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EQ 5 – Residual Income Valuation

Worked examples are fully visible. Check-yourself items are study aids you can reveal one at a time.

An income statement is built to report what is left for the owners. It deducts the cost of borrowed money as interest expense, but it deducts nothing at all for the money the owners themselves have tied up in the business. That silence is convenient for accounting and misleading for valuation. A company can report a healthy profit and still be a poor use of shareholder capital, because the shareholders could have earned a return elsewhere at the same level of risk.

Residual income closes that gap. It is net income after subtracting an explicit charge for the opportunity cost of equity. The charge is the required rate of return on equity, sometimes called the cost of equity, applied to the equity capital employed. Because the cost of equity is a marginal cost, it measures what an additional unit of equity must earn, whether that equity is retained from profits or raised by issuing new shares.

Equity capital times the required return on equity gives the charge.

The idea is old. Alfred Marshall wrote about it in 1890, General Motors was using it to judge business segments in the 1920s, and it has since travelled under a long list of names: economic profit, abnormal earnings, excess earnings and economic value added. The version examined here is the general residual income model, built from publicly available accounting data and non-proprietary adjustments, and used to estimate the intrinsic value of common stock.

Two ways to reach the same number

There are two computational routes, and they agree under conditions that are worth stating precisely. The first route starts from net income, which already carries the cost of debt inside it, and deducts only the equity charge. The second route takes the perspective of every provider of capital: it starts from net operating profit after taxes, which carries no financing cost at all, and deducts a total capital charge covering both debt and equity.

The two agree when two conditions hold. First, the marginal after-tax cost of debt used in the capital charge equals the after-tax interest expense already embedded in net income. Second, the weights used to build the capital charge come from the book values of debt and equity rather than from market values. Change either assumption and the two routes separate.

Figure 1: Two routes to the same residual income

Both chains end at the same figure when the marginal after-tax cost of debt matches the interest expense in net income and the capital weights are book weights.

Example 1 · Worked

Axis Manufacturing Company, Inc. (AXCI) is a very small company by market capitalisation. Its balance sheet carries €2 million of assets, funded in equal halves by debt and by equity. The pre-tax cost of debt is 7%, and interest is tax deductible in this jurisdiction, so the after-tax cost of debt is 4.9%. In jurisdictions where corporate interest is not deductible, the after-tax cost of debt equals the pre-tax cost. The cost of equity capital is 12%. AXCI reports earnings before interest and taxes (EBIT) of €200,000 and faces a tax rate of 30%.

AXCI income statement

Line	Amount (€)
EBIT	200,000
Less: interest expense	70,000
Pre-tax income	130,000
Less: income tax expense	39,000
Net income	91,000

1. Compute residual income by deducting an equity charge from net income.

Solution. Interest expense is 7% of the €1,000,000 of debt, or €70,000, so pre-tax income is €130,000, tax at 30% is €39,000 and net income is €91,000.

Equity charge = Equity capital × Cost of equity = €1,000,000 × 12% = €120,000.

Residual income = €91,000 − €120,000 = (€29,000).

AXCI is profitable in an accounting sense and unprofitable in an economic sense. Earnings of €91,000 did not cover the €120,000 that the owners could have earned elsewhere on the same capital at the same risk.

2. Compute residual income from the capital charge perspective and reconcile the two routes.

Solution. Build the total capital charge from both sources of finance:

Equity charge = 0.12 × €1,000,000 = €120,000.
Debt charge = 0.07 × (1 − 0.30) × €1,000,000 = €49,000.
Total capital charge = €169,000.

Net operating profit after taxes (NOPAT) is €200,000 less 30% tax, or €140,000. Residual income is €140,000 − €169,000 = (€29,000), the same answer.

The reconciliation is simple arithmetic. Moving from net income to NOPAT adds back after-tax interest expense of €49,000, calculated as €70,000 × (1 − 30%). Moving from the equity charge to the capital charge adds the after-tax debt charge of €49,000. The same amount is added to both sides, so the difference is unchanged.

Reconciling the two routes

Route 1 line	€	Bridge	Route 2 line	€
Net income	91,000	Add back interest, after tax, of 49,000	Net operating profit after tax	140,000
Less: equity charge	120,000	Add the debt element of the charge, 49,000	Less: capital charge	169,000
Residual income	(29,000)		Residual income	(29,000)

3. Express the same conclusion as a comparison of returns rather than as a comparison of money amounts.

Solution. Scale both the profit and the charge by total capital of €2,000,000.

After-tax net operating return on total assets, also called return on invested capital (ROIC) = €140,000 ÷ €2,000,000 = 7%.
Effective capital charge = €169,000 ÷ €2,000,000 = 8.45%.

The company earns 1.45 percentage points less than it must pay for the capital it uses. Multiplying that shortfall by beginning capital reproduces the money figure: (0.07 − 0.0845) × €2,000,000 = (€29,000).

The return form of the capital charge route.

One simplification runs through this reading. Every calculation assumes clean surplus accounting, the condition that earnings capture every change in the book value of equity other than transactions with owners. It also assumes financing consists only of common equity and debt. Where a company also has preferred stock, preferred dividends are deducted from net income before the equity charge for common shareholders is applied. The clean surplus assumption is examined in detail in the accounting sections later in this lesson.

If residual income is positive, the company is generating more than the cost of the capital it consumes, and it is creating value. If residual income is negative, it is destroying value. Holding everything else constant, higher residual income should go with a higher valuation and a higher price-to-book ratio. The link runs in a straight line from the economics to the market price, and the AXCI figures show it working.

Example 2 · Worked

Return to AXCI from Example 1. Add the following assumptions: the equity initially trades at book value of €1 million across 100,000 shares, so both book value per share and the share price are €10; earnings continue indefinitely at the current level; and all net income is paid out as dividends.

1. What share price is consistent with the residual income AXCI actually earns?

Solution. Earnings per share are €91,000 ÷ 100,000 = €0.91. Value is being destroyed at a rate of €29,000 a year, which is €29,000 ÷ 100,000 = €0.29 per share. That destruction is a perpetuity, so discount it at the 12% cost of equity:

Present value of the value destroyed = €0.29 ÷ 0.12 = €2.42.

Share price = €10 − €2.42 = €7.58.

2. Confirm that price using the earnings yield, and state the implied price-to-book ratio.

Solution. For a company with no growth, the earnings yield (E/P) estimates the expected rate of return. The stock is fairly priced when that yield equals the required return on equity:

€0.91 ÷ P = 12%, so P = €0.91 ÷ 0.12 = €7.58333.

Total equity market value = €7.58333 × 100,000 = €758,333. Against book value per share of €10, the price-to-book ratio is €7.58333 ÷ €10 = 0.7583.

Negative residual income drives the shares to a discount to book value. Positive residual income would drive them to a premium. That is the whole economic content of the price-to-book ratio in a residual income framework.

What residual income models are used for

Equity valuation is only one application. Residual income has been applied to individual stocks and to whole indexes, including the Dow Jones Industrial Average. It is also widely used inside companies to measure the internal performance of divisions and business segments, and to set executive compensation. Indeed, the commercial packages built around the concept are marketed mainly for internal performance measurement and compensation rather than for security valuation.

Economic value added

Economic value added (EVA) is the best known commercial implementation. The acronym is a trademark of Stern Stewart & Co. and is generally associated with a specific set of accounting adjustments proposed by that firm. Structurally, EVA follows the capital charge route of Example 1.

C% is the percentage cost of capital and TC is total capital, both after the EVA adjustments.

NOPAT and total capital are determined under generally accepted accounting principles and then adjusted. The more common adjustments include the following.

Spending on research and development is capitalised and amortised instead of expensed, so the expense net of estimated amortisation is added back when NOPAT is computed.
An inventory reserve arising from last-in, first-out costing is added back to capital, and the increase in that reserve is added back into NOPAT.
Only cash taxes count as an expense, so deferred taxes are removed.
Leases classified as operating are recharacterised as capital leases, and non-recurring items are adjusted.
Where a strategic investment is not expected to earn a return straight away, the capital charge on it is deferred to a later date.

Those adjustments mean that EVA will generally land on a different number from the plain calculation in Example 1. General residual income valuation also considers the effect of accounting methods on reported results, but the analyst adjustments used there usually differ from the specific EVA set.

Market value added

A company must generate economic profit over time for its market value to rise above the accounting cost of the capital invested in it. Market value added captures that gap directly.

Positive economic profit should show up as a positive market value added.

Whether these value-added measures explain returns better than conventional measures is unsettled. Peterson and Peterson (1996) reported a slightly stronger correlation with stock returns for value-added measures than for conventional profitability ratios. Working at Merrill Lynch, Bernstein and Pigler (1997), and then Bernstein, Bayer and Pigler (1998), reported no predictive advantage at all over simpler yardsticks such as growth in earnings.

Why the name changes so often

Residual income, economic profit, abnormal earnings, excess earnings and economic value added all describe the same deduction: profit after a charge for every unit of capital, equity included. The label usually signals the context. Economic profit is the language of economics. Abnormal earnings is the language of forecasting, where the presumption is that competition drives long-run returns to the cost of capital, so anything above that is abnormal. The excess earnings method is the language of United States tax valuation of small businesses, in use since the 1920s. The discounted abnormal earnings model and the Edwards–Bell–Ohlson model are the academic names, after the researchers who formalised the approach. None of these implies a different calculation in principle; economic value added does, because it comes bundled with a prescribed set of accounting adjustments.

