Gordon Growth Model
- [b] Calculate and explain the meaning of the value of a common stock using the dividend discount model (DDM) for single or multiple holding periods.
- [c] Compute the worth of common stock utilizing the Gordon growth formula and clarify the presumptions supporting the model.
- [f] Compute and explain the implied growth rate of dividends utilizing the Gordon growth model and the present market value of the stock.
- [g] Calculate and interpret PVGO and the component of P/E related to PVGO.
- [h] Calculate and explain the meaning of forward and backward P/E ratios based on the Gordon growth model.
- [d] Compute the price of a noncallable perpetual preferred stock with a fixed interest rate.
- [e] Explain the benefits and drawbacks of using the Gordon growth model and provide reasons for its choice in valuing a company’s common stock.
- 00:34 – Fundamentals of the Gordon Growth Model (GGM)
- 02:16 – Assumptions and Limitations of GGM
- 03:04 – Example: Calculating Intrinsic Value with GGM
- 05:10 – Characteristics of the Gordon Growth Model
- 05:54 – Application: Implied Growth Rate
- 08:49 – Application: Justified P/E Ratio
- 13:47 – Application: Present Value of Growth Opportunities (PVGO)
- 16:26 – Application: Preferred Stock Valuation
- 17:44 – Strengths and Limitations of the GGM
Fundamentals of the Gordon Growth Model (GGM)
The Gordon Growth Model (GGM), also known as the Constant-Growth Dividend Discount Model, is a method for valuing a stock based on the assumption that its dividends will grow at a constant rate indefinitely.
It simplifies the general Dividend Discount Model (DDM), which states that a stock’s value is the present value of all its future dividends:
\[ PV_{stock} = \sum_{t=1}^{\infty} \frac{D_t}{(1+r)^t} = \frac{D_1}{(1+r)^1} + \frac{D_2}{(1+r)^2} + \dots \]
Forecasting infinite dividends is impossible. The GGM solves this by assuming a constant growth rate (g) for dividends from a certain point forward. If we assume this constant growth starts from the first period, the formula simplifies to a growing perpetuity.
The GGM Formula
The intrinsic value of a stock today ( \( V_0 \) ) is calculated as the next period’s dividend ( \( D_1 \) ) divided by the difference between the required rate of return (r) and the constant dividend growth rate (g).
\[ V_0 = \frac{D_1}{r – g} \]
Where:
- \( V_0 \): The intrinsic value of the stock today.
- \( D_1 \): The expected dividend in the next period (one year from now). This is a crucial component.
- r: The required rate of return on the stock (often estimated using CAPM).
- g: The constant, perpetual growth rate of dividends.
Often, you are given the most recently paid dividend ( \( D_0 \) ). To find \( D_1 \), you must grow \( D_0 \) by the growth rate, g:
\[ D_1 = D_0(1+g) \]
Therefore, the GGM formula can also be written as:
\[ V_0 = \frac{D_0(1+g)}{r – g} \]
Exam Tip: Always be careful whether the question provides \( D_0 \) (the dividend just paid) or \( D_1 \) (the dividend to be paid next year). Using the wrong one is a common mistake.
Assumptions and Limitations of GGM
The GGM is built on several key assumptions which also represent its limitations:
- Dividends are the correct metric: The model assumes that dividends are the appropriate measure of shareholder wealth. This makes it unsuitable for companies that do not pay dividends (e.g., many early-stage growth companies).
- Constant Dividend Growth: The dividend growth rate, g, is assumed to be constant and perpetual. This is a strong assumption that may not hold in reality, especially for companies with unpredictable growth patterns.
- Constant Required Return: The required rate of return, r, is also assumed to be constant over time.
- r > g: The required rate of return must be strictly greater than the dividend growth rate. If \( g \geq r \), the denominator becomes zero or negative, resulting in a nonsensical (infinite or negative) valuation.
Example: Calculating Intrinsic Value with GGM
Problem
XYZ stock recently paid a dividend of $5. The dividend payout is expected to grow at 4% per year indefinitely. The required rate of return is 9%. Calculate the intrinsic value of this stock today.
Solution
- Identify the inputs:
- Most recent dividend ( \( D_0 \) ) = $5.00
- Constant growth rate (g) = 4% or 0.04
- Required rate of return (r) = 9% or 0.09
- Calculate the next period’s dividend ( \( D_1 \) ): \[ D_1 = D_0(1+g) = \$5.00 \times (1 + 0.04) = \$5.20 \]
- Apply the GGM formula: \[ V_0 = \frac{D_1}{r – g} = \frac{\$5.20}{0.09 – 0.04} = \frac{\$5.20}{0.05} = \$104 \]
The intrinsic value of XYZ stock is $104.
Characteristics of the Gordon Growth Model
The value derived from the GGM is highly sensitive to the inputs, particularly the spread between r and g.
- As the difference between r and g (the spread) narrows, the stock’s value increases.
- As the difference between r and g (the spread) widens, the stock’s value decreases.
This sensitivity highlights that small changes in the estimates for the required return or the growth rate can lead to large differences in the calculated intrinsic value.
Application: Implied Growth Rate
The GGM formula can be rearranged to solve for any of its four variables if the other three are known. A common application is to solve for the implied growth rate (g) that is priced into the current market stock price.
Starting with \( V_0 = \frac{D_0(1+g)}{r – g} \), we can algebraically rearrange it to solve for g:
\[ g = \frac{V_0 \times r – D_1}{V_0 + D_0} \]
Or, more intuitively, by rearranging the simpler form:
\[ V_0 (r-g) = D_1 \]
\[ r – g = \frac{D_1}{V_0} \]
\[ g = r – \frac{D_1}{V_0} \]
This shows the growth rate implied by the market is the required rate of return minus the expected dividend yield.
Example: Implied Growth Rate
The market price of XYZ stock is $130. The most recent dividend was $5, and the required rate of return is 9%. What is the implied growth rate that investors are pricing in?
- Identify Inputs:
- Market Price ( \( V_0 \) ) = $130
- Most recent dividend ( \( D_0 \) ) = $5
- Required rate of return (r) = 9% or 0.09
- Rearrange the GGM formula to solve for g: \[ V_0 = \frac{D_0(1+g)}{r – g} \] \[ \$130 = \frac{\$5(1+g)}{0.09 – g} \] \[ \$130(0.09 – g) = \$5(1+g) \] \[ \$11.7 – 130g = \$5 + 5g \] \[ \$6.7 = 135g \] \[ g = \frac{\$6.7}{135} \approx 0.0496 \text{ or } 5.0\% \]
The market is pricing in an implied perpetual growth rate of approximately 5.0%. Since this is higher than the analyst’s estimated 4% growth, it explains why the market price ($130) is higher than the analyst’s calculated intrinsic value ($104).
Application: Justified P/E Ratio
The GGM can be used to derive formulas for the justified price-to-earnings (P/E) ratio, which is the P/E ratio based on the company’s fundamentals rather than its current market price. We can calculate both a justified leading (forward) P/E and a justified trailing (historical) P/E.
Justified Leading P/E Ratio (P₀/E₁)
This ratio uses the next period’s expected earnings ( \( E_1 \) ). We start with the GGM and divide both sides by \( E_1 \).
\[ \frac{V_0}{E_1} = \frac{D_1 / E_1}{r – g} \]
Since \( D_1 / E_1 \) is the expected dividend payout ratio (which is equal to 1 minus the retention ratio, b), the formula becomes:
\[ \text{Justified Leading P/E} = \frac{P_0}{E_1} = \frac{1-b}{r-g} \]
Justified Trailing P/E Ratio (P₀/E₀)
This ratio uses the last period’s earnings ( \( E_0 \) ). We divide the GGM by \( E_0 \).
\[ \frac{V_0}{E_0} = \frac{D_0(1+g) / E_0}{r – g} \]
Since \( D_0 / E_0 \) is the historical dividend payout ratio (1 – b), the formula becomes:
\[ \text{Justified Trailing P/E} = \frac{P_0}{E_0} = \frac{(1-b)(1+g)}{r-g} \]
Notice that the justified trailing P/E is simply the justified leading P/E multiplied by (1+g). The trailing P/E will be higher than the leading P/E for a growing company because the denominator (earnings) is smaller.
