Discounted cash flow techniques: formula sheet

Key formulas

Net present value

NPV = Σ [CFt ÷ (1 + r)^t], t = 0 to n

CF0 is normally the initial outflow (negative). r is the cost of capital or a project-specific rate.

NPV decision rule

Accept if NPV > 0; for mutually exclusive projects, choose the highest positive NPV

With no capital rationing, accepting all positive-NPV projects maximises wealth.

IRR by interpolation

IRR ≈ L + [NPV_L ÷ (NPV_L − NPV_H)] × (H − L)

L is the lower rate with positive NPV (NPV_L), H is the higher rate with negative NPV (NPV_H). The result is an estimate, because NPV is not a straight line against the rate. Choose two rates close together for accuracy.

IRR decision rule

Accept if IRR > cost of capital

Reliable only for a single project with conventional cash flows. For mutually exclusive projects, use NPV or the incremental IRR.

Incremental IRR

IRR of (larger project cash flows − smaller project cash flows)

If this exceeds the cost of capital, the larger project is better. It gives the same choice as NPV.

Number of IRRs

Maximum number of IRRs = number of sign changes in the cash flows

Conventional flows (− then +) have one sign change. More changes can mean multiple IRRs.

Perpetuity PV

PV = annual cash flow ÷ r

Use for level cash flows that go on indefinitely, with the first one a year from now. Not valid for growth.

Relevant cash flow test

Include if: future AND incremental AND cash

Exclude sunk costs, depreciation, apportioned fixed overheads and financing flows (interest, dividends).

Nominal cash flow

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

h is the inflation rate for that item. Use specific rates where the question gives them.

Fisher equation

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

Use it to move between real and nominal rates. Do not simply add the rates.

Working capital cash flow

Cash flow in year t = −(WC needed at end of t − WC needed at end of t−1)

A rise is an outflow. A fall, including the final release, is an inflow. Inflate the balance if prices rise.

Tax on operating cash flow

Tax = pre-tax operating cash flow × tax rate

Apply it in the same year or one year later, as the question states.

Capital allowance tax saving

Tax saving = capital allowance × tax rate

Reducing-balance allowance = rate × tax written-down value (TWDV) at the start of the year.

Balancing allowance or charge

TWDV − disposal proceeds (positive = allowance, negative = charge)

Normally claimed in the final year in place of the normal allowance.

NPV

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

Use k as the money rate for nominal flows. Accept if NPV > 0.

MIRR (terminal value method)

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

TV of inflows = each inflow compounded to year n at the cost of capital (reinvestment rate). PV of outflows = outflows discounted to time 0 at the cost of capital. n = project life in years.

MIRR (ACCA present value form)

MIRR = (PV of returns ÷ PV of investment)^(1/n) × (1 + re) − 1

PV figures are both discounted at the cost of capital. re is the reinvestment rate, normally the cost of capital. This gives the same answer as the terminal value method.

MIRR decision rule

Accept if MIRR > cost of capital

For a single project with conventional cash flows this agrees with a positive NPV.

Payback period

Years before full recovery + (unrecovered cost at start of year ÷ cash flow in that year)

Uses undiscounted cash flows. Assumes inflows arise evenly through the year.

Discounted payback period

Same as payback, but using present values of cash flows at the cost of capital

Discount every cash flow first, build the cumulative PV column, then interpolate.

Discounted cash flow

PV = cash flow ÷ (1 + r)^t

Used for both discounted payback and the MIRR present value.

Profitability index (single-period)

PI = NPV ÷ capital outlay in the rationed period

Some texts use (PV of inflows ÷ outlay) = PI + 1. Rankings are the same. State which one you use.

Divisible projects

Rank by PI, fund in order, fund the last project in part: fraction = remaining funds ÷ its outlay

Total NPV = sum of NPVs of full projects + fraction × NPV of the last project.

Indivisible projects

Choose the feasible combination with the highest total NPV, with total outlay ≤ funds available

Check mutually exclusive projects. Leftover funds are assumed to earn only the cost of capital, so they add zero NPV.

