Dividend Discount Model with Changing Growth Rate

The Dividend Discount Model (DDM) is a powerful tool for valuing stocks, particularly when it comes to understanding the potential future value of a company’s dividends. This model becomes even more insightful when considering a changing growth rate, which is a common scenario in the real world. In this article, we will explore the intricacies of the Dividend Discount Model with a variable growth rate, uncovering how adjustments in growth expectations can affect stock valuations. We will delve into detailed mathematical formulas, real-world applications, and illustrative examples to provide a comprehensive understanding of how to leverage this model effectively.

Understanding the Dividend Discount Model

The Dividend Discount Model (DDM) is based on the principle that the value of a stock is the present value of all future dividends it will pay. In its simplest form, the model assumes a constant dividend growth rate. However, in reality, dividend growth rates can vary over time due to changes in a company's performance, economic conditions, and other factors. To accommodate this, the model can be adapted to account for changing growth rates, which allows for a more accurate valuation.

The Gordon Growth Model (Constant Growth)

The basic form of the Dividend Discount Model is the Gordon Growth Model (GGM), which assumes that dividends grow at a constant rate indefinitely. The formula is:

$P_0=\frac{D_0(1+g)}{r-g}$P0​=r−gD0​(1+g)​

where:

• $P_0$P0​ = Current stock price
• $D_0$D0​ = Most recent dividend payment
• $g$g = Constant growth rate of dividends
• $r$r = Discount rate or required rate of return

Introducing Variable Growth Rates

When dividends grow at varying rates, the model needs to be adjusted to reflect these changes. The two-stage dividend discount model is a common approach for handling such variability. This model divides the valuation into two distinct periods: a high-growth phase and a stable-growth phase.

Two-Stage Dividend Discount Model

In the two-stage model, the valuation process is as follows:

1. Calculate the Present Value of Dividends During the High-Growth Phase

   This phase involves a high rate of dividend growth that is expected to last for a specific period. The dividends during this phase are calculated using:

   $P_1=\frac{D_0(1+g_1)}{(1+r{)}^1}+\frac{D_1(1+g_1)}{(1+r{)}^2}+\dots +\frac{D_n(1+g_1)}{(1+r{)}^n}$P1​=(1+r)1D0​(1+g1​)​+(1+r)2D1​(1+g1​)​+…+(1+r)nDn​(1+g1​)​

   where:

   • $D_i$Di​ = Dividend in year $i$i
   • $g_1$g1​ = Growth rate during the high-growth phase

2. Calculate the Terminal Value at the End of the High-Growth Phase

   The terminal value represents the value of all dividends beyond the high-growth phase, assuming a constant growth rate in perpetuity. This is calculated using the Gordon Growth Model:

   $TV=\frac{D_n(1+g_2)}{r-g_2}$TV=r−g2​Dn​(1+g2​)​

   where:

   • $g_2$g2​ = Growth rate after the high-growth phase

3. Discount the Terminal Value to Present Value

   The terminal value must be discounted back to its present value:

   $P{V}_{TV}=\frac{TV}{(1+r{)}^n}$PVTV​=(1+r)nTV​

4. Sum the Present Value of Dividends and Terminal Value

   Finally, add the present value of dividends during the high-growth phase and the discounted terminal value to determine the stock’s current value:

   $P_0={\sum }_{i=1}^n\frac{D_i(1+g_1)}{(1+r{)}^i}+\frac{TV}{(1+r{)}^n}$P0​=∑i=1n​(1+r)iDi​(1+g1​)​+(1+r)nTV​

Practical Application: An Example

To illustrate, let’s use an example of a company that is expected to have a high dividend growth rate of 10% for the next 5 years, after which the growth rate is expected to stabilize at 3%. Assume the current dividend is $2 per share, and the discount rate is 8%.

1. Calculate Present Value of Dividends for the First 5 Years

   D1​D2​D3​D4​D5​​=2×(1+0.10)=2.20=2.20×(1+0.10)=2.42=2.42×(1+0.10)=2.66=2.66×(1+0.10)=2.93=2.93×(1+0.10)=3.22​

   Present value of dividends:

   PVD1​PVD2​PVD3​PVD4​PVD5​​=(1+0.08)12.20​=2.04=(1+0.08)22.42​=1.88=(1+0.08)32.66​=1.75=(1+0.08)42.93​=1.62=(1+0.08)53.22​=1.50​

   Total present value of dividends:

   $P{V}_D=2.04+1.88+1.75+1.62+1.50=8.79$PVD​=2.04+1.88+1.75+1.62+1.50=8.79

2. Calculate the Terminal Value

   TVPVTV​​=0.08−0.033.22×(1+0.03)​=0.053.32​=66.40=(1+0.08)566.40​=45.21​

3. Sum the Present Values

   $P_0=8.79+45.21=54.00$P0​=8.79+45.21=54.00

   The current value of the stock, considering the changing growth rate, is $54.00.

Conclusion

The Dividend Discount Model with a changing growth rate provides a more nuanced and realistic valuation of stocks compared to the constant growth model. By segmenting the dividend growth into different phases and adjusting for varying growth rates, investors can better reflect the real-world dynamics affecting a company's future dividends. This model is particularly useful for companies with varying growth trajectories and can be a valuable tool in the investment decision-making process.

Key Takeaways

• Adjusting for Variable Growth Rates: The two-stage model helps in accurately valuing stocks with changing dividend growth rates.
• Practical Application: Real-world scenarios often involve multiple growth phases, making the two-stage model essential for precise valuations.
• Comprehensive Analysis: By incorporating both high-growth and stable-growth phases, investors can better assess the potential value of their investments.