Problem 76 - Elementary and Intermediate Algebra

Question

assume that there are no deposits or withdrawals. Comparison of Compounding Methods. An initial deposit of \(\$ 30,000\) grows at an annual rate of \(8 \%\) for 20 years. Compare the final balances resulting from annual compounding and continuous compounding.

Step-by-Step Solution

Answer

Continuous compounding results in a higher balance (\$148,590) than annual compounding (\$139,800) over 20 years.

Step 1: Understand Annual Compounding

Annual compounding means the interest is calculated and added to the balance once per year. The formula for annual compounding is given by:\[ A = P (1 + r)^t \]where:- \( A \) is the future value of the investment/loan, including interest.- \( P \) is the principal investment amount (initial deposit). Here, \( P = 30000 \).- \( r \) is the annual interest rate (in decimal form). Here, \( r = 0.08 \).- \( t \) is the number of years the money is invested/borrowed for. Here, \( t = 20 \).

Step 2: Calculate Final Balance with Annual Compounding

Using the formula from Step 1, calculate the final balance:\[ A = 30000 (1 + 0.08)^{20} \]Evaluate:\[ A = 30000 (1.08)^{20} \]Now, calculate \((1.08)^{20}\):\[ (1.08)^{20} \approx 4.66 \]Then:\[ A = 30000 \times 4.66 \approx 139800 \]So, with annual compounding, the balance grows to approximately \$139,800.

Step 3: Understand Continuous Compounding

Continuous compounding calculates interest based on the formula that assumes that interest is being calculated and added to the balance an infinite number of times in any given time period. The formula for continuous compounding is:\[ A = P e^{rt} \]where:- \( e \) is Euler’s number, approximately equal to 2.71828.The other variables are the same as in step 1.

Step 4: Calculate Final Balance with Continuous Compounding

Using the continuous compounding formula:\[ A = 30000 e^{0.08 \times 20} \]This simplifies to:\[ A = 30000 e^{1.6} \]Calculate \( e^{1.6} \):\[ e^{1.6} \approx 4.953 \]Then:\[ A = 30000 \times 4.953 \approx 148590 \]So, with continuous compounding, the balance grows to approximately \$148,590.

Step 5: Compare Results

Now compare the two results:- Annual Compounding: \\(139,800- Continuous Compounding: \\)148,590Continuous compounding results in a higher balance after 20 years compared to annual compounding.

Key Concepts

Annual Compounding

When money is left in an account and interest is added to the balance once a year, it grows through a process known as annual compounding. This method uses the formula:

• \(A = P (1 + r)^t \)

To break this down:- \(A\) is the amount of money accumulated after n years, including the interest.- \(P\) represents the principal, which is the initial amount of money, here \(\$30,000\).- \(r\) is the annual interest rate expressed as a decimal, which is \(0.08\) or \(8\%\) in this case.- \(t\) represents the time in years, here \(20\) years.
Every year, the interest is calculated on the current balance, which means that each year's interest is slightly higher than the last. This method is simple but can have a very satisfying result over a long period like 20 years.

Continuous Compounding

In continuous compounding, interest is calculated assuming it is added an infinite number of times continuously over any time period. This might sound complex, but it can be quite powerful in generating returns. The formula for continuous compounding is:

• \(A = P e^{rt}\)

In this equation:- \(e\) stands for Euler's number, a special constant about 2.71828.- All other variables are the same as before: \(P = 30,000\), \(r = 0.08\), and \(t = 20\).
Because the compounding happens continuously, the interest starts working immediately rather than waiting until year's end. Continuous compounding often gives a bit more return than periodic compounding types like annual, because the interest piles up infinitesimally at all times.

Interest Rate Mathematics

Understanding interest rate mathematics is crucial for figuring out how investments grow over time. It involves using different formulas depending on how often interest is compounded. Here are some quick insights:

• Annual Compounding adds interest once per year and uses the formula \(A = P(1 + r)^t\).
• Continuous Compounding accrues interest constantly using the formula \(A = P e^{rt}\).

With both techniques, the core idea is the same: money grows based on an interest percentage over time. However, the method of compounding affects the final amount.
Interest rate calculations involve understanding exponential growth since money grows exponentially rather than linearly. This means small changes in the interest rate or time period can lead to significant differences in accrued interest over time.