Single-Stage vs. Multi-Stage Residual Income Models

Understanding Residual Income

Residual income (RI) is often described as what’s left over after covering the shareholder’s “cost of equity.” In more formal terms, residual income for a given period is:

Here, r is the cost of equity, capturing the return investors expect for the risk they’re taking. If a firm generates net income above that required return, it has “residual” gains that boost intrinsic value.

I remember the first time I applied an RI model. Honestly, I was helping a friend evaluate a small chain of coffee shops—quite the local hotspot. You’d think coffee shops would be all about fancy cappuccinos, but the residual income approach gave us an insider’s look at how they produced returns beyond shareholders’ expectations. That’s the magic of RI: it peels away the fluff and focuses on value created or destroyed relative to equity capital.

Single-Stage Residual Income Model

A single-stage residual income model assumes that growth (in earnings, payouts, and book values) continues at a constant rate forever. This approach is straightforward—though it can be a bit of a stretch for rapidly changing or newly public companies.

When we say “single-stage,” we often assume:

• A stable return on equity (ROE).
• A stable retention rate (or a stable payout ratio).
• A constant growth rate (g).
• A constant cost of equity (r).

Under these assumptions, the intrinsic value of equity today can be approximated by:

• \( B_0 \): Current book value per share.
• \( ROE \): Return on equity.
• \( r \): Cost of equity.
• \( g \): Perpetual growth rate of residual income.

Essentially, \(B_0\) is the baseline value (the book value of equity right now). The fraction \(\frac{(ROE - r),B_0}{(r - g)}\) captures the present value of all future residual income, assuming it grows at a constant rate g.

Simple Numeric Example

Suppose:

• \( B_0 = 100 \)
• \( ROE = 12% \)
• \( r = 10% \)
• \( g = 4% \)

Using the single-stage formula:

1. Calculate the equity “spread”: \( ROE - r = 0.12 - 0.10 = 0.02 \) (2%).
2. Multiply by the current book value: \( 0.02 \times 100 = 2 \).
3. Divide by \((r - g)\): \( 0.10 - 0.04 = 0.06 \).
4. Therefore, the present value of future RI = \( \frac{2}{0.06} = 33.33 \).
5. Add \( B_0 \): \( 100 + 33.33 = 133.33 \).

So the total estimated intrinsic value is approximately 133.33 per share. Notice how a modest 2% spread can lead to significant added value when g is near r.

Best Situations for Single-Stage

• The firm is mature, with stable earnings growth and a predictable payout policy.
• The industry has reached relatively steady competition—like many utilities or established consumer staples.
• Analyst resources are limited, and a rough but quick measure is needed.

Multi-Stage Residual Income Model

Life is rarely that simple (or constant). Firms face evolving competitive pressures, shifting capital structures, and changes in product demand. That’s where the multi-stage RI model shines.

In a multi-stage approach, you forecast residual income explicitly for several years—maybe a high-growth phase—then apply a continuing or terminal value for the stable phase afterward. Conceptually, you split the valuation timeline like so:

1. High-Growth Phase: Forecast residual income year by year, when ROE and growth may be relatively high.
2. Transition Phase: Gradually settle into a more modest ROE and growth.
3. Stable/Continuing Phase: Assume a constant (or near-constant) growth in perpetuity, similar to the single-stage model.

Mathematically, the multi-stage formula extends the summation of discounted RI terms and then attaches a terminal value. Typically, you’ll see something like:

where:

TV\(_T\) is the terminal value at the end of year \(T\) based on a single-stage formula or another assumption of stable growth:

Visual Flow

Below is a simple Mermaid diagram illustrating how single-stage and multi-stage concepts branch out. It’s a bit of a minimalist view, but it highlights the differing perspectives:

    flowchart LR
	  A["Single-Stage (Constant Growth)"] --> B["Stable ROE, stable g"]
	  C["Multi-Stage (Evolving Growth)"] --> D["Explicit forecast <br/> plus terminal value"]

Practical Multi-Stage Example

Imagine a tech company in its early growth years. Suppose:

• For the first 3 years, ROE is 18% and grows quickly. After that, it stabilizes to 12%.
• The cost of equity (r) is 10%.
• Book value is 100 initially.
• We project net income, book value, and use these to calculate residual income year by year for three years. Then, from year 4 onward, we assume a stable scenario.

Let’s illustrate a hypothetical (simplified) forecast snippet:

Year 1:
• Book value at the start: 100.
• Projected net income: 18.
• Equity charge: \(0.10 \times 100 = 10\).
• RI = \(18 - 10 = 8\).
• Present Value of RI: \(\frac{8}{1.10}\approx 7.27\).
• End-of-year book value: \(B_1 = 100 + (18 - \text{dividends})\). Suppose a 0% payout to fuel growth, so \(B_1 = 118\).

