📖 DDM：多阶段增长模型

Equity Investments

I. Lesson Focus

This lesson explains how to construct and apply two-stage, three-stage, and H-model variants of the dividend discount model (DDM) to estimate the intrinsic value of equity when dividend growth is not constant. Candidates must master the formulas, assumptions, calculation mechanics, and limitations of each model, including correct treatment of the terminal value and the critical $g_s < r$ constraint.

II. The Problem

Most companies do not grow dividends at a single constant rate forever. A typical firm experiences a high-growth phase driven by competitive advantage, followed by a transition to a stable-growth phase limited by the economy’s long-term growth rate. Applying the single-stage Gordon growth model to such firms produces large valuation errors. Multistage DDM solves this by explicitly forecasting dividends during the high-growth (and transition) period and then capitalizing a terminal value at the onset of stable growth.

III. Core Logic of Multistage DDM

The single-stage Gordon model is $V_0 = \frac{D_1}{r-g}$. Multistage models split value into (1) the present value of expected dividends during the explicitly forecast period and (2) the present value of the terminal value at the end of that period.

Terminal value at time $n$ is usually calculated with the Gordon model:
$$TV_n = \frac{D_{n+1}}{r - g_s}$$
where $g_s$ is the stable perpetual growth rate. The stock’s intrinsic value is:
$$V_0 = \sum_{t=1}^{n} \frac{D_t}{(1+r)^t} + \frac{TV_n}{(1+r)^n}$$

IV. Two-Stage DDM

The two-stage model assumes $n$ years of high growth at constant rate $g_h$, followed by immediate transition to perpetual growth at $g_s < r$.

$$V_0 = \sum_{t=1}^{n} \frac{D_0(1+g_h)^t}{(1+r)^t} + \frac{\frac{D_0(1+g_h)^n(1+g_s)}{r-g_s}}{(1+r)^n}$$

Key assumptions:
- $g_h$ may exceed $r$ provided it lasts only a finite number of years.
- $g_s$ must be sustainable (typically 3%–5%, close to long-run GDP growth).
- The cost of equity $r$ is usually estimated via CAPM: $r = r_f + \beta(r_m - r_f)$.

V. Three-Stage DDM

The three-stage model adds a transitional period during which growth declines linearly from $g_h$ to $g_s$. It is conceptually closest to the real corporate life cycle.

Growth rates in the transition phase can be interpolated:
$$g_t = g_h - \frac{(g_h - g_s)}{n_2} \times t$$
Each dividend is forecasted and discounted individually. Terminal value is calculated at the end of the transition period using $g_s$. This model is preferred for firms whose competitive advantage is gradually eroding.

VI. The H-Model

The H-model is a simplified two-stage variant that assumes growth declines linearly from $g_h$ to $g_s$ over $2N$ years while the payout ratio remains constant. The closed-form formula is:
$$V_0 = \frac{D_0(1+g_s) + D_0 \times N \times (g_h - g_s)}{r - g_s}$$
where $N$ equals half the length of the transition period.

Advantages: computationally efficient, no need to forecast every interim dividend.
Limitations: assumes constant payout during transition; less accurate when payout ratio changes materially.

VII. Model Selection and Practical Considerations

• High-ROE, low-payout growth companies → two-stage or three-stage DDM.
• Mature firms with gradually declining growth → H-model for speed.
• $g_s$ must be less than $r$; exceeding long-run economic growth is unrealistic.
• For non-dividend-paying firms, switch to free-cash-flow-to-equity (FCFE) or residual-income models once dividends are expected to begin.

Worked Cases

Case 1: Basic Two-Stage DDM

ABC Corp pays a current dividend $D_0 = 2.00$. Dividends are expected to grow at 18% for the next 5 years, then at a stable 4% forever. The required return $r = 11\%$. Calculate intrinsic value.

Step-by-step solution
1. High-growth dividends:
$D_1 = 2.00 \times 1.18 = 2.36$
$D_2 = 2.7848$, $D_3 = 3.2861$, $D_4 = 3.8776$, $D_5 = 4.5756$.

1. Terminal value at $t=5$:
   $D_6 = 4.5756 \times 1.04 = 4.7586$
   $TV_5 = \frac{4.7586}{0.11-0.04} = 67.98$.

2. Present values:
   PV of dividends (years 1–5) = 13.85
   PV of $TV_5 = 67.98 / 1.11^5 = 40.32$

Intrinsic value $V_0 = 13.85 + 40.32 = 54.17$.

