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11-CS-1 Engineering Economics · December 2013

Question 1 of 5: Effective Interest Rates

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National Exams — December 2013 — 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 are given below; standard compound-interest factors are used and minor rounding is immaterial.

Question 1: 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.

Both quotations are nominal statements attached to a stated compounding frequency, so neither can be compared with the other as printed. Investment 1 quotes 6% over a six-month period that is itself the compounding period, so within one half-year no interest is earned on interest and the quoted figure is already effective for that period. Investment 2 quotes 3% over a three-month compounding period, so a half-year contains two compoundings and a full year four. The whole question turns on restating both offers on one common effective basis before any comparison is made.

(a) Effective Semi-Annual Rates

$$\text{Inv 1 (6\%/6 mo):}\ i_{sa} = \boxed{6.00\%}\quad(\text{already semi-annual})$$
$$\text{Inv 2 (3\%/quarter):}\ i_{sa} = (1.03)^{2}-1 = \boxed{6.09\%}$$

(b) Effective Annual Rates

$$\text{Inv 1:}\ (1.06)^{2}-1 = 1.1236-1 = \boxed{12.36\%}$$
$$\text{Inv 2:}\ (1.03)^{4}-1 = 1.125509-1 = \boxed{12.55\%}$$

(c) Which to Choose

Investment 2 has the higher effective annual rate (12.55% > 12.36%), so CFC should choose Investment 2.

(d) Quarterly Rate for Indifference

Set the effective annual rate of Investment 2 equal to that of Investment 1:

$$(1+i)^{4} = (1.06)^{2} = 1.1236 \;\Rightarrow\; i = 1.1236^{1/4}-1 = 1.02956-1 \approx \boxed{2.96\%\ \text{per quarter}}$$

At about 2.96% per quarter, Investment 2 would return the same 12.36% effective annual rate as Investment 1, and neither would be preferred. The check is direct: $(1.029563)^{4} = 1.1236$, which is exactly $(1.06)^{2}$, so a dollar placed in either investment grows to the same $1.1236 after one year. Note that the indifference rate is below the 3% actually offered, which is simply the other side of part (c): because quarterly compounding works a little harder than semi-annual compounding, Investment 2 can afford to quote a slightly lower periodic rate and still match Investment 1 over the year.

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