Question 2 of 13: 2 (5 marks, compulsory)

Question 1.2 (5 marks, compulsory)

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. Discount (interest) rate $i=10\%$ per year; two future cash-flow years $n=5$ and $n=15$; cash flow $F=100$ million (each case).

Find. The single-payment present-worth discount factor $(P/F,10\%,n)$ for $n=5,15$, and the discounted (present) value of the 100-million cash flow in each case.

Approach. A discount rate converts a future cash flow to its present-day equivalent value by compounding the time value of money backwards; apply the standard single-payment present-worth factor $(1+i)^{-n}$.

1. 1.2.1 – Discounting factor. The discount rate is the annual rate of return an investor requires (or could earn elsewhere) to be indifferent between a dollar today and a dollar received later; it converts a future cash flow to its present-day equivalent: $$(P/F,i,n) = \dfrac{1}{(1+i)^n}$$ $$n=5:\quad (P/F,10\%,5)=\dfrac{1}{(1.10)^5}=\boxed{0.6209}$$ $$n=15:\quad (P/F,10\%,15)=\dfrac{1}{(1.10)^{15}}=\boxed{0.2394}$$
2. 1.2.2 – Discounted values at year zero. Multiply the future cash flow by its factor: $$P_5 = 100\times0.6209=\boxed{62.09\text{ million}}\qquad P_{15}=100\times0.2394=\boxed{23.94\text{ million}}$$ A dollar 15 years out is worth barely a third of the same dollar 5 years out at this discount rate – the compounding is highly non-linear.

1.2.3 – Effect on pit design and scheduling. Because far-future cash flows are worth so little today, discounting drives the pit optimizer and the mine scheduler toward front-loading value: high-grade, low-strip-ratio ore is scheduled as early as practically possible (even at the cost of a smaller, less “metal-efficient” ultimate pit), waste stripping is deferred wherever the slope/access constraints allow, and phased (pushback) pit designs are preferred over a single large pit precisely because they delay the waste-heavy outer pushbacks until later, discounted years. A high discount rate can even make an NPV-optimal pit smaller than the tonnage-maximizing ultimate pit, because the marginal outer benches (low grade, high strip, mined only in year 15+) contribute almost nothing in present-value terms while still consuming capital and stripping cost today.

1.2.4 – Discount rate as a risk proxy. Yes, in a limited way: raising the discount rate used in an NPV/DCF evaluation is the standard way analysts embed country or sovereign policy risk (expropriation, royalty/tax changes, permitting reversals) into an otherwise deterministic cash-flow model – a project in a jurisdiction perceived as unstable is evaluated at a higher hurdle rate (e.g. 12–15% instead of 8–10%), which penalizes long-dated cash flows (the years most exposed to a future policy change) far more than near-term ones. This is a coarse, single-number proxy, however: it cannot distinguish a small probability of total asset loss from a general increase in operating cost, and it does not model the timing or magnitude of a specific policy event – scenario/decision-tree analysis or explicit sovereign-risk premia are the more rigorous alternatives used alongside a discount-rate adjustment.