# Absolute Valuation: The Dividend Discount Model

The Dividend Discount Model (DDM) is an absolute valuation method that attempts to calculate a company’s intrinsic value based on the theory that stock prices are reflective of the sum of all discounted future dividend payments.

The DDM ignores current market conditions, macroeconomic trends, and relative valuation methods and instead relies on dividend payout factors and expected market returns. The result of the DDM is the maximum an investor should be willing to spend on one share of a stock. If the result is higher than the current stock price, the stock is considered undervalued and may qualify as a buy.

This post will explore the general DDM model as well as specific versions of it tailored for different assumptions about a company’s future growth. I also will highlight some of the issues in using dividend discount models and summarize the results of studies that have examined its efficacy.

## What is the Dividend Discount Model?

While the dividend discount model (DDM) has many different forms, it is an absolute valuation method that attempts to calculate a company’s intrinsic value based on the theory that its current price is the worth the sum of its discounted future dividend payments.

Some analysts have turned away from the dividend discount model in favor of the discounted cash flow model on the premise that the DDM is outmoded. However, much of the intuition that drives the DCF model is embedded in the DDM and there are still companies and instances where the dividend discount model remains a useful tool for estimating a company’s value.

Disclaimer: Throughout this post I will use various companies' stock prices and dividends as inputs in formulas. The recommendation of any formula is not necessarily representative of my opinion. A dividend discount model is just one quantitative tool in the big universe of stock valuation methodologies and should only be one part of your investment analysis. Since the DDM requires multiple assumptions and predictions, it may not be the sole best way to evaluate a particular security.

Nothing in this post constitutes a recommendation that any particular security, portfolio of securities, transaction, or investment strategy is suitable for any specific person. I can not and will not advise you personally concerning the nature, potential, value, or suitability of any particular security, portfolio of securities, transaction, investment strategy, or any other matter.

### The General Model

Investors typically receive two types of cash flows from the purchase, ownership, and sale of a stock: dividends, and the difference between purchase price and expected sale price. In dividend paying companies, the expected sale price is itself reflective of expectations of future dividends. Therefore, the value of a stock is the present value of dividends in perpetuity.

$$\text{Value of Stock} = \sum_{t=1}^{t=\infty}\frac{\text{E}(\text{DPS}_t)}{(1+k_e)^t}$$

Where:

• E(DPSt) = Expected Dividends per Share
• ke = Cost of Equity

The foundation of the model lies in the present value rule which states that “the value of any asset is the present value of expected future cash flows discounted at a rate appropriate to the riskiness of the cash flows” (Damodaran, "Chapter 13: Dividend Discount Models", 2012, p. 323).

There are two inputs in this model: expected dividends and cost of equity.

• Expected Dividends - Typically calculated by adjusting current dividends with future expected growth rates and payout ratios.

• Cost of Equity - Also known as an investor’s required rate of return. This is determined differently in different models. More on calculating cost of equity in Appendix A.

The model is flexible enough to allow for time-varying discount rates, where the time variation is caused by expected changes in interest rates or risk across time (Damodaran, 2012).

Like any valuation methodology, the DDM should only be one part of your investment analysis. Don’t buy a stock just because the dividend discount model says that it’s cheap, and similarly, don’t avoid a stock just because the model says it is expensive. There are dozens of other valuation techniques and subjective analyses that can and should be taken into account before the purchase or sale of a security.

## Variations

It is impossible to predict the exact future dollar value of dividends through infinity. Several variations of the DDM have been developed for scenarios with different assumptions about a company’s future growth:

### The Zero Growth Model

The zero-growth model is a variation of the dividend discount model that posits a company will continue to pay the exact same dividend forever (i.e. a zero percent dividend growth rate).

The formula for the zero-growth model is:

Intrinsic Value of Stock = DPS1 / ke

Where:

• DPS1 = Next Year’s Estimated Dividend
• ke = Cost of Equity   a.k.a. (Required Rate of Return)

Example:

Exxon Mobil (XOM) is expected to pay a dividend of $3.48 in 2019 (Morningstar, n.d.). Let’s assume you calculate your required rate of return at 8.0%. Intrinsic Value of Exxon Mobil =$3.48 / .08
IV of XOM = $43.50 XOM was trading at$79.03 as of 2/28/2019. Assuming XOM never increases their dividend, the intrinsic value of the stock is $43.50 indicating that, as of 2/28/2019, XOM was overvalued. Now let’s assume you calculated a 4.0% rate of return; the intrinsic value would now be$87.00 making XOM appear undervalued. To calculate your required rate of return see Appendix A at the end of this post.

Embedded below is a zero-growth dividend discount model from Excel for you to play around with:

The zero-growth model is infrequently used except as a ‘worst case scenario’ tool. If you are investing in a company, you typically want that company to grow. Even if it is a mature company with a strong dividend history, you would at least expect dividends to keep up with inflation.

The zero-growth model can therefore be used to say ‘even if the company has limited upside, I do not think they will ever cut their dividend.’ This formula would then show you the max you should pay for a company that can not or will not increase their dividend.

### The Constant Growth Model(The Gordon Growth Model)

The Gordon growth model (GGM) is arguably the most common form of the DDM. The beauty of this model lies in in its simplicity. However, therein also lies its biggest weakness.

The GGM is sometimes referred to as “the stable model” because one of its primary assumptions is that dividend growth is constant over time. The Gordon growth model is best used “to value a firm that is in 'steady state' with dividends growing at a rate that can be sustained forever” (Damodaran, 2012).

The formula for the Gordon growth model is:

$$\text{Value of Stock} = \frac{\text{DPS}_1}{k_e-g}$$

Where:

• DPS1 = Next Year’s Estimated Dividend
• ke = Cost of Equity   a.k.a. (Required Rate of Return)
• g = Estimated Future Dividend Growth Rate

This model is best used for:

• firms with stable growth rates

• firms which pay out dividends that are high and approximate FCFE.

• firms with stable leverage.

The Gordon growth model is best suited for firms growing at a rate comparable to or lower than the nominal growth in the economy and which have well established dividend payout policies that they intend to continue into the future. The dividend payout of the firm has to be consistent with the assumption of stability, since stable firms generally pay substantial dividends. In particular, this model will under estimate the value of the stock in firms that consistently pay out less than they can afford and accumulate cash in the process (Damodaran, 2012).

Aswath Damodaran, a professor at the Stern School of Business at New York University, has categorized what he calls “obvious candidates for the Gordon Growth Model” (Damodaran, n.d., p. 3). Damodaran’s categories and reasoning:

• Regulated Companies, such as utilities, because

• their growth rates are constrained by geography and population to be close to the growth rate in the economy in which they operate.

• they pay high dividends, largely again as a function of history

• they have stable leverage (usually high)

• Large financial service companies, because

• their size makes its unlikely that they will generate extraordinary growth

• Free cash flows to equity are difficult to compute

• they pay large dividends

• they generally do not have much leeway in terms of changing leverage

• Real estate investment trusts, because

• they have to pay out 95% of their earnings as dividends

• they are constrained in terms of investment policy and cannot grow at high rates

What is a sustainable growth rate?

While the Gordon growth model is a simple and powerful approach to valuing equity, its use is limited to firms that are growing at a stable rate. There are two insights to keep in mind when estimating a stable growth rate.

First, since the growth rate in the firm's dividends is expected to last forever, the firm's other measures of performance (including earnings) can also be expected to grow at the same rate (Damodaran, 2012).

To see this in action, consider the consequences of a firm with earnings increasing at 5% per year and dividends increasing at 7% per year. Over time the dividends will exceed net earnings resulting in a payout ratio over 100%. On the other hand, if earnings are growing at a faster rate than dividends the payout ratio will converge towards zero, which is also not a steady state.

Thus, though the model's requirement is for the expected growth rate in dividends, analysts should be able to substitute in the expected growth rate in earnings and get precisely the same result, if the firm is truly in steady state (Damodaran, 2012).

