23-CS-1 Engineering Economics · December 2013
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
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 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.
Depreciation $=20000/7=\$2{,}857$/yr (years 1–7). After-tax savings $=3700(0.60)=\$2{,}220$/yr (years 1–9). Depreciation shield $=2857(0.40)=\$1{,}143$/yr (years 1–7). Salvage after tax (book value $0$, fully taxed) $=3000(0.60)=\$1{,}800$ at year 9.
Three separate time spans therefore have to be carried, and mixing them up is the commonest way to get this question wrong. The $20,000 purchase is not deductible in itself, so it sits at $t=0$ untouched by tax. The quality-improvement savings are ordinary taxable income and run for the full nine-year service life, so they are reduced by the 40% tax rate and discounted over nine years. The capital allowance runs only over the seven-year depreciation life the tax rules impose, so the shield stops at year 7 even though the device keeps saving money for two more years. Finally, because straight-line depreciation over seven years to a zero salvage writes the book value down to nothing, the entire $3,000 received on disposal is recaptured depreciation, taxed as ordinary income rather than treated as a tax-free return of capital. Each stream is then discounted at the after-tax MARR of 11%, which is the rate the question already states on an after-tax basis, so no further adjustment to the discount rate is needed.
The after-tax PW is negative, so the investment should NOT be made at an 11% after-tax MARR.
The standard shortcut for an approximate after-tax rate of return is to find the before-tax rate of return from the raw cash flows and scale it by the fraction of income the firm keeps after tax, $i_{AT} \approx i_{BT}(1-T)$. Before tax, the device costs $20,000, saves $3,700 a year for 9 years and returns $3,000 at the end:
Trial rates give $PW_{BT}(12\%) = -20000 + 3700(5.32825) + 3000(0.36061) \approx +\$796$ and $PW_{BT}(13\%) = -20000 + 3700(5.13166) + 3000(0.33288) \approx -\$14$. Interpolating:
Cross-check against the exact after-tax rate, which is the rate that zeroes the after-tax stream of part (a): $PW(8\%)\approx+\$719$ and $PW(9\%)\approx-\$110$, so the exact after-tax IRR is $8\% + \frac{719}{719+110}(1\%) \approx 8.9\%$. The shortcut understates it by about a point. Scaling the whole before-tax rate by $(1-T)$ in effect taxes the recovery of the capital as well as the return on it, whereas in the exact analysis the $1,143/yr depreciation shield over years 1–7 gives that tax back. Both figures lie well below the 11% MARR, so the difference does not affect the decision.
The approximate after-tax IRR (about 7.8%) is below the 11% after-tax MARR, so reject the investment. This agrees with the negative present worth in part (a), and with the exact after-tax IRR of about 8.9%, which is also below 11%.
