Capital Structure: formula sheet

Key formulas

WACC

WACC = wd × rd × (1 − t) + wp × rp + we × re

Weights are market-value proportions (or target weights if given) and must sum to 1. Drop the preferred term if the firm has none.

After-tax cost of debt

rd(1 − t), where rd = YTM

Use the current YTM, not the coupon rate. Tax applies only to debt.

Cost of preferred stock

rp = Dp ÷ Pp

For non-callable, non-convertible perpetual preferred. Dp is the annual dividend and Pp is the current price.

Cost of equity: CAPM

re = rf + β × (E(Rm) − rf)

(E(Rm) − rf) is the equity risk premium. If the question gives the premium directly, multiply it by β.

Cost of equity: DDM (Gordon growth)

re = D1 ÷ P0 + g, with D1 = D0 × (1 + g)

Assumes constant dividend growth. Check whether the question gives D0 or D1.

Cost of equity: bond yield plus risk premium

re = rd + risk premium

Here rd is the firm's pre-tax yield on its long-term debt. The premium is a judgemental add-on for equity risk.

Market-value weight

wi = market value of source i ÷ total market value of all capital

Market value of equity = shares outstanding × price. Book values are not used for weights.

MM Proposition I, no taxes

V(L) = V(U)

Firm value is independent of capital structure. Assumes no taxes, no distress costs, perfect markets.

MM Proposition II, no taxes

r(e) = r(0) + (r(0) − r(d)) × (D ÷ E)

r(0) is the cost of capital of the unlevered firm (also the WACC). Cost of equity rises linearly with D/E.

MM Proposition I, with taxes

V(L) = V(U) + t × D

t × D is the present value of the tax shield, assuming permanent debt at a constant tax rate t.

MM Proposition II, with taxes

r(e) = r(0) + (r(0) − r(d)) × (1 − t) × (D ÷ E)

Equity cost rises with leverage, but less steeply than without taxes.

WACC

WACC = (D ÷ V) × r(d) × (1 − t) + (E ÷ V) × r(e)

Constant with no taxes; declines as D/V rises with taxes.

Value of levered firm with taxes (MM Proposition I)

V(L) = V(U) + t × D

Assumes permanent debt and no distress costs. t is the tax rate and D is the market value of debt.

Value with distress costs (static trade-off)

V(L) = V(U) + PV(tax shield) − PV(costs of financial distress)

Optimal debt is where the value is highest.

Expected cost of financial distress

Expected cost = P(distress) × cost of distress

Both terms tend to rise with leverage.

Optimal structure condition

Marginal benefit of tax shield = Marginal expected cost of distress

WACC is lowest and firm value highest at this point.

Pecking order of financing

Internal funds → New debt → New equity

Ranked from least to most information cost. Equity is the last resort.

Equity issue signal

Equity issue → signal of possible overvaluation → share price falls

Applies to announcements of new equity issuance by existing public firms.

Debt issue signal

Debt issue → neutral or less negative than equity (pecking order)

This is why debt ranks above equity. A positive reading comes from the separate signaling argument: only firms confident of meeting mandatory payments take on debt.

Buyback or dividend increase signal

Buyback / dividend rise → management confidence

Credible because management expects cash flows to sustain them and dividend cuts are penalized.

Free cash flow hypothesis

Free cash flow = cash left after funding all positive-NPV projects

Excess free cash invites wasteful spending. Debt service reduces this cash and disciplines managers.

Theory contrast

Pecking order: no target debt ratio. Static trade-off: optimal target debt ratio

Pecking order debt ratio reflects funding history, not a chosen optimum.

Debt-to-equity ratio

D/E = total debt ÷ total equity

Common way to express a target. Say whether you use book or market values; the exam usually states which.

Weights in the capital structure

wd = D ÷ (D + E); we = E ÷ (D + E); wd + we = 1

Target weights, not current weights, are normally used for WACC.

