Question

Suppose that a sum \(S_{0}\) is invested at an annual rate of return \(r\) compounded continuously. $$ \begin{array}{l}{\text { (a) Find the time } T \text { required for the original sum to double in value as a function of } r .} \\ {\text { (b) Determine } T \text { ifr }=7 \% \text { the achieved if the initial investment is to double in }} \\ {\text { glears. }}\end{array} $$

Short answer

Question: Determine the time T required for the original sum to double in value at an annual rate of return of 7% compounded continuously. Answer: Approximately 9.9 years.

Step-by-step solution

Set up the equation to find the time T

We want the original sum to double, which means that the final amount \(A\) will be \(2S_0\). We can plug these values into the formula for continuous compound interest to set up the equation as: $$ 2S_0 = S_0e^{rT} $$

Solve for T as a function of r

Now we need to find the value of \(T\) for the given equation. To do so, we can first cancel out the initial investment \(S_0\) from both sides, as it should not affect the time required to double the investment: $$ 2 = e^{rT} $$ Take the natural logarithm of both sides to isolate \(T\): $$ \ln(2) = \ln(e^{rT}) $$ Utilize the property \(\ln(a^b) = b \cdot \ln(a)\) to get: $$ \ln(2) = rT \cdot \ln(e) $$ Since \(\ln(e) = 1\), the equation becomes: $$ T = \frac{\ln(2)}{r} $$

Substitute r = 7% to find T

We are given that \(r = 7\%\). To find the time required to double the investment at this rate, we substitute \(r = 0.07\) into the formula from step 2: $$ T = \frac{\ln(2)}{0.07} $$ Calculate the value of T: $$ T \approx 9.9 \text{ years} $$ In conclusion, the time required for the original sum to double in value at an annual rate of return of \(7\%\) compounded continuously is approximately \(9.9\) years.

Key concepts

Rate of Return

Understanding the rate of return is fundamental when you're learning about investments. It tells you how much money you earn on an investment as a percentage of the initial investment. When an investment provides a return every year, this is known as the annual rate of return. Continuous compound interest takes this a step further, where the interest earned is calculated and added to the investment balance continuously, at every moment.

This means that the interest is not just added at the end of the year, but constantly, allowing your investment to grow at every instant. It's an acceleration of the concept that money can earn money. Mathematically, the formula to calculate the future value of an investment compounded continuously is given by: \[A = S_0 e^{rT}\] where \(A\) is the amount of money accumulated after time \(T\), including interest, \(S_0\) is the principal sum (the initial amount of money), \(e\) is Euler's number (approximately equal to 2.71828), and \(r\) is the annual interest rate.

Natural Logarithm

The natural logarithm, commonly represented by \(\ln(x)\), is a mathematical function that is the inverse of the exponential function with base \(e\). When you raise \(e\) to a power, you get a number, and taking the natural logarithm of that number gives you back the power to which \(e\) was raised.

Using our previous formula, \(2S_0 = S_0e^{rT}\), when you want to isolate the variable \(T\), you must use the natural logarithm to 'undo' the exponential function. This process allows us to solve for time without the complexity of dealing with the exponentials directly. Indeed, a natural logarithm is a powerful tool in finance, especially when considering continuous compound interest.

Exponential Growth

Exponential growth refers to an increase in quantity that is proportional to the current amount. In finance, continuous compound interest is a prime example of exponential growth as the amount of interest earned grows exponentially with the principal sum.

The magic of exponential growth is that it can seem slow at the start but can rapidly lead to large increases over time. This principle is especially relevant when talking about investments, as the continual reinvestment of returns can substantially increase the future value of the initial investment. Mathematically, this type of growth is often represented by the exponential function \(e^{rt}\), where \(e\) represents the base of the natural logarithm and underscores the relationship between exponential growth and continuous compounding.

Time to Double Investment

A common financial goal is to know: 'How long will it take for my investment to double?' The 'Rule of 72' offers a simple way to estimate this time for compound interest that accrues annually by dividing 72 by the annual percentage rate. However, for continuous compounding, the time to double can be precisely calculated using the natural logarithm.

As illustrated in the exercise, we find the time to double (\(T\)) by isolating \(T\) in the formula \(2 = e^{rT}\) to get \(T = \frac{\ln(2)}{r}\). This formula is indispensable because it provides an exact figure instead of an estimate. For example, with a 7% rate of return, it would take approximately 9.9 years for an investment to double when compounded continuously, demonstrating not just the power of compound interest but also the infinite potential of patience and time in investing.