Cost of Capital: Advanced Topics: formula sheet

Key formulas

Weighted average cost of capital

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

Weights wd, wp, we are target market-value proportions and must sum to 1. rd is the pre-tax cost of debt; t is the marginal tax rate.

After-tax cost of debt

rd(after-tax) = rd × (1 − t)

Use the current market yield on the firm's debt, not the old coupon rate.

Cost of equity (CAPM)

re = Rf + β × (Rm − Rf)

Rm − Rf is the equity risk premium. Use the beta that matches the project or firm risk.

Weights from debt-to-equity ratio

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

Use when the vignette gives D/E. If given D/V, then we = 1 − wd.

Break point for marginal cost of capital

Break point = amount of capital at which a component's cost changes ÷ weight of that component

Example: if cheap debt is limited to ₹40 crore and debt weight is 40%, the break point is ₹100 crore of total capital.

After-tax cost of debt

rd(1 − t)

rd is the pre-tax cost (YTM). t is the marginal tax rate. Use this in WACC.

Bond price from YTM

P = Σ [PMT ÷ (1 + y/m)^k] + FV ÷ (1 + y/m)^N

Solve for y to find YTM. m is periods per year. N is the total number of periods.

Annualising a periodic YTM

Annual YTM (bond-equivalent) = periodic yield × m

For semiannual bonds, double the six-month yield. This is the stated annual rate, not the effective rate.

Approximate YTM

YTM ≈ [C + (FV − P) ÷ N] ÷ [(FV + P) ÷ 2]

A quick check only. Use the calculator for the exam answer.

Debt rating approach yield

Yield ≈ benchmark yield + credit spread for the rating

Match rating and maturity. Spread comes from the vignette's exhibit.

Floating rate cost

rd = reference rate + quoted spread

Use the current or expected reference rate, not the rate at issue.

Cost of preferred stock

rp = Dp ÷ P

Dp is the annual fixed dividend; P is the current price. No tax adjustment. If floatation cost applies, use P × (1 − f).

CAPM cost of equity

re = Rf + β × (E(Rm) − Rf)

(E(Rm) − Rf) is the equity risk premium. Use the risk-free rate that matches the cash flow horizon.

Dividend discount model cost of equity

re = D1 ÷ P0 + g

D1 = D0 × (1 + g). Assumes constant growth forever.

Sustainable growth

g = b × ROE

b is the earnings retention rate = 1 − payout ratio.

Bond yield plus risk premium

re = YTM of company's long-term debt + risk premium

The risk premium is a judgment, commonly a few percentage points.

Regression beta

Rᵢ = α + β × Rₘ + ε; β = Cov(Rᵢ, Rₘ) ÷ Var(Rₘ)

Beta is the slope of the stock's returns regressed on market returns.

Blume adjusted beta

Adjusted β = (2/3) × raw β + (1/3) × 1.0

Pulls the raw beta toward 1. The 2/3 and 1/3 weights are the standard version; use other weights only if the question gives them.

Unlever beta (debt beta of zero)

β_asset = β_equity ÷ [1 + (1 − t) × (D ÷ E)]

Uses the comparable firm's own tax rate and D/E. This is the standard form when debt is assumed to have zero beta.

Relever beta

β_equity = β_asset × [1 + (1 − t) × (D ÷ E)]

Use the target's tax rate and target D/E.

Country risk premium

CRP = Sovereign yield spread × (σ_equity index ÷ σ_sovereign bond market)

Sovereign spread is the emerging-market government bond yield minus a comparable developed-market government bond yield.

Cost of equity with CRP

r_e = Rf + β × (ERP + CRP)

Some versions use r_e = Rf + β × ERP + CRP. Follow the form the question uses or implies.

CAPM

E(Ri) = Rf + βi × [E(Rm) − Rf]

One-factor benchmark. The bracket is the market risk premium.

Fama-French model

E(Ri) = Rf + β_mkt × [E(Rm) − Rf] + β_SMB × E(SMB) + β_HML × E(HML)

E(SMB) and E(HML) are factor premiums, not returns on one stock. Use the sign of each beta and premium.

Carhart model

E(Ri) = Rf + β_mkt × MRP + β_SMB × SMB + β_HML × HML + β_WML × WML

Fama-French plus the momentum factor (winners minus losers).

Pastor-Stambaugh model

E(Ri) = Rf + β_mkt × MRP + β_SMB × SMB + β_HML × HML + β_LIQ × LIQ

Adds a liquidity factor to the Fama-French factors. Versions often also include the momentum factor, so follow the factors listed in the exhibit.

Build-up method

re = Rf + ERP + size premium + specific-company premium

No beta in the basic form. Used for private companies.

Expanded CAPM

re = Rf + β × ERP + size premium + specific-company premium

Beta is usually from comparable public firms and adjusted for leverage.

