Capital Budgeting: formula sheet

Key formulas

Initial outlay

Initial outlay = Cost of new asset + Installation and freight + Increase in net working capital − After-tax proceeds from sale of old asset

If the old asset is sold at a loss or gain against book value, adjust the tax effect on that loss or gain.

Annual operating cash flow (profit method)

CFAT = (Sales − Operating costs excluding depreciation − Depreciation) × (1 − t) + Depreciation

Operating costs exclude interest. Here t is the tax rate.

Annual operating cash flow (shield method)

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

Gives the same answer as the profit method. Handy for speed.

Depreciation tax shield

Tax shield = Depreciation × t

Applies only if the firm has enough taxable profit to absorb the depreciation.

Straight-line depreciation

Depreciation = (Cost − Salvage value) ÷ Life

Use the method given in the question.

After-tax salvage value

After-tax salvage = Sale price − t × (Sale price − Book value)

If sale price is below book value, the loss reduces tax and so adds to cash.

Terminal cash flow

Terminal CF = After-tax salvage value + Working capital released

Add this to the final-year operating cash flow.

Incremental cash flow for replacement

Incremental CFAT = CFAT of new asset − CFAT of old asset

Compare yearly, including the change in depreciation.

Payback period (equal annual inflows)

Payback = Initial investment ÷ Annual cash inflow

Use only when yearly inflows are the same. Cash inflow means profit after tax plus depreciation (and other non-cash charges).

Payback period (unequal inflows)

Payback = Full years before recovery + (Unrecovered amount at start of the year ÷ Cash inflow of that year)

Work from cumulative cash inflows. Assumes inflows arise evenly within the year.

Discounted payback period

Same as payback, but using Present value of inflows = Cash inflow × PV factor

Cumulate discounted inflows until they equal the initial outlay.

Accounting rate of return (on average investment)

ARR = Average annual profit after tax ÷ Average investment × 100

Average investment = (Initial investment + Salvage value) ÷ 2. Some books add working capital that is released at the end; follow the question.

Accounting rate of return (on initial investment)

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

Use this when the question says initial or original investment.

Average annual profit

Average profit = Sum of profits after depreciation and tax over the life ÷ Number of years

Profit is after depreciation. Cash inflow is not the same as profit.

Payback reciprocal

Payback reciprocal = Annual cash inflow ÷ Initial investment × 100

A rough estimate of IRR only if the life is at least twice the payback and inflows are equal.

Net Present Value

NPV = Σ [Ct ÷ (1 + k)^t] − C0

Ct is the cash inflow in year t, k is the cost of capital, C0 is the initial outlay. Accept if NPV > 0.

Present value using factors

PV = Cash flow × PV factor at k% for year t

In exams, PV factors are usually given in the question. Use them as given.

Profitability Index

PI = PV of cash inflows ÷ PV of cash outflows

Also written as 1 + (NPV ÷ initial outlay) when the whole outflow occurs at time zero. Accept if PI > 1.

IRR condition

Σ [Ct ÷ (1 + r)^t] = C0, that is NPV at r = 0

r is the IRR. Accept if r > cost of capital.

IRR for equal annual inflows (starting point)

Annuity factor = Initial outlay ÷ Annual inflow

Look up this factor in the annuity table for the given number of years. The rate nearest to it is your trial rate.

IRR by interpolation

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

L is the lower trial rate (PV above outlay), H is the higher trial rate (PV below outlay). The two rates should be 1% to 5% apart.

Modified IRR

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

Terminal value is the future value of all inflows at the cost of capital at the end of year n. Accept if MIRR > cost of capital.

Decision rules

NPV > 0, PI > 1, IRR > k: accept. NPV < 0, PI < 1, IRR < k: reject.

For mutually exclusive projects, NPV is the preferred criterion because it measures absolute wealth added.

Net Present Value

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

k is the cost of capital. Accept if NPV > 0. Among mutually exclusive projects, choose the highest NPV.

Internal Rate of Return

IRR is the rate r at which Σ [CFt ÷ (1 + r)^t] = Initial outlay, i.e. NPV = 0

Accept if IRR > cost of capital. IRR is a rate, not a rupee gain, so it ignores scale.

Profitability Index

PI = PV of cash inflows ÷ Initial outlay = 1 + (NPV ÷ Initial outlay)

PI > 1 means NPV > 0. Useful for ranking under capital rationing.

Crossover rate

Crossover rate = rate at which NPV of project A = NPV of project B (IRR of the incremental cash flows A − B)

If cost of capital is below the crossover rate, the methods may conflict. Above it, they agree.

Decision rule for conflict

Mutually exclusive projects: choose the highest positive NPV

Reason: NPV assumes reinvestment at the cost of capital and maximises shareholder wealth.