The second insight to keep in mind is what growth rate is reasonable as a stable growth firm. Within in the GGM, it is assumed that the infinite dividend growth rate can not exceed the growth rate for the overall economy (GNP) in which the firm operates. Some analysts may project a firm will grow faster but rarely by more than a small amount (1-2%) (Damodaran, n.d).

If a firm is likely to maintain a few years of 'above-stable' growth rates, an approximate value for the firm can be obtained by adding a premium to the stable growth rate, to reflect the above-average growth in the initial years. Even in this case, the flexibility that the analyst has is limited. The sensitivity of the model to growth implies that the stable growth rate cannot be more than 1% or 2% above the growth rate in the economy. If the deviation becomes larger, the analyst will be better served using a two-stage or a three-stage model to capture the 'super-normal' or 'above-average' growth and restricting the Gordon growth model to when the firm becomes truly stable (Damodaran, 2012).

Uncertainty arises because not all analysts will agree on a growth rate if they agree a firm is a “stable growth firm.”

1. Analysts may use different benchmark growth rates with varying estimates of inflation and real growth in the economy given the inherent uncertainty.

2. While the growth rate of the firm can not be greater than the economy, it can be less. Firms can becomes smaller over time relative to the economy.

For more on how to calculate sustainable growth rates see Appendix B.

Does a stable growth rate have to be constant over time?
The Gordon growth model assumes dividends will increase at the same rate forever. That's a difficult assumption to meet, especially given the volatility of earnings.

If a firm has an average dividend growth rate that is close to a stable growth rate, the GGM can be used with little impact on intrinsic value. Therefore, cyclical firms expected to have y/y swings in growth rates but with steady averages can be valued using the Gordon growth model without a significant loss of generality.

The rationale lies in the logic that since dividends are smoothed even when earnings are volatile, they are less likely to be affected by y/y changes in earnings growth. Secondly, the mathematical effects of using an average growth rate rather than a constant growth rate are small enough to essentially be negligible.

Other Things to Consider with the Gordon Growth Model:

• The company is, by default, a good value if the dividend growth rate is greater than or equal to your required rate of return.

• For companies that have a strong history of dividend growth, many analysts will assume the historic growth rate will continue unless the company has stated otherwise.

• If the company has not declared a dividend for next year, many analysts will assume it will grow from last year’s dividend at a rate consistent with its historical growth rate, unless the company has stated otherwise. (Motley Fool, 2016)

Issues with the Gordon Growth Model:

The first limitation of the Gordon growth model is in its sensitivity to its inputs.

The Gordon growth model is a simple and convenient way of valuing stocks but it is extremely sensitive to the inputs for the growth rate. Used incorrectly, it can yield misleading or even absurd results, since, as the growth rate converges on the discount rate, the value goes to infinity (Damodaran, 2012).

As the growth rate approaches the cost of equity, the value per share approaches infinity. If the growth rate exceeds the cost of equity, the value per share becomes negative (Damodaran, 2012).

Let’s look at an example of how a decrease in the dividend growth rate can disproportionately lower the intrinsic value of the stock. Walmart (WMT) is expected to pay a dividend of $2.98 in 2019. With a dividend growth rate at 5% and a required rate of return of 7% the intrinsic value of WMT is$149.00. If the dividend growth rate decreases by 10% to 4.5% the intrinsic value drops to $119.20. That’s a 20% decrease in value from a 10% decrease in the dividend growth assumption. (Chen, 2019) Proctor & Gamble Co. (PG) is one of the best dividend stocks in U.S. history. PG has increased their dividend a staggering 62 years in a row with no apparent signs of slowing down. Proctor & Gamble quarterly dividends over time (Source: P&G Investor Relations) Even in a great dividend stock like this, the growth rate is not consistent. Here is Proctor & Gamble’s dividend increase in percent over time: PG is a behemoth (market cap$250B+) and a very solid dividend king and still its dividends are not consistent. Last year the dividend increased 4.0%. PG has, on average, increased their dividend:

• Over the last 5 years: 3.0%

• Over the last 10 years: 5.5%

• Over the last 20 years: 8.1%

• All time: 9.7%

When plugging growth rates into the GGM, should we expect another year at 4% growth? Or is it more likely to revert to it historical average of 9.7%? If management is (hypothetically) saying a dividend increase of 5% can be expected for next several years, do you trust them? How accurate is their track record? Further, if you do trust the 5%, how long do you think it’s sustainable? Maybe forever, but maybe not. These are the kinds of questions you will have to answer when using the Gordon Growth Model.

For more on how to calculate sustainable growth rates see Appendix B.

Example:

Let’s consider Public Service Enterprise Group (PEG). PEG is “an energy company with a diversified business mix [whose] operations are located primarily in the Northeastern and Mid-Atlantic United States.” (PEG, 2018) We will be analyzing PEG as of 12/31/2018.

Rationale for using the model:

• PSE&G is in stable growth; based upon size and the area that it serves. Its rates are also regulated. It is unlikely that the regulators will allow profits to grow at extraordinary rates.

• The firm is in a stable business and regulation is likely to restrict expansion into new businesses.

• PEG is in relatively stable financial leverage (2.15 debt/equity)

• PEG pays out dividends that are roughly equal to FCFE.

• Average Annual FCFE from 2016-2018 = $918.50 million • Average Annual Dividends from 2016 to 2018 =$849.75 million

• Dividends as % of FCFE = 92.51%

• Background Information:

• Earnings per share in 2018 = $2.83 • Dividend Payout Ratio in 2018 = 63.60% • Dividends per share in 2018 =$1.80

• Return on Equity = 10.00%

• Cost of Equity (using CAPM):

• PEG Beta = 0.44

• Risk-Free Rate = 2.66%

• Market Risk Premium = 7.61%

• Cost of Equity = 2.66 + .44 * (10.27 - 2.66) = 6.01%

• We estimate the expected growth rate from fundamentals:

• Method 1: Expected Growth Rate = (1 - Payout Ratio) * ROE

• Expected Growth Rate = (1-.6360) * .1000 = 3.64%

• Method 2: Expected Growth Rate = PRAT

• Expected Growth Rate = 0.1483 * 0.364 * 0.2139 * 3.152 = 3.64%

We can now calculate an intrinsic value of PEG as:

Intrinsic Value of Stock= ($1.80 * 1.0364) / (0.0601 - 0.0364) IV =$1.87 / 0.0237
IV = $78.71 On 12/31/2018 PEG was trading at$52.05, meaning that PEG may have been undervalued.

Embedded below is a Gordon growth model from Excel for you to play around with. It’s pre-loaded with the PEG data we used above.

Our value for PEG is different from the market price, and this is likely to be the case with almost any company that you value.

There are three possible explanations for this deviation. One is that you are right and the market is wrong. While this may be the correct explanation, you should probably make sure that the other two explanations do not hold – that the market is right and you are wrong or that the difference is too small to draw any conclusions (Damodaran, 2012).

To examine the magnitude of the difference between the market price and your estimate of value, you can hold the other variables constant and change the growth rate in your valuation until the value converges on the price (Damodaran, 2012).

The implied growth rate that provides the current price:

$52.05 =$1.87 * (1 + g) / (6.01% - g)

Solving for g:

g = (.0601 * 52.05 - 1.87) / (52.05 + 1.87) = 2.33%

Since we estimated growth from fundamentals, we can use the implied growth rate to estimate an implied return on equity.

• Implied Return on Equity = Implied Growth Rate / Retention Ratio

• ROE = Net Income / Shareholder Equity

• Retention Ratio = g / ROE

Implied Return on Equity = .0233% / (0.0364 / ($1,438 /$14,377))
Implied Return on Equity = .0233% / (0.0364 / .1000)
Implied Return on Equity = .0233% / .364
Implied Return on Equity = 6.40%

The document below is an interactive excel chart. (The chart is much easier to see and use on a computer. I apologize to mobile users.)

