Advanced Capital Budgeting Decisions: formula sheet

Key formulas

Net Present Value

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

k is the cost of capital. Accept if NPV > 0. Rank mutually exclusive projects by higher NPV.

Internal Rate of Return (interpolation)

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

L is the lower trial rate (NPV positive), H the higher trial rate (NPV negative). Accept if IRR > cost of capital. The result is an approximation.

Profitability Index

PI = PV of cash inflows ÷ PV of cash outflows

PI > 1 is the same as NPV > 0. Equivalent to 1 + (NPV ÷ outlay) when there is a single initial outlay.

Modified IRR

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

Terminal value = Σ CFt × (1 + r)^(n − t), with r the reinvestment rate. Outflows are discounted at the finance rate. n is the project life.

Payback period

Payback = Years before full recovery + (Unrecovered outlay ÷ Cash flow of the recovery year)

Uses cumulative cash flows. Assumes cash flows are spread evenly within the year. For discounted payback, use cumulative present values.

Accounting-based check: Average rate of return

ARR = Average annual accounting profit ÷ Average (or initial) investment

Use the base the question states. Uses profit, not cash, and ignores time value.

Annual after-tax operating cash flow (CFAT)

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

Same as (Revenue − Cash costs) × (1 − t) + Depreciation × t. Interest is excluded.

Depreciation tax shield

Tax shield = Depreciation × tax rate

Depreciation itself is non-cash. Only the tax saved is a cash benefit.

After-tax value of a cash item

After-tax cash = Pre-tax amount × (1 − t)

Use for taxable inflows and deductible costs, including opportunity costs such as forgone rent.

Initial outlay

Outlay = Asset cost + Installation and other capitalised costs + Initial working capital − After-tax proceeds of any old asset sold

Sunk costs are never included.

Terminal cash flow

Terminal = Salvage value − Tax on gain (or + tax saved on loss) + Working capital released

Tax on salvage depends on the gain or loss over book value, as the question assumes.

Fisher relationship

(1 + Nominal rate) = (1 + Real rate) × (1 + Inflation rate)

Real rate = (1 + Nominal) ÷ (1 + Inflation) − 1. Do not just subtract inflation unless told to approximate.

Inflated cash flow

Nominal CF in year n = Real CF × (1 + inflation)^n

Apply the specific inflation rate for each item if rates differ for revenue and costs.

Change in working capital

Cash flow = −(Working capital needed this year − Working capital at last year-end)

A rise is an outflow. The balance is released at the end of the project life.

NPV at risk-adjusted discount rate

NPV = Σ [CFt ÷ (1 + kr)^t] − Initial outlay, where kr = Rf + risk premium

Use kr for every year. A riskier project gets a higher kr. Accept if NPV > 0.

Certainty equivalent coefficient

α(t) = Certain (CE) cash flow ÷ Risky expected cash flow, with 0 ≤ α ≤ 1

A lower α means the management is less sure of that year's cash flow.

NPV by certainty equivalent method

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

Discount at the risk-free rate only. Do not use the RADR here, or you count risk twice.

Sensitivity (percentage change)

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

The variable with the larger result is the more critical one.

Break-even change in a variable

Allowable fall in an inflow-type variable (%) = NPV ÷ PV of that variable × 100

For the outlay, allowable rise (%) = NPV ÷ Initial outlay × 100. Holds when only that one variable changes and the NPV is positive.

Expected NPV across scenarios

Expected NPV = Σ (Probability of scenario × NPV of scenario)

Probabilities across scenarios must add up to 1.

Equivalence of RADR and CE

α(t) = [(1 + Rf) ÷ (1 + kr)]^t, for a constant risk premium

Shows that RADR implies a CE factor that falls geometrically each year.

Expected NPV

E(NPV) = Σ pᵢ × NPVᵢ

Probabilities across all outcomes must add up to 1.

Expected cash flow and ENPV

E(CFₜ) = Σ p × CFₜ; ENPV = Σ E(CFₜ) ÷ (1 + k)ᵗ − Initial outlay

Use this when probabilities are given on yearly cash flows, not on NPVs.

Standard deviation

σ = √[Σ pᵢ × (xᵢ − x̄)²]

x̄ is the expected value. Weight each squared deviation by its probability.

