Stocks

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Stocks

Marcela Valenzuela

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Stock Issuance

⇒ Why do firms issue stocks?

• To raise capital, finance investments, acquire othercompanies, repurchase debt.

⇒ Where do firms issue stock?

• Sales of shares to raise new capital are said to occur in theprimary market.

• However, such sales occur relatively infrequently and mosttrades take place on the stock exchange, where investors buyand sell existing shares.

• Stock exchanges are really secondary markets.

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Stock Market Indexes

⇒ The Standard and Poor’s 500 Index (S&P 500) is a stockmarket index based on the market capitalizations of 500 largecompanies having common stock listed on the NYSE or NASDAQ.

⇒ The FTSE 100 Index is a stock market index based on themarket capitalizations of 100 large companies having commonstock listed on the London Stock Exchange.

⇒ The Selective Stock Price Index (Indice de Precios Selectivo deAcciones, or IPSA) is composed of the 40 most heavily tradedstocks in Santiago Stock Exchange.

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Stock Market Indexes

1

510

50100

5001000

SP

−500

1860 1880 1900 1920 1940 1960 1980 2000

recession and markets downrecession and markets up

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Transactions Involving Stock

⇒ Buy (Long Position)

• Savings motive.

• Expect stock to increase in value.

⇒ Sell

• Liquidity needs.

• Expect stock to decline in value.

⇒ Short Sell (Sell stock without first owning it)

• Borrow stock from your broker with the promise to return itat some later date.

• Sell the borrowed stock.

• Repurchase it at a later date to return it to your broker.

• The short seller is responsible for paying the dividend.

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Types of Orders

⇒ Market Orders

• Buy or sell at the current market price.

⇒ Limit Orders

• Buy or sell at a pre-specified price.

• Limit by time period.

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Valuing Common Stocks

⇒ Expected Return (r): The percentage return that an investorexpects to get from a specific investment over a set period of time.

⇒ Suppose that: the current price of a share is P0, the expectedprice at the end of a year is P1, and the expected dividend pershare is Div1, then:

r =Div1 + P1 − P0

P0

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Valuing Common Stocks

⇒ (E) If a company is selling for $150 per share today and isexpected to sell for $170 one year now. If investors expect thedividend per share to be equal to $10, the expected retun is:

r =10 + 170− 150

150= 20%

⇒ We can break the return formula into two parts: dividend yieldand capital appreciation:

r =Div1P0︸︷︷︸

Dividend Yield

+P1 − P0

P0︸ ︷︷ ︸Capital Gain Rate

⇒ Typically capital gains are the larger fraction of returns.

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Valuing Common Stocks

⇒ The discount rate r is the rate of return investors can expect toearn on securities with similar risk.

⇒ Group of stocks with the same risk have to offer the sameexpected return.

⇒ If it does not hold? This is a condition of market equilibrium. Ifit did not hold, the share would be overpriced or underpriced, andinvestors would rush to sell or buy it. The flood of seller or buyerswould force the price to adjust.−→ All securities in an equivalent risk class are priced to offer thesame expected return

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Valuing Common Stocks

⇒ The value of a stock is equal to the stream of cash paymentsdiscounted at the rate of return that investors expect to receive onother securities with equivalent risks.

⇒ Stocks do not have a fixed maturity; their cash paymentsconsist of an indefinite stream of dividends.

⇒ The discounted-cash-flow (DCF) formula for the present valueof a stock is:

PV =∞

∑t=1

Divt(1 + r)t

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Valuing Common Stocks

PV =∞

∑t=1

Divt(1 + r)t

⇒ The price an investor is willing to pay for a share of stockdepends upon:

• Magnitude and timing of expected future dividends.

• Risk of the stock.

⇒ However, the cash payoff to owners of common stocks comes intwo forms: cash dividends and capital gains/losses. Where arecapital gains and losses?

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Valuing Common Stocks

⇒ If investors have forecasts of dividends, prices and the expectedreturn offered by other equally risky stocks, they can predicttoday’s price:

P0 =Div1 + P1

1 + r

⇒ What about P1, the next year’s price?

P1 =Div2 + P2

1 + r

⇒ Plugin P1 in the today’s price expression:

P0 =Div11 + r

+Div2

(1 + r)2+

P2

(1 + r)2

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Valuing Common Stocks

⇒ Continuing indefinitely gives:

P0 =Div11 + r

+Div2

(1 + r)2+ · · ·+ DivH

(1 + r)H+

PH

(1 + r)H

P0 =H

∑t=1

Divt(1 + r)t

+PH

(1 + r)H

⇒ As H approaches infinity, the price of a stock equals the presentvaue of its cash flows.

