Investment Decisions: formula sheet

Key formulas

Initial outlay (year 0)

Initial outlay = Cost of asset + Installation and transport + Net working capital required (+ opportunity cost of any existing asset used) − Sale proceeds of old asset replaced (adjusted for tax)

Include all cash paid to get the project ready. Do not include sunk costs.

Depreciation

Annual depreciation = (Cost − Salvage value) ÷ Life, for straight line

Use the method the question states. Depreciation is a non-cash charge.

Operating cash flow (PAT method)

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

Add depreciation back after computing tax. Interest is excluded.

Operating cash flow (shortcut)

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

Same answer as above. Depreciation gives a tax shield of Depreciation × t.

Terminal cash flow

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

Tax on salvage depends on the book value and the tax rule given in the question.

Working capital cash flow

Cash flow from working capital = −(Increase in NWC) in the year it rises; +Release of NWC at the end

Only the change in working capital each year is a cash flow.

Payback period (equal annual inflows)

Payback = Initial investment ÷ Annual cash inflow

Use cash inflow after tax, not accounting profit. Add back depreciation.

Payback period (uneven inflows)

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

Assumes inflows arrive evenly through the year.

Discounted payback period

Same as payback, but use PV of inflow = Cash inflow × PV factor at cost of capital

Cumulate present values, not raw cash flows.

Payback reciprocal

Payback reciprocal = (Average annual cash inflow ÷ Initial investment) × 100 = (1 ÷ Payback period) × 100

Good approximation of IRR only if life is at least twice the payback and inflows are even.

Decision rule

Accept if payback ≤ maximum acceptable period; among alternatives choose the shortest

The cut-off is set by management, not by a formula.

ARR on average investment

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

Use this when the question asks for average investment. If the question does not name the base, state which base you have used.

ARR on initial investment

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

Use when the question asks for it or defines ARR this way.

Average annual profit

Average profit = Σ (annual profits after depreciation and tax) ÷ Number of years

If profits are given before depreciation, deduct depreciation first.

Average investment (with salvage)

Average investment = (Initial investment + Salvage value) ÷ 2

Valid for straight-line depreciation when there is no working capital.

Average investment (with working capital)

Average investment = (Cost + Salvage value) ÷ 2 + Working capital

Working capital stays tied up throughout the project, so it is added in full and not halved.

Decision rule

Accept if ARR ≥ target rate; for ranking, choose the highest ARR

Mutually exclusive projects: pick the higher ARR if it is above the cut-off.

Present value of a single cash flow

PV = Cash flow in year n ÷ (1 + r)^n

r is the discount rate. In exams you usually multiply by the PV factor given in the question instead.

Net Present Value

NPV = Σ [Ct ÷ (1 + r)^t] − C0

Ct is the net cash inflow in year t. C0 is the initial outflow. Include working capital and salvage value in the right years.

NPV with annuity

NPV = (Annual cash inflow × Annuity PV factor) − Initial outflow

Use only when inflows are equal each year.

Profitability Index

PI = PV of cash inflows ÷ PV of cash outflows

If outflows occur only at time 0, the denominator is the initial investment.

Relationship between NPV and PI

PI = 1 + (NPV ÷ PV of outflows)

Holds when outflows are all at time 0. NPV > 0 means PI > 1.

Decision rule for a single project

Accept if NPV > 0 (PI > 1); reject if NPV < 0 (PI < 1)

At NPV = 0 (PI = 1) the project earns exactly the required rate, so you are indifferent.

Decision rule for mutually exclusive projects

Choose the project with the highest positive NPV

Compare projects on the same discount rate and the same time basis.

IRR definition

Σ [CFt ÷ (1 + r)^t] − Initial outflow = 0

The r that satisfies this is the IRR. t runs from 1 to n.

Interpolation formula

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

L is the lower rate (positive NPV) and H is the higher rate (negative NPV). Keep the gap between L and H small, ideally 1% to 5%.

Equal annual inflows: first guess

Factor = Initial outlay ÷ Annual cash inflow

Look up this factor in the annuity table on the row for n years. The column closest to it gives your first trial rate.

Uneven inflows: rough first guess (heuristic only)

Factor = Initial outlay ÷ Average annual inflow

This is only a rough heuristic, not a dependable method. It ignores the timing of the inflows and can be well off when they are very uneven. Use the nearest annuity factor only to pick a first trial rate, then check the NPV and adjust.

MIRR

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

Terminal value is each inflow compounded to the end of year n at the reinvestment rate. Outflows are discounted to year 0 at the finance rate (usually the cost of capital).

IRR decision rule

Accept if IRR > cost of capital; reject if IRR < cost of capital

At IRR equal to cost of capital, the project is at break-even. For mutually exclusive projects, prefer NPV if the two methods conflict.

Profitability Index

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

Use this to rank divisible projects under a single-period budget. Where the outlay is all at time 0, both forms give the same value.

Capital rationing rule (divisible)

Rank by PI (highest first). Fund in order until the budget ends. Take the last project in proportion.

