Question

In Exercises, determine the principal \(P\) that must be invested at interest rate \(r\), compounded continuously, so that \(\$ 1,000,000\) will be available for retirement in \(t\) years. $$ r=7.5 \%, t=40 $$

Short answer

To have \$1,000,000 in 40 years with a continuously compounded interest rate of 7.5%, the required initial investment or principal will approximate to \$73,473.34.

Step-by-step solution

Convert the Percentage Interest Rate to Decimal

Normally, the interest rate is usually given in a percentage format. However, for the calculation, the rate is needed in a decimal format. Therefore, convert the interest rate from percentage to decimal. The conversion is done by dividing the percentage rate by 100, that is, \(r = 7.5\% = 0.075\).

Rearrange the Continously Compounded Interest Formula

We will rearrange the continuously compounded interest formula \(A = Pe^{rt}\) to solve for the principal, \(P\). This can be achieved by dividing both sides of the formula by \(e^{rt}\) to get \(P = \frac{A}{e^{rt}}\).

Substitute Given Values into the Re-arranged Formula

Substitute the given values into our newly rearranged formula: \(t = 40\) years, \(r = 0.075\), and \(A = \$1,000,000\). The calculation will now be \(P = \frac{1,000,000}{e^{(0.075*40)}}\).

Evaluate the Expression

Finally, evaluate the expression using the exponential function to find the value of \(P\). The value of \(P\) will be the principal that needs to be invested currently to have \$1,000,000 after 40 years compounded continuously at a 7.5% interest rate.

Key concepts

Exponential Function

Exponential functions are a key concept in mathematics and finance, especially when dealing with compound interest, such as in the given exercise. An exponential function is a mathematical expression where a constant base is raised to a variable exponent. In finance, this is often represented as \( e^{rt} \), where \( e \) is the base of the natural logarithm, \( r \) is the rate of interest, and \( t \) is time.
This function models situations where growth or decay happens at a continuous rate. For compound interest, it's used to calculate the amount of money accumulated over time when interest is continuously compounded.
Continuous compounding assumes interest is calculated and added to the principal at every possible instant. Hence, it results in the highest possible return on investment over time. In the formula \( A = Pe^{rt} \), \( A \) stands for the amount of money accumulated after \( t \) years, \( P \) is the principal amount invested initially, and \( r \) is the rate of interest in decimal form.

Principal Amount

The principal amount \( P \) is crucial in any investment scenario. It is the initial sum of money invested or loaned before any interest is added. In the context of our exercise, determining the principal means finding out how much money one needs to invest now to reach a specified future goal, using a known interest rate and time period.
The formula for continuous compounding, \( A = Pe^{rt} \), can be manipulated to isolate \( P \), allowing us to solve for this initial investment. The rearranged formula \( P = \frac{A}{e^{rt}} \) is used, where \( A \) is the target amount, \( e^{rt} \) is the exponential growth factor over time, at the specified rate \( r \), and time \( t \). By substituting the known values for \( A \), \( r \), and \( t \), one can calculate the principal amount that needs to be invested to achieve a future value of \$1,000,000 after 40 years.

Interest Rate Conversion

Interest rate conversion is an essential step in financial calculations involving percentages. Interest rates are often expressed as percentages, but for calculations, these need to be converted to decimal form to be used in mathematical formulas.
To convert a percentage rate to a decimal, you simply divide the percentage by 100. For example, a 7.5% interest rate becomes \( 0.075 \) in decimal form. Understanding this conversion is critical because it ensures that the rate is compatible with the mathematical operations involved in exponential functions and other related calculations.
Once converted, this interest rate can be utilized in the continuously compounded interest formula \( A = Pe^{rt} \) to determine how much the investment will grow over a certain period of time. Accurately converting interest rates provides the foundation for calculating expected returns and planning long-term financial goals effectively.