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CFA Level II Exam · Residual Income Valuation

Sustainable Growth, ROE and Implied Metrics in Residual Income Valuation

Updated 7 October 2026 · Fact-checked

Justified P/B is the price-to-book ratio implied by the residual income model. With constant growth g, it equals (ROE − g) ÷ (r − g). Sustainable growth is g = b × ROE, where b is the earnings retention rate. Rearrange the formula to solve for implied g or implied required return r.

Understand Sustainable Growth, ROE and Implied Metrics

Residual income (RI) is earnings above the charge for equity capital: RI = net income − r × beginning book value. The RI model says value equals current book value plus the present value of expected future RI. So a firm is worth more than book only when it is expected to earn more than its cost of equity.

If RI grows at a constant rate g forever, the sum collapses to a simple form. Value = B0 + RI1 ÷ (r − g). Constant RI growth at g requires book value and earnings to grow at g as well, which is consistent with a stable ROE and g = b × ROE. Since RI1 = (ROE − r) × B0, you can divide by B0 and get the justified P/B = 1 + (ROE − r) ÷ (r − g) = (ROE − g) ÷ (r − g). This is the link between the RI model and the P/B multiple.

Read the formula as a story. If ROE = r, then P/B = 1, whatever the growth. If ROE > r, growth adds value, so P/B > 1 and rises with g. If ROE < r, growth destroys value, so P/B < 1 and falls as g rises. This holds when r > g.

Sustainable growth is the growth a firm can fund from retained earnings without changing its ROE or leverage: g = b × ROE, where b = 1 − dividend payout ratio. A higher ROE or a higher retention rate gives higher g. The exam often asks you to compute g this way, then feed it into the P/B formula.

Because the formula links four variables (P/B, ROE, g, r), you can solve for any one if you know the other three. Solving for g gives the implied growth rate. Solving for r gives the implied required return. These are the same algebra moves you use in the Gordon growth model.

Key formulas to remember

Residual income
RI = NI − r × B(t−1) = (ROE − r) × B(t−1)
B(t−1) is beginning book value. r is the cost of equity.
Single-stage RI value
V0 = B0 + (ROE − r) × B0 ÷ (r − g)
Assumes RI grows at constant g forever and r > g. This requires book value and earnings to grow at g too, consistent with a stable ROE and g = b × ROE.
Justified P/B
P/B = (ROE − g) ÷ (r − g)
Equals 1 when ROE = r. Above 1 when ROE > r.
Sustainable growth rate
g = b × ROE = (1 − payout) × ROE
b is the retention rate. Assumes stable ROE and no new equity issued.
Implied growth rate
g = (P/B × r − ROE) ÷ (P/B − 1)
Rearranged from the justified P/B formula. Requires P/B ≠ 1.
Implied required return
r = g + (ROE − g) ÷ (P/B)
Use the market P/B to back out the return the market is demanding.

How to solve Sustainable Growth, ROE and Implied Metrics questions

Use this method for any question that links P/B, ROE, growth and required return in an RI setting.

  1. 1Identify what is asked: justified P/B, value per share, implied g, implied r, or sustainable g.
  2. 2Pull ROE, r, payout or retention, and book value from the vignette. Check whether ROE uses beginning or average equity and use what the vignette states.
  3. 3If growth is not given, compute g = b × ROE where b = 1 − payout ratio.
  4. 4Check r > g. If not, the constant-growth formula does not apply.
  5. 5Write P/B = (ROE − g) ÷ (r − g) and fill in the known values. Rearrange before substituting if you are solving for g or r.
  6. 6If value per share is wanted, multiply justified P/B by book value per share.
  7. 7Sense-check: ROE > r should give P/B > 1. ROE < r should give P/B < 1.
  8. 8Compare with market P/B if asked for a view: market P/B above justified suggests overvaluation, on these inputs.

Quickest way: Plug into the P/B formula and test the sign

When to use it: Use when the item set gives ROE, r and either g or payout, and asks for P/B, implied g or implied r.

  1. Compute g = (1 − payout) × ROE if needed.
  2. Calculate (ROE − g) and (r − g) as decimals.
  3. Divide to get P/B. Check it sits above 1 only if ROE > r.
  4. For implied g, use g = (P/B × r − ROE) ÷ (P/B − 1).
  5. For implied r, use r = g + (ROE − g) ÷ (P/B).
  6. Eliminate answer options on the wrong side of 1 before doing any long arithmetic.

Common mistakes in Sustainable Growth, ROE and Implied Metrics

  • Using the dividend growth rate or earnings growth blindly instead of b × ROE when the question gives payout and ROE.

    Students see a growth figure in the vignette and grab it without checking it is the sustainable rate asked for.

