24-Pet-B3 Petroleum Geology, December 2015, Q21

Question 21

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. A company buys a shale gas property for $50,000,000 (start of Year 1). It undertakes additional drilling investment of $10,000,000 at the start of each of Years 3, 6 and 9. The desired Hurdle Rate for sale of the property at the end of Year 10 is 15% (inflation and taxes ignored).

Find. The complete Future Value table (Investment, Cumulative Investment, Expected Value at start of year, Return at end of year, all in k$) for Years 1–10, and the minimum sale value the property must reach at the end of Year 10 to meet the 15% hurdle rate.

Fig. Q21-1 — cash-flow diagram of the four investment outlays (down arrows) and the minimum required terminal sale value (up arrow) that exactly earns 15%/yr on the whole stream.

Approach. Grow a single running “Expected Value” balance year by year at the 15% hurdle rate, adding each new investment the instant it is made (start of Years 1, 3, 6, 9), then read the balance plus its final year's return as the minimum terminal sale value.

Set up the recursion. With Expected Value at the start of year t written EV(t) and Return earned during year t written R(t):$$EV(t) = EV(t-1) + R(t-1) + \text{Investment}(t), \qquad R(t) = EV(t)\times 15\%$$starting from EV(1) = Investment(1) = 50,000 (thousand dollars; no prior return to add in Year 1).
Tabulate all 10 years. Applying the recursion with Investment(3) = Investment(6) = Investment(9) = 10,000 (thousand dollars) and Investment(t) = 0 otherwise gives the completed table below.
Minimum sale value at Year 10. $$\text{Minimum value} = EV(10) + R(10) = 231{,}484.1 + 34{,}722.6 = \boxed{266{,}206.7\text{ k\$} \approx \$266.2\text{ million}}$$
Independent cross-check. Each investment, made at the start of year t, compounds at 15% for (11 − t) return-events through the end of Year 10, so the same total can be built directly as a sum of four independently-compounded lump sums:$$50{,}000(1.15)^{10} + 10{,}000(1.15)^{8} + 10{,}000(1.15)^{5} + 10{,}000(1.15)^{2} = \boxed{266{,}206.7\text{ k\$}}$$matching the recursive table build-up to within 0.1 thousand dollars, confirming the year-by-year build-up is internally consistent.

Completed Future Value table (all values k$)
Year	Investment	Cumulative Investment	Expected Value (start of yr)	Return (end of yr, 15%)
1	50,000	50,000	50,000.0	7,500.0
2	0	50,000	57,500.0	8,625.0
3	10,000	60,000	76,125.0	11,418.8
4	0	60,000	87,543.8	13,131.6
5	0	60,000	100,675.3	15,101.3
6	10,000	70,000	125,776.6	18,866.5
7	0	70,000	144,643.1	21,696.5
8	0	70,000	166,339.6	24,950.9
9	10,000	80,000	201,290.5	30,193.6
10	0	80,000	231,484.1	34,722.6
Minimum to Meet Hurdle Rate (end of Yr 10)				266,206.7

Final results
Quantity	Value
Total capital invested (undiscounted)	80,000 k$
Minimum sale value to meet 15% hurdle (Yr 10)	266,206.7 k$ (≈ $266.2 million)
Implied multiple on total invested capital	3.33× over 10 years

Topic: Future value of a series of unequal, unevenly-timed cash outlays

Key relations: $FV = \sum_i C_i(1+i)^{n-t_i}$ for each outlay $C_i$ made at the START of year $t_i$, compounded through $n$ return-events by the valuation date; equivalently, an Expected-Value balance that itself earns the hurdle rate and absorbs new investment the instant it occurs.

Why this works: A “start of year = time zero” investment convention means the FIRST year's investment earns a FULL year of return by year-end, which is why the Year-1 outlay compounds for all 10 return periods (exponent 10, not 9) through the end of Year 10.

Common pitfall: Off-by-one exponent errors from mismatching the start-of-year-investment / end-of-year-return timing convention — always cross-check a recursive table build against a closed-form compounding sum, as done in Step 4 above.

Source: Thompson & Wright, Oil Property Evaluation, Ch. on time value of money.

Question 21 of 22

Question 21