Rationing rule

Choose the feasible set with the highest total NPV subject to Σ outlays ≤ budget

For divisible projects, rank by PI. For indivisible projects, compare combinations.

Risk-adjusted discount rate

RADR = Risk-free rate + Risk premium

Use RADR to discount the expected (uncertain) cash flows. Higher risk means a higher premium.

NPV using RADR

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

CFt is the expected cash flow in year t.

Certainty equivalent coefficient

α = Certain cash flow ÷ Risky cash flow

α lies between 0 and 1. It usually falls as the year gets further away or the risk gets higher.

NPV using certainty equivalent

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

Discount at the risk-free rate Rf only. The risk is already removed from the cash flows.

Expected NPV

Expected NPV = Σ (Probability × NPV of the outcome)

Probabilities across all outcomes must add up to 1. Used in scenario analysis and decision trees.

Standard deviation of NPV

σ = √[Σ P × (NPV − Expected NPV)²]

A larger σ means higher risk.

Coefficient of variation

CV = σ ÷ Expected NPV

Risk per rupee of expected return. Use it to compare projects of different size. Lower CV is better.

Break-even change in a variable (sensitivity)

% change that makes NPV zero = NPV ÷ PV of that item × 100

For cash inflows, the PV of the inflows. For the outlay, the outlay itself. A smaller % means the NPV is more sensitive to that variable.

Quick revision

• Use incremental, after-tax cash flows; ignore sunk costs and include opportunity costs.
• Depreciation is not a cash flow, but its tax shield is: depreciation × tax rate.
• Payback period = time to recover the initial investment; it ignores cash flows after payback and the time value of money.
• ARR = average annual accounting profit ÷ investment (initial or average, as the question says).
• NPV = Σ (cash inflow ÷ (1 + k)^t) − initial outlay; accept if NPV > 0.
• PI = present value of inflows ÷ initial outlay; accept if PI > 1.
• IRR is the rate at which NPV = 0; accept if IRR exceeds the cost of capital.
• Find IRR by trial rates and then interpolate: lower rate + (NPV at lower rate ÷ difference in NPVs) × rate gap.
• For mutually exclusive projects with conflicting ranks, prefer the higher NPV.
• Conflicts arise from differences in project size, timing of cash flows or project life.
• Under capital rationing, choose the combination of projects with the highest total NPV within the budget.
• Risk tools: certainty equivalent adjusts cash flows; risk-adjusted discount rate adjusts the rate; sensitivity changes one variable at a time.

Common mistakes

• Confusing capital budgeting with a cash budget or working capital planning. Fix: Capital budgeting is about long-term investment in fixed assets or projects. A cash budget covers short-term cash inflows and outflows.
• Mixing up mutually exclusive and independent proposals. Fix: Ask: if I accept A, can I still accept B? If yes, they are independent. If no, they are mutually exclusive. If B is possible only when A is accepted, B is contingent.
• Treating depreciation as a cash outflow, or forgetting to add it back Fix: After computing PAT, always add depreciation back. Or use the shield method: after-tax cash profit plus depreciation × t.
• Including interest in the cash flows Fix: Leave out interest and financing costs. The cost of capital used as the discount rate covers them.
• Using profit instead of cash inflow in payback. Fix: Add back depreciation to profit after tax before computing payback. Use profit only for ARR.
• Forgetting to deduct depreciation when finding profit for ARR. Fix: If cash inflows are given, subtract depreciation to get profit before using ARR.
• Discounting the initial outlay at year 1 instead of treating it as year 0. Fix: The outlay made today has a factor of 1. Subtract it directly from the total PV of inflows.
• Using accounting profit instead of cash flow, or deducting depreciation from the inflow. Fix: Add back depreciation to profit after tax to get cash inflow, unless the question already gives net cash flows.
• Choosing the project with the higher IRR when it conflicts with NPV. Fix: For mutually exclusive projects, always choose the higher NPV, since it shows the rupee gain to shareholders.
• Saying the conflict arises because one method is wrong. Fix: State the cause: differences in scale, timing or life, and the different reinvestment assumptions (cost of capital for NPV, IRR for IRR).

Exam tips

• Define first, then explain. A clear definition with the key features earns easy marks.
• Write the process stages in order and give one line on each, rather than a bare list.
• For types of decisions, use headed points: nature, relationship between proposals and availability of funds.
• Name the evaluation techniques in the process answer, then move on unless the question asks for calculations.
• Use rupee examples, such as an Indian manufacturer replacing a machine, to make theory answers concrete.
• Draw a Year 0 to Year n table and show each component. ICSI answers earn marks for method even if arithmetic slips.
• State your assumptions in one line, for example that interest is ignored and working capital is fully recovered.
• Read the question for the depreciation method and tax treatment of the sale of assets before you start.