Application: Present Value of Growth Opportunities (PVGO)
The value of a stock can be partitioned into two components:
- The value of the company if it had zero growth (i.e., its assets in place). This is calculated as a simple perpetuity of its earnings: \( \frac{E_1}{r} \).
- The Present Value of Growth Opportunities (PVGO), which represents the additional value created by reinvesting earnings into projects with returns greater than the required rate of return.
The total intrinsic value is the sum of these two parts:
\[ V_0 = \text{Value of Assets in Place} + \text{PVGO} \]
\[ V_0 = \frac{E_1}{r} + \text{PVGO} \]
Therefore, PVGO can be calculated as:
\[ \text{PVGO} = V_0 – \frac{E_1}{r} \]
Example: PVGO
ZoomPlus Corp is a tech company with shares trading at $120. It has expected earnings of $3 per share and a required return of 15%. Determine the proportion of the company’s leading P/E ratio attributable to PVGO.
- Calculate PVGO per share:
- Value of Assets in Place = \( \frac{E_1}{r} = \frac{\$3}{0.15} = \$20 \)
- PVGO = \( V_0 – \frac{E_1}{r} = \$120 – \$20 = \$100 \)
- Calculate the Leading P/E and the P/E attributable to PVGO:
- Firm’s Leading P/E = \( \frac{P_0}{E_1} = \frac{\$120}{\$3} = 40x \)
- P/E due to PVGO = \( \frac{\text{PVGO}}{E_1} = \frac{\$100}{\$3} = 33.3x \)
- Determine the proportion: \[ \text{Proportion} = \frac{\text{P/E due to PVGO}}{\text{Firm’s Leading P/E}} = \frac{33.3}{40} = 0.8325 \text{ or } 83.3\% \]
Over 83% of ZoomPlus Corp’s leading P/E ratio is attributable to its future growth opportunities.
Application: Preferred Stock Valuation
Preferred stock typically pays a fixed dividend that does not grow. This is a special case of the GGM where the growth rate (g) is zero.
The GGM formula \( V_0 = \frac{D_1}{r – g} \) simplifies to the perpetuity formula:
\[ V_{pref} = \frac{D_{pref}}{r} \]
Where:
- \( V_{pref} \): The value of the preferred stock.
- \( D_{pref} \): The fixed annual dividend of the preferred stock.
- r: The required rate of return on the preferred stock.
Example: Preferred Stock
XYZ company has preferred stock issued at $100 par that pays a fixed 7% dividend each year. Based on a required rate of return of 9%, what is the intrinsic value of the stock today?
- Calculate the annual dividend: \[ D_{pref} = \text{Par Value} \times \text{Dividend Rate} = \$100 \times 7\% = \$7 \]
- Apply the preferred stock valuation formula: \[ V_{pref} = \frac{D_{pref}}{r} = \frac{\$7}{0.09} = \$77.78 \]
The intrinsic value of the preferred stock is $77.78.
Strengths and Limitations of the Gordon Growth Model
| Strengths | Limitations |
|---|---|
| Well-suited for the valuation of stable, mature, dividend-paying firms. | Cannot be easily applied to non-dividend-paying firms. |
| Can be used for valuing market indices. | Valuations are very sensitive to estimates of growth rates (g) and required return (r). |
| Can be rearranged to determine implied growth rates, justified P/E ratios, and PVGO. | Unpredictable growth patterns (non-constant growth) may result in unreliable valuations. |
| Easily communicated and explained due to its straightforward approach. | |
| Can be used to supplement other, more complex valuation methods. |
Summary
The Gordon Growth Model is a foundational tool in equity valuation. It provides a simple way to estimate the intrinsic value of a stock assuming constant dividend growth. While its assumptions limit its applicability, especially for non-dividend-paying or high-growth companies, its various applications—such as calculating implied growth rates, justified P/E ratios, and PVGO—make it a versatile and insightful model for financial analysis. Understanding its mechanics, assumptions, and characteristics is essential for any finance professional.
