Under the DDM, the value of a common stock is the present value of all future dividends. If the stock is sold at some point in the future, its value at that time will be the present value of all future dividends. In fact, the buyer is paying for the remaining dividend stream. If a stock does not pay dividend for some early years, investors expect at some point in the future the firm will start to pay dividends. Thus, valuation of stocks paying no dividends uses the same DDM approach, except that some of the early dividends are 0.

Single Holding Period

Assume an investor wants to buy the stock, hold it for one year, and then sell it. To determine the value of the stock using DDM, we must estimate the dividend to be received during the period (D₁), the expected sale price at the end of the holding period (P₁), and the investor's required rate of return (r). Then:

V₀ = D₁/(1 + r) + P₁/(1 + r)

If we assume that the stock is fairly valued by the market (its market price P₀ equals the intrinsic value V₀), we can express the expected holding-period return on the stock in the following way:

r = (D₁ + P₁)/P₀ - 1 = D₁/P₀ + (P₁ - P₀)/P₀

The first term in the expression above is the dividend yield and the second term is the expected price appreciation. Therefore, the expected holding period return is a sum of dividend yield and stock price appreciation.

Notice that expected holding period return and required rate of return are two distinct concepts. We use the required rate of return as a discount rate in DCF models. Required rate of return is calculated using one of the three models that we have described before. Expected holding-period return, on the other hand, is the rate implied by the DCF model itself provided that the assumption of fair market pricing holds. If the market price of a company's stock does not equal its intrinsic value, expected holding-period return will differ from the required rate of return. This difference is called alpha or expected abnormal return. Investors look for positive alphas (expected holding period return minus required rate of return), since these investments earn more than other assets with similar risk. By definition, there are few investments with positive alphas if the market is efficient. When market price of a stock converges to its intrinsic value, expected holding-period return converges to the required rate of return, and alpha is eliminated.

Example

Consider the following information about a public firm.

• Current stock price: $24.
• Required rate of return: 12%.
• Expected dividend of the next period: $1.
• Expected market price at the end of next period: $26.

The expected holding period return is (26 + 1) / 24 - 1 = 12.5%.
Alpha: 12.5% - 12% = 0.5%. This indicates that the stock is attractive for investors with a single-period horizon.

Multiple Holding Periods

If you anticipate holding the stock for several years and then selling it, the valuation estimate is harder. You must forecast several future dividend payments and estimate the sale price of the stock several years in the future.

V₀ = D₁/(1 + r)¹ + D₂/(1 + r)² + D₃/(1 + r)³ + ... + D_n/(1 + r)ⁿ + P_n/(1 + r)ⁿ

The intrinsic value of a company therefore equals a sum of discounted dividends and discounted terminal-period price of the company's stock. The terminal-period price itself depends on the stream of dividends after the terminal period. Consequently, an alternative expression for the multiple-period dividend discount model is the discounted infinite stream of dividends:

V₀ = Σ(D_t / (1 + r)^t)
Σ: (t=1 to infinite)

Example:

A stock paid $10 dividend last year. Dividends are expected to grow 30% for year 1 and 2, 15% for year 3 to 5, and then 5% from year 6. The required rate of return on equity is 10%.

• The present value of the first stage supernormal growth: V₁ = $10 x (1 + 0.3)/(1 + 0.1)¹ + $10 x (1 + 0.3)² / (1 + 0.1)² = $25.8.
• The present value of the second stage supernormal growth: V₂ = $10 (1 + 0.3)² x (1 + 0.15)/(1 + 0.1)³ + $10 (1 + 0.3)² x (1 + 0.15)²/(1 + 0.1)⁴ + $10 (1 + 0.3)² x (1 + 0.15)³/(1 + 0.1)⁵ = $45.9.
• The terminal value of constant growth at the end of the 5th year: terminal value = $10 x (1 + 0.3)² x (1 + 0.15)³ x (1 + 0.05) / (10% - 5%) = $539.8.
• The present value of constant value: V_{constant growth} = $539.8/(1 + 0.1)⁵ = $335.3.
• The total value of the stock: $25.8 + $45.9 + $335.3 = $407.