Sensitivity Analysis in Residual Income Models

The residual income model is mathematically elegant but fragile: a 1% shift in cost of equity or a change in how quickly ROE mean-reverts to cost of capital can swing intrinsic value by 20–40%. Sensitivity analysis exposes these vulnerabilities and reveals which assumptions are load-bearing.

The anatomy of residual income model fragility

The residual income model (RIM) values equity as the sum of three components:

1. Current book value (the balance sheet starting point)
2. Present value of forecast-period excess returns (explicitly projected years, e.g., 5–10 years)
3. Present value of terminal-period excess returns (perpetuity or perpetual growth, beyond the forecast)

The formula:

Intrinsic Value = Book Value₀ + Σ[RI_t / (1 + ke)^t] + [Terminal RI / (ke − g)] / (1 + ke)^n

Where:

• RI_t = Residual Income in year t = (ROE_t − ke) × Book Value_t−1
• ke = Cost of equity (discount rate)
• g = Terminal growth rate
• n = End of forecast period

The problem: the model’s output is hypersensitive to three deeply uncertain inputs: the cost of equity, the fade trajectory of ROE, and the terminal ROE level. Small errors in any of these propagate through both the forecast period and the terminal value, and terminal value often represents 60–80% of total intrinsic value.

Worked example: a base-case build

Company: Stable industrial manufacturer Current Book Value (2025): $50 per share Forecast period: 5 years (2026–2030) Terminal assumption: ROE fades to cost of equity by year 6, held constant thereafter

Base-case parameters:

• Cost of equity: 9%
• Forecast-period ROE: starts at 14%, fades linearly to 9% by year 6
• Terminal growth: 2% (long-run GDP growth)
• Book value growth: 6% annually (earnings retained)

Forecast-period residual income:

Terminal value (at end of 2030, discounted back to 2025):

Starting book value in year 6 = $66.90 × 1.06 = $70.91

At terminal, ROE = ke = 9%, so residual income = 0. But the model assumes growth, so perpetuity is:

Terminal RI (year 6) = 0 (because ROE = ke)

However, this is a nuance: under the Gordon growth model formulation, if growth > 0 and ROE = ke, the terminal value contribution is still zero in steady state. Some practitioners use a different assumption: ROE stays above ke in perpetuity. Let’s recalculate assuming terminal ROE = 11% (a 200 basis-point spread above ke):

Terminal RI (year 6) = (0.11 − 0.09) × $70.91 = $1.42

PV of terminal perpetuity = [$1.42 / (0.09 − 0.02)] / (1.09)^5 = [$1.42 / 0.07] / 1.5386 = $20.29 / 1.5386 = $13.18

Intrinsic Value = $50.00 + $5.78 + $13.18 = $68.96 per share

This is the base case. Now we stress-test the key assumptions.

Sensitivity to cost of equity

Increasing ke from 9% to 10% (a 100 basis-point increase):

Decreasing ke to 8%:

Lesson: A 100 basis-point swing in cost of equity produces a −6% to +9% range around the base case. The effect is asymmetric because lower ke increases both the present value factor and the perpetuity denominator denominator, amplifying the sensitivity. For a stock trading at $70, this narrows the range to $64.80–$75.30 — a material band of uncertainty.

Sensitivity to terminal ROE and fade rate

In the base case, terminal ROE = 11%. Let’s test:

Terminal ROE = 9% (fades completely to cost of equity):

• Terminal RI = 0
• PV of terminal RI = 0
• Intrinsic Value = $50.00 + $5.78 + $0 = $55.78 (−19% from base)

Terminal ROE = 13% (assumes sustained competitive advantage):

• Terminal RI (year 6) = (0.13 − 0.09) × $70.91 = $2.84
• PV of terminal RI = [$2.84 / 0.07] / 1.5386 = $26.36
• Intrinsic Value = $50.00 + $5.78 + $26.36 = $82.14 (+19% from base)

This −19% to +19% swing underscores the critical role of terminal assumptions. Terminal value dominates long-term valuations, and small differences in assumed competitive advantage drive massive swings in intrinsic value. A company trading at $70 could theoretically be worth $55–$82 depending on whether you assume mean reversion to cost of capital or sustained outperformance.

Sensitivity to fade rate (speed of ROE mean-reversion)

In the base case, ROE fades linearly over 5 forecast years. What if fade is faster?

Fast fade (3 years): ROE declines 14% → 9% in 3 years instead of 5.

Forecast-period RI shrinks (fewer years of high RI):

• PV of forecast RI = ~$4.20 (down from $5.78)
• Terminal value is unchanged (same terminal ROE)
• Intrinsic Value ≈ $50 + $4.20 + $13.18 = $67.38 (−2% from base)

Slow fade (10 years): Extend forecast to 10 years; ROE declines more gradually.

More years of above-cost-of-capital returns:

• PV of forecast RI = ~$8.50 (up from $5.78)
• Terminal value slightly lower (later discount factor)
• Intrinsic Value ≈ $50 + $8.50 + $11.80 = $70.30 (+2% from base)

A 7-year swing in fade timing moves intrinsic value roughly ±2%. The impact is smaller than cost-of-equity sensitivity because it affects only the forecast period, not the perpetuity denominator.

Multi-variable sensitivity table

In practice, analysts build a matrix testing multiple assumptions simultaneously. Example 2D table: Cost of equity vs. Terminal ROE

Reading across the base row (ke = 9%):

• Downside (terminal ROE = cost of equity): $55.78 (−19%)
• Base case (terminal ROE = 11%): $68.96
• Upside (terminal ROE = 13%): $82.14 (+19%)

An analyst might define a fair-value range using conservative, base, and optimistic cases:

• Conservative: ke = 9.5%, Terminal ROE = 9% → ~$52
• Base: ke = 9%, Terminal ROE = 11% → ~$69
• Optimistic: ke = 8.5%, Terminal ROE = 12% → ~$77

This bracketing approach acknowledges the model’s intrinsic uncertainty rather than false precision.

Why sensitivity matters

The residual income model is theoretically appealing: it cleanly decomposes value into book value, forecast excess returns, and terminal excess returns. But its output depends critically on unreliable estimates:

• Cost of equity is not directly observable; CAPM estimates vary by 1–2% depending on risk-free rate, beta, and market risk premium assumptions.
• Terminal ROE is a pure forecast, often 5–20+ years out. Market conditions, competition, and technological disruption are unknowable.
• Fade rate is a modeler’s judgment call with no empirical anchor.

Sensitivity analysis is not a defect of the model; it is essential discipline. It forces the analyst to identify which assumptions move the needle and to justify them with conviction. A valuation that is insensitive to reasonable changes in key assumptions is either the result of lucky offsetting errors or a sign that the model has been overfit to the current price.

See also