Project Evaluation: formula sheet

Key formulas

Net cash outlay (initial investment)

Initial outlay = Cost of asset + Installation and transport costs + Increase in working capital − Sale proceeds of old asset (adjusted for tax effect, if replacing)

Include only incremental cash flows. Ignore sunk costs already spent.

Decision rule (accept or reject)

Accept if the project meets the firm's criterion, for example NPV > 0 or IRR > cost of capital

Detailed techniques are covered in the later topics on payback, ARR, NPV, PI and IRR.

Four-fold appraisal

Technical + Commercial + Financial + Economic viability

Use this as the answer skeleton when the question asks for stages of appraisal.

Initial outlay (new project)

Initial outlay = Cost of asset + Freight and installation + Increase in net working capital

Capitalised costs are part of the depreciable base. Working capital is not depreciated.

Initial outlay (replacement)

Net outlay = Cost of new asset + Increase in working capital − Sale proceeds of old asset ± Tax effect on sale of old asset

A taxable gain adds to the outlay as tax paid. A loss that can be set off reduces the outlay as tax saved. State your assumption.

Annual depreciation (SLM)

Depreciation = (Cost + Installation − Salvage value) ÷ Life in years

Use the method the question gives. For WDV, depreciation = opening book value × rate.

Operating cash flow (NOPAT method)

OCF = (Sales − Cash operating costs − Depreciation) × (1 − t) + Depreciation

Here t is the tax rate. Interest is excluded.

Operating cash flow (tax shield method)

OCF = (Sales − Cash operating costs) × (1 − t) + Depreciation × t

Gives the same answer as the NOPAT method. Use it as a check.

Terminal cash flow

Terminal CF = Salvage value − Tax on gain (or + tax saved on loss) + Recovery of working capital

Gain or loss is measured against the book value at the end of the project.

Tax on sale of asset

Tax effect = (Sale price − Book value) × t

Positive means tax payable. Negative means tax saved, if the loss can be set off against other income.

Incremental cash flow

Incremental CF = Cash flow with project − Cash flow without project

Use this for replacement and expansion decisions.

Payback period (even inflows)

Payback = Initial investment ÷ Annual cash inflow

Use only when every year's cash inflow is equal.

Payback period (uneven inflows)

Payback = Years fully recovered + (Unrecovered cost at start of the final year ÷ Cash inflow of that year)

Build the cumulative cash inflow column first. Assumes inflows arise evenly within the year.

Discounted cash inflow

PV = Cash inflow × 1 ÷ (1 + r)^n

r is the cost of capital, n is the year. Use PV factors if given.

Discounted payback

Same as uneven payback, but using cumulative present values

Compare against the initial outlay, not against undiscounted flows.

ARR on initial investment

ARR = Average annual profit after depreciation and tax ÷ Initial investment × 100

Use the basis the question states.

ARR on average investment

Average investment = (Initial investment + Salvage value) ÷ 2; ARR = Average annual profit ÷ Average investment × 100

If extra working capital is needed, add it to both the initial and the average investment, as the question's treatment directs.

Average annual profit

Average profit = Total profit after depreciation and tax over life ÷ Number of years

Profit is accounting profit, not cash flow.

Net Present Value

NPV = Σ [Ct ÷ (1 + k)^t] − C0

Ct is the cash inflow in year t, k is the discount rate, C0 is the initial outlay. Accept if NPV > 0.

Profitability Index

PI = PV of cash inflows ÷ PV of cash outflows (initial outlay)

Also PI = 1 + NPV ÷ outlay when the outlay is all at time zero. Accept if PI > 1.

Net PI

Net PI = PI − 1 = NPV ÷ outlay

Accept if Net PI > 0.

IRR definition

Σ [Ct ÷ (1 + r)^t] − C0 = 0

r is the IRR. Accept if IRR > cost of capital (required rate).

Interpolated IRR

IRR = L + [NPV at L ÷ (NPV at L − NPV at H)] × (H − L)

L is the lower trial rate with positive NPV, H is the higher rate with negative NPV. The result is an approximation.

Annuity shortcut for IRR

Annuity factor = Initial outlay ÷ Annual inflow

For equal annual inflows, find the rate whose annuity factor for the project life equals this value.

Modified IRR (MIRR)

MIRR = (Terminal value of inflows ÷ PV of outflows)^(1/n) − 1

Terminal value compounds inflows to year n at the reinvestment rate. Outflows are discounted at the financing rate. It removes the assumption that inflows are reinvested at the IRR.

Net present value

NPV = Σ [Cash flow in year t ÷ (1 + r)^t] − Initial outlay

r is the cost of capital. Use NPV as the final decision rule for mutually exclusive projects.

IRR condition

Σ [Cash flow in year t ÷ (1 + IRR)^t] − Initial outlay = 0

Accept if IRR is greater than the cost of capital. IRR is unreliable for ranking exclusive projects.

Crossover rate

Rate r at which NPV(A) = NPV(B), i.e. IRR of (B − A) cash flows

Below this rate the NPV ranking and IRR ranking can conflict. Above it they agree.

Profitability index

PI = Present value of cash inflows ÷ Initial outlay = 1 + (NPV ÷ Outlay)

Use to rank divisible projects under single-period capital rationing.

Present value annuity factor

PVAF(r, n) = [1 − (1 + r)^−n] ÷ r

Used to find the EAA.

Equivalent annual annuity

EAA = NPV ÷ PVAF(r, n)

Choose the project with the higher EAA when lives are unequal and projects are repeatable.

Capital rationing rule

Maximise Σ NPV subject to Σ Outlay ≤ Budget

Divisible: rank by PI and fill the budget, taking a fraction of the last project. Indivisible: compare feasible combinations.

Risk-adjusted discount rate

r(adj) = Risk-free rate + Risk premium

The premium is higher for riskier projects. Use r(adj) for every year's cash flow.

NPV under RADR

NPV = Σ [CFt ÷ (1 + r(adj))^t] − Initial outlay

Cash flows stay as expected values. Only the rate changes.

Certainty equivalent coefficient

α = Certain cash flow ÷ Risky (expected) cash flow

α lies between 0 and 1. A lower α means higher risk. CE cash flow = α × expected cash flow.

NPV under certainty equivalent

NPV = Σ [αt × CFt ÷ (1 + Rf)^t] − Initial outlay

Discount at the risk-free rate Rf, not the cost of capital, because risk is already removed.

Expected NPV

E(NPV) = Σ [Pi × NPVi]

Pi is the probability of scenario i. The probabilities must add up to 1.

Standard deviation of NPV

σ = √ Σ [Pi × (NPVi − E(NPV))²]

Measures the absolute spread of outcomes around the expected NPV.

Coefficient of variation

CV = σ ÷ E(NPV)

Risk per rupee of expected return. Lower CV means less risk per rupee. Use it when expected NPVs differ.

Sensitivity measure

Sensitivity = % change in NPV ÷ % change in the variable

The variable with the largest value is the most critical one.

Decision tree rollback

EMV at chance node = Σ [P × payoff]; at decision node choose the highest EMV

Start at the right-hand end. Subtract any cost on a branch before comparing.

Debt Service Coverage Ratio (DSCR)

DSCR = (PAT + Depreciation + Interest on term loan + other non-cash charges) ÷ (Interest on term loan + Repayment of principal in the year)

Use the year's figures. Interest is added back in the numerator because it is also part of debt service in the denominator. Some texts use PAT + depreciation + interest; follow the question's given data.

Average DSCR

Average DSCR = Σ (annual cash available for debt service) ÷ Σ (annual debt service) over the loan period

Total cash available divided by total debt service. It is not the simple mean of yearly ratios unless the question says so.