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Example

⇒ XYZ Company is forecast to pay dividends of $3, $3.24, and$3.5 over the next three years, respectively. At the end of threeyears you anticipate selling your stock at a market price of $94.48.What is the price of the stock given a 12% expected returns?

PV =3

1 + 0.12+

3.24

(1 + 0.12)2+

3.5 + 94.48

(1 + 0.12)3= $75

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Constant Dividend Growth

⇒ Dividends grow at rate g : where Div1 is the dividend at the endof the first period.

P0 =Div11 + r

+Div1(1 + g)

(1 + r)2+ · · ·+ Div1(1 + g)t−1

(1 + r)t+ · · ·

⇒ Applying the growing perpetuity formula:

P0 =Div1r − g

note g < r

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Market Capitalization Rate

⇒ It is often helpful to twist the previous formula around and useit to estimate the market capitalization rate r , given P0 andestimates of Div1 and g :

r =Div1P0

+ g

⇒ The expected return equals the dividend yield plus the expectedrate of growth in dividends g .

⇒ This formula rest on the assumption of constant dividendgrowth in perpetuity. This may be acceptable for maturecompanies with low to moderate growth rates.

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Example

⇒ Company Z’s earnings and dividends per share are expected togrow indefinitely by 5% a year. If next year’s dividend is $10 andthe market capitalization rate is 8%, what is the current stockprice?

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Example

⇒ Current stock price:

P0 =10

0.08− 0.05= $333.33

⇒ The rate of return is 8% per year:

• For example, the prices in year 1, year 2, and year 3.:

Year 1 Year 2 Year 3

Divt $10 $10.5 $11.03Pricet 350 367.5 385.88

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Example

• From year 0 to year 1: 10+350−333.33333.33 = 0.08

• From year 1 to year 2: 10.5+367.5−350350 = 0.08

• From year 2 to year 3: 11.03+385.88−367.50367.50 = 0.08

⇒ A two-year inverstor expects 8% in each of the first 2 periods.Similarly, a three-year investor expects 8% in each of the first threeyears.

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Zero Growth

If the company does not grow at all and all of its earnings are paidout as dividends to shareholders. The firm simply produces aconstant stream of dividends.

⇒ Here we assume that:

Div1 = Div2 = · · ·

⇒ Applying the perpetuity formula:

P0 =Div11 + r

+Div2

(1 + r)2+ · · ·

P0 =Div1r

⇒ If dividends are constant, then the expected return on equity,r = Div

P0, is equal to the Dividend Yield.

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Exercise

⇒ Consider the following three stocks:

1 Stock A is expected to provide a dividend of $10 a shareforever.

2 Stock B is expected to pay a dividend of $5 next year.Thereafter, dividend growth is expected to be 4% a yearforever.

3 Stock C is expected to pay a dividend of $5 next year.Thereafter, dividend growth is expected to be 20% a year forfive years (i.e., until year 6) and zero thereafter.

If the market capitalization rate for each stock is 10%, which stockis the most valuable?

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Dividends vs Investment Growth

⇒ Stock prices increase with (1) dividends Div1 and (2) growth g .There is a trade-off:

• More dividends −→ less $ for investment −→ less growth andviceversa.

⇒ Consider Dividend Payout Ratio (PR) as the fraction ofearnings that the firm pays in dividends each year.

PRt =DivtEPSt

,

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Dividends vs Investment Growth

⇒ EPS are earnings per share. If the company has Nsharest ofshares outstanding at year t and annual earnigns Et , then:

Divt =Et

Nsharest∗ PRt ,

⇒ To increase dividends, the firm must:

• increase total earnings (E ).

• decrease shares outstanding.

• increase the dividend payout rate (PR).

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Growth Rate in Dividends g

⇒ The growth rate in dividends, g , is not zero if some earningsare not paid out as dividends. That is, only if some earnings areretained.

⇒ Hence, it is important to consider the Plowback Ratio:

⇒ Plowback Ratio: Fraction of earnings retained by the firm.

Plowback ratio = 1− payout ratio = 1− PRt ,

then,

Divt = (1− Plowback Ratio) x EPSt

⇒ How can we obtain the growth g?

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Growth Rate in Dividends g

⇒ Growth can be derived from applying the return on equity tothe percentage earnings plowed back into operations:

g = return on equity x plowback ratio

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Growth Rate in Dividends g

⇒ The change in the Book Equity (BE) depends on the plowbackratio and earnings.

