Valuation using Dividend Discount Models
The constant growth model, multi-stage growth models
1 Introduction
When you purchase a share of stock, what exactly are you buying? At its core, you are acquiring a claim on a portion of a company’s future cash flows—specifically, the dividends that company will distribute to its shareholders over time. This fundamental insight forms the basis of dividend discount models (DDM), one of the most important frameworks in equity valuation.
The central idea is elegantly simple: the intrinsic value of a stock equals the present value of all the cash flows you expect to receive from owning it. For equity investors, those cash flows come in the form of dividends. While this concept is straightforward, applying it in practice requires us to grapple with several challenging questions. How do we forecast future dividends? What discount rate should we use? How do we handle the fact that stocks can pay dividends forever?
In this lecture, we will develop progressively sophisticated approaches to answering these questions. We begin with the general dividend discount model, then explore special cases that make the mathematics more tractable—including the constant-growth model and multi-stage growth models. Along the way, we will connect these valuation techniques to concepts you have already learned, particularly the Capital Asset Pricing Model (CAPM), which provides a principled way to estimate the discount rate. By the end of this lecture, you will have a versatile toolkit for estimating the intrinsic value of dividend-paying stocks.
2 The General Dividend Discount Model
The intrinsic value of a firm’s equity at time \(t\), which we denote \(V_t\), equals the present value of all expected future dividends:
\[V_t = \frac{D_{t+1}}{1+r} + \frac{D_{t+2}}{(1+r)^2} + \frac{D_{t+3}}{(1+r)^3} + \cdots\]
Here, \(D_{t+1}\), \(D_{t+2}\), and so on represent the dividends expected in each future period, and \(r\) is the discount rate. This formula tells us something profound: a stock’s value today depends entirely on what cash flows it will generate in the future, appropriately discounted for the time value of money and risk.
We can also express this model with a finite horizon. If we plan to hold the stock for \(H\) periods and then sell it, the valuation formula becomes:
\[V_t = \frac{D_{t+1}}{1+r} + \frac{D_{t+2}}{(1+r)^2} + \cdots + \frac{D_{t+H} + P_{t+H}}{(1+r)^H}\]
The term \(P_{t+H}\) represents the price at which we expect to sell the stock after \(H\) periods. This is often called the terminal value of the stock. Notice that if we think carefully about what determines \(P_{t+H}\), we realize it must equal the present value (at time \(t+H\)) of all dividends from period \(t+H+1\) onward. This means the two formulations are mathematically equivalent—the terminal value simply packages up all the dividends beyond our explicit forecast horizon.
2.1 The Discount Rate
The discount rate \(r\) in the dividend discount model—sometimes called the market capitalization rate—represents the expected return that investors require to hold the stock. From the company’s perspective, this same rate represents its cost of equity, the return it must offer shareholders to compensate them for bearing risk.
How do we estimate this required return? The most common approach uses the Capital Asset Pricing Model (CAPM):
\[r = R_f + \beta \cdot (E[R_m] - R_f)\]
In this expression, \(R_f\) is the risk-free rate, \(\beta\) measures the stock’s systematic risk relative to the market, and \(E[R_m] - R_f\) is the market risk premium. While CAPM is not the only option—alternatives include the Fama-French Three-Factor Model or the Arbitrage Pricing Theory—it remains the most widely used approach in practice.
The discount rate matters enormously. A small change in \(r\) can produce large swings in the estimated intrinsic value, particularly for stocks with long duration (those whose value depends heavily on cash flows far in the future). This sensitivity underscores why careful estimation of the required return is essential for valuation.
3 The Constant-Growth Model
The general dividend discount model requires us to forecast dividends indefinitely into the future—a daunting task. The constant-growth model (also known as the Gordon Growth Model) offers a simplification that makes the problem tractable by assuming dividends grow at a constant rate \(g\) forever.
