth model uses this base because it naturally models situations of constant percentage growth over time, making it ideal for financial applications where money compounds continuously.

In our exercise, the term \( e^{0.57} \) encapsulates the concept of continuous growth over 15 years at an annual rate of 3.8%. The outcome of this calculation, approximately 1.7690, illustrates the growth factor applied to the present value to yield the future value. Knowing how to interpret and compute exponential functions allows investors to predict and understand the full scope of their investments under continuous compounding.