CMA Final · Strategic Financial Management

Evaluation of Risky Proposals for Investment Decisions: formula sheet

Full chapter guide

Key formulas

Expected value of cash flow: E(CF) = Σ (Pᵢ × CFᵢ); Pᵢ is the probability of outcome i. The probabilities must add up to 1.
Standard deviation of cash flow: σ = √[ Σ Pᵢ × (CFᵢ − E(CF))² ]; Measures spread around the expected value. A higher σ means higher stand-alone risk for the same mean.
Coefficient of variation: CV = σ ÷ Expected value; Risk per rupee of expected return. Use it to compare projects of different size or different means.
Risk-adjusted discount rate: RADR = Risk-free rate + Risk premium; A riskier project gets a higher rate. Covered in detail in its own topic.
Certainty equivalent coefficient: α = Certain cash flow ÷ Risky expected cash flow; α lies between 0 and 1 for a risk-averse decision maker. Multiply expected cash flows by α, then discount at the risk-free rate.
Risk-adjusted discount rate: RADR (k) = Rf + risk premium; Under CAPM the premium is β × (Rm − Rf). Questions may give the premium directly or ask you to build it.
NPV using RADR: NPV = Σ [E(CFt) ÷ (1 + k)^t] − Initial outlay; Use expected (uncorrected) cash flows. Do not also scale them by coefficients.
CE coefficient: α(t) = Certain cash flow ÷ Expected risky cash flow; Lies between 0 and 1 for a risk-averse decision-maker. Lower means more risk.
NPV using certainty equivalent: NPV = Σ [α(t) × E(CFt) ÷ (1 + Rf)^t] − Initial outlay; Discount at the risk-free rate only. The initial outlay is usually certain, so its coefficient is 1 unless stated.
Equivalence of the two methods: α(t) = [(1 + Rf) ÷ (1 + k)]^t; Holds when both methods give the same present value and k is constant. Useful to find an implied coefficient.
NPV: NPV = Σ [CFt ÷ (1 + k)^t] − Initial investment; Base-case NPV is the starting point for every sensitivity or scenario question.
Break-even annual cash flow (level inflows): Break-even cash flow = Initial investment ÷ Annuity factor (k, n); Valid when inflows are equal each year and there is no salvage value. Otherwise equate PV of inflows to the outflow.
Sensitivity of NPV to cash inflows: % fall in inflows that makes NPV zero = NPV ÷ PV of inflows × 100; Applies when the whole inflow stream changes by the same percentage.
Sensitivity of NPV to initial investment: % rise in investment that makes NPV zero = NPV ÷ PV of investment × 100; Use PV of the outflow; if the investment is all at time 0, this is the investment itself.
Scenario NPV and expected NPV: Expected NPV = Σ (Probability of scenario × NPV of scenario); Probabilities must add to 1. Use only if probabilities are given.
Range of NPV: Range = Best-case NPV − Worst-case NPV; A simple measure of how widely the outcome can vary.
Expected value: E(X) = Σ pᵢ × Xᵢ; Probabilities must sum to 1. Applies to cash flows or to NPVs.
Expected NPV: ENPV = Σ [E(CFₜ) ÷ (1 + r)ᵗ] − Initial outlay; Initial outlay is treated as certain unless the question gives its distribution.
Standard deviation of a distribution: σ = √[Σ pᵢ × (Xᵢ − E(X))²]; Use deviations from the mean, weighted by probability.
σ of NPV, independent yearly cash flows: σ(NPV) = √[Σ σₜ² ÷ (1 + r)²ᵗ]; Square the discount factor. Add variances, then take the root.
σ of NPV, perfectly correlated cash flows: σ(NPV) = Σ [σₜ ÷ (1 + r)ᵗ]; Add the discounted standard deviations. This gives the highest σ.
Coefficient of variation: CV = σ ÷ ENPV; Lower CV means lower risk per rupee of expected NPV. Meaningful when ENPV is positive.
Z value for probability under normal distribution: Z = (X − ENPV) ÷ σ; For P(NPV < 0), use X = 0. Read the area from the normal table.
Expected value at a chance node: EV = Σ (probability × value of branch); Probabilities on the branches leaving the node must sum to 1. Use PV of the branch values at the same date.
Value at a decision node: Value = highest of (value of each alternative after its own outlay or proceeds); Choose the maximum for NPV or profit. Cross out (prune) the rejected branches.
Present value of a later value: PV = Value at time t ÷ (1 + r)^t; Discount the rolled-back value to time 0 before deducting the initial outlay.
Expected NPV of a strategy: Expected NPV = PV of expected inflows (with best later decisions) − initial outlay; Accept if it is positive. Between mutually exclusive strategies, pick the higher expected NPV.
Joint probability of a path: P(path) = product of the branch probabilities along the path; Use it when the question asks for the probability of a particular outcome or for NPV along each path.
Expanded (strategic) NPV: Expanded NPV = Base NPV (without option) + Value of real option; Accept the project if expanded NPV is positive, even when base NPV is negative.
Expected NPV from simulation: Mean NPV = Σ NPVi ÷ n; n is the number of trials. Also report standard deviation and P(NPV < 0).
Abandonment decision: Abandon if salvage value > PV of continuing cash flows; Compare at the decision date using the same discount rate.
Value of abandonment option: Option value = Σ Probability × max(Salvage value − PV of continuing, 0); Applies to each outcome at the decision date, then discount to today.
Expansion option payoff: Payoff = max(PV of expansion inflows − Expansion outlay, 0); Exercise only when the payoff is positive.
Option analogy: Underlying = PV of project inflows; Exercise price = investment outlay; Time = decision window; Volatility = uncertainty of inflows; Use when Black-Scholes is asked for a real option. Higher volatility and longer time raise option value.
Expected NPV: ENPV = Σ Pi × NPVi; Pi is the probability of each outcome. Probabilities must add up to 1.
Standard deviation of a project: σ = √[Σ Pi × (NPVi − ENPV)²]; Measures total risk of one project, in rupees.
Coefficient of variation: CV = σ ÷ ENPV; Risk per rupee of expected NPV. Lower is better. Use it only when ENPV is positive.
Portfolio expected NPV: ENPVp = w1 × ENPV1 + w2 × ENPV2; If both projects are taken in full, weights are 1 and ENPVp is the plain sum.
Portfolio SD of two projects: σp = √(w1²σ1² + w2²σ2² + 2 × w1 × w2 × ρ12 × σ1 × σ2); With both projects in full (w = 1): σp = √(σ1² + σ2² + 2ρ12σ1σ2).
Covariance: Cov12 = ρ12 × σ1 × σ2; If covariance is given, use it in place of ρσ1σ2.
Limiting cases of correlation: ρ = +1: σp = σ1 + σ2 | ρ = 0: σp = √(σ1² + σ2²) | ρ = −1: σp = |σ1 − σ2|; The last holds for two projects taken in full. ρ always lies between −1 and +1.