Under this assumption, if the next dividend is \(D_{t+1}\), then \(D_{t+2} = D_{t+1}(1+g)\), \(D_{t+3} = D_{t+1}(1+g)^2\), and so on. The infinite sum in the general model collapses to a remarkably simple formula:
\[V_t = \frac{D_{t+1}}{r - g}\]
We can also write this in terms of the dividend just paid:
\[V_t = \frac{D_t \cdot (1 + g)}{r - g}\]
This formula is the present value of a growing perpetuity—a stream of payments that grows at rate \(g\) forever. For the formula to make sense, we need \(r > g\); otherwise, the present value would be infinite (or negative). This mathematical requirement has economic intuition: if dividends grew faster than the discount rate forever, the stock would be infinitely valuable, which cannot be true in equilibrium.
The constant-growth model is powerful because of its simplicity. With just three inputs—the next dividend, the growth rate, and the discount rate—we can estimate intrinsic value. However, this simplicity comes at a cost. Very few firms actually grow at a constant rate forever. Most companies go through phases: rapid growth when young, slower growth as they mature, and sometimes decline. The constant-growth model works best for mature, stable companies with predictable dividend policies, such as utilities or consumer staples firms. For high-growth companies, we need a more flexible approach.
Example 1: Constant-Growth Valuation
Assume Microsoft (MSFT) just paid a dividend of $2 per share, and these dividends will grow at 6% per year in perpetuity. You have estimated that MSFT has a market beta of 1.1, and the market risk premium is 7% per year. The yield on a 1-year T-bill is 0.1% (0.001). What is the estimated intrinsic value of one share of MSFT?
Step 1: Calculate the required return using CAPM.
\[r = R_f + \beta \cdot (E[R_m] - R_f) = 0.001 + 1.1 \times 0.07 = 0.001 + 0.077 = 0.078 = 7.8\%\]
Step 2: Calculate the next expected dividend.
\[D_{t+1} = D_t \times (1 + g) = \$2.00 \times 1.06 = \$2.12\]
Step 3: Apply the constant-growth formula.
\[V_t = \frac{D_{t+1}}{r - g} = \frac{\$2.12}{0.078 - 0.06} = \frac{\$2.12}{0.018} = \$117.78\]
The estimated intrinsic value of one share of MSFT is $117.78.
4 Multi-Stage Growth Models
For most companies, assuming constant dividend growth forever is unrealistic. Young, fast-growing firms typically experience high growth rates initially, which then moderate as the company matures and the industry becomes more competitive. Multi-stage growth models accommodate this reality by allowing different growth rates in different periods.
In a multi-stage growth model, we explicitly forecast dividends for some finite number of periods \(H\) (the high-growth phase), after which we assume dividends grow at a constant rate \(g\) forever (the stable-growth phase). The valuation formula becomes:
\[V_t = \frac{D_{t+1}}{1+r} + \frac{D_{t+2}}{(1+r)^2} + \cdots + \frac{D_{t+H}}{(1+r)^H} + \frac{D_{t+H} \cdot (1 + g)/(r - g)}{(1+r)^H}\]
The first \(H\) terms represent the present value of dividends during the high-growth phase, which we forecast individually. The final term is the present value of the terminal value—itself calculated using the constant-growth formula applied to dividends from period \(t+H+1\) onward, then discounted back to time \(t\).
There are several common approaches for estimating the dividends during the high-growth phase. You might extrapolate from the firm’s own recent dividend growth, apply the average growth rate observed for comparable companies, or use the average growth rate for the industry as a whole. The choice depends on what information you find most reliable and relevant for the company you are valuing.
4.1 Estimating the Terminal Growth Rate
The terminal growth rate \(g\) has an outsized influence on the valuation because it determines the value of dividends extending infinitely into the future. Small changes in this assumption can dramatically alter your estimate of intrinsic value, so careful thought is warranted.
