Problem 34 - A Graphical Approach to College Algebra

Question

The interest rate stated by a financial institution is sometimes called the nominal rate. If interest is compounded, the actual rate is, in general, higher than the nominal rate, and is called the effective rate. If \(r\) is the nominal rate and \(n\) is the number of times interest is compounded annually, then $$R=\left(1+\frac{r}{n}\right)^{n}-1$$ is the effective rate. Here, \(R\) represents the annual rate that the investment would earn if simple interest were paid. Estimate the effective rate if the nominal rate is \(4.5 \%\) and interest is compounded daily \((n=365)\)

Step-by-Step Solution

Answer

The effective rate is approximately 4.602%.

Step 1: Identify Given Values

We are provided that the nominal rate \( r = 4.5\% = 0.045 \) and the compounding frequency \( n = 365 \). We will use these values to calculate the effective interest rate \( R \).

Step 2: Apply the Formula

The effective interest rate \( R \) is given by the formula: \[ R = \left(1 + \frac{r}{n}\right)^{n} - 1 \] Substituting the given values into the formula: \[ R = \left(1 + \frac{0.045}{365}\right)^{365} - 1 \]

Step 3: Calculate Intermediate Result

First, calculate the daily interest rate: \[ \frac{0.045}{365} \approx 0.000123 \]. Now, the next part of the calculation becomes: \[ 1 + 0.000123 \approx 1.000123 \].

Step 4: Perform Exponentiation

Raise the result of the sum to the power of \( 365 \): \[ (1.000123)^{365} \]. Use a calculator to perform the exponentiation, which yields approximately: \[ (1.000123)^{365} \approx 1.04602 \].

Step 5: Subtract One to Find Effective Rate

Finally, subtract 1 from the previous result to find the effective rate: \[ 1.04602 - 1 = 0.04602 \]. This corresponds to an effective rate of \( 4.602\% \).

Key Concepts

Nominal Rate

The nominal rate is essentially the interest percentage stated by a financial institution without taking compounding into account. It serves as a basic indicator of the interest an investor might expect, yet it doesn't reflect the actual earnings. For instance, a nominal rate of 4.5% suggests that’s what you would earn annually if the interest is calculated only once a year. However, when compounding comes into play—compounding means the interest is calculated periodically and added to the account balance—the real earnings can be different. The nominal rate is a simple representation and doesn’t account for the complexities of compounding. For someone calculating returns or costs on an investment or loan, using just the nominal rate might not provide a complete picture unless it is paired with the compounding frequency.

Effective Rate

The effective rate provides a clearer picture of what one earns or pays annually, considering the compounding of interest. This rate, often referred to as the annual equivalent rate, accounts for how often interest is compounded and provides a more realistic earnings measure. To calculate the effective rate, you would use the formula:\[ R = \left(1 + \frac{r}{n}\right)^{n} - 1 \]Where

• \( R \) is the effective rate,
• \( r \) is the nominal rate, and
• \( n \) is the number of compounding periods per year.

By integrating this method, investors and borrowers can better understand the true return or cost that includes compounding. For example, a nominal rate of 4.5%, when compounded daily, results in a slightly higher effective rate of about 4.602%, illustrating how compounding adds value over time.

Interest Compounded Daily

Compounding is the process in which interest is earned on both the initial principal and the accumulated interest from previous periods. When interest is compounded daily, it means the interest is calculated and added to the balance every day. This frequent compounding can lead to more interest earnings over time compared to less frequent compounding. Using the compounded daily scenario would look like taking the nominal rate and dividing it by 365 (the number of days in a year) to find the daily rate. For someone with a nominal rate of 4.5%, the daily rate would be approximately 0.000123. This small daily addition might seem negligible at first glance but considering it for 365 days significantly increases the total earnings. Therefore, daily compounding benefits the investor by maximizing the interest accumulation, making the whole process a testament to the power of compound interest. It's an essential concept for those aiming for optimized growth in investments and savings.