The horizontal green line in the chart below is the current stock price and the blue line is the intrinsic value of a stock at various dividend growth rates.

The red line is the implied growth rate calculated above. It can be thought of as the “justification growth rate.” That is, the dividend growth rate required to justify the stock’s current price. We can see that at price of $52.50, the GGM would signal this to be a good buy if 2.33%+ dividend increases could be expected forever. Play around with the P, D, and r variables to see the implied growth rate for different stocks. The Gordon growth model doesn’t have the flexibility of variability in dividend increases. The number you input forces the model to assume that the dividend will grow at that rate forever, a tough task. As we will see later in the post, other dividend discount models allow for different periods of growth (i.e. I think the dividend will grow at 10% for 5 years and then 5% every year thereafter). ### The Variable Growth Model(The Multistage Model) The zero-growth and constant-growth models do not allow for any shift in expected growth rates. The multistage dividend discount model builds on the constant-growth model by applying varying growth rates to the calculation. Variable growth DDM’s are much closer to reality, by assuming that the company will experience different growth phases. (Vaidya, 2019) It is assumed in a variable-growth dividend discount model, that the investor is holding the stock through all of the periods. (CFI, n.d.-b) There are multiple versions of the the variable growth DDM. Some have two stages, some are so complicated as to assume growth rates for each year individually. In this post we’ll be focusing on the three most common multistage dividend discount models: the ### Two-Stage Model The two-stage dividend discount model allows for only two stages of growth: an initial phase with non-stable growth, and a subsequent steady state where the growth rate is stable and is expected to remain so for the long term. While, in most cases, the growth rate during the initial phase is higher than the stable growth rate, the model can be adapted to value companies that are expected to post low or even negative growth rates for a few years and then revert back to stable growth (Damodaran, 2012). Value of the Stock = PV of Dividends during extraordinary phase + PV of terminal price. $$P_0=\sum_{t=1}^{t=n}\frac{\text{DPS}_t}{(1+\text{k}_{e,hg})^t}+\frac{\text{P}_n}{(1+\text{k}_{e,hg})^n}\text{where P}_n = \frac{\text{DPS}_{n+1}}{(\text{k}_{e,st}-\text{g}_n)}$$ where, • DPSt = Expected dividends per share in year t • ke = Cost of Equity • hg: High Growth period • st: Stable Growth period • Pn = Price (terminal value) at the end of year n • g = Extraordinary growth rate for the first n years • gn = Steady state growth rate forever after year n In the case where the extraordinary growth rate (g) and payout ratio are unchanged for the first n years, this formula can be simplified. $$\text{P}_0=\frac{\text{DPS}_0*(1+\text{g})*\Bigl(1-\frac{(1+\text{g})^n}{(1+\text{k}_{e,hg})^n}\Bigr)}{\text{k}_{e,hg}-\text{g}}+\frac{\text{DPS}_{n+1}}{(\text{k}_{e,st}-\text{g}_n)(1+\text{k}_{e,hg})^n}$$ where the inputs are as defined above. Similar to the Gordon growth rate model, the terminal growth rate for the firm (gn) must be comparable to the nominal growth rate in economy. In addition, the payout ratio has to be consistent with the estimated growth rate. If the growth rate is expected to drop significantly after the initial growth phase, the payout ratio should be higher in the stable phase than in the growth phase. A stable firm can pay out more of its earnings in dividends than a growing firm (Damodaran, 2012). One way to estimate this new payout ratio is: Expected Growth = Retention Ratio * Return on Equity. Manipulating the formula yields the following stable period payout ratio: $$\text{Stable payout ratio} = \frac{\text{Stable growth rate}}{\text{Stable period return on equity}}$$ For example, a firm with a 6% stable growth rate and an ROE of 16% will have a stable payout ratio of 37.50%. “The other characteristics of the firm in the stable period should be consistent with the assumption of stability” (Damodaran, 2012). For example, it’s reasonable to assume that a firm in a high-growth period may have a beta of 2.0. It’s unreasonable, however, to assume that beta would remain unchanged when the firm enters the stable period. For reference, stable period betas are most frequently between 0.8 and 1.2. Similarly, the firm’s ROE, which can be higher during the initial growth phase, should lower to levels commensurate with a firm in the stable growth phase. For reference, to estimate a stable ROE, you can use the industry average or the firm’s average during previous stable growth periods. Limitations of the Two-Stage Model One of the bigger challenges with the two stage model is in defining the length of the extraordinary growth period. By definition, the value of an investment will be greater the longer the period of extraordinary growth. However, it is difficult to convert qualitative considerations into a specific time period to accurately encompass the high-growth period. The second challenge with the two-stage model lies in the assumption that “the growth rate is high during the initial period and is transformed overnight to a lower stable rate at the end of the period” (Damodaran, 2012). While it is possible for a firm to experience this sudden transformation of growth rates, it’s much more realistic, and much more common, to assume that the shift from the high to low growth rates happens gradually over time. This issue will be addressed with the H-Model and Three-Stage Model below. Finally, the focus on dividends in this model can “lead to skewed estimates of value for firms that are not paying out what they can afford in dividends.” (Damodaran, 2012) Specifically, this model will underestimate the value of firms that accumulate cash and pay out comparatively little in dividends. Works Best For: Since the two-stage dividend discount model is based upon two clearly delineated growth stages, high growth and stable growth, it is best suited for firms which are in high growth and expect to maintain that growth rate for a specific time period, after which the sources of the high growth are expected to disappear. One scenario, for instance, where this may apply is when a company has patent rights to a very profitable product for the next few years and is expected to enjoy super-normal growth during this period. Once the patent expires, it is expected to settle back into stable growth. Another scenario where it may be reasonable to make this assumption about growth is when a firm is in an industry which is enjoying super-normal growth because there are significant barriers to entry (either legal or as a consequence of infra-structure requirements), which can be expected to keep new entrants out for several years (Damodaran, 2012). The assumption that the growth rate drops precipitously from its level in the initial phase to a stable rate also implies that this model is more appropriate for firms with modest growth rates in the initial phase. For instance, it is more reasonable to assume that a firm growing at 12% in the high growth period will see its growth rate drops to 6% afterwards than it is for a firm growing at 40% in the high growth period (Damodaran, 2012). Finally, the model works best for firms that maintain a policy of paying out most of residual cash flows – i.e, cash flows left over after debt payments and reinvestment needs have been met – as dividends (Damodaran, 2012). Example: I think a good candidate for the two stage model may be Amgen, Inc. (AMGN). Amgen is an American biopharmaceutical firm offering products for the treatment of oncology, hematology, cardiovascular, inflammation, bone health, and neuroscience. Amgen completed two acquisitions in 2012 and 2013 and have integrated them quite successfully, strengthening their market share in oncology. Amgen, in 2018, released the first FDA-approved treatment for migraines by blocking the calcitonin gene-related peptide (CGRP). That’s only one example. “In the past five years, Amgen has launched nine products, including two in new therapeutic areas… Amgen has several interesting candidates in its pipeline, which represent a significant commercial potential… Amgen also has an intriguing lineup of early and mid-stage programs, which can contribute to growth in the long term.” (Zacks, 2019) A Rationale for using the Model: • Why two-stage? Amgen has short-term protection from patents but over the next decade, biosimilars are going to pose a larger and larger threat. Two of Amgen’s larger products, Enbrel and Repatha, have faced softness in sales. Finally, the FDA requested additional safety information for Evenity and Kanjinti, delaying their release. It should be noted that Amgen was able to quickly and sufficiently prove the safety of these two therapies. We will assume that the firm will continue to grow but restrict the growth period to 5 years. • Why dividends? Amgen has a reputation for paying high dividends (current yield of 3.02%, 22.94% CAGR dividend increase) and it has not accumulated large amounts of cash over the last decade ($27.03B to $29.3B from 2014 to 2019). Background Information: • Earnings per share in 2018:$12.70
• Dividends per share in 2018: $5.64 • Payout Ratio in 2018:$5.64 / $12.70 = 44.38% • Return on equity 2013-2018 average: 29.11% Cost of Equity: • AMGN Beta: 1.26 (10-year beta vs. S&P 500) • Risk Free Rate: 3.02% (30 Year Treasury Yield as of 12/31/2018) (U.S. Department of the Treasury) • Market Return: 10.27% (Historical Return 1926-2017) (Vanguard, n.d.) • Cost of Equity: 3.02% + 1.26 * (10.27% - 3.02%) = 12.155% To estimate the expected growth in earnings per share over the five-year high growth period, we use the retention ratio in the most recent financial year (2018) and the return on equity of 29.11%. Expected growth rate = Retention ratio * Return on Equity Expected Growth Rate = (1 - 0.4438) * 0.2911 Expected Growth Rate = 16.19% In the stable growth period, we assume the beta will regress to 1, lowering the cost of equity to 10.27%. We'll assume the growth rate of the firm will be 3.16% (The average US growth rate from 1947 to 2019) (FRED, n.d.) and that the ROE will increase to 41.62%. The retention ratio in stable growth is calculated: Retention ratio in stable growth g / ROE Retention ratio = 3.16% / 41.62% Retention ratio = 7.59% The payout ratio in the stable period is therefore 92.41% Estimating the value: The first component of value is the present value of the expected dividends during the high growth period. Based upon the current earnings ($12.70), the expected growth rate (16.19%) and the expected dividend payout ratio (44.38%), the expected dividends can be computed for each year in the high growth period.