Weighted average flotation cost

F = Σ (wi × fi)

wi is the share of financing from source i and fi is its flotation cost as a percentage of funds raised. Use the project's financing weights.

Flotation cost in currency

Flotation cost = F × investment amount

The default in this page's worked example: apply F directly to the investment amount (the $50 million). Only if the vignette says you must raise enough to net the investment I after fees, use the gross-up: amount raised = I ÷ (1 − F), and the cost is that amount minus I. Follow the vignette's wording.

Adjusted NPV

NPV (adjusted) = NPV (before flotation) − flotation cost

The cost is a time-zero outflow. Leave WACC unchanged. Assume fees are not tax-deductible unless the vignette says otherwise.

Unlevering beta (Hamada form)

βasset = βequity ÷ [1 + (1 − t) × D/E]

Assumes debt beta is zero. Use the pure-play firm's own D/E and tax rate.

Relevering beta

βequity = βasset × [1 + (1 − t) × D/E]

Use the project's target D/E and tax rate.

CAPM cost of equity

ke = Rf + β × (market risk premium)

Use the relevered project beta for a project-specific cost of equity.

WACC

WACC = (E/V) × ke + (D/V) × kd × (1 − t)

Convert D/E to weights: D/V = (D/E) ÷ (1 + D/E), E/V = 1 ÷ (1 + D/E).

Quick revision

• WACC = wd × rd × (1 − t) + wp × rp + we × re, using target market-value weights.
• Cost of debt is the current market yield for new borrowing, not the historical coupon rate.
• Only debt gets the tax shield: multiply by (1 − t).
• Cost of preferred stock = dividend ÷ price, since it is a perpetuity.
• CAPM: re = Rf + β × (equity risk premium).
• Unlever the comparable's beta, then relever at the target firm's debt-to-equity ratio.
• Higher leverage raises equity beta.
• CRP = sovereign yield spread × (annualized equity index volatility ÷ annualized sovereign bond volatility). It can go inside the beta bracket: re = Rf + β × [E(Rmkt) − Rf + CRP]. Some treatments add it outside: re = Rf + β × [E(Rmkt) − Rf] + CRP. Follow the method the vignette states.
• Multifactor models add factor risk premia times factor sensitivities to the risk-free rate.
• Build-up: risk-free rate + equity risk premium + size premium + specific risk premium, used mainly for private firms.
• Flotation costs are best handled as an adjustment to the project's cash outflow, not by raising the cost of capital.
• The marginal cost of capital typically rises as more capital is raised, with breakpoints where a component's cost increases.

Common mistakes

• Using book-value weights from the balance sheet Fix: Use target weights, or current market values. Book values are correct only if the question says so.
• Forgetting to apply (1 − t) to the cost of debt Fix: Apply the tax shield to debt only. Preferred dividends and equity returns get no tax adjustment.
• Using the coupon rate as the cost of debt. Fix: The cost of debt is the current market yield. Use the coupon only as the PMT input to find YTM.
• Forgetting to double or halve the periodic yield for semiannual bonds. Fix: Set N to the number of periods and multiply the periodic I/Y by the number of periods per year to get the annual YTM.
• Using the market return as the equity risk premium in CAPM. Fix: Subtract the risk-free rate from the market return first, or use the formula Rf + β(Rm − Rf).
• Using D0 instead of D1 in the DDM. Fix: Always compute D1 = D0 × (1 + g) before dividing by price.
• Using the target's D/E and tax rate when unlevering the comparable's beta. Fix: Label each figure Comparable or Target. Unlever with Comparable data, relever with Target data.
• Forgetting the (1 − t) factor. Fix: Always write 1 + (1 − t) × D/E and check that a tax rate is given.
• Multiplying the beta by the full market return instead of the market risk premium. Fix: Use the premium, E(Rm) − Rf, when the exhibit gives market return and Rf.
• Ignoring the sign of SMB, HML or WML premiums. Fix: Copy the sign from the exhibit and multiply carefully. A negative premium times a positive beta is a negative term.

Exam tips

• Read for the word target. If the vignette states a target structure, use it, even when an exhibit shows different current weights.
• Expect distractor inputs: coupon rate, book values, pre-tax cost and historical returns. Pick the market-based, forward-looking ones.
• For marginal cost questions, sort the break points from lowest to highest and pick the range that contains the capital amount asked about.
• Judgement questions ask when WACC is appropriate for a project. Answer by comparing project risk and financing mix with the firm's average.
• There is no penalty for wrong answers, so never leave a question blank.
• The vignette often gives both a coupon and a market price. Always compute YTM from the price. The coupon is a distractor for the cost of debt.
• Check whether the question wants pre-tax or after-tax. Many wrong options are the other one.
• Check payment frequency before you key N and multiply the periodic rate back to an annual figure.