Partial NPV = (funds used ÷ full outlay) × NPV of that project.

Capital rationing rule (indivisible)

Choose the feasible combination with the highest total NPV, with total outlay ≤ budget

A combination that uses the whole budget is not always the best. Compare NPVs, not PIs.

Present value annuity factor

PVAF(r, n) = [1 − (1 + r)^−n] ÷ r

Usually given in the PV annuity table. Use the factor for the project's own life.

Equivalent Annual Annuity

EAA = NPV ÷ PVAF(r, n)

Use for mutually exclusive projects of unequal lives. Higher EAA is better.

Equivalent Annual Cost

EAC = PV of total costs ÷ PVAF(r, n)

Use when projects have only costs, such as choosing between two machines. Lower EAC is better.

Risk-adjusted discount rate

RADR = Risk-free rate + Risk premium

Discount the normal (uncertain) cash flows at RADR. Higher risk means a higher premium.

Certainty equivalent coefficient

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

α lies between 0 and 1. Lower α means the management is more risk averse for that year.

NPV by certainty equivalent

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

Discount at the risk-free rate Rf, not at the cost of capital.

Expected NPV

E(NPV) = Σ [p × NPV]

Probabilities must add up to 1.

Standard deviation of NPV

σ = √ Σ [p × (NPV − E(NPV))²]

Measures total spread of outcomes around the expected NPV.

Coefficient of variation

CV = σ ÷ E(NPV)

Risk per rupee of expected return. Lower CV is better when comparing projects.

SD of project NPV, independent cash flows

σNPV = √ Σ [σt² ÷ (1 + i)^(2t)]

Use when cash flows of different years are independent. i is the risk-free rate.

SD of project NPV, perfectly correlated cash flows

σNPV = Σ [σt ÷ (1 + i)^t]

Use when cash flows in each year move together perfectly.

Decision tree rollback

Value at a chance node = Σ (probability × value of branch); at a decision node = best branch

Start at the right-hand end and move left.

Quick revision

• Use incremental, after-tax cash flows, not accounting profit, for NPV, IRR and payback.
• Ignore sunk costs. Include opportunity costs and working capital changes.
• Depreciation is non-cash. It matters only through its tax shield: depreciation × tax rate.
• Payback = years to recover the initial outlay. Accept if it is within the cut-off. It ignores cash flows after payback.
• Discounted payback uses present values of cash flows, so it is never shorter than simple payback. It may not exist if the discounted flows never recover the outlay.
• ARR = average accounting profit ÷ average (or initial) investment. State which base you used.
• NPV = Σ present value of inflows − 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.
• MIRR compounds inflows to the end at the reinvestment rate, then finds the rate that equates this to the outlay.
• For mutually exclusive projects with conflicting ranks, NPV is the preferred method.
• With capital rationing, rank by PI for divisible projects and pick the combination with the highest total NPV for indivisible ones.
• For unequal lives, compare using equivalent annual NPV or cost, or a common life.

Common mistakes

• Confusing capital budgeting with working capital management. Fix: Capital budgeting deals with long-term assets and multi-year benefits. Working capital deals with short-term assets and liabilities.
• Listing the steps of the process in the wrong order. Fix: Learn the sequence: identification, screening, evaluation, selection and approval, implementation, performance review. Write them in that order.
• Including sunk costs such as a market survey already paid for. Fix: Ask: does this cash change if I accept or reject? If not, leave it out.
• Deducting depreciation as a cash outflow, or forgetting to add it back. Fix: Deduct it only to compute tax, then add it back. Or use the shortcut CFAT = (S − C)(1 − t) + Dep × t.
• Using accounting profit instead of cash inflow Fix: Add back depreciation and other non-cash charges to get cash inflow before using it.
• Interpolating with the wrong year's cash flow Fix: Divide the unrecovered balance by the inflow of the year in which recovery happens.
• Using profit before depreciation. Fix: ARR uses accounting profit. Always deduct depreciation, and tax if given, before averaging.
• Using initial investment when the question wants average investment. Fix: Underline the base the question names. If it is not named, show average investment and state that you have used it.
• Including interest on borrowed funds in the cash flows Fix: Leave financing costs out. The discount rate already reflects the cost of funds.
• Forgetting to add back depreciation Fix: Depreciation is non-cash. Cash flow after tax = PAT + depreciation, but still deduct the tax effect correctly.

Exam tips

• Theory questions on this topic are often short notes or 5-mark explanations. Prepare a ready list of features, importance, steps and difficulties.
• Write the steps in order and name each one. Examiners check the sequence.
• Learn the three project types (independent, mutually exclusive, contingent) with one example each, because MCQs often test them.
• Link your answer to wealth maximisation to show you understand why the topic matters.
• Revise this topic before the numerical topics (NPV, IRR, payback), since it gives you the language for interpretation.
• Always write a working note for depreciation and tax. Step marks are given even if the final figure is wrong.
• Read the question for words like already spent, allocated, or interest. These usually signal items to exclude.
• Check for working capital and salvage value in the last line of the question. They are easy to miss.