Question

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

Short answer

It takes approximately 11.58 years for monthly compounding and 11.55 years for continuous compounding.

Step-by-step solution

Understand the Problem

We need to determine the time required for an investment to double in value at a 6% interest rate. We'll solve this independently for monthly compounding and continuous compounding.

Determine the Formula for Compounding Monthly

When an investment is compounded monthly, we use the formula \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]where \(A\) is the future value, \(P\) is the present value (or principal), \(r\) is the annual interest rate (expressed as a decimal), \(n\) is the number of compounding periods per year, and \(t\) is the time in years. Set \(A = 2P\) because the investment needs to double.

Apply the Formula for Compounding Monthly

Using the formula: \( A = P \left(1 + \frac{0.06}{12}\right)^{12t} \), and setting \( A = 2P \), we get:\[2P = P \left(1 + \frac{0.06}{12}\right)^{12t}\]\[2 = \left(1 + \frac{0.06}{12}\right)^{12t}\]Refine to find \(t\) using logarithms:\[\log 2 = 12t \cdot \log \left(1 + \frac{0.06}{12}\right)\]Solve for \(t\):\[t = \frac{\log 2}{12 \cdot \log \left(1 + \frac{0.06}{12}\right)}\]

Calculate the Time for Compounding Monthly

Calculate the numerical value:\[\log 2 \approx 0.3010\quad \text{and}\quad \log \left(1 + \frac{0.06}{12}\right) \approx 0.0026\]Therefore:\[t = \frac{0.3010}{12 \cdot 0.0026} \approx 11.58\, \text{years}\]

Determine the Formula for Continuous Compounding

When a rate is compounded continuously, we use the formula \[ A = Pe^{rt} \]where \(e\) is the base of natural logarithms. Set \(A = 2P\) to find out how long it takes for the investment to double.

Apply the Formula for Continuous Compounding

Using the formula: \( 2P = Pe^{0.06t} \), we simplify:\[2 = e^{0.06t}\]Take the natural logarithm of both sides to find \(t\):\[\ln 2 = 0.06t\]Solve for \(t\):\[t = \frac{\ln 2}{0.06}\]

Calculate the Time for Continuous Compounding

Calculate the numerical value:\[\ln 2 \approx 0.6931\]Thus:\[t = \frac{0.6931}{0.06} \approx 11.55\, \text{years}\]

Key concepts

Logarithms

Logarithms are incredibly useful when it comes to solving exponential equations, particularly in finance with compound interest problems. The logarithm essentially helps to "undo" an exponentiation, allowing us to solve for the unknown variable.

For example, in problems where we need to determine how long it takes for an investment to double, we might start with an equation where both sides are exponentiated. To isolate the variable representing time, we apply a logarithm to both sides. This process moves the exponent down to a coefficient, making it simpler to solve for the variable.

A common version used in finance is the natural logarithm, often represented as "ln," which is based on the constant "e" (approximately 2.718). This is handy in continuous compounding scenarios. For instance, when we see formulas like \(e^{rt}\), we use the natural logarithm to find the time \(t\) when the amount \(A\) is doubled.

• Logarithms convert multiplication into addition, making it easier to handle exponents.
• In the formula \(\log 2 = 12t \cdot \log \left(1 + \frac{0.06}{12}\right)\), logarithms help to solve for \(t\) easily.
• They provide insight into the growth patterns of investments under exponential conditions.

Continuous Compounding

Continuous compounding is an advanced interest compounding technique where the frequency of compounding is essentially infinite. Instead of earning interest quarterly, monthly, or even daily, interest is compounded constantly, adding the smallest possible quantities of interest at every moment.

To achieve this, the formula \(A = Pe^{rt}\) is used, where \(P\) is the principal, \(r\) is the rate, \(t\) is the time, and \(A\) is the amount of money accumulated after time \(t\). This formula utilizes the mathematical constant \(e\), which unfolds naturally in various natural growth and decay processes.

Take the scenario of doubling your investment with continuous compounding: by setting \(2P = Pe^{0.06t}\), we can simplify and solve for \(t\). By taking the natural logarithm (\(\ln\)) of both sides, we obtain \(\ln 2 = 0.06t\), and solve \(t = \frac{\ln 2}{0.06}\). As calculations show, this translates to about 11.55 years.

• This method aligns closely with natural processes, representing constant growth without discrete breaks.
• Continuous compounding often yields slightly more favorable returns compared to other compounding methods like monthly or yearly compounding.

Monthly Compounding

Monthly compounding refers to the scenario where interest is calculated and added to the principal balance on a monthly basis. This increases the principal each month, allowing for subsequent interest calculations to be made on this new, higher principal.

The formula used is \(A = P \left(1 + \frac{r}{n}\right)^{nt}\) where \(n\) is the number of compounding periods per year. For monthly compounding, \(n\) would be 12. An example calculation might be using \(r = 0.06\) and \(n = 12\), which means each month's rate is \(\frac{0.06}{12}\).

The steps to solve such a problem involve setting \(A = 2P\) to find out how long money takes to double, leading to \(\left(1 + \frac{0.06}{12}\right)^{12t} = 2\). Solving this with logarithms gives us the time \(t\) it will take. In this specific situation, the result shows time \(t\) to be roughly 11.58 years.

• Monthly compounding results in more interest accrued over a year compared to simpler yearly compounding at the same rate.
• It offers a more frequent accumulation of interest, which can benefit many savers and investors.

Time Value of Money

The principle of the time value of money (TVM) is fundamental in finance, capturing the concept that money available today is worth more than the same amount in the future due to its potential earning capacity. This notion is essential when dealing with interest, investments, and compounding.

Money today can be invested to earn interest and increase in value over time, which is why time calculation plays a crucial role in determining how beneficial an investment can be. The concept of TVM is often encapsulated in compound interest formulas because they allow us to project the growth of money over time, considering both principal and interest earned.

When evaluating an investment plan, knowing how long it will take for the money to double plays directly into the TVM. For instance, when you determine it takes around 11.58 years for an investment to double with monthly compounding, it signifies how powerful the interest rate and frequency of compounding are in enhancing the value of your money.

• TVM dictates that today's money can grow to be worth more tomorrow through investments.
• It forms the basis for understanding different compounding methods and their respective financial impacts.