Residual Income Valuation: formula sheet

Key formulas

Equity charge

Equity charge = cost of equity (r) × beginning book value of equity (B₀)

Use beginning-of-period book value unless the question says otherwise.

Residual income

RI = net income − equity charge = NI − r × B₀

Also called economic profit or abnormal earnings.

RI using ROE

RI = (ROE − r) × B₀, with ROE = NI ÷ B₀

RI is positive only if ROE exceeds the cost of equity.

Economic value added

EVA = NOPAT − (WACC × total capital)

Capital is debt plus equity. NOPAT = EBIT × (1 − tax rate).

Market value added

MVA = market value of the firm − capital invested

Equals the present value of expected future EVA.

Residual income

RIt = NIt − r × Bt−1

r is the cost of equity. Bt−1 is beginning-of-period book value of equity. The product is the equity charge.

Residual income using ROE

RIt = (ROEt − r) × Bt−1

Equivalent to the first form when ROE = NIt ÷ Bt−1 (beginning equity).

General residual income model

V0 = B0 + Σ [RIt ÷ (1 + r)^t], t = 1 to ∞

V0 is intrinsic value of equity. Per share, use book value per share and EPS-based residual income.

Single-stage (constant growth) model

V0 = B0 + RI1 ÷ (r − g)

Requires g < r. g is the constant growth rate of residual income. RI1 = (ROE − r) × B0 when ROE is applied to beginning equity.

Multistage model with terminal value

V0 = B0 + Σ [RIt ÷ (1 + r)^t] for t = 1 to T + [RIT+1 ÷ (r − g)] ÷ (1 + r)^T

Terminal value at time T equals RIT+1 ÷ (r − g). It is discounted back T periods. With persistence factor ω, RIT+1 ÷ (1 + r − ω) can be used.

Clean surplus relation

Bt = Bt−1 + NIt − Dt

Change in book value equals net income less dividends, with no other items bypassing the income statement.

Residual income

RI(t) = NI(t) − r × B(t−1)

Equivalent: RI(t) = (ROE(t) − r) × B(t−1). B is book value of equity.

Persistence fade

RI(T+n) = ω^n × RI(T)

Also RI(t+1) = ω × RI(t). ω is between 0 and 1 in the usual exam case.

Terminal value of continuing RI at time T

PV at T of RI after T = RI(T+1) ÷ (1 + r − ω)

Here RI(T+1) = ω × RI(T). Check the sign: the denominator is 1 + r − ω, not r − ω.

Multistage RI value

V0 = B0 + Σ RI(t) ÷ (1 + r)^t + [RI(T+1) ÷ (1 + r − ω)] ÷ (1 + r)^T

Sum runs t = 1 to T. Discount the terminal value by T years.

Special cases of the terminal value

ω = 1: persistent RI, PV = RI(T+1) ÷ r. ω = 0: PV = 0

With ω = 1 the denominator 1 + r − ω becomes r, so it is a perpetuity.

Residual income

RI = NI − r × B(t−1) = (ROE − r) × B(t−1)

B(t−1) is beginning book value. r is the cost of equity.

Single-stage RI value

V0 = B0 + (ROE − r) × B0 ÷ (r − g)

Assumes RI grows at constant g forever and r > g. This requires book value and earnings to grow at g too, consistent with a stable ROE and g = b × ROE.

Justified P/B

P/B = (ROE − g) ÷ (r − g)

Equals 1 when ROE = r. Above 1 when ROE > r.

Sustainable growth rate

g = b × ROE = (1 − payout) × ROE

b is the retention rate. Assumes stable ROE and no new equity issued.

Implied growth rate

g = (P/B × r − ROE) ÷ (P/B − 1)

Rearranged from the justified P/B formula. Requires P/B ≠ 1.

Implied required return

r = g + (ROE − g) ÷ (P/B)

Use the market P/B to back out the return the market is demanding.

Clean surplus relation

B(t) = B(t−1) + NI(t) − D(t)

B is book value of equity, NI is net income, D is dividends net of issuance and repurchases. Holds only if no dirty surplus items exist.

Book value with dirty surplus

B(t) = B(t−1) + NI(t) + OCI(t) − D(t)

OCI is the dirty surplus amount that goes directly to equity.

