FIN 300

Dividend Discount Model
Lecture 7

Topics Covered



  • Dividend Discount Model
    • DDM with Constant Growth
    • Building on the DDM
    • DDM with Two-Stage Growth
  • Valuation Ratios
  • Common Stock Features

Dividend Discount Model



  • A dividend is a cash payment to the stockholder
  • Dividends usually occur quarterly
  • The value of the stock can be modelled as the present value of the dividends over time
  • A model is a description of reality.
      DDM does not always describe the stock value very well.
Perpetual Level Dividend

$Value_{\text{Stock}}= \dfrac{D}{(1+R)}+\dfrac{D}{(1+R)^2}+\dfrac{D}{(1+R)^3}+\dfrac{D}{(1+R)^4}+\dfrac{D}{(1+R)^5}+\dots$


$Value_{\text{Stock}}= \dfrac{D}{R}$
Example: Level (Annual) Dividend

Suppose a stock had an annual dividend of $\$1$ that was to be paid in perpetuity. If the discount rate is $10\%$, what is the stock worth according to the DDM?


$Value_{\text{Stock}}= \dfrac{D}{R}$$= \dfrac{\$1}{0.10}$$=\$10$
Example: Quarterly Dividend

Suppose a stock had a quarterly dividend of $\$0.25$ that was to be paid in perpetuity. If the discount rate is $10\%$ annually, what is the stock worth according to the DDM?

Step 1: Get quarterly discount rate

$R_q = (1+0.10)$$^{\frac{1}{4}}$$-1$ $ = 0.02411$



Step 2: Calculate perpetuity with quarterly dividend and quarterly discount rate:

$Value_{\text{Stock}}= \dfrac{D}{R}$$= \dfrac{\$0.25}{0.02411}$$=\$10.36$

DDM with Constant Growth

The dividend discount model values the stock as a perpetuity of dividends.


We can add growth to this model:

$Value_{\text{Stock}}= \dfrac{D_1}{R-g}$$=\dfrac{D_0 \times (1+g)}{R-g}$

g is the rate of dividend growth
Example: Quarterly Dividend with Growth

A stock stock paid a quarterly dividend of $\$0.25$. The dividend is expected to grow at a rate of $0.5\%$ quarterly, forever. What is the stock's value under DDM if $R=2.411\%$.

$Value_{\text{Stock}}=\dfrac{D_0 \times (1+g)}{R-g}$

$=\dfrac{\$0.25 \times (1+0.005)}{0.02411-0.005}$$=\$13.15$

Building on the DDM

  • The Dividend Discount Model can be elaborated
  • We are not restricted to a single growth rate
  • Consider the case where a dividend will be initiated in 5 quarters.
  • Once it begins, it will be $\$0.25$ per quarter forever.

With $R=2.411\%$, the stock's price in 4 quarters should be:

$P_{\space 4 \space quarters} = \dfrac{\$0.25}{0.02411} = \$10.36$

What is $P_0$, the price today?
The price today [with no interim dividends] is the discounted future price:

$P_0 = \dfrac{P_{\space 4\space quarters}}{(1+R_q)^{4}}$


$P_0 = \dfrac{\$10.36}{(1+0.02411)^{4}}$$=\$9.42$

Now consider a stock that pays a dividend of $D_0$ each quarter. Starting in 9 quarters the dividend will grow at a rate of $g$ per quarter.


$P_{0}= \dfrac{D_0}{(1+R)}+\dfrac{D_0}{(1+R)^2}+\dfrac{D_0}{(1+R)^3}+\dfrac{D_0}{(1+R)^4} \\ \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space +\dfrac{D_0}{(1+R)^5}+\dfrac{D_0}{(1+R)^6}+\dfrac{D_0}{(1+R)^7}+\dfrac{D_0}{(1+R)^8} \\ \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space + \dfrac{D_0 \times (1+g)}{(1+R)^9}+ \dfrac{D_0 \times (1+g)^2}{(1+R)^{10}}+\dots$

In quarter 8, the growing dividend perpetuity will be worth:

$P_{Q8}=\dfrac{D_0 \times (1+g)}{r-g}$

Today, that's only worth:

$\dfrac{\frac{D_0 \times (1+g)}{r-g}}{(1+R)^8}$$ = \dfrac{P_{Q8}}{(1+R)^8}$


$P_{0}= \dfrac{D_0}{(1+R)}+\dfrac{D_0}{(1+R)^2}+\dfrac{D_0}{(1+R)^3}+\dfrac{D_0}{(1+R)^4} \\ \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space +\dfrac{D_0}{(1+R)^5}+\dfrac{D_0}{(1+R)^6}+\dfrac{D_0}{(1+R)^7}+\dfrac{D_0}{(1+R)^8} \\ \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space + \dfrac{P_{Q8}}{(1+R)^8}$




$Annuity_{\text{Quarters 1-8}}= D_0 \times \left[ \dfrac{1 - \dfrac{1}{(1+R)^8}}{R} \right] $

$P_{0}= D_0 \times \left[ \dfrac{1 - \dfrac{1}{(1+R)^8}}{R} \right] + \dfrac{P_{Q8}}{(1+R)^8}$

This represents a simple annuity of dividends for 8 quarters, plus the stock price in 8 quarters, discounted to present value.
Zero growth dividends followed by a growth stage:

$P_{0}= D_0 \times \left[ \dfrac{1 - \dfrac{1}{(1+R)^t}}{R} \right] + \dfrac{P_{t}}{(1+R)^t}$

$P_{t}=\dfrac{D_0 \times (1+g)}{R-g}$

DDM with Two-Stage Growth

  • We can add growth to the first stage annuity
  • The first stage becomes:

    • $PV_{first \space stage}= D_0 \times (1+g_1) \times \left[ \dfrac{1 - \left( \dfrac{1+g_1}{1+R} \right)^t }{R-g_1} \right] $
The DDM with two stage growth becomes:

$Value_{stock}= D_0 \times (1+g_1) \times \left[ \dfrac{1 - \left( \dfrac{1+g_1}{1+R} \right)^t}{R-g_1} \right] + \dfrac{P_{t}}{(1+R)^t}$

$P_{t}=\dfrac{D_0 \times (1+g_1)^t \times (1+g_2)}{R-g_2}$

The Required Return

  • Usually we can observe the stock's price
  • We can find the required rate of return using the DDM
$P_0 = \frac{D_1}{R-g}$

$R = \frac{D_1}{P_0}+g$

$R = Dividend \space Yield+ Capital \space Gains \space Yield$
Example:

$P_0=\$25, R=12\%, g=1\%$

Valuation Using Multiples

  • Sometimes stocks don't pay dividends
  • We could look at the Free Cash Flow [Not in this course]
  • We can also compare Earnings or Sales ratios
  • Assume similar earnings risk and market efficiency


$P_t=PE_{Benchmark} \times EPS_t$

$P_t=\text{Price-Sales}_{Benchmark} \times EPS_t$

Example: Firm A has an EPS of $\$2.21$ per share. The benchmark PE is 18. Is the share price of $\$36$ overvalued?

Common Stock Features

  • Voting
    • Cummalative Voting [Good for minority holders]
    • Straight Voting
  • Proxy Voting
  • Dual Class Shares
      Some shares can't vote.
  • Dividends
      Proportional to holdings

Summary



  • Dividend Discount Model
    • DDM with Constant Growth
    • Building on the DDM
    • DDM with Two-Stage Growth
  • Valuation Ratios
  • Common Stock Features