1. Comparison of 3 models of discounted dividend, net cash flow and residual income
All three models rely on discounting future cash flows back to the present to determine the value of a company. However, each method takes a different view on determining future cash flows.
| Discounted Dividend Flow Model | Net cash flow model | Residual Income Model | |
| Definition | This model identifies the dividend stream paid to shareholders as the company’s future cash flow. | This model defines the future cash flows of the company as the net cash flows that the company has left after paying its capital requirements and the costs associated with its debt obligations. (FCFE) | This model defines a company’s future cash flows as the portion of the company’s earnings that exceeds the investor’s required rate of return. |
| Strengths | Theoretically reasonable Dividends will be less volatile than earnings |
Applies to most companies, regardless of whether they pay dividends or not | Can be applied to companies with negative net cash flow |
| Weaknesses | Difficult to apply to companies that do not pay dividends Only suitable when evaluating the investment perspective of small investors who do not have control |
Companies with large capital needs may experience negative cash flow | The level of accuracy depends largely on the transparency of the management system |
| Application | Applies to companies that regularly pay dividends or have a dividend payment policy equivalent to profits Suitable for analytical applications for retail investors |
Applicable to companies that do not regularly pay dividends, companies whose net cash flow corresponds to the level of profit Suitable when applying investment analysis from the perspective of major shareholders |
Applies to companies that do not regularly pay dividends and have negative net cash flow. Applicable to companies with high levels of transparency in management |
2. Discounted Dividend Stream Model
If an investor wants to buy a stock and hold it for one year, the value of that stock today is the present value of the expected dividends received from that stock plus the present value of the expected selling price in one year.
In which:
- V0 = value of 1 share today, at time t = 0
- P1 = is the expected price per share at time t = 1
- D1 = is the expected dividend per share for Year 1
- r = required rate of return on the stock
For the n-period model, the value of a stock is the present value of the expected dividends in n periods plus the present value of the expected price in n periods (at time t = n).
- V0 = value of 1 share today, at time t = 0
- Pn = is the expected price per share at time t = n
- Dn = is the expected dividend per share for Year n
- r = required rate of return on the stock
If we extend the holding period indefinitely, that value simply becomes the present value of the infinite stream of dividends, expressed by the formula DDM by John Burr Williams (1938).
To use DDM, the forecasting problem must be simplified. There are two broad approaches, each with some variations.
a. Future dividends can be forecasted by assigning the future dividend stream to one of a number of stylized growth models.
b. A finite number of dividends can be individually forecasted up to the end point, usually 3 to 10 years in the future.
3. Gordon Growth Model
3.1. Gordon growth model
The Gordon growth model (GGM), developed by Gordon and Shapiro (1956) and Gordon (1962), assumes that dividends grow indefinitely at a constant rate.
In which:
- D0 = recently paid dividends
- r = required rate of return on the stock
- g = dividend growth rate
3 assumptions of the Gordon Growth Model:
- The company is expected to pay dividends, D1, in the next 1 year
- Dividends grow indefinitely at a constant rate, g (which may be less than 0).
- The growth rate g is less than the required rate of return r.
Some other notes:
- A company’s growth forecast can be compared to the growth rate of the economy to determine whether it can continue indefinitely.
- It is unrealistic to assume that any firm can continue to grow indefinitely at a rate greater than the long-run growth rate of real gross domestic product (GDP) plus the long-run rate of inflation.
- Overall, forecasting a dividend growth rate above 5% forever is questionable.
3.2. Growth in the Gordon growth model
The Gordon growth model involves a set of relationships regarding the growth rate of dividends, earnings, and stock value.
→ Both dividends, value, and earnings grow at rate g (holding r and payout ratio constant)
→ g is the value ratio or capital appreciation ratio (sometimes also called the capital gains ratio)
Dividend Yield: 
= r – g (from equation (1)) → The dividend yield remains constant. (also because both dividends and price are expected to increase at the same rate g)
Capital Gains Yield: g (as discussed above) → Capital Gains Yield remains unchanged.
Total return = Dividend yield + Capital gains yield = r- g + g = r
3.3. Share buybacks and implied growth rates
3.3.1. Share repurchase
Share buybacks involve reducing the number of shares outstanding, all else being equal. They are generally less predictable than cash dividends from companies with a determinable dividend policy.
When companies buy back shares, analysts have two methods to calculate the value:
- Focus on the total payout (i.e. dividends plus buybacks) and use that in their valuation model.
In which: SRt is bought back in time t
- Focus on the cash dividend discount model after adjusting for the number of shares outstanding.
Where: g and r are currently taken into account when buying back shares
3.3.2. Implied dividend growth rate
The Gordon growth model consists of four variables, so if we know any three of them, we can solve for the fourth. Here, we calculate the implied growth rate of dividends (g).
