Investment appraisal techniques: formula sheet

Key formulas

Relevant cash flow test

Relevant = future + incremental + cash

If a flow fails any one of the three tests, leave it out.

Incremental cash flow

Cash flow with the project − cash flow without the project

Use this to test any doubtful item, such as overheads or lost contribution.

Working capital cash flow

Cash flow in year t = working capital needed at end of year t − working capital in place at start of year t

An increase is an outflow. A decrease is an inflow. Release the remaining balance in the final year.

Opportunity cost of a resource

If the resource would be replaced, relevant cost = replacement cost. If not, relevant cost = higher of net resale value and the net benefit from the next best use.

For a resource already owned and not needed elsewhere, use the net resale value or the lost contribution from the next best use, whichever is higher.

Tax effect of capital allowances

Tax saving = capital allowance × tax rate

Timing depends on the tax rules given in the question.

Payback period (interpolation)

Payback = full years before recovery + (unrecovered outlay at start of the recovery year ÷ cash flow in that year)

Assumes cash flows arise evenly through each year. Use actual cash flows, not profits.

Present value of a cash flow

PV = cash flow × 1 ÷ (1 + r)ⁿ

In the exam you normally take the discount factor from the tables provided.

Discounted payback

Discounted payback = full years before recovery + (unrecovered outlay in PV terms ÷ PV of cash flow in the recovery year)

Use the same cost of capital as for NPV. Cumulate present values, not undiscounted cash flows.

Constant annual inflow shortcut

Payback = initial outlay ÷ annual cash inflow

Only valid for simple payback when every year's inflow is the same. Do not use it for discounted payback.

Decision rule

Accept if payback ≤ target period

Compare projects by choosing the shortest payback, but only for projects that meet the target.

ARR using average investment

ARR = average annual accounting profit ÷ average investment × 100%

Average investment = (initial investment + residual value) ÷ 2.

ARR using initial investment

ARR = average annual accounting profit ÷ initial investment × 100%

Gives a lower figure than the average version when there is a residual value less than cost. Use only if the question says so or you state it.

Average annual accounting profit

(total cash inflows − total cash outflows − total depreciation) ÷ number of years

Equivalent to average annual cash flow less average annual depreciation. Depreciation = (cost − residual value) ÷ life for straight-line.

Decision rule

Accept if ARR > target ARR

For mutually exclusive projects, the higher ARR is preferred on this measure alone.

Net present value

NPV = Σ [CFt ÷ (1 + r)^t] − initial investment

CFt is the net cash flow in year t and r is the discount rate. Time 0 cash flows are not discounted.

Decision rule

NPV > 0 → accept; NPV < 0 → reject

For mutually exclusive projects, choose the highest positive NPV. With capital rationing, use a profitability index instead.

Inflating a cash flow

Inflated cash flow = current price cash flow × (1 + i)^n

Use the specific inflation rate for each item (sales, costs, wages) if given. n is the number of years from now.

Nominal and real rates (Fisher)

(1 + nominal rate) = (1 + real rate) × (1 + inflation rate)

Multiply, do not just add the rates. Rearrange to find the real rate: (1 + nominal) ÷ (1 + inflation) − 1.

Tax on operating cash flow

Tax = taxable cash profit × tax rate

Taxable cash profit is revenue less cash costs. Timing is given in the question, often the same year or one year later.

Tax saving from capital allowances

Tax saving = allowance × tax rate

Reducing balance allowance = tax written down value × allowance rate. On disposal, a balancing allowance or charge = written down value − sale proceeds.

Working capital cash flow

Cash flow = −(this year's working capital − last year's working capital)

Only the increase is a cash outflow. A decrease is an inflow. All remaining working capital is released at the end of the project.

Annuity factor

Annuity factor = [1 − (1 + r)^−n] ÷ r

Use for equal cash flows in years 1 to n. Tables are given in the exam.

IRR interpolation formula

IRR = A + [NPV at A ÷ (NPV at A − NPV at B)] × (B − A)

A is the lower rate and gives a positive NPV. B is the higher rate and gives a negative NPV. The bottom line is the two NPVs added together, ignoring the minus sign.

Decision rule for one project

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

Valid for conventional cash flows (an outflow followed by inflows). At equality the NPV is zero and the project is marginal.

Annuity shortcut for a first guess

Annuity factor = initial investment ÷ annual cash inflow

Use this when inflows are level. Find the factor in the annuity table for the project life. The matching rate is a good first estimate of the IRR.

Margin of safety on the discount rate

IRR − cost of capital

Shows how far the cost of capital can rise before the project's NPV turns negative.

Present value of a single cash flow

PV = CF × 1 ÷ (1 + r)ⁿ

Discount factor for year n at rate r. Use tables where possible.

Annuity factor

AF = [1 − (1 + r)⁻ⁿ] ÷ r

For n years of equal end-year cash flows. Tables give the same value.

PV of an annuity

PV = annual cash flow × annuity factor (n years, r%)

First cash flow is at the end of year 1.

PV of a perpetuity

PV = cash flow ÷ r

First cash flow is one year from the valuation date.

Deferred annuity

PV = cash flow × (AF to final year − AF to the year before the start)

Example: cash flows in years 4 to 10 use AF(10) − AF(3).

