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11-CS-1 Engineering Economics · May 2014

Question 1 of 5: Credit-Card Effective Interest Rates

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National Exams — May 2014 — 11-CS-1 Engineering Economics. Open book; non-communicating calculator permitted. Any four of the five questions constitute a complete paper; all questions are of equal value. Fully worked solutions to all five questions follow; standard compound-interest factors are used and minor rounding is immaterial.

Question 1: Credit-Card Effective Interest Rates (25 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(a) Effective Annual Rates

$$\text{Smart Visa:}\ \left(1+\tfrac{0.24}{365}\right)^{365}-1 = \boxed{27.11\%}$$
$$\text{Principal:}\ \left(1+\tfrac{0.25}{52}\right)^{52}-1 = \boxed{28.33\%}$$
$$\text{Acrobat:}\ \left(1+\tfrac{0.26}{12}\right)^{12}-1 = \boxed{29.33\%}$$

(b) Effective Semi-Annual Rates

Using $i_{sa} = (1+i_a)^{1/2}-1$ (or half the compounding periods):

$$\text{Smart Visa:}\ (1.27115)^{1/2}-1 = \boxed{12.75\%};\quad \text{Principal:}\ \boxed{13.28\%};\quad \text{Acrobat:}\ \boxed{13.72\%}$$

(c) Which to Prefer

These are charges on overdue balances, i.e. rates you would pay. The borrower prefers the lowest effective rate, so Smart Visa is preferable (27.11%, the lowest), followed by Principal Card (28.33%) and then Acrobat Express (29.33%). Note that the ranking is the same as the ranking by nominal rate here, but only by arithmetic accident: Smart Visa compounds most frequently of the three, which works against it, and its 3.11-point gap between nominal and effective rate is still smaller than Acrobat's 3.33 points simply because that premium grows with the size of the nominal rate faster than the extra compounding frequency can offset over this range. The comparison is only valid once all three are placed on the same effective basis.

(d) Break-Even Rate for Smart Visa

Find the nominal rate $r$ (compounded daily) giving the same effective annual rate as Principal Card (28.33%):

$$\left(1+\tfrac{r}{365}\right)^{365} = 1.28326 \;\Rightarrow\; 1+\tfrac{r}{365} = 1.28326^{1/365} \;\Rightarrow\; r \approx \boxed{24.95\%}$$

At about 24.95% compounded daily, Smart Visa would match Principal Card's 28.33% effective annual rate. Note that the break-even nominal rate (24.95%) is below Principal Card's own nominal 25%, because daily compounding is more frequent than weekly.

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