Question 2 of 11: Discount Rate and Pit Scheduling

Question 1.2: Discount Rate and Pit Scheduling (7 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 $i=10\%$/yr; future cash-flow years $n=5$ and $n=15$; cash flow $F=\$100$ million in each case.

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

Approach. A discount rate converts a future cash flow to its present-day equivalent 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 value: $$(P/F,i,n)=\dfrac{1}{(1+i)^n}$$ $$n=5:\ (P/F,10\%,5)=\dfrac{1}{1.10^5}=\boxed{0.6209}\qquad n=15:\ (P/F,10\%,15)=\dfrac{1}{1.10^{15}}=\boxed{0.2394}$$
2. 1.2.2 – Discounted values at year zero. $$P_5=100\times0.6209=\boxed{\$62.09\text{M}}\qquad P_{15}=100\times0.2394=\boxed{\$23.94\text{M}}$$ 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 mine scheduler toward front-loading value: high-grade, low strip-ratio ore (typically the near-surface supergene-enriched cap on a copper porphyry, or the higher-grade epithermal vein sets) is scheduled as early as practically possible, waste stripping is deferred wherever slope/access allows, and phased (pushback) pit designs are preferred over a single monolithic pit precisely because they delay the waste-heavy outer pushbacks into later, more heavily discounted years. A high discount rate can even make the NPV-optimal pit SMALLER than the tonnage-maximizing ultimate pit, because low-grade, high-strip outer benches mined only in year 15+ contribute almost nothing in present-value terms while still consuming capital and stripping cost today – directly relevant to the Q2 Phase-1/Phase-2 sequencing, where Phase 1 is mined first partly because its cash flow is worth more, discounted, than deferring it to expose Phase 2 sooner.

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 rise in operating cost, and it says nothing about the TIMING or magnitude of a specific policy event – scenario/decision-tree analysis or an explicit sovereign-risk premium layered on top of the base discount rate are the more rigorous alternatives used alongside it in practice.