annual and effective annual interest rate

Question

Can anyone help me to find the answer to the question below? I am taking the Engineering Economics exam and need to be sure about the correctness of my answer. Here is the question:

Smart Visa, Principal Card, and Acrobat Express are 3 credit card companies that charge different interest on overdue balances. Smart Visa charges 24% compounded daily, Principal Card charges 25% compounded weekly, and Acrobat Express charges 26% compounded monthly. (hint: 1 year could be 365 days or 52 weeks).

1. What is the effective annual interest rate charged by each company?
2. What is the effective semi-annual interest rate charged by each company?
3. Which credit card company would you prefer?
4. How much should the interest rate be for Smart Visa in order to break-even with Principal Card?

Answer 1

Hint: The effective annual rate may be computed as $\left(1+\frac{r}{n}\right)^n-1$ where $r$ is the stated annual rate, and $n$ is the number of compoundings per year.

Answer 2

1. Using the formula for annual effective rate: $\displaystyle i_{effect}= \bigg(1+ \frac{i}{n}\bigg)^n-1$, we may compute the following.

Smart Visa: \begin{align}i_{eff} &= \bigg(1 + \frac{0.24}{365} \bigg)^{365}-1 \\ &=0.271149 \\ &\approx 0.27 \end{align}

Principle Card: \begin{align}i_{eff} &= \bigg(1 + \frac{0.25}{52} \bigg)^{52}-1 \\ &=0.283256 \\ &\approx0.28 \end{align}

Acrobat Express: \begin{align}i_{eff} &= \bigg(1+ \frac{0.26}{12} \bigg)^{12} -1 \\ &= 0.293334 \\ &\approx0.29\end{align}

2. This is computed in exactly the same way, but instead of using $n$, we use $2n$.

3. After calculating everything, you want to compare them in terms of interest rates annually and semi-annually. Since you have to pay the interest, you would obviously wish to pick the one with the lowest interest rate.