Deferred perpetuity

PV = (cash flow ÷ r) × discount factor for the year before the first cash flow

If the first flow is in year 5, discount the perpetuity value back 4 years.

Growing perpetuity

PV = first cash flow ÷ (r − g)

Needs r > g. The first cash flow is the one arising at the end of year 1.

Annuity due (first payment at time 0)

PV = cash flow × (1 + AF for n − 1 years)

The time 0 payment has a factor of 1.0.

Profitability index

PI = NPV ÷ capital invested in the rationed period

Use the outlay in the year capital is limited, usually Year 0. Some texts use PV of inflows ÷ outlay, which gives a different number, so follow the question's definition. The ranking is the same for single-period rationing.

Divisible projects rule

Rank by PI, fund highest first, take a fraction of the last project

Total NPV = sum of full NPVs + fraction × NPV of the part-funded project.

Indivisible projects rule

Choose the combination within the budget with the highest total NPV

Do not rank by PI alone. Leftover capital may be better used by a different combination.

Fraction of a project

Fraction = capital remaining ÷ project's full outlay

NPV earned = fraction × full NPV.

Equivalent annual cost

EAC = PV of costs over one cycle ÷ annuity factor (n years, r%)

Use costs as positive numbers. Resale value reduces the PV of costs. Lowest EAC is best.

Annuity factor

AF = [1 − (1 + r)^−n] ÷ r

The exam gives annuity tables. Use the factor for the cycle length n, not the number of years you plan to operate.

Replacement cycle decision

Choose the cycle length n with the lowest EAC

Calculate PV of costs for each n, then divide by AF for that n.

Assumptions of EAC

Identical replacement, same costs, infinite chain

State these if asked to discuss limitations. Inflation, technology change and capital rationing weaken them.

Quick revision

• Relevant cash flows are future, incremental and cash. Ignore sunk costs, depreciation and allocated overheads.
• Include opportunity costs and working capital changes. Working capital is usually recovered at the end.
• Payback ignores cash flows after the payback point and, in its simple form, the time value of money.
• Discounted payback uses present values, so it is longer than simple payback.
• ARR = average annual accounting profit ÷ investment (initial or average). It uses profit, not cash.
• NPV rule: accept if NPV is positive. It shows the increase in wealth in money terms.
• IRR = a + [NPV_a ÷ (NPV_a − NPV_b)] × (b − a), using two rates a and b.
• Accept a project if its IRR is above the cost of capital. Interpolation gives only an estimate.
• Perpetuity present value = annual cash flow ÷ discount rate.
• Annuity factor for years 1 to n is found from tables. For years 2 to n, subtract the year 1 factor.
• Profitability index = NPV ÷ capital invested. Rank by it when capital is limited and projects are divisible.
• Equivalent annual cost = PV of costs ÷ annuity factor for the asset life. Choose the lower figure.

Common mistakes

• Including depreciation as a cost of the project. Fix: Remove it. Use the actual cash outlay at Year 0. Only capital allowances matter, and only for their tax saving.
• Including interest or the cost of finance in the cash flows. Fix: Leave financing out of the cash flows. The discount rate already reflects the required return on finance.
• Using profits instead of cash flows Fix: Payback uses cash flows only. Remove depreciation and other non-cash items. Include relevant tax and working capital cash flows if the question gives them.
• Dividing the whole outlay by the year's cash flow in the recovery year Fix: Divide only the balance still unrecovered at the start of that year by that year's cash flow.
• Using cash flow instead of accounting profit Fix: For ARR always deduct depreciation from the cash flows before averaging. Depreciation is a non-cash item in NPV but part of profit in ARR.
• Dividing by the wrong investment base Fix: Underline the base in the question. If none is given, use average investment and state that choice.
• Using a real discount rate with inflated cash flows, or the nominal rate with real cash flows. Fix: State the basis of the rate and the cash flows first. Convert with (1 + nominal) = (1 + real) × (1 + inflation) so both match.
• Including depreciation, interest or sunk costs as cash flows. Fix: Ask: is it future, incremental and cash? If not, leave it out. Show the capital allowance only as a tax saving.
• Using two rates that both give a positive NPV or both give a negative NPV Fix: If NPV is positive, go up. If it is negative, go down. Always have one of each before you interpolate. Using two same-sign NPVs is extrapolation and is less reliable.
• Subtracting the NPVs incorrectly in the denominator Fix: The denominator is the positive NPV minus the negative NPV, which means the two absolute values added together. For example, 3,560 − (−5,520) = 9,080.

Exam tips

• In objective questions, expect a list of figures with several irrelevant ones. Cross them out first, then calculate.
• Look for words like already paid, committed, allocated, depreciation and interest. They usually signal an item to exclude.
• In Section C, set out a clear cash flow table by year, with working capital and tax on separate lines. Marks are given for each correct line.
• Where you exclude an item, say why in one short phrase. Examiners reward the reasoning.
• Check the timing of working capital and tax. A one-year slip changes the discounting and costs marks.
• In a Section A or B objective question, check whether the question wants simple or discounted payback before you start. The two answers differ, and the wrong one scores zero because marking is all or nothing.
• Check the answer format: years to two decimals, or years and months. Convert a decimal fraction of a year to months by multiplying by 12.
• In Section C, set out a neat table with columns for year, cash flow, discount factor, PV and cumulative PV. Markers can give credit for correct method even if there is an arithmetic slip.