Year 2:
• Book value at the start: 118.
• Net income: \(\text{ROE} \times 118 = 0.18 \times 118 \approx 21.24\).
• Equity charge: \(0.10 \times 118 = 11.8\).
• RI = \(21.24 - 11.8 \approx 9.44\).
• Present Value of RI: \(\frac{9.44}{1.10^2} \approx 7.80\).
• End-of-year book value: \(B_2 = 118 + (21.24 - 0) = 139.24\).

Continue similarly for Year 3. Then at Year 4, we assume the firm’s ROE transitions to 12%, with a stable growth of, say, 4%. You’d compute the terminal value and discount it all back. It can be a bit tedious by hand, but that’s precisely why we rely on either spreadsheets or quick Python scripts.

Quick Python Example

Below is a tiny illustration of how you might sum up the residual incomes in Python:

 1import math
 2
 3r = 0.10  # cost of equity
 4initial_book = 100
 5ROE_high = 0.18
 6years_high = 3
 7ROE_stable = 0.12
 8g_stable = 0.04
 9
10book_value = initial_book
11pv_ri = 0
12for t in range(1, years_high+1):
13    net_income = ROE_high * book_value
14    equity_charge = r * book_value
15    ri = net_income - equity_charge
16    pv_ri += ri / ((1 + r) ** t)
17    book_value += net_income  # assume zero dividend for simplicity
18
19ri_year4 = (ROE_stable * book_value) - (r * book_value)
20tv = (ri_year4 * (1 + g_stable)) / (r - g_stable)  # continuing value from year 4 onward
21pv_tv = tv / ((1 + r) ** years_high)
22
23value_est = initial_book + pv_ri + pv_tv
24print(f"Value Estimate: {value_est:.2f}")

Depending on your numbers, this yields a certain total value that might exceed what a single-stage model would project for a fast-growing tech firm. The multi-stage approach better captures the spike in early years.

Choosing Between Single-Stage and Multi-Stage

Sometimes you might say, “Uh, sure, but which one do I pick?” The choice often depends on how stable the firm’s operations are:

• If the firm’s in a stable industry with consistent margins (like some big consumer goods or utilities), a single-stage model can be perfectly fine.
• If the firm’s still navigating big expansions or reorganizations—like biotech exploring new patents or a social media startup scaling user base—a multi-stage model is usually more realistic.

Key Assumptions and Consistency

Regardless of the model you choose, be consistent:

• Align your ROE, cost of equity, and growth rate assumptions.
• Don’t assume a 20% ROE forever if your cost of equity is just 8% with a stable economy—this could lead to overvaluing.
• Carefully reflect new capital injections or share buybacks in book value forecasts.
• Revisit inflation assumptions, especially if you’re modeling over many years.

Sensitivity Analysis

RI models can be sensitive to small changes in:

• The cost of equity (r).
• Growth rate (g).
• ROE estimates.

Try tweaking each assumption by, say, ±1% to see how drastically the final valuation changes. If a slight change in r or g leads to wild swings, it’s a good clue that your modeling or assumptions may be too aggressive or uncertain.

Common Pitfalls and Best Practices

• Overly Optimistic Growth: Especially in multi-stage models, it’s tempting to assume high growth for too long.
• Ignoring Off-Balance-Sheet Items: Make sure your book value is correct—underestimating liabilities can inflate your numbers.
• Mishandling Terminal Value: Terminal value can make up a big chunk of total value. Small errors can be huge in final valuations.
• Mistimed Residual Income Calculations: Watch out for whether you’re using beginning-of-period vs. end-of-period book values consistently.

Exam Tips and Wrap-Up

• Practice setting up different growth phases carefully. The exam will likely give you data for multiple periods and test your ability to forecast net income, book value, and RIs step by step.
• Keep your timeline clear—mixing up year-end or year-beginning can lead to confusion.
• Label each step in calculations so you can easily check your work.
• Watch for hidden data in vignettes, like an upcoming expansion or a share repurchase. Those details might impact your book value or cost of equity.
• Time constraints matter: master the quick approach for single-stage on stable companies, but be ready to pivot to multi-stage logic if the vignette indicates variable growth.

Residual income valuation is incredibly powerful. It complements other equity models (like FCFE or Dividend Discount Models) by focusing on how net income surpasses (or falls short of) the equity cost. In a sense, it’s all about whether the firm creates genuine economic profit for its shareholders.

References and Further Reading

• CFA Institute Level II Curriculum (Residual Income Readings)
• Damodaran, A. (2012). “Investment Valuation: Tools and Techniques for Determining the Value of Any Asset.”
• Koller, T., Goedhart, M., & Wessels, D. (2020). “Valuation: Measuring and Managing the Value of Companies.”
• NYU Stern’s website (Real-world data sets and modeling examples): http://pages.stern.nyu.edu/~adamodar/

Test Your Knowledge: Single-Stage vs. Multi-Stage Residual Income Models