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23-CS-1 Engineering Economics · May 2013

Question 5 of 6: Circuit Tester — After-Tax Analysis

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

National Exams — May 2013 — 11-CS-1 Engineering Economics. Open book; non-communicating calculator permitted. Any five of the six questions constitute a complete paper; all questions are of equal value. Fully worked solutions to all six questions are given below. Standard compound-interest factors are used throughout; minor rounding differences are immaterial.

Question 5: Circuit Tester — After-Tax Analysis (20 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.

Set-up: After-Tax Cash Flows

Depreciation (straight-line, 7 yr, $0 salvage) $= 16000/7 = \$2{,}285.7$/yr for years 1–7. Tax rate $t=0.45$.

(a) After-Tax Present Worth (i = 10%)

$$PW = -16000 + 1650(P/A,10\%,8) + 1028.6(P/A,10\%,7) + 1100(P/F,10\%,8)$$
$$= -16000 + 1650(5.3349) + 1028.6(4.8684) + 1100(0.46651)$$
$$= -16000 + 8802.6 + 5007.5 + 513.2 \approx \boxed{-\$1{,}677}$$

The after-tax PW is negative, so the investment should NOT be made at a 10% after-tax MARR.

(b) Approximate After-Tax IRR

The approximate method scales the before-tax rate of return by the retention rate, $i_{\text{after}} \approx i_{\text{before}}(1-t)$. The before-tax IRR uses the raw cash flows ($16,000 out; $3,000/yr for 8 years; $2,000 salvage):

$$-16{,}000 + 3{,}000\,(P/A,i,8) + 2{,}000\,(P/F,i,8) = 0 \;\Rightarrow\; i_{\text{before}} = 11.51\%$$
$$i_{\text{after}} \approx 11.51\%\,(1-0.45) = \boxed{6.33\%}$$

For comparison, the exact after-tax IRR is the rate that zeroes the after-tax cash flows built above. Those give $PW(7\%)\approx+\$36$ and $PW(8\%)\approx-\$569$, so

$$\text{IRR}_{\text{exact}} \approx 7\% + \frac{36}{36+569}(1\%) = \boxed{7.06\%}$$

(c) Decision by IRR

Both figures—the approximate after-tax IRR of 6.33% and the exact after-tax IRR of 7.06%—are below the 10% after-tax MARR, so reject the investment. This is consistent with the negative after-tax present worth in (a).

(d) Approximate versus Exact IRR

No—not always. The approximation $i_{\text{after}} \approx i_{\text{before}}(1-t)$ assumes tax simply shrinks every cash flow by the factor $(1-t)$. It does not: the depreciation tax shield is an extra inflow that no before-tax cash flow contains, and it has its own timing (here 7 years of capital allowance against an 8-year project), while the salvage is taxed as recaptured depreciation rather than scaled. The approximation therefore understates the true after-tax return here—6.33% against the exact 7.06%, a gap of about 0.7 percentage points.

In this case both lie well below the 10% MARR, so the two analyses give the same reject decision. But the gap is large enough to change the answer near the hurdle: at a MARR of 7% the approximate method would say reject while the exact after-tax analysis says accept. Whenever the return is close to the MARR—or the depreciation life differs materially from the service life—the exact after-tax analysis must be used.