Lesson 3 · Equity AnalysisCFA L1

Residual Income Valuation and Value Drivers

Free cash flows are often negative and uninformative about economic performance, so practice has shifted to earnings-based valuation. The residual income (economic profit) model values equity as current book value plus the present value of future abnormal earnings — and reconciles exactly with the DCF and dividend models. This lesson builds it from the clean-surplus relation, links it to free cash flow, and ends with steady-state growth, the P/E relationship, and the value drivers behind every model.

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Why earnings-based models The residual income model Firm-level economic profit FCF ↔ economic profit Steady state & growth The P/E relationship Value drivers & Rappaport

Earnings are less volatile than cash flows and easier to model — profitability (ROE) tends to mean-revert under competition. Residual-income models show how a firm creates value each period, yet, under consistent assumptions, deliver a valuation identical to DCF or the DDM.

Learning outcomes

Derive the residual income model from the clean-surplus relation.
Express residual income via the ROE–r_e and ROIC–WACC spreads.
Link free cash flow to economic profit and value equity both ways.
Use steady-state relationships and the sustainable growth rate.
Connect the DGM to the P/E ratio and identify the value drivers.

Why earnings-based models?

Forecasting net dividends ("wealth distribution") is hard, and DCF ("value creation") struggles when free cash flows are negative (e.g. a fast-investing firm like Ocado) — annual FCF then says little about economic performance. So the trend is toward economic profit / residual income (RI) models, which relate intrinsic value to earnings and book value.

The residual income valuation model

Start from the clean-surplus relation (CSR): all changes in equity flow through (comprehensive) earnings, $b_t-b_{t-1}=e_t-d_t$. "Normal" earnings on a marginal (zero-NPV) firm would be $r_e\,b_{t-1}$; residual income is the excess:

Residual income → the valuation model

$$ RI_t=e_t-r_e\,b_{t-1}=(\text{ROE}_t-r_e)\,b_{t-1} $$

$$ MV_{E,0}=b_0+\sum_{t\ge1}\frac{RI_t}{(1+r_e)^t} \qquad\xrightarrow{RI_{t+1}=(1+g)RI_t}\qquad MV_{E,0}=b_0+\frac{RI_1}{r_e-g} $$

Substituting the dividend from the CSR into the DDM and telescoping (assuming the PV of book assets in the far future → 0) collapses to this elegant result: value = current book value + PV of future abnormal earnings. Its great advantage is the reliable, available anchor of current book value $b_0$. (The Ohlson model, covered later, uses an alternative RI-decay assumption with $g<0$.)

Worked example

Highgate plc by residual income

Book value $b_0=500$; cost of equity 9%; forecast year-1 earnings 108; abnormal earnings grow at 5%.

Show solution

Normal earnings $=b_0\times9\%=45$, so $RI_1=108-45=63$. With 5% growth: $MV_{E,0}=500+\frac{63}{0.09-0.05}=\textbf{2,075}$ — identical to the DCF value from Lesson 2. Same economics, anchored on book value rather than projected cash flows.

b₀ 500 + RI₁ 63/(0.09−0.05) = 2,075

Firm-level economic profit

Charging a capital cost on opening invested capital gives the firm-level RI — also called economic profit (the basis of Stern-Stewart's EVA®):

Economic profit and the spread

$$ RIF_t=\text{NOPAT}_t-\text{WACC}\cdot IC_{t-1}=(\text{ROIC}_t-\text{WACC})\cdot IC_{t-1} $$

where $\text{ROIC}_t=\text{NOPAT}_t/IC_{t-1}$. Value is created only when the return on capital exceeds its cost — a positive economic spread. (Recall invested capital $IC=OA-OL=$ fixed + intangibles + net operating working capital $=$ book equity + book debt − non-operating cash.)