One common approach is to benchmark the terminal growth rate against the growth rate of the overall economy. The logic is that no company can grow faster than the economy forever—if it did, it would eventually become larger than the entire economy, which is impossible. In the United States, nominal GDP has grown at roughly 6% per year over the past 70 years. Many analysts use this figure, or something slightly lower, as an upper bound for the terminal growth rate. If you believe a company will merely keep pace with the economy in the long run, 6% is a reasonable choice. If you think it will grow somewhat slower than the economy, you might use 3-4%.
Example 2: Two-Stage Growth Valuation
Assume Microsoft (MSFT) just paid a dividend of $2 per share. Dividends will grow at 30% per year for the next 5 years, then at 6% per year in perpetuity (matching the historical average growth rate of U.S. GDP). MSFT has a market beta of 1.1, the market risk premium is 7% per year, and the 1-year T-bill yield is 4%. What is the estimated intrinsic value of one share of MSFT?
Step 1: Calculate the required return using CAPM.
\[r = R_f + \beta \cdot (E[R_m] - R_f) = 0.04 + 1.1 \times 0.07 = 0.04 + 0.077 = 0.117 = 11.7\%\]
Step 2: Forecast dividends for the high-growth phase (years 1-5).
| Year | Calculation | Dividend |
|---|---|---|
| 1 | \(\$2.00 \times 1.30^1\) | $2.60 |
| 2 | \(\$2.00 \times 1.30^2\) | $3.38 |
| 3 | \(\$2.00 \times 1.30^3\) | $4.39 |
| 4 | \(\$2.00 \times 1.30^4\) | $5.71 |
| 5 | \(\$2.00 \times 1.30^5\) | $7.43 |
Step 3: Calculate the terminal value at the end of year 5.
The terminal value is the present value (at year 5) of all dividends from year 6 onward:
\[TV_5 = \frac{D_6}{r - g} = \frac{D_5 \times (1+g)}{r - g} = \frac{\$7.43 \times 1.06}{0.117 - 0.06} = \frac{\$7.88}{0.057} = \$138.18\]
Step 4: Discount all cash flows to the present.
| Year | Cash Flow | Discount Factor | Present Value |
|---|---|---|---|
| 1 | $2.60 | \(1/(1.117)^1 = 0.8953\) | $2.33 |
| 2 | $3.38 | \(1/(1.117)^2 = 0.8015\) | $2.71 |
| 3 | $4.39 | \(1/(1.117)^3 = 0.7176\) | $3.15 |
| 4 | $5.71 | \(1/(1.117)^4 = 0.6424\) | $3.67 |
| 5 | $7.43 + $138.18 = $145.61 | \(1/(1.117)^5 = 0.5751\) | $83.75 |
Step 5: Sum the present values.
\[V_t = \$2.33 + \$2.71 + \$3.15 + \$3.67 + \$83.75 = \$95.61\]
The estimated intrinsic value of one share of MSFT is $95.61.
4.2 The Fundamental Growth Rate
Another approach to estimating long-run dividend growth is to derive it from the company’s fundamental characteristics. The fundamental growth rate (also called the sustainable growth rate) links dividend growth to the firm’s profitability and reinvestment policy:
\[g = ROE \times b\]
Here, \(ROE\) is the return on equity—net income divided by book equity—and \(b\) is the retention ratio (also called the plowback ratio), which measures the fraction of earnings the company retains rather than paying out as dividends:
\[b = 1 - \text{Dividend Payout Ratio} = 1 - \frac{\text{Dividends}}{\text{Net Income}}\]
The intuition behind this formula is straightforward. Each year, the company retains a fraction \(b\) of its earnings. If it can reinvest those retained earnings at a rate of return equal to \(ROE\), then book equity—and by extension, dividends—will grow at rate \(g = ROE \times b\).
This approach has the advantage of grounding the growth rate in observable financial data. However, it assumes that the company’s ROE and retention ratio remain stable over time, which may not hold, especially during transitions between growth phases. As with other inputs, you can use the firm’s own historical data, or average values for comparable companies or the industry as a whole.