Year EPS DPS Present Value
1 $14.76$6.55 $5.84 2$17.15 $7.61$6.05
3 $19.92$8.82 $6.26 4$19.92 $10.27$6.49
5 $26.90$11.94 $6.72 Sum$31.35

The present value is computed using the cost of equity of 12.19% for the high-growth period.

The cumulative present value of dividends during high growth = $31.35. The present value of the dividends can be computed: $$\text{PV of Dividends} = \frac{(5.64 * (1 + .1619))\Bigl(1 - \frac{(1 + .1619)^5}{1+.12155}\Bigr)}{(0.12155-0.1619)} = 31.35$$ The price (terminal value) at the end of the high growth phase (the end of year 5) can be estimated using the constant growth model. $$\text{Terminal Price} = \frac{\text{Expected Dividends per share}_ {n+1}}{\text{k}_{e,st}-\text{g}_n}$$ Expected Earnings per share =$12.70 * (1.1619^5) * 1.0316 = $27.74 Expected Dividends per share = EPS * Stable Period Payout Ratio Expected Dividends per share =$27.74 *0.9241 = $25.64 Terminal Price =$25.64 / (0.1027 -.0316) = $360.58 The present value of the terminal price is: $$\text{PV of Terminal Price} = \frac{360.58}{(1.12155)^5}= 202.88$$ The cumulated present value of dividends and the terminal price can then be calculated. $$P_0 = \frac{(5.64 * (1 + .1619))\biggl(1 - \frac{(1 + .1619)^5} + \frac=31.35+202.88= 234.26$$ AMGN was trading at$194.67 on 12/31/2018 (the time period of this analysis). The two-stage model would indicate that at that time, AMGN was undervalued.

Alright, that was a lot of math, but we finally got our answer. Now instead of recreating all of those lines of math you can use the excel model below.

### H Model

The H Model is a two-stage model for growth, but unlike the two-stage model above, the initial high-growth stage is not constant but declines linearly over the high-growth period until the stable growth rate is reached. This model was published by Fuller and Hsia in the Financial Analysts Journal in 1984 (Fuller & Hsia, 2018).

The principal assumption is that the earnings growth rate starts at an initially high rate (ga) and declines linearly over the extraordinary growth period (which is assumed to last 2H periods) to a stable growth rate (gn). It also assumes that the dividend payout and cost of equity are constant over time and are not affected by the shifting growth rates.

The value of the expected dividends in the H Model can be written:

$$P_0 = \frac$$

where:

• P0 = Intrinsic value of the firm per share
• DPSt = DPS in year t
• ke = Cost of equity
• ga = Growth rate initially
• gn = Growth rate at end of 2H years for infinity

Limitations of the H Model:

One of the limitations is the precipitous drop in the growth rate between the high and stable periods. This model avoids that issue but does so at the cost of of assuming the growth rate will decline at a steady, linear pace. This is generally more forgiving to small deviations but large deviations can cause problems.

The other major limitation of this model is the assumption that the payout ratio will remain constant through both phases of growth. Typically we would assume that as growth rates decline, payout ratios increase.

Works best for:

The allowance for a gradual decrease in growth rates over time may make this a useful model for firms which are growing rapidly right now, but where the growth is expected to decline gradually over time as the firms get larger and the differential advantage they have over their competitors declines. The assumption that the payout ratio is constant, however, makes this an inappropriate model to use for any firm that has low or no dividends currently. Thus, the model, by requiring a combination of high growth and high payout, may be quite limited in its applicability (Damodaran, n.d.).

Proponents of the model would argue that using a steady state payout ratio for firms which pay little or no dividends is likely to cause only small errors in the valuation (Damodaran, n.d.).

### Three-Stage Model

The three-stage dividend discount model is a combination of the two-stage and H models. The three stage models starts with a period of high growth, a transitional period of declining growth and then a third period of low and stable growth forever. The three-stage model is the most general because it does not impose any restrictions on the payout ratio.

The value of the stock is the present value of the expected dividends during the high growth and transitional periods and the terminal price at the start of the stable growth phase. Or, mathematically:

$$P_0 = \sum_$$

where:

• EPSt = Earnings per share in year t
• DPSt = Dividends per share in year t
• ga = Growth rate in high growth phase (lasts n1 periods)
• gn = Growth rate in stable growth phase
• a = Payout ratio in high growth phase
• n = Payout ratio in stable growth phase
• ke = Cost of equity in high growth (hg), transition (t), and stable growth (st)

Assumptions

The three stage model removes many of the limitations imposed by other variations of the dividend discount model. In return, however, the three-stage model requires many more inputs including year-specific payout ratios, growth rates and betas.

For firms where there is substantial noise in the estimation process, the errors in these inputs can overwhelm any benefits that accrue from the additional flexibility in the model (Damodaran, 2012).

Works best for:

The flexibility of this model make it a good choice for many firms that are expected to have changes in growth rates, payout policies, or risk.

It is best suited for firms which are growing at an extraordinary rate now and are expected to maintain this rate for an initial period, after which the differential advantage of the firm is expected to deplete leading to gradual declines in the growth rate to a stable growth rate (Damodaran, 2012).

Example:

I think a good company for the three stage model is Comcast (CMCSA). Comcast is the largest cable provider in the United States and that is likely what most Americans associate them with. However, Comcast’s assets are far deeper and further reaching than that.

• Comcast owns NBCUniversal which operates the broadcast networks NBC and Telemundo as well as dozens of other cable networks including MSNBC, CNBC, the Golf Channel, E!, and USA Network. NBCUniversal had a 33% stake in Hulu which was recently sold to Disney (DIS).

• Comcast owns DreamWorks Animation which has released massive animated films such as Shrek, Madagascar, Kung-Fu Panda, How to Train Your Dragon, and The Boss Baby.

• Comcast owns Spectacor, a sports entertainment company, that owns the Philadelphia Flyers NHL team.

• Comcast has also partnered with Verizon to launch Xfinity mobile a mobile carrier and cell-service provider.

I believe Comcast has a large core group of users that are unlikely to ditch cable in the next few years even as more streaming providers enter the market. Comcast owns several business that can be expected to generate steady cash for the next several years.

Why the Three-Stage Model?

