📖 DDM：固定增长模型（Gordon Growth）

Equity Investments

I. Lesson Focus

This lesson focuses on the Gordon Growth Model (also known as the constant-growth Dividend Discount Model), the simplest and most widely used form of DDM. Candidates must master its derivation, application, assumptions, limitations, and sensitivity to inputs. The model is frequently tested in equity valuation vignettes, particularly for mature firms with stable dividend policies.

II. The Problem

You are valuing a mature utility company that has increased its dividends at a steady 4% annual rate for the past five years and is expected to continue doing so indefinitely. Given the most recent dividend D0 = $2.50, and your required rate of return r = 9%, how do you convert this information into a reasonable estimate of the stock’s intrinsic value? The Gordon Growth Model solves exactly this problem for companies expected to grow dividends at a constant rate forever. It is the foundational constant-growth DDM and appears regularly on the CFA Level I exam.

III. Core Logic of the Dividend Discount Model (DDM)

The intrinsic value of a stock equals the present value of all expected future dividends, analogous to bond valuation. The general DDM formula is:

$$V_0 = \sum_{t=1}^{\infty} \frac{D_t}{(1+r)^t}$$

where: - $V_0$ = current intrinsic value per share - $D_t$ = expected dividend per share in period t - $r$ = required rate of return on equity (cost of equity)

When dividends are assumed to grow at a constant perpetual rate g, the infinite series simplifies directly into the Gordon Growth Model.

IV. Gordon Growth Model Formula and Assumptions

The Gordon Growth Model rests on three critical assumptions: 1. Dividends grow at a constant rate g forever. 2. g is less than r (otherwise the present value becomes infinite and the model is invalid). 3. The firm has reached a stable-growth stage.

The closed-form formula is:

$$V_0 = \frac{D_1}{r - g} = \frac{D_0(1+g)}{r - g}$$

where $D_1 = D_0(1+g)$ is next year’s expected dividend.

This expression is the present-value formula for a growing perpetuity. The model is appropriate only for mature companies with stable and predictable dividend policies, such as utilities or consumer staples. Analysts commonly estimate g using long-run GDP growth, inflation plus real growth, or the sustainable growth rate $g = b \times ROE$, where b is the retention ratio (1 – dividend payout ratio). The required return r is typically estimated via the CAPM: $r = r_f + \beta(r_m - r_f)$.

V. Economic Interpretation and Limitations

The model implies that in stable growth, a firm’s ROE should be close to its cost of equity; otherwise the assumed g may be unrealistic. Because the denominator is (r – g), the model is extremely sensitive to small changes in either input. It cannot be applied to high-growth firms, firms that do not pay dividends, or firms whose growth is expected to change over time. In those cases, analysts must use multistage DDM variants.

VI. Practical Valuation Steps

1. Identify the most recent dividend (D0) or the next expected dividend (D1).
2. Estimate a realistic perpetual growth rate g.
3. Estimate the cost of equity r using CAPM or another method.
4. Verify that g < r.
5. Compute $V_0 = \frac{D_1}{r-g}$.
6. Compare $V_0$ with the current market price to determine over- or undervaluation.

Worked Cases

Case 1: Basic Valuation

ABC Corp. paid a most recent dividend D0 = $3.20. Dividends are expected to grow at 4% perpetually. The required return is 10%. Calculate the intrinsic value.

Solution:
$D_1 = 3.20 \times 1.04 = 3.328$
$V_0 = \frac{3.328}{0.10 - 0.04} = \frac{3.328}{0.06} = 55.47$

If the market price is below $55.47, the stock appears undervalued.

Case 2: Implied Growth Rate (Common Exam Task)

A stock trades at $68. The most recent dividend was $2.50, and the required return is 9%. Assuming the Gordon model holds, what perpetual growth rate is implied by the current market price?

Solution:
$68 = \frac{2.50(1+g)}{0.09 - g}$
$68(0.09 - g) = 2.50 + 2.50g$
$6.12 - 68g = 2.50 + 2.50g$
$3.62 = 70.5g$
$g = 3.62 / 70.5 \approx 0.0513$ or 5.13%

Case 3: Scenario and Sensitivity Analysis

XYZ Corp. paid D0 = $1.80. An analyst prepares three scenarios:
- Optimistic: g = 5%, r = 10%
- Base: g = 4%, r = 9.5%
- Pessimistic: g = 3%, r = 11%

Calculate intrinsic values and comment on sensitivity.

Solution:
Optimistic: $V_0 = \frac{1.80 \times 1.05}{0.10 - 0.05} = \frac{1.89}{0.05} = 37.80$
Base: $V_0 = \frac{1.80 \times 1.04}{0.095 - 0.04} = \frac{1.872}{0.055} \approx 34.04$
Pessimistic: $V_0 = \frac{1.80 \times 1.03}{0.11 - 0.03} = \frac{1.854}{0.08} = 23.18$

A 1% increase in g or a 0.5% decrease in r produces large changes in value, illustrating the model’s high sensitivity to input assumptions.

Traps

Key Formulas

• $V_0 = \frac{D_1}{r - g} = \frac{D_0(1+g)}{r - g}$
• $D_t = D_0(1+g)^t$
• Sustainable growth: $g = b \times ROE$ (b = retention ratio)
• Cost of equity: $r = r_f + \beta(r_m - r_f)$
• Expected return decomposition: $r = \frac{D_1}{P_0} + g$ (dividend yield + capital gains yield)

Practice Questions

Q1. The core assumption of the Gordon Growth Model is that:
A. Dividend growth rates decline over time
B. Dividends grow at a constant rate g forever and g < r
C. The company pays no dividends
D. The model applies only to high-growth firms

Q2. A stock’s most recent dividend D0 is $4, g = 5%, and r = 11%. Its intrinsic value is closest to:
A. $36.36
B. $42.00
C. $70.00
D. $80.00

Q3. If g increases from 4% to 5% while r remains 10%, the stock’s value will:
A. Decrease
B. Remain unchanged
C. Increase by approximately 25%
D. Increase by approximately 33%

Q4. Which type of company is most suitable for direct application of the Gordon Growth Model?
A. High-growth technology start-up
B. Mature regulated electric utility
C. Highly cyclical automobile manufacturer
D. Biotech firm that has never paid dividends

Q5. A stock trades at $68, next year’s dividend D1 = $3.40, and r = 9%. The market-implied perpetual growth rate g is closest to:
A. 3.0%
B. 4.0%
C. 5.0%
D. 6.0%

Q6. In the Gordon model, holding other factors constant, an increase in the retention ratio b will:
A. Decrease g and decrease value
B. Increase g; value may rise or fall depending on whether ROE > r
C. Always increase value
D. Render the model inapplicable

Q7. A firm has ROE = 12% and a dividend payout ratio of 40%. Its sustainable growth rate g is:
A. 4.8%
B. 7.2%
C. 12.0%
D. 30.0%

Q8. Which situation is most likely to cause severe overvaluation when using the Gordon model?
A. Underestimating r
B. Overestimating g when g approaches r
C. Using D0 instead of D1
D. Both B and C

Answers

Takeaways

• Core valuation formula: $V_0 = \frac{D_1}{r-g}$; always use next period’s dividend.
• Strict requirement: g must be less than r and the firm must be in stable growth.
• Model is highly sensitive to changes in r and g; small input errors cause large valuation swings.
• Sustainable growth rate $g = b \times ROE$ is a key tool for estimating a realistic g.
• Apply only to mature, dividend-paying firms with stable policies; multistage models are needed otherwise.
• Exam frequently tests implied growth rates, implied required returns, and sensitivity analysis.