Dividend Discount Models (Single-Stage, Multistage)

Introduction and Overview

Dividend Discount Models (DDMs) hold a special place in equity valuation because, at their core, they’re beautifully intuitive: the value of a share of stock is simply the sum of all the future dividends you expect to receive, discounted to today’s dollars (or euros, yen, etc.). Sounds pretty elegant, right? In practice, though, applying DDMs can feel a bit more complicated—particularly if you’re juggling changing growth rates, uncertain discount rates, and real-world market conditions.

But I remember the first time I heard about the Dividend Discount Model, I thought: “Wait—so all I need is the future dividends, my required return, and a growth rate? Easy.” Then I found out about multiple valuation stages, terminal value assumptions, and how the model can go haywire if your growth rate is too close to your required rate of return. That’s when I realized: “Ah, it’s not always that simple.”

Still, whether you’re studying for the CFA Level I exam or looking to ground your equity valuations in time-tested principles, the DDM forms a bedrock. So let’s start with the basics—Single-Stage (often known as the Gordon Growth Model) and then progress to Multistage models. We’ll get into key formulas, essential assumptions, pitfalls, and methods to incorporate real-world constraints.

The Single-Stage (Gordon Growth) Model

Core Concept

The Single-Stage Dividend Discount Model (DDM), commonly referred to as the Gordon Growth Model, assumes that a firm’s dividends grow at a constant, unchanging rate forever. I know that can sound a bit unrealistic, but many mature companies with stable operations—think large utility companies—can come close to this scenario, at least over extended periods.

Mathematically, the Single-Stage DDM states:

where:
• \(V_0\) = Current intrinsic value of the stock.
• \(D_1\) = Expected dividend at the end of the first year.
• \(r\) = Required rate of return (often derived from CAPM or other risk/return models).
• \(g\) = Expected constant growth rate of dividends.

Key Assumptions

• Dividends grow at a constant rate \(g\), indefinitely.
• \(r > g\), since if \(g \geq r\), the model breaks down or implies an infinite valuation.
• The company’s payout policy remains consistent and does not drastically shift over time.

To illustrate how this model flows, here’s a small Mermaid diagram:

    flowchart LR
	A["Forecast Annual Dividend (D1)"] --> B["Estimate Required Return (r)"]
	B["Estimate Required Return (r)"] --> C["Estimate Growth (g)"]
	C["Estimate Growth (g)"] --> D["Compute Intrinsic Value: D1 ÷ (r - g)"]

And that’s it. On the surface, it’s straightforward: one formula, three main variables. But real-life complexities—like unpredictable dividend policies or big changes in capital structure—mean you need to approach the Gordon Growth Model with caution.

Practical Example of the Single-Stage Model

Let’s do a quick numeric example, to make this more concrete.

Suppose a utility firm’s current annual dividend is $2.00 per share. Management and analysts forecast that dividends will increase by 3% every year. Meanwhile, you decide your required rate of return, \(r\), is 7%.

• The dividend one year from now, \(D_1\), will be:

So, if you trust these assumptions (a big “if,” right?), you might conclude the stock is worth $51.50 per share. If it’s trading at $45, maybe you see value; if it’s trading at $60, you might pass.

Potential Pitfalls and Limitations

1. Growth Rate Close to Required Return
   If \(g\) is 6.5% and \(r\) is 7%, you wind up dividing by something that’s just 0.5%. That can create a massive valuation, and you have to ask yourself: “Does it really make sense for the company’s growth to be so high forever?” Possibly not.

2. Dividend Policy Changes
   Some firms aren’t consistent with their dividend payouts. They might decide to invest heavily in growth, cutting dividends, or they might do buybacks. This can effectively disrupt the “steady growth rate” assumption.

3. Irrelevance to Non-Dividend Payers
   Let’s be honest: many fast-growing tech companies either pay no dividend or pay a negligible amount. The single-stage model sort of breaks down there. We might come back to them with something like a multistage approach… or maybe a different valuation metric entirely.

4. Macroeconomic Shifts
   If an economic downturn forces companies to freeze or reduce dividends, that constant growth rate you baked in might suddenly be out the window.

You can see how even a simple formula requires a thoughtful approach. But let’s not throw the baby out with the bathwater. For stable, mature companies with proven track records of dividend growth, the single-stage model can offer a good ballpark estimate—and a quick one. Just remember it’s built on quite a few assumptions that might shift out from under you.

