Chapter 7. Stocks and Stock Valuation. LEARNING OBJECTIVES. 1. Explain the basic characteristics of common stock. 2. Define the primary market and the secondary market. 3. Calculate the value of a stock given a history of dividend payments.
Stocks and Stock Valuation
cash flow of the company.
Known as “residual” rights.
Owner’s Equity = Assets - Liabilities
Example: Stock price with known dividends and sale price.
Agnes wants to purchase common stock of New Frontier Inc. and hold it for 4 years. The directors of the company just announced that they expect to pay an annual cash dividend of $4.00 per share for the next 5 years. Agnes believes that she will be able to sell the stock for $40 at the end of four years. In order to earn 12% on this investment, how much should Agnes pay for this stock?
Price = $40.00 x 0.635518 + $4.00 x 3.03734
Price = $25.42 + $12.15 = $37.57
Method 2. Using a financial calculator
Mode: P/Y=1; C/Y = 1
Key: N I/Y PV PMT FV
Input: 4 12 ? 4 40
4 variations of a dividend pricing model have been used to value common stock
1. The constant dividend model with an infinite horizon
2. The constant dividend model with a finite horizon
3. The constant growth dividend model with a finite horizon
4. The constant growth dividend model with an infinite horizon
Assumes that the firm is paying the same dividend
amount in perpetuity.
Div1 = Div2 = Div3 = Div4 = Div5…= Div∞
Recall a perpetuity
PV = PMT/r
where r the required rate and PMT is the cash flow.
Thus, for a stock that is expected to pay the same dividend forever we have its price as:
Price = Dividend/Required rate ofreturn
Example: Quarterly dividends forever
Let’s say that the Peak Construction Company is paying a quarterly dividend of $0.50 and has decided to pay the same amount forever. If Joe wants to earn an annual rate of return of 12% on this investment, how much should he offer to buy the stock at?
Quarterly dividend = $0.50
Quarterly rate of return = Annual rate/4= 12%/4 = 3%
PV = Quarterly dividend/Quarterly rate of return
Price = 0.50/.03 = $16.67
Example: Constant dividends with finite holding period.
Let’s say that the Peak Construction Company is paying an annual dividend of $2.00 and has decided to pay the same amount forever.
Joe wants to earn an annual rate of return of 12% on this investment, and plans to hold the stock for 5 years, with the expectation of selling it for $20 at the end of 5 years.
What is his offer price for the stock?
Annual dividend = $2.00 = PMT
Selling Price = $20 = FV
Annual rate of return = 12% = I/Y
Horizon or Holding Period = 5 = N
Offer Price (PV) = PV of dividend stream over 5 years
+ PV of Year 5 price
Mode: P/Y=1; C/Y = 1
Key: N I/Y PV PMT FV
Input: 5 12.0 ? 2.00 20.00
Known as the Gordon model (after its developer, Myron Gordon).
Estimate is based on the discounted value of an infinite stream of future dividends that grow at a constant rate, g.
where r is the required rate of return and new owner gets the next dividend payment (Div1 or Div0 x (1 + g)1)
Example: Constant growth rate, infinite horizon (with growth rate given).
Let’s say that the Peak Construction Company just paid its shareholders an annual dividend of $2.00 (DIV0)and has announced that the dividends would grow at an annual rate of 8% forever. If investors expect to earn an annual rate of return of 12% on this investment how much would they offer to buy the stock for?
Div0 = $2.00; g=8%; r=12% with infinite horizon
Div1= Div0 x (1+g)
Div1=$2.00 x (1.08) or Div1=$2.16
P0 = Div1 / (r-g) = $2.16 / (0.12 - 0.08) = $54.00
Price0 = $54.00
Note: r and g are in decimal format
EXAMPLE: Constant growth rate, infinite horizon (with growth rate estimated from past history).
Let’s say that you are considering an investment in the common stock of QuickFix Enterprises and are convinced that its last paid dividend of $1.25 will grow at its historical average growth rate from here on. Using the past 10 years of dividend history and a required rate of return of 14%, calculate the price of QuickFix’s common stock.
