Problem 33 - 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. Find the effective rate to the nearest hundredth of a percent if the nominal rate is \(3 \%\) and interest is compounded quarterly.

Step-by-Step Solution

Answer

The effective rate is approximately 3.03%.

Step 1: Understand the Given Information

The problem gives us a nominal interest rate of 3% per annum, which means \( r = 0.03 \). Since the interest is compounded quarterly, \( n = 4 \). The task is to calculate the effective rate using the formula given.

Step 2: Write Down the Formula

The formula to find the effective rate \( R \) is given by:\[R = \left(1 + \frac{r}{n}\right)^{n} - 1\]

Step 3: Substitute the Values into the Formula

Substitute \( r = 0.03 \) and \( n = 4 \) into the formula:\[R = \left(1 + \frac{0.03}{4}\right)^{4} - 1\]

Step 4: Simplify Inside the Parentheses

Calculate \( \frac{0.03}{4} \):\[\frac{0.03}{4} = 0.0075\]Now, substitute back into the formula:\[R = \left(1 + 0.0075\right)^{4} - 1\]

Step 5: Calculate the Power

Calculate the expression inside the parentheses raised to the power of 4:\[(1.0075)^{4} \approx 1.030339\]

Step 6: Subtract 1 to Find the Effective Rate

Subtract 1 from the result of the power calculation to get the effective rate:\[R = 1.030339 - 1 \approx 0.030339\]

Step 7: Convert the Effective Rate to Percentage

Convert the effective rate from decimal form to a percentage by multiplying by 100:\[R \approx 0.030339 \times 100 = 3.0339\]

Step 8: Round to the Nearest Hundredth

Round the effective rate to the nearest hundredth of a percent:\[\approx 3.03\%\]

Key Concepts

Nominal Rate

The nominal rate is essentially the named or stated interest rate on a financial product or loan. When a bank or financial institution declares an annual interest rate of, say, 3%, they are typically referring to the nominal rate. It is called 'nominal' because it doesn’t take into account how often the interest is actually applied or compounded over time, which can significantly affect the total amount of interest accrued.

- **Stated Interest**: The nominal rate is the interest rate agreed upon or advertised by the financial institution.
- **Benchmark**: It serves as a base to calculate other rates, like the effective rate, when the interest is compounded.
While the nominal rate gives a quick sense of the interest you might expect to earn or pay, it is vital to understand how often interest is compounded to get an accurate picture of the actual rate.

Compounded Interest

Compounded interest is the interest on a loan or deposit calculated based on both the initial principal and the accumulated interest from previous periods. Interest can be compounded at intervals such as annually, semi-annually, quarterly, monthly, or even daily. The more frequently interest is compounded, the more interest will be accrued, which directly benefits an investor through higher returns, or costs more to a borrower.

- **Formula Understanding**: The compound interest formula often involves the nominal rate and the compounding periods. In our exercise where interest is compounded quarterly, interest calculations include four periods in a year.
- **Growth Factor**: Compounding causes exponential growth of an invested amount, especially over long periods.
The concept of compounding showcases the potential of investments to grow over time due to multiplying effects.

Annual Rate

The annual rate, often referred to as the effective rate, represents how much an investment grows or how much interest you end up paying over the course of one year when compounding is considered. The effective rate provides a truer picture of the financial product's interest when compared to the nominal rate, making it more useful for comparing different financial products.

- **Effective Calculation**: Using the formula \[ R = \left(1 + \frac{r}{n}\right)^{n} - 1 \], you can determine the effective rate by accounting for how often interest is applied. In the exercise, for a 3% nominal rate compounded quarterly, the effective rate calculated rounds to 3.03%.
- **Comparative Utility**: The annual or effective rate is ideal for comparing different financial investments, as it adjusts the nominal rate to include compounding effects.
This annualized perspective aids in making informed decisions between various savings accounts, loans, or investment opportunities.