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Summary C9, C10, C12, C14, C15, C16, C17 & C18 9,3
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Corporate fiaacaa maiagemeitChapter 9: Vaaucig stoaks
9.1 The dcvcdeid-dcsaouit modea (see exampae 9.1 at page 311)
The Law of Oie Prcae implies that to value any security, we must determine the expected cash fows
an investor will receive from owning it. There are two tme horiionss One year investors and longer
tme horiions.
A One-Year Investor
There are two potental sources of cash fows from owning a stocks frm pay outs (dividend) and
selling the shares by the inital investor on a later moment. The total amount of receiving’s depends
on the tme horiion.
Investor buys stock at current market price for the share (P0)
Dividend is paid end of the year (Div1)
Investor sells stocks at the end of the year (P1)
0……………………………………………………1
-P0 Div1 + P1
Of course, Div1 and P1 aren’t known with certainty. These values are based on the expectatons of
the investor at the moment he/she buys stocks. Given these expectatons, the investor will be willing
to buy the stock at today’s prices as long as the NPV of the transacton is not negatve. That is, as long
as the current price does not exceed the present value of the expected future dividend and sale
price. Because these cash fows are risky, we cannot compute their present value using the risk-free
interest rate. Instead, we must discount them base on the equcty aost of aapctaa (Re), for the stock,
which is the expected return of other investments available in the market with equivalent risk to the
frm’s shares. Doing so leads to the following conditons under which an investor would be willing to
buy the stocks
P0 < Div1 + P1 (Outcome is bigger than P0)
1 + Re
Similarly, for an investor to be willing to sell the stock, she must receive at least as much today as the
present value she would receive if she waited to sell next years
P0 > Div1 + P1 (Outcome is smaller than P0)
1 + Re
But because for every buyer of the stock there must be a seller, BOTH equatons must hold, and
therefore the stock price should satsfy
P0 = Div1 + P1 (Outcome is equal to P0)
1 + Re
In other words, in a compettve market, buying or selling a share of stock must be a iero-NPV
investment opportunity.
Dividend Yields, Capital Gains, and Total Returns
,We can reinterpret equaton .1 if we multply by (1 + Re), divide by P0, and subtract 1 from both
sidess
Totaa Returi (9.2):
Div1 + P1 Div1 P1 - P0
Re = -1= +
P0 P0 P0
Div1/P0 = Dividend Yield
(P1-P0)/P0 = Capital Gain Rate
Dcvcdeid Ycead: is the expected annual dividend of the stock divided by its current price. The dividend
yield is the percentage return the investor expects to earn from the dividend paid by the stock.
Capctaa gaci: is the diference between the expected sale price and purchase price for the stock (P1-
P0). We divide the capital gain by the current stock price to express the capital gain as a percentage
return, called the aapctaa gaci rate
The sum of the dividend yield and the capital gain rate is called the totaa returi of the stock. Thus,
the formula states that the stock’s total return should equal the equity cost of capital. In other
words, the expected total return of the stock should equal the expected return of the other
investments available in the market with equivalent risk.
The result in .2 is what we should expects the frm must pay its shareholders a return
commensurate with the return they can earn elsewhere while taking the same risk. If the stock
ofered a higher return than other securites with the same risk, investors would sell those other
investments and buy the stock instead. This actvity would drive up the stock’s current prices,
lowering its dividend yield and capital gain rate untl .2 holds true. If the stock ofered a lower
expected return, investors would sell the stock and drive down its current prices untl .2 holds true
again.
Multiyear Investor
.1 depends upon the expected stock price in one year, P1. But suppose we planned to hold the
stock for two years. Then we would receive dividends in both year 1 and 2 before selling the stock.
