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2024 Curriculum CFA Program Level II Equity Investments


Common stock represents an ownership interest in a business. A business in its operations generates a stream of cash flows, and as owners of the business, common stockholders have an equity ownership claim on those future cash flows. Beginning with John Burr Williams (1938), analysts have developed this insight into a group of valuation models known as discounted cash flow (DCF) valuation models. DCF models—which view the intrinsic value of common stock as the present value of its expected future cash flows—are a fundamental tool in both investment management and investment research.

Although the principles behind discounted cash flow valuation are simple, applying the theory to equity valuation can be challenging. Four broad steps in applying DCF analysis to equity valuation are:

  • choosing the class of DCF model—equivalently, selecting a specific definition of cash flow;

  • forecasting the cash flows;

  • choosing a discount rate methodology; and

  • estimating the discount rate.

In our coverage of this topic, we take the perspective that dividends—distributions to shareholders authorized by a company’s board of directors—are an appropriate definition of cash flows. The class of models based on this idea is called dividend discount models, or DDMs. The basic objective of any DDM is to value a stock. The variety of implementations corresponds to different ways to model a company’s future stream of dividend payments. The steps of choosing a discount rate methodology and estimating the discount rate involve the same considerations for all DCF models, so they have been presented separately in an earlier discussion.

The sections are organized as follows: We first provide an overview of present value models. We then provide a general statement of the dividend discount model. Forecasting dividends, individually and in detail, into the indefinite future is not generally practicable, so the dividend-forecasting problem is usually simplified. One approach is to assign dividends to a stylized growth pattern. In the subsequent section, we focus on the simplest pattern—dividends growing at a constant rate forever (the constant growth or “Gordon growth” model). We then explain that for some companies, it is more appropriate to view earnings and dividends as having multiple stages of growth. We present multistage dividend discount models along with spreadsheet modeling. We lay out the determinants of dividend growth rates in the last section and conclude with a summary.

Learning Outcomes

The member should be able to:

  1. compare dividends, free cash flow, and residual income as inputs to discounted cash flow models and identify investment situations for which each measure is suitable;

  2. calculate and interpret the value of a common stock using the dividend discount model (DDM) for single and multiple holding periods;

  3. calculate the value of a common stock using the Gordon growth model and explain the model’s underlying assumptions;

  4. calculate the value of non-callable fixed-rate perpetual preferred stock;

  5. calculate and interpret the implied growth rate of dividends using the Gordon growth model and current stock price;

  6. calculate and interpret the present value of growth opportunities (PVGO) and the component of the leading price-to-earnings ratio (P/E) related to PVGO;

  7. calculate and interpret the justified leading and trailing P/Es using the Gordon growth model;

  8. describe strengths and limitations of the Gordon growth model and justify its selection to value a company’s common shares;

  9. explain the growth phase, transition phase, and maturity phase of a business;

  10. explain the assumptions and justify the selection of the two-stage DDM, the H-model, the three-stage DDM, or spreadsheet modeling to value a company’s common shares;

  11. describe terminal value and explain alternative approaches to determining the terminal value in a DDM;

  12. calculate and interpret the value of common shares using the two-stage DDM, the H-model, and the three-stage DDM;

  13. explain the use of spreadsheet modeling to forecast dividends and to value common shares;

  14. estimate a required return based on any DDM, including the Gordon growth model and the H-model;

  15. calculate and interpret the sustainable growth rate of a company and demonstrate the use of DuPont analysis to estimate a company’s sustainable growth rate;

  16. evaluate whether a stock is overvalued, fairly valued, or undervalued by the market based on a DDM estimate of value.


We have provided an overview of DCF models of valuation, discussed the estimation of a stock’s required rate of return, and presented in detail the dividend discount model.

  • In DCF models, the value of any asset is the present value of its (expected) future cash flows

    V 0 = t = 1 n CF t ( 1 + r ) t ,

    where V 0 is the value of the asset as of t = 0 (today), CF t is the (expected) cash flow at time t, and r is the discount rate or required rate of return. For infinitely lived assets such as common stocks, n runs to infinity.

  • Several alternative streams of expected cash flows can be used to value equities, including dividends, free cash flow, and residual income. A discounted dividend approach is most suitable for dividend-paying stocks in which the company has a discernible dividend policy that has an understandable relationship to the company’s profitability and the investor has a non-control (minority ownership) perspective.

  • The free cash flow approach (FCFF or FCFE) might be appropriate when the company does not pay dividends, dividends differ substantially from FCFE, free cash flows align with profitability, or the investor takes a control (majority ownership) perspective.

  • The residual income approach can be useful when the company does not pay dividends (as an alternative to a FCF approach) or free cash flow is negative.

  • The DDM with a single holding period gives stock value as

    V 0 = D 1 ( 1 + r ) 1 + P 1 ( 1 + r ) 1 = D 1 + P 1 ( 1 + r ) 1 ,

    where D 1 is the expected dividend at Time 1 and V 0 is the stock’s (expected) value at Time 0. Assuming that V 0 is equal to today’s market price, P 0, the expected holding-period return is

    r = D 1 + P 1 P 0 1 = D 1 P 0 + P 1 P 0 P 0 .

