Lesson 2 · Equity AnalysisCFA L1

Discounted Cash Flow Models and the Cost of Capital

Dividends are hard to forecast, so analysts shift focus from value distribution to value creation — the cash a business generates. This lesson reformats the balance sheet into invested capital and net debt, defines the firm- and equity-level free cash flows (FCFF / FCFE), builds the two DCF models, confronts the WACC circularity, and develops the adjusted present value (APV) approach that handles changing leverage through the unlevered cost of capital.

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Reformatting the balance sheet FCFF, FCFE & FCFD DCF: firm vs equity WACC & its circularity Unlevered cost of capital The APV model PDMP vs ADMP

Cash flows are generated by assets, financed by a mix of capital providers, so valuation must keep clear which assets produce which cash flows for which owners. We switch notation to reflect this: MVE (market value of equity) for value, r_e (cost of equity) for the discount rate.

Learning outcomes

Reformat the balance sheet into invested capital and net financial obligations.
Define FCFF, FCFE and FCFD and estimate them from the statements.
Value a firm by the firm-level (WACC) and equity-level (cost of equity) DCF.
Explain the WACC circularity and the unlevered cost of capital.
Apply the APV model under passive and active debt-management policies.

Reformatting the balance sheet

Start from the balance-sheet identity $OA+FA=OL+FO+\text{Equity}$ (operating assets, financial assets, operating liabilities, financial obligations). Rearranged into a valuation-friendly form:

Reformatted balance sheet

$$ \underbrace{(OA-OL)}_{IC}=\underbrace{(FO-FA)}_{NFO}+\text{Equity} \quad\Rightarrow\quad IC=NFO+\text{Equity} $$

Invested capital (IC) (= net operating assets) is the capital tied up in operations; net financial obligations (NFO) = financial obligations − financial assets ("net debt", conventionally taken at book value). Reformatting matters because non-operating assets have different accounting and tax treatments (marketable securities marked-to-market, illiquid stakes at cost, interest tax-deductible), so each is better valued separately and netted against financial obligations.

FCFF, FCFE and FCFD

As a going concern, only free cash flow can be distributed — operating cash less the reinvestment needed to keep growing. Free cash flow to the firm (FCFF) is the after-tax cash flow as if the firm held only operating assets and were all-equity financed:

FCFF from the statements

$$ \text{NOPAT}=\text{NI}+\text{NFE}, \qquad \text{FCFF}=\text{NOPAT}-\Delta IC $$

$$ \text{FCFF}=\text{CFO}-(\text{LTInv}+\text{NCE})+\text{net interest}\times(1-\tau)\ \text{[US GAAP]} $$

NOPAT = net operating profit after tax (net income + net after-tax financial expense NFE); ΔIC is total new investment (long-term + working capital). The interest add-back is needed under US GAAP (CFO is stated after interest) but not UK GAAP (CFO is before). FCFF splits between the two capital providers:

The cash-flow split

$$ \text{FCFF}=\text{FCFE}+\text{FCFD} $$

$$ \text{FCFE}=\text{net dividends}+\text{excess cash}, \quad \text{FCFD}=\text{Int}(1-\tau)+(\text{debt repaid}-\text{new debt}) $$

DCF: firm level vs equity level

By no-arbitrage, the value of invested capital discounts FCFF at the WACC; equity value is then found indirectly by subtracting net debt, or directly by discounting FCFE at the cost of equity:

The two DCF models

$$ MV_{IC,0}=\sum_{t\ge1}\frac{E_0[\text{FCFF}_t]}{(1+\text{WACC})^t}, \qquad MV_{E,0}=MV_{IC,0}-MV_{NFO,0}\ \text{(indirect)} $$

$$ MV_{E,0}=\sum_{t\ge1}\frac{E_0[\text{FCFE}_t]}{(1+r_e)^t}\ \text{(direct)} $$

All the multi-stage models from Lesson 1 apply (explicit forecast + terminal value), and in theory the dividend, FCFE and FCFF approaches reconcile to the same equity value when assumptions are consistent (note the growth rate $g$ need not be the same under FCFF and FCFE).

Worked example

Highgate plc — a firm-level DCF

100% equity-financed; cost of capital 9%. Assets: £400m fixed + £100m working capital. EBITD = 35% of opening assets; depreciation = 10% of opening fixed assets (tax-allowable); investment = replace depreciation + 5% expansion; all FCF paid as dividends; tax 20%. Value the firm.