Comcast’s recent acquisitions and venture capital arm have the potential to push the company with several years of above average growth. However, I maintain that the future is still about streaming. Xfinity on Demand is one of the largest streaming providers but does not create its own content. As companies pull their content for exclusive use on their own services Comcast will be left with infrastructure but few movies and TV shows to offer. In Comcast’s corner, there is no good alternative to cable for watching live sports. I think Comcast will maintain this benefit for some years before being dethroned by a startup.

It is my belief that Comcast will begin to lose its competitive advantages and then slowly begin to lose customers and market share until it reaches a stable level where it likely to stay for infinity (for the purposes of the model).

Background Information:

• Earnings per share in 2018: $2.64 • Dividends per share in 2018:$0.78
• Payout Ratio in 2018: $0.78 /$2.64 = 29.55%
• Return on equity 2013-2018 average: 19.42%

Cost of Equity (High-Growth Period):

• CMCSA Beta: 0.99 (10-year beta vs. S&P 500)
• Risk Free Rate: 3.02% (30 Year Treasury Yield as of 12/31/2018) (U.S. Department of the Treasury)
• Market Return: 10.27% (Historical Return 1926-2017) (Vanguard, n.d.)
• Cost of Equity: 3.02% + 0.99 * (10.27% - 3.02%) = 10.20%

Cost of Equity (Stable-Growth Period):

• CMCSA Beta: 1.22 (Telecommunication Services Sector Avg Beta) (Damodaran, 2019)
• Risk Free Rate: 3.02% (30 Year Treasury Yield as of 12/31/2018) (U.S. Department of the Treasury)
• Market Return: 10.27% (Historical Return 1926-2017) (Vanguard, n.d.)
• Cost of Equity: 3.02% + 1.22 * (10.27% - 3.02%) = 11.87%

During the transition period, the cost of equity will increase/decrease lineraly from ke,hg to ke,st. (In this case the cost of equity would increase lineraly from 10.20% ro 11.87%).

The expected growth rate during the high-growth period is estimated using the current ROE of 19.42% and current payout ratio of 29.55%.

Expected Growth Rate = Retention Ratio * ROE
EGR = (1 - 29.55%) * (19.42%) = 13.68%

During the transition period, the growth rate will decline lineraly from 13.68% to 3.16% (The average US growth rate from 1947 to 2019) (FRED, n.d.). To estimate the payout ratio in stable growth, we assume a ROE of 11.45% (GuruFocus, 2019).

Stable Period Payout Ratio = 1 - (g / ROE)
PR = 1 - (3.16% / 11.45%) = 72.40%

During the transition phase, the payout ratio increases linerarly from 29.55% to 72.40%.

Year Expected Growth EPS Payout Ratio DPS Cost of Equity Present Value
High-Growth Stage
1 13.68% $3.00 29.55%$0.89 10.18% $0.80 2 13.68%$3.41 29.55% $1.01 10.18%$0.83
3 13.68% $3.88 29.55%$1.15 10.18% $0.86 4 13.68%$4.41 29.55% $1.30 10.18%$0.88
5 13.68% $5.01 29.55%$1.48 10.18% $0.91 Transition Stage 6 12.63%$5.65 33.83% $1.91 10.34%$1.07
7 11.58% $6.30 38.12%$2.40 10.51% $1.21 8 10.53%$6.96 42.40% $2.95 10.68%$1.35
9 9.47% $7.62 46.67%$3.56 10.85% $1.47 10 8.42%$8.26 50.97% $4.21 11.02%$1.56
11 7.37% $8.87 55.26%$4.90 11.19% $1.64 12 6.32%$9.43 59.54% $5.62 11.36%$1.68
13 5.26% $9.93 63.83%$6.34 11.53% $1.70 14 4.21%$10.35 68.12% $7.05 11.70%$1.69
15 3.16% $10.68 72.40%$7.73 11.87% $1.86 Note: Since the costs of equity changes each year, the present value has to be calculated using the cumulated cost of equity. For example in year 9, the present value of dividends is: $$\text{PV of Year 9 Dividends} = \frac = 1.47$$ The terminal price at the end of year 15 can be calculated based upon the earnings per share in year 16, the stable growth rate of 3.16%, a cost of equity of 11.87%, and a payout ratio of 72.40%. $$\text{Terminal Price} = \frac{0.1187 - 0.0316} = 88.79$$ The components of value are as follows: • Present Value of Dividends in High Growth Phase:$4.29

• Present Value of Dividends in Transition Phase: $15.23 • Present Value of Terminal Price at End of Transition:$21.35

• Value of CMCSA Stock: $40.86 Comcast (CMCSA) was trading at$34.05 on 12/31/2018.

Embedded below is a three-stage dividend discount model in Excel. Play around with the blue cells and a stock pick of your own or download the document for future reference.

## Issues in Using the Dividend Discount Model

The primary attraction to the dividend discount model lies in its simplicity and its intuitive logic. The DDM remains however the target of some analysts who are incredulous because of perceived limitations. The DDM, some claim, is not actually useful in stock valuation except in a few cases of extraordinary stable, high-dividend paying, blue chip stocks. This section will examine of the areas where the dividend discount model is perceived to fall short.

1) The DDM is not able to properly value non-dividend paying or low dividend paying stocks.

Some analysts claim that the DDM can not be used to value a stock that pays low or no dividends. In fact, as long as the dividend payout ratio is adjusted to reflect changes in the expected growth rate, a reasonable value can be obtained even for non-dividend paying firms.

Thus, a high-growth firm, paying no dividends currently, can still be valued based upon dividends that it is expected to pay out when the growth rate declines. If the payout ratio is not adjusted to reflect changes in the growth rate, however, the dividend discount model will underestimate the value of non-dividend paying or low-dividend paying stocks (Damodaran, 2012).

2) The DDM is too conservative in estimating value.

A common critique of the DDM is that its estimation of value is consistently too conservative. This criticism is based on the notion that value is determined by more than the present value of expected dividends.

For instance, it is argued that the dividend discount model does not reflect the value of 'unutilized assets'. There is no reason, however, that these unutilized assets cannot be valued separately and added on to the value from the dividend discount model. Some of the assets that are supposedly ignored by the dividend discount model, such as the value of brand names, can be dealt with simply within the context of the model (Damodaran, 2012).

A more legitimate criticism of the model is that it does not incorporate other ways of returning cash to stockholders (such as stock buybacks). If you use the modified version of the dividend discount model, this criticism can also be countered (Damodaran, 2012).

3) The DDM is contrarian by nature.

The dividend discount model is considered by many to be a contrarian model. As the market rises, fewer and fewer stocks, they argue, will be found to be undervalued using the dividend discount model.

This is not necessarily true. If the market increase is due to an improvement in economic fundamentals, such as higher expected growth in the economy and/or lower interest rates, there is no reason, a priori, to believe that the values from the dividend discount model will not increase by an equivalent amount. If the market increase is not due to fundamentals, the dividend discount model values will not follow suit, but that is more a sign of strength than weakness. The model is signaling that the market is overvalued relative to dividends and cashflows and the cautious investor will pay heed (Damodaran, 2012).

## Tests of the Dividend Discount Model

The ultimate test of a model lies in how well it works at identifying undervalued and overvalued stocks.

The dividend discount model has been tested and the results indicate that it does, in the long term, provide for excess returns. It is unclear, however, whether this is because the model is good at finding undervalued stocks or because it proxies for well- know empirical irregularities in returns relating to price-earnings ratios and dividend yields (Damodaran, 2012).

Sorensen and Williamson conducted a simple study of the DDM, where they valued 150 stocks from the S&P 400 in December 1980.

They used the difference between the market price at that time and the model value to form five portfolios based upon the degree of under or over valuation. The returns on these five portfolios were estimated for the following two years (January 1981-January 1983) and excess returns were estimated relative to the S&P 500 Index using the betas estimated at the first stage and the CAPM (Damodaran, 2012).

The most undervalued portfolio had a positive excess return of 16% per year, while the most overvalued portfolio had a negative excess return of 15% per year.