Over the long run, a company earning more than its cost of capital should sell above book value, and one earning less should sell below it. The residual income model turns that statement into a valuation equation with two components: the book value of equity that already exists, and the present value of the residual income the company is expected to add to it.

When the object being valued is a single share rather than total shareholders equity, every input is expressed per share, so earnings per share replaces net income.

Equation 3. Book value per share today plus the present value of expected per-share residual income.

In this notation V₀ is the value of a share today, B₀ is current per-share book value of equity, B_t is expected per-share book value at time t, r is the required rate of return on equity, E_t is expected earnings per share for period t, and RI_t is expected per-share residual income, equal to E_t − rB_t−1.

Note which book value the equity charge uses. It is the book value at the beginning of the period, because that is the capital the shareholders had committed while the period earnings were being produced. Whenever earnings per share exceed that per-share equity charge, residual income is positive; whenever they fall short, it is negative.

Example 3 · Worked

David Smith is evaluating expected residual income for the Canadian National Railway Company (CNR) as of the end of January 2019. He builds the required return from the capital asset pricing model, taking an adjusted beta of 1.02 against the TSX 300 Index, a government bond yield of 1.75% at the 10-year point, and an equity risk premium put at 7.5%. He gathers the following, in Canadian dollars, as of the close on 1 February 2019.

Canadian National Railway inputs (CAD)

Item	Value
Current market price	109.12
Book value per share as of 31 December 2018	24.32
Consensus earnings estimate, FY 2019 (ending December)	6.23
Consensus earnings estimate, FY 2020	6.96
Annualised dividend per share forecast, FY 2019	2.15
Annualised dividend per share forecast, FY 2020	2.32

What is the forecast per-share residual income for the fiscal years ended December 2019 and December 2020?

Solution. First the required return, from the capital asset pricing model:

r = 1.75% + (1.02 × 7.5%) = 9.40%.

Then roll the book value forward and take the equity charge off each year of earnings. The source exhibit applies a FY 2020 dividend of 2.31 rather than the 2.32 shown in the forecast list; the figures below follow the exhibit.

Forecast residual income for CNR (CAD)

Item	2019	2020
Beginning book value (B_t−1)	24.32	28.40
Earnings per share forecast (E_t)	6.23	6.96
Less dividend forecast (D_t)	2.15	2.31
Change in retained earnings (E_t − D_t)	4.08	4.65
Forecast ending book value per share	28.40	33.05
Cost of equity	× 0.094	× 0.094
Per-share equity charge (rB_t−1)	2.29	2.67
Less equity charge from EPS	6.23 − 2.29	6.96 − 2.67
Per-share residual income	3.94	4.29

The 2019 equity charge is 24.32 × 0.094 = 2.29, and the 2020 charge is 28.40 × 0.094 = 2.67.

Deriving the model from the dividend discount model

The residual income model is not a rival theory of value. It is the dividend discount model rewritten, and the derivation is short enough to be worth following once because it explains exactly where the book value term comes from.

Start with the dividend discount model, then impose the clean surplus relation, which states that ending book value equals beginning book value plus earnings less dividends.

The clean surplus relation, holding apart from transactions with owners.

Rearranging that relation gives the dividend as earnings less the increase in book value, D_t = E_t + B_t−1 − B_t. Substitute this expression for the dividend into every term of the dividend discount model and collect terms. The book value figures in successive numerators partially cancel across periods, leaving B₀ outside the summation and a numerator of E_t − rB_t−1 inside it. That is Equation 3.

Because residual income in any year can also be written as the abnormal rate of return multiplied by the capital on which it is earned, the model has a second and equally common form. Return on equity here uses beginning book value in the denominator, not the average book value that financial statement analysis normally uses.

Equation 4. Identical to Equation 3, since RI_t = (ROE_t − r)B_t−1.

Apart from the required rate of return, every input to the model comes from accounting data. That is the source of both its convenience and its vulnerability, a theme the accounting sections return to.

Example 4 · Worked

Bugg Properties is expected to earn $2.00, $2.50 and $4.00 per share over the next three years, and to pay dividends of $1.00, $1.25 and $12.25. The final dividend is a liquidating dividend: analysts expect Bugg to cease operations after Year 3. Book value stands at $6.00 per share today, and shareholders require 10%.

1. Build the schedule of per-share book value and residual income across the three years.

Solution. Roll book value forward with the clean surplus relation and take 10% of beginning book value as the equity charge.

Bugg Properties: book value and residual income ($)

Item	Year 1	Year 2	Year 3
Beginning book value per share (B_t−1)	6.00	7.00	8.25
Net income per share (EPS)	2.00	2.50	4.00
Dividends per share paid	1.00	1.25	12.25
Retained during the year	1.00	1.25	−8.25
Closing book value per share	7.00	8.25	0.00
Less per-share equity charge (rB_t−1)	0.60	0.70	0.825
Residual income	1.40	1.80	3.175

2. Estimate the value of the stock using Equation 3.

Solution. Discount each year of residual income at 10% and add current book value.

V₀ = 6.00 + 1.40 ÷ 1.10 + 1.80 ÷ (1.10)² + 3.175 ÷ (1.10)³
= 6.00 + 1.2727 + 1.4876 + 2.3854 = $11.15.

3. Confirm the estimate using the discounted dividend approach.

Solution. Discount the three dividends, the last of which liquidates the company.

V₀ = 1.00 ÷ 1.10 + 1.25 ÷ (1.10)² + 12.25 ÷ (1.10)³
= 0.9091 + 1.0331 + 9.2036 = $11.15.

The two models agree exactly, as they must when the inputs are consistent and clean surplus holds.

4. Reproduce the residual income figures using the return on equity form in Equation 4.

Solution. Divide each year of earnings by the corresponding beginning book value, subtract the required return and multiply the abnormal rate of return back by beginning book value.

Bugg Properties: the return on equity form ($ and decimals)

Item	Year 1	Year 2	Year 3
Earnings per share	2.00	2.50	4.00
Divided by opening book value per share	÷ 6.00	÷ 7.00	÷ 8.25
Return on equity	0.3333	0.3571	0.4848
Less the required return	− 0.1000	− 0.1000	− 0.1000
Abnormal rate of return (ROE − r)	0.2333	0.2571	0.3848
Times opening book value per share	× 6.00	× 7.00	× 8.25
Residual income	1.400	1.800	3.175

The residual income figures match those from the earnings form exactly, so the valuation is again $11.15.

Where the value is recognised

Example 4 carries a second lesson that matters more in practice than the algebra. Both models produce $11.15, but they get there on very different timetables. The residual income model attributes $6.00 of the $11.15 to the opening book value, before the first period has even begun. That is 53.81% of total value, fixed by the balance sheet rather than by a forecast. The dividend discount model attributes $9.2036 of the same $11.15 to the present value of the final period, the liquidating dividend three years out. Section eight develops the consequence: a model that front-loads value recognition is less exposed to error in distant forecasts.

Applying Equation 4 to a real company requires three things and no more: an opening book value per share, a path for return on equity, and a required rate of return. Everything else follows mechanically. Each year the forecast return on equity applied to beginning book value gives earnings; earnings less dividends roll book value forward; the abnormal rate of return multiplied by beginning book value gives residual income; and the discounted residual income stream is added to the opening book value.

The company in the following example paid no dividend at the valuation date, which makes a dividend discount valuation awkward because the analyst would have to guess when a dividend might be initiated. That is precisely the situation in which a residual income model earns its place.

Example 5 · Worked

Robert Sumargo, an equity analyst, is valuing Alphabet Inc. Class C shares in mid 2019, when a recent closing price is $1,037.39. Alphabet had a fairly high return on equity over the previous 10 years, and consensus estimates for the next two fiscal years imply a high return on equity continuing. Sumargo does not expect that level to be sustainable. He works with the following assumptions.

The capital asset pricing model gives a required rate of return of approximately 8.2%.
Book value per share on 31 December 2018 was $255.40.
Return on equity is expected to be 20.2% in 2019, then to decline by 0.5 percentage points each year under competitive pressure until it reaches the required return. In 2043 return on equity will be 8.2%, and residual income from that year onward will be zero.
No dividend is paid or expected in the foreseeable future, so all earnings are reinvested. Share repurchases are expected to offset new share issuance approximately.

1. Show the first year of the schedule in full.

Solution. Beginning book value per share is $255.40 and forecast return on equity is 20.2%, so forecast earnings per share are 255.40 × 0.202 = $51.59. No dividend is paid and clean surplus holds, so ending book value is 255.40 + 51.59 = $306.99.

The equity charge is 255.40 × 0.082 = $20.94, so residual income is 51.59 − 20.94 = $30.65. The same figure comes directly from the abnormal rate of return: 255.40 × (0.202 − 0.082) = $30.65.

Discounting one year at 8.2% gives 30.65 ÷ 1.082 = $28.33.

2. Continue the schedule to 2042 and value the shares.

Solution. Repeat the same four steps each year, reducing forecast return on equity by 0.5 percentage points annually. No forecast is needed beyond 2042 because residual income is zero from 2043. Add the present values of residual income to the opening book value per share of $255.40. The estimated value is $972.25, below the recent closing price of $1,037.39.

