Effective Interest Rates: 11-CS-1 Engineering Economics, December 2015, Q1

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.

(a) Effective Quarterly Rates

\text{Inv 1 (2\%/mo):}\ (1.02)^{3}-1 = \boxed{6.12\%};\qquad \text{Inv 2 (6\%/quarter):}\ \boxed{6.00\%}

(b) Effective Annual Rates

\text{Inv 1:}\ (1.02)^{12}-1 = \boxed{26.82\%};\qquad \text{Inv 2:}\ (1.06)^{4}-1 = \boxed{26.25\%}

(c) Which to Choose

Investment 1 has the higher effective annual rate (26.82% > 26.25%), so CFI Inc. should choose Investment 1. The margin is small but real: 0.58 percentage points a year, worth roughly $580 a year on every $100,000 invested.

Note that the two nominal quotes are identical — 12 × 2% = 24% per year for the first investment and 4 × 6% = 24% per year for the second — so a comparison made on the nominal rates would find the two investments indistinguishable and decide nothing. The whole difference is the compounding frequency, which is exactly what parts (a) and (b) are designed to expose.

(d) Monthly Rate for Indifference

"Neither investment preferred over the other after one year" means the two must grow a dollar by the same factor over twelve months. Investment 2 turns a dollar into $(1.06)^4$ after four quarterly compoundings, so the required monthly rate $i$ is the twelfth root of that same factor:

(1+i)^{12} = (1.06)^{4} = 1.262477 \;\Rightarrow\; i = 1.262477^{1/12}-1 \approx \boxed{1.96\%\ \text{per month}}

The answer is below the 2% actually offered, which is the expected direction: Investment 1 is currently the better of the two, so its rate would have to be cut to 1.96% before the investor became indifferent. A rate above 2% here would be a sign that parts (a) and (b) had been read the wrong way round.