A.3.3 The Cost of Capital

Basic Prerequisites: 02 · Cash-Flow Forecasting.

1. Intuition & Practical Objective

The discount rate is the price of risk. It answers: what return do the providers of capital demand for handing me their money instead of buying something safe? Get it wrong and the entire valuation shifts - a $1%$ error in WACC moves a stable-growth value by tens of percent (page 05).

There are two rates because there are two claimants:

Cost of equity $k_{e}$ - the return shareholders require. Estimated with the CAPM (or a multi-factor model): $k_{e} = r_{f} + β (ERP)$ .
Cost of debt $k_{d} (1 - t)$ - the interest rate lenders charge, after the tax shield (interest is tax-deductible, so the government subsidises debt).
WACC - the market-value-weighted blend, which is the correct discount rate for FCFF.

The essential discipline: the discount rate must match the cash flow. Equity flows → $k_{e}$ . Firm flows → WACC.

The one-sentence essence. “WACC $= k_{e} \frac{E}{D + E} + k_{d} (1 - t) \frac{D}{D + E}$ - a market-value-weighted blend of the required returns of every capital provider, and the only correct rate for unlevered FCFF.”

2. Mathematical Ground Truth & Derivations

2.1 CAPM and the cost of equity

k_{e} = r_{f} + β (ERP), β = \frac{Cov ( R _{i} , R _{m} )}{Var ( R _{m} )} .

Three inputs, three estimation problems (Damodaran Ch 7–8):

$r_{f}$ : the default-free rate matched to the valuation horizon (a 10-year government bond for a 10-year cash-flow horizon).
ERP $= E [R_{m}] - r_{f}$ : the market risk premium, estimated from history, surveys, or the implied-forward method; for non-US firms add a country risk premium.
$β$ : regression beta vs a market index, or (better) a bottom-up beta built from comparable firms’ unlevered betas and re-levered to the target structure.

Bottom-up beta (re-levering): with unlevered (asset) beta $β_{U}$ , tax rate $t$ and target debt-to-equity $D / E$ ,

β_{L} = β_{U} [1 + (1 - t) \frac{D}{E}] .

2.2 Cost of debt and the tax shield

k_{d} (1 - t) = pre-tax borrowing rate \times (1 - marginal tax rate) .

For a rated firm the pre-tax rate is the risk-free rate plus a default spread implied by the rating; for an unrated firm, use the interest-coverage ratio to synthesise a rating.

2.3 WACC

K_{c} = k_{e} \frac{E}{D + E} + k_{d} (1 - t) \frac{D}{D + E} .

Weights are market values, not book values (book weights undervalue equity and distort the blend). Note the circularity: WACC needs market values of E and D, but the valuation produces the value of E. The standard resolutions are to iterate to a fixed point, or to use target/industry weights (Damodaran Ch 15, “Market Value Weights and Circular Reasoning”).

Why WACC is the right rate for FCFF: the interest tax shield is embedded in $k_{d} (1 - t)$ . That is exactly why FCFF starts from $EBIT (1 - t)$ - before interest - so the shield is counted once, in the discount rate, never in the cash flow.

3. Computational Implementation - from beta to WACC

Stdlib only. Reproduces Damodaran’s Tube Investments cost of equity ( $21.30%$ ) and WACC ( $15.60%$ ), the Illustration 2.1 WACC ( $9.94%$ ), and builds a bottom-up beta with a leverage sensitivity.

Both Damodaran checks reproduce exactly ( $21.30%$ / $15.60%$ / $9.94%$ ). The leverage table shows the M&M-with-taxes logic: re-levering raises $k_{e}$ (equity gets riskier) but the after-tax debt is cheap, so WACC falls - the theoretical case for debt. The decline is not a free lunch: this static model omits the financial-distress and agency costs that eventually reverse it, which is why the “optimal” capital structure is an interior minimum, not “all debt”.

4. Failure Modes & First-Principles Breakdowns

Book-value weights. Using book $D / (D + E)$ instead of market values systematically mis-weights the blend; because book equity ignores growth and intangibles, it typically overstates the debt weight and understates WACC or vice-versa. Always use market values or targets.
Beta mismatches. A regression beta from a thin, illiquid stock is noise; a beta measured against the wrong index (e.g. a local index for a global cash flow) mis-specifies risk. Prefer bottom-up betas and match the index to the cash flow’s currency/exposure.
Erp and $r_{f}$ mismatch. Mixing a nominal ERP with a real $r_{f}$ , or a US ERP with emerging-market cash flows (no country premium), corrupts the rate. Damodaran’s fix for Tube Investments added a $5.23%$ country premium to the $4%$ mature-market premium.
Double-counting the tax shield. If you use after-tax $k_{d}$ in WACC and add interest back into FCFF, you count the shield twice. FCFF is pre-interest; WACC carries the shield. Pick one place.
Ignoring the circularity. WACC depends on the equity value it is used to compute. Naively plugging in pre-valuation market weights can be inconsistent for a firm whose value you expect to change - iterate or use target weights.

5. References

Damodaran, Investment Valuation
Berk & DeMarzo, Corporate Finance
Modigliani & Miller (1958), “The Cost of Capital, Corporation Finance and the Theory of Investment”, AER
Koller et al. (McKinsey), Valuation

6. Connected Graph Bridges

Back: 02 · Cash-Flow Forecasting · Index Hub
Forward: 04 · Terminal Value & EV→Equity
Base: Statistics & Inference (beta = OLS slope) · Modern Portfolio Theory (CAPM’s origin)
Risk: Credit Risk & the Merton Model (default spreads → cost of debt)

Kwant Atlas

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