Many studies have replicated results showing that identifying undervalued stocks generates positive alpha on a risk adjusted basis. However, there are some caveats to be aware of before generalizing the results. “The dividend discount model outperforms the market over five-year time periods, but there have been individual years where the model has significantly under performed the market” (Damodaran, 2012).

Further, it is unclear how much the model adds in value to investment strategies that use PE ratios or dividend yields to screen stocks. Jacobs and Levy (2016) indicate that the marginal gain is relatively small.

Attribute Average Excess Return per Quarter
Dividend Discount Model 0.06% per quarter
Low P/E Ratio 0.92% per quarter
Book / Price 0.01% per quarter
Cashflow / Price 0.18% per quarter
Sales / Price 0.96% per quarter
Dividend Yield -0.51% per quarter

This suggests that using low PE ratios to pick stocks adds 0.92% to your quarterly returns, whereas using the dividend discount model adds only a further 0.06% to quarterly returns. “If, in fact, the gain from using the dividend discount model is that small, screening stocks on the basis of observables (such as PE ratio or cashflow measures) may provide a much larger benefit in terms of excess returns” (Damodaran, 2012).

Ping McLemore, George Woodward, and Tom Zwirlein wrote a paper called “Back-tests of the Dividend Discount Model Using Time-varying Cost of Equity” published in the Journal of Applied Finance in 2015. The trio examined stocks from 1989 to 2008 using the DDM in an attempt to measure prediction error.

One of the primary conclusions is that the DDM appears to undervalue stocks on average. This relates to Issue 2 in the section above.

Shorter duration models on average underestimate actual stock prices by 8.8% or more. The underestimation rises to 54.1% in models based on longer streams of dividends. Another way to view these results is that estimated equity discount rates are too high which leads to lower estimates of intrinsic value. For many firms in our sample, if the cost of equity estimates were lower, the intrinsic value estimates would be higher and prediction error would be reduced (McLemore, Woodward, & Zwirein, 2016).

Another key finding is that the DDM works best for shorter time periods, where less assumptions need to be made. When trying to estimate a cost of equity or payout ratio in 20 years, there are simply too many things that could lead that estimate to be wildly inaccurate.

Our findings indicate the DDM works better when intrinsic value estimates are made over relatively short periods from one to five years. Prediction errors in these shorter models vary from an average of 8.8% in one year time horizon models to 14.1% in five year models. Prediction error increases in models of six years and greater reaching a maximum of over 54% in the twenty year time horizon models we use in the analysis. The results suggest shorter estimation periods may be preferable to longer models (McLemore, Woodward, & Zwirein, 2016).

Tax Disadvantages from High Dividends

Investors who use the DDM heavily are likely to create portfolios with high dividend yields. High yields can create tax disadvantages if dividends are taxed at a rate greater than capital gains or if there is a substantial tax timing liability associated with dividends. The excess returns uncovered in the studies above are pre-tax. Personal taxes may significantly reduce or even eliminate excess returns.

In summary, the dividend discount model's impressive results in studies looking at past data have to be considered with caution. For a tax-exempt investment, with a long time horizon, the dividend discount model is a good tool, though it may not be the only one, to pick stocks. For a taxable investor, the benefits are murkier, since the tax consequences of the strategy have to be considered. For investors with shorter time horizons, the dividend discount model may not deliver on its promised excess returns, because of the year-to-year volatility in its performance (Damodaran, 2012).

## Appendix

### Appendix A: Cost of Equity Calculation

Cost of equity is the rate of return a shareholder requires for investing equity into a business.

The rate of return an investor requires is based on the level of risk associated with the investment, which is measured as the historical volatility of returns. A firm uses cost of equity to assess the relative attractiveness of investments, including both internal projects and external acquisition opportunities. Companies typically use a combination of equity and debt financing, with equity capital being more expensive. (CFI, n.d.-a)

Cost of equity can be calculated using the the dividend capitalization model, and the capital asset pricing model (CAPM) both explained briefly below.

The Dividend Capitalization Model:

The dividend capitalization model has the advantage of being simple and straightforward. It’s limited by the fact that it can only work on dividend-paying companies and also assumes that dividends will grow at a constant rate forever, a dangerous assumption. The dividend capitalization model is also very sensitive to changes in the dividend growth rate, and does not consider the riskiness of the investment.

The formula for the dividend capitalization is:

ke = (D1 / P0) + g

Where:

• ke = Cost of Equity
• D1 = Dividends per Share Next Year
• P0 = Current Share Price
• g = Dividend Growth Rate

Below is a dividend capitalization model in Excel with data on Coca Cola (KO). You can edit the D1, P0, and g cells to see the impact on the cost of equity.

Maybe you feel that looking at a company in a vacuum isn’t effective and you would rather use market conditions to estimate your cost of equity. If so you would want to use the CAPM.

The Capital Asset Pricing Model (CAPM):

The CAPM takes into account the riskiness of an investment relative to the overall market. However, because the model relies heavily on historical information, if the future is unlike the past, the model can be misleading. The formula for the CAPM is:

ERi = Rf + βi * (ERm - Rf)

Where:

• ERi = Expected Return of Investment
• Rf = Risk-Free Rate
• βi = Beta of the Investment
• ERm = Expected Return of the Market
• (ERm - Rf) = Market Risk Premium

There is much debate about what should be used as a risk-free rate and what should be used as market returns. Commonly used variables are the 10-year treasury yield and the long-term US equity returns:

• Rf = 2.07% (10 Year Treasury Yield) (U.S. Department of the Treasury)
• ERm = 10.27% (Historical Return 1926-2017) (Vanguard, n.d.)
• (ERm - Rf) = 8.20%

Example:

You can calculate required return with the CAPM by using the assumptions above and a stock’s beta. Apple Inc. (AAPL), for example, currently has a beta of 0.90. The CAPM would calculate a minimum required return of:

2.07% + 0.90 * (10.27% - 2.07%)
2.07 + 0.90 * (8.20%)
2.07% + 7.38%
Cost of Equity = 9.45%

Below is an embedded excel document of the CAPM with data on Coca Cola (KO). You can edit the β, Rf, and ERm cells to see the impact on the cost of equity.

After choosing either the dividend capitalization model or or capital asset pricing model, you would have your cost of equity, or your required minimum annual returns. If, for example, your goal as an investor is to produce an annual rate of return of 9.45% you would enter 9.45% (or .0945) in the formulas and excel tables above.

### Appendix B: Calculating Sustainable Growth Rates

Several formulas in this post as well as the the dividend capitalization model in Appendix A use a stock’s dividend growth rate as one of the inputs. How should you think about that value? You can calculate this using averages or fundamentals.

Sustainable Growth Rates - Using Averages

Using averages, there are two main ways to calculate a growth rate: the average annual growth rate (AAGR) and the compound annual growth rate (CAGR).

Using Coca-Cola’s actual dividends from 2014-2018, we can see how to calculate both the AAGR and the CAGR. (Note that in this case the two methods equal the same value. This is not always the case. Play around with the dividend row to see when AAGR’s and CAGR’s differ).

Maybe you feel that the dividend history of the company you are evaluating is not representative of your expectations of future performance. In that case, there are two other ways to calculate a sustainable future dividend growth rate. (Wilkinson, 2014)

Sustainable Growth Rates - Using Fundamentals

Formula 1:

SGR = (1-d) x ROE

Where:

• d = Dividend Payout Ratio (i.e. Dividends / Earnings)

• ROE = Return on Equity (i.e. Net Income / Shareholders’ Equity)

Example:

Costco (COST) paid $689 million in dividends in their fiscal year 2018 with a total net income of$3.134 billion. (Costco, 2018) Therefore COST, at the end of fiscal 2018, had a dividend payout ratio of 21.98%.

