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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?
daOnlyBG
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Vahid Garshasbi
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    What have you done so far? Do you know the definitions for each of the interest rates you're asked to find? – daOnlyBG Nov 06 '14 at 21:20

2 Answers2

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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.

paw88789
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  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}

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

  2. 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.

user860374
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  • Thank you Dillon, what if the question gives you the interest rate per a shorter period of time? For example if the interest rate is 5% per 3 months then I shall consider the annual interest rate as 4X5%=20% ?! – Vahid Garshasbi Nov 07 '14 at 22:16
  • Only a pleasure :) – user860374 Nov 07 '14 at 22:16
  • Thank you Dillon, what if the question gives you the interest rate per a shorter period of time? For example if the interest rate is 5% per 3 months then I shall consider the annual interest rate as 4X5%=20% ?! – Vahid Garshasbi Nov 09 '14 at 16:42
  • @VahidGarshasbi . You will still use the same formula 5%, but instead of $n$, you will have $4n$, since if the interest is every 3 months, that means it gets calculated $12/3 =4$ times per year. – user860374 Nov 09 '14 at 18:18
  • Thanks so much for useful information – Vahid Garshasbi Nov 10 '14 at 02:03