  • The expression for the DDM for any given finite holding period n and the general expression for the DDM are, respectively,

    V 0 = t = 1 n D t ( 1 + r ) t + P n ( 1 + r ) n and V 0 = t = 1 D t ( 1 + r ) t .

  • There are two main approaches to the problem of forecasting dividends. First, an analyst can assign the entire stream of expected future dividends to one of several stylized growth patterns. Second, an analyst can forecast a finite number of dividends individually up to a terminal point and value the remaining dividends either by assigning them to a stylized growth pattern or by forecasting share price as of the terminal point of the dividend forecasts.

  • The Gordon growth model assumes that dividends grow at a constant rate g forever, so that Dt = Dt– 1(1 + g). The dividend stream in the Gordon growth model has a value of

    V 0 = D 0 ( 1 + g ) r g , or V 0 = D 1 r g where r > g .

  • The value of non-callable fixed-rate perpetual preferred stock is V 0 = D/r, where D is the stock’s (constant) annual dividend.

  • Assuming that price equals value, the Gordon growth model estimate of a stock’s expected rate of return is

    r = D 0 ( 1 + g ) P 0 + g = D 1 P 0 + g .

  • Given an estimate of the next-period dividend and the stock’s required rate of return, the Gordon growth model can be used to estimate the dividend growth rate implied by the current market price (making a constant growth rate assumption).

  • The present value of growth opportunities is the part of a stock’s total value, V 0, that comes from profitable future growth opportunities in contrast to the value associated with assets already in place. The relationship is V 0 = E 1/r + PVGO, where E 1/r is defined as the no-growth value per share.

  • The leading price-to-earnings ratio (P 0/E 1) and the trailing price-to-earnings ratio (P 0/E 0) can be expressed in terms of the Gordon growth model as, respectively,

    P 0 E 1 = D 1 / E 1 r g = 1 b r g and P 0 E 0 = D 0 ( 1 + g ) / E 0 r g = ( 1 b ) ( 1 + g ) r g .

    The foregoing expressions give a stock’s justified price-to-earnings ratio based on forecasts of fundamentals (given that the Gordon growth model is appropriate).

  • The Gordon growth model may be useful for valuing broad-based equity indexes and the stock of businesses with earnings that are expected to grow at a stable rate comparable to or lower than the economy’s nominal growth rate.

  • Gordon growth model values are very sensitive to the assumed growth rate and required rate of return.

  • For many companies, growth falls into phases. In the growth phase, a company enjoys an abnormally high growth rate in earnings per share, called supernormal growth. In the transition phase, earnings growth slows. In the mature phase, the company reaches an equilibrium in which such factors as earnings growth and the return on equity stabilize at levels that can be sustained long term. Analysts often apply multistage DCF models to value the stock of a company with multistage growth prospects.

  • The two-stage dividend discount model assumes different growth rates in Stage 1 and Stage 2:

    V 0 = t = 1 n D 0 ( 1 + g S ) t ( 1 + r ) t + D 0 ( 1 + g S ) n ( 1 + g L ) ( 1 + r ) n ( r g L ) ,

    where gS is the expected dividend growth rate in the first period and gL is the expected growth rate in the second period.

  • The terminal stock value, Vn , is sometimes found with the Gordon growth model or with some other method, such as applying a P/E multiplier to forecasted EPS as of the terminal date.

  • The H-model assumes that the dividend growth rate declines linearly from a high supernormal rate to the normal growth rate during Stage 1 and then grows at a constant normal growth rate thereafter:

    V 0 = D 0 ( 1 + g L ) r g L + D 0 H ( g S g L ) r g L = D 0 ( 1 + g L ) + D 0 H ( g S g L ) r g L .

  • There are two basic three-stage models. In one version, the growth rate in the middle stage is constant. In the second version, the growth rate declines linearly in Stage 2 and becomes constant and normal in Stage 3.

  • In addition to valuing equities, the IRR of a DDM, assuming assets are correctly priced in the marketplace, has been used to estimate required returns. For simpler models (such as the one-period model, the Gordon growth model, and the H-model), well-known formulas may be used to calculate these rates of return. For many dividend streams, however, the rate of return must be found by trial and error, producing a discount rate that equates the present value of the forecasted dividend stream to the current market price.

  • Multistage DDM models can accommodate a wide variety of patterns of expected dividends. Even though such models may use stylized assumptions about growth, they can provide useful approximations.

  • Dividend growth rates can be obtained from analyst forecasts, statistical forecasting models, or company fundamentals. The sustainable growth rate depends on the ROE and the earnings retention rate, b: g = b × ROE. This expression can be expanded further, using the DuPont formula, as

    g = Net income Dividends Net income × Net income Sales × Sales Total assets × Total assets Shareholders' equity .

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