11-CS-1 Engineering Economics · May 2019
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
National Exams — May 2019 — 11-CS-1 Engineering Economics. Three hours; open book; any non-communicating calculator permitted. Any four of the five questions constitute a complete exam paper and each question is of equal value (25 marks). Fully worked solutions to all five follow.
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.
Each card quotes a nominal annual rate $r$ compounded $m$ times a year, so its interest rate per compounding period is $r/m$ and the effective annual rate is $i_a=(1+r/m)^m-1$.
| Card | Nominal rate | Periods per year, $m$ | Rate per period, $r/m$ | Effective annual rate |
|---|---|---|---|---|
| Silver Visa | 24%, daily | 365 | 0.065753% | 27.11% |
| Fast Card | 25%, weekly | 52 | 0.480769% | 28.33% |
| Mach Express | 26%, monthly | 12 | 2.166667% | 29.33% |
Note that the ordering by effective rate is the same as the ordering by nominal rate here: the extra compounding frequency of the daily card is not enough to overcome its 2-point lower nominal rate. That is not guaranteed in general, which is exactly why the conversion must be done rather than assumed.
A half-year contains $m/2$ compounding periods, so $i_{sa}=(1+r/m)^{m/2}-1$. Equivalently $i_{sa}=(1+i_a)^{1/2}-1$, which is the quicker route once part (a) is done:
Each semi-annual rate is a little more than half its annual counterpart's nominal equivalent but a little less than half the effective annual rate, because $(1+i_{sa})^2=1+i_a$ compounds the half-year rate once.
These rates are charged on overdue balances — money the cardholder owes. A borrower therefore prefers the lowest effective rate, so the preference is Silver Visa, at 27.11% effective annually (12.75% effective semi-annually). Over a year, a $1,000 overdue balance costs $271 on Silver Visa against $283 on Fast Card and $293 on Mach Express.
(Had these been rates earned on deposits, the preference would reverse and Mach Express would be best. The comparison itself is neutral; it is the direction of the cash flow that decides which end of the ranking is wanted.)
"Break-even" means Silver Visa costs a borrower exactly what Fast Card costs, i.e. the two effective annual rates are equal. Silver Visa compounds daily, so solve for its nominal rate $r$:
Silver Visa could raise its quoted rate from 24% to about 24.95% compounded daily before a borrower became indifferent between it and Fast Card's 25% compounded weekly. The break-even nominal rate is below Fast Card's 25% because daily compounding is more frequent, so a smaller nominal rate produces the same effective cost.