Example 3: Fundamental Growth Rate Valuation
Use the same inputs as Example 2, except estimate the perpetual growth rate using Microsoft’s fundamental growth rate instead of the 6% GDP growth assumption. In the most recent fiscal year, MSFT had book equity of $600 billion, net income of $100 billion, and paid $40 billion in dividends. What is the estimated intrinsic value of one share of MSFT?
Step 1: Calculate the fundamental growth rate.
First, calculate ROE: \[ROE = \frac{\text{Net Income}}{\text{Book Equity}} = \frac{\$100\text{B}}{\$600\text{B}} = 0.1667 = 16.67\%\]
Next, calculate the retention ratio: \[b = 1 - \frac{\text{Dividends}}{\text{Net Income}} = 1 - \frac{\$40\text{B}}{\$100\text{B}} = 1 - 0.40 = 0.60 = 60\%\]
Finally, calculate the fundamental growth rate: \[g = ROE \times b = 0.1667 \times 0.60 = 0.10 = 10\%\]
Step 2: Recall the required return from Example 2.
\[r = 11.7\%\]
Step 3: Use the same dividend forecasts for years 1-5 (30% growth).
| Year | Dividend |
|---|---|
| 1 | $2.60 |
| 2 | $3.38 |
| 3 | $4.39 |
| 4 | $5.71 |
| 5 | $7.43 |
Step 4: Calculate the terminal value using the fundamental growth rate.
\[TV_5 = \frac{D_5 \times (1+g)}{r - g} = \frac{\$7.43 \times 1.10}{0.117 - 0.10} = \frac{\$8.17}{0.017} = \$480.76\]
Step 5: Discount all cash flows to the present.
| Year | Cash Flow | Discount Factor | Present Value |
|---|---|---|---|
| 1 | $2.60 | 0.8953 | $2.33 |
| 2 | $3.38 | 0.8015 | $2.71 |
| 3 | $4.39 | 0.7176 | $3.15 |
| 4 | $5.71 | 0.6424 | $3.67 |
| 5 | $7.43 + $480.76 = $488.19 | 0.5751 | $280.77 |
Step 6: Sum the present values.
\[V_t = \$2.33 + \$2.71 + \$3.15 + \$3.67 + \$280.77 = \$292.63\]
The estimated intrinsic value of one share of MSFT is $292.63.
Note: The dramatic difference between this estimate ($292.63) and Example 2’s estimate ($95.61) illustrates how sensitive valuations are to the terminal growth rate assumption. A 10% terminal growth rate versus 6% more than triples the estimated value.
5 Using Multiples to Estimate Terminal Value
So far, we have estimated terminal value using the constant-growth formula, which requires us to commit to a perpetual growth rate. An alternative approach uses valuation multiples to estimate what the stock will be worth at the end of our explicit forecast horizon. This hybrid method combines the dividend discount approach for near-term cash flows with relative valuation for the terminal value.
In the general valuation formula:
\[V_t = \frac{D_{t+1}}{1+r} + \frac{D_{t+2}}{(1+r)^2} + \cdots + \frac{D_{t+H} + P_{t+H}}{(1+r)^H}\]
we can estimate the terminal price \(P_{t+H}\) by applying a valuation multiple to some fundamental metric projected \(H\) periods into the future. For example, if we use a price-to-book (P/B) multiple:
\[P_{t+H} = \text{PB Multiple} \times \text{Book Equity per Share}_{t+H}\]
Similarly, we could use a price-to-earnings (P/E) multiple applied to projected earnings, or an enterprise value to EBITDA (TEV/EBITDA) multiple for the entire firm. The choice of multiple depends on what you believe is most relevant and reliably estimated for the company in question.
This approach has several advantages. First, it sidesteps the need to specify a perpetual growth rate, which is inherently uncertain. Second, by using market-derived multiples from comparable companies or the target’s own historical trading range, we incorporate information about how the market currently values similar firms. However, the approach also has limitations: it embeds current market sentiment into the valuation, which may be irrational, and it requires projecting both the relevant fundamental metric and the appropriate multiple into the future.
