Single-Stage Residual Income Model: Example and Formula

The single-stage residual income model is the simplest version of residual income valuation. It assumes a firm will sustain a constant level of residual income (or grow it at a constant rate) indefinitely, then discounts that to a present value and adds it to current book value. A worked example shows how each input flows into the formula, making the logic transparent and revealing which assumptions drive the valuation most.

The single-stage formula

The perpetual-growth residual income model is:

$$V_0 = BV_0 + \frac{RI_1}{r_e - g}$$

where:

• V₀ = intrinsic value per share today
• BV₀ = book value of equity per share
• RI₁ = expected residual income per share in the next year
• r_e = cost of equity (the required return)
• g = perpetual growth rate of residual income

Residual income itself is:

$$RI = (ROE - r_e) \times BV$$

The model says: intrinsic value equals the book value shareholders already have, plus the present value of all future excess returns.

If a firm earns exactly its cost of equity (ROE = r_e), then RI = 0, and intrinsic value equals book value. If it earns more (ROE > r_e), value is created. If it earns less, value is destroyed.

Worked example: A stable utility

Consider a regulated electric utility with stable cash flows and predictable growth.

Given data:

• Current book value of equity: $1,000 million
• Number of shares outstanding: 100 million
• Book value per share: $10
• Expected net income next year: $120 million
• Cost of equity (estimated via CAPM): 8%
• Perpetual growth rate (assumed): 3% (aligned with GDP growth)

Step 1: Calculate expected ROE

Expected ROE = Expected NI ÷ BV₀ = $120M ÷ $1,000M = 12%

Step 2: Calculate residual income in year 1

RI₁ (total) = (ROE − r_e) × BV₀ = (12% − 8%) × $1,000M = $40 million

RI₁ per share = $40M ÷ 100M shares = $0.40 per share

Step 3: Calculate present value of perpetual residual income

PV(RI) = RI₁ ÷ (r_e − g) = $40M ÷ (8% − 3%) = $40M ÷ 5% = $800 million

Or on a per-share basis: PV(RI) per share = $0.40 ÷ (8% − 3%) = $0.40 ÷ 5% = $8.00 per share

Step 4: Add to book value to get intrinsic value

V₀ = BV₀ + PV(RI) = $1,000M + $800M = $1,800 million total

Or per share: V₀ = $10 + $8 = $18 per share

Interpretation: The utility’s intrinsic value is $18 per share, compared to a current book value of $10. The $8 premium reflects the present value of the utility’s ability to earn a 4 percentage-point spread (ROE − cost of equity) indefinitely. If the stock trades at $18, it is fairly valued. If it trades at $15, it is undervalued. If it trades at $20, it is overvalued.

Sensitivity to key assumptions

The valuation is highly sensitive to three inputs: cost of equity, growth rate, and the ROE spread.

Sensitivity to cost of equity

Cost of equity is often the most uncertain input. Let’s rerun the example with different assumptions:

A 1 percentage-point rise in cost of equity shrinks the intrinsic value by 13–20%. This makes sense: if shareholders’ required return rises, future excess returns are discounted more heavily. In a rising-rate environment, residual income models deflate sharply.

Sensitivity to growth rate

Growth rate feeds the denominator. A higher growth rate raises intrinsic value because residual income is expected to persist longer and compound.

A 2 percentage-point rise in perpetual growth (from 3% to 5%) lifts intrinsic value by $5.33 per share, a 30% increase. This illustrates the model’s sensitivity to long-term growth assumptions. Small errors in growth can swing the valuation significantly.

Sensitivity to the ROE spread

The width of the ROE-minus-cost-of-equity spread drives the magnitude of residual income.

A 1 percentage-point widening of the spread (from 3% to 4%) adds $2 per share of value. This is why competitive advantage and pricing power matter so much in residual income valuation: they determine the sustainable spread.

Sanity-checking with price-to-book

In the worked example, intrinsic value of $18 per share on book value of $10 yields a price-to-book ratio of 1.8×. Is this reasonable?

For a stable utility with 4 percentage-points of excess return, a P/B of 1.8× is plausible. Growth stocks with 8–10 percentage-point spreads might justify P/B ratios of 3×–5×. Value stocks with 1–2 percentage-point spreads might trade at 1.2×–1.4×. The model output should align with intuition and peer multiples.

If the RI model yields a P/B of 0.5× for a stable business with a positive spread, something is wrong with the assumptions (too-high cost of equity, too-low growth rate, or an incorrectly stated book value).

When the perpetual-growth assumption breaks down

The single-stage model assumes the ROE and growth rate are both constant forever. This is rarely true in practice:

• High-growth companies (biotech, early-stage tech) likely see higher ROE and growth early, then a decline as the firm matures and competition intensifies. A single-stage model overstates their value.
• Cyclical firms (airlines, autos) have ROE and growth that vary dramatically with the economic cycle. Assuming a perpetual average spreads the risks but obscures the timing.
• Regulatory transitions can shift ROE (new regulations, rate caps, subsidies changes). A single stage can’t capture a step change.

For these firms, a two-stage or three-stage model is more appropriate: forecast explicit years of higher growth and ROE, then assume a mature perpetual level. But those models are more complex and require more assumption-setting.

Relationship to net present value of growth

A useful reframing: the residual income model can be thought of as net present value of growth (NPVG) plus the current book value.

$$V_0 = BV_0 + NPVG$$

where NPVG is the present value of all growth in earnings above the cost of capital. This perspective clarifies why book value is the anchor: it represents the value if the firm earns exactly its cost of capital (zero excess growth). Value is created by growth that exceeds the cost of capital.

Building a sensitivity table for presentations

When presenting a residual income valuation, a two-way sensitivity table is standard. For the utility example:

Each cell is the intrinsic value per share. This table shows the valuation across a plausible range of cost-of-equity and growth assumptions. A reader can see both the base-case estimate and how value shifts if assumptions change. The table should highlight the base case (8% cost of equity, 3% growth, yielding $18 per share) so the point estimate is clear amid the sensitivity.

Common mistakes in single-stage application

1. Confusing book value growth with residual income growth. If book equity grows because the firm retains earnings, the model should reflect that explicitly in later years, not assume perpetual flat RI. A true perpetual-growth model requires an assumption about how much of earnings is retained and reinvested.

2. Using current earnings instead of sustainable earnings. If the firm is in a cyclical peak year, next-year earnings may be lower. Use a normalized or forward estimate, not a historical average that may embody outdated conditions.

3. Forgetting that g should not exceed the long-term growth of the broader economy. A firm cannot grow 8% forever in a 3% economy—it would eventually become larger than the world. g should be anchored to realistic long-term drivers (GDP growth, demographic growth, market-share assumptions).

4. Ignoring the cost of equity calculation. If the cost-of-equity estimate is off by 1–2 percentage points (plausible, given estimation error in beta or risk premiums), the valuation can swing 20–30%. Sensitivity analysis is not optional.

See also