Net social benefit

Net social benefit = Social benefits − Social costs (both at shadow prices)

Discount at the social discount rate to get social NPV.

Shadow price

Shadow price = Market price × Conversion factor

The conversion factor is below 1 when the market price overstates the social value, and above 1 when it understates it.

Social NPV

Social NPV = Σ [Net social benefit in year t ÷ (1 + r)^t], t = 0 to n

r is the social discount rate. Accept the project if social NPV is positive.

Quick revision

• NPV = present value of cash inflows − present value of cash outflows; accept if NPV is greater than zero.
• PI = present value of inflows ÷ present value of outflows; accept if PI is greater than 1.
• IRR is the discount rate at which NPV is zero; accept if IRR is greater than the cost of capital.
• Payback is the time taken to recover the initial investment; it ignores cash flows after that point and the time value of money in its simple form.
• ARR = average accounting profit ÷ investment (initial or average, as the question states); it uses profit, not cash flow.
• Use incremental, after-tax cash flows only; ignore sunk costs and include opportunity costs.
• Depreciation is a non-cash charge. Either add it back in full to profit after tax (which already captures its tax saving), or compute cash flow before depreciation and tax, deduct tax on it by multiplying by (1 − t), then add depreciation × t. Do not do both.
• Recover working capital and salvage value in the final year.
• For mutually exclusive projects with conflicting ranks, NPV is generally preferred.
• Under capital rationing, select the combination of projects with the highest total NPV within the budget.
• Sensitivity analysis changes one variable at a time; scenario analysis changes several together.
• Social cost benefit analysis weighs benefits and costs to society, not only to the firm.

Common mistakes

• Treating capital budgeting as the same thing as working capital management. Fix: Capital budgeting deals with long-term assets and returns over several years. Working capital deals with day-to-day current assets and liabilities.
• Calling financial appraisal the only test of a project. Fix: State that technical, commercial, financial and economic viability are all required. A project that fails any one needs revision or rejection.
• Subtracting depreciation as a cash outflow, or forgetting to add it back Fix: Deduct depreciation only to find tax. Then add it back, or use OCF = (S − C) × (1 − t) + Dep × t.
• Including interest in operating cash flows Fix: Exclude financing costs. The discount rate covers them. Including interest double counts the cost of funds.
• Using profit instead of cash inflow for payback. Fix: Add back depreciation to profit after tax to get cash inflow.
• Forgetting to deduct depreciation when computing ARR. Fix: ARR uses accounting profit after depreciation and tax.
• Forgetting to subtract the initial outlay and reporting the present value of inflows as NPV. Fix: Write NPV = PV of inflows − outlay as the last line of every table.
• Using PI = NPV ÷ outlay and calling it PI. Fix: PI is inflows ÷ outlay. NPV ÷ outlay is Net PI, which is PI − 1.
• Recommending the project with the higher IRR when it conflicts with NPV for mutually exclusive projects. Fix: Follow NPV when the two conflict. State that IRR assumes reinvestment at the IRR, while NPV assumes the cost of capital.
• Comparing raw NPVs of projects with different lives. Fix: Convert each NPV to an EAA using the annuity factor for that project's life and compare the EAAs.

Exam tips

• Use the four-fold appraisal as a fixed skeleton for any question on viability. It keeps your answer structured.
• In case questions, tie each fact to a stage by name. Marks go to application, not just recall.
• Keep definitions short and spend the time on features, types and the link to the case.
• Remember the distinction between mutually exclusive, independent and contingent projects. It is a common short-note topic.
• Always end with a conclusion or recommendation, even for theory questions.
• Show the Year 0 calculation as a separate working note. Examiners give marks for each component of the outlay.
• Write your assumptions in one line, such as tax set-off on loss, working capital recovered at the end, and depreciation method used. This earns marks when the question is silent.
• If the question gives interest or financing details, mention that you excluded them and why.