Linear programming (multi-period)

Maximise Σ(NPV of project j × xj) subject to Σ(outlay of j in year t × xj) ≤ funds in year t, and 0 ≤ xj ≤ 1

For indivisible projects, xj must be 0 or 1. Divisible projects let xj be any fraction from 0 to 1.

Shadow price

Shadow price = extra NPV from one more unit of a binding constraint's funds

A constraint with spare funds has a shadow price of zero.

APV

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

Accept the project if APV > 0. Side effects can be negative, such as issue costs.

Ungearing an equity beta

βa = βe × Ve ÷ [Ve + Vd(1 − t)]

Assumes the debt beta is zero and debt is permanent. If the question gives a debt beta, use the extended formula from the formula sheet approach: βa = [βe × Ve + βd × Vd(1 − t)] ÷ [Ve + Vd(1 − t)].

Ungeared cost of equity

Keu = Rf + βa × (Rm − Rf)

Use βa from the proxy company. Use this rate for the base-case cash flows.

Annual tax shield on interest

Tax shield = Debt × interest rate × tax rate

Discount at the pre-tax cost of debt unless the question says otherwise. Match the shield's life to the debt's life.

Subsidised loan benefit

Annual benefit = Loan × (market rate − subsidised rate) × (1 − t)

Discount at the market pre-tax cost of debt. The tax shield on the actual interest paid is a separate item.

Issue costs

Net issue cost = Issue cost × (1 − t) if tax-deductible

Issue costs are usually based on the gross amount raised. Check whether tax relief is allowed. Use the cost as stated if the question gives no relief.

CAPM cost of equity

Ke = Rf + βe × (Rm − Rf)

Rm − Rf is the equity risk premium. Use the regeared equity beta for the project.

Asset beta (ungearing)

βa = [Ve × βe + Vd(1 − T) × βd] ÷ [Ve + Vd(1 − T)]

Ve and Vd are market values of equity and debt of the proxy. T is the tax rate. Use the proxy's own gearing and tax.

Asset beta when debt beta is zero

βa = βe × Ve ÷ [Ve + Vd(1 − T)]

The usual simplification in exam questions. Equivalent to βe ÷ [1 + (1 − T) × Vd/Ve].

Equity beta (regearing)

βe = βa × [Ve + Vd(1 − T)] ÷ Ve when βd = 0

Use the target gearing, not the proxy's gearing.

Regearing with a debt beta

βe = βa + (βa − βd) × Vd(1 − T) ÷ Ve

Rearranged from the asset beta formula. Use it when a debt beta is given.

Debt beta from the cost of debt

βd = (Kd − Rf) ÷ (Rm − Rf)

Only if the question tells you to treat Kd as the CAPM return on debt. Kd is the pre-tax cost of debt.

After-tax cost of debt

Kd after tax = Kd × (1 − T)

For irredeemable debt. For redeemable debt, find the pre-tax yield by IRR first, then apply (1 − T) if tax relief is available.

Risk-adjusted WACC

WACC = Ke × Ve ÷ (Ve + Vd) + Kd(1 − T) × Vd ÷ (Ve + Vd)

Weights are market values at the gearing you regeared to. Convert a D:E ratio into proportions of total value.