Valuation of Alphabet using the residual income model ($ per share)

Year	Projected EPS	Ending book value per share	Forecast ROE on beginning book value	Equity charge	Residual income	PV of book value and RI
Opening		255.40				255.40
2019	51.59	306.99	20.20%	20.94	30.65	28.33
2020	60.48	367.47	19.70%	25.17	35.30	30.16
2021	70.55	438.02	19.20%	30.13	40.42	31.91
2022	81.91	519.93	18.70%	35.92	45.99	33.56
2023	94.63	614.56	18.20%	42.63	51.99	35.06
2024	108.78	723.34	17.70%	50.39	58.38	36.39
2025	124.41	847.75	17.20%	59.31	65.10	37.50
2026	141.57	989.32	16.70%	69.52	72.06	38.36
2027	160.27	1,149.60	16.20%	81.12	79.15	38.94
2028	180.49	1,330.08	15.70%	94.27	86.22	39.20
2029	202.17	1,532.25	15.20%	109.07	93.11	39.13
2030	225.24	1,757.50	14.70%	125.64	99.60	38.68
2031	249.56	2,007.06	14.20%	144.11	105.45	37.85
2032	274.97	2,282.03	13.70%	164.58	110.39	36.62
2033	301.23	2,583.25	13.20%	187.13	114.10	34.99
2034	328.07	2,911.33	12.70%	211.83	116.25	32.94
2035	355.18	3,266.51	12.20%	238.73	116.45	30.50
2036	382.18	3,648.69	11.70%	267.85	114.33	27.67
2037	408.65	4,057.35	11.20%	299.19	109.46	24.49
2038	434.14	4,491.48	10.70%	332.70	101.43	20.97
2039	458.13	4,949.61	10.20%	368.30	89.83	17.17
2040	480.11	5,429.73	9.70%	405.87	74.24	13.11
2041	499.53	5,929.26	9.20%	445.24	54.30	8.86
2042	515.85	6,445.11	8.70%	486.20	29.65	4.47
Total						972.25

The projected dividend is 0.00 in every year and the cost of equity is 8.20% throughout, so both columns are omitted. This schedule was built in a spreadsheet, so figures may differ slightly from calculator results because of rounding.

Two features of the schedule deserve attention. Earnings per share rise every year, yet residual income peaks in 2035 at $116.45 and falls thereafter, because the shrinking spread between return on equity and the required return eventually outweighs the growing capital base. The present value column peaks earlier still, in 2028 at $39.20, since discounting works against the later years.

The clean surplus assumption inside the schedule

The book value column in that schedule was rolled forward as beginning book value plus net income minus dividends. That is the clean surplus relation in action, and it is an assumption rather than an observed fact. Under both International Financial Reporting Standards and United States generally accepted accounting principles, several items of income and expense bypass the income statement and change the book value of equity directly. Changes in the market value of certain securities are the standard example. Items that bypass the income statement are dirty surplus items and are reported as other comprehensive income, so that comprehensive income equals net income plus other comprehensive income.

Strictly, a residual income model should be built on income measured under clean surplus accounting, meaning all items of income and expense. Where an analyst can reliably estimate material future departures from clean surplus, an adjustment to net income may be appropriate. The last two sections of this lesson deal with that problem in detail.

The general model makes no assumption about the growth of earnings or dividends. If constant growth is imposed, however, the model collapses into a compact expression that shows what actually drives residual income, and it lands directly on the price-to-book ratio.

Begin with the justified price-to-book ratio based on forecasted fundamentals, which follows from the Gordon constant growth dividend discount model together with the sustainable growth rate relation g = b × ROE, where b is the earnings retention rate.

Justified price-to-book from forecasted fundamentals.

The same expression rearranged, with the premium over book value isolated.

The second form is the more revealing. The ratio is one plus a term that depends entirely on the gap between return on equity and the required return. If the justified price is the intrinsic value, so that P₀ = V₀, multiplying through by book value gives the single-stage residual income model.

Equation 5. The constant-growth, or single-stage, residual income model.

Read the two terms separately. B₀ is what the company owns net of what it owes, as the accounts measure it. The second term is the present value of the expected stream of economic profit, the extra value the company creates by earning more on that capital than the capital costs. Three cases follow immediately, and they are worth committing to memory because a great many examination questions are only asking which of the three applies.

If return on equity exceeds the required return, residual income is positive, intrinsic value exceeds book value and the justified price-to-book ratio is above one.
If return on equity equals the required return, residual income is zero and intrinsic value is exactly book value per share.
If return on equity falls below the required return, residual income is negative, intrinsic value is below book value and the justified price-to-book ratio is below one.

Figure 2: Justified price-to-book as a function of the return gap

The whole premium or discount to book value is generated by the gap between return on equity and the required return. Nothing else in the constant-growth model moves the ratio.

Two qualifications belong here. First, the interpretation of B₀ as the fair value of net assets holds only in an idealised world. Both major accounting frameworks permit some liabilities to sit off the balance sheet, and neither reports many corporate assets at fair value, although the international direction of travel is toward fair value accounting, particularly for financial assets. The collapse of Enron Corporation in the United States is the standing reminder of how much off-balance-sheet financing can hide. Second, the model has no view on growth by itself: growth raises value only when return on equity exceeds the required return, and destroys it when the reverse is true.

Tobin’s q

Tobin’s q is a close relative of the price-to-book ratio. It sets the market value of debt plus equity against what it would cost to replace the whole asset base.

A whole-firm analogue of the price-to-book ratio.

The differences from the price-to-book ratio are worth stating explicitly. The numerator covers total capital, debt as well as equity, rather than equity alone. The denominator uses total assets rather than equity. And those assets are measured at replacement cost rather than at historical accounting cost, so the ratio takes account of inflation. All else equal, Tobin’s q is higher the more productive the company assets are. Tobin theorised that q would average one across all companies, because the economic rents earned by assets would average to zero. The practical obstacle is data: replacement costs are rarely disclosed. Where market values or replacement costs of assets are available, they are generally more useful in a valuation than historical costs.

The single-stage residual income model is Equation 5 used directly. It assumes a constant return on equity and a constant earnings growth rate for ever. Like the Gordon growth dividend discount model to which it is related, it can be run in either direction: supply the inputs and solve for value, or supply the market price and solve for the growth rate the market appears to be assuming.

Example 6 · Worked

Joseph Yoh is weighing an investment in Koninklijke Philips N.V. The shares carry a book value of €13.22 each and trade at €35.40. Looking well past the current cycle, Yoh settles on a sustainable return on equity of 12%, growth of 6.75% and a cost of equity of 8.5%.

1. What is the intrinsic value per share under a single-stage residual income model?

Solution. Apply Equation 5.

V₀ = 13.22 + [(0.12 − 0.085) ÷ (0.085 − 0.0675)] × 13.22
= 13.22 + (0.035 ÷ 0.0175) × 13.22
= 13.22 + 2 × 13.22 = €39.66.

The premium over book value is exactly twice book value, because the abnormal return of 3.5 percentage points is twice the 1.75 percentage point spread between the required return and the growth rate.

2. Yoh is comfortable with his other inputs but wants to know what growth rate the market is assuming. Solve for the implied growth rate using the current price of €35.40.

Solution. Set intrinsic value equal to the market price and solve for g.

35.40 = 13.22 + [(0.12 − 0.085) ÷ (0.085 − g)] × 13.22

Subtract book value and divide by it: (35.40 − 13.22) ÷ 13.22 = 22.18 ÷ 13.22 = 1.6778, which is the required multiple of book value in the premium term.
Then 0.035 ÷ (0.085 − g) = 1.6778, so 0.085 − g = 0.035 ÷ 1.6778 = 0.02086.
g = 0.085 − 0.02086 = 6.41%.

The market is implying slower residual income growth than Yoh forecasts, which is consistent with his estimated value of €39.66 exceeding the price of €35.40.

At a price of €35.40 against book value of €13.22, the shares trade at almost 2.7 times book value, and the reason is that return on equity exceeds the cost of equity. Had return on equity equalled the cost of equity, the shares would be worth book value. Had it been lower, residual income would be negative and the shares would be worth less than book value. When a company has no prospect of covering its cost of capital at all, liquidating it and redeploying the assets may be the value-maximising course.

Why one stage is rarely enough

The weakness of the single-stage model is the assumption that an excess return above the cost of equity persists indefinitely. Competition rarely allows that. An abnormally high return on equity attracts entrants, competition intensifies and returns fall for everyone in the industry. An abnormally low return on equity drives companies out, through bankruptcy or otherwise, and returns rise for the survivors. The empirical regularity is mean reversion: return on equity drifts back toward an industry or market level, and residual income drifts toward zero.

Research has taken that regularity seriously. Lee and Swaminathan (1999) and Lee, Myers and Swaminathan (1999) valued the Dow 30 with a residual income model in which return on equity fades to the industry mean over time, and found the model better able to predict future returns than traditional price multiples. The practical response is a multistage model, which is the subject of the next section.

A multistage residual income model forecasts residual income explicitly over a horizon the analyst can actually see, then attaches a terminal value that stands for everything beyond it. Continuing residual income is the residual income earned after the forecast horizon, and choosing an assumption about it is the central judgement in this section.