Example 4: Terminal Value Using a P/B Multiple
Use the same inputs as Example 2, except estimate the terminal value using a price-to-book multiple instead of the constant-growth formula. In the most recent fiscal year, MSFT had book equity of $600 billion and 7.6 billion shares outstanding. You estimate that its P/B ratio in 5 years will be 10. What is the estimated intrinsic value of one share of MSFT?
Step 1: Recall the required return from Example 2.
\[r = 11.7\%\]
Step 2: Use the same dividend forecasts for years 1-5 (30% growth).
| Year | Dividend |
|---|---|
| 1 | $2.60 |
| 2 | $3.38 |
| 3 | $4.39 |
| 4 | $5.71 |
| 5 | $7.43 |
Step 3: Calculate current book equity per share.
\[\text{Book Equity per Share}_t = \frac{\$600\text{B}}{7.6\text{B shares}} = \$78.95\]
Step 4: Project book equity per share to year 5.
We need to estimate how book equity will grow over the next 5 years. Since book equity grows through retained earnings, we can use the retention ratio and ROE from Example 3. Recall that \(ROE = 16.67\%\) and \(b = 60\%\), giving a book equity growth rate of \(g_{book} = ROE \times b = 10\%\).
\[\text{Book Equity per Share}_5 = \$78.95 \times (1.10)^5 = \$78.95 \times 1.6105 = \$127.19\]
Step 5: Calculate the terminal price using the P/B multiple.
\[P_5 = \text{PB Multiple} \times \text{Book Equity per Share}_5 = 10 \times \$127.19 = \$1,271.90\]
Step 6: Discount all cash flows to the present.
| Year | Cash Flow | Discount Factor | Present Value |
|---|---|---|---|
| 1 | $2.60 | 0.8953 | $2.33 |
| 2 | $3.38 | 0.8015 | $2.71 |
| 3 | $4.39 | 0.7176 | $3.15 |
| 4 | $5.71 | 0.6424 | $3.67 |
| 5 | $7.43 + $1,271.90 = $1,279.33 | 0.5751 | $735.79 |
Step 7: Sum the present values.
\[V_t = \$2.33 + \$2.71 + \$3.15 + \$3.67 + \$735.79 = \$747.65\]
The estimated intrinsic value of one share of MSFT is $747.65.
Note: This estimate is much higher than Examples 2 and 3 because we assumed a P/B multiple of 10, which implies the market will continue to value MSFT at a substantial premium to book value. The choice of terminal multiple is as important as the choice of terminal growth rate in the constant-growth approach.
6 Key Takeaways
The dividend discount model rests on a fundamental insight that every finance student should internalize: the value of any financial asset equals the present value of the cash flows it generates. For equity, those cash flows are dividends—the distributions that shareholders receive from the company’s profits. This principle connects equity valuation to broader concepts in finance, including the time value of money, risk and return, and the cost of capital.
The constant-growth model transforms this general principle into a practical tool by assuming dividends grow at a constant rate forever. The resulting formula—intrinsic value equals the next dividend divided by the difference between the required return and the growth rate—is elegant but limited. It works well for mature, stable firms but fails to capture the dynamics of companies experiencing changing growth rates.
Multi-stage growth models provide the flexibility needed for most real-world valuations. By explicitly forecasting dividends during a high-growth phase and then applying the constant-growth formula for the terminal value, we can accommodate firms transitioning from rapid growth to maturity. The key insight here is that terminal value often dominates the calculation, which means our assumptions about long-run growth and discount rates matter enormously.
Whether we estimate terminal value using the constant-growth formula or valuation multiples, we face fundamental uncertainty about the future. The GDP-based approach to terminal growth provides a reasonable anchor, while the fundamental growth rate links growth to observable financial metrics like ROE and the retention ratio. Using multiples incorporates market information but embeds current sentiment into our estimate. No approach is perfect, and sensitivity analysis—examining how valuation changes as we vary key assumptions—is essential.