Comprehensive income

CI(t) = NI(t) + OCI(t)

Use CI in place of NI so that the clean surplus relation is restored.

Residual income

RI(t) = E(t) − r × B(t−1)

E is earnings (use comprehensive income when dirty surplus exists), r is the required return on equity, B(t−1) is beginning book value.

Residual income value of equity

V0 = B0 + Σ RI(t) ÷ (1 + r)^t

B0 should be the adjusted book value; RI should be built from adjusted earnings.

Residual income

RIₜ = Eₜ − (r × Bₜ₋₁) = (ROEₜ − r) × Bₜ₋₁

E is net income, B is beginning book value, r is cost of equity. Positive RI means value creation.

Residual income valuation

V₀ = B₀ + Σ [RIₜ ÷ (1 + r)ᵗ]

Add terminal value if the forecast is finite. Intrinsic value is book value plus the PV of RI.

Clean surplus relation

Bₜ = Bₜ₋₁ + Eₜ − Dₜ

Book value changes only through earnings and dividends. Violations distort RI.

Value versus book

V₀ > B₀ when expected RI has positive present value; V₀ < B₀ when it is negative

This links RI to P/B: justified P/B above 1 means the firm is expected to earn above its cost of equity.

Quick revision

• Equity charge = cost of equity × beginning book value of equity.
• Residual income = net income − equity charge.
• RI also equals (ROE − cost of equity) × beginning book value.
• Intrinsic value = current book value + present value of expected future RI.
• Positive RI means the firm earns above its cost of equity, so value exceeds book value.
• Book value rolls forward as beginning book value + net income − dividends under clean surplus.
• Persistence factor between 0 and 1 shows how fast abnormal RI fades; higher means slower fade.
• Sustainable growth rate = retention rate × ROE.
• Terminal value choice often drives most of the estimate, so check the assumption in the vignette.
• Clean surplus is violated when items bypass net income and go straight to equity, so adjust earnings or book value.
• RI suits firms with no dividends or negative free cash flow, if accounting quality is good.
• Weak accounting quality or heavy non-recurring items make RI less reliable.

Common mistakes

• Using ending book value for the equity charge. Fix: Use beginning-of-period book value unless the question states otherwise.
• Using WACC to compute residual income. Fix: RI uses the cost of equity against net income. EVA uses WACC against NOPAT.
• Using ending book value for the equity charge Fix: The charge uses beginning-of-year equity. Year 1 uses B0. Year 2 uses B1, which you must roll forward.
• Forgetting to add book value Fix: Always write V0 = B0 + PV(RI). Make B0 the first term in your working.
• Using r − ω in the denominator instead of 1 + r − ω. Fix: RI declines by factor ω, so the growth rate is ω − 1. That gives r − (ω − 1) = 1 + r − ω.
• Forgetting to multiply RI(T) by ω before applying the formula. Fix: The numerator is RI(T+1) = ω × RI(T). Write that line explicitly.
• Using the dividend growth rate or earnings growth blindly instead of b × ROE when the question gives payout and ROE. Fix: Read the question wording. If it asks for growth consistent with retention and ROE, compute g = b × ROE.
• Using the payout ratio as b. Fix: b is retention = 1 − payout. Write it out before multiplying.
• Treating net income as the only source of change in book value Fix: Always check for OCI items. Roll forward with NI + OCI − dividends, or use comprehensive income in RI.
• Using reported net income in RI when OCI is large and keeping book value unchanged Fix: Scan every exhibit for OCI. Use comprehensive income to keep earnings and book value consistent.

Exam tips

• Read which rate the vignette gives. Cost of equity points to residual income. WACC points to EVA.
• Check the sign of RI by comparing ROE with r before calculating.
• Watch the units and the year: beginning book value, not ending.
• Remember that MVA is the present value of future EVA, a common conceptual question.
• Check the timing words in the vignette. Beginning book value drives the equity charge, and many wrong options come from using ending book value.
• Compare ROE with r before calculating. This tells you whether the value should be above or below book and removes options fast.
• When a terminal value is given or implied, write the discount period beside it. Most multistage errors are timing errors.
• Expect a conceptual question on the model versus the DDM: it suits firms with no dividends or negative free cash flow, but depends on accounting quality and clean surplus.