Risk analysis in capital budgeting

A project's NPV is calculated from forecasts, and every forecast is uncertain. Sales may fall short, costs may rise, a competitor may cut prices, a regulation may change. The discount rate, developed in lessons 10 and 11, compensates investors for the kind of risk that cannot be diversified away. But managers also need to understand the project's own risks: which assumptions drive the result, how bad things could get, and what they can do if events turn out differently. This lesson covers the main tools: sensitivity analysis, scenario analysis, break-even analysis, operating leverage, Monte Carlo simulation, decision trees and real options. None of them replaces NPV; each helps decide how much to trust it.

The base case

Throughout this lesson we use one project. It requires an investment of $2.0 million, lasts five years and has no salvage value. The firm expects to sell 40,000 units a year at $60 each, with variable costs of $35 a unit and fixed cash costs of $300,000 a year. To keep the arithmetic transparent, assume the investment is deducted for tax in equal amounts of $400,000 a year over five years (lesson 8 showed how the actual Canadian and US rules differ). The tax rate is 25 % and the discount rate 11 %.

Sensitivity analysis

Sensitivity analysis changes one input at a time, holding the others at their base-case values, and records how NPV responds. It shows which variables matter most, and therefore where better information or tighter management is most valuable. Results are often shown as a tornado chart, with the variables sorted by the width of their NPV range. The weakness of sensitivity analysis is that variables rarely move one at a time: in a recession, volumes and prices may fall together. And the ranges used must be realistic; a variable looks important simply if it is given a wide range.

Scenario analysis

Scenario analysis changes several variables together in a consistent story. A pessimistic scenario might combine lower volume, lower price, higher unit costs and higher fixed costs, as might happen if a strong competitor entered; an optimistic one the reverse. Scenarios capture the correlation between variables that sensitivity analysis ignores, and they force managers to think through how the business would actually behave. Their weakness is that they give only a few points, with no measure of how likely each is.

Break-even analysis

Break-even analysis asks how low sales can fall before the project stops paying its way. The accounting break-even is the volume at which accounting profit is zero: fixed costs plus depreciation, divided by the contribution margin per unit. It is a misleading target, because a project that only breaks even in accounting terms loses money in present value: it returns the original investment with no return on it. The NPV break-even, also called financial break-even, is the volume at which NPV is zero. It requires the operating cash flow to cover the equivalent annual cost of the investment.

Operating leverage

Projects with high fixed costs and low variable costs are more sensitive to volume than projects with the opposite structure. The degree of operating leverage (DOL) measures this: the percentage change in operating profit for a 1 % change in sales volume. A highly automated plant has high operating leverage: profits soar when volume is strong and collapse when it is weak. Choosing a cost structure, for example whether to automate, outsource or lease, is therefore partly a choice about risk.

Monte Carlo simulation

Monte Carlo simulation extends scenario analysis to thousands of scenarios. The analyst specifies a probability distribution for each uncertain input, and correlations between them where relevant; the computer draws random values, calculates the NPV for each draw, and builds a distribution of outcomes. The result shows the expected NPV, the spread of outcomes and the probability of a negative NPV. Simulation is widely used in mining, energy, pharmaceuticals and real estate. Two cautions apply. First, the output is only as good as the input distributions, which are often guesses. Second, the distribution of NPV is not a reason to change the discount rate or to reject a project just because it might lose money: investors care about systematic risk, which the discount rate already reflects, while the spread of a single project's NPV mostly reflects risk that diversified shareholders can avoid.

# Monte Carlo simulation of the project's NPV
import random, statistics
random.seed(201)
annuity = (1 - 1.11 ** -5) / 0.11
def npv(units, price, var_cost, fixed):
    ocf = (units * (price - var_cost) - fixed) * 0.75 + 400_000 * 0.25
    return -2_000_000 + ocf * annuity

results = []
for _ in range(20_000):
    units = random.gauss(40_000, 5_000)
    price = random.gauss(60, 3)
    var_cost = random.gauss(35, 1.5)
    fixed = random.uniform(280_000, 320_000)
    results.append(npv(units, price, var_cost, fixed))
results.sort()
print(f"mean NPV {statistics.mean(results):,.0f}")
print(f"standard deviation {statistics.stdev(results):,.0f}")
print(f"5th percentile {results[1000]:,.0f}   95th percentile {results[19000]:,.0f}")
print(f"probability NPV < 0: {sum(v < 0 for v in results) / len(results):.1%}")

mean NPV 310,134
standard deviation 508,804
5th percentile -487,409   95th percentile 1,184,340
probability NPV < 0: 27.9%

Decision trees

Many projects are not all-or-nothing commitments made once. Managers can gather information, wait, expand, scale back or abandon as events unfold. A decision tree maps out the sequence of decisions and uncertain events, with probabilities and cash flows, and is solved backwards: at each decision point, choose the branch with the highest value; at each chance point, take the probability-weighted average; discount back to the present.

Real options

The test-market example shows a general point: flexibility has value. Opportunities to change course after information arrives are called real options, by analogy with financial options. The main types are the option to expand if a project does well, the option to abandon and recover salvage value if it does badly, the option to delay until uncertainty is resolved, and the option to switch inputs, outputs or locations. A standard NPV calculation that assumes the firm will follow a fixed plan whatever happens undervalues projects with such options. Real options also explain some apparently odd behaviour, such as a firm investing in a small, negative-NPV pilot project because it opens the door to a large follow-on investment.

Managing forecast risk

Formal risk analysis cannot fix biased forecasts. Studies of large projects, such as Bent Flyvbjerg's research on infrastructure, find systematic cost overruns and benefit shortfalls, driven by optimism and by incentives to get projects approved. Practical safeguards include asking where the project's positive NPV comes from (what advantage lets the firm earn more than competitors?), comparing forecasts with a reference class of similar past projects, separating the people who propose projects from those who review them, and conducting post-completion audits. CON-310 and PMP courses discuss the management side of keeping projects on budget.