Finally, remember that the discount rate connecting all these calculations comes from asset pricing models like CAPM. The required return reflects both the time value of money (captured by the risk-free rate) and compensation for bearing systematic risk (captured by beta times the market risk premium). Understanding how these components fit together gives you a complete framework for equity valuation: forecast cash flows, estimate risk, discount appropriately, and arrive at intrinsic value.
7 Key Formulas Summary
| Concept | Formula | When to Use |
|---|---|---|
| General DDM | \(V_t = \sum_{s=1}^{\infty} \frac{D_{t+s}}{(1+r)^s}\) | Conceptual foundation; rarely applied directly |
| Finite Horizon DDM | \(V_t = \sum_{s=1}^{H} \frac{D_{t+s}}{(1+r)^s} + \frac{P_{t+H}}{(1+r)^H}\) | When you have explicit dividend forecasts and a terminal value estimate |
| Constant-Growth Model | \(V_t = \frac{D_{t+1}}{r - g} = \frac{D_t(1+g)}{r-g}\) | Mature, stable companies with predictable dividend growth |
| Multi-Stage Model | \(V_t = \sum_{s=1}^{H} \frac{D_{t+s}}{(1+r)^s} + \frac{D_{t+H}(1+g)/(r-g)}{(1+r)^H}\) | Companies transitioning from high growth to stable growth |
| CAPM (Discount Rate) | \(r = R_f + \beta(E[R_m] - R_f)\) | Estimating the required return for equity |
| Fundamental Growth Rate | \(g = ROE \times b\) | Deriving growth from profitability and reinvestment |
| Retention Ratio | \(b = 1 - \frac{\text{Dividends}}{\text{Net Income}}\) | Calculating the plowback ratio for growth estimation |
| Terminal Value (Multiples) | \(P_{t+H} = \text{Multiple} \times \text{Fundamental}_{t+H}\) | When market-based valuation is preferred for terminal value |
8 Practice Problems
Practice Problem 1: Constant-Growth Valuation
Johnson & Johnson (JNJ) recently paid an annual dividend of $4.76 per share. Analysts expect dividends to grow at 5% per year indefinitely. JNJ has a beta of 0.65, the current risk-free rate is 3.5%, and the market risk premium is 6%. What is the intrinsic value of one share of JNJ using the constant-growth model?
Step 1: Calculate the required return using CAPM.
\[r = R_f + \beta \cdot (E[R_m] - R_f) = 0.035 + 0.65 \times 0.06 = 0.035 + 0.039 = 0.074 = 7.4\%\]
Step 2: Calculate the next expected dividend.
\[D_{t+1} = D_t \times (1 + g) = \$4.76 \times 1.05 = \$5.00\]
Step 3: Apply the constant-growth formula.
\[V_t = \frac{D_{t+1}}{r - g} = \frac{\$5.00}{0.074 - 0.05} = \frac{\$5.00}{0.024} = \$208.33\]
The estimated intrinsic value of one share of JNJ is $208.33.
Practice Problem 2: Two-Stage Growth Valuation
Tesla (TSLA) just initiated a dividend of $0.50 per share. Analysts expect dividends to grow at 40% per year for the next 4 years as the company matures, then at 5% per year in perpetuity. TSLA has a beta of 2.0, the risk-free rate is 4%, and the market risk premium is 7%. What is the intrinsic value of one share of TSLA?
Step 1: Calculate the required return using CAPM.
\[r = R_f + \beta \cdot (E[R_m] - R_f) = 0.04 + 2.0 \times 0.07 = 0.04 + 0.14 = 0.18 = 18\%\]
Step 2: Forecast dividends for the high-growth phase (years 1-4).
| Year | Calculation | Dividend |
|---|---|---|
| 1 | \(\$0.50 \times 1.40^1\) | $0.70 |
| 2 | \(\$0.50 \times 1.40^2\) | $0.98 |
| 3 | \(\$0.50 \times 1.40^3\) | $1.37 |
| 4 | \(\$0.50 \times 1.40^4\) | $1.92 |
Step 3: Calculate the terminal value at the end of year 4.
\[TV_4 = \frac{D_5}{r - g} = \frac{D_4 \times (1+g)}{r - g} = \frac{\$1.92 \times 1.05}{0.18 - 0.05} = \frac{\$2.02}{0.13} = \$15.51\]
Step 4: Discount all cash flows to the present.
| Year | Cash Flow | Discount Factor | Present Value |
|---|---|---|---|
| 1 | $0.70 | \(1/(1.18)^1 = 0.8475\) | $0.59 |
| 2 | $0.98 | \(1/(1.18)^2 = 0.7182\) | $0.70 |
| 3 | $1.37 | \(1/(1.18)^3 = 0.6086\) | $0.83 |
| 4 | $1.92 + $15.51 = $17.43 | \(1/(1.18)^4 = 0.5158\) | $8.99 |
Step 5: Sum the present values.
\[V_t = \$0.59 + \$0.70 + \$0.83 + \$8.99 = \$11.11\]
The estimated intrinsic value of one share of TSLA is $11.11.
Practice Problem 3: Fundamental Growth Rate Valuation
Procter & Gamble (PG) just paid a dividend of $3.76 per share. Dividends are expected to grow at 8% for the next 3 years, then at the fundamental growth rate in perpetuity. PG has book equity of $50 billion, net income of $15 billion, and paid $9 billion in dividends. The company has a beta of 0.45, the risk-free rate is 3%, and the market risk premium is 6%. With 2.4 billion shares outstanding, what is the intrinsic value of one share of PG?
Step 1: Calculate the required return using CAPM.
\[r = R_f + \beta \cdot (E[R_m] - R_f) = 0.03 + 0.45 \times 0.06 = 0.03 + 0.027 = 0.057 = 5.7\%\]
Step 2: Calculate the fundamental growth rate.
\[ROE = \frac{\$15\text{B}}{\$50\text{B}} = 0.30 = 30\%\]
\[b = 1 - \frac{\$9\text{B}}{\$15\text{B}} = 1 - 0.60 = 0.40 = 40\%\]
\[g = ROE \times b = 0.30 \times 0.40 = 0.12 = 12\%\]
Wait—the fundamental growth rate (12%) exceeds the required return (5.7%), which would make the constant-growth formula invalid. This indicates the company’s current ROE and retention policy are unsustainable at this discount rate. Let’s assume the terminal growth rate moderates to 4% (below r) as the company matures.
Let \(g_{terminal} = 4\%\).
Step 3: Forecast dividends for the high-growth phase (years 1-3).
| Year | Calculation | Dividend |
|---|---|---|
| 1 | \(\$3.76 \times 1.08^1\) | $4.06 |
| 2 | \(\$3.76 \times 1.08^2\) | $4.39 |
| 3 | \(\$3.76 \times 1.08^3\) | $4.74 |
Step 4: Calculate the terminal value at the end of year 3.
\[TV_3 = \frac{D_3 \times (1 + g_{terminal})}{r - g_{terminal}} = \frac{\$4.74 \times 1.04}{0.057 - 0.04} = \frac{\$4.93}{0.017} = \$290.00\]
Step 5: Discount all cash flows to the present.
| Year | Cash Flow | Discount Factor | Present Value |
|---|---|---|---|
| 1 | $4.06 | \(1/(1.057)^1 = 0.9461\) | $3.84 |
| 2 | $4.39 | \(1/(1.057)^2 = 0.8951\) | $3.93 |
| 3 | $4.74 + $290.00 = $294.74 | \(1/(1.057)^3 = 0.8468\) | $249.59 |
Step 6: Sum the present values.
\[V_t = \$3.84 + \$3.93 + \$249.59 = \$257.36\]
The estimated intrinsic value of one share of PG is $257.36.
