How do you value a company using residual income when ROE is expected to change over time?

Question

AcadiFi · Accepted Answer

When ROE is expected to change, you build a multi-stage residual income model with explicit forecasts for each period where ROE differs, then apply a terminal value assumption for the steady-state period.

**Multi-Stage RI Formula:**

V_0 = B_0 + Σ [RI_t / (1+r)^t] + TV / (1+r)^T

Where RI_t = (ROE_t - r) × B_{t-1}

**Worked Example -- Silverline Pharmaceuticals (fictional)**

Given: B_0 = $20/share, r = 11%, retention ratio = 60%

| Year | ROE | RI = (ROE - r) × B_{t-1} | Book Value Growth |
|---|---|---|---|
| 1 | 22% | (0.22 - 0.11) × $20.00 = $2.20 | B_1 = $20 + 0.60 × 0.22 × $20 = $22.64 |
| 2 | 18% | (0.18 - 0.11) × $22.64 = $1.58 | B_2 = $22.64 + 0.60 × 0.18 × $22.64 = $25.09 |
| 3 | 14% | (0.14 - 0.11) × $25.09 = $0.75 | B_3 = $25.09 + 0.60 × 0.14 × $25.09 = $27.20 |
| 4+ | 11% | (0.11 - 0.11) × B = $0.00 | RI = 0, no excess returns |

**Valuation:**

PV of RI_1 = $2.20 / 1.11 = $1.98
PV of RI_2 = $1.58 / 1.11^2 = $1.28
PV of RI_3 = $0.75 / 1.11^3 = $0.55
Terminal value = $0 (ROE converges to cost of equity)

**V_0 = $20.00 + $1.98 + $1.28 + $0.55 = $23.81**

\`\`\`mermaid
flowchart TD
    A[Start: High ROE > r] --> B[Year 1-3: Positive Residual Income]
    B --> C[Competition Erodes ROE]
    C --> D[ROE Converges to Cost of Equity]
    D --> E[RI = 0 in Terminal Period]
    E --> F[Value = Book Value + PV of Excess RI Period]
\`\`\`

The declining ROE pattern reflects competitive erosion of economic moats. If you believe ROE will persist above r indefinitely (strong moat), use a persistence factor (omega between 0 and 1) in the terminal value: TV = RI_T × omega / (1 + r - omega). An omega of 0.5 means RI decays by 50% each year beyond the explicit forecast period.

How do you value a company using residual income when ROE is expected to change over time?

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