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# The Dividend Discount Model

## Single Holding Period

The cash flows to an investor who holds shares are the dividends paid and the market price of the share when they sell the share. For example, suppose an investor expects to hold the share for one year. In that case, the value of the share today is the present value of the anticipated dividend plus the present value of the anticipated selling price in one year.

$$V_0= \frac{D_1}{(1+r)^1} + \frac{P_1}{(1+r)^1} = \frac{(D_1+ P_1)}{(1+r)^1}$$

Where:

$$V_{0}=$$ The value of a share today, at $$t = 0$$.

$$P_{1}=$$ The anticipated price per share at $$t = 1$$.

$$D_{1}=$$ The expected dividends per share to be paid at the end of the year at $$t = 1$$.

$$r =$$ The discount rate of the stock.

#### Example: Single Period DDM

An investor expects a company to pay a dividend of $1.50 at the end of the year. The investor anticipates selling the share for$38.00 immediately after. With a required rate of return of 9%, the value of the stock is closest to:

\begin{align*}V_0&=\frac{(D_1+ P_1)}{(1+r)^1}\\ \\ &= \frac{(1.50+ 38.00)}{(1.09)}=36.24\end{align*}

## Multiple Period DDM

Suppose an investor plans to hold the share for two years. In that case, the value of the share is the sum of the present value of the expected dividends at the end of year 1, the present value of the expected dividends at the end of year 2, and the present value of the expected selling price at the end of year 2.

\begin{align*}V_0 &= \frac{D_1}{(1+r)^1} + \frac{D_2}{(1+r)^2} +\frac{P_2}{(1+r)^2} \\ \\ &= \frac{D_1}{(1+r)^1} + \frac{(D_2+ P_2)}{(1+r)^2}\end{align*}

For n periods, the share value is the sum of the present value of the expected dividends for the n periods and the present value of the expected price at period t = n.

\begin{align*}V_0&= \frac{D_1}{(1+r)^1} +⋯+ \frac{D_n}{(1+r)^n} +\frac{P_n}{(1+r)^n} \\ \\ & = ∑_{(t=1)}^n\frac{D_t}{(1+r)^t} + \frac{P_n}{(1+r)^n} \end{align*}

With a finite holding period, the DDM finds the value of a stock as the sum of:

1. The present value of the expected dividends throughout the holding period.
2. The present value of the expected share price at the end of the holding period

If the holding period reaches the indefinite future, the share’s value is the present value of all expected future dividends.

\begin{align*} V_0&= \frac{D_1}{(1+r)^1} +⋯+ \frac{D_n}{(1+r)^n} \\ \\&= ∑_{(t=1)}^∞\frac{D_t}{(1+r)^t} \end{align*}

#### Example: Multiple Period DDM

ABC Inc. expects dividends of $2 and$2.5 at the end of the next two years, respectively. The expected stock price at the end of year 2 is \$48. If the required rate of return is 15%, then the value of ABC’s share today is closest to:

#### Solution

\begin{align*}V_0&= \frac{D_1}{(1+r)^1} +⋯+ \frac{D_n}{(1+r)^n} +\frac{P_n}{(1+r)^n}\\ \\&=\frac{\2}{1.15}+\frac{(\2.5+\48)}{(1.15)^2} =\39.92\end{align*}

Forecasting dividends into the indefinite future is a challenge due to the uncertainty of the variables involved. There are two approaches used to solve this:

i. The forecasted dividends can be assigned several growth patterns. These patterns are:

• Constant growth (the Gordon growth model).
• Two distinct stages of growth (the two-stage model and the H-model).
• Three definite stages of growth (the three-stage growth model).

The DDM value of the share is calculated as the sum of the discounted values of future dividends.

ii. Forecasting a finite number of dividends up to a terminal point. The forecasted period depends on the predictability of the company’s earnings. After this period:

• The remaining dividends from the terminal point can be assigned a growth pattern.
• The analyst can forecast the share price at the terminal point.

The DDM value of the share is calculated by discounting the dividends and the forecasted terminal share price, if any.

## Question

Which of the following is the least likely method of estimating the value of a share using an indefinite number of future dividends?

1. The Gordon growth model.
2. The two-stage model.
3. The present value of future dividends and terminal price.

#### Solution

The present value of future dividends and terminal price is a method used to estimate the value of a share assuming a finite number of dividends.

A is incorrect: The Gordon growth model assumes a constant growth of indefinite dividends into the future to estimate the value of a share.

B is incorrect: The two-stage model is an approach for estimating the value of a share that assumes an indefinite number of future dividends.

LOS 23 (b) Calculate and interpret the value of common stock using the dividend discount model (DDM) for single and multiple holding periods.

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