The Multistage Dividend Discount Model

When to Use Multistage Models

Let’s say we have a firm that’s just emerging from a big technology transition. Maybe for the next few years, management says they’ll grow dividends at 12%—which is pretty high. But eventually, the advantage will narrow, competition will come in, and you expect the growth rate to settle at 4%.

The single-stage approach doesn’t handle that well, because it demands a constant \(g\). To handle more realistic scenarios, we turn to the Multistage Dividend Discount Model (often called the “Two-Stage” or “Three-Stage” DDM, though you can theoretically have as many stages as you want).

Structure and Formula

In a two-stage model, we typically break up the firm’s life into two periods:

1. High-growth period (Stage 1): Dividends are forecast to grow at a higher rate, \(g_1\), for a certain number of years (often noted as \(n\) years).
2. Long-term stable growth period (Stage 2): After year \(n\), dividends grow at a more sustainable, constant rate, \(g_2\).

At the end of the high-growth period, we often compute a “terminal value,” which represents the present value of all dividends from Year \(n+1\) onward, assuming the stable growth rate. Then, we discount the entire stream of dividends—including the terminal value—back to the present.

Symbolically, for a two-stage model:

where:

• \(D_t\) are the dividends in each of the next \(n\) years, growing at higher rate \(g_1\).
• \(TV_n\) is the terminal value at the end of year \(n\), often estimated by applying the Gordon Growth Model at that point:

Below is a quick Mermaid diagram illustrating the valuation process conceptually:

    flowchart TB
	A["Forecast Dividends (Years 1 - n) with High Growth"] --> B["Forecast Dividend at Year n+1 with Stable Growth"]
	B["Forecast Dividend at Year n+1 with Stable Growth"] --> C["Compute Terminal Value = D_(n+1) / (r - g_2)"]
	C["Compute Terminal Value"] --> D["Discount Each Dividend and Terminal Value to Present"]

Practical Example of a Two-Stage DDM

Let’s say you have a company that just launched a blockbuster product. You estimate a 10% dividend growth for the next three years. After that, you believe competition will catch up, and the growth will normalize to 4% indefinitely. The stock’s current dividend is $1.00, and your required rate of return is 8%.

1. Forecast Dividends in High-Growth Phase

   • \(D_1 = 1.00 \times (1 + 0.10) = 1.10\)
   • \(D_2 = 1.10 \times (1 + 0.10) = 1.21\)
   • \(D_3 = 1.21 \times (1 + 0.10) = 1.331\)

2. Forecast Dividend in Stable Growth (Year 4 onward)
   Now from Year 4 onward, growth is 4%. So:

   • \(D_4 = 1.331 \times (1 + 0.04) = 1.38524\) (just for context when we do the terminal value).

3. Compute Terminal Value at the End of Year 3
   Using the Gordon Growth formula for the stable period:

   $$ TV_3 = \frac{D_4}{r - g_2} = \frac{1.38524}{0.08 - 0.04} = \frac{1.38524}{0.04} = 34.631. $$

4. Discount All Cash Flows to the Present
   Now discount each of the first three dividends plus the terminal value (which is added to the Year 3 dividend stream) back to \(t=0\) at the 8% required return:

   $$ V_0 = \frac{1.10}{1.08} + \frac{1.21}{(1.08)^2} + \frac{1.331 + 34.631}{(1.08)^3}. $$
   • \(PV(D_1) = \frac{1.10}{1.08} \approx 1.0185.\)
   • \(PV(D_2) = \frac{1.21}{(1.08)^2} \approx \frac{1.21}{1.1664} \approx 1.037.\)
   • \(PV(D_3 + TV_3) = \frac{1.331 + 34.631}{(1.08)^3} \approx \frac{35.962}{1.2597} \approx 28.55.\)

   Summing them up:

   $$ V_0 \approx 1.0185 + 1.037 + 28.55 \approx 30.6055. $$

So you’d estimate the firm is worth about $30.61 per share based on your two-stage dividend assumptions.

Insights on Multistage Models

• You can break the high-growth period into multiple mini-stages if you believe the growth path will taper more gradually (a three-stage model, for instance).
• The terminal value can dwarf the sum of the earlier phases. You’ll notice that 34.631 (the terminal value) contributed almost the entire chunk of the final discounted figure. Always pay attention to your terminal value assumptions.
• The discount rate \(r\) and the stable growth rate \(g_2\) must remain realistic (i.e., \(g_2\) less than \(r\)).

