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23-CS-1 Engineering Economics · May 2017

Question 1 of 5: Financing-Company Effective Rates

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National Exams — May 2017 — 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: Financing-Company Effective 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

For a nominal rate $r$ compounded $m$ times a year the period rate is $r/m$ and the effective annual rate is $i_a=(1+r/m)^{m}-1$. Using the paper's hint, $m=366$ days for A, $52$ weeks for B and $12$ months for C:

$$\text{A:}\ \left(1+\tfrac{0.10}{366}\right)^{366}-1 = 0.105156 = \boxed{10.52\%}$$
$$\text{B:}\ \left(1+\tfrac{0.1025}{52}\right)^{52}-1 = 0.107826 = \boxed{10.78\%};\qquad \text{C:}\ \left(1+\tfrac{0.105}{12}\right)^{12}-1 = 0.110203 = \boxed{11.02\%}$$

The spread between the three nominal rates (0.50 percentage points) is larger than the extra compounding gains, so the ranking by effective rate follows the ranking by nominal rate here.

(b) Effective Quarterly Rates

A quarter contains $366/4 = 91.5$ days, $52/4 = 13$ weeks or $12/4 = 3$ months, so compound each company's own period rate over one quarter — equivalently, take $i_q=(1+i_a)^{1/4}-1$ from part (a), which gives the same figures:

$$\text{A:}\ \left(1+\tfrac{0.10}{366}\right)^{91.5}-1 = \boxed{2.53\%};\quad \text{B:}\ \left(1+\tfrac{0.1025}{52}\right)^{13}-1 = \boxed{2.59\%};\quad \text{C:}\ \left(1+\tfrac{0.105}{12}\right)^{3}-1 = \boxed{2.65\%}$$

Check: $(1.0253116)^4-1 = 0.10516$, which reproduces Company A's effective annual rate, as it must.

(c) Which to Choose

These are loan rates, so the entrepreneurs want the lowest effective cost—Company A (10.52%), whose lower nominal 10% compounded daily still beats the higher nominal rates compounded less often.

(d) Break-Even Rate for Company A

Break-even means Company A's effective annual rate equals Company B's, $10.7826\%$. Solve for the nominal daily rate $r$ by taking the $366$th root:

$$\left(1+\tfrac{r}{366}\right)^{366} = 1.107826 \;\Rightarrow\; r = 366\left(1.107826^{1/366}-1\right) = 0.102413 = \boxed{10.24\%\ \text{compounded daily}}$$

So Company A could raise its quoted rate from 10.00% to about 10.24% before the entrepreneurs would become indifferent between A and B — a 0.24-point margin, which is the practical value of A's more frequent compounding being offset by its lower nominal rate.

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