8–8 Constant Growth Rate, g
A stock is trading at $80 per share. The stock is expected to have a year_x0002_end dividend of
$4 per share (D1 5 $4), and it is expected to grow at some
constant rate g throughout time. The stock’s required rate of return is 14%
(assume the market is in equilibrium with the required return equal to the
expected return). What is your forecast of g?
Using the constant growth model
Required return = Expected Dividend/Price + g
14% = 4/80 + g
g = 9.00%
Crisp Cookware’s common stock is expected to pay a dividend of $3 a share
at the end of this year (D1 5 $3.00); its beta is 0.8; the risk-free rate is 5.2%;
and the market risk premium is 6%. The dividend is expected to grow at some
constant rate g, and the stock currently sells for $40 a share. Assuming the
market is in equilibrium, what does the market believe will be the stock’s
price at the end of 3 years (i.e., what is P⁄ 3)?
We first calculate the growth rate g
Using the dividend model
P0 = D1/(Ke-g)
Here
P0 = current market price = $40
Ke = required return, which we can calculate using the CAPM equation
Required return = Rf + (Rm-Rf) X beta = 5.2% + 6%X0.8 = 10%
D1 = expected dividend = $3.00
We get
40 = 3/(10%-g) = 2.5%
Solving gives g as 2.5%
Growth rate g is the capital gains yield implying that the stock price will
increase by this growth rate
Price at the end of three years P3 = P0 X (1+g)^3
= 40 X (1.025)^3 = $43.08
What is the required rate of return on a preferred stock with a $50 par value,
a stated annual dividend of 7% of par, and a current market price of (a) $30,
(b) $40, (c) $50, and (d) $70 (assume the market is in equilibrium with the
required return equal to the expected return)?
Stated annual dividend is 7% of par ($50)
D0 = $50*.07 = 3.50
Vps = Dps/rps
a) $30
30 = 3.50/rps
Rps = 11.67%
b) $40
40 = 3.50/rps
Rps = 8.75%
c) $50
Rps = 7%
50 = 3.50/rps
d) $70
Rps = 5%
70 = 3.50/rps
Assume that the average firm in your company’s industry is expected to grow
at a constant rate of 6% and that its dividend yield is 7%. Your company is
about as risky as the average firm in the industry and just paid a dividend
(D0
) of $1. You expect that the growth rate of dividends will be 50% during
the first year (g0,1 5 50%) and 25% during the second year (g1,2 5 25%). After
Year 2, dividend growth will be constant at 6%. What is the required rate of
return on your company’s stock? What is the estimated value per share of
your firm’s stock?
D0 = $1,
rs = 7% + 6% = 13%,
g1 = 50%,
g2 = 25%,
gn= 6%
Find the Dividends up to Year 3
D0 = 1
D1 = 1(1.50) = 1.5
D2 = 1.5(1.25) = 1.875
D3 = 1.875(1.06) = 1.9875
PV of Dividends
Year 1 PVDiv = D1/(1+rs)^1
PvDiv = 1.5/(1.13) = 1.32
Year 2 PVDiv = D2/(1+rs) ^2
PvDiv = 1.875/ (1.13^2) = 1.47
Year 3 PvDiv 3= 1.9875/ 0.07 * (1.13^2 ) =22.23
Adding all the present values to find the value of
stock
P0=1.32 + 1.47+22.24
Estimated value per share of the stock = $25.03
Simpkins Corporation does not pay any dividends because it is expanding
rapidly and needs to retain all of its earnings. However, investors expect
Simpkins to begin paying dividends, with the first dividend of $0.50 com_x0002_ing 3 years from today. The
dividend should grow rapidly—at a rate of 80%
per year—during Years 4 and 5. After Year 5, the company should grow at a
constant rate of 7% per year. If the required return on the stock is 16%, what
is the value of the stock today (assume the market is in equilibrium with the
required return equal to the expected return)?
D0 = 0
D1 = 0
D2 = 0
D3 = 0.50
D4 = 0.50(1.8) = 0.90
D5 = 0.90(1.8) = 1.62
D6 = 1.62(1.07) = 1.7334
Price at Year 5 or P5
P5 = D6/(rs - g)
P5 = 1.7334/ (0.16-.07)
P5 = 19.26
P(0)
1.5886939
PV of P5
9.1699367
P0
10.76