Residual income valuation

You built the excess-return model for a bank because a bank leaves you no choice, but nothing about the model is specific to banking. Strip away the deposit franchise and the capital constraint and the same algebra values any company at all, under the name residual income. It's worth building properly, because it's the one model that starts from a number already audited and printed, instead of from a forecast you invented.

What is the residual income formula?

Residual income is what's left of earnings after the shareholders' own capital has been charged for. It has two equivalent definitions:

RI_t = Net income_t − r × Book equity_{t−1}
     = (ROE_t − r) × Book equity_{t−1}          r = cost of EQUITY

You get the second line from the first by writing net income as ROE_t × Book equity_{t−1}, and it's the form to think in: the spread, times the shareholders' money at work. Accounting profit asks whether the company made money. Residual income asks whether it made more than the money cost. That's the only question that decides whether book value should trade above or below one times.

That leads to an identity worth proving for yourself. The claim:

V₀ = Book equity₀ + Σ RI_t / (1+r)^t

The proof is three lines and worth writing out once, because it shows you which accounting assumption the model rests on. Start from the dividend discount model, V₀ = Σ D_t/(1+r)^t. Now impose clean surplus: book equity changes only through earnings and distributions, so B_t = B_{t−1} + NI_t − D_t, and therefore D_t = NI_t − B_t + B_{t−1}. Substitute that into the dividend series and collect terms. Every B_t appears twice, once discounted at t and once at t+1, and the pair collapses to B_{t−1} × r/(1+r)^t plus a residual B_0. What's left is B₀ + Σ (NI_t − r B_{t−1})/(1+r)^t, which is the identity. The dividend series and the residual-income series are the same series rearranged, so they can't disagree. If your two spreadsheets disagree, one of them has a book value that doesn't roll forward.

Worked example: Vindhya Fasteners

Take a synthetic Indian non-financial to see all three routes tie. Vindhya Fasteners: book equity ₹240 crore, ROE a stable 18%, cost of equity 12%, growth 6% forever. Retention is g/ROE = 33.33%, so payout is 66.67%.

• Residual income route. RI₁ = (18% − 12%) × 240 = ₹14.40 crore, growing at 6%. V = 240 + 14.40/(0.12 − 0.06) = 240 + 240 = **₹480 crore**.
• Dividend route. E₁ = 0.18 × 240 = ₹43.20 crore, then D₁ = 43.20 × 0.6667 = ₹28.80 crore, so V = 28.80/0.06 = **₹480 crore**.
• Justified P/B route. (ROE − g)/(r − g) = (18 − 6)/(12 − 6) = 2.00×, and 2.00 × 240 = **₹480 crore**.

Read the split before moving on. Half the value is book equity you can see on a balance sheet, and half is a claim that a spread will persist. So the model tells you how much of the price is a forecast, and that's its real gift. A DCF hides the same information inside a terminal value. Residual income puts it in the first line.

Constant ROE forever is a fiction for a non-financial, because competition arrives. It's the same fade you met in the terminal-value work, wearing entity clothing. So forecast residual income explicitly while ROE fades, then take a continuing value. Vindhya again, with ROE fading linearly from 18% to 13% over five years, retention held at one third, r = 12%:

What continuing value should you use?

The explicit residual income is worth ₹34.90 crore. Year 6's residual income, if ROE holds at 13%, is (13% − 12%) × 308.72 = ₹3.09 crore. What happens after that is the continuing assumption, and there are four standard ones. The general form uses a persistence factor ω between 0 and 1, the fraction of this year's residual income that survives into next year:

Continuing value at T = RI_{T+1} / (1 + r − ω)

• ω = 1, residual income persists at its current level for ever. CV₅ = 3.09/0.12 = ₹25.73 crore, PV ₹14.60, so V₀ = 240 + 34.90 + 14.60 = **₹289.50 crore**, a P/B of 1.21×. This is the assumption that says the moat never erodes further.
• ω = 0, residual income vanishes after the forecast. CV₅ = 3.09/1.12 = ₹2.76 crore, PV ₹1.56, V₀ = **₹276.46 crore**, P/B 1.15×. This is the competitive assumption, that ROE reverts to the cost of equity the moment you stop looking.
• 0 < ω < 1, residual income decays toward zero. At ω = 0.60, CV₅ = 3.09/0.52 = ₹5.94 crore, PV ₹3.37, V₀ = **₹278.27 crore**, P/B 1.16×. Take this as the base case for a normal industrial company, because the empirical fade in ROE is real and reasonably fast.
• Residual income settles at a permanent positive level, so a P/B premium survives indefinitely. Reserve this for a company whose advantage you can name, and argue for the named advantage instead of the number.

