Problem 32 - 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 $35,360.

Step 1: Identify the Formula for Continuous Compounding

The formula for calculating the future value with continuous compounding is given by:\[ FV = PV \times e^{rt} \]where \( FV \) is the future value, \( PV \) is the present value or initial investment, \( r \) is the annual interest rate (as a decimal), and \( t \) is the time in years. \( e \) is the base of the natural logarithm, approximately equal to 2.718.

Step 2: Substitute the Known Values into the Formula

Substitute the given values into the continuous compounding formula:- \( PV = 20,000 \)- \( r = 0.038 \) (since \( 3.8\% = 0.038 \))- \( t = 15 \)The equation becomes:\[ FV = 20,000 \times e^{0.038 \times 15} \]

Step 3: Calculate the Exponent

Compute the exponent \( 0.038 \times 15 \):\[ 0.038 \times 15 = 0.57 \]

Step 4: Compute \( e^{0.57} \)

Use a calculator to find the value of \( e^{0.57} \). Approximating, we find:\[ e^{0.57} \approx 1.768 \]

Step 5: Calculate the Future Value

Substitute \( e^{0.57} \approx 1.768 \) into the formula to find the future value:\[ FV = 20,000 \times 1.768 \]Calculate the result:\[ FV \approx 35,360 \]

Key Concepts

Future Value Calculation

When we speak about future value, we are discussing the amount of money an investment will grow to over a certain period. This value depends on the rate at which the investment grows. In continuous compounding, the future value can be calculated using the formula:

• \[ FV = PV \times e^{rt} \]
• Here, \( FV \) is the future value, \( PV \) is the present value or initial investment, \( r \) is the annual interest rate (as a decimal), and \( t \) is the time in years.

This formula employs exponential growth, making it essential for calculations where the compounding is continuous. By continuously compounding the interest, the investment theoretically earns interest at each instant of time. This results in a higher future value compared to other compounding methods like annually or quarterly.

Exponential Growth

Exponential growth is a process where a quantity increases by a constant percentage over equal time periods. In finance, this is how investments that earn compound interest grow over time. With continuous compounding, investments increase their value at a rate proportional to their current value.

The use of the number \( e \) in the future value formula represents this exponential growth. The constant \( e \) is about 2.718 and is a cornerstone of natural exponential functions. When calculating exponential growth, determining \( e^{rt} \) is key. Plugging in our specific values, \( e^{0.038 \times 15} \), the growth factor we get is approximately 1.768. This means the original investment grows by about 76.8% over 15 years with continuous compounding at an interest rate of 3.8%.

Interest Rate

The interest rate is a crucial component in determining both the rate of growth and the final future value of an investment. When it is "compounded continuously," this means that interest is added repeatedly at every possible moment.

• The interest rate is often expressed as a percentage, and in our calculations, it must be converted to a decimal.
• For example, 3.8% becomes 0.038.

An understanding of how the interest rate affects growth can help in making better financial decisions. A higher interest rate will lead to a greater future value in the long run, amplifying the benefits of continuous compounding - essentially letting both time and the powerful effect of compounding work in your favor.