Coefficient of variation

CV = σ ÷ Expected NPV

Use to compare projects with different ENPVs. Lower CV means lower relative risk. Meaningful only when ENPV is positive.

σ of NPV, independent yearly cash flows

σ(NPV) = √[Σ σₜ² ÷ (1 + k)²ᵗ]

Use only when cash flows of different years are independent.

σ of NPV, perfectly correlated cash flows

σ(NPV) = Σ σₜ ÷ (1 + k)ᵗ

Use when a good year is always followed by a good year.

Probability of a negative NPV (normal assumption)

Z = (0 − ENPV) ÷ σ

Read the area from the normal table. State the normal-distribution assumption.

Joint (path) probability in a tree

P(path) = P(first branch) × P(second branch | first branch)

Multiply along the path. Never add.

Rolling back

Chance node value = Σ p × value of branch; Decision node value = highest branch value

Deduct costs incurred on a branch before comparing.

Profitability index (PI)

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

Use present value of future inflows over the outlay. If outflows also occur in later years, use their present value in the denominator.

NPV and PI link

NPV = PV of inflows − PV of outflows; PI = 1 + (NPV ÷ Outlay)

NPV > 0 means PI > 1. Some books define PI as NPV ÷ Outlay; read the question and stay consistent.

Divisible projects rule

Rank by PI (highest first); accept in order until budget ends; take a fraction of the last project = Balance funds ÷ Its outlay

Total NPV = full NPVs of accepted projects + fraction × NPV of the part-accepted project. Valid for a single-period budget.

Indivisible projects rule

Select the combination with Σ outlay ≤ budget that gives the highest Σ NPV

Check feasible combinations. Do not stop at PI ranking. Leftover funds can be assumed to earn only their cost, so they add no NPV unless stated.

Linear programming form

Maximise Σ (NPVj × xj) subject to Σ (Cjt × xj) ≤ Bt for each period t; 0 ≤ xj ≤ 1 (divisible) or xj = 0 or 1 (indivisible)

Cjt is the cash outlay of project j in period t, and Bt is the budget available in period t. Add extra constraints for mutually exclusive or contingent projects.

Annuity factor

PVAF(r, n) = [1 − (1 + r)^−n] ÷ r

Use the table value if the question gives one. Use the same rate throughout.

Equivalent annual cost (EAC)

EAC = PV of all costs over the life ÷ PVAF(r, n)

PV of costs = initial outlay + PV of operating costs + PV of any later repairs − PV of salvage. Choose the lowest EAC.

Equivalent annual benefit (EAB)

EAB = NPV ÷ PVAF(r, n)

Use when the project has inflows. Choose the highest EAB, only among projects with positive NPV.

Replacement chain NPV (finite)

Chain NPV = NPV × [1 + (1 + r)^−n + (1 + r)^−2n + ...] up to the common life

Each repetition's NPV is discounted back by the number of years elapsed before it starts.

Infinite replacement NPV

NPV∞ = NPV × (1 + r)^n ÷ [(1 + r)^n − 1]

Applies when the project is repeated forever with the same cash flows. It ranks projects the same way as EAB.

Adjusted Present Value

APV = Base-case NPV + PV of financing side effects

Base-case NPV uses Ku. Side effects include tax shield, issue costs, subsidy benefit.

Base-case NPV

Base-case NPV = Σ [CFt ÷ (1 + Ku)^t] − Initial outlay

Cash flows are unlevered: no interest deducted, taxed as if no debt.

Annual interest tax shield

Tax shield = Interest × Tax rate

Discount at the cost of debt (Kd) unless the question says otherwise.

Subsidised loan benefit

Benefit = Loan amount − PV of repayments at market Kd (after-tax adjusted if asked)

Use the market rate for discounting the concessional loan's cash flows.

Ku using asset beta (CAPM)

Ku = Risk-free rate + Asset beta × Market risk premium

Use the unlevered or asset beta when given. Follow the method the question specifies.

Unlevering the beta

Asset beta = Equity beta × E ÷ (E + D × (1 − t)); with no tax, Asset beta = Equity beta × E ÷ (E + D)

E and D are market values of equity and debt, t is the tax rate. Unlever the equity beta first, then use the asset beta in CAPM to get Ku.

