Problem 30 - Precalculus with Calculus Previews

Question

If (7) is solved for \(P\), that is, \(P=S e^{-r t}\), we obtain the amount that should be invested now at an annual rate \(r\) of interest in order to be worth \(S\) dollars after \(t\) years. We say that \(P\) is the present value of the amount \(S\). What is the present value of \(\$ 100,000\) at an annual rate of \(3 \%\) compounded continuously for 30 years?

Step-by-Step Solution

Answer

The present value is approximately $40,656.97.

Step 1: Understanding the Formula

The formula given is: \(P = S e^{-r t}\). Here, \(P\) represents the present value, \(S\) is the future value, \(r\) is the annual interest rate, and \(t\) is the time in years.

Step 2: Identify Given Values

We are given \(S = 100,000\), \(r = 0.03\) (since 3% as a decimal is 0.03), and \(t = 30\) years.

Step 3: Plug in Values into the Formula

Substitute the given values into the formula: \[P = 100,000 \times e^{-0.03 \times 30}\]

Step 4: Calculate the Exponential

Calculate the exponent: \(-0.03 \times 30 = -0.9\). Then calculate the exponential function: \(e^{-0.9}\).

Step 5: Find the Exponential Value

Use a calculator to find \(e^{-0.9} \approx 0.40656966\).

Step 6: Calculate Present Value

Multiply the future value \(100,000\) by the exponential result: \[P = 100,000 \times 0.40656966 \approx 40,656.97\]

Step 7: Conclusion

Therefore, the present value \(P\) is approximately $40,656.97.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where the variable appears in the exponent. They are widely used in growth and decay scenarios. In finance, exponential functions are essential for understanding compound interest. This is because money grows exponentially over time when it is invested at a certain interest rate.
In the formula for present value calculation, \(P = S e^{-r t}\), the exponential function \(e^{x}\) is used. This function is inherently linked to the mathematical constant "e," which is approximately equal to 2.71828. The exponential function \(e^{x}\) becomes crucial in continuous compounding, since it models how amounts grow continuously rather than at discrete intervals.

• \(e^{x}\) increases rapidly for positive \(x\).
• For negative \(x\), such as \(e^{-0.9}\), it decreases, which corresponds to discounting in financial contexts.

Understanding exponential functions helps us see why money grows at a certain rate when compounded continuously, rather than just at certain points (e.g., yearly). As such, the concept allows us to accurately calculate present and future values.

Continuous Compounding

Continuous compounding involves the concept of constantly reinvesting earned interest back into the principal amount. This means interest earns interest continuously rather than at discrete intervals, like monthly or annually. In mathematical terms, it leads to using the exponential function in calculations.
The formula for continuous compounding is related to the expression \(A = Pe^{rt}\). However, when we reverse this to find present value, we use \(P = S e^{-rt}\), as seen in our problem. Here, the exponential function \(e^{-rt}\) serves to discount the future value \(S\) back to its present value.

• Continuous compounding results in slightly higher returns than compounding at discrete intervals.
• The more frequently interest is compounded, the closer it mimics continuous compounding.

This method of compounding is theoretical but serves as a useful model for understanding interest rate behavior and maximizing investment returns over time.

Interest Rate Calculations

Interest rate calculations are fundamental to understanding how investments grow over time. They help you determine how much an amount will be worth in the future or what you need to invest now to reach a future financial goal.
When we calculate interest rates, especially with continuous compounding, we usually express the annual rate as a decimal. For example, 3% becomes 0.03. The interest rate in our present value formula \(P = S e^{-rt}\) shows how we discount future amounts.

• With continuous compounding, small differences in the interest rate can lead to significant changes in the amount.
• The formula helps make direct comparisons between different investment options by seeing how rates affect compound growth.

By understanding these calculations, you can make informed financial decisions, choosing the right investments and seeing exactly how they can appreciate over time.