Note: This example illustrates an important practical issue—computed fundamental growth rates can exceed reasonable discount rates, requiring judgment to select a sustainable terminal growth rate.
Practice Problem 4: Terminal Value Using Multiples
Apple (AAPL) just paid a dividend of $0.96 per share. Dividends are expected to grow at 15% per year for the next 6 years. Apple has book equity of $60 billion and 15.5 billion shares outstanding. You estimate its P/B ratio will be 40 in 6 years (reflecting its strong brand and ecosystem). Apple’s beta is 1.25, the risk-free rate is 4%, and the market risk premium is 6%. If book equity grows at 8% per year, what is the intrinsic value of one share of AAPL?
Step 1: Calculate the required return using CAPM.
\[r = R_f + \beta \cdot (E[R_m] - R_f) = 0.04 + 1.25 \times 0.06 = 0.04 + 0.075 = 0.115 = 11.5\%\]
Step 2: Forecast dividends for years 1-6 (15% growth).
| Year | Calculation | Dividend |
|---|---|---|
| 1 | \(\$0.96 \times 1.15^1\) | $1.10 |
| 2 | \(\$0.96 \times 1.15^2\) | $1.27 |
| 3 | \(\$0.96 \times 1.15^3\) | $1.46 |
| 4 | \(\$0.96 \times 1.15^4\) | $1.68 |
| 5 | \(\$0.96 \times 1.15^5\) | $1.93 |
| 6 | \(\$0.96 \times 1.15^6\) | $2.22 |
Step 3: Calculate current book equity per share.
\[\text{Book Equity per Share}_t = \frac{\$60\text{B}}{15.5\text{B shares}} = \$3.87\]
Step 4: Project book equity per share to year 6.
\[\text{Book Equity per Share}_6 = \$3.87 \times (1.08)^6 = \$3.87 \times 1.5869 = \$6.14\]
Step 5: Calculate the terminal price using the P/B multiple.
\[P_6 = 40 \times \$6.14 = \$245.60\]
Step 6: Discount all cash flows to the present.
| Year | Cash Flow | Discount Factor | Present Value |
|---|---|---|---|
| 1 | $1.10 | \(1/(1.115)^1 = 0.8969\) | $0.99 |
| 2 | $1.27 | \(1/(1.115)^2 = 0.8044\) | $1.02 |
| 3 | $1.46 | \(1/(1.115)^3 = 0.7214\) | $1.05 |
| 4 | $1.68 | \(1/(1.115)^4 = 0.6470\) | $1.09 |
| 5 | $1.93 | \(1/(1.115)^5 = 0.5803\) | $1.12 |
| 6 | $2.22 + $245.60 = $247.82 | \(1/(1.115)^6 = 0.5204\) | $128.97 |
Step 7: Sum the present values.
\[V_t = \$0.99 + \$1.02 + \$1.05 + \$1.09 + \$1.12 + \$128.97 = \$134.24\]
The estimated intrinsic value of one share of AAPL is $134.24.
9 Ask an LLM
Here are three questions you might ask an AI assistant to deepen your understanding of these concepts:
- How would the valuation of a company change if it uses share buybacks instead of dividends to return cash to shareholders? Can the dividend discount model be adapted for firms that don’t pay dividends?
- What are the key assumptions behind CAPM that might not hold in practice, and how would violations of these assumptions affect our discount rate estimates in dividend discount models?
- Walk me through a sensitivity analysis for a two-stage DDM: how much does the intrinsic value change if I vary the terminal growth rate by ±1% and the discount rate by ±1%? What does this tell me about valuation uncertainty?
- Why might a company’s actual stock price differ significantly from the intrinsic value estimated using a dividend discount model? What market factors or model limitations could explain this gap?
- How do professional analysts decide between using a constant-growth model, a multi-stage model, or a multiples-based terminal value when valuing a specific company?