Question 3 of 6: Rate of Return

Question 3: Rate of Return (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.

Given. Cash flows to the oil company: nothing at $t=0$; net income $+\$50{,}000$ in each of years 1–4; the water-flood investment $-\$650{,}000$ at year 5; secondary-recovery income $+\$100{,}000$ in each of years 6–20. The net present worth at a 10 % minimum acceptable rate of return (MARR) is stated as $\$227{,}000$.

Find. (a) whether to accept the arrangement; (b) the investor's (discounted-cash-flow) rate of return.

Approach. Part (a) is a net-present-worth test at the 10 % MARR (accept if $\text{NPW}>0$); part (b) seeks the discounted-cash-flow rate of return — the interest rate that drives NPW to zero — which we test by scanning NPW across a wide range of rates and applying Descartes' rule of signs to the cash-flow sequence.

(a) Accept or reject?

1. Confirm the stated NPW at 10 %. Using single-payment and uniform-series present-worth factors, $$\text{NPW}_{10\%}= 50\text{k}\,(P/A,10,4) - 650\text{k}\,(P/F,10,5) + 100\text{k}\,(P/A,10,15)(P/F,10,5)$$ $$= 50\text{k}(3.170) - 650\text{k}(0.6209) + 100\text{k}(7.606)(0.6209)= \$158{,}500-\$403{,}600+\$472{,}400=\boxed{+\$227{,}000}$$ which reproduces the value given in the problem.
2. Apply the acceptance criterion. Since $\text{NPW}_{10\%}=+\$227{,}000>0$, the venture earns more than the 10 % minimum rate; on the NPW criterion the lease-and-flood arrangement should be accepted — and attractively so, since the company commits no capital of its own up front.

(b) The investor's rate of return

1. Define the rate of return. The investor's (DCF) rate of return is the interest rate $i^\*$ at which $\text{NPW}(i^\*)=0$. Set the same NPW expression to zero and search for a root.
2. Scan NPW versus interest rate. Evaluating the NPW expression:

   i	10%	15%	20%	25%	30%	35%	40%	50%	100%
   NPW (k$)	227	110	56	32	21	17.9	17.8	21	30

   The NPW never reaches zero: it falls to a shallow minimum of about $+\$17{,}800$ near 38–40 % and rises again, staying strictly positive at every non-negative interest rate.
3. Explain with Descartes' rule of signs. The cash-flow sequence is $0,\,+,+,+,+,\,-,\,+,+,\dots,+$ — two sign changes (the $-\$650$k at year 5 is bracketed by positive flows), so there are at most two positive real roots. Here there are none: because the company receives income (years 1–4) before it invests anything, this is a non-conventional, "financing-type" cash flow for which no single finite rate of return exists. $$\boxed{\text{No real positive DCF rate of return exists; } \text{NPW}(i)>0 \text{ for all } i\ge 0.}$$
4. Give the board a usable answer. The practical meaning is that the return is effectively unbounded — the oil company puts up none of its own money and is cash-positive throughout except at year 5, which the surrounding income more than covers. When the ordinary rate-of-return criterion is indeterminate like this, Peters & Timmerhaus direct the decision back to NPW, which is decisively positive ($+\$227{,}000$ at 10 %). As a reference figure the board can quote the return earned on the only capital genuinely at risk — the year-5 water-flood: recovering $\$650{,}000$ from the $\$100{,}000/\text{yr}$ secondary income of years 6–20 requires $(P/A,i,15)=6.5$, i.e. $i\approx 13\%$, comfortably above the 10 % MARR.