COST also had total stockholder equity of $12.799 billion for a return on assets of 24.49%. This method would give KO a sustainable growth rate of: SGR = (1.00 - 0.2198) * 0.2449 0.7802 * 0.2449 SGR = 19.11% A sustainable growth rate of 19.11% seems high but compared to COST’s 10-year dividend CAGR of 15.07% it may be more reasonable than at first glance. Formula 2: SGR = PRAT Where: • P = Profit Margin (i.e. Net Profit / Revenue) • R = Retention Rate (i.e. 1 - Dividend Payout Ratio) • A = Asset Turnover Ratio (i.e. Sales Revenue / Total Assets) • T = Assets-to-Equity Ratio (i.e. Total Assets / Shareholders’ Equity) To continue with COST: • Revenue =$141.576 billion

• Net Income = $3.134 billion • Dividend Payout Ratio = 21.98% • Total Assets =$40.380 billion

• Shareholders’ Equity = $12.799 billion So solving for PRAT: • Profit Margin = 2.21%. • Retention Rate = 78.02% • Asset Turnover Ratio = 3.51x • Assets-to-Equity Ratio = 3.15x Finally: SGR = 2.21% * 78.02 * 3.51 * 3.15 SGR = 19.06% Both methods for calculating SGR came within 0.05% of each other, which should lend some confidence to the calculation. Both measures are about ~4% higher than the 10-year dividend CAGR. If you feel Costco’s future looks brighter than it’s past, maybe one of these measures will serve you better than the AAGR or CAGR. ### Appendix C: How to Account for Stock Buybacks All of the models above use dividends as the sole way of returning cash to shareholders. In recent years, especially since the 1990’s, firms in the United States have increasingly turned to stock buybacks as a way to return cash to stockholders. What are the implications for the dividend discount model? What are the implications for the dividend discount model? Focusing strictly on dividends paid as the only cash returned to stockholders exposes us to the risk that we might be missing significant cash returned to stockholders in the form of stock buybacks (Damodaran, 2012). The simplest way to incorporate stock buybacks into a dividend discount model is to add them to the dividends and compute a modified payout ratio: $$\text{Modified dividend payout ratio} = \frac{\text{Dividends + Stock Buybacks}}{\text{Net Income}}$$ This adjustment is simple and straightforward. However, the resulting ratio for any one year can be highly skewed. Stock buybacks, unlike dividends, are not smoothed out. A firm may repurchase$5 billion in shares in one year and then not repurchase any for the next 4 years.

Consequently, a much better estimate of the modified payout ratio can be obtained by looking at the average value over a four or five year period. In addition, firms may sometimes buy back stock as a way of increasing financial leverage (Damodaran, 2012).

We can adjust for this by netting out new debt issued from the calculation above:

$$\text{Modified dividend payout ratio} = \frac{\text{Dividends + Stock Buybacks - Long Term Debt Issues}}{\text{Net Income}}$$

Adjusting the payout ratio to include stock buybacks will impact the ROE, the estimated growth rate and ultimately the terminal value of a stock.

The new growth rate is simply:

Modified Growth Rate = (1 - Modified Payout Ratio) * ROE

The book value of a stock is reduced by the market value of the repurchased shares. A firm that buys back a large quantity of stock can reduce it’s book value, and therefor increase their ROE, significantly. Using that ROE may overstate the value of the firm.

We can correct for this by adding back share repurchases to the book equity and recalculating the ROE. In doing so, we can sometimes yield a more reasonable ROE for new investments.

Consider our earlier valuation of Amgen, Inc. (AMGN) in the two-stage model above. We used dividends as the basis for our projections. Over the last 5 years, Amgen has bought back a significant amount of stock. The table below summarizes the dividends and buybacks over the last five years.

2018 2017 2016 2015 2014 Total
Net Income $8,394$1,979 $7,722$6,939 $5,158$30,192
Dividends $3,507$3,365 $2,998$2,396 $1,851$14,117
Buybacks $17,920$3,351 $2,910$1,785 ($48)$25,918
Dividends + Buybacks $21,427$6,716 $5,908$4,181 $1,803$40,035
Payout Ratio 41.78% 170.04% 38.82% 34.53% 35.89% 46.76%
Modified Payout Ratio 255.27% 339.36% 76.51% 60.25% 34.96% 132.60%

Summing up the total cash returned to shareholders, we have a modified payout ratio of 132.60%. If we substitute this payout ratio in the expected growth rate equation we calculate:

(Modified) Expected Growth Rate = (1 - Modified Payout Ratio) * ROE
=(1 - 1.326) * .2911
= -0.326 * 2911
Modified EGR = -0.0949
= -9.49%

Changing this value lowers the terminal value of the stock from $234.26 to$73.68. This is a dramatic reduction in value that attempts to reflect that Amgen Inc. may not have as many profitable new investments at a 30% ROE.

### Appendix D: Valuing A Market using the Dividend Discount Model

All of the dividend discount models above have been used to value a specific company. There’s no reason that the dividend discount model couldn’t be used to value a secotr, industry, or even the entire market.

To value a market we would need to make the following assumptions regarding the variables:

Old Variable New Variable
Market Price of Stock Cumulative market value of all stocks in the sector or market
Expected Dividends Cumulated dividends of all stocks (could be expanded to include stock buybacks)
Expected Growth Rate Growth rate in cumulated earnings of the index
Beta No need for beta. Use 1.00 because you're looking at a whole market.

You could use a two-stage model, where the growth rate is greater than the growth rate of the economy, but

you should be cautious about setting the growth rate too high or the growth period too long because it will be difficult for cumulated earnings growth of all firms in an economy to run ahead of the growth rate in the economy for extended periods (Damodaran, 2012).

Valuing the S&P 500 Using a Dividend Discount Model: December 31, 2018

As of December 31st 20018, the S&P 500 (^GSPC) was trading at $2,506.85. The dividend yield was 2.08% (YCharts, n.d.-b) and stock buybacks were 3.83% (YCharts, n.d.-a)(First Trust Advisors, 2019). Barron’s is expecting a dividend CAGR of 4.2% from 2017 to 2027. The 10-year treasury was 2.69% which comes out to a cost of equity of 10.27%. (U.S. Department of the Treasury) The expected dividends and stock buybacks on the index for the next 9 years can be estimated from the current dividends and expected growth of 4.2%. Current Yield = 5.91% * 2,506.85 =$148.15

Expected Dividends Present Value
1 $154.37$139.99
2 $160.86$132.29
3 $167.61$125.01
4 $174.65$125.01
5 $181.99$111.62
6 $189.63$105.48
7 $197.59$99.67
8 $205.89$94.19
9 $214.54$89.00

The present value is calculated by discounting back the dividends at the cost of equity of 10.27%. To estimate the terminal value, we estimate dividends in year 10 on the index:

Expected dividends in year 10 = $214.54 * (1.042) =$223.55
Terminal value of the index = $223.55 / (0.1027 - 0.042) =$3,682.88
PV of the terminal value = $3,682.88 / (1.042^9) =$2,543.19
Value of index = $1,015.38 +$2,543.19 = $3,558.57 Based upon this, we would have concluded that the index was undervalued at$2,506.85.

### Appendix E: Value of Growth

Investors pay a price premium when purchasing shares in companies with high growth potential. This premium takes the form of higher price-earnings or price-book value ratios.

While no one will contest the proposition that growth is valuable, it is possible to pay too much for growth. In fact, empirical studies that show low price-earnings ratio stocks earning return premiums over high price-earnings ratio stocks in the long term supports the notion that investors overpay for growth (Damodaran, 2012).

Here, we will use the two-stage dividend discount model to examine the value of growth and it provides a benchmark that can be used to compare the actual prices paid for growth.