    Fix: Read the question wording. If it asks for growth consistent with retention and ROE, compute g = b × ROE.

  • Using the payout ratio as b.

    Both are percentages and the symbols look alike.

    Fix: b is retention = 1 − payout. Write it out before multiplying.

  • Writing P/B = (r − g) ÷ (ROE − g), flipping the fraction.

    The formula resembles Gordon growth and students misremember the order.

    Fix: Check by testing: if ROE > r, P/B must exceed 1, so ROE − g goes on top.

  • Concluding P/B is above 1 whenever growth is positive.

    Growth is assumed to always add value.

    Fix: Growth adds value only when ROE > r. If ROE < r, higher growth lowers P/B.

  • Mixing percentages and decimals, or using ROE from the wrong equity base.

    Time pressure and several ratios in the exhibits.

    Fix: Convert to decimals first and use the ROE definition the vignette gives.

  • Solving for implied g without rearranging, or forgetting to divide by (P/B − 1).

    The rearrangement is longer than other growth formulas.

    Fix: Derive once: P/B × (r − g) = ROE − g, so g × (1 − P/B) = ROE − P/B × r, giving g = (P/B × r − ROE) ÷ (P/B − 1).

Worked examples

Example 1

Vignette: Kestrel Holdings has an ROE of 14%, a dividend payout ratio of 40% and a cost of equity of 10%. Book value per share is 50. Questions: (1) What is the sustainable growth rate? (2) What is the justified P/B? (3) What is the justified value per share?

Show the solution
  1. Retention b = 1 − 0.40 = 0.60.
  2. g = 0.60 × 14% = 8.4%.
  3. Check r > g: 10% > 8.4%, so the formula applies.
  4. P/B = (0.14 − 0.084) ÷ (0.10 − 0.084) = 0.056 ÷ 0.016 = 3.5.
  5. Value per share = 3.5 × 50 = 175.

Answer: (1) g = 8.4%. (2) Justified P/B = 3.5. (3) Justified value = 175 per share.

Example 2

Vignette: Orlan Corp has a market P/B of 2.0, an ROE of 12% and a cost of equity of 9%. Questions: (1) What constant growth rate is implied by the market P/B? (2) Separate scenario: ignore the 9% stated cost of equity. If growth is 4% and P/B is 2.0, with ROE still 12%, what required return is implied?

Show the solution
  1. (1) g = (P/B × r − ROE) ÷ (P/B − 1).
  2. P/B × r = 2.0 × 0.09 = 0.18.
  3. Numerator = 0.18 − 0.12 = 0.06. Denominator = 2.0 − 1 = 1.0.
  4. g = 6.0%. Check: (0.12 − 0.06) ÷ (0.09 − 0.06) = 0.06 ÷ 0.03 = 2.0. Correct.
  5. (2) This is a different scenario, so r is not the 9% from part (1). r = g + (ROE − g) ÷ (P/B) = 0.04 + (0.12 − 0.04) ÷ 2.0.
  6. = 0.04 + 0.04 = 8.0%.
  7. Check: (0.12 − 0.04) ÷ (0.08 − 0.04) = 0.08 ÷ 0.04 = 2.0. Correct.

Answer: (1) Implied growth = 6.0%. (2) In the separate scenario with 4% growth, implied required return = 8.0%.

Exam tips

  • Always test your answer against ROE versus r. This catches inverted formulas in seconds.
  • Look for payout or retention data in the vignette. It is a signal that g = b × ROE is expected.
  • For implied g or r questions, rearrange first, then substitute. Verify by plugging your answer back into the P/B formula.
  • Convert implied values into a view: if implied growth looks unrealistically high compared with sustainable g, the stock may be overvalued on those inputs.
  • No penalty for wrong answers, so never leave a question blank.

Sustainable Growth, ROE and Implied Metrics: frequently asked questions

What is the justified P/B in the residual income model?

It is the P/B the model supports given ROE, required return and growth. With constant growth, it equals (ROE − g) ÷ (r − g). Multiply by book value per share to get intrinsic value.

How do I calculate the sustainable growth rate from ROE?

Multiply the retention rate by ROE: g = b × ROE, where b = 1 − payout ratio. It assumes a stable ROE and no new equity issued.

How do I find the implied growth rate from the residual income model?

Set justified P/B equal to the market P/B and solve for g. The result is g = (P/B × r − ROE) ÷ (P/B − 1). Check it by substituting back.

What is the relationship between ROE, cost of equity and P/B?

If ROE equals r, P/B is 1. If ROE exceeds r, P/B is above 1 and rises with growth. If ROE is below r, P/B is below 1 and falls as growth rises.