Problem 15 - Precalculus with Limits

Question

Determine the time necessary for \(P\) dollars to double when it is invested at interest rate \(r\) compounded (a) annually, (b) monthly, (c) daily, and (d) continuously. r=10 \%

Step-by-Step Solution

Answer

The time required for an initial amount (P) to double when invested at a 10% interest rate, compounded annually, monthly, daily, and continuously can be found by using the formulas pertaining to each method of compounding. These include compound interest formulas for annual, monthly, and daily instances, and the continuous compound interest formula. Solve each respective equation for 't' and get the desired times.

Step 1: Understand Compounded Annually

When the interest is compounded annually, the formula is \(P = P_0 (1 + r)^t\), where \(P\) is the final amount, \(P_0\) is the initial principal balance, \(r\) is the interest rate (in decimal), and \(t\) is time (in years). The target here is to find the time \(t\) required for the principal \(P_0\) to double, i.e., \(P = 2P_0\). So, plugging the values, we get \(2 = (1 + 0.10)^t\). Solving this, we find the appropriate time.

Step 2: Understand Compounded Monthly

When the interest is compounded monthly, we adjust the formula: \(P = P_0 (1 + r/n)^{nt}\) where \(n\) is the number of times interest is applied per time period. Setting \(P = 2P_0\) and solving for \(t\) with \(n = 12\) (since it's monthly), you get the required time.

Step 3: Understand Compounded Daily

Similarly, for daily interest, we set \(n = 365\) in our formula \(P = P_0 (1 + r/n)^{nt}\) and calculate the time \(t\) for \(P = 2P_0\).

Step 4: Understand Compounded Continuously

For continuously compounded interest, we use the formula \(P = P_0e^{rt}\) where \(e\) is the base of natural logarithms (~2.718). So, setting \(P = 2P_0\) and solving for \(t\) in this scenario provides us with the time required when interest is compounded continuously.

Key Concepts

Doubling Time

Understanding doubling time is essential in the world of finance and investments. It refers to the period required for an investment to grow to twice its size. This concept is instrumental when considering how quickly you can expect your money to grow depending on various interest compounding frequencies.
Doubling time can vary significantly based on different compounding methods, such as annually, monthly, daily, or continuously. Each method uses a distinct formula to calculate how fast your initial investment will double in value. Comprehending these formulas allows you to make informed decisions and maximize the growth of your investments.

Annual Compounding

Annual compounding is a straightforward method where interest is calculated and added to the principal only once per year. The formula used to determine the doubled investment is:

• \(P = P_0 (1 + r)^t\)

Given this formula, when you want your investment to double, you set \(P = 2P_0\), known as the doubling condition. For a 10% annual interest rate, you plug the values into the equation to find the time \(t\) when \(2 = (1 + 0.10)^t\).
Using logarithms, this equation can be solved to reveal how many years it will take for your initial investment to double with annual compounding.

Continuous Compounding

Continuous compounding applies the concept of extremely frequent compounding. In theory, it implies the interest is compounded every infinitesimally small moment, resulting in a higher return compared to other compounding methods.
The formula used for continuous compounding is:

• \(P = P_0e^{rt}\)

Here, \(e\) is a mathematical constant approximately equal to 2.718. To find the doubling time, you set \(P = 2P_0\) and solve \(2 = e^{0.1t}\) for \(t\).
This method often results in quicker doubling times if compared to other compounding frequencies, due to its nature of continuously adding earned interest.

Monthly Compounding

Monthly compounding involves calculating and adding interest to the principal twelve times a year, offering more frequent compounding than annually. The formula here is:

• \(P = P_0 (1 + r/12)^{12t}\)

To determine the time required for your money to double, substitute \(P = 2P_0\) into the formula and solve for \(t\). This requires handling \(2 = (1 + 0.10/12)^{12t}\).
Due to the more frequent addition of interest, monthly compounding generally results in a quicker doubling time compared to annual compounding.

Daily Compounding

Daily compounding calculates and adds interest each day, making it the most frequent among these methods besides continuous compounding. The process is governed by:

• \(P = P_0 (1 + r/365)^{365t}\)

To find how long it takes for your investment to double with daily compounding, again, set \(P = 2P_0\) and solve \(2 = (1 + 0.10/365)^{365t}\).
Due to compounding interest extremely frequently, this method is likely to offer a fairly quick doubling time, very close to what continuous compounding provides.