Nonconstant growth stock valuation example

Stock Valuation: Key Concepts and Skills

Understand how stock prices depend on future dividends and dividend growth
Be able to compute stock prices using the dividend growth model
Understand how corporate directors are elected
Understand how stock markets work
Understand how stock prices are quoted
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Outline

Common Stock Valuation
Some Features of Common and Preferred Stocks
The Stock Markets
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Cash Flows for Stockholders

If you buy a share of stock, you can receive cash in two ways
The company pays dividends
You sell your shares, either to another investor in the market or back to the company
As with bonds, the price of the stock is the present value of these expected cash flows
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One-Period Example

Suppose you are thinking of purchasing the stock of Moore Oil, Inc. You expect it to pay a $2 dividend in one year, and you believe that you can sell the stock for $14 at that time. If you require a return of 20% on investments of this risk, what is the maximum you would be willing to pay?
Compute the PV of the expected cash flows
Price = (14 + 2) / (1.2) = $13.33
Or FV = 16; I/Y = 20; N = 1; CPT PV = -13.33
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Two-Period Example

Now, what if you decide to hold the stock for two years? In addition to the dividend in one year, you expect a dividend of $2.10 in two years and a stock price of $14.70 at the end of year 2. Now how much would you be willing to pay?
PV = 2 / (1.2) + (2.10 + 14.70) / (1.2)2 = 13.33
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Three-Period Example

Finally, what if you decide to hold the stock for three years? In addition to the dividends at the end of years 1 and 2, you expect to receive a dividend of $2.205 at the end of year 3 and the stock price is expected to be $15.435. Now how much would you be willing to pay?
PV = 2 / 1.2 + 2.10 / (1.2)2 + (2.205 + 15.435) / (1.2)3 = 13.33
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Developing The Model

You could continue to push back the year in which you will sell the stock
You would find that the price of the stock is really just the present value of all expected future dividends
So, how can we estimate all future dividend payments?
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Estimating Dividends:
Special Cases

Constant dividend
The firm will pay a constant dividend forever
This is like preferred stock
The price is computed using the perpetuity formula
Constant dividend growth
The firm will increase the dividend by a constant percent every period
The price is computed using the growing perpetuity model
Supernormal growth
Dividend growth is not consistent initially, but settles down to constant growth eventually
The price is computed using a multistage model
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Zero Growth

If dividends are expected at regular intervals forever, then this is a perpetuity and the present value of expected future dividends can be found using the perpetuity formula
P0 = D / R
Suppose stock is expected to pay a $0.50 dividend every quarter and the required return is 10% with quarterly compounding. What is the price?
P0 = .50 / (.1 / 4) = $20
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Dividend Growth Model

Dividends are expected to grow at a constant percent per period.
P0 = D1 /(1+R) + D2 /(1+R)2 + D3 /(1+R)3 + …
P0 = D0(1+g)/(1+R) + D0(1+g)2/(1+R)2 + D0(1+g)3/(1+R)3 + …
With a little algebra and some series work, this reduces to:
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DGM – Example 1

Suppose Big D, Inc., just paid a dividend of $0.50 per share. It is expected to increase its dividend by 2% per year. If the market requires a return of 15% on assets of this risk, how much should the stock be selling for?
P0 = .50(1+.02) / (.15 - .02) = $3.92
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DGM – Example 2

Suppose TB Pirates, Inc., is expected to pay a $2 dividend in one year. If the dividend is expected to grow at 5% per year and the required return is 20%, what is the price?
P0 = 2 / (.2 - .05) = $13.33
Why isn’t the $2 in the numerator multiplied by (1.05) in this example?
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Stock Price Sensitivity to Dividend Growth, g

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Stock Price Sensitivity to Required Return, R

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Example 8.3 Gordon Growth
Company - I

Gordon Growth Company is expected to pay a dividend of $4 next period, and dividends are expected to grow at 6% per year. The required return is 16%.
What is the current price?
P0 = 4 / (.16 - .06) = $40
Remember that we already have the dividend expected next year, so we don’t multiply the dividend by 1+g
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Example 8.3 – Gordon Growth
Company - II

What is the price expected to be in year 4?
P4 = D4(1 + g) / (R – g) = D5 / (R – g)
P4 = 4(1+.06)4 / (.16 - .06) = 50.50
What is the implied return given the change in price during the four year period?
50.50 = 40(1+return)4; return = 6%
PV = -40; FV = 50.50; N = 4; CPT I/Y = 6%
The price is assumed to grow at the same rate as the dividends
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Nonconstant Growth
Problem Statement

Suppose a firm is expected to increase dividends by 20% in one year and by 15% in two years. After that, dividends will increase at a rate of 5% per year indefinitely. If the last dividend was $1 and the required return is 20%, what is the price of the stock?
Remember that we have to find the PV of all expected future dividends.
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Nonconstant Growth
Example Solution

Compute the dividends until growth levels off
D1 = 1(1.2) = $1.20
D2 = 1.20(1.15) = $1.38
D3 = 1.38(1.05) = $1.449
Find the expected future price
P2 = D3 / (R – g) = 1.449 / (.2 - .05) = 9.66
Find the present value of the expected future cash flows
P0 = 1.20 / (1.2) + (1.38 + 9.66) / (1.2)2 = 8.67
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Using the DGM to Find R

Start with the DGM:
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Finding the Required Return - Example

Suppose a firm’s stock is selling for $10.50. It just paid a $1 dividend, and dividends are expected to grow at 5% per year. What is the required return?
R = [1(1.05)/10.50] + .05 = 15%
What is the dividend yield?
1(1.05) / 10.50 = 10%
What is the capital gains yield?
g =5%
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Table 8.1 - Stock Valuation Summary

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Features of Common Stock

Voting Rights
Proxy voting
Classes of stock
Other Rights
Share proportionally in declared dividends
Share proportionally in remaining assets during liquidation
Preemptive right – first shot at new stock issue to maintain proportional ownership if desired
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Dividend Characteristics

Dividends are not a liability of the firm until a dividend has been declared by the Board
Consequently, a firm cannot go bankrupt for not declaring dividends
Dividends and Taxes
Dividend payments are not considered a business expense; therefore, they are not tax deductible
The taxation of dividends received by individuals depends on the holding period
Dividends received by corporations have a minimum 70% exclusion from taxable income
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Features of Preferred Stock

Dividends
Stated dividend that must be paid before dividends can be paid to common stockholders
Dividends are not a liability of the firm, and preferred dividends can be deferred indefinitely
Most preferred dividends are cumulative – any missed preferred dividends have to be paid before common dividends can be paid
Preferred stock generally does not carry voting rights
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Stock Market

Dealers vs. Brokers
New York Stock Exchange (NYSE)
Largest stock market in the world
License holders (1,366)
Commission brokers
Specialists
Floor brokers
Floor traders
Operations
Floor activity
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NASDAQ

Not a physical exchange – computer-based quotation system
Multiple market makers
Electronic Communications Networks
Three levels of information
Level 1 – median quotes, registered representatives
Level 2 – view quotes, brokers & dealers
Level 3 – view and update quotes, dealers only
Large portion of technology stocks
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Work the Web Example

Electronic Communications Networks provide trading in NASDAQ securities
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Sample Quote
What information is provided in the stock quote?

Click on the web surfer to go to Bloomberg for current stock quotes.

Reading Stock Quotes

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g-R

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00.050.10.150.2

Growth Rate

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