Specific investment decisions (lease or buy, asset replacement, capital rationing): formula sheet

Key formulas

Post-tax cost of debt

Kd (post-tax) = pre-tax borrowing rate × (1 − tax rate)

Use this to discount all lease and buy cash flows. Use the rate for borrowing to buy the asset, not the WACC.

Tax relief on lease payments

Tax saved = lease payment × tax rate

Assume rentals are tax deductible unless told otherwise. Check whether tax is paid in the same year or a year later.

Tax saved from capital allowances

Tax saved = capital allowance × tax rate

Use the allowance pattern the question gives, for example 25% reducing balance or straight line. Use the rates given in the question.

Balancing allowance or charge

Tax written-down value − disposal proceeds

A positive result is a balancing allowance (extra tax relief). A negative result is a balancing charge (extra tax payable).

Net cost of buying

PV(purchase cost) − PV(tax saved on allowances) − PV(disposal proceeds)

Disposal proceeds are a cash inflow for the buyer. A lessee gets none.

Net cost of leasing

PV(lease payments) − PV(tax saved on payments)

Rentals paid in advance start at time 0. Rentals paid in arrears start at time 1.

Decision rule

Choose the option with the lower present value cost

Equivalent: compute the lease-minus-buy difference and lease if it is negative.

Equivalent annual cost

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

Use the annuity factor for the length of the cycle being tested. Use a PV of costs, so the EAC is a positive cost.

PV of costs for one cycle

PV = purchase cost + PV of running costs (after tax if given) − PV of resale value (and tax benefits treated as reductions in cost)

Resale proceeds at the end of the cycle reduce the cost. Capital allowance tax savings also reduce it.

Decision rule

Choose the replacement cycle with the lowest EAC

Only valid when costs are the only difference and the asset is replaced indefinitely by identical assets.

Annuity factor

AF = [1 − (1 + r)^−n] ÷ r

Normally taken from the tables provided in the exam.

Profitability index (PI)

PI = NPV ÷ initial capital outlay

Use the capital that is rationed, usually the time 0 outlay. Some texts use PV of inflows ÷ outlay, which gives a different number, so follow the question's definition. The ranking is the same either way.

Divisible project selection

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

Fraction = funds left ÷ full outlay of that project. NPV from it = fraction × full NPV.

Indivisible project selection

Choose the feasible combination with the highest total NPV

A combination is feasible if total outlay ≤ funds available. Ranking by PI may not give the best answer.

Decision rule

Only projects with NPV > 0 are considered

Reject negative-NPV projects even if capital is left over.

Objective function

Maximise Σ (NPV of project j × xj)

xj is the fraction (or units) of project j undertaken. For divisible, non-repeatable projects, 0 ≤ xj ≤ 1.

Funds constraint for each year t

Σ (cash outflow of project j in year t × xj) ≤ funds available in year t

Write one constraint per year in which funds are limited.

Non-negativity and upper limits

xj ≥ 0 and, if projects cannot be repeated, xj ≤ 1

Do not forget these. They often decide which corner is optimal.

Shadow price of a constraint

Shadow price = increase in maximum NPV ÷ increase in the funds limit

Zero if the constraint has slack. It is valid only within a range where the same constraints stay binding.

Profitability index (single period only)

PI = NPV ÷ capital invested in the rationed period

Valid for one scarce period only. Do not use it when two or more years are limited.

Quick revision

• Lease or buy: compare the present value cost of each option and choose the lower one.
• The lease or buy discount rate is normally the after-tax cost of borrowing, because it is a financing decision.
• Include tax relief on lease payments and on capital allowances, with the correct timing.
• Equivalent annual cost = present value cost of one cycle ÷ annuity factor for the cycle length.
• Choose the asset or replacement cycle with the lowest equivalent annual cost.
• Equivalent annual cost assumes the asset is replaced in the same way indefinitely, so costs are repeated.
• Profitability index = NPV ÷ initial investment, used to rank divisible projects when capital is limited in one period.
• Divisible projects: rank by profitability index and fund the last project partly. Indivisible projects: ranking by profitability index is not reliable, so test the feasible combinations directly for the highest total NPV.
• Indivisible projects cannot be split. Non-repeatable projects cannot be undertaken more than once. Mutually exclusive projects mean only one of a group can be chosen.
• Multi-period rationing cannot be solved by simple ranking. Linear programming is the technique used; check the current FM study guide for the depth you need.
• A shadow price is the extra NPV from one more unit of a scarce resource, valid only within a limited range.
• Always add a brief written conclusion that states the decision and one limitation.

Common mistakes

• Discounting at the WACC or the pre-tax cost of debt Fix: Use the post-tax cost of debt: borrowing rate × (1 − tax rate). Write one line explaining that leasing is a substitute for debt.
• Putting lease payments in the wrong year Fix: Underline the timing in the question. Payments in advance start at time 0, so use 1 + the annuity factor for the remaining years.
• Choosing the cycle with the lowest total PV of costs. Fix: Always divide by the annuity factor and compare EACs.
• Using the wrong annuity factor, such as the 3-year factor for a 2-year cycle. Fix: Label each row with n and write the factor next to it before dividing.
• Ranking by NPV instead of PI for divisible projects. Fix: When capital is limited, rank by NPV per unit of scarce capital. Always compute PI.
• Using PI ranking alone for indivisible projects. Fix: Test combinations. Ranking can leave funds unused and give a lower total NPV.
• Ranking projects by profitability index when two or more years are rationed. Fix: Check how many years are limited. If more than one, set up LP. Mention that PI ignores the interaction between years.
• Forgetting upper limits such as x ≤ 1 for projects that cannot be repeated. Fix: Read the question for 'cannot be repeated' or 'divisible'. Write the limit down in the formulation before solving.

Exam tips

• Write the post-tax discount rate calculation at the top of your answer and give one line of reasoning. Markers look for it.
• Mark the timing words in the question: in advance, in arrears, tax paid one year later. Most lost marks come from timing.
• Present the answer as a clear table by year, with a total PV for each option, then a one-sentence decision.
• In Section A and B objective questions, expect a short question on the discount rate or the difference between operating and finance leases. Answers are all or nothing, so check that the option matches the exact wording.
• In written parts, give both financial and non-financial points: maintenance, obsolescence, flexibility, the statement of financial position effect under IFRS 16 (right-of-use asset and lease liability, with short-term and low-value exemptions) and the lessor's interest rate.
• In Section C, set out a clear table by cycle length. Marks are given for the PV of each cycle, the annuity factor used and the EAC, so show each one.
• In objective test questions, look for the line that asks for the lowest cost. Calculate the EAC for each option quickly and watch for options that are the unconverted PV.
• If tax and inflation appear, deal with them in the cash flows first: inflate, apply tax, then discount. Keep the rate consistent with the cash flows.