Converting D/E to debt weight

wd = (D/E) ÷ (1 + D/E)

For example, D/E of 0.5 gives wd = 0.5 ÷ 1.5 = 33.3%.

WACC with target weights

WACC = wd × rd × (1 − t) + we × re

Use after-tax cost of debt and target weights.

Trade-off logic

More debt: bigger interest tax shield, but higher expected distress costs

The optimal level is where the marginal benefit equals the marginal cost.

Quick revision

• Capital structure is the mix of debt and equity (and other sources) used to fund the firm.
• WACC = (E ÷ V) × rE + (D ÷ V) × rD × (1 − t), using market-value weights where possible.
• The after-tax cost of debt is rD × (1 − t), so debt interest is tax-deductible in the standard setup.
• M&M Proposition I without taxes: firm value is independent of capital structure.
• M&M Proposition II without taxes: cost of equity rises as the debt-to-equity ratio rises, and WACC stays constant.
• M&M with taxes: debt adds value equal to the tax shield, so value of levered firm = value of unlevered firm + tax rate × debt (for permanent debt).
• Financial distress costs include direct costs such as legal fees and indirect costs such as lost customers and suppliers.
• Static trade-off theory: the optimal debt level balances the tax shield benefit against the expected cost of financial distress.
• Pecking order theory: firms prefer internal funds first, then debt, then new equity, because of information asymmetry.
• Issuing equity can signal that managers think shares are overvalued, which can lower the share price.
• Target capital structure is the mix management aims for over time, and the actual mix can differ from it in the short run.
• Factors that support more debt include stable cash flows, tangible assets and a higher tax rate; the opposite factors favour less debt.

Common mistakes

• Using pre-tax cost of debt in WACC Fix: Always multiply debt cost by (1 − t) before weighting. Apply it to debt only.
• Using book value weights Fix: Use market values unless the question gives target weights. If only book debt is given as a proxy for market debt, use it as stated, but equity still uses market capitalisation.
• Saying the WACC falls when debt is added in the no-tax case. Fix: Remember that the cost of equity rises to offset the cheaper debt exactly. WACC stays at r(0).
• Leaving out (1 − t) in Proposition II with taxes. Fix: Check the question for a tax rate. If there is one, include (1 − t) in the D/E term.
• Saying the tax shield alone gives an optimal structure at the highest debt Fix: Always add the distress and agency cost side. The optimum comes only from the trade-off.
• Treating distress costs as only legal fees of bankruptcy Fix: Include indirect costs: lost customers, supplier terms, staff departures and forced asset sales. These are often larger.
• Thinking pecking order has a target debt ratio. Fix: Remember that pecking order debt levels are a by-product of funding needs and internal cash, not a chosen optimum.
• Placing equity before debt in the order. Fix: The order is about information costs, not risk. Equity is most sensitive to managers' private information, so it comes last.
• Using current market weights in WACC when the question gives a target structure. Fix: Use the target weights stated. Current weights matter only if no target is given.
• Treating D/E as if it were the debt weight. Fix: Convert with wd = (D/E) ÷ (1 + D/E) before using WACC.

Exam tips

• Read the tax rate and note which source it applies to. Most WACC traps are the missing (1 − t) on debt.
• When the stem gives amounts and prices for each source, build weights from market values. Do not use the book values shown in a balance sheet extract.
• Check whether the dividend is D0 or D1 before using the DDM. The wording 'just paid' or 'recently paid' means D0.
• Use the answer range as a check. WACC must sit between the lowest and highest component cost, which often removes one or two of the three options.
• Be ready to choose between CAPM and DDM conceptually. CAPM needs beta and a risk premium. The DDM needs price, dividend and a stable growth rate.
• Always check if taxes are mentioned. The no-tax and with-tax results are opposite for firm value and WACC.
• Know the directions cold: cost of equity rises with leverage in both cases, value only rises with taxes.
• Questions with three options often include the no-tax answer as a trap when taxes are given. Eliminate it first.