CMA Final · Strategic Financial Management

Investment Decisions, Project Planning and Control: formula sheet

Full chapter guide

Key formulas

Net Present Value (NPV): NPV = Σ [Ct ÷ (1 + k)^t] − C0, for t = 1 to n; Accept if NPV > 0 for an independent project. For mutually exclusive projects of similar scale, choose the highest NPV.
Payback period: Payback = Initial investment ÷ Annual cash inflow (for even inflows); For uneven inflows, add cumulative inflows until the investment is recovered. It ignores cash flows after payback and, in the simple form, the time value of money.
Profitability index: PI = Present value of cash inflows ÷ Initial outlay; PI > 1 means NPV > 0. Useful for ranking under capital rationing.
Incremental cash flow rule: Relevant cash flow = Cash flow with the project − Cash flow without the project; Include opportunity costs and working capital changes. Exclude sunk costs. Ignore financing flows when discounting at the cost of capital.
Incremental cash flow: Incremental cash flow = Cash flows with the project − Cash flows without the project; Test every item with this. If it does not change because of the project, leave it out.
Initial outlay: Initial outlay = Cost of asset + Installation and freight + Increase in working capital − Net proceeds of old asset (if replacing); Do not include sunk costs. For a replacement, adjust the old asset's sale proceeds for tax on gain or tax saved on loss.
Operating cash flow (tax method): OCF = (Revenue − Cash operating costs − Depreciation) × (1 − t) + Depreciation; t is the tax rate. Interest is excluded.
Shortcut form of OCF: OCF = (Revenue − Cash operating costs) × (1 − t) + Depreciation × t; Same answer as above. The second part is the depreciation tax shield.
Depreciation tax shield: Tax shield = Depreciation × t; Applies when the firm has enough taxable profit to use the deduction. Use the depreciation method given in the question.
Terminal cash flow: Terminal cash flow = Working capital released + Salvage value − Tax on gain (or + Tax saved on loss); Gain or loss = Salvage value − Tax book value of the asset.
Opportunity cost: Opportunity cost = Best alternative cash flow forgone, after tax; Include it as a cost of the project. Example: rent forgone on a space the project will use.
Change in working capital: ΔNWC = Increase in current assets − Increase in current liabilities; Outflow when it rises, inflow when it is released.
Payback period: Payback = Years before full recovery + (Unrecovered cost at start of the year ÷ Cash inflow of that year); For even inflows: Initial outlay ÷ Annual cash inflow. Undiscounted cash flows.
Discounted payback: Same as payback, but use present values: PV of inflow = Cash inflow ÷ (1 + r)^t; Always at least as long as simple payback for a positive discount rate. Needs a cost of capital.
ARR: ARR = Average annual accounting profit after depreciation (and tax) ÷ Investment × 100; Investment may be the initial or the average investment. Use whichever the question states. Average investment = (Initial cost + Salvage value) ÷ 2.
Net present value: NPV = Σ [CFt ÷ (1 + r)^t] − Initial outlay; Accept if NPV > 0. For mutually exclusive projects choose the highest NPV.
Internal rate of return: IRR: the r at which Σ [CFt ÷ (1 + r)^t] = Initial outlay. Interpolation: IRR = Lower rate + [NPV at lower rate ÷ (NPV at lower rate − NPV at higher rate)] × (Higher rate − Lower rate); Accept if IRR > cost of capital. For even inflows, outlay ÷ inflow gives the annuity factor to look up in the table.
Profitability index: PI = PV of cash inflows ÷ PV of cash outflows = 1 + (NPV ÷ Initial outlay); Accept if PI > 1. Useful for ranking under a budget limit.
MIRR: MIRR = (Terminal value of inflows ÷ PV of outflows)^(1 ÷ n) − 1, where Terminal value = Σ CFt × (1 + reinvestment rate)^(n − t); PV of outflows is at the finance rate (usually the cost of capital). n is the project life in years.
Crossover (Fisher's) rate: Rate at which NPV of two projects is equal; solve using the incremental cash flows (A − B) and find the IRR of the difference; Below this rate, NPV and IRR rankings can conflict. Above it, they agree.
Profitability index: PI = PV of cash inflows ÷ Initial outlay = 1 + (NPV ÷ Initial outlay); Some books define it as NPV ÷ outlay. Check the question and state your definition. Both give the same ranking.
Capital rationing objective: Maximise Σ NPV of selected projects, subject to Σ outlay ≤ budget; For indivisible projects, test combinations. For divisible projects, rank by PI and part-fund the last project.
Annuity factor: PVAF(r, n) = [1 − (1 + r)^(−n)] ÷ r; Use the table value given in the question if provided.