BEt − BEt−1 = EPSt x Plowback Ratio

⇒ Earnings depend on the Return on Equity:

EPSt = BEt−1 x ROE

⇒ Then,

BEt − BEt−1 = BEt−1 x ROE x Plowback Ratio

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Growth Rate in Dividends g

⇒ This implies that:

BEt − BEt−1BEt−1

= ROE x Plowback Ratio

⇒ Recalling that:

Divt = (1− Plowback Ratio) x EPSt , and

EPSt = BEt−1 x ROE

⇒ Then,

Divt = (1− Plowback Ratio) x ROE x BEt−1,

Divt−1 = (1− Plowback Ratio) x ROE x BEt−2

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Growth Rate in Dividends g

⇒ Growth g is:

Divt −Divt−1Divt−1

=(1− Plowback Ratio) x ROE x(BEt−1 − BEt−2)

(1− Plowback Ratio) x ROE xBEt−2

=BEt−1 − BEt−2

BEt−2= ROE x Plowback Ratio

and

EPSt − EPSt−1EPSt−1

=Divt−Divt−1

1−Plowback RatioDivt−1

1−Plowback Ratio

= ROE x Plowback Ratio

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Growth Rate in Dividends g

⇒ Hence,

g = ROE x Plowback Ratio

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Growth Opportunities

⇒ Consider a company that does not grow at all. The companydoes not plow back any earnings and all of them are paid out asdividends. The firm simply produces a constant stream ofdividends.

⇒ Recalling the perpetuity formula, the return is equal to thedividend yield. Since all the earnings are paid out as dividends, theexpected return is also equal to the earnings per share divided bythe share price (EPS ratio). In math. words:

r =Div1P0

=EPS1P0

⇒ The price:

P0 =Div1r

=EPS1r

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Exercise

⇒ Recalling Company Z: The earnings and dividends per share areexpected to grow indefinitely by 5 percent a year. If next year’searnings are $15, dividend is $10 and the market capitalization rateis 8%, what is the current stock price?

P0 =10

0.08− 0.05= $333.33

What if the growth of this company would stop after year 4 andstarting with year 5 it will pay out all earnings as dividends?

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Example

⇒ First company forecasts to pay a $8.33 dividend next year,which represents 100% of its earnings. This will provide investorswith a 15% expected return. Second company decides to plowback 40% of the earnings at the firms current return on equity of25%. What are the values of the stocks of both companies?

• First company (no growth)

P0 =8.33

0.15= $55.53

• Second company (with growth)

g = 0.25 x 0.4 = 0.1

P0 =5.00

0.15− 0.1= $100

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Example

• If the company did not plowback some earnings, the stockprice is $55.53.

• If the company did plowback some earnings then the pricerose to $100.

• The difference between these two numbers is called thePresent Value of Growing Opportunities (PVGO = $44.47)

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Growth Opportunities

⇒ Many companies have growth opportunities. In this case, wecan think of stock price as the sum of two terms:

P0 =EPS1r

+ PVGO.

⇒ PVGO is the net present value of investments that the firm willmake in order to grow. This is the additional value if the firmretains earnings in order to fund new projects.

• EPS1/r is the capitalized value of earnings per share that thefirm would generate under a no-growth policy. This is thevalue of the firm if it simply distributes all earnings to thestockholders.

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Example

⇒ Remember the example about the company Z growing at anannual 5% with the next year’s dividend at $10 and the marketcapitalization rate at 8%

⇒ If this company were to redistribute all its earnings, it couldmaintain a level dividend stream of $15 a share. How much is themarket actually paying per share for growth opportunities?

P0 =15

0.08+ PVGO = $333.33

therefore, PVGO=$145.83.

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Growth Opportunities

⇒ Recalling:

P0 =EPS1r

+ PVGO

⇒ Rearranging terms, we can obtain the earnings to price ratio:

EPS1P0

= r

(1− PVGO

P0

)

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Exercise

1 Complete the table.

2 Assume that the opportunity cost of capital is 12%. Calculatethe value of the company’s stock (i.e. at year zero).

3 What part of that value reflects the discounted value of P3,the price forecasted for year 3?

4 What part of P3 reflects the present value of growthopportunities (PVGO) after year 3?

5 Suppose that competition will catch up with the company byyear 4, so that it can earn only its cost of capital on anyinvestments made in year 4 or subsequently. What is thestock worth now under this assummption?

1 2 3 4

Book equity at start of year 10.00Earnings per share, EPSReturn on equity, ROE 0.25 0.25 0.16 0.16Payout ratio 0.2 0.2 0.5 0.5Dividends per shareGrowth rate of dividends