ACCA Applied Skills · Financial Management

Adjusting for risk and uncertainty in investment appraisal: formula sheet

Full chapter guide

Key formulas

Expected value (EV): EV = Σ (probability × outcome); Used when probabilities are known, so it applies to risk. It is a long-run average. It may not be an outcome that can actually happen in a one-off project.
Definition of risk: Risk = several possible outcomes, with probabilities that can be estimated; Quote this in definition questions.
Definition of uncertainty: Uncertainty = outcomes where probabilities cannot be estimated; Typical when there is no past data, for example a new market or a long project life.
Probabilities rule: Σ probabilities = 1; Check this before calculating an EV. If the probabilities do not add to 1, something is missing.
Sensitivity margin (general): Sensitivity margin = NPV ÷ PV of the cash flow affected × 100%; Use the PV of the specific cash flow being flexed, over its whole life. Use a positive NPV.
Initial investment: Margin = NPV ÷ initial investment × 100%; The investment is already at time 0, so its PV is the amount itself.
Selling price: Margin = NPV ÷ PV of sales revenue × 100%; Price changes flow straight to profit, so use revenue (not contribution).
Sales volume: Margin = NPV ÷ PV of contribution × 100%; Volume changes revenue and variable costs together, so use contribution.
Variable or fixed costs: Margin = NPV ÷ PV of that cost × 100%; Use the PV of the cost line only. A rise in cost lowers NPV.
Discount rate: Margin = IRR − cost of capital (in percentage points); This is usually measured as an absolute difference, not a percentage of NPV.
Expected value: EV = Σ (probability × outcome); Probabilities must add up to 1 (100%). Use it for cash flows or for NPVs.
Expected NPV: ENPV = Σ (probability × NPV of each outcome); Gives the same answer as discounting the expected cash flows, provided the cash flow structure is linear and the discount rate is the same in every outcome.
Joint probability (independent events): P(A and B) = P(A) × P(B); Only valid if A and B are independent. If the second event depends on the first, use the conditional probability instead: P(A and B) = P(A) × P(B given A).
Probability of a negative NPV: P(NPV < 0) = sum of the probabilities of all outcomes with NPV < 0; A simple way to show risk alongside the ENPV.
Decision tree roll-back: Value at a chance node = Σ (probability × value of branch); at a decision node take the best branch; Work from right to left. Discount future values back to the same date.
Risk adjusted discount rate: Project discount rate = Base cost of capital + Risk premium; The base rate is normally the WACC for average-risk projects. The premium is a judgement and is usually given in the question.
Net present value: NPV = Σ [Cash flow in year t ÷ (1 + r)^t] − Initial investment; Use the risk adjusted r. Accept if NPV is positive.
Payback period: Payback = Years before full recovery + (Unrecovered cost at start of that year ÷ Cash flow in that year); Assumes cash flows arise evenly through the year. Compare with the required maximum period.
Discounted payback: Discount each cash flow at the cost of capital, then apply the payback method to the discounted cash flows; Use cumulative discounted cash flows. If they never turn positive, the NPV is negative and the project is not recovered.
Expected NPV from scenarios: E(NPV) = Σ (probability of scenario × NPV of scenario); Probabilities across all scenarios must total 1. Use only if probabilities have been given or estimated.
Probability of a negative NPV (scenarios): P(NPV < 0) = sum of probabilities of scenarios with negative NPV; Only meaningful with a few discrete scenarios. With three scenarios it is a rough guide.
Random number allocation: Allocate two-digit random numbers 00–99 in proportion to probability: probability 0.3 = 30 numbers; Ranges must be cumulative, must not overlap and must cover 00–99.
NPV in each trial: NPV = Σ (cash flow in year t × discount factor in year t) − initial investment; Simulation repeats this calculation many times with different random inputs.

Quick revision

Risk: probabilities of outcomes are known or can be estimated. Uncertainty: they cannot.
Sensitivity (%) = NPV ÷ PV of the cash flow affected × 100. For sales volume use the PV of contribution, for selling price use the PV of sales revenue, and for a cost use the PV of that cost. Use after-tax figures where tax applies. The smaller the percentage, the more sensitive the project.
For the discount rate, sensitivity is found through the IRR, the rate at which NPV is zero. Compare the IRR with the cost of capital to see how much the rate can rise.
Sensitivity shows how much a variable can change before NPV hits zero. It ignores probabilities and changes one variable at a time.
Expected value = Σ (probability × outcome). Probabilities must add to 1.
An expected value is a long-run average. It may not match any single outcome, so it is weak for a one-off project.
Expected values give no measure of spread, so they do not show how risky the project is.
A higher discount rate for riskier projects applies the same risk premium every year. This implicitly compounds risk over time, so later cash flows are discounted more heavily. That may not be realistic for every project. The risk premium itself is also subjective.
Payback favours quick returns, so it reduces exposure to later uncertainty, but it ignores cash flows after the cut-off.
Scenario analysis changes several variables together, such as best, most likely and worst cases.
Worst-case analysis shows the lowest possible NPV but says nothing about how likely it is.
Simulation uses random values from probability distributions to produce a range of NPVs. It is powerful but costly and relies on good inputs.
In written answers, recommend a decision and give a reason linked to your figures.

Common mistakes

Saying risk is about bad outcomes and uncertainty is about good outcomes. Fix: In FM, risk means variability with known probabilities, and it covers both good and bad outcomes. Uncertainty means probabilities are unknown.
Using the two words as if they mean the same thing in a written answer. Fix: Define each term first, then use it consistently. Say what is known and what is not.
Dividing NPV by the undiscounted cash flow. Fix: Always divide by the PV of that cash flow, using the same discount factor as in the NPV.
Using revenue for a volume margin, or contribution for a price margin. Fix: Price changes affect revenue only, so use PV of revenue. Volume changes affect revenue and variable costs, so use PV of contribution.
Using probabilities that do not add up to 1, or using percentages and decimals mixed together. Fix: Add the probabilities for each variable before you start. If they do not total 100%, re-read the question.
Adding probabilities of joint events instead of multiplying them. Fix: For 'both happen' with independent events, multiply. Check that all joint probabilities in your table add to 1.
Multiplying the base rate by the premium instead of adding it. Fix: Add percentage points unless the question says otherwise. 10% with a 4% premium is 14%.
Using the old WACC for discount factors after deciding to adjust. Fix: Write the adjusted rate at the top of your working and use only that rate.
Describing scenario analysis as changing one variable at a time. Fix: Remember the contrast: sensitivity changes one variable and holds the rest constant; scenario and simulation change several together.
Saying simulation gives the correct decision or the best project. Fix: Say it shows a range and the probability of outcomes. The decision still depends on management's attitude to risk.

Exam tips

Start any definition answer by stating the single distinguishing feature: probabilities known (risk) or not known (uncertainty).
In objective questions, read each option closely. Examiners often swap the two definitions in the wrong options.
In written answers, link every source of uncertainty to the scenario. A generic list earns fewer marks than two or three well-explained, relevant points.
Always give a limitation after using a technique. Examiners reward the comment on what the technique does not show.
Remember that this topic is the setting for the rest of the chapter. Know which technique matches risk and which matches uncertainty.
Show the PV of each cash flow as a separate line before dividing. Method marks in Section C depend on clear workings.
In objective questions, read which variable is flexed. The wrong base (revenue vs contribution) is the usual trap, because each distractor is the margin for a different base.
State the direction of change with each margin: revenue falls, costs rise.