Value with real option

Expanded NPV = Static NPV + Option value

Option value is a probability-weighted gain, or a binomial or Black-Scholes value.

Abandonment option payoff

Value at decision point = Max (PV of continuing, Salvage value)

Like a put option with salvage value as the exercise price.

Expansion option payoff

Value at decision point = Max (0, PV of expansion inflows − Expansion cost)

Like a call option; exercise only if positive.

Quick revision

• NPV = Σ cash flow ÷ (1 + k)^t minus initial outlay; accept if NPV > 0.
• Use incremental, after-tax cash flows only; ignore sunk costs and include opportunity costs.
• Depreciation is not a cash flow; include only its tax shield = depreciation × tax rate.
• Add back working capital recovery and salvage value (after tax) in the final year.
• Profitability index = PV of inflows ÷ initial outlay; accept if PI > 1.
• For mutually exclusive projects, NPV is the safer guide when it conflicts with IRR.
• Expected NPV = Σ probability × NPV of each outcome.
• Decision trees are solved by rolling back from the last decision to the first.
• Under capital rationing, pick the combination with the highest total NPV within the budget.
• EAC = NPV of costs ÷ annuity factor for the asset's life; choose the lower EAC.
• APV = base-case NPV (all-equity) + PV of financing side effects such as interest tax shield.
• Real options such as expand, abandon or defer add value that plain NPV misses.

Common mistakes

• Choosing the project with the higher IRR when the projects are mutually exclusive and NPV ranks them the other way. Fix: Follow NPV for mutually exclusive projects. State the cause of the conflict (scale or timing) and the crossover rate if given.
• Using the wrong discount rate, such as the IRR or the coupon rate, instead of the cost of capital. Fix: Use the rate the question labels as the cost of capital or required return. Underline it before you start.
• Deducting interest from cash flows and also discounting at the cost of capital Fix: Leave interest out of project cash flows. The cost of capital already compensates for financing.
• Treating depreciation as a cash outflow, or ignoring it altogether Fix: Deduct it only to find tax, then add it back. Its only cash effect is the tax shield, Depreciation × t.
• Discounting CE cash flows at the risk-adjusted rate or the cost of capital. Fix: In the CE method risk is already removed from the cash flow. Discount at the risk-free rate only.
• Applying the CE factor to the discounted cash flow instead of the cash flow. Fix: Multiply the cash flow by α first, then discount. Both orders give the same figure for a single year, but write the formula in the correct order for the marks.
• Taking the expected value of the cash flows and then using it as if it were the NPV, without discounting or deducting the outlay. Fix: Discount each year's expected cash flow, add them, and then deduct the outlay. Alternatively, find each scenario's NPV first and then average.
• Adding probabilities along a path in a tree instead of multiplying. Fix: Multiply along a path to get the joint probability. Add only the probabilities of different paths that end in the same outcome.
• Using PI ranking alone for indivisible projects Fix: For indivisible projects, always compare total NPV across feasible combinations. PI ranking can leave funds idle and miss a better mix.
• Ranking by NPV instead of PI for divisible projects Fix: Under a budget limit you want the most NPV per rupee of outlay. Rank by PI when projects are divisible.

Exam tips

• In written answers, show the table of cash flows and discount factors first. Marks are given for each step even if the final figure is off.
• When the question gives a single IRR trial rate or a table, use only those values. Do not switch to your own calculator figures midway.
• If two mutually exclusive projects are given, calculate NPV and IRR for both and comment on the conflict. The comment on cause and the final NPV-based recommendation usually carry marks.
• For MCQ case scenarios, check the decision rule first. For example, if NPV is positive then PI is above 1 and IRR is above the cost of capital for conventional flows. This can remove options without calculation.
• State assumptions where the question is silent, such as year-end cash flows and the reinvestment rate used for MIRR.
• Draw a year-wise table with rows for outlay, operating CFAT, working capital, salvage and tax, even if the question does not ask. Marks are given for each item.
• Write a one-line note for each exclusion (sunk cost, interest, allocated overhead). Examiners reward stated assumptions.
• If the question gives no tax treatment for salvage, state your assumption clearly before you apply it, and apply it consistently.