We must estimate g from the dividend history of the firm…
QuickFix Enterprises’ Annual Dividends
Required rate of return = 14%
Compound growth rate “g” = (FV/PV)1/n -1
Where FV = $1.25; PV = 0.50; n = 9 (number of dividend changes)
g = (1.25/0.50)1/9 – 1 = 10.7173% (or on calculator, g = I/Y)
Div1 = Div0(1+g)= $1.25 x (1.107173)= $1.383966
P0 = Div1/(r-g) = $1.383966 / (0.14 - 0.107173) = $42.16
Investor expects to hold a stock for a limited number of years,
Company’s dividends are growing at a constant rate.
Following formula is used to value the stock…
Note: This formula would lead to the same price estimate as the Gordon model, if it is assumed that the growth rate of dividends and the required rate of return of the next owner, (after n years) remain the same.
Example: Constant growth, finite horizon.
The QuickFix Company just paid a dividend of $1.25 and analysts expect the dividend to grow at its compound average growth rate of 10.7173% forever.
If you plan on holding the stock for just 7 years, and you have an expected rate of return of 14%, how much would you pay for the stock?
Assume that the next owner also expects to earn 14% on his or her investment.
We can solve this in 2 ways.
Method 1: Use the constant growth, finite horizon formula
Method 2: Use the Gordon Model since g is constant forever, and both investors have the same required rates of return
Method 1 Use the following equation:
Price at end of year 7 or P7 = Div8/(r-g)
Div0 = $1.25; g =10.7173%; Div8 = D0(1+g)8 = $1.25 x (1.107173)8 = $2.8225
P7=2.8225 /(0.14 - 0.107173)= $85.98
Price0 = $42.16 x 0.1849684 + 34.36 = $42.16
Method 2: Use the Gordon Model
P0 = D0(1+g)/(r-g)
P0 = $1.25 x (1.107173)/(0.14 - 0.107173)
P0 = $42.16
Example: Non-constant dividend pattern
The Rapid Growth Company is expected to pay a
dividend of $1.00 at the end of this year. Thereafter, the dividends are expected to grow at the rate of 25% per year for 2 years, and then drop to 18% for 1 year, before settling at the industry average growth rate of 10% indefinitely.
If you require a return of 16% to invest in a stock of this risk level, how much would you be justified in paying for this stock?
D1=$1.00; g1=25%; n1=2; g2=18%; n2=1; gc=10%; r=16%
Step 1. Calculate the annual dividends expected in Years 1-4, using the appropriate growth rates.
D1 = $1.00; D2 = $1.00 x (1.25) = $1.25;
D3= $1.25 x (1.25) = $1.56; D4 = $1.56 x (1.18) = $1.84;
Step 2. Calculate the price at the start of the constant growth phase using the Gordon model.
P4 = D4 x (1+g)/(r-g) = $1.84 x (1.10)/(0.16 - 0.10)
= $2.02/.06 = $33.73
Step 3. Discount the annual dividends in Years 1-4 and the Price at the end of Year 4, back to Year 0 using the required rate of return as the discount rate, and add them up to solve for the current price.
P0 = $1.00/(1.16) + $1.25/(1.16)2 + $1.56/(1.16)3 + $1.84/(1.16)4 + $33.73/(1.16)4
P0 = $0.862 + $0.928 + $0.999 + $1.016 + $18.629
P0 = $22.43
Pays constant dividend as long as the stock is outstanding.
Typically has infinite maturity, but some are convertible into common stock at some pre-determined ratio.
Have “preferred status” over common stockholders in the case of dividend payments and liquidation payouts.
Dividends can be cumulative or non-cumulative
To calculate the price of preferred stock, we use the PV of a perpetuity equation, i.e. Price0 = PMT/r
PMT = Annual dividend (dividend rate * par value); and
r = investor’s required rate of return.
Example 8: Pricing preferred stock.
The Mid-American Utility Company’s preferred stock pays an annual dividend of 8% per year on its par value of $60. If you want to earn 10% on your investment how much should you offer for this preferred stock?
Annual dividend = .08*$60 = $4.80
Price = $4.80/0.10 = $48.00
Market in which security prices are current and fair to all traders.
Transactions costs are minimal.
There are two forms of efficiency: