Cost of capital and evaluating investment projects: formula sheet

Key formulas

Gearing ratio (debt to equity)

Gearing = D ÷ E

D and E should be stated on a consistent basis, preferably market values. Some texts use D ÷ (D + E). State the definition you use.

MM Proposition I (no tax)

V_L = V_U

Value of a levered firm equals value of an unlevered firm. Holds under perfect markets, no taxes and no distress costs.

MM Proposition II (no tax)

k_e = k_0 + (k_0 − k_d) × D ÷ E

Cost of equity rises linearly with gearing. k_0 is the overall cost of capital. Assumes risk-free-style debt cost k_d is constant.

MM Proposition I (with corporate tax)

V_L = V_U + T × D

T is the corporate tax rate. Assumes permanent debt and no distress costs. T × D is the present value of the tax shield.

Annual tax shield on interest

Tax shield = T × interest

Interest is assumed tax deductible.

After-tax cost of debt

k_d(1 − T)

Use when interest is tax deductible and the firm pays tax.

Trade-off theory

V_L = V_U + PV(tax shield) − PV(distress costs)

Optimal gearing is where the marginal tax benefit equals the marginal expected distress cost.

Dividend growth model (cost of equity)

Ke = D1 ÷ P0 + g

Assumes dividends grow at a constant rate g forever and g < Ke. P0 is the ex-dividend price. D1 = D0 × (1 + g).

Share price from dividend growth

P0 = D1 ÷ (Ke − g)

Same model, rearranged. Use it to find price when Ke is known.

Estimating growth from past dividends

g = (D_n ÷ D_0)^(1/n) − 1

Use the number of growth periods n, which is one fewer than the number of dividends listed.

Gordon growth with retention

g = b × r

b is the proportion of earnings retained. r is the return on retained funds. Assumes both stay constant.

CAPM

Ke = Rf + β × (Rm − Rf)

Rf is the risk-free rate. Rm − Rf is the market risk premium. β measures systematic risk only.

Cost of irredeemable debt (before tax)

Kd = I ÷ P0

I is annual interest. P0 is the ex-interest market price.

After-tax cost of debt

Kd(after tax) = Kd × (1 − t)

Applies when the company pays tax and can use the interest deduction. t is the corporate tax rate.

Cost of redeemable debt

P0 = Σ I(1 − t) ÷ (1 + Kd)^k + R ÷ (1 + Kd)^n

Solve for Kd using interpolation or trial rates. Use I(1 − t) for after-tax cost. Redemption value R is usually not taxed.

WACC

WACC = [E ÷ (E + D)] × ke + [D ÷ (E + D)] × kd × (1 − t)

E and D are market values of equity and debt. ke is the cost of equity, kd the pre-tax cost of debt, t the corporate tax rate. Extend with more terms if there is preference capital.

Market value of equity

E = number of shares × current share price

Use ex-dividend prices. Do not use nominal value or book value.

Cost of equity (CAPM)

ke = rf + β × (rm − rf)

rf is the risk-free rate, rm the expected market return, β the equity beta.

Cost of equity (dividend growth model)

ke = D1 ÷ P0 + g

D1 is next year's dividend, P0 the current ex-dividend price, g the constant growth rate. Assumes constant growth forever.

After-tax cost of debt

kd(1 − t) where kd is the redemption yield

Find the yield from the bond price, coupon and redemption value. This is the rate r with price = PV of coupons and redemption at r. Assumes interest is fully tax-deductible and the firm pays tax.

Weight of a source

weight = market value of the source ÷ total market value of all finance

Weights must add up to 1.

Net present value

NPV = Σ [ C_t ÷ (1 + r)^t ], t = 0 to n

C_t is the net relevant cash flow at time t (outflows negative). r is the required return per period. The initial outlay at t = 0 is not discounted.

NPV decision rule

Accept if NPV > 0; reject if NPV < 0

For mutually exclusive projects, choose the highest positive NPV. NPV = 0 means the project just earns the required return.

IRR definition

Σ [ C_t ÷ (1 + IRR)^t ] = 0

Solved by trial and error, interpolation, or a calculator or spreadsheet function.

IRR decision rule

Accept if IRR > required return r

Reliable for a single conventional project. Use with care when ranking projects or when there are several sign changes.

Linear interpolation for IRR

IRR ≈ r1 + (r2 − r1) × NPV1 ÷ (NPV1 − NPV2)

Use two rates giving NPVs of opposite sign, one positive (NPV1 at r1) and one negative (NPV2 at r2). The result is an estimate. Keep the two rates close together.

Annuity discount factor

a(n) = [1 − (1 + r)^(−n)] ÷ r

Use for level cash flows at the end of each period for n periods.

Perpetuity present value

PV = C ÷ r

For a level cash flow C at the end of each period, forever.

Payback period

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

Assumes cash flows arrive evenly through the year when interpolating. State this assumption.

Discounted cash flow

PV of cash flow in year t = CFt ÷ (1 + i)^t

Use the cost of capital i. Then apply the payback method to cumulative PVs.

Accounting rate of return (initial investment)

ARR = average annual accounting profit ÷ initial investment

Profit is after depreciation. Average profit = total profit over life ÷ number of years.

Accounting rate of return (average investment)

ARR = average annual accounting profit ÷ [(initial investment + residual value) ÷ 2]

Use this only when the question asks for average investment. Always state the definition used.

Average annual profit

Average profit = (total net cash inflows − total depreciation) ÷ n

Total depreciation = initial cost − residual value, if depreciated straight line.

NPV for comparison

NPV = Σ CFt ÷ (1 + i)^t − initial outlay

Accept if NPV > 0.

Net present value

NPV = Σ CFₜ ÷ (1 + r)ᵗ − initial outlay

CFₜ is the net cash flow at time t after tax. Use the risk-adjusted r if one is given.

Risk-adjusted discount rate

r = risk-free rate + β × (market return − risk-free rate)

CAPM-based rate for a project of similar risk. Alternatively, add a stated premium to the base rate.

Sensitivity margin

Sensitivity (%) = NPV ÷ PV of the cash flows affected by the variable × 100

This is the percentage fall in that variable that takes NPV to zero. Use the PV of the cash flows that depend on the variable.

Profitability index

PI = NPV ÷ capital invested in the rationed period

Use for divisible projects under single-period rationing. Some texts use PV of inflows ÷ outlay, which is PI + 1. State which you use.

Real and nominal rates

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

Discount real cash flows at the real rate and nominal cash flows at the nominal rate. Never mix them.

Expected NPV from scenarios

E(NPV) = Σ pᵢ × NPVᵢ

pᵢ are the probabilities of the scenarios. They must sum to 1.

Quick revision

• WACC weights use market values of each source, not book values, where they are available.
• Interest on debt is tax-deductible, so the after-tax cost of debt is lower than the pre-tax cost.
• Equity is riskier than debt for the investor, so the cost of equity is higher than the cost of debt.
• NPV = Σ of discounted cash flows minus the initial outlay. Accept if NPV > 0.
• IRR is the discount rate at which NPV = 0. Accept if IRR exceeds the required return.
• Use cash flows, not accounting profits, in NPV and IRR. Ignore sunk costs and include opportunity costs.
• Do not include financing costs in project cash flows, because the discount rate already covers them.
• Payback ignores the time value of money and cash flows after the payback date.
• Discounted payback fixes the time value issue but still ignores later cash flows.
• ARR uses accounting profit, so it depends on accounting policies and is not a cash measure.
• Mutually exclusive projects: rank by NPV, as IRR can mislead when scales or timing differ.
• With capital rationing, choose the mix of projects that gives the highest total NPV within the budget.

Common mistakes

• Saying MM proves that capital structure never matters in real life. Fix: State that the no-tax result holds only in perfect markets. Real firms face taxes, distress costs and information gaps.
• Using the pre-tax cost of debt when tax applies. Fix: Check the question for a tax rate. If interest is deductible, use k_d(1 − T).
• Using D0 instead of D1 in the dividend growth model. Fix: Always check which dividend is stated. If it is the dividend just paid, compute D1 = D0 × (1 + g) before dividing by P0.
• Forgetting to add g after D1 ÷ P0. Fix: Write Ke = yield + growth every time. The return has an income part and a growth part.
• Using the pre-tax cost of debt in WACC. Fix: Always write kd × (1 − t) in the formula line before substituting numbers. Tick it off as a checklist item.
• Using book values for weights. Fix: Use market values unless the question says only book values are available. Equity market value = shares × price.
• Including sunk costs, such as money already spent on research. Fix: Ask whether the cash flow changes depending on the decision. If it has already been spent and cannot be recovered, leave it out.
• Deducting depreciation or interest as a cash flow. Fix: Use cash only. Depreciation is non-cash. Financing costs are covered by the discount rate, so deducting interest double counts them.
• Computing payback with undiscounted flows when the question asks for discounted payback. Fix: Always build a PV row first. Cumulative PV should be smaller than cumulative cash flow in every year if i > 0.
• Using profit instead of cash flow for payback, or cash flow instead of profit for ARR. Fix: Payback uses cash flows. ARR uses accounting profit after depreciation. Write this beside your working.

Exam tips

• Always state the assumptions behind MM before giving its result. Markers award marks for them.
• In written answers, compare sources on at least cost, risk, control and tax. Use a short bullet for each.
• For recommendation questions, tie your choice to the firm's facts in the case, such as stable cash flows or high existing gearing.
• In multiple-choice questions, check whether tax is mentioned. It changes the MM answer.
• Show the formula and each step in numerical parts, even for simple calculations.
• Read whether the price is cum or ex dividend or interest. Adjust it if needed before using the formula.
• Show the formula in symbols first, then your numbers. Written answers give method marks even if the arithmetic slips.
• In discussion parts, explain why Ke is above Kd: equity holders carry more risk and are paid last.