3.4. Present value of growth opportunities (PVGO)
The value of a stock includes:
- Value of the firm without reinvestment of profits
- Present value of growth opportunity
a. The value of the company without reinvestment of profits
- An increase in shareholder wealth occurs only when reinvested earnings are earned more than the opportunity cost of capital (ROE > r) – that is, when investing in projects with positive NPVs.
- A firm with no projects with positive expected NPV is defined as a no-growth firm (g = 0).
Companies should distribute all of their earnings as dividends:
b. Present value of growth opportunity:
PVGO (value-to-go), the present expected total value of future income-generating reinvestment opportunities, can be defined as the sum of the forecasted net present values of future projects.
1. Components of expected P/E according to PVGO
In which:
= The company’s P/E value is not growing</p >
= Components of P/E from the fundamental growth association
4. Multi-stage discounted dividend model
4.1. 3 stages of business
While the basic GGM assumes continuous growth, most companies go through a growth model that consists of three stages as follows:
- Early growth stage: Company has strong revenue growth, little or no dividends, and heavy reinvestment
- Transition stage: Revenues and dividends increase but at low rates because competitive growth reduces profit opportunities and requires more reinvestment
- Maturity stage: Revenues are growing at a steady but slower rate and payout ratios are stabilizing due to reinvestment in line with depreciation and maintenance requirements for assets.
| Variable | Growth phase | ||
| Early Stage | Conversion | Mature | |
| Revenue growth | Very High | Above average but declining | Stable over long periods |
| Capital Investment | Large request | Reduce | Stable over long periods |
| Profit Margin | High | Above average but declining | Stable over long periods |
| FCFE | Sound | Possible and growing | Stable over long periods |
| ROE & required return | ROE > r | ROE approaches r | ROE = r |
| Dividend payout ratio | Low or zero | Increase | Stable over long periods |
| Matching Model | 3-phase model | 2-phase model | Gordon Growth Model |
4.2. Multi-tier discount model
For most companies, the Gordon growth model’s assumption of constant and perpetual dividend growth is unrealistic.
→ We need more realistic multi-stage growth models to estimate value for companies with multiple growth stages in the future:
- 2 Stage Dividend Discount
- H Model
- 3-phase model
- Spreadsheet Modeling
4.2.1. 2 stages of dividend discount
The two-stage DDMis the most basic multi-stage model, in which we assume the firm grows at a high rate for a relatively short period of time ( first stage) and then returns to a long-term permanent growth rate (second stage).
Example:
Suppose a company is expected to grow at 15% until its patent expires in 4 years, then immediately return to its long-term growth rate of 3% forever.
This stock should be modeled using a two-stage model, with dividends growing at 15% before the patent expires and 3% thereafter.
4.2.2. H Model
The problem with the basic two-stage DDM: it is often unrealistic to assume that a stock will grow strongly for a short period, then immediately fall back to its long-term level.
→ The H model is a version of the two-stage DDM, in which the dividend growth rate is assumed to decline from an abnormal rate to a mature growth rate during Stage 1.
Example of H model:
Consider a company that is currently growing at 15 and its growth rate will decrease by 3% each year until it reaches 3% at the end of year four.
4.2.3. 3-Stage Growth Model
The three-stage DDMis suitable for companies that are expected to have three distinct stages of earnings growth. It is a slightly more complex evolution of the two-stage model.
Example of three-phase DDM:
Suppose we predict that a biotech company will have a stellar growth rate of 25% for three years, then 15% for five years, and finally decline to a long-term plateau of 3%.
In addition, in Stage 2, the growth rate can also decrease linearly to the long-term, stable growth rate in Stage 3.
4.2.4. Spreadsheet modeling
In fact, we can use a spreadsheet to model any dividend growth model we want with different growth rates for each year because the spreadsheet does all the calculations.
The spreadsheet model can be applied to companies where we have a lot of information and can predict different growth rates for different periods.
Example:
| Growth Rate | |||||||
| Year | 1 | 2 | 3 | 4 | 5 | 6 | 7 and later |
| Scenario 1 | 20% | 19% | 13% | 5% | 5% | 5% | 5% |
| Scenario 2 | 20% | 19% | 13% | 11% | 5% | 5% | 5% |
| Scenario 3 | 20% | 19% | 13% | 11% | 8% | 7% | 5% |
4.3. Final value
Terminal value is the estimated value of an investment at a specific point in the future. It represents the present value of all expected future cash flows beyond a certain period of time, usually when the company is assumed to grow at a stable and sustainable rate.
No matter which dividend discount model we use, we have to estimate the terminal value at some point in the future.
Example:
4.3.1. Alternative methods for determining final values
There are two ways to estimate the final value:
- Using the Gordon Growth Model
At some point in the future, we assume that dividends will start growing at a constant, long-term rate. → The terminal value at that point is simply the value derived from the Gordon growth model.
- Using the market multiples approach
We use market price multiples, specifically the P/E ratio at the forecast date, and then estimate the final value when the P/E multiplies with estimated earnings.