One convenient finite-horizon specification assumes that at the end of horizon T the company trades at some premium over book value, P_T − B_T.

Equation 6. The final term is the present value of the terminal premium over book value.

Equation 7. The same model written with the abnormal rate of return.

The longer the explicit forecast period, the more likely it is that residual income has already converged to zero by the terminal date, in which case the final term can be set to zero. Over shorter horizons a premium has to be forecast.

A distinctive feature of residual income terminal values

In a residual income valuation the current book value usually accounts for a large share of total value, and the terminal value often does not, for two reasons: book value is typically large relative to any single year of residual income, and return on equity is fading toward the cost of equity, which drives residual income toward zero anyway. That is the opposite of what happens in a dividend discount or discounted cash flow valuation, where the present value of the terminal value routinely dominates the total.

Analysts commonly adopt one of four assumptions about continuing residual income:

it drops to zero in the terminal year and stays there;
it carries on indefinitely at a positive level;
it decays to zero as return on equity drifts back to the cost of equity; or
it settles at whatever level follows from return on equity reverting to some mean.

Figure 3: Fading return on equity and the residual income it leaves behind

As the forecast return on equity converges on the cost of equity, the abnormal rate of return shrinks and residual income approaches zero, which is why the terminal value carries less weight here than in other present value models.

Example 7 · Worked

Diana Rosato, CFA, is assessing Zenlandia Chemical Company, a fictitious specialty chemicals manufacturer, as a possible holding. Her file as of August 2020 contains the following.

Current price is ZL$95.6 and the cost of equity is 12%.
Return on equity has ranged from 18% to 22.9% over 2015–2019, and the only year below 20% in that period was 2016.
The company paid a cash dividend of ZL$2.9995 in 2019, and book value per share was ZL$28.8517 at the end of 2019.
Rosato forecasts earnings per share of ZL$7.162 for 2020 and ZL$8.356 for 2021, with dividends of ZL$2.9995 and ZL$3.2995.
She expects return on equity of 25% from 2022 through 2026, then 20% through 2039, with an earnings retention ratio of 60% after 2021.
After 2039 return on equity will be 12%, so residual income and the terminal value are both zero.

1. Verify the first two forecast years and the first year of the constant retention period.

Solution. For 2020 the equity charge is 28.8517 × 0.12 = ZL$3.4622, so residual income is 7.1620 − 3.4622 = ZL$3.6998, and its present value is 3.6998 ÷ 1.12 = ZL$3.30. Book value rolls to 28.8517 + 7.1620 − 2.9995 = ZL$33.0142, and the implied return on beginning equity is 7.1620 ÷ 28.8517 = 24.82%.

For 2021, the equity charge is 33.0142 × 0.12 = ZL$3.9617 and residual income is 8.3560 − 3.9617 = ZL$4.3943, discounted two years to ZL$3.50. Book value rolls to 38.0707 and the return on beginning equity is 25.31%.

From 2022 the schedule is generated rather than forecast item by item. Net income is 25% of beginning book value: 38.0707 × 0.25 = ZL$9.5177. With 60% retained, the dividend is 40% of net income, or ZL$3.8071. The equity charge is 38.0707 × 0.12 = ZL$4.5685, so residual income is ZL$4.9492, discounted three years to ZL$3.52.

2. Value the shares, assuming a terminal value of zero.

Solution. Add the present values of residual income from 2020 through 2039 to the 2019 book value of ZL$28.8517. The total is ZL$86.41, against a market price of ZL$95.6.

Price sits above the estimate, so the market must be pricing in some mix of stronger residual income through 2039, a positive premium at the horizon, and a cheaper cost of equity. Taking her own forecasts seriously, Rosato calls the shares overvalued.

Zenlandia Chemical residual income schedule (ZL$)

Year	Book value	Projected income	Dividend per share	Forecast ROE on beginning equity (%)	Cost of equity (ZL$)	Residual income	PV of residual income
2019	28.8517						28.85
2020	33.0142	7.1620	2.9995	24.82	3.4622	3.6998	3.30
2021	38.0707	8.3560	3.2995	25.31	3.9617	4.3943	3.50
2022	43.7813	9.5177	3.8071	25.00	4.5685	4.9492	3.52
2023	50.3485	10.9453	4.3781	25.00	5.2538	5.6916	3.62
2024	57.9008	12.5871	5.0349	25.00	6.0418	6.5453	3.71
2025	66.5859	14.4752	5.7901	25.00	6.9481	7.5271	3.81
2026	76.5738	16.6465	6.6586	25.00	7.9903	8.6562	3.92
2027	85.7626	15.3148	6.1259	20.00	9.1889	6.1259	2.47
2028	96.0541	17.1525	6.8610	20.00	10.2915	6.8610	2.47
2029	107.5806	19.2108	7.6843	20.00	11.5265	7.6843	2.47
2030	120.4903	21.5161	8.6065	20.00	12.9097	8.6065	2.47
2031	134.9492	24.0981	9.6392	20.00	14.4588	9.6392	2.47
2032	151.1431	26.9898	10.7959	20.00	16.1939	10.7959	2.47
2033	169.2802	30.2286	12.0914	20.00	18.1372	12.0914	2.47
2034	189.5938	33.8560	13.5424	20.00	20.3136	13.5424	2.47
2035	212.3451	37.9188	15.1675	20.00	22.7513	15.1675	2.47
2036	237.8265	42.4690	16.9876	20.00	25.4814	16.9876	2.47
2037	266.3657	47.5653	19.0261	20.00	28.5392	19.0261	2.47
2038	298.3296	53.2731	21.3093	20.00	31.9639	21.3093	2.47
2039	334.1291	59.6659	23.8664	20.00	35.7996	23.8664	2.47
Present value							86.41
Terminal premium							0.00

The cost of equity is 12.00% in every year. The 2019 row carries the opening book value of ZL$28.8517, shown in the final column as 28.85 because it enters the valuation directly and needs no discounting.

Notice what happens from 2027 onward. Once return on equity is 20% and the payout ratio is 40%, residual income equals exactly the dividend, because the equity charge takes 12 of the 20 percentage points and the remaining 8 points equal 40% of 20%. Book value then grows at the sustainable rate of 0.60 × 20% = 12%, which is exactly the discount rate, so the present value of residual income settles at a constant ZL$2.47 a year.

Fading to an industry mean

Lee and Swaminathan (1999) and Lee, Myers and Swaminathan (1999) forecast residual income explicitly for three years and then let return on equity fade to the industry mean, estimating terminal value as the terminal-year residual income capitalised in perpetuity. Their justification is that any growth in earnings after the horizon is value neutral. An analyst taking that route needs a view on where the industry mean sits, and on any trend in it.

United States sector returns on equity

Sector	Return on equity (%)
Utilities	8.18
Energy	8.81
Basic Materials	11.14
Financial	12.76
Healthcare	19.95
Consumer Goods	19.96
Transportation	21.49
Industrial Goods	23.16
Retail	23.37
Consumer Non-cyclicals	26.59
Technology	28.97

Based on data from CSIMarket on 5 August 2019. The spread across sectors, from 8.18% in Utilities to 28.97% in Technology, is why a single economy-wide fade target is unsatisfactory.

The persistence factor

Dechow, Hutton and Sloan (1999) formalised the fade with a persistence factor, written here as w and denoted by the Greek letter omega in the original research, which takes a value between zero and one.

Equation 8. The persistence factor w governs how slowly residual income decays after the horizon.

A persistence factor of one means residual income does not fade at all and continues at the same level in perpetuity. A persistence factor of zero means residual income stops entirely after the explicit forecast horizon. The higher the factor, the larger the final-stage stream and the higher the valuation, all else equal. In a large sample of company data from 1976 to 1995, Dechow and co-authors estimated a persistence factor of 0.62, which Bauman (1999) interpreted as residual income decaying at an average rate of 38% a year. Persistence clearly varies by company: a business with a strong market leadership position should decay more slowly.

What points toward a slow or a fast fade

Fades faster	Fades more slowly
Accounting rates of return at an extreme level	A low dividend payout ratio
Special items, such as non-recurring charges, at extreme levels	An industry record of persistent excess returns
Accounting accruals at extreme levels

Characteristics identified by Dechow, Hutton and Sloan (1999). Extreme values of any kind tend to revert, which is the common thread on the left.

Example 8 · Worked

Rosato returns to Zenlandia Chemical. Her supervisor questions the assumption that there will be no premium over book value at the end of the forecast period. Recall that residual income in 2039 is ZL$23.8664, the cost of equity is 12%, and the horizon is 20 years from the 2019 valuation date.

1. Value the shares assuming continuing residual income is a perpetuity of the 2039 amount.

Solution. Capitalise the final-year residual income and discount it back 20 years.

Terminal value = ZL$23.8664 ÷ 0.12 = ZL$198.8867.
Present value = ZL$198.8867 ÷ (1.12)²⁰ = ZL$20.6179.

Adding this to the earlier estimate of ZL$86.41, which assumed a zero terminal value, gives ZL$107.03. Because the market price of ZL$95.6 is below that figure, market participants are implying lower continuing residual income than the supervisor assumption, a lower interim return on equity, or both. On these new forecasts the shares are undervalued.