Show solution

By building replacement of depreciation into investment, the 5% expansion is net growth, so FCF (and profit) grow at $g=5\%$. Year-1 free cash flow works out to £83m. As a growing perpetuity:

$$ MV_{IC,0}=\frac{\text{FCF}_1}{r-g}=\frac{83}{0.09-0.05}=\textbf{£2,075m}. $$

FCF₁ = 83, g = 5% → value £2,075m

The WACC and its circularity

The firm-level discount rate blends the costs of equity and debt by market-value weights:

WACC

$$ \text{WACC}=\frac{MV_E}{MV_{IC}}\,r_e+\frac{MV_D}{MV_{IC}}\,(1-\tau)\,r_d $$

The circularity problem

The WACC needs the market value of equity (MVE) as a weight — but MVE is exactly what we are trying to estimate. Worse, in a world with frictions the WACC changes with leverage (U-shaped in the MM economy), and even a firm with a target gearing ratio may take years to reach it. So a single constant WACC is an uneasy assumption.

The unlevered cost of capital

Modigliani–Miller's escape: imagine the unlevered (all-equity) firm with an unlevered cost of capital $r_u$. Its "asset beta" depends only on the business, not on financing, so $r_u$ is independent of leverage. Because debt is cheaper, $r_u\ge\text{WACC}$, and MM's central result is:

Value additivity (MM)

$$ \text{Value of levered firm}=\text{Value of unlevered firm}+\text{PV of debt tax shields} $$

This lets us value the unlevered firm at a constant $r_u$ and add the financing effects separately — which copes with varying leverage that a constant WACC cannot.

The adjusted present value (APV) model

Myers (1974): MV_L = MV_u + PV of all financing side effects. Discount the unlevered firm's FCFF at $r_u$ (the asset-beta rate), then add tax relief and financing costs modelled explicitly in cash-flow terms and discounted at an appropriate rate. From the equality of asset risk and claim risk, the levered cost of equity is:

Levered vs unlevered cost of equity

$$ r_e=r_u+Z\cdot\frac{MV_D}{MV_E}\,(r_u-r_d), \qquad Z=\begin{cases}1-\tau & \text{passive debt (MM)}\\ 1 & \text{active debt (Ruback)}\end{cases} $$

Passive vs active debt management

PDMP vs ADMP

Passive (PDMP) — a fixed dollar amount of (perpetual) debt; tax shields are as safe as the debt, so discount them at $r_d$: $MV_{txa}=\tau MV_D$, and $MV_L=MV_u+\tau MV_D$. This gives MM Proposition 2: $r_e=r_u+(1-\tau)\frac{MV_D}{MV_E}(r_u-r_d)$. Active (ADMP, Ruback 2002) — debt is kept proportional to firm value (rebalanced), so tax shields share the firm's risk and are discounted at $r_u$: $MV_{txa}=\tau r_d MV_D/r_u$, and $r_e=r_u+\frac{MV_D}{MV_E}(r_u-r_d)$. Here $r_u$ is the pre-tax WACC, and $MV_L=\frac{\text{FCFF}}{r_u}+\frac{\tau r_d MV_D}{r_u}$.

Worked example

Recapitalisation — APV both ways

An unlevered firm earns after-tax operating profit of £30m forever (investment = replacement). Asset beta 1.0, $r_f=2\%$, MRP 4%, tax 28%. It issues £100m of debt (beta 0.2) to pay a special dividend. Value the levered firm under PDMP and ADMP.

Show solution

$r_u=2\%+1.0(4\%)=6\%$ so $MV_u=30/0.06=£500\text{m}$; $r_d=2\%+0.2(4\%)=2.8\%$. PDMP: tax shield $=\tau MV_D=0.28\times100=£28\text{m}$ → $MV_L=528$, equity $=528-100=£428\text{m}$. ADMP (Ruback): tax shield $=\tau r_d MV_D/r_u=0.28\times0.028\times100/0.06=£13.07\text{m}$ → $MV_L=513.07$, equity $=£413.07\text{m}$. The tax assets (£28m / £13m) are small — ~95% of value is the unlevered firm. APV is the tool of choice when leverage shifts sharply (LBOs, private equity); ADMP is the more conservative assumption.

PDMP: MV_L 528, equity 428 · ADMP: MV_L 513, equity 413

Key takeaway

Reformat the balance sheet into invested capital and net debt, then value the operating business by its free cash flow. FCFF (all-equity, operating) discounts at the WACC for the value of invested capital, less net debt for equity; FCFE discounts at the cost of equity directly — and the two reconcile with the dividend model. Because the WACC is circular and leverage-dependent, the unlevered cost of capital $r_u$ and the APV model (value of the unlevered firm plus the PV of tax shields) are the cleaner tools when gearing changes — with the tax-shield discount rate ($r_d$ under passive, $r_u$ under active debt policy) setting the relationship between $r_e$ and $r_u$.

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