Question 4 of 27

Question 1.4 (5 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. Capital cost = $500 million; uniform-series present-worth factors (P/A,10%,n): PWF ₅=3.791, PWF ₁₅=7.606.

Find. (1.4.1) definition of discount rate; (1.4.2) the level annual cash flow A that retires the $500M debt over 5 and over 15 years; (1.4.3)/(1.4.4) qualitative discussion.

1.4.1 — discount rate. The discount rate i is the annual rate at which future cash flows are converted to present-day value, reflecting the investor’s required return (cost of capital plus a risk premium) — it is the rate used to compute the present worth factors below.

Approach. The capital, $500M, is an ordinary annuity present value; the level annual cash flow A required equals Capital ÷ PWF for the chosen term, since P = A×(P/A,i,n).

1. 5-year retirement. $$A_5 = \dfrac{P}{(P/A,10\%,5)} = \dfrac{\$500\text{M}}{3.791}$$ Substituting: $$\boxed{A_5 \approx \$131.89\text{ million/yr}}$$
2. 15-year retirement. $$A_{15} = \dfrac{P}{(P/A,10\%,15)} = \dfrac{\$500\text{M}}{7.606}$$ Substituting: $$\boxed{A_{15} \approx \$65.74\text{ million/yr}}$$

1.4.3 — effect of discount rate on retirement. A HIGHER discount rate shrinks the present-worth factor for a given n, so the level annual cash flow A required to retire the same $500M in the same n years INCREASES — equivalently, at a fixed achievable annual cash flow, a higher discount rate lengthens the time needed to retire the debt, because more of each year’s cash flow is “consumed” discounting future money back to the present. Discount rate and required debt-service cash flow move together.

1.4.4 — discount rate and political/regional risk. Yes, in part: raising the discount rate (adding a country/political risk premium on top of the base cost of capital) is the standard way analysts penalize a project’s NPV for elevated sovereign or regulatory risk, and it correctly captures the TIME VALUE cost of that risk (cash flows further out are discounted harder). It is a blunt instrument, however — it applies the SAME extra penalty to every future year uniformly, whereas real political/regulatory risk (expropriation, royalty changes, permit reversal) is often a discrete, sudden, asymmetric event rather than a smoothly compounding one; scenario analysis or decision-tree/real-options methods complement (not replace) a risk-adjusted discount rate for this purpose.