Equivalent annual annuity: EAA = NPV ÷ PVAF(r, n); n is the project's own life. Choose the project with the higher EAA.
Equivalent annual cost: EAC = PV of all costs ÷ PVAF(r, n); Use when projects only have costs. Choose the lower EAC.
Replacement chain NPV: Chain NPV = NPV + NPV ÷ (1 + r)^n + NPV ÷ (1 + r)^(2n) + ... over the common life; Common life is the lowest common multiple of the project lives. Add each repeat's NPV discounted back to today.
Risk-adjusted discount rate: RADR = Risk-free rate + Risk premium; NPV = Σ [CFt ÷ (1 + RADR)^t] − Initial outlay; Use the higher rate for riskier projects. Cash flows are the expected (uncertain) ones.
Certainty equivalent coefficient: α(t) = Certain cash flow ÷ Risky expected cash flow; α lies between 0 and 1. It usually falls for later years as uncertainty grows.
NPV by certainty equivalent: NPV = Σ [α(t) × CFt ÷ (1 + Rf)^t] − Initial outlay; Discount at the risk-free rate only. Using RADR here counts risk twice.
Expected NPV: E(NPV) = Σ P(i) × NPV(i); P(i) are the probabilities of the scenarios and they must add up to 1.
Standard deviation of NPV (scenarios): σ = √[ Σ P(i) × (NPV(i) − E(NPV))² ]; Use the NPV of each scenario, not the cash flow.
Standard deviation of NPV (yearly cash flows, independent): σ(NPV) = √[ Σ σt² ÷ (1 + r)^(2t) ]; Valid when cash flows of different years are independent. Use the risk-free rate unless the question says otherwise.
Standard deviation of NPV (perfectly correlated): σ(NPV) = Σ [ σt ÷ (1 + r)^t ]; Valid only when cash flows of all years are perfectly correlated.
Coefficient of variation: CV = σ ÷ E(NPV); Risk per rupee of expected NPV. Compare projects with different sizes. Lower is better. Meaningful only when E(NPV) is positive.
Probability of NPV below zero (normal assumption): Z = (0 − E(NPV)) ÷ σ; read the area from the normal table; Use only when the question tells you to assume NPV is normally distributed.
Sensitivity of a variable: % change that makes NPV zero = NPV ÷ PV of that variable's cash flows × 100; A smaller percentage means NPV is more sensitive to that variable.
Decision tree roll-back: EMV at a chance node = Σ (probability × value of branch); at a decision node choose the highest EMV; Work from right to left. Deduct outlays on the branch where they occur.
Initial outlay in replacement: Cost of new asset − Sale proceeds of old asset − Tax saved on loss on sale (or + Tax payable on gain); Add any extra working capital. Use only cash items and the tax effect of the sale.
Incremental annual cash flow (CFAT): (Savings or extra profit before depreciation − Incremental depreciation) × (1 − t) + Incremental depreciation; The same result: Savings × (1 − t) + Incremental depreciation × t. Here t is the tax rate.
Replacement NPV: NPV = PV of incremental CFAT + PV of incremental terminal cash flows − Net initial outlay; Replace if NPV > 0. For unequal lives, use the equivalent annual approach.
Post-tax cost of debt: Kd (post-tax) = Interest rate × (1 − t); The usual discount rate in lease versus buy, because lease and loan flows are debt-like.
Cost of leasing: PV of rentals × (1 − t); Take rentals after tax. Adjust the timing if rentals are paid at the start of the year.
Cost of buying: Purchase price − PV of depreciation tax shield (Depreciation × t) − PV of salvage after tax; Add PV of after-tax maintenance or insurance if the buyer bears them and the lessor does not.
Net advantage of leasing: NAL = Net cost of buying − Net cost of leasing; Lease if NAL > 0. All figures at present value.
Nominal and real rate: (1 + nominal rate) = (1 + real rate) × (1 + inflation rate); The relation is multiplicative. The simple sum is only an approximation.
Abandonment rule: Abandon if Salvage value (after tax) > PV of future cash flows from continuing; Test it at each possible year-end. Compare at the same date.
PERT expected time: te = (to + 4tm + tp) ÷ 6; to = optimistic, tm = most likely, tp = pessimistic time.
Activity variance and standard deviation: σ² = [(tp − to) ÷ 6]²; σ = (tp − to) ÷ 6; Project variance is the sum of variances of critical-path activities only.
Project standard deviation: σ(project) = √Σσ² of critical activities; Assumes activities are independent.
Probability of completion by a date: Z = (Due date − Expected project time) ÷ σ(project); Read probability from the normal table.
Slack / float: Slack = LST − EST = LFT − EFT; Critical activities have zero slack.
Cost overrun: Cost overrun % = (Actual cost − Budgeted cost) ÷ Budgeted cost × 100; Time overrun is computed the same way on durations.
Crash cost per period: (Crash cost − Normal cost) ÷ (Normal time − Crash time); Crash the cheapest critical activity first.

