Problem 362 - Calculus Volume 1

Question

True or False? If true, prove it. If false, find the true answer. If a bank offers annual interest of 7.5\(\%\) or continuous interest of \(7.25 \%,\) which has a better annual yield?

Step-by-Step Solution

Answer

False. Continuous interest of 7.25\(\%\) has a better annual yield of about 7.54\(\%\).

Step 1: Calculate Annual Yield with Annual Compounding

The annual compounding yield is straightforward and is given by the interest rate: 7.5\(\%\). This means that at the end of one year, the principal amount will increase by 7.5\(\%\).

Step 2: Calculate Annual Yield with Continuous Compounding

For continuous compounding, we use the formula: \( A = Pe^{rt} \), where \( r = 0.0725 \), \( t = 1 \) year, and \( P = 1 \) (initial principal for calculation purposes). The annual yield is given by \( e^{r} - 1 = e^{0.0725} - 1 \). Calculating this, \( e^{0.0725} \approx 1.0754 \). Thus, the yield is about 7.54\(\%\).

Step 3: Compare the Two Yields

The annual yield for annual compounding is 7.5\(\%\), while for continuous compounding, it is approximately 7.54\(\%\). By comparing these percentages, it is evident that continuous compounding yields a slightly higher return.

Key Concepts

Annual Interest

Annual interest is the percentage increase in your initial investment over one year, due to interest being added at the end of the year. This type of interest is simple to understand. You invest a principal amount and, at the end of the year, you receive a specified percentage of that amount as interest.

• For instance, with an annual interest rate of 7.5%, if you invest $100, you would have $107.50 at the end of the year.
• The interest is calculated just once per year, making it straightforward to predict the growth of your investment.
• This method does not consider any additional compounding within the year.

Understanding annual interest is crucial for making informed financial decisions, as it helps you gauge how much your investment will grow over a specific period without additional contributions.

Continuous Compounding

Continuous compounding means that the interest is calculated and added not just annually or semi-annually, but constantly, at every possible instant of time. It's as if your investment grows continuously.
The formula to compute the amount of an investment with continuous compounding is given by:\[A = Pe^{rt}\]Where:

• \(A\) is the amount of money accumulated after a certain period,
• \(P\) is the principal amount,
• \(e\) is the base of the natural logarithm, approximately equal to 2.71828,
• \(r\) is the annual interest rate (as a decimal), and
• \(t\) is the time the money is invested for, in years.

For example, if you invest \(100 at a continuous interest rate of 7.25%, using the formula with \(t = 1\) year, your investment grows to approximately \)107.54, giving you a yield of 7.54%. This shows that with continuous compounding, even a slightly lower nominal rate than the annual rate could result in a higher yield.
Continuous compounding can be advantageous because it maximizes the potential return by making use of the natural growth of investments as time passes.

Annual Yield

Annual yield is a measure of the total return on an investment over one year. It considers both the interest rate and the compounding method, which affects the growth of the investment.
This concept helps investors understand which investment offers the best returns:

• For annual compounding, the annual yield is typically the same as the interest rate, here 7.5%.
• For continuous compounding, the yield is calculated using the formula \(e^{r} - 1\). Here, it resulted in a yield of about 7.54%.

Comparing annual yields is essential for evaluating investments with different interest rates and compounding methods.
This decision-making process ensures that you make the most profitable investment choices based on potential returns.