📖 DDM：零增长模型

Equity Investments

I. Lesson Focus

This lesson introduces the zero-growth dividend discount model (DDM), the simplest form of equity valuation using expected dividends. Candidates must master the derivation of the perpetuity formula, its relationship to the Gordon growth model, appropriate application conditions, limitations, and sensitivity to input assumptions. The focus is on calculating intrinsic value, solving for implied required returns, and recognizing when the model is or is not suitable.

II. The Problem

Consider a mature utility company that pays a constant annual dividend of $1.80 per share with no expected growth in perpetuity. Given a current market price of $32, is the stock overvalued or undervalued? The zero-growth DDM provides the direct solution for valuing stocks with permanently fixed cash distributions. It serves as the foundational building block for all subsequent multi-stage DDM variations tested at CFA Level I.

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

The DDM states that the intrinsic value of a stock equals the present value of all expected future dividends received by the investor. The general expression 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 in period $t$ - $r$ = required rate of return on equity

When dividends are expected to remain constant forever (zero growth), the infinite series simplifies to the closed-form perpetuity formula, which is the zero-growth model.

IV. Derivation and Formula of the Zero-Growth Model

If the expected dividend is a constant amount $D$ each year and $r > 0$, the model becomes:

$$V_0 = \frac{D}{r}$$

This is mathematically identical to the present value of a perpetuity.
Key Assumptions: - Dividends remain fixed in perpetuity ($g = 0$) - The firm is a going concern with infinite life - The required return $r$ is positive and constant - Dividend policy remains unchanged indefinitely

The model is most applicable to highly mature firms with negligible growth prospects and extremely stable dividend policies, such as certain utilities, regulated industries, and some REITs.

V. Relationship to the Gordon Growth Model

The zero-growth DDM is a special case of the Gordon (constant-growth) model. When the perpetual growth rate $g = 0$, the Gordon formula collapses to the zero-growth version:

$$V_0 = \frac{D_1}{r - g} \quad \text{reduces to} \quad V_0 = \frac{D_1}{r} \text{ when } g = 0$$

Mastering the zero-growth case is therefore essential for understanding single-stage, two-stage, and multi-stage DDM applications.

VI. Advantages and Limitations

Advantages: - Extremely simple to compute - Clearly illustrates the discounted cash flow principle - Appropriate for stable, mature businesses

Limitations: - Truly permanent zero dividend growth is rare in practice - Extremely sensitive to errors in estimating $r$ - Inappropriate for high-growth or cyclical companies - Ignores reinvestment opportunities and future growth potential

Worked Cases

Case 1: Basic Valuation

A company is expected to pay a fixed annual dividend of $2.40 per share forever. Investors require an 8% return. The current market price is $28. Calculate the intrinsic value and determine whether the stock is attractive.

Solution: $$V_0 = \frac{2.40}{0.08} = 30$$

Since the intrinsic value of $30 exceeds the market price of $28, the stock is undervalued and should be considered for purchase.

Case 2: Solving for the Implied Required Return

A stock currently trades at $45 and has paid a stable $3.15 dividend annually for the past five years, with the same level expected to continue indefinitely. Calculate the market’s implied required rate of return.

Solution: $$r = \frac{D}{V_0} = \frac{3.15}{45} = 0.07 = 7\%$$

The market is currently requiring a 7% return on this security.

Case 3: Scenario and Sensitivity Analysis

A bank’s preferred stock pays a fixed annual dividend of $1.60. An analyst’s estimate of the required return ranges from 6% to 9%. Compute intrinsic values across this range. If the current market price is $22, at what required-return levels would the stock appear overvalued?

Solution: - At $r = 6\%$: $V_0 = 1.60 / 0.06 \approx 26.67$ - At $r = 8\%$: $V_0 = 1.60 / 0.08 = 20.00$ - At $r = 9\%$: $V_0 = 1.60 / 0.09 \approx 17.78$

At a market price of $22: - If the analyst uses 6%, the stock is undervalued (26.67 > 22) - If the analyst uses 8%, the stock is overvalued (20 < 22)

This demonstrates that valuation conclusions are highly sensitive to the choice of $r$.

Traps

Key Formulas

• Zero-growth DDM: $V_0 = D / r$
• Special case of Gordon model: when $g = 0$, $V_0 = D_1 / (r - g)$ simplifies to $D / r$
• Implied required return: $r = D / V_0$
• Present value of perpetuity: $PV = C / r$
• $r$ must be positive; otherwise the model is undefined

Practice Questions

Q1. The zero-growth model is most appropriate for valuing which type of company?
A. High-tech start-ups
B. Rapidly growing consumer-goods firms
C. Utility companies with stable fixed dividends
D. Cyclical manufacturing companies

Q2. A stock pays a constant annual dividend of $1.50 with a required return of 10%. Its intrinsic value is closest to:
A. $10
B. $15
C. $1.50
D. Cannot be calculated

Q3. A stock trades at $36 and pays a stable annual dividend of $2.88. The market’s implied required return is closest to:
A. 6%
B. 8%
C. 10%
D. 12%

Q4. Which statement about the zero-growth DDM is least accurate?
A. It is the Gordon growth model when $g = 0$
B. The formula is $V_0 = D_1 / r$
C. It can be applied to all dividend-paying companies
D. Valuation is highly sensitive to the required-return estimate

Q5. If the required return rises from 8% to 10% while the fixed dividend remains $2, the intrinsic value will:
A. Increase by 25%
B. Decrease by 20%
C. Remain unchanged
D. Increase by 10%

Q6. A REIT distributes a fixed $4.50 annual dividend. An analyst estimates a 7.5% required return. At a market price of $55 the REIT is:
A. Undervalued
B. Overvalued
C. Fairly valued
D. Cannot be determined

Q7. Which of the following is not a required assumption of the zero-growth model?
A. The company operates in perpetuity
B. Dividends exhibit zero growth
C. The company is in a high-growth phase
D. The required return is stable

Q8. A firm pays a $2 dividend expected to remain constant forever. Using a 12% required return, the per-share value is closest to:
A. $16.67
B. $24.00
C. $14.29
D. $12.00

Answers

Takeaways

• The zero-growth DDM formula is $V_0 = D / r$, mathematically a perpetuity
• It is the Gordon growth model evaluated at $g = 0$
• Valuation is extremely sensitive to the accuracy of the required-return estimate
• Use the model only for mature companies with permanently stable dividends
• Although computationally simple, its real-world applicability is narrow; it forms the foundation for multi-stage models
• Always use the next expected dividend $D_1$ and ensure $r > 0$