Free cash flow ↔ economic profit

The two views are algebraically linked. Since $\text{FCFF}_t=\text{NOPAT}_t-\Delta IC_t$, substituting the economic-profit definition gives the equity value either way:

Economic-profit valuation of invested capital

$$ \text{FCFF}_t=RIF_t+(1+\text{WACC})IC_{t-1}-IC_t $$

$$ MV_{IC,0}=IC_0+\sum_{t\ge1}\frac{RIF_t}{(1+\text{WACC})^t}, \qquad MV_{E,0}=(IC_0-MV_{NFO,0})+\sum_{t\ge1}\frac{RIF_t}{(1+\text{WACC})^t} $$

So invested capital is worth its book value plus the PV of future economic profit. The same WACC circularity from Lesson 2 applies to this indirect route. (RI models assume opening book value $b_{t-1}$ — required for the DCF reconciliation — though firms change through the year, so some practitioners use mid-year conventions.)

Steady state and sustainable growth

With a constant retention ratio $q$ (so $d_1=(1-q)e_1$) and constant ROE $=k$, the clean-surplus relation makes book value, earnings and dividends all grow at the same rate:

Sustainable growth

$$ b_1=b_0(1+qk), \qquad g=qk=\text{retention ratio}\times\text{ROE} $$

From the DGM to the P/E ratio

Substituting the sustainable growth rate into the dividend growth model and dividing by forward earnings yields the forward P/E as a function of three things — the cost of equity, the payout/retention and the ROE:

Forward P/E

$$ P_0=\frac{(1-q)e_1}{r_e-qk} \quad\Rightarrow\quad \frac{P_0}{e_1}=\frac{1-q}{r_e-qk}=f(r_e,\ q,\ \text{ROE}) $$

Worked example

Highgate plc on a P/E basis

Earnings 108, book 500, FCF 83, growth 5%. Find the implied forward P/E and check the value.

Show solution

ROE $k=108/500=21.6\%$; retention $q=(108-83)/108=23.15\%$ (so $g=qk=5\%$ ✓). Forward P/E $=\frac{1-0.2315}{0.09-0.05}=19.21$, and $19.21\times108=\textbf{2,075}$ — the same value yet again. Three models (DCF, residual income, P/E) reconcile because they share the same assumptions.

ROE 21.6%, q 23.15%, P/E 19.21 → 19.21×108 = 2,075

Value drivers and the Rappaport model

Every model reduces to a few value drivers. At equity level, $P_0/e_1=f(r_e,g,\text{ROE})$; at firm level, $MV_{IC,0}/\text{NOPAT}_1=f(\text{WACC},g,\text{ROIC})$. These can be decomposed further by DuPont (ROE = operating margin × asset turnover × leverage). The Rappaport model forecasts FCFF directly from value-driver ratios:

Rappaport value-driver FCFF

$$ \text{FCFF}_{t+1}=\underbrace{\text{OM}(1-\tau)}_{\text{after-tax margin}}(1+g)S_t-\underbrace{\text{IR}}_{IC/\text{sales}}\,g\,S_t $$

Worked example

Henkel AG — value-driver FCFF

Sales €16,428m; after-tax operating margin (NOPAT/sales) 11.58%; investment ratio (IC/sales) 81%; growth 3.4%; tax 24.8%. Estimate next-year FCFF.

Show solution

$\text{FCFF}=11.58\%(1.034)(16{,}428)-81\%(3.4\%)(16{,}428)=\textbf{€1,515m}$. The first term is the after-tax operating cash the grown sales throw off; the second is the capital the growth consumes. Holding the driver ratios constant lets you roll this forward each year.

11.58%·1.034·16,428 − 81%·3.4%·16,428 = €1,515m

All of these use accounting numbers

Whether you estimate residual income directly or use value drivers to forecast FCFF, the inputs are accounting figures — so all the methods can be distorted by accounting, depending on asset replacement patterns, growth, and reporting quality. Be alert to the effects and restate figures before analysing where needed.

Key takeaway

The residual income model values equity as book value + PV of abnormal earnings $\big(b_0+\sum RI_t/(1+r_e)^t\big)$, where $RI=(\text{ROE}-r_e)b$; at the firm level it becomes economic profit $(\text{ROIC}-\text{WACC})\cdot IC$. It reconciles exactly with DCF and the DDM — value is created only by a positive spread of return over cost of capital. In steady state, growth is retention × ROE, which makes the forward P/E a function of $r_e$, payout and ROE — the same handful of value drivers that the Rappaport model uses to forecast FCFF directly. And because every input is an accounting number, every model is only as clean as the accounting behind it.

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