Problem 24 - Calculus and its applications

Question

Following the birth of a child, a parent wants to make an initial investment \(P_{0}\) that will grow to \(\$ 30,000\) by the child's 20 th birthday. Interest is compounded continuously at \(6 \% .\) What should the initial investment be?

Step-by-Step Solution

Answer

The initial investment should be approximately $9037.89.

Step 1: Understanding the Problem

We know that the investment needs to grow to $30,000 by the child's 20th birthday with continuous compounding at a rate of 6%. Our task is to calculate the initial investment, denoted as \( P_0 \). We'll use the formula for continuous compounding interest: \[ A = P_0 e^{rt} \], where \( A \) is the amount after time \( t \), \( r \) is the rate, and \( t \) is the time in years.

Step 2: Setting Up the Equation

We assign the values where \( A = 30000 \), \( r = 0.06 \), and \( t = 20 \). Substitute these into the continuous compounding formula: \[ 30000 = P_0 e^{0.06 imes 20} \] which simplifies to \[ 30000 = P_0 e^{1.2} \].

Step 3: Solving the Equation for \( P_0 \)

Rearrange the formula to solve for \( P_0 \): \[ P_0 = \frac{30000}{e^{1.2}} \]. Calculate \( e^{1.2} \) using a calculator, which is approximately 3.3201169.

Step 4: Calculating the Initial Investment

Plug the value of \( e^{1.2} \) into the equation: \[ P_0 = \frac{30000}{3.3201169} \].So, \( P_0 \approx 9037.89 \). Thus, the initial investment should be approximately $9037.89.

Key Concepts

Continuous Compounding

When we talk about continuous compounding, it's important to understand how it differs from other types of interest compounding. Typically, interest is compounded at certain intervals, such as annually, semi-annually, or monthly. However, continuous compounding takes this concept to the extreme, by compounding the interest at every possible moment.

This means that the investment is constantly earning interest on both the initial principal and the accumulating interest. The formula used to calculate the future value of an investment with continuous compounding is \( A = P_0 e^{rt} \), where:

• \( A \) is the amount of money accumulated after n years, including interest.
• \( P_0 \) is the initial principal balance (the amount that you start with).
• \( r \) is the annual interest rate (expressed as a decimal).
• \( t \) is the time in years.
• \( e \) is the base of the natural logarithm, approximately equal to 2.71828.

Continuous compounding is a powerful tool in finance, allowing investments to grow rapidly over time, especially over long periods.

Exponential Growth

Exponential growth is a process that increases quantity over time, at a rate proportional to the current value. It’s a crucial concept in understanding investments with continuous compounding.

The mathematical nature of exponential growth means that as time goes on, the quantity grows faster and faster. For investments, this growth is driven by the interest rate. As the interest is continuously compounded, the amount of interest earned gets larger with each passing moment.

This type of growth can be visualized as a curve that starts off gradual and becomes very steep as time progresses. In our exercise, a 6% interest rate implies that the investment gets multiplied by a factor of \( e^{0.06} \) annually, compounded continuously, which results in a substantially larger growth by the 20th year.

Understanding exponential growth is essential for appreciating why early investments and longer investing periods can significantly increase returns.

Investment Calculation

Investment calculation involving continuous compounding requires a precise method of solving for the initial amount, especially when you know what the future sum should be. In our exercise, the challenge was to determine how much money \( P_0 \) should be invested today to ensure it grows to \\(30,000 in 20 years.

The formula \( A = P_0 e^{rt} \) was rearranged to solve for \( P_0 \):

• First, substitute the known values: \( A = 30000 \), \( r = 0.06 \), \( t = 20 \).
• Ensure that the equation is solved correctly: \( 30000 = P_0 e^{1.2} \).
• Solve for \( P_0 \): \( P_0 = \frac{30000}{e^{1.2}} \).
• Calculate \( e^{1.2} \) which is approximately 3.3201169.
• Finally, compute the initial investment: \( P_0 \approx 9037.89 \).

This means that an initial investment of approximately \\)9037.89 is required. This calculation exemplifies how, with a known future amount, we can calculate the present necessary investment using the continuous compounding model.