Notice how little the four assumptions move the answer here: ₹276 to ₹290, a 5% spread. That's no coincidence. It's the model's structural advantage over a DCF. Most of the value is already on the balance sheet, so the part that depends on a guess is small. The mirror image is the model's structural weakness, and the accounting below deals with it. If the balance sheet is wrong, everything is wrong, and no continuing assumption saves you.

What is dirty surplus, and how does it break the model?

The proof above needed one accounting condition and one only: B_t = B_{t−1} + NI_t − D_t. Modern accounting violates it routinely, because a family of gains and losses runs through other comprehensive income and goes straight into equity without ever crossing the income statement. Foreign-currency translation differences, remeasurement gains and losses on defined-benefit plans, changes in the fair value of certain investments and the effective portion of cash-flow hedges all behave this way under both Ind AS and IFRS. US GAAP has a similar list.

Run the break yourself. Take Vindhya's year 3, and suppose the annual report shows a translation loss of ₹18 crore in other comprehensive income. The book value rolls forward ₹18 crore short of what net income minus dividends predicts. Now build the residual-income series off reported net income and use the reported closing book value. You've just counted ₹18 crore of destroyed value nowhere at all. It left equity, and your earnings series never saw it go. On the base case above, the PV of the omission is 18/1.12³ = ₹12.81 crore, or 4.6% of the ₹278.27 crore valuation. That's a lot of value to lose in one line of a statement most readers skip.

The repair is mechanical. Use comprehensive income in place of net income in the residual-income series. Or, equivalently, force the book roll-forward to tie and treat the plug as an earnings item. The diagnostic takes thirty seconds on any annual report: opening equity, plus profit for the year, minus dividends, should equal closing equity. When it doesn't, the statement of changes in equity names the line responsible. That line is either a real economic loss you must charge, or a transaction with owners, such as a rights issue, a buyback or a share-based-payment credit. A transaction with owners isn't an earnings item at all, and it belongs outside the series.

Which adjustments matter most with residual income?

Because the model starts from book value, the adjustments a DCF can ignore carry real weight here. Four of them do most of the work.

Dirty-surplus items, as above. Charge them, or reconcile the roll-forward and say why not.

Goodwill. An acquisitive company carries the price it paid for other people's businesses on its own balance sheet. Its book equity is inflated relative to an organic peer, and its ROE is correspondingly depressed. If you're consistent, the model misvalues neither company, because a bigger B₀ is offset by smaller residual income and the identity still holds. What you mustn't do is compare the two companies' ROEs and call one a better business. To judge the capital allocator, compute both on a goodwill-inclusive basis, since the acquirer really did spend that money. To judge the operating business, compute both on a goodwill-excluded basis. Say which you did.

Non-recurring items. Residual income is a series. Leave a one-off restructuring charge from year 2 in the series and it gets capitalized into the valuation by whatever your continuing assumption does with it. Strip real one-offs out of the earnings path but leave them in the book value, because the cash really did leave. The asymmetry looks odd, and it's correct.

Intangibles. This is the big one, and it cuts against the model. A software or consumer-brand company expenses the investment that built its franchise, so its book equity is a fraction of the economic capital in the business, its ROE looks impossible, and residual income carries almost all of the valuation. The model hasn't broken, but its advantage has evaporated. You're back to forecasting a spread for decades, with no balance-sheet anchor to speak of. Capitalizing research and development or brand-building spend restores some of the anchor and is worth doing, with the mechanics from your returns-on-capital work. Even then, a company whose book equity is 10% of its market value is a company where residual income is a DCF wearing a disguise. Use the model where the balance sheet means something, which is most of the industrial, financial and capital-intensive world. Where it doesn't, say so.

All of this is the equity-side twin of economic profit, which your value-drivers work proved equivalent to a DCF. The two reconcile to the rupee on a consistent leverage assumption. The bank version is the same model with a regulatory capital constraint bolted onto the growth rate.