Quick revision

Risk means outcomes with known or estimable probabilities; uncertainty means probabilities cannot be reliably assigned.
Risk-adjusted discount rate raises the rate for riskier projects; it adjusts the denominator.
Certainty equivalent converts uncertain cash flows to certain ones; it adjusts the numerator.
Certainty equivalent factor = certain cash flow ÷ expected risky cash flow; it lies between 0 and 1 for a risk-averse decision maker.
Sensitivity analysis changes one variable at a time and observes the effect on NPV.
Scenario analysis changes several variables together, such as best, expected and worst cases.
Expected NPV = Σ (probability × NPV of each outcome).
Standard deviation = √Σ p × (x − mean)²; it measures absolute risk.
Coefficient of variation = standard deviation ÷ expected NPV; use it to compare projects of different size.
Decision trees are solved backward, from the final outcomes to the first decision.
Monte Carlo simulation draws random values from assumed distributions to build a distribution of NPV.
Real options value the flexibility to expand, delay, abandon or switch a project.

Common mistakes

Using risk and uncertainty as exact synonyms in a question that asks for the difference. Fix: When asked for the difference, write that risk has known or estimable probabilities and uncertainty does not. Add one example for each.
Choosing the project with the higher expected NPV without looking at spread. Fix: Always compute σ or CV when probabilities are given and comment on both return and risk before concluding.
Discounting certainty equivalent cash flows at the RADR instead of the risk-free rate. Fix: CE already removes the risk from the cash flow. Discount at the risk-free rate only. Using the RADR would count risk twice.
Applying the CE coefficient to the initial outlay when it was not given. Fix: Treat the outlay as certain (coefficient 1) unless the question gives a coefficient for year 0.
Changing several variables at once and calling it sensitivity analysis. Fix: In sensitivity analysis change one variable and hold the rest at base values. Changing a combined set of inputs is scenario analysis.
Dividing the NPV by the wrong base, for example by the annual cash flow instead of the PV of inflows. Fix: Use NPV ÷ PV of inflows for cash-flow sensitivity and NPV ÷ PV of the investment for investment sensitivity.
Adding yearly standard deviations without discounting them Fix: Discount σₜ by (1 + r)ᵗ for correlated flows, or discount σₜ² by (1 + r)²ᵗ for independent flows.
Discounting σ² with (1 + r)ᵗ instead of (1 + r)²ᵗ Fix: Square the discount factor whenever you discount a variance. Take the root only at the end.
Working left to right and choosing the best branch at the first decision without looking ahead. Fix: Always roll back from the right. A first-stage choice is judged by the best value of everything after it.
Forgetting to deduct the outlay on an alternative before comparing it with others at a decision node. Fix: At each decision node, write net value = expected inflows − outlay for every alternative, then compare.

Exam tips

In theory questions, write the difference between risk and uncertainty in two clear lines and add one example each. This scores quickly.
In case-based MCQs, match the stated event to its source: a rival price cut is competitive risk, a rate hike is market risk, rupee movement on an overseas project is international risk.
In numerical questions, show expected value, σ and CV in a table. Step marks are given even if one figure goes wrong.
Always end with a recommendation that mentions both return and risk. A bare number is not a decision.
Read whether the question asks for stand-alone, corporate or market risk. The answer changes with the view.
Check the wording to find the method. Premium or higher rate means RADR; coefficient or sure amount means CE. Do not mix them in one calculation.
Write the discount rate in the first line of your answer. Marks are often given for the correct rate.
The MCQs test concepts: which rate is used in CE, what a lower coefficient means, and which method assumes risk rises each year. Revise these in one line each.