Estimating the Value of Growth:

Value of Any Firm = Extraordinary Growth + Stable Growth + Assets in Place

Mathematically speaking,

$$P_0=\left(\frac{\text{DPS}_0*(1+g)*\left(1-\frac{(1+g)^n}{(1+\text{k}_{e,hg})^n}\right)}{\text{k}_{e,hg}-g}+\frac{\text{DPS}_{n+1}}{(\text{k}_{e,st}-g_n)(1+\text{k}_{e,hg})^n}-\frac{\text{DPS}_1}{(\text{k}_{e,st}-g_n)}\right) +\left(\frac{\text{DPS}_1}{(\text{k}_{e,st}-g_n)}-\frac{\text{DPS}_0}{\text{k}_{e,st}}\right)+\frac{\text{DPS}_0}{\text{k}_{e,st}}$$

Where:

• DPSt = Expected dividends per share in year t
• ke = Required rate of return
• Pn = Price at the end of year n
• g = Growth rate during high growth stage
• gn = Growth rate forever after year n

Value of Extraordinary Growth: Value of the firm with exrtaordinary growth in first n years - value of the firm as a stable growth firm

Value of Stable Growth: Value of the firm as a stable growth firm - value of firm with no growth

Assets in Place: Value of the firm with no growth

In making these estimates, though, we have to remain consistent. For instance, to value assets in place, you would have to assume that the entire earnings could be paid out in dividends, while the payout ratio used to value stable growth should be a stable period payout ratio. (Damodaran, 2012)

In our example of the two-stage model we found the intrinsic value of Amgen, Inc. (AMGN) to be $234.26 as of 12/31/2018. Using the formula above we can calculate the value of the assets in place, and the value of the stable growth. Subtracting these from the current stock price gives us the premium we are paying for extraordinary growth. We will first calculate the value of the assets in place using current earnings and assuming that all earnings are paid out as dividends. We will use the stable cost of equity as the discount rate. Value of Assets in Place = Current EPS / Stable Period Cost of Equity Value of Assets in Place =$12.70 / .1027 = $123.66 To estimate the cost of stable growth we will use the stable period growth rate, and the stable period payout ratio. Value of Stable Growth = (Current EPS * Stable Payout Ratio * (1 + Growth Rate)) / (Stable Period Cost of Equity - Growth Rate) - Value of Assets in Place Value of Stable Growth = (($12.70 * .9241 (1 + .0316)) / (.1027 - .0316)) - 123.66
Value of Stable Growth = $170.28 -$123.66 = $46.62 Therefore the value paid for growth is: Value of Extraordinary Growth = Current Stock Price - Value of Assets in Place - Value of Stable Growth Value of Extraordinary Growth =$234.26 - $123.66 -$46.62
Value of Extraordinary Growth = \$63.98

Determinations of the Value of Growth:

1. Growth rate during extraordinary period:

The higher the growth rate in the extraordinary period, the higher the estimated value of growth will be. “Conversely, the value of high growth companies can drop precipitously if the expected growth rate is reduced, either because of disappointing earnings news from the firm or as a consequence of external events” (Damodaran, 2012).

2. Length of the extraordinary growth period:

The longer the extraordinary growth period, the greater the value of growth will be.

3. Profitability of projects:

The profitability of projects determines both the growth rate in the initial phase and the terminal value. “As projects become more profitable, they increase both growth rates and growth period, and the resulting value from extraordinary growth will be greater” (Damodaran, 2012).

4. Riskiness of the firm/equity:

The riskiness of a firm determines the discount rate at which cashflows in the initial phase are discounted. “Since the discount rate increases as risk increases, the present value of the extraordinary growth will decrease” (Damodaran, 2012).

## References

### Works Cited:

• CFI. (n.d.-a). Cost of Equity - Formula, Guide, How to Calculate Cost of Equity. Retrieved from https://corporatefinanceinstitute.com/resources/knowledge/finance/cost-of-equity-guide/
• CFI. (n.d.-b). Dividend Discount Model - Definition, Formulas and Variations. Retrieved from https://corporatefinanceinstitute.com/resources/knowledge/valuation/dividend-discount-model/
• Chen, J. (2019, June 03). Dividend Discount Model – DDM. Retrieved from https://www.investopedia.com/terms/d/ddm.asp
• Costco. (2018, September 2). Costco - Form 10-K. Retrieved from https://www.sec.gov/Archives/edgar/data/909832/000090983218000013/cost10k9218.htm
• Damodaran, A. (2012). Chapter 13. In Investment valuation: Tools and techniques for determining the value of any asset (3rd ed., pp. 323-349). Hoboken, New J.: John Wiley et Sons.
• Damodaran, A. (2019, January). Betas by Sector (US). Retrieved from http://pages.stern.nyu.edu/~adamodar/New_Home_Page/datafile/Betas.html
• Damodaran, A. (n.d.). Dividend Discount Model. Retrieved from http://people.stern.nyu.edu/adamodar/pdfiles/ddm.pdf
• First Trust Advisors. (2019, March 26). S&P 500 Index Stock Buybacks. Retrieved from https://www.ftportfolios.com/retail/blogs/marketcommentary/index.aspx?ID=4719&Print=Y
• FRED. (n.d.). Real Gross Domestic Product. Retrieved from https://fred.stlouisfed.org/graph/?g=eUmi.
• Fuller, R. J., & Hsia, C. (2018, December 31). A Simplified Common Stock Valuation Model. Retrieved from https://tandfonline.com/doi/abs/10.2469/faj.v40.n5.49
• GuruFocus. (2019, March). Comcast ROE %. Retrieved from https://www.gurufocus.com/term/ROE/CMCSA/ROE/Comcast Corp
• Jacobs, B. I., & Levy, K. N. (2016, October 06). Disentangling Equity Return Regularities: New Insights and Investment Opportunities. Retrieved from https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1608012
• McLemore, P., Woodward, G., & Zwirein, T. (2016, September 16). Back-Tests of the Dividend Discount Model Using Time-Varying Cost of Equity. Retrieved from https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2838993
• Morningstar. (n.d.). Exxon Mobil Corporation (XOM). Retrieved from https://www.morningstar.com/stocks/xnys/xom/dividends
• Motley Fool. (2016, September 30). What is the Dividend Discount Model? Retrieved from https://www.fool.com/knowledge-center/what-is-the-dividend-discount-model.aspx
• PEG. (2018, December 31). Public Service Enterprise Group - Form 10-K. Retrieved from https://www.sec.gov/Archives/edgar/data/81033/000078878419000005/pseg201810k.htm
• U.S. Department of the Treasury. (n.d.). Daily Treasury Yield Curve Rates. Retrieved from https://www.treasury.gov/resource-center/data-chart-center/interest-rates/Pages/TextView.aspx?data=yield
• Vaidya, D. (2019, April 16). Dividend Discount Model. Retrieved from https://www.wallstreetmojo.com/dividend-discount-model/
• Vanguard. (n.d.). Historical Index Risk/Return (1926-2017). Retrieved from https://advisors.vanguard.com/VGApp/iip/advisor/csa/analysisTools/portfolioAnalytics/historicalRiskReturn
• Wilkinson, J. (2014, May 30). Sustainable Growth Rate. Retrieved from https://strategiccfo.com/sustainable-growth-rate/
• YCharts. (n.d.-a). S&P 500 Market Cap. Retrieved from https://ycharts.com/indicators/sandp_500_market_cap
• YCharts. (n.d.-b). S&P 500 Dividend Yield. Retrieved from https://ycharts.com/indicators/sandp_500_dividend_yield_ttm
• Zacks. (2019, July 22). Amgen Inc. (AMGN). Retrieved from https://research2.fidelity.com/fidelity/research/reports/pdf/getReport.asp?feedID=17129&docTag=AMGN&versionTag=3RGQFTAGD9D0MCI0MHVEMJNLE7

### Bibliography:

• http://www.pginvestor.com/Splits-and-Dividend-History
• https://www.treasury.gov/resource-center/data-chart-center/interest-rates/pages/textview.aspx?data=yield
• https://www.treasury.gov/resource-center/data-chart-center/interest-rates/pages/textview.aspx?data=yield
• https://www.investopedia.com/terms/c/capm.asp
• https://investinganswers.com/dictionary/c/capital-asset-pricing-model-capm
Erik Cianci