Required Return and Sustainable Growth Rate
- [i] Calculate a necessary rate of return using various dividend discount models, such as the Gordon Growth Model and the H-Model.
- [p] Compute and interpret a firm’s sustainable growth rate and apply DuPont analysis to estimate the sustainable growth rate.
Timecode Reference
- 00:16 – Implied Required Rate of Return
- 00:44 – Implied Return using Gordon Growth Model (GGM)
- 02:10 – Implied Return using the H-Model
- 03:17 – Implied Return using the Two-Stage DDM (Trial & Error)
- 06:11 – Sustainable Growth Rate (SGR)
- 07:02 – The PRAT Model (DuPont Decomposition)
- 08:32 – PRAT Analysis Example
Discounted Dividend Valuation: Required Return and Growth Rate
This lesson covers two key applications derived from Discounted Dividend Models (DDMs):
- Calculating the implied required rate of return from a stock’s current market price.
- Estimating a company’s sustainable growth rate (SGR).
1. Implied Required Rate of Return
The DDM framework can be used not only to find a stock’s intrinsic value but also to determine the rate of return implied by its current market price. This is essentially the stock’s Internal Rate of Return (IRR), assuming the market price equals the present value of all future dividends.
If we subscribe to the Efficient Market Hypothesis (EMH), this calculated IRR can be considered a good estimate of the market’s required rate of return for that stock.
Implied Return using Gordon Growth Model (GGM)
The Gordon Growth Model has four variables: V₀ (value/price), D₀ (current dividend), r (required return), and g (growth rate). Given any three, we can solve for the fourth.
The GGM formula is:
\[ V_0 = \frac{D_0(1+g)}{r-g} = \frac{D_1}{r-g} \]
To find the implied required return (r), we can rearrange the formula:
\[ r = \frac{D_1}{V_0} + g \]
Example: XYZ Stock
- Given:
- Current Market Price (V₀) = $130
- Last Dividend (D₀) = $5.00
- Constant Growth Rate (g) = 4%
- Objective: Calculate the implied required return (r).
- Step 1: Calculate the next dividend (D₁). \[ D_1 = D_0(1+g) = \$5.00(1+0.04) = \$5.20 \]
- Step 2: Use the rearranged GGM formula. \[ r = \frac{\$5.20}{\$130} + 0.04 \] \[ r = 0.04 + 0.04 = 0.08 \]
- Result: The implied required rate of return is 8%.
Implied Return using the H-Model
The same principle of rearranging the formula can be applied to the H-Model.
The H-Model formula is:
\[ V_0 = \frac{D_0(1+g_L) + D_0 \times H \times (g_S – g_L)}{r-g_L} \]
Example: Channai Corp
- Given:
- Current Market Price (V₀) = $60
- Last Dividend (D₀) = $3.00
- Initial Short-term Growth Rate (gₛ) = 8%
- Final Long-term Growth Rate (gₗ) = 5%
- Half-life of high-growth period (H) = 2 years (since the decline happens over 4 years)
- Objective: Calculate the implied required return (r).
- Step 1: Plug the values into the H-Model formula. \[ \$60 = \frac{\$3(1+0.05) + \$3 \times 2 \times (0.08 – 0.05)}{r-0.05} \]
- Step 2: Simplify the numerator. \[ \$60 = \frac{\$3.15 + \$6 \times (0.03)}{r-0.05} \] \[ \$60 = \frac{\$3.15 + \$0.18}{r-0.05} = \frac{\$3.33}{r-0.05} \]
- Step 3: Rearrange and solve for r. \[ r – 0.05 = \frac{\$3.33}{\$60} \] \[ r – 0.05 = 0.0555 \] \[ r = 0.0555 + 0.05 = 0.1055 \]
- Result: The implied required rate of return is 10.55%.
Implied Return using the Two-Stage DDM (Trial & Error)
For more complex models like the general two-stage DDM, algebraically solving for ‘r’ is difficult because ‘r’ appears in the denominator of multiple terms with different powers. In an exam context, you will be given multiple-choice answers, allowing you to use a trial-and-error approach.
Example: Monstrous Tech
- Given:
- Current Market Price (V₀) = $62.30
- Last Dividend (D₀) = $1.00
- Stage 1: Growth (gₛ) = 15% for 2 years.
- Stage 2: Growth (gₗ) = 6% perpetually thereafter.
- Choices for r: A) 8%, B) 10%, C) 12%
- Objective: Find the implied required return (r).
- Strategy: Test the middle value (10%) first. Calculate the stock’s value using r=10% and compare it to the market price.
- Step 1: Calculate the dividends and terminal value using a test rate of r = 10%.
- \( D_1 = \$1.00 \times 1.15 = \$1.15 \)
- \( D_2 = \$1.15 \times 1.15 = \$1.3225 \)
- \( D_3 = \$1.3225 \times 1.06 = \$1.40185 \)
- Terminal Value at end of Year 2 (V₂): \[ V_2 = \frac{D_3}{r-g_L} = \frac{\$1.40185}{0.10-0.06} = \frac{\$1.40185}{0.04} = \$35.05 \]
- Step 2: Calculate the stock’s value (NPV) using the financial calculator’s cash flow function with I=10%.
