Correia 10E · Ch 10 · Capital budgeting: risk analysis
One project · two places to put risk · four traps
Pricing project risk
Once you have measured a project's risk, you must charge for it somewhere in the NPV equation. Put it in the denominator and you have a risk-adjusted discount rate. Put it in the numerator and you have certainty equivalents. Each choice looks harmless on one project and changes the shape of the whole firm over time. Move the sliders and watch where the value goes.
The map
Two places in the equation to charge for risk
Risk-adjusted discount rate (RADR)
NPV = Σ E(CFt)(1 + rf + RP)t − I₀
Risk sits in the denominator. Expected cash flows stay as forecast, and a risk premium RP is added to the risk-free rate. Used by most firms because RP can be read off market data (CAPM, betas, industry hurdle rates).
Certainty equivalents (CE)
NPV = Σ αt × E(CFt)(1 + rf)t − I₀
Risk sits in the numerator. Each risky cash flow is cut to the certain amount management would swap it for (α between 0 and 1), then discounted at the risk-free rate only. Time and risk are kept apart.
Lesson 1 · Over-adjusting
Double the risk premium, never the whole rate
A project twice as risky as average needs twice the reward for risk. The risk-free part of the rate pays only for waiting, and waiting does not get riskier. Doubling the cost of capital charges twice for time as well.
Inputs
How the rate is built
Risk-free: pays for timeRisk premium: pays for riskExtra charged by the wrong method
Lesson 2 · RADR versus certainty equivalents
A risk premium in the rate compounds every year
Every RADR secretly contains a set of certainty-equivalent factors: αt = [(1 + rf) ÷ (1 + RADR)]t. Because the premium compounds, α shrinks every year whether or not the later cash flows are really riskier. Project Y from Example 10.6 is loaded (cost R200m, expected R150m then R200m, rf 10%, management's α of 0.90 and 0.80).
Inputs
What R1 of expected cash is worth as certain cash
α implied by the RADR (compounding)Management's own α for years 1 and 2
Lesson 3 · One WACC for every project
A single hurdle rate slowly makes the firm riskier
The firm below runs two divisions, as in the textbook: a low-risk division A (β 0.5) and a high-risk division B (β 1.5). Its WACC is just the weighted average of the two. Each dot is a project proposal. The security market line is the correct hurdle for each project's own beta; the flat WACC line is the shortcut. Wherever the two lines disagree, the shortcut makes the wrong call.
A new mining process saves money now but leaves an uncertain environmental restoration bill at the end. The instinct is to discount the "risky" bill at a higher rate. For an outflow that does the opposite of being careful: the higher the rate, the smaller the present value of the cost, and the better the project looks.
Inputs
Present value of the same restoration bill
The trade-offs
Every choice helps one dimension and hurts another
Risk-adjusted discount rate
+Rests on observable market data (betas, hurdle rates), so it is defensible to a board.
+One number per risk class is easy to apply across a capital budget.
−The premium compounds, so long-dated cash flows are punished hardest. Long-term strategic projects (plant, mines, infrastructure) are systematically undervalued.
−Mixes time and risk in one figure, which hides how much of the "discount" is really a risk charge.
Certainty equivalents
+Keeps time (rf) and risk (α) apart, so each year's risk is judged on its own merits.
+Forces management to state its risk appetite openly, year by year.
−No market guide for α. Two managers can pick different factors for the same project.
−Open to optimism bias: a keen sponsor can nudge α towards 1.
Single firm-wide WACC
+Short term: simple, consistent, and every accepted project beats the "cost of money".
+Right answer whenever a project has the firm's average risk.
−Long term: high-beta projects slip through and safe ones are turned away, so the firm's beta and cost of capital creep upward without any board decision.
−The safe division is starved of capital it should have received.
Exam and boardroom pointers
When a question says a project is "twice as risky", double the premium: rf + 2 × (k − rf). Doubling k is a classic lost mark.
Under certainty equivalents, discount at the risk-free rate. Using the WACC after cutting the cash flows charges for risk twice.
For a diversified firm, use the division's or industry's beta (pure play, unlever and relever), not the group WACC.
For an uncertain outflow, a lower rate is the conservative choice. Say so explicitly in a written answer.
The textbook itself notes that most firms use RADRs despite the compounding problem. A strong answer weighs whether a compounding premium suits the specific project's risk profile.