CFA Level I · CFA Level I Exam

Capital Investments and Capital Allocation: formula sheet

Full chapter guide

Key formulas

Four steps of capital budgeting: Generate ideas → Analyze proposals → Create the capital budget → Monitor and post-audit; Know the order. Questions often ask which step an activity belongs to.
NPV decision rule: NPV = Σ CFt ÷ (1 + r)^t, for t = 0 to n; accept if NPV > 0; CF0 is normally negative. r is the project's required rate of return. NPV is the main criterion in the principles.
Incremental cash flow principle: Incremental CF = CF with the project − CF without the project; Excludes sunk costs. Includes opportunity costs and externalities.
Project categories: Replacement (maintenance or cost reduction), Expansion, New product or service, Regulatory/safety/environmental, Other; Mandatory projects are usually the regulatory, safety and environmental group.
Incremental cash flow: Incremental CF = CF with the project − CF without the project; Include opportunity costs and externalities. Exclude sunk costs and financing costs.
Initial outlay: Outlay = FCInv + NWCInv − Sal₀ + T(Sal₀ − B₀); FCInv = fixed capital investment including installation; NWCInv = increase in net working capital; Sal₀ = sale proceeds of the old asset; B₀ = its book value; T = tax rate. The last two terms apply only to replacements. The tax term is a cost when there is a gain and a saving when there is a loss.
Operating cash flow (two forms): CF = (S − C − D)(1 − T) + D = (S − C)(1 − T) + T × D; S = incremental sales, C = incremental cash operating costs, D = depreciation. Both forms give the same answer.
Depreciation tax shield: Tax shield = T × D; Real cash saving each year, as long as the firm has taxable income to shelter.
Terminal cash flow: TNOCF = Sal_T + NWCInv − T(Sal_T − B_T); Sal_T = salvage at the end; B_T = book value then; NWCInv = working capital recovered. Add it to the final year's operating cash flow.
Straight-line depreciation: D = (Cost − Salvage for depreciation) ÷ Useful life; Cost includes installation if it is capitalized.
Net present value: NPV = CF₀ + Σ [CFₜ ÷ (1 + r)ᵗ], t = 1 to n; CF₀ is the initial outlay and is negative. r is the required rate of return.
Internal rate of return: 0 = CF₀ + Σ [CFₜ ÷ (1 + IRR)ᵗ]; IRR is the rate that sets NPV to zero. Solve it on the calculator.
NPV decision rule: Accept if NPV > 0; reject if NPV < 0; For independent projects, accept every project with positive NPV. For mutually exclusive projects, choose the highest positive NPV.
IRR decision rule: Accept if IRR > required return; reject if IRR < required return; Gives the same answer as NPV for a standalone project with conventional cash flows.
Crossover rate: Crossover rate = IRR of the difference in cash flows (Project A − Project B); At this rate both projects have equal NPV. Subtract so the first cash flow is negative.
Conventional cash flow pattern: One outflow first, then only inflows; Non-conventional patterns, with more than one sign change, can give multiple IRRs or no IRR.
Payback period: Years before full recovery + (unrecovered amount at start of year ÷ cash flow in that year); Assumes cash flows arrive evenly through the recovery year. Uses undiscounted cash flows.
Discounted payback: Same as payback, but use PV of each cash flow = CF ÷ (1 + r)^t; Always longer than or equal to simple payback for positive rates.
Average accounting rate of return: AAR = Average net income ÷ Average book value of investment; Average net income is after depreciation. The denominator follows the question's definition: often (initial investment + salvage value) ÷ 2 under straight-line depreciation, or sometimes the initial investment.
Profitability index: PI = PV of future cash flows ÷ Initial investment = 1 + NPV ÷ Initial investment; Accept if PI > 1; reject if PI < 1. Equivalent to NPV > 0 for a single project.
Net present value: NPV = Σ CFt ÷ (1 + r)^t, for t = 0 to n; CF0 is usually negative. Accept if NPV > 0. For mutually exclusive projects, pick the highest positive NPV.
Internal rate of return: IRR is the r that makes NPV = 0; Accept an independent project with conventional cash flows if IRR > required rate of return.
Crossover rate: Crossover rate: the r at which NPV(A) = NPV(B); Find it as the IRR of the difference in cash flows. The sign of the difference does not affect the crossover rate, so you can use A − B or B − A.
Multiple IRR condition: Number of real IRRs above −100% ≤ number of sign changes in the cash flows; This is an upper limit, not an exact count. Conventional flows have one sign change, so at most one such IRR. More sign changes allow several IRRs, or none.
Decision rule for mutually exclusive projects: Choose the project with the highest NPV, if NPV > 0; Use NPV when the rankings conflict.
NPV including real options: Project NPV = NPV (based on DCF, no options) − cost of options + value of options; Option value is never negative. The cost of options is the up-front cost of acquiring or creating the option, not the later exercise cost of expansion, which is already inside the option value. If no separate cost of acquiring the option is given, the formula reduces to base NPV plus option value.
Abandonment option (put-like): Payoff at decision date = max(salvage value, value of continuing cash flows); It sets a floor on project value. Worth more when salvage value is high and future cash flows are uncertain.
Expansion option (call-like): Payoff = max(0, value of expanded project − cost of expansion); Management expands only if the expansion NPV is positive at the decision date.
Timing option rule: Invest now only if NPV now ≥ value of waiting; Waiting has value when information will arrive. It is costly if competitors can enter or cash flows are lost.
Effect of volatility: Higher volatility of the underlying → higher option value; Because the holder can ignore bad outcomes, more uncertainty raises the value of the right.
Profitability index: PI = PV of future cash flows ÷ initial investment = 1 + NPV ÷ initial investment; Use PI to rank divisible projects under a single-period budget limit. PI above 1 means NPV above 0.
Capital rationing objective: Maximize Σ NPV of the chosen projects, subject to Σ investment ≤ budget; For indivisible projects, test feasible combinations. Do not simply fund the highest PI first.
Expected NPV from scenarios: Expected NPV = Σ (probability of scenario × NPV in scenario); Probabilities must sum to 1. Expected NPV is a probability-weighted average, not the base-case NPV.
Risk-adjusted discount rate: Project rate = base required return + risk premium for the project's extra risk; A higher rate lowers NPV. Do not also cut the cash flows for the same risk, or you double count.
Hard vs soft rationing: Hard = external, binding limit. Soft = internal, flexible limit.; The test usually asks which one the firm faces and whether it can exceed the budget.
Economic income: Economic income = Cash flow + (Ending market value − Beginning market value) = Cash flow − Economic depreciation; Cash flow plus the change in market value; the change is the negative of economic depreciation.
Economic depreciation: Economic depreciation = Beginning market value − Ending market value; Can be negative if market value rises.
Economic profit (residual income): EP = NOPAT − (WACC × Capital) = EBIT(1 − t) − WACC × Capital; Positive EP means the project earns more than its capital cost. Capital is measured at the beginning of the period.
NPV and economic profit: NPV = Σ EP_t ÷ (1 + WACC)^t, for t = 1 to N; Holds when the capital charge uses beginning-of-period capital and the initial investment is part of the capital base, so the outlay is not a separate term.
Market value added: MVA = Market value of firm − Capital invested = PV of all future economic profits; Shows value created over the capital provided; it equals the sum of the NPVs of the firm's projects.
Claims valuation: Firm value = Value of debt + Value of equity; Project value is split among claimants.
Stock price effect: Change in price per share ≈ Unanticipated NPV ÷ Shares outstanding; Applies when the project was not already anticipated by the market and there are no new financing effects.

