Problem 29 - Calculus Early Transcendentals: Pearson New International Edition

Question

How long does it take money to double in value for the specified interest rate? (a) \(6 \%\) compounded monthly (b) \(6 \%\) compounded continuously

Step-by-Step Solution

Answer

(a) 11.58 years for monthly compounding; (b) 11.55 years for continuous compounding.

Step 1: Understanding the Problem

We need to determine the time it takes for an investment to double in value under different compounding methods. We will use the formula for compound interest to solve this problem.

Step 1: Doubling Time for Compound Interest

For monthly compounding, use the formula \( A = P \left(1 + \frac{r}{n}\right)^{nt} \), where \( A \) is the final amount, \( P \) is the principal amount, \( r \) is the annual interest rate, \( n \) is the number of times interest is compounded per year, and \( t \) is the time in years. For doubling, \( A = 2P \).

Step 2: Set Up the Equation for Monthly Compounding

Using the specific values, we have \( 2P = P \left(1 + \frac{0.06}{12}\right)^{12t} \). Simplify to \( 2 = \left(1 + \frac{0.06}{12}\right)^{12t} \).

Step 3: Solve for \( t \) (Monthly Compounding)

Take natural logarithm on both sides to isolate \( t \): \( \ln(2) = 12t \ln\left(1 + \frac{0.06}{12}\right) \). Solve for \( t \): \( t = \frac{\ln(2)}{12 \ln\left(1 + \frac{0.06}{12}\right)} \approx 11.58 \text{ years} \).

Step 4: Doubling Time for Continuous Compounding

For continuous compounding, use the formula \( A = Pe^{rt} \), where \( e \) is the base of the natural logarithm. For doubling, \( A = 2P \), leading to \( 2P = Pe^{0.06t} \). Simplify to \( 2 = e^{0.06t} \).

Step 5: Solve for \( t \) (Continuous Compounding)

Take natural logarithm on both sides: \( \ln(2) = 0.06t \). Solve for \( t \): \( t = \frac{\ln(2)}{0.06} \approx 11.55 \text{ years} \).

Key Concepts

Doubling Time

Doubling time refers to the duration required for an investment or value to grow to twice its original size. This concept is key to understanding how different compounding interest mechanisms work to amplify growth. The Rule of 72 is a simple way to estimate the doubling time: divide 72 by the annual interest rate percentage. For instance, at a 6% interest rate, an approximate doubling time would be 72/6 = 12 years. However, more precise calculations consider how interest is compounded.

• Monthly compounding and continuous compounding have unique formulations.
• Both methods provide precise predictions but yield slightly different results.

Considering the compounding frequency is essential for accurate financial planning.

Continuous Compounding

Continuous compounding is a special scenario where interest is calculated and added at every possible instant. This is achieved through the mathematical constant 'e' (approximately 2.71828), which represents the base of the natural logarithm. The formula for continuous compounding is: \[ A = Pe^{rt} \]where:

• \(A\) is the final amount.
• \(P\) is the principal amount.
• \(r\) is the annual interest rate as a decimal.
• \(t\) is the time in years.

In our exercise, to find the doubling time, we set:\[ 2P = Pe^{0.06t} \]which simplifies to:\[ 2 = e^{0.06t} \]By isolating \(t\), through natural logarithms, the result is:\[ t = \frac{\ln(2)}{0.06} \approx 11.55 \text{ years} \]This duration is slightly shorter than monthly compounding, illustrating how continuous compounding maximizes growth.

Monthly Compounding

In monthly compounding, interest is calculated and added to the principal monthly, thus increasing the total amount more frequently. This compounding type utilizes the formula:\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]where:

• \(A\) is the final amount.
• \(P\) is the principal amount.
• \(r\) is the annual interest rate as a decimal.
• \(n\) is the number of compounding periods per year (12 for monthly).
• \(t\) is the time in years.

For the doubling problem, substituting the values gives:\[ 2P = P \left(1 + \frac{0.06}{12}\right)^{12t} \]which simplifies to:\[ 2 = \left(1 + \frac{0.06}{12}\right)^{12t} \]Using logarithms to solve for \(t\):\[ t = \frac{\ln(2)}{12 \ln\left(1 + \frac{0.06}{12}\right)} \approx 11.58 \text{ years} \]Although both continuous and monthly compounding achieve similar results, monthly compounding slightly lengthens the doubling period, due to less frequent interest accumulation. Understanding these differences is crucial for financial projections and decision making.