Quick revision

Use incremental, after-tax cash flows only; ignore sunk costs and allocated overheads that do not change.
Include opportunity costs and the working capital needed, and recover working capital at the end.
Do not include interest in project cash flows when discounting at the cost of capital; the rate already reflects financing.
Depreciation is not a cash flow, but it creates a tax shield: depreciation × tax rate.
NPV = present value of cash inflows − present value of cash outflows; accept if NPV > 0.
Accept on IRR if it is higher than the required rate, but check conflicts with NPV for mutually exclusive projects.
Profitability index = present value of inflows ÷ initial outlay; accept if PI > 1.
Payback ignores time value and cash flows after the payback period; discounted payback fixes only the first problem.
When funds are rationed, choose the combination of projects with the highest total NPV within the budget.
For unequal lives, compare using the equivalent annual approach or a common life, not raw NPV.
Sensitivity analysis changes one variable at a time; scenario analysis changes several together.
Post-audit compares actual results with the plan to learn and correct future estimates.

Common mistakes

Treating capital budgeting as only NPV and IRR calculation. Fix: Always mention identification, screening, appraisal, selection, planning, control and post-audit when the question asks about the process.
Including sunk costs, such as past feasibility study fees, in project cash flows. Fix: Include only costs that change because of the decision. Include opportunity costs, exclude sunk costs.
Including sunk costs such as a feasibility study or market survey already paid for Fix: Ask whether the cash changes if you accept or reject. Money already spent does not change, so leave it out.
Deducting depreciation as a cash cost and not adding it back Fix: Depreciation only reduces tax. Add it back after tax, or use the shortcut: Depreciation × t.
Treating payback or ARR as a time-value-adjusted measure, or ignoring cash flows after the payback period Fix: State its limits in the answer: no time value, ignores later flows. Use it only as a secondary screen.
Calculating ARR on cash flow instead of accounting profit Fix: For ARR, deduct depreciation from the cash flow, then tax if stated, to get accounting profit. Check whether investment is initial or average.
Ranking by PI and stopping, even when projects are indivisible. Fix: With indivisible projects, always check whether another combination uses the budget better and gives a higher total NPV.
Comparing raw NPVs of projects with different lives. Fix: Whenever mutually exclusive projects have different lives, convert to EAA or use a replacement chain first.
Discounting certainty equivalent cash flows at the risky rate (RADR). Fix: CE flows already carry the risk adjustment. Discount them at the risk-free rate.
Taking the average of the cash flows as the answer to a probability question and skipping NPV of each scenario. Fix: Compute NPV for each scenario, then the expected NPV, then the standard deviation of NPV.

Exam tips

Write the stages in order. Examiners give credit for sequence and for linking each stage to the case facts.
In MCQs, watch the wording: mutually exclusive, independent and contingent are tested as definitions and as small cases.
In case answers, mention non-financial appraisal before moving to numbers, then end with a clear recommendation.
If a question gives sunk costs or allocated overheads, say clearly that they are excluded and why.
Link control to post-audit when the question asks about the full cycle.
Start every answer by listing relevant and irrelevant items in one or two lines. Examiners often award marks for correctly excluding sunk costs and interest.
Show the depreciation tax shield as a separate line. It is easy to check, and it protects your marks if you make an arithmetic slip elsewhere.
Always put working capital in both Year 0 and the final year. In MCQs, check whether the question says it is recovered.