Applying the Dividend-Discount Model with Changing Growth Rates

The Dividend-Discount Model (DDM) is a cornerstone of valuation in finance, primarily used to estimate the value of a stock based on its expected future dividends. While the traditional DDM assumes a constant growth rate for dividends, real-world scenarios often involve varying growth rates. This article delves into how to adapt the DDM to account for changing growth rates, providing a detailed methodology and practical examples to enhance understanding.

Introduction to the Dividend-Discount Model

The Dividend-Discount Model is based on the premise that the value of a stock is the present value of all its future dividend payments. For a simple, constant-growth DDM, the formula is:

$\text{Stock Value}=\frac{D_0(1+g)}{r-g}$Stock Value=r−gD0​(1+g)​

where $D_0$D0​ is the current dividend, $g$g is the growth rate of dividends, and $r$r is the discount rate.

However, this model becomes more complex when dividends are expected to grow at different rates over time. To address this, a multi-stage DDM can be employed, which incorporates varying growth rates over different periods.

Multi-Stage Dividend-Discount Model

A multi-stage DDM allows for different growth rates over different stages of a company's life cycle. Typically, this model is divided into three stages:

1. Initial High-Growth Stage: This stage represents a period where the company is expected to grow at an above-average rate. This period often corresponds to the company’s early years or periods of significant expansion.

2. Transition Stage: In this stage, the company’s growth rate starts to stabilize but is not yet constant. It represents the phase where growth is decelerating from the initial high rate to a more sustainable level.

3. Stable Growth Stage: The final stage assumes a constant growth rate for dividends, reflecting a mature company with stable earnings and dividends.

Step-by-Step Application of Multi-Stage DDM

1. Estimate Dividends for Each Stage: Forecast dividends for each stage based on expected growth rates. For example:

   • High-Growth Stage: If a company is expected to grow its dividends by 20% annually for the first five years, calculate the dividends for each year during this period.
   • Transition Stage: Next, forecast dividends for the transition period where growth rates might decrease gradually.
   • Stable Growth Stage: Finally, estimate dividends for the stable growth period with a constant growth rate.

2. Calculate Present Value for Each Stage:

   • High-Growth Stage: Discount the dividends for this stage back to the present value using the discount rate.
   • Transition Stage: Similarly, discount the dividends for this stage to the present value.
   • Stable Growth Stage: Calculate the present value of dividends in this stage using the Gordon Growth Model (constant growth DDM) and then discount this value to the present.

3. Sum the Present Values: The value of the stock is the sum of the present values of all dividends from each stage. This total gives a more accurate estimate of the stock's worth considering its varying growth phases.

Practical Example: Multi-Stage DDM Calculation

Let's apply the multi-stage DDM to a hypothetical company, XYZ Corp.

• Current Dividend (D0): $3.00
• Discount Rate (r): 8%
• Growth Rates:
  • High-Growth Stage: 20% for 5 years
  • Transition Stage: 10% for the next 5 years
  • Stable Growth Stage: 5% thereafter

1. Calculate Dividends for Each Stage:

• High-Growth Stage Dividends:

  • Year 1: $3.00 × (1 + 0.20) = $3.60
  • Year 2: $3.60 × (1 + 0.20) = $4.32
  • Year 3: $4.32 × (1 + 0.20) = $5.18
  • Year 4: $5.18 × (1 + 0.20) = $6.21
  • Year 5: $6.21 × (1 + 0.20) = $7.45

• Transition Stage Dividends:

  • Year 6: $7.45 × (1 + 0.10) = $8.20
  • Year 7: $8.20 × (1 + 0.10) = $9.02
  • Year 8: $9.02 × (1 + 0.10) = $9.92
  • Year 9: $9.92 × (1 + 0.10) = $10.92
  • Year 10: $10.92 × (1 + 0.10) = $12.01

• Stable Growth Stage Dividends:

  • Starting Dividend: $12.01 (Year 11)
  • Growth Rate: 5%

2. Present Value Calculations:

• High-Growth Stage:

  • PV of Year 1 Dividend: $\frac{3.60}{(1+0.08{)}^1}=3.33$(1+0.08)13.60​=3.33
  • PV of Year 2 Dividend: $\frac{4.32}{(1+0.08{)}^2}=3.21$(1+0.08)24.32​=3.21
  • PV of Year 3 Dividend: $\frac{5.18}{(1+0.08{)}^3}=3.10$(1+0.08)35.18​=3.10
  • PV of Year 4 Dividend: $\frac{6.21}{(1+0.08{)}^4}=2.87$(1+0.08)46.21​=2.87
  • PV of Year 5 Dividend: $\frac{7.45}{(1+0.08{)}^5}=2.65$(1+0.08)57.45​=2.65
  • Total PV for High-Growth Stage: $15.21

• Transition Stage:

  • PV of Year 6 Dividend: $\frac{8.20}{(1+0.08{)}^6}=5.29$(1+0.08)68.20​=5.29
  • PV of Year 7 Dividend: $\frac{9.02}{(1+0.08{)}^7}=4.90$(1+0.08)79.02​=4.90
  • PV of Year 8 Dividend: $\frac{9.92}{(1+0.08{)}^8}=4.55$(1+0.08)89.92​=4.55
  • PV of Year 9 Dividend: $\frac{10.92}{(1+0.08{)}^9}=4.23$(1+0.08)910.92​=4.23
  • PV of Year 10 Dividend: $\frac{12.01}{(1+0.08{)}^{1}0}=3.94$(1+0.08)1012.01​=3.94
  • Total PV for Transition Stage: $22.91

• Stable Growth Stage:

  • Terminal Value (Year 10): $\frac{12.01\times (1+0.05)}{(0.08-0.05)}=400.33$(0.08−0.05)12.01×(1+0.05)​=400.33
  • PV of Terminal Value: $\frac{400.33}{(1+0.08{)}^{10}}=187.97$(1+0.08)10400.33​=187.97

3. Sum of Present Values:

• Total PV of Stock: $15.21 (High-Growth) + $22.91 (Transition) + $187.97 (Stable Growth) = $226.09

The calculated value of XYZ Corp, based on varying growth rates, is approximately $226.09 per share.

Conclusion

Adapting the Dividend-Discount Model to account for changing growth rates provides a more nuanced valuation of a stock. By breaking down the dividend growth into stages and applying different growth rates and present value calculations, investors can achieve a more accurate reflection of a company's worth. This approach not only accommodates varying growth expectations but also aligns better with real-world financial scenarios where companies often undergo periods of rapid growth followed by stabilization.

Key Takeaways:

• Use Multi-Stage DDM for Variable Growth: Incorporate different growth rates for different periods to better reflect a company's lifecycle.
• Detailed Calculations Required: Accurately forecast and discount dividends for each stage to determine the present value of a stock.
• Practical Example: Applying the model to a real-world example demonstrates the effectiveness of this approach in valuation.

By mastering the multi-stage Dividend-Discount Model, investors can enhance their valuation techniques and make more informed investment decisions.