Case 2: Three-Stage DDM with Transition

XYZ Corp’s $D_0 = 1.50$. Growth is 20% for 3 years, then declines linearly to 5% over the following 3 years, after which it remains at 5%. $r = 12\%$.

Growth path in transition: 17%, 14%, 11%. After explicit forecast, $D_7 = D_6 \times 1.05$.
PV of explicit dividends ≈ 9.87.
$TV_6 = \frac{D_7}{0.12-0.05} = 68.14$, PV of TV ≈ 34.51.
Total value $V_0 = 9.87 + 34.51 = 44.38$.

Case 3: H-Model Application

MNO Corp’s $D_0 = 3.00$, $g_h = 15\%$, $g_s = 5\%$, transition length = 8 years so $N = 4$, $r = 11\%$.

$$V_0 = \frac{3.00(1+0.05) + 3.00 \times 4 \times (0.15-0.05)}{0.11-0.05} = \frac{3.15+12}{0.06} = 253.33$$

A full two-stage spreadsheet yields approximately 251.8, confirming the H-model’s reasonable approximation when payout is relatively stable.

Traps

Key Formulas

• Two-stage DDM: $V_0 = \sum_{t=1}^{n}\frac{D_t}{(1+r)^t} + \frac{TV_n}{(1+r)^n}$
• Terminal value: $TV_n = \frac{D_{n+1}}{r-g_s}$
• H-model: $V_0 = \frac{D_0(1+g_s)+D_0 \times N \times (g_h-g_s)}{r-g_s}$
• Sustainable growth: $g = ROE \times b$
• Constraint: $g_s < r$ and $g_s$ ≈ long-run nominal GDP growth

Practice Questions

Q1. Which of the following is NOT a valid general assumption of multistage DDM?
A. High-growth rate may temporarily exceed the required return
B. Stable growth rate must be less than the required return
C. Payout ratio must remain constant throughout the transition phase
D. Terminal value must be discounted back to present

Q2. In a two-stage DDM, $D_0=1.8$, $g_h=25\%$ for 4 years, $g_s=5\%$, $r=12\%$. The terminal value at the end of year 4 is closest to:
A. 38.5
B. 52.4
C. 61.7
D. 75.2

Q3. In the H-model, the parameter $N$ represents:
A. Total years of high growth
B. One-half the length of the transition period
C. Years of stable growth
D. Total forecast horizon

Q4. When a firm currently pays no dividend but is expected to begin paying dividends in the future, the most appropriate approach is to:
A. Apply the Gordon model immediately
B. Use multistage DDM starting from the first expected dividend payment
C. Conclude intrinsic value is zero
D. Rely exclusively on P/E multiples

Q5. The primary advantage of a three-stage model over a two-stage model is that it:
A. Requires fewer calculations
B. Better captures the gradual decline in growth from high to stable levels
C. Eliminates the need to estimate the required return
D. Never requires a terminal value

Q6. An H-model valuation produces $V_0=85$ while a full two-stage model produces 82. The most likely reason is:
A. The H-model overstates the contribution of growth during transition
B. Incorrect cost of equity
C. Changing payout ratio during transition
D. Stable growth rate set too high

Q7. Which growth rate is most appropriate for the stable stage $g_s$?
A. Firm’s 10-year historical average of 12%
B. Analyst’s forecast next-year growth of 18%
C. Country’s long-run real GDP growth plus inflation = 4.5%
D. Industry average ROE × retention ratio = 9%

Q8. If growth immediately equals the required return after the high-growth phase:
A. Terminal value becomes infinite and the model fails
B. Value equals only the PV of high-growth dividends
C. The H-model can still be used without adjustment
D. The analyst should immediately switch to a residual-income model

Answers

Takeaways

• Multistage DDM value = PV(explicit dividends) + PV(terminal value).
• Two-stage is appropriate for distinct high-growth then perpetual stable phases.
• Three-stage adds a linear transition and is conceptually most realistic.
• H-model is a convenient approximation; remember $N$ is half the transition years.
• $g_s$ must be sustainably below $r$ and aligned with long-run economic growth.
• Terminal value must always be discounted to today; forgetting this is the most common exam trap.