Quick revision

Use incremental, after-tax cash flows only; ignore sunk costs and include opportunity costs and externalities.
Financing costs are not in project cash flows; they are captured in the discount rate.
NPV = Σ CFt ÷ (1 + r)^t, with CF0 as the initial outlay. Accept if NPV > 0.
Accept on IRR when IRR is greater than the required rate of return.
For mutually exclusive projects with conflicting ranks, choose the higher NPV.
Conflicts arise from differences in project size or in the timing of cash flows.
Payback ignores time value and cash flows after the payback period; discounted payback fixes the first but not the second.
Profitability index = PV of future cash flows ÷ initial investment; accept if PI > 1.
Changes in net working capital are cash flows, usually recovered at the project's end.
Real options include timing, sizing, flexibility and fundamental options; they add value to base NPV.
Under capital rationing, pick the combination of projects with the highest total NPV within the budget.
Sensitivity changes one input, scenario analysis changes several together, and simulation uses probability distributions.

Common mistakes

Including sunk costs, such as a feasibility study already paid for, in project cash flows. Fix: Ask whether the cash flow changes if you accept or reject. Money already spent does not change, so exclude it.
Putting interest or financing costs into the project cash flows. Fix: Financing costs are reflected in the discount rate. Including them in cash flows double counts them.
Including sunk costs such as a past study fee or old equipment's original cost. Fix: Ask: would this cash flow change if we accept the project? If not, leave it out.
Subtracting depreciation and stopping, so the cash flow is just after-tax profit. Fix: Add depreciation back, or use (S − C)(1 − T) + T × D. Depreciation only matters through its tax effect.
Forgetting to subtract the initial outlay and reporting the present value of inflows as the NPV. Fix: Always include CF₀ with its negative sign. NPV = PV of inflows − outlay. Check that your answer is plausible against the size of the outlay.
Entering the outlay as a positive number in the calculator. Fix: Press the +/− key when you enter CF0. If IRR returns an error or a strange number, check your signs first.
Saying discounted payback considers all cash flows. Fix: Remember it still ignores cash flows after the payback point. Only NPV and IRR use all cash flows.
Using PV of all cash flows including the outlay in the PI numerator. Fix: PI numerator is PV of future inflows only. NPV is that minus the outlay.
Choosing the project with the higher IRR when the projects are mutually exclusive Fix: For mutually exclusive projects, rank by NPV. IRR ignores project size and timing.
Saying a project with several sign changes always has several IRRs Fix: Sign changes set an upper limit on the number of real IRRs above −100%. The project may have fewer, or none.

Exam tips

Memorize the four steps in order and the purpose of the post-audit. Process questions are usually direct.
Learn the five project categories and link each to a one-word action: replace, cut costs, expand, launch, comply.
On principle questions, scan options for sunk costs, financing costs and ignored externalities. The two wrong choices usually contain one of these errors.
All questions have three options and no penalty for wrong answers, so never leave one blank. Eliminate first, then guess if needed.
This topic is conceptual, so expect no heavy calculation. Save time here for the NPV and IRR questions.
Read the question for what the cash flow is asked at: time 0, a middle year or the final year. Many wrong options are right numbers for the wrong year.
Mark sunk cost, opportunity cost and externality items first. A conceptual question often asks only which item should be included, and the answer is the one that changes with the decision.
Check whether installation is capitalized, since it raises depreciation as well as the outlay.