Quick revision

• Accept a project if NPV > 0 at the appropriate discount rate; NPV shows the change in shareholder wealth.
• IRR is the rate at which NPV = 0; compare it with the required return, and use interpolation for an estimate only.
• For mutually exclusive projects, NPV is the preferred method when NPV and IRR conflict.
• Relevant cash flows are future, incremental cash flows; exclude sunk costs, depreciation and apportioned fixed overheads.
• Include opportunity costs, working capital changes and tax effects of capital allowances, with the timing the question gives.
• Use a money discount rate with money cash flows and a real rate with real cash flows; do not mix them.
• MIRR = (terminal value of inflows ÷ present value of outflows)^(1/n) − 1. This form applies when the outflows occur at time 0 and the inflows are compounded to year n at the reinvestment rate. The reinvestment rate is often taken as the cost of capital. Outflows are discounted to today at the finance (cost of capital) rate.
• Discounted payback uses discounted cash flows. At a positive discount rate, for conventional projects, it is longer than simple payback, and it may not be achieved at all if the discounted inflows never recover the outlay. It also ignores cash flows after the cut-off.
• Single-period rationing with divisible projects: rank by profitability index (NPV ÷ capital used) and fund in order.
• Multi-period rationing with divisible projects uses linear programming, and the shadow prices show the value of extra capital. Indivisible projects need integer programming, which is a different technique, and shadow prices do not apply in the same way.
• APV = base-case NPV + PV of financing side effects such as tax shield and issue costs. Discount base-case flows at the ungeared cost of equity (Keu), found by ungearing a proxy company's beta (Modigliani and Miller with tax). The tax shield on debt is typically discounted at the pre-tax cost of debt (Kd). Other side effects are discounted at rates that reflect their risk.
• Use a project-specific rate when the project's business risk or financing differs from the company's; ungear a proxy beta and regear it for the project.

Common mistakes

• Choosing the project with the higher IRR when projects are mutually exclusive Fix: Rank on NPV. A smaller project with a higher IRR can add less wealth than a larger project with a lower IRR. Use the incremental IRR if asked.
• Interpolating IRR using two rates that both give positive (or both negative) NPVs Fix: Check that one NPV is positive and one is negative. If not, try another rate. Extrapolating outside the two rates gives a poorer estimate.
• Discounting nominal cash flows at a real rate (or real flows at a money rate). Fix: Label every rate and cash flow as real or nominal before you start. Convert with the Fisher equation if they do not match.
• Including depreciation, interest or sunk costs. Fix: Apply the test future, incremental and cash. Say in one line why you excluded each item, as the marker rewards this.
• Compounding inflows at the IRR instead of the cost of capital Fix: MIRR uses the cost of capital (or the stated reinvestment rate) for compounding. Check the question for a different reinvestment rate.
• Using the wrong n Fix: If the last inflow is at the end of year 4 and the investment is at time 0, n = 4. Draw a timeline.
• Ranking by NPV or IRR instead of PI when funds are limited. Fix: When capital is the scarce resource, rank by NPV per unit of that capital. Use IRR only for comment, not for the ranking.
• Using PI ranking alone for indivisible projects. Fix: After the PI ranking, test other combinations. Use PI only as a starting point.
• Discounting the operating cash flows at WACC instead of Keu. Fix: In APV the base case is all-equity. Always discount operating cash flows at the ungeared cost of equity.
• Forgetting to ungear the proxy beta, or ungearing it with the wrong debt value. Fix: If the beta comes from a geared company, ungear it using market values of equity and debt, with Vd(1 − t) in the denominator.

Exam tips

• In AFM, state the decision rule and its link to shareholder wealth. Marks are given for the reasoning as well as the figures.
• When asked to compare NPV and IRR, cover both strengths and weaknesses. Use points such as scale and timing conflicts, multiple IRRs and different reinvestment assumptions, and apply them to the scenario's cash flows.
• If cash flows change sign more than once, say so early and show NPV at more than one rate if time allows. This supports your point about multiple IRRs.
• Always add a closing recommendation with caveats. Mention the accuracy of forecasts, the discount rate and capital rationing. These show commercial judgement for the professional skills marks.
• Keep the workings tidy: a cash flow table, discount factors and a labelled NPV for each rate. Part marks are given for method.
• Write the tax timing, the inflation basis and the type of discount rate at the top of your answer. Markers give credit for correct assumptions even when the arithmetic slips.
• In a scenario question, add a short comment on the key judgements, such as the opportunity cost of the asset or the risk in the working capital release. This earns professional skills marks for analysis and commercial acumen.
• Show the allowance schedule separately, with the balancing figure labelled. Method marks are often awarded for the TWDV and the tax saving timing.