- CF₀ = 0
- C01 = 1.15 (Dividend for year 1)
- C02 = 1.3225 + 35.05 = 36.3725 (Dividend for year 2 + Terminal Value)
- I = 10
- CPT NPV = $31.10
- Step 3: Analyze the result.
- The calculated value ($31.10) is much lower than the market price ($62.30).
- To get a higher present value (NPV), we need a lower discount rate (r).
- Therefore, the correct answer must be 8%.
- Step 4 (Verification): Recalculate with r = 8%.
- New Terminal Value (V₂) at r=8%: \[ V_2 = \frac{\$1.40185}{0.08-0.06} = \frac{\$1.40185}{0.02} = \$70.09 \]
- Using the CF function with I=8%:
- CF₀ = 0
- C01 = 1.15
- C02 = 1.3225 + 70.09 = 71.4125
- I = 8
- CPT NPV = $62.29
- This value is approximately equal to the market price of $62.30, confirming that 8% is the correct implied required return.
2. Sustainable Growth Rate (SGR)
The sustainable growth rate (SGR) is the rate at which a company can grow its earnings and dividends without changing its financing policy (i.e., its debt-to-equity ratio) or issuing new equity.
It is a fundamental way to estimate the long-term growth rate (g) for use in DDM models.
The basic formula is:
\[ SGR = b \times ROE \]
- \( b \) = Earnings Retention Ratio = \( 1 – \text{Dividend Payout Ratio} \) = \( \frac{\text{Net Income} – \text{Dividends}}{\text{Net Income}} \)
- \( ROE \) = Return on Equity = \( \frac{\text{Net Income}}{\text{Total Equity}} \)
The PRAT Model (DuPont Decomposition of SGR)
The SGR formula can be expanded using the DuPont decomposition of ROE. This provides deeper insight into the drivers of a company’s growth. This expanded formula is often called the PRAT Model.
The formula is:
\[ SGR = \left( \frac{\text{Net Income} – \text{Div}}{\text{Net Income}} \right) \times \left( \frac{\text{Net Income}}{\text{Sales}} \right) \times \left( \frac{\text{Sales}}{\text{Total Assets}} \right) \times \left( \frac{\text{Total Assets}}{\text{Equity}} \right) \]
This breaks down into four key drivers:
- Profit Margin: \( \frac{\text{Net Income}}{\text{Sales}} \) – Measures operating efficiency.
- Retention Ratio: \( \frac{\text{Net Income} – \text{Div}}{\text{Net Income}} \) – Measures how much profit is reinvested.
- Asset Turnover: \( \frac{\text{Sales}}{\text{Total Assets}} \) – Measures asset use efficiency.
- T – Financial Leverage: \( \frac{\text{Total Assets}}{\text{Equity}} \) – Measures use of debt financing.
Note: The product of Profit Margin and Asset Turnover is the Return on Assets (ROA). The first two components (Profit Margin and Asset Turnover) relate to the company’s performance, while the last two (Retention Ratio and Financial Leverage) relate to its financing decisions.
PRAT Analysis Example
Let’s compare two competitors, Company A and Company B, to understand their growth drivers.
Given Data
| Company | A | B |
|---|---|---|
| Revenue | $1,200 | $2,000 |
| Net Income | $120 | $500 |
| Dividend | $6 | $200 |
| Assets (beginning) | $15,000 | $10,000 |
| Equity (beginning) | $3,000 | $8,000 |
PRAT Calculation & Analysis
| Company | A | B | Formula |
|---|---|---|---|
| Profit Margin | 0.100 | 0.250 | Net Income / Revenue |
| Asset Turnover | 0.080 | 0.200 | Revenue / Assets |
| Retention Ratio | 0.950 | 0.600 | 1 – (Dividend / Net Income) |
| Leverage | 5.000 | 1.250 | Assets / Equity |
| SGR | 3.8% | 3.8% | Product of the four ratios above |
| ROA (Performance) | 0.8% | 5.0% | Profit Margin x Asset Turnover |
Comments on Growth Drivers
- Although both companies have the same sustainable growth rate of 3.8%, the sources of their growth are very different.
- Company A’s growth is driven by financing decisions. It has a very high retention ratio (95%) and high financial leverage (5.0). Its operating performance (ROA) is very low at 0.8%.
- Company B’s growth is driven by performance. It has a much higher profit margin (25%) and asset turnover (0.20), leading to a strong ROA of 5%. It relies less on retention and leverage to achieve its growth.
- An analyst would likely view Company B’s growth as higher quality and more sustainable because it stems from strong operational performance rather than aggressive financing policies.
Summary
This lesson demonstrated how to work backwards from a stock’s market price to find the implied required rate of return (r) using various DDM models like the GGM and H-Model. For more complex models, a trial-and-error approach with a financial calculator is necessary. Additionally, the lesson covered the calculation of the Sustainable Growth Rate (SGR), a key input for DDM models. The SGR can be deconstructed using the PRAT model, which allows analysts to identify the fundamental drivers of a company’s growth, distinguishing between performance-driven and financing-driven growth.