0…………………………1…………………………2
-P0 Div1 Div2 + P2
Setng the stock price equal to the present value of the future cash fows in this case implies ( .3)s
P0 = Div1 + Div2 + P2
1 + Re (1+Re)^2
One-year and multyear investors do not value the stock diferently. While a one-year investor does
not care about the dividend and stock price in year 2 directly, she will dare about them indirectly
because they will afect the price for which she can sell the stock at the end of year 1. Example, one-
year investor sells to another one-year investors
,P1 = Div2 + P2
1 + Re
Substtutng this expression for P1 into .1, we get the same result as shown in .3s
Div1 + P1 div1 1 (Div2 + P2) Div1 div2+P2
P0 = = + ( ) = +
1 + Re 1+Re 1+Re (1 + Re) 1+Re (1+Re)^2
The dividend-Discount Model Equatons
We can contnue doing so for any number of years. ^2 becomes N=years
P0 = Div1 + Div2 + Div3
1 + Re (1+Re)^2 (1+Re)^N
9.2 Appaycig the Dcvcdeid-Dcsaouit Modea
It is hard to estmate the dividend, especially when N becomes higher. A common assumpton is that
the dividend growths a constant rate. In this secton, we will consider the implicatons of the
assumpton for stock prices and explore the trade-of between dividends and growth.
Constant dividend growth
Constant rate = g
0……………..1………………...2…………………..3
-P0 Div1 Div1(1+g) Div1(1+g)^2
Because the expected dividends are a constant growth perpetuity we can calculate the present value.
Coistait dcvcdeid growth modea:
P0 = Div1 ( .6)
Re-g
According to the CDGM, the value of the frm depends on the dividend level for the coming year,
divided by the equity cost of capital adjusted by the expected growth rate of dividends.
For another interpretaton of .6 note that we can rearrange it as followss
Re = Div1 + g ( .7)
P0
Comparing .7 with .2, we see that g equals the expected capital gain rate. In other words, with
constant expected dividend growth, the expected growth rate of the share prices matches the
growth rate of dividends.
Dividends versus Investment and Growth
, In .6 the frm’s share price increases with the current dividend level, div1, and the expected growth
rate, g. To maximiie its share price, a frm would like to increase both these quanttes. O en,
however, the frm faces a trade-ofs Increasing growth may require investment, and money spent on
investment cannot be used to pay dividends. We can use the CDGM to gain insight into this trade-of.
A scmpae modea of Growths
If we defne a frm’ s dcvcdeid payout rate as the fracton of its earnings that the frm pays as
dividends each year, than we can write the frm’s dividend per share at date, t, as followss
Divt = EarningsT X Dividend Payout RateT ( .8)
Shares OutstandingT
(EPSt)
That is, the dividend each year is the frm’s earnings per share (EPS) multplied by its dividend payout
rate. Thus, the frm can increase its dividend in three wayss
1. It can increase its earnings (net income)
2. It can increase its dividend payout rate
3. It can decrease its shares outstanding
A frm can do one of two things with its earningss Pay out to investors or it can retain and reinvest
them.
Change in earnings = New Investments X Return on New Investment ( . )
New investments equals earnings multplied by the frm’s reteitioi rate,i the fracton of current
earnings that the frm retainss
New investment = Earnings X retenton rate ( .10)
Substtutng .10 and . and dividing by earnings gives an expression for the growth rate of
earningss
Earnings growth rate = Change in Earnings = Retenton rate X return of new investment ( .11)
Earnings
If the frm chooses to keep its dividend payout rate constant, then the growth in dividends will equal
the growth of earningss
G = retenton rate X return on new investment ( .12)
This growth rate is sometmes referred to as the frm’s sustaciabae growth rate, the rate at which it
can grow using only retained earnings.
Proftabae growth:
.12 shows that a frm can increase its growth rate by retaining more of its earnings. However, if the
frm retains more earnings, it will be able to pay out less of those earnings and, according to .8, will
have to reduce its dividend. If a frm wants to increase its share price, should it cut its dividend and
invest more or should it cut investment and increase its dividend? Not surprisingly, the answer
depends on the proftability of the frm’s investments.
Changing growth rates:
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