Question

Prove the basic continual compounded interest equation. Assuming an initial deposit of \(P_{0}\) and an interest rate of \(r\), set up and solve an equation for continually compounded interest.

Short answer

Future value \\(A\\) after time \\(t\\) is \\(A = P_0 e^{rt}\\).

Step-by-step solution

Understand Continual Compounding

Continual compounding means that interest is being added to the principal balance continuously. Unlike other compounding intervals (yearly, monthly, etc.), here, interest grows mathematically as a limit.

Basic Function Setup

Start with the concept that the future value after a short time period, \("]\Delta{t}\), with a principal \(P_0\) and rate \(r\) is \(P_0 + rP_0\Delta{t}\). When compounding is continuous, this approximation leads to the formulation of an exponential growth function.

Derive the Formula

To derive the formula for continuously compounded interest, we need to consider the limit as \(\Delta{t}\) approaches zero. This is expressed as: \[\lim_{n \to \infty} \left(1 + \frac{r}{n}\right)^{nt} = e^{rt}\] Thus, if \(A\) represents the future value, \(A = P_0 e^{rt}\).

Result and Explanation

The resulting formula for the future value of an investment \(P_0\) after time \(t\) with a continual compounding rate \(r\) is expressed as \(A = P_0 e^{rt}\), where \(e\) is the base of the natural logarithm, approximately equal to 2.71828. Therefore, at any given time, you can calculate the accumulated balance with the exponential growth model.

Key concepts

Exponential Growth in Continual Compounding

Exponential growth is a powerful concept that explains how quantities increase at a rate proportional to their current value. In the context of continually compounded interest, this means that the amount of money grows at a consistent rate, leading to more growth as the principal amount increases. When you invest a sum of money with continuous compounding, the growth can be described mathematically by the formula:\[ A = P_0 e^{rt} \]where:- \( A \) is the future value of the investment- \( P_0 \) is the initial principal balance- \( e \) is the base of the natural logarithm (approximately 2.71828)- \( r \) is the interest rate- \( t \) is the time the money is invested forAs the growth is not limited to specific periods (such as monthly or yearly), money in the account continually earns interest on both the initial amount and the accumulated interest from earlier periods. This results in the exponential compounding formula, which maximizes growth and can significantly increase the amount of money over time.

Understanding the Natural Logarithm and its Role

The natural logarithm is a mathematical concept that arises when dealing with exponential growth and continual compounding. The natural logarithm, usually represented as \( \ln(x) \), is the inverse of the exponential function involving \( e \). In the formula for continually compounded interest \( A = P_0 e^{rt} \), \( e \) is a fundamental constant approximately equal to 2.71828. This number serves as the base for natural logarithms. It is particularly useful in financial calculations because it simplifies the mathematics of growth processes that are continuous in nature.For example, if you need to solve for the rate \( r \) in a compounding interest problem, you might take the natural logarithm of both sides:\[ \ln(A) = \ln(P_0 e^{rt}) \]This allows you to linearize the equation by using the property \( \ln(ab) = \ln(a) + \ln(b) \) and solve for \( r \). The natural logarithm thus plays a crucial role in making complex financial calculations more manageable.

Interest Rate in the Context of Continual Compounding

An interest rate in continual compounding represents the effective rate at which money grows over time. In other words, it's the speed at which your investment increases because of compounding. Since every tiny fraction of a second contributes to the increase in value, the interest rate \( r \) becomes pivotal in determining how rapidly your money grows.Unlike discrete interest rates that apply at interval breaks (like annually or quarterly), a continuously compounded interest rate affects the principal balance at every moment. In our key formula \( A = P_0 e^{rt} \), \( r \) is not just a static figure, but a dynamic one that highlights the continuous nature of compounding.To understand its effect, consider small fluctuations in \( r \) — a slight increase leads to a much larger amount due to the exponential nature of the compounding process. Therefore, when examining investments or comparing financial products, understanding the implication of the interest rate in this context is crucial for maximizing returns.