2. Value the shares instead on the assumption that residual income decays after 2039, using a persistence parameter of 0.60.

Solution. Apply the final term of Equation 8, with T equal to 20 and terminal-year residual income of ZL$23.8664. The 1.12 factor here reflects a 12% growth rate, calculated as the retention ratio multiplied by return on equity, (0.60)(20%) = 0.12.

Present value of the terminal value = 23.8664 ÷ [(1 + 0.12 − 0.60)(1.12)¹⁹] = 23.8664 ÷ (0.52 × 8.612762) = ZL$5.33.

This is added to ZL$83.93, which is the book value plus the present value of residual income over the first 19 years, that is the ZL$86.41 total from Example 7 less the ZL$2.47 present value attributable to 2039. The two components sum to ZL$89.26. The source reports the total as ZL$86.26; the components as printed add to ZL$89.26. Either figure sits below the market price of ZL$95.6, so the conclusion is unchanged: if residual income does not persist at a stable level past 2039 and instead deteriorates, the shares are modestly overvalued at ZL$95.6.

3. Summarise how sensitive the valuation is to the continuing residual income assumption.

Solution. The three assumptions produce ZL$86.41 with a zero terminal value, ZL$89.26 with a decaying stream at a persistence factor of 0.60, and ZL$107.03 with a perpetuity of the final-year amount. The perpetuity assumption adds ZL$20.62 to the estimate, roughly a quarter of the zero-terminal-value figure, and it is the only one of the three that turns the verdict from overvalued to undervalued. The terminal assumption is not a detail.

Terminal value from a market multiple

The terminal residual value in Equations 6 and 7 is P_T − B_T, the terminal price less the terminal book value. Nothing constrains how the terminal price is estimated. It can come from a dividend discount model, a price–earnings multiple or a price-to-book multiple. The next example takes the last of these.

Example 9 · Worked

Andreea Popescu is valuing URS Holdings with a two-stage residual income model on this set of assumptions: opening book value per share is €15.00; the cost of equity is 7.95%; over six years, earnings per share run at a quarter of opening book value; the company distributes 30% of earnings as cash each year; and at the six-year mark the shares trade at 1.80 times book value.

1. Calculate per-share book value and residual income over the forecast horizon.

Solution. Four mechanical rules generate every row. Net income equals a quarter of opening book value. The dividend takes 30% of that net income. Closing book value is opening book value plus net income less the dividend. Residual income is net income less 7.95% of opening book value. Year 1 is representative: net income is 0.25 × 15.000 = 3.750, dividends are 0.30 × 3.750 = 1.125, ending book value is 15.000 + 3.750 − 1.125 = 17.625, and residual income is 3.750 − (0.0795 × 15.000) = 3.750 − 1.1925 = 2.558, whose present value is 2.558 ÷ 1.0795 = 2.369.

Residual income for URS Holdings (€ per share)

Year	Beginning book value	Net income	Dividends	Ending book value	Residual income	PV of residual income
1	15.000	3.750	1.125	17.625	2.558	2.369
2	17.625	4.406	1.322	20.709	3.005	2.579
3	20.709	5.177	1.553	24.334	3.531	2.807
4	24.334	6.083	1.825	28.592	4.149	3.055
5	28.592	7.148	2.144	33.595	4.875	3.325
6	33.595	8.399	2.520	39.475	5.728	3.620
Sum						17.755

2. Estimate the value per share using Equation 6.

Solution. The terminal premium is the terminal price less terminal book value, and the terminal price is 1.8 times terminal book value, so the premium is simply 0.8 times terminal book value.

Terminal premium = (1.8 × 39.475) − 39.475 = 31.580.
Present value at 7.95% over six years = 31.580 ÷ (1.0795)⁶ = 19.956.

Value per share = 15.000 + 17.755 + 19.956 = €52.711.

3. Confirm the estimate with a discounted dividend valuation.

Solution. Discount the six dividends and the terminal price at 7.95%.

Dividend discount valuation of URS Holdings (€ per share)

Year	Dividends	PV of dividends
1	1.125	1.042
2	1.322	1.134
3	1.553	1.235
4	1.825	1.344
5	2.144	1.463
6	2.520	1.592
Sum of six years of dividends		7.810
Terminal price (1.8 × B_T)	71.054	44.901
Value per share		52.711

The two approaches agree at €52.711, but they allocate the total very differently: the dividend model puts 44.901 of the 52.711 into the terminal price, while the residual income model puts 15.000 into book value that already exists.

Valuation models built on discounting dividends or free cash flows are exactly as theoretically sound as the residual income model. They differ in what they discount and therefore in when they recognise value. The dividend discount model and the free cash flow to equity model forecast cash flows to shareholders and discount them at the cost of equity. The free cash flow to the firm model forecasts cash available to all providers of capital and discounts at the weighted average cost of capital. The residual income model does something different: it starts from a balance sheet figure, the book value of equity, and adds the present value of the economic profit expected on top of it.

In theory the total is the same in every case. Given fully consistent assumptions and the same required return, book value plus expected residual income, expected dividends, and expected free cash flow all produce the same present value. What differs is the timing of recognition, and that difference has real practical consequences.

Example 10 · Worked

A company will earn $1.00 per share for ever and pays out all earnings as dividends. Book value per share is $6.00 and the required rate of return on equity is 10%.

1. Value the stock with the dividend discount model.

Solution. The dividend is a perpetuity, so its present value is D ÷ r.

V₀ = $1.00 ÷ 0.10 = $10.00 per share.

2. Calculate the level of per-share residual income earned each year.

Solution. All net income is paid out, so nothing is retained and book value per share stays at $6.00 for ever. Every year:

RI_t = E_t − rB_t−1 = $1.00 − 0.10($6.00) = $1.00 − $0.60 = $0.40.

3. Value the stock with the residual income model.

Solution. Value equals current book value per share plus the present value of the residual income stream, which is itself a perpetuity.

V₀ = $6.00 + $0.40 ÷ 0.10 = $6.00 + $4.00 = $10.00.

The two models agree, as they must.

4. Compare where each model places the value, year by year.

Solution. Tabulate the present value of each component under both models.

Value recognition in the dividend discount and residual income models ($)

Year	Dividend	PV of dividend	Book value or residual income	PV of book value or residual income
0			6.00	6.000
1	1.00	0.909	0.40	0.364
2	1.00	0.826	0.40	0.331
3	1.00	0.751	0.40	0.301
4	1.00	0.683	0.40	0.273
5	1.00	0.621	0.40	0.248
6	1.00	0.564	0.40	0.226
7	1.00	0.513	0.40	0.205
8	1.00	0.467	0.40	0.187
Total		10.00		10.00

Both columns continue indefinitely and both sum to $10.00. The rows shown are the first eight of an infinite series.

5. Quantify the difference in timing.

Solution. In the residual income model the opening book value of $6.00 is 60% of the total present value of $10, and it is recognised at time zero, before any forecast is required. In the dividend discount model less than $1.00 of the $10 comes from the first year, and the first five dividends together contribute $0.909 + $0.826 + $0.751 + $0.683 + $0.621 = $3.79, or about 38% of the total.

Figure 4: Where each model places the value

The two columns have the same total height because both models give the same value. The residual income model rests most of that total on a balance sheet figure that requires no forecast, while the dividend model rests it on a distant terminal value.

That is the practical advantage. In a dividend discount or free cash flow valuation, a large fraction of total value is typically the present value of an expected terminal value, and substantial uncertainty usually surrounds it. Residual income valuations are far less sensitive to the terminal value estimate, and in some contexts the terminal value can reasonably be set to zero altogether. Deriving value from the earlier part of the forecast horizon, where forecasting error is smallest, is one of the main reasons the model is worth having.

Strengths of residual income models

Terminal values do not make up a large portion of total present value, relative to other models.
The models use accounting data that is already published and readily available.
They can be applied to companies that pay no dividends, or that have no positive expected near-term free cash flow.
They can be used when cash flows are unpredictable.
They focus attention on economic profitability rather than on accounting profitability.

Weaknesses of residual income models

They rest on accounting data, which management can manipulate.
The accounting inputs may need significant adjustment before they are usable.
They require either that the clean surplus relation holds or that the analyst makes an appropriate adjustment where it does not.
Using accounting income assumes that interest expense reflects the cost of debt capital appropriately.

Choosing when to use the model

A residual income model is most appropriate when a company pays no dividend or pays an unpredictable one; when expected free cash flow is negative within the horizon the analyst can forecast comfortably; or when great uncertainty surrounds the terminal value under an alternative present value approach. It is least appropriate when there are significant departures from clean surplus accounting, or when the key determinants of residual income, book value and return on equity, are themselves unpredictable.

Because every present value model descends from the same theoretical root, fully consistent forecasts of earnings, cash flow, dividends, book value and residual income through a complete set of pro forma financial statements, discounted at the same required return, must give the same value. In practice the items cannot all be forecast with equal confidence, and that is what decides the choice. A company with near-term negative free cash flow and an uncertain terminal value suits residual income. A company with positive and predictable cash flow that pays no dividend suits discounted free cash flow.