5. Calculate the value of stocks using the multi-stage discount model
5.1. Stock value according to the 2-stage model
Review: The two-stage constant growth rate model is based on the assumption that a company will have an initial period of high growth, followed by a maturity or stability period, in which the growth rate will be lower but sustainable.
In which:
- gs = short-term growth rate
- gl = long-term growth rate
- r = must return
- n = length of high growth period
5.2. Valuation of non-dividend-paying enterprises
The value of a firm that currently pays no dividends is a simplified version of the two-stage DDM, where the firm pays no dividends in the first stage.
→ The value of the company is simply the present value of the terminal value calculated at the time when dividends are expected to start being paid.
5.3. Business valuation using the H Model
Review: The H-model estimates the value of a firm assuming that its initially high growth rate declines linearly over a given period of time.
In which:
- VS12 = present value of the company’s dividend stream if it grows at
forever - VS1 is an approximate value of the cumulative value added to the stock due to its abnormal growth in Years 1 to 2H
- gs is the short-term growth rate
- gl is the long-term growth rate
- r is required to return
- H = n/2 is the half-life during the years of high growth.
5.4. Business valuation using the 3-stage growth model
Review: There are two popular versions of the three-stage DDM, distinguished by how the second stage is modeled.
First version (general three-stage model): The firm is assumed to have three distinct growth stages, and the growth rate of the second stage is usually zero.
Second version: The growth rate in the middle (second) stage is assumed to decrease linearly with the adult growth rate: essentially, the second and third stages are considered as the H model.
Where: H’ = (nk) /2 is the half-life calculated in years of high growth period (phase 2).
6. Spreadsheet model for forecasting dividends and valuing common stocks
6.1. Tabular model
Problems with two-, three-stage DDM and the H-model:
- A company’s dividends (or cash flow) do not typically grow at a steady rate over a long period of time.
- When dividend changes are predictable, it is clear that there may be more than two or three periods of change involved.
- There are often idiosyncratic events that, even if predictable, do not fit exactly into any of the patterns required by these models.
In practice, spreadsheets are more likely to be used than any of the above stylized models when pricing equity securities due to their inherent flexibility and computational accuracy.
6.2. Steps to implement the tabular model
There are 4 steps to implement the tabular model as follows:
- Establish the cash flow or dividend base: In the case of dividends, this is usually the amount paid in the previous year or some normalized amount based on the company’s projected earnings.
- Estimates changes in the company’s dividends in the near future and projects future cash dividends based on these estimates.
- Estimating the normal growth that will occur at the end of the supernormal growth period: This allows for estimating the terminal value, which represents the cash flows (i.e. the value of the company if sold at this point) that will be received at the end of the supernormal growth period.
- Discount all expected dividends and terminal values back to today to estimate the company’s current value.
7. Estimating the required return based on the dividend discount model
Based on any DDM, a given current price, and all the DDM inputs except the required return, the IRR can be calculated.
→ Such IRR has been used as an estimate of the required return (expected return of an efficient market).
7.1. Required return with the Gordon growth model
If the dividend growth rate is constant forever, we can use the Gordon growth model to calculate the implied expected return given the expected dividend, the current market price, and the expected growth rate:
→ Required Return: 
7.2. Required return with the H model
The H model can be rewritten in terms of r and used to solve r for other model inputs:
→ Required Return: 
7.3. Required return with 2-stage growth model
Using the general two-stage model is more difficult because we have to solve for r using an iterative process since there is no closed-form solution.
8. Dupont Analysis
8.1. Stable growth rate
The sustainable growth rate (SGR) is the rate at which earnings (and dividends) can continue to grow indefinitely, assuming that the company’s debt-to-equity ratio remains unchanged and that the company does not issue new equity.
SGR is a simple function of earnings retention rate and return on equity:
SGR = bx ROE
In which:
b is the profit retention ratio = 1 – dividend payout ratio
ROE is return on equity
- The lower (higher) the ROE, the lower (higher) the SGR, all other factors remaining constant.
- The lower (higher) b, the lower (higher) SGR, all other values remaining constant.
→ This relationship is called the dividend shift of earnings.
SGR is important because it tells us how fast a company can grow with internally generated capital.
8.2. Dupont Analysis
ROE can be estimated using the DuPont formula, which presents the relationship between net profit margin, total asset turnover, and financial leverage as determinants of ROE:
In which:
NI is net income / A is total assets
E is equity /S is sales
Other things being equal, we can see that the growth of a company’s earnings (and dividends) is a function of its ROE and its retention ratio:
- The above formula is also known as the PRAT model, where SGR is a function of the firm’s financial decisions (retention ratio (R), financial leverage ( T)) and the company’s operating efficiency (profit margin (P), asset turnover ratio (A)).
- If the actual growth rate is forecast to be greater than the SGR, the firm will have to issue equity unless it increases its retention ratio, profit margin, total asset turnover, or leverage.
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