23-CS-1 Engineering Economics · May 2017
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
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 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.
The part is worth 12 marks and explicitly says use a rate of return method, so the decision must be reached on rates. For mutually exclusive alternatives that means the incremental procedure: order by increasing first cost (2, 3, 1), check that the cheapest clears the MARR on its own, then buy each successive increment only if that increment itself earns more than 12%. Every alternative here begins with an outflow, so each increment below is an ordinary investment and the normal “accept if the rate exceeds the MARR” test applies (a financing increment, which would open with a receipt, reverses that test).
Step 1 — is the cheapest alternative acceptable on its own? Solve $-5-1.5(P/A,i,5)+20(P/F,i,5)=0$. The expression is $+0.238$ at $i=14\%$ and $-0.389$ at $i=16\%$, so the root lies between; solving gives $i^{*}_{2}=\boxed{14.73\%}$. Since $14.73\% > 12\%$, Investment 2 is acceptable and becomes the current best, and do-nothing is discarded.
Step 2 — increment 2 to 3: $-\$2$M now, $-\$0.5$M/yr for five years, $+\$9$M at year 5.
$21.06\% > 12\%$, so the extra $2M of capital earns well above the hurdle rate: advance to Investment 3.
Step 3 — increment 3 to 1: $-\$2$M now, $-\$1$M/yr, $+\$9$M at year 5.
$8.76\% < 12\%$, so the last $2M fails to earn the MARR and that step is rejected. Bright Star should select Investment 3.
Cross-check by present worth at 12%, using $(P/A,12\%,5)=3.60478$ and $(P/F,12\%,5)=0.56743$ (all figures in millions):
The two routes agree, as they must: $PW_3-PW_2=+\$1.30$M confirms the accepted increment and $PW_1-PW_3=-\$0.50$M the rejected one. Investment 3 is the recommended choice.
No. All three investments share the same five-year life, so $AW = PW\,(A/P,12\%,5)$ with one identical positive factor for each. Multiplying the three present worths by the same positive number cannot reorder them, so Annual Worth also selects Investment 3. Incremental ROR, the method used in part (a), is consistent with both: an increment is accepted exactly when its present worth (and hence its annual worth) at the MARR is positive. All three criteria must agree when they are applied over a common period at a common MARR.
A rate-of-return method is recommended (i) when the result is wanted as a single percentage that can be set beside the MARR, the cost of capital or a borrowing rate and communicated to management and non-specialists; (ii) when the MARR is uncertain or disputed, because the ROR is the break-even discount rate and shows directly how much margin a project has; and (iii) for judging whether a single project is acceptable at all. It is not a safe ranking tool for mutually exclusive alternatives unless the incremental procedure is used, as in part (a), and it can give multiple or no rates on non-conventional cash flows with several sign changes.
No. A standalone rate of return measures efficiency per dollar invested, not the amount of value created, so a small alternative can post the highest percentage while a larger one adds more worth. Mutually exclusive alternatives must therefore be ranked by incremental ROR, which always agrees with maximum present worth at the MARR.
On this paper the two criteria happen to coincide: the standalone rates are $i^{*}_{1}=14.72\%$, $i^{*}_{2}=14.73\%$ and $i^{*}_{3}=16.59\%$, so Investment 3 carries both the highest standalone return and the largest present worth, and the given data contain no counter-example to point at. The ranking is nonetheless not driven by size — the smallest alternative (2) edges out the largest (1) on standalone return while creating only $0.94M of value against $1.75M. A genuine counter-example is one keystroke away: raise Investment 2's year-5 return from $20M to $22M and its standalone rate becomes 17.75%, the highest of the three, yet $PW_2=+\$2.08$M still trails $PW_3=+\$2.25$M and Investment 3 still wins the incremental comparison.