Two further uses are worth noting. Like the dividend discount and free cash flow models, a residual income model can be used to establish justified market multiples: divide the estimated value by earnings to obtain a justified price–earnings ratio, for instance. And it can be run alongside another model as a consistency check. Where two appropriate models produce widely different values, the inconsistency usually lies in the assumptions rather than in the models, and the analyst has to establish which set of assumptions is mutually consistent and which model best fits the company.

Does earlier recognition make the model more accurate

Not by itself. The value of a share is whatever it is; no rearrangement of the algebra changes it. What earlier recognition changes is the exposure of the estimate to forecast error. If the analyst is wrong about year 25, a dividend discount valuation in which most of the value sits in the terminal figure moves a great deal, whereas a residual income valuation in which 60% of the value is an audited balance sheet number moves comparatively little. The trade-off is that residual income transfers the exposure onto the quality of the accounting: book value has to mean something, and net income has to capture all changes in it. A model is not more accurate because it front-loads recognition; it is differently vulnerable, and the analyst should choose the vulnerability that is easier to manage for the company in hand.

Bauman (1999) pointed out that the two components of the residual income model balance each other, provided the clean surplus relation is respected. A company making aggressive accounting choices reports higher book values and, later, lower earnings. In the model, the present value of the difference in future income exactly offsets the initial difference in book value, so the valuation is unaffected. That is an elegant property and it is why the model tolerates a good deal of accounting variation.

The property fails in two ways in practice. First, the clean surplus relation does not always hold, because both major accounting frameworks let items bypass the income statement and go straight to equity. Second, analysts often use past earnings to predict future earnings, so a distortion in reported earnings propagates into the forecast rather than reversing.

Figure 5: The clean surplus relation and the route around it

Items routed around the income statement still land in equity, so book value remains correct. It is net income, and therefore return on equity and residual income, that is understated or overstated.

What bypasses the income statement

The clearest example is a change in the market value of investments classified as available for sale under United States generally accepted accounting principles, or as equity instruments measured at fair value through other comprehensive income under International Financial Reporting Standards. Both frameworks carry these on the balance sheet at market value, but the unrealised movement in that value lands inside other comprehensive income instead of on the income statement itself.

Comprehensive income covers every movement in equity over a period apart from what the owners put in and what is paid out to them. It equals net income plus other comprehensive income. Items that commonly take the second route include the following.

Certain pension adjustments.
Part of the gain or loss on certain hedging instruments.
Unrealised movements in the fair value of some financial instruments.
Adjustments arising on translation of foreign currency.
Movements in revaluation surplus on property, plant and equipment or on intangibles, permitted under IFRS but not under US GAAP.
On certain classes of liability, the portion of a fair value change caused by a shift in the credit risk of that liability, again IFRS only.

In every one of these cases the book value of equity is stated correctly, because it includes accumulated other comprehensive income. It is net income that is wrong for residual income purposes, and with it return on equity, which has net income in its numerator, and therefore residual income itself.

Two practical points follow. Historical return on equity is best measured in aggregate, with net income over total shareholders equity, and not on a per-share basis, because issuing and repurchasing shares distorts the per-share version. And, as Frankel and Lee (1999) noted, bias enters the valuation only if the present value of the clean surplus violations does not net to zero. Reductions in income in some periods may be offset by increases in others. The work is to comb the equity section of the balance sheet, alongside the statement of comprehensive income and the statement of changes in equity, form a view on whether the amounts are likely to cancel out, and judge the effect on future return on equity.

Example 11 · Worked

Two statements of changes in shareholders equity for the year ended 31 December 2018 are under review. The first, prepared under IFRS, is for Nokia Corporation, a supplier of network equipment, software and services to telecommunications network companies. The second, prepared under US GAAP, is for SAP AG, headquartered in Germany, a worldwide provider of enterprise application software.

Columns holding items that bypassed the income statement in 2018

Company	Columns outside retained earnings	Currency translation effect (€ millions)
Nokia Corporation	Share issue premium; translation differences; fair value and other reserves; reserve for invested unrestricted equity	341
SAP AG	Share premium; other components of equity	887

For SAP the €887 million figure is the movement in other components of equity, which includes the translation adjustment for the year. Sources: company statements of changes in shareholders equity.

How should these amounts be interpreted for a residual income valuation?

Solution. Both amounts are positive adjustments to shareholders equity, so each would have increased reported income had it passed through the income statement instead. Reported net income is therefore lower than comprehensive income in both cases, and historical return on equity computed from net income understates the return actually earned on the equity base.

If an analyst has used that historical return on equity as the starting point for a residual income valuation, and expects the pattern of positive translation adjustments to continue, an upward adjustment to the estimate of future return on equity may be warranted. The qualification is important: exchange rate movements can reverse, and a translation gain in one period is quite capable of becoming a translation loss in the next. That is exactly the situation Frankel and Lee described, where violations may net to zero over time and no adjustment is needed.

How the forecasting assumption interacts with other comprehensive income

The worked examples so far took actual beginning equity and a forecast return on equity, and multiplied one by the other to obtain forecast net income. Since equity includes accumulated other comprehensive income, an assumption about future other comprehensive income feeds straight through into forecast net income and hence into residual income. The effect is easy to underestimate until it is laid out.

Example 12 · Worked

A hypothetical company reports, in year t − 1, net income of $120 on beginning equity of $1,000, a return of 12%. It pays no dividends, so ending retained earnings are $120. It also reports other comprehensive income of −$100, a loss, so accumulated other comprehensive income closes at −$100 and ending total equity is $1,020. The required return is 10%.

Three forecasts for years t and t + 1 all assume a return on equity of 12% applied to beginning book value, and all assume no dividends. They differ only in what they assume about future other comprehensive income. Forecast A assumes none in either year. Forecast B assumes the same −$100 in each year. Forecast C assumes the prior loss reverses in year t, so accumulated other comprehensive income returns to zero.

Three forecasts with different assumptions about comprehensive income ($)

Item	Actual t − 1	A: t	A: t + 1	B: t	B: t + 1	C: t	C: t + 1
Beginning total equity	1,000.00	1,020.00	1,142.40	1,020.00	1,042.40	1,020.00	1,242.40
Net income	120.00	122.40	137.09	122.40	125.09	122.40	149.09
Other comprehensive income	(100.00)			(100.00)	(100.00)	100.00
Ending accumulated other comprehensive income	(100.00)	(100.00)	(100.00)	(200.00)	(300.00)
Ending total equity	1,020.00	1,142.40	1,279.49	1,042.40	1,067.49	1,242.40	1,391.49
Equity charge at 10%	100.00	102.00	114.24	102.00	104.24	102.00	124.24
Residual income	20.00	20.40	22.85	20.40	20.85	20.40	24.85

Common stock is 1,000.00 throughout and there are no liabilities, so total equity equals total assets in every column.

1. All three forecasts assume the same 12% return on equity and full retention. What growth rate in residual income should that imply, and which forecast delivers it?

Solution. With every dollar of earnings retained, a constant 12% return on equity implies that both net income and residual income grow at 12%. Only Forecast A delivers it.

Forecast A: net income moves from $122.40 to $137.09, up 12%, and residual income moves from $20.40 to $22.85, up 12% as well.
Forecast B: residual income grows from $20.40 to $20.85, an increase of 2.2%.
Forecast C: residual income grows from $20.40 to $24.85, an increase of 21.8%.

The other comprehensive income assumption changes the equity base on which the assumed return on equity is applied, so it silently changes the growth rate of the very quantity being valued.

2. Take Forecast B and instead measure return on equity and residual income on a comprehensive income basis. What happens?

Solution. On a comprehensive income basis, the return earned in year t − 1 was ($120 − $100) ÷ $1,000 = $20 ÷ $1,000 = 2.0%, not 12%.

Carrying that basis into year t: forecast comprehensive income is $22.40, being net income plus other comprehensive income; the equity charge is 10% of beginning equity of $1,020, or $102; and residual income is $22.40 − $102 = −$79.60.

Against the positive $20.40 obtained when the clean surplus violation is ignored, the swing is exactly $100 in a single year. A forecast of return on equity or of net income that ignores clean surplus violations distorts residual income, and unless the present value of those distortions nets to zero it distorts the valuation too.

The implication for practice is direct. If future other comprehensive income is expected to be significant relative to net income, and the year-to-year amounts are not expected to net to zero, the analyst should try to bring those items into the forecast so that residual income is closer to what it would be under clean surplus. Where possible that means making explicit assumptions about future amounts of other comprehensive income rather than leaving them out.

Example 13 · Worked

Per-share forecasts for Mannistore, Inc., a hypothetical operator of a chain of retail stores, are given below. The cost of equity capital is 10%.

Forecasts for Mannistore, Inc. ($ per share)

Variable	Year 1	Year 2	Year 3	Year 4	Year 5
Shareholders equity_t−1	8.58	10.32	11.51	14.68	17.86
Plus net income	2.00	2.48	3.46	3.47	4.56
Less dividends	0.26	0.29	0.29	0.29	0.38
Less other comprehensive income	0.00	1.00	0.00	0.00	0.00
Equals shareholders equity_t	10.32	11.51	14.68	17.86	22.04

The Year 2 entry of 1.00 on the other comprehensive income line is a deduction from equity, that is a loss of $1.00 routed around the income statement.

1. The forecast terminal price at the end of Year 5 is $68.40. Estimate the value per share using the dividend discount model.

