Problem 25 - Calculus and its applications

Question

Following the birth of their child, the Irwins want to make an initial investment \(P_{0}\) that will grow to \(\$ 40,000\) by the child's 20 th birthday. Interest is compounded continuously at \(5.3 \% .\) What should the initial investment be?

Step-by-Step Solution

Answer

The Irwins should invest approximately \( \$13,854.84 \).

Step 1: Understand the Problem

We need to find the initial investment amount \( P_0 \) that will grow to \( \$40,000 \) in 20 years with a continuous compounding interest rate of 5.3%. The formula for continuous compounding is \( A = P_0 \cdot e^{rt} \) where \( A \) is the future value, \( r \) is the interest rate, and \( t \) is the time in years.

Step 2: Identify Known Values

From the problem, the future value \( A = \$40,000 \), the interest rate \( r = 0.053 \) as a decimal, and the time \( t = 20 \) years.

Step 3: Set Up the Equation

Insert the known values into the continuous compounding formula: \( 40000 = P_0 \cdot e^{0.053 \cdot 20} \). To find \( P_0 \), we need to solve for it.

Step 4: Calculate \( e^{rt} \)

First, calculate the exponent: \( rt = 0.053 \times 20 = 1.06 \). Then compute \( e^{1.06} \). Using a calculator, we find that \( e^{1.06} \approx 2.888 \).

Step 5: Solve for \( P_0 \)

Rearrange the equation to solve for \( P_0 \): \( P_0 = \frac{40000}{e^{1.06}} \approx \frac{40000}{2.888} \). Calculate this division to find \( P_0 \approx 13854.84 \).

Key Concepts

Compound Interest Formula

Continuous compounding is a powerful concept in finance that allows an investment to grow at a more rapid rate than regular, compounded interest. The formula used for continuous compounding is:
\[ A = P_0 \cdot e^{rt} \]
Here:

• \( A \) is the future value of the investment or the amount of money you aim to achieve after a certain period.
• \( P_0 \) represents the initial investment or the principal amount you start with.
• \( e \) is the base of natural logarithms, which is approximately equal to 2.71828.
• \( r \) is the annual interest rate expressed as a decimal. For example, 5.3% becomes 0.053.
• \( t \) stands for the time the money is invested for, in years.

Using this formula, the compounding occurs continuously, theoretically leading to constant growth. Understanding this formula is crucial because it shows how small changes in the rate or time can significantly impact the future value of your investment.

Initial Investment Calculation

Calculating the initial investment \( P_0 \) is a crucial step in planning your financial future. Suppose you know how much you want your investment to grow to. In that case, you can find out how much you need to invest initially using the continuous compounding formula rearranged as:
\[ P_0 = \frac{A}{e^{rt}} \]
Given the future value \( A \), you can determine \( P_0 \) by dividing \( A \) by \( e^{rt} \). This calculation tells you the exact amount you need to invest today to reach your desired financial goal when your investment grows continuously.For instance, in the Irwins' scenario:

• Future Value \( A = \\(40,000 \)
• Interest Rate \( r = 0.053 \)
• Time \( t = 20 \text{ years} \)

By solving \( P_0 = \frac{40000}{e^{1.06}} \), this division results in an initial investment of approximately \\)13,854.84. This illustrates the power of continuous compounding, allowing relatively modest initial sums to grow substantially over time.

Future Value Determination

Determining the future value of an investment is essential for evaluating different financial options and making informed decisions. The future value \( A \) represents what your investment will grow to over time, considering the effects of interest. For continuous compounding, using:
\[ A = P_0 \cdot e^{rt} \]
helps us understand how initial amounts and interest rates influence expected returns.To determine the future value effectively:

• Identify your initial investment \( P_0 \).
• Know the continuous interest rate \( r \), expressed as a decimal.
• Determine the duration \( t \) over which the investment will grow.

For long-term plans, such as saving for education or retirement, continuous compounding provides a realistic expectation of how money can grow. Knowing how to calculate future value empowers investors to make wiser choices regarding when and where to invest their money.