Best Practices and Common Pitfalls

1. Estimating Growth Rates

• High-growth vs. stable growth: applying the wrong growth rate to the stable phase is a classic mistake. If you assume 8% growth forever, but your required return is 9%, be prepared for a massive inflated value.
• Macroeconomic ties: consider inflation, real GDP growth, and industry conditions. If the economy is only growing at 2% real, can your company sustain a 12% dividend growth for decades?

2. Estimating the Discount Rate

• CAPM Example:

3. Potential COVID/Global Conflict Twist

• Once the world discovered how quickly external shocks can slam supply chains and business demand, it became clear that even stable dividend payers might have “blips.” Build in scenario analysis—maybe a moderate scenario, a worse scenario, and a best-case scenario.

4. Double Counting or Inconsistent Forecasts

• If a firm is repurchasing shares aggressively, that may change how quickly dividends might grow on a per-share basis. Make sure your assumptions about share count, payout ratio, and earnings growth are all consistent.

Interrelationships with Other Valuation Concepts

Dividend Discount Models are a major cornerstone, but they’re not the only game in town. You might recall from other parts of Chapter 9 that you can value equity using Free Cash Flow to Equity (FCFE) or by comparing Price Multiples (like P/E or EV/EBITDA). In practice, analysts compare these approaches. If, say, your DDM-based valuation yields $50 but your P/E-based approach and FCFE approach both suggest $40, that discrepancy could be a red flag—or a buying opportunity, depending on your assumptions.

Diagram: Single-Stage vs. Multistage at a Glance

For a little extra clarity, here’s a side-by-side snapshot:

    flowchart LR
	A["Single-Stage DDM"] -- Simple, stable growth -->B["V = D1 / (r - g)"]
	C["Multistage DDM"] -- Changing growth rates -->D["Sum of PV of each dividend + Terminal Value"]

Both hinge on discounting dividends but differ in their assumptions about growth. Single-stage is straightforward but risky if growth is not truly stable. Multistage is more realistic for growth that eventually tapers—though it’s more complex.

Exam Relevance and Tips

• The CFA Level I exam often tests your understanding of the single-stage (Gordon Growth) model. You might be asked to plug in the numbers, interpret the result, or identify incorrect uses of the model (like \(g > r\)).
• You can also face questions on two-stage or even three-stage DDM. They’ll typically give you growth rates in different phases, plus a discount rate. You’ll need to carefully forecast dividends, compute a terminal value, and discount everything back.
• Time Management: You might see item sets or short answer questions requiring swift, methodical calculations. Practice your arithmetic so you don’t get bogged down.
• Clear Step-by-Step: They might present tricky details, like a partial-year dividend. Pay attention: sometimes you see “the next dividend will be paid in exactly one year,” or “the last dividend was just paid,” which can change the formula for \(D_1\).

Real-World Anecdote

A friend of mine once used a single-stage DDM on a high-growth biotech startup—big no-no. The firm had never paid dividends but was planning to start in three years. He assumed a near-constant growth rate based on super-optimistic projections. The result was a borderline ridiculous valuation way above the market price. After the share price later dropped and the firm cut its future dividend outlook, he realized that the single-stage approach was ill-suited. The key lesson: pick the right model for the right scenario.

Conclusion

Dividend Discount Models—both single-stage and multistage—remain bedrock tools for anyone analyzing equities from a dividend perspective. The single-stage (Gordon Growth) approach is elegantly simple but only relevant when dividends grow at a stable rate that’s comfortably below the required return. Multistage models address the realistic idea that growth rates evolve over a company’s life cycle.

Meanwhile, remember that your model is only as good as your inputs—particularly your growth projections and required rate of return. A small tweak in growth can shift valuations dramatically. These models offer a framework, not a crystal ball. For the CFA exam, focus on mastering the formulas, understanding which model is appropriate when, and practicing enough so you can do the calculations (and interpret the results) confidently under time pressure.

References

• CFA Institute Program Curriculum, Equity Valuation Readings.
• Pinto, Jerald E. (CFA). “Equity Asset Valuation.” Wiley.
• Damodaran, Aswath. “Investment Valuation.” NYU Stern: http://pages.stern.nyu.edu/~adamodar/

Test Your Knowledge: Dividend Discount Models Essentials