Solution. Discount the five dividends and the terminal price at 10%.

V₀ = 0.26 ÷ 1.10 + 0.29 ÷ (1.10)² + 0.29 ÷ (1.10)³ + 0.29 ÷ (1.10)⁴ + 0.38 ÷ (1.10)⁵ + 68.40 ÷ (1.10)⁵ = $43.59.

2A. Estimate the value using the residual income model, with residual income based on net income without adjustment.

Solution. Residual income is net income less beginning shareholders equity multiplied by the 10% cost of equity.

Residual income from unadjusted net income ($ per share)

Item	Year 1	Year 2	Year 3	Year 4	Year 5
RI = NI − (SE_t−1 × r)	1.14	1.45	2.30	2.00	2.77

Solution, continued. Using Equation 6, with the terminal premium equal to the terminal price less the Year 5 book value of $22.04:

V₀ = 8.58 + 1.14 ÷ 1.10 + 1.45 ÷ (1.10)² + 2.30 ÷ (1.10)³ + 2.00 ÷ (1.10)⁴ + 2.77 ÷ (1.10)⁵ + (68.40 − 22.04) ÷ (1.10)⁵
= 8.58 + 35.84 = $44.42.

2B. Repeat with residual income based on net income plus other comprehensive income.

Solution. Only Year 2 changes, because the $1.00 loss is deducted from that year of income: 2.48 − 1.00 − (10.32 × 0.10) = $0.45.

Residual income from net income adjusted for other comprehensive income ($ per share)

Item	Year 1	Year 2	Year 3	Year 4	Year 5
RI = (NI + OCI) − (SE_t−1 × r)	1.14	0.45	2.30	2.00	2.77

Solution, continued. The valuation becomes

V₀ = 8.58 + 1.14 ÷ 1.10 + 0.45 ÷ (1.10)² + 2.30 ÷ (1.10)³ + 2.00 ÷ (1.10)⁴ + 2.77 ÷ (1.10)⁵ + (68.40 − 22.04) ÷ (1.10)⁵
= 8.58 + 35.01 = $43.59.

3. Interpret the two answers.

Solution. The first calculation omits the adjustment for the clean surplus violation and overstates value by $0.83, giving $44.42. The second includes the adjustment and returns $43.59, which matches the dividend discount estimate exactly. The dividend discount model was never affected, because dividends were forecast directly and did not depend on how income was routed. The residual income model was affected, because it depends on income being complete.

4. Suppose no terminal price forecast is available and a Gordon growth terminal value is appropriate instead. Growth in net income and dividends from t = 5 to t = 6 is estimated at 8%. Predict Year 6 residual income and its growth rate.

Solution. Year 6 net income is $4.56 × 1.08 = $4.92. The equity charge is beginning book value of $22.04 multiplied by the 10% cost of capital, or $2.204, so residual income is 4.92 − 2.204 = $2.72.

The growth rate in residual income is 2.72 ÷ 2.77 − 1, which is approximately −2%. Net income and dividends grow at a positive 8% while residual income shrinks, because the equity charge is levied on a book value that has grown faster than earnings. Do not assume that the growth rate in residual income equals the growth rate in earnings or dividends; in general it does not.

Where the analyst has no basis for explicit assumptions about future other comprehensive income, the correct response is not to ignore the issue but to stay aware of the potential effect on residual income and to adjust the return on equity estimate in light of it. If the distortion cannot be handled at all, an alternative valuation model is the better answer.

Two quantities drive residual earnings: book value of equity and return on equity. Both come from the accounts, so the analyst has to know what the accounts leave out and what they measure at something other than fair value. A reliable book value requires identifying and scrutinising significant off-balance-sheet assets and liabilities, and restating reported items to fair value where that is possible. The financial statement footnotes are where these usually surface.

Operating leases are the most common example. They do not change the amount of equity, because the off-balance-sheet assets offset the off-balance-sheet liabilities, but they can change the assessment of future earnings and therefore the residual income component of value. Inventory carried at last-in, first-out cost is another: it may need restating to current value, although LIFO is not permitted under IFRS. Items commonly reviewed for balance sheet adjustment include the following, and the list is not exhaustive.

Leases classified as operating.
Intangible assets.
Inventory.
Allowances and reserves, bad debt provisions among them.
Deferred tax balances on both sides of the balance sheet.

Aggressive capitalisation and the return on equity it produces

Consider a company with $1,000,000 of book value that reports $200,000 of earnings before taxes after expensing an outlay of $50,000. Ignoring taxes, its return on equity is 20%. Had it capitalised that outlay instead of expensing it, earnings would be $250,000 and book value $1,050,000, giving a return on equity of 23.81%. Capitalising an expenditure overstates both current earnings and current book value at the same time.

The capitalised item will eventually be amortised or written off, reducing realised future earnings, but analyst expectations often lean on historical data, so the inflated return on equity can be carried forward into the forecast and the model will then overestimate value. Where capitalisation persists at a company of stable size, return on equity can decline over time as net income normalises while book value stays overstated. Where the company is growing and the expenditure in question is growing with it, return on equity can stay high indefinitely. Since the model runs almost entirely on accounting inputs, aggressive methods such as accelerating revenue or deferring expenses can produce real valuation errors.

Intangible assets

Specifically identifiable intangibles that can be separated from the entity and sold are generally included in book value of equity, and where they have a finite useful life they are amortised as an expense. The complication is that intangibles often are not recognised as assets at all unless they are acquired. Advertising expenditure can build a highly valuable brand, but the advertising is expensed and the brand appears nowhere on the balance sheet unless the company owning it is bought.

Example 14 · Worked

Two companies, Alpha and Beta, report the following, in thousands of euros. Each pays out all net income as dividends, so growth is zero and the clean surplus relation holds. Alpha has a return on equity of 12% and Beta has 15%, both expected to continue indefinitely, and each has a required rate of return of 10%. The fair market value of each company property, plant and equipment equals its book value.

Alpha and Beta before any transaction (€ thousands)

Balance sheet and earnings	Alpha	Beta
Cash	1,600	100
Plant and equipment	3,400	900
Assets in total	5,000	1,000
Shareholders equity	5,000	1,000
Annual net income	600	150

1. Value each company separately in a residual income framework.

Solution. Apply Equation 5 with g = 0. Using total book value rather than per-share figures gives the same answer.

Alpha: V₀ = 5,000 + [(0.12 − 0.10) ÷ (0.10 − 0.00)] × 5,000 = €6,000.
Beta: V₀ = 1,000 + [(0.15 − 0.10) ÷ (0.10 − 0.00)] × 1,000 = €1,500.

Combined, the two are worth €7,500. Both are worth more than book value because both earn more than the required return, and neither balance sheet reflects that excess.

2. Alpha buys Beta for €1,500 in cash. The €500 paid above Beta reported assets of €1,000 is the fair value of an exclusive 10-year licence held by Beta, whose original cost was an immaterial application fee. Beta property, plant and equipment is already at its fair value of €1,000. Value the combined company with the licence amortised over 10 years.

Solution. The entire purchase price is allocated to identifiable assets, so no goodwill arises. Alpha cash becomes €1,600 + €100 − €1,500 = €200; property, plant and equipment becomes €4,300; the licence is carried at €500; total assets and equity are unchanged at €5,000, because the purchase was made in cash and no new shares were issued.

Combined expected net income is €600 + €150 − €50 of amortisation = €700, so the expected return on equity is 700 ÷ 5,000 = 14%.

V₀ = 5,000 + [(0.14 − 0.10) ÷ 0.10] × 5,000 = €7,000.

3. Why is the combined company apparently worth €500 less than the two separate companies, and what is the fix?

Solution. Nothing economic has changed. If a fair price was paid, the combined value should not fall. The drop is an artefact of the amortisation of the licence, which depresses net income and therefore return on equity.

Adding the amortisation back before computing return on equity gives net income of €750 and a return on equity of 15%, so

V₀ = 5,000 + [(0.15 − 0.10) ÷ 0.10] × 5,000 = €7,500,

which equals the sum of the two separate valuations.

4. Would the answer differ if Alpha had paid with newly issued stock rather than cash?

Solution. No. The currency used to settle a transaction should not change total value. With €1,500 of newly issued stock, Alpha cash stays at €1,700, property, plant and equipment is €4,300, the licence is €500, and total assets and equity are €6,500. Projected earnings excluding licence amortisation are €750, so projected return on equity is 750 ÷ 6,500 = 11.538%.

V₀ = 6,500 + [(0.11538 − 0.10) ÷ 0.10] × 6,500 = €7,500.

Book value of equity is higher and return on equity is lower, and the two effects offset exactly. The lesson generalises: if an acquirer overpays, the overpayment will show up as reduced future residual income rather than disappearing.

Research and development

Research and development is the other intangible that needs care. Under US GAAP it is generally expensed directly, with limited exceptions such as the capitalisation of software development costs after product feasibility is established. Under IFRS some development costs can be capitalised and amortised. Either way the spending is reflected in return on equity, and therefore in residual income, over the long term. Unproductive research spending lowers residual income through the outlay itself. Productive research spending should generate higher revenue that offsets the outlay over time. For a mature company on a continuing basis, return on equity should reflect the productivity of research spending without any adjustment.

