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23-CS-1 Engineering Economics · May 2014

Question 1 of 5: Credit-Card Effective Interest Rates

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Notes on this paper

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%). The effective ranking happens to match the nominal ranking here, but that is not guaranteed: Smart Visa compounds most often, which lifts its effective rate 3.11 points above its nominal 24%, while Acrobat's monthly compounding adds 3.33 points to its 26%. The one-point gaps between the nominal rates are large enough that the compounding premiums do not reorder the three; with nominal rates closer together, daily compounding could make the lowest-quoted card the dearest. 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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