Problem 75 - Elementary and Intermediate Algebra

Question

assume that there are no deposits or withdrawals. Comparison of Compounding Methods. An initial deposit of \(\$ 5,000\) grows at an annual rate of \(8.5 \%\) for 5 years. Compare the final balances resulting from annual compounding and continuous compounding.

Step-by-Step Solution

Answer

Annual compounding: $7,512.50; Continuous compounding: $7,646.50. Continuous compounding gives a higher balance.

Step 1: Understanding the Problem

We have an initial deposit of $5,000 that grows at an annual interest rate of 8.5% for 5 years. We need to determine the final balances for two types of compounding methods: annual and continuous.

Step 2: Annual Compounding Formula

The formula for annual compounding is:\[A = P \left(1 + \frac{r}{n}\right)^{nt}\]Where:- \(A\) is the final amount,- \(P = 5000\) is the initial principal,- \(r = 0.085\) is the annual interest rate,- \(n = 1\) is the number of compounding periods per year,- \(t = 5\) is the time in years.

Step 3: Calculate Final Balance for Annual Compounding

Substitute the values into the annual compounding formula:\[A = 5000 \left(1 + \frac{0.085}{1}\right)^{5} = 5000 \times (1.085)^5\]Compute to find:\[A \approx 5000 \times 1.5025 \approx 7512.50\]The final balance with annual compounding is approximately $7,512.50.

Step 4: Continuous Compounding Formula

The formula for continuous compounding is:\[A = Pe^{rt}\]Where \(e\) is the base of the natural logarithm (approximately 2.71828). The other symbols have the previously defined meanings.

Step 5: Calculate Final Balance for Continuous Compounding

Substitute the known values into the continuous compounding formula:\[A = 5000 \times e^{0.085 \times 5} = 5000 \times e^{0.425}\]Compute to find:\[A \approx 5000 \times 1.5293 \approx 7646.50\]The final balance with continuous compounding is approximately $7,646.50.

Step 6: Comparison of the Results

The final balance with annual compounding is about $7,512.50, while with continuous compounding it is about $7,646.50. Thus, continuous compounding yields a higher final amount.

Key Concepts

Annual Compounding

In the world of finance, annual compounding is a common method used to calculate the interest earned on an investment over time. It involves calculating interest once a year on the initial principal, as well as on any interest that has been added to the principal from previous periods. Hence, it's often referred to as interest on interest.

To compute the final balance with annual compounding, the formula used is:

• \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]

Where:

• \(A\) is the final amount,
• \(P\) is the initial principal (in this case, \(5,000),
• \(r\) represents the annual interest rate (8.5% or 0.085),
• \(n\) is the number of times the interest is compounded per year (which is 1 for annual compounding),
• and \(t\) is the time the money is invested for (5 years).

Using the formula, you calculate how much the original deposit will grow after each year and find that over a period of 5 years, your initial \)5,000 will grow to approximately $7,512.50.

Continuous Compounding

Continuous compounding calculates interest as if it were being added an infinite number of times in a year. This method results in the highest possible returns because the interest is continuously being calculated and added back to the principal. The continuous compounding formula helps in understanding the growth of money if the compounding were to occur constantly.

The formula for continuous compounding is:

• \[ A = Pe^{rt} \]

Here:

• \(A\) is the final amount,
• \(P\) is the initial principal (again, \(5,000),
• \(r\) is the annual interest rate (0.085),
• and \(t\) is the time in years (5 years).

The constant \(e\) is the base of the natural logarithm, approximately equal to 2.71828.

By plugging the values into the formula, you compute that your initial \)5,000 investment will grow to about $7,646.50 after 5 years. This amount is slightly higher than what you'd earn with annual compounding, showcasing the benefits of continuous growth.

Interest Rate Calculations

Interest rate calculations are a key aspect in evaluating the best methods to grow your investments. In the context of this exercise, there are two primary formulas to consider: one for annual compounding and the other for continuous compounding.

With annual compounding, interest is calculated once a year, and each year's earned interest is added back to the principal. This interest-on-interest effect accelerates the growth of the investment.

On the other hand, continuous compounding works on the principle of reinvesting interest instantly and continuously. As such, it results in the maximum possible growth due to the exponential effect of constant reinvestment of interest.

Here's a quick reminder of the formulas:

• Annual Compounding: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
• Continuous Compounding: \[ A = Pe^{rt} \]

While continuous compounding generally leads to a larger return, the differences compared to annual compounding might be marginal depending on the timeframe and interest rate involved. Nonetheless, understanding these calculations empowers better decision-making when it comes to choosing the right investment method.