Lundholm and Sloan (2007) made the underlying point precisely: adding back an omitted asset and then amortising it has no effect on valuation under a residual income model. The adjustment raises estimated value by adding the asset to book value at time zero, and lowers it by an equal amount through the present value of the future amortisation plus the present value of the periodic capital charge on the asset. What does change is the profile of return on equity through time. Expensing research produces an immediately lower return on equity than capitalising it, and then a slightly higher return on equity in later years, once the capitalised amount is being amortised. Because return on equity appears in so many expressions derived from the model, and is often used to forecast net income, the profile matters even when the level of value does not.

Non-recurring items

A residual income forecast should be built on recurring items. Companies frequently report non-recurring charges inside earnings, and leaving them there leads to over- or underestimates of future residual earnings. No adjustment to book value is needed for these items, because non-recurring gains and losses are already reflected in the value of the assets in place. Hirst and Hopkins (2000) observed that non-recurring items sometimes arise from accounting rules and sometimes from strategic management decisions, and emphasised reading the financial statement notes and other disclosures for items that may need adjustment, such as:

restructuring charges;
discontinued operations;
changes in accounting;
unusual items; and
extraordinary items, a category that survives under US GAAP but not under IFRS.

There is a judgement to make here. Where management records a restructuring or unusual charge in every single period, the item is arguably an ordinary operating expense and may need no adjustment at all. Separately, a non-operating gain is sometimes netted, wrongly, against an operating expense line such as selling, general and administrative costs. If material, that treatment can usually be uncovered by careful reading of the footnotes and press releases, and such items should more often than not be removed from operating earnings when forecasting residual income.

Other aggressive practices

Accelerating revenue into the current period and deferring expenses to a later one both increase earnings and book value simultaneously. A company might ship unordered goods to customers at the year end, recording revenue and a receivable; or capitalise rather than expense a cash payment, lowering expenses and raising assets. The opposite manipulation is the cookie jar reserve, in which excess losses or expenses are booked in an earlier period, often alongside an acquisition or a restructuring, and then released to reduce expenses and lift income in later periods. Reserves deserve careful examination when residual earnings are being assessed, and the integrity of management is part of the assessment of the inputs.

International considerations

Accounting standards differ across countries, which produces different measures of book value and earnings and suggests that models built on accrual accounting data might travel less well internationally than other present value models. The evidence is more encouraging than that suggests. Frankel and Lee (1999) applied a simple residual income model, without any of the adjustments discussed above, and found it accounted for 70% of the cross-sectional variation of stock prices across 20 countries. They identified three primary considerations for international application:

whether reliable earnings forecasts can be obtained at all;
whether accounting rules of poor quality delay the recognition of value changes; and
whether the clean surplus assumption is violated systematically.

The model should work best where earnings forecasts exist, clean surplus violations are limited and recognition is not delayed. Because unadjusted data already gave good explanatory power, adjusting reported data for clean surplus and other violations ought to make international comparisons more comparable still. Where clean surplus violations exist, accounting choices delay recognition, or the disclosures simply do not permit an adjustment, the residual income model is not appropriate and a model less dependent on accounting data, such as a free cash flow to equity model, should be used instead.

Convergence has reduced the problem over time. As of 2019, according to the AICPA, an association representing the accounting profession, approximately 120 nations and reporting jurisdictions permitted or required IFRS for domestic listed companies, and approximately 90 countries had fully conformed with IFRS as promulgated by the IASB and included a statement acknowledging that conformity in audit reports. Elsewhere, national standard setters are still working to close the remaining gaps between their own rules and IFRS. Even inside a single framework, however, companies make choices and estimates that affect valuation, so the analytical work described in these last two sections does not go away.

Check yourself

1. A company has equity capital of $400 million, a cost of equity of 11%, and reports net income of $38 million. Is it creating or destroying value, and by how much?

The equity charge is $400 million × 11% = $44 million, so residual income is $38 million − $44 million = −$6 million. The company is profitable in an accounting sense and is destroying $6 million of value a year in an economic sense. Its shares should trade below book value, all else equal.

2. Under what two conditions do the net income route and the capital charge route to residual income give the same answer?

First, the marginal after-tax cost of debt used in the capital charge must equal the after-tax interest expense already deducted in arriving at net income. Second, the weights used to compute the capital charge must be book value weights for debt and equity rather than market value weights. Relax either condition and the two routes diverge.

3. A company has book value per share of $20, a sustainable return on equity of 9%, a cost of equity of 9% and a growth rate of 4%. What is its intrinsic value per share and its justified price-to-book ratio?

Return on equity exactly equals the cost of equity, so residual income is zero in every period and the premium term vanishes: V₀ = 20 + [(0.09 − 0.09) ÷ (0.09 − 0.04)] × 20 = $20. The justified price-to-book ratio is 1.00. Note that the growth rate is irrelevant here: growth adds value only when there is an abnormal return to grow.

4. An analyst values a company at $60 a share, of which $34 is current book value. Another analyst values the same company with a dividend discount model and finds that 70% of the value comes from the terminal value. Which estimate is more exposed to an error in the year 15 forecast, and why?

The dividend discount estimate. Its terminal value carries 70% of the total, so an error in the distant forecast moves most of the answer. In the residual income estimate, $34 of $60, or roughly 57%, is a balance sheet figure that requires no forecast at all, so only the remaining $26 is exposed. This is the practical case for residual income when terminal values are uncertain. It is not a claim that the residual income estimate is more accurate: it shifts the exposure onto the quality of the accounting instead.

5. A company reports a large foreign currency translation gain in other comprehensive income each year for five consecutive years. An analyst forecasts return on equity from reported net income. Which direction is the residual income estimate biased, and when would no adjustment be needed?

Reported net income excludes the gains, so historical return on equity understates the economic return and the residual income forecast is biased downward, understating value. No adjustment would be needed if the present value of the clean surplus violations netted to zero, that is if the translation gains were expected to be offset by translation losses in other periods. A persistent one-directional pattern is precisely the case where that does not hold.

6. In a multistage model, why does the choice between a zero terminal value and a perpetuity of terminal-year residual income usually matter less here than the equivalent choice in a dividend discount model?

Because current book value normally accounts for a large share of the residual income estimate, and because return on equity is usually assumed to fade toward the cost of equity, which drives residual income toward zero by the terminal date anyway. The terminal component is therefore a smaller slice of a total that is already largely fixed. In a dividend discount valuation the terminal figure frequently dominates. Note that “less” is not “not at all”: in Zenlandia the perpetuity assumption added ZL$20.62 to an ZL$86.41 estimate and flipped the verdict.

7. An acquirer buys a target for cash, allocates the entire premium to an identifiable licence, and amortises it. Its post-deal return on equity falls and a residual income valuation of the combined company drops below the sum of the two standalone values. What has gone wrong?

Nothing economic. If a fair price was paid, the combined value cannot be lower. The amortisation charge depresses net income and therefore the return on equity used in the constant-growth model, while book value is unchanged because the purchase was made in cash. Adding the amortisation back before computing return on equity restores the valuation to the sum of the standalone values. Genuine value destruction from overpaying would show up instead as lower future residual income.

End of lesson.

Back to 05 Equity Investments

EQ 5 – Residual Income Valuation

Overview

Residual income and the charge for equity capital

Two ways to reach the same number

The residual income model and its link to the dividend discount model

Deriving the model from the dividend discount model

Where the value is recognised

Valuing a company with the general residual income model

The clean surplus assumption inside the schedule

Fundamental determinants and the justified price-to-book ratio

Tobin’s q

Single-stage valuation and the implied growth rate

Why one stage is rarely enough

Multistage models and continuing residual income

A distinctive feature of residual income terminal values

Fading to an industry mean

The persistence factor

Terminal value from a market multiple

Comparison with dividend discount and free cash flow models

Strengths of residual income models

Weaknesses of residual income models

Choosing when to use the model

Clean surplus violations and other comprehensive income

What bypasses the income statement

How the forecasting assumption interacts with other comprehensive income

Balance sheet, intangible and international accounting considerations

Aggressive capitalisation and the return on equity it produces

Intangible assets

Research and development

Non-recurring items

Other aggressive practices

International considerations

EQ 5 – Residual Income Valuation

Overview

Residual income and the charge for equity capital

Two ways to reach the same number

From residual income to share price, EVA and market value added

What residual income models are used for

Economic value added

Market value added

The residual income model and its link to the dividend discount model

Deriving the model from the dividend discount model

Where the value is recognised

Valuing a company with the general residual income model

The clean surplus assumption inside the schedule

Fundamental determinants and the justified price-to-book ratio

Tobin’s q

Single-stage valuation and the implied growth rate

Why one stage is rarely enough

Multistage models and continuing residual income

A distinctive feature of residual income terminal values

Fading to an industry mean

The persistence factor

Terminal value from a market multiple

Comparison with dividend discount and free cash flow models

Strengths of residual income models

Weaknesses of residual income models

Choosing when to use the model

Clean surplus violations and other comprehensive income

What bypasses the income statement

How the forecasting assumption interacts with other comprehensive income

Balance sheet, intangible and international accounting considerations

Aggressive capitalisation and the return on equity it produces

Intangible assets

Research and development

Non-recurring items

Other aggressive practices

International considerations