Stock Valuation• Understand how stock prices depend on future

dividends and dividend growth• Estimates of Parameters in the Dividend-Discount

Model• Compute present value of stock prices using the

dividend growth model• Understand how growth opportunities affect stock

values• Understand the PE ratio• Understand how stock markets work

• Preferred stock• Efficient Market Hypothesis (EMH)

Common Stock: Owners, Directors, and Managers

• Represents ownership.

• Ownership implies control.

• Stockholders elect directors.

• Directors hire management.

• Since managers are “agents” of shareholders, their goal should be: Maximize stock price.

Different Approaches for Valuing Common Stock

• Dividend growth model

• Using the multiples of comparable firms

• Free cash flow method

The Present Value of Common Stocks

• Dividends versus Capital Gains

• Valuation of Different Types of Stocks– Zero Growth– Constant Growth– Differential Growth

Case 1: Zero Growth

• Assume that dividends will remain at the same level forever

rP

rrrP

Div

)1(

Div

)1(

Div

)1(

Div

0

33

22

11

0

321 DivDivDiv Since future cash flows are constant, the value of a zero

growth stock is the present value of a perpetuity:

If g = 0, the dividend stream is a perpetuity.

2.00 2.002.00

0 1 2 3rs=13%

P0 = = = $15.38.PMT

r

$2.00

0.13^

Stock Value = PV of Dividends

Case 2What is a constant growth stock?

One whose dividends are expected togrow forever at a constant rate, g.

P0 =^

(1+r)1 (1+r)2 (1+r)3 (1+r)∞

D1 D2 D3 D∞+ + +…+

Case 2For a constant growth stock:

D1 = D0(1+g)1

D2 = D0(1+g)2

Dt = D0(1+g)t

If g is constant and less than rs, then:

P0 = ^ D0(1+g)

r - g=

D1

r - g

Intrinsic Stock Value D0 = 2.00, rs = 13%, g = 6%.

Constant growth model:

= = $30.29.0.13 - 0.06

$2.12 $2.12

0.07

P0 = ^ D0(1+g)

r - g=

D1

r - g

Case 3: Differential Growth• Assume that dividends will grow at different rates

in the foreseeable future and then will grow at a constant rate thereafter.

• To value a Differential Growth Stock, we need to:– Estimate future dividends in the foreseeable

future.– Estimate the future stock price when the stock

becomes a Constant Growth Stock (case 2).– Compute the total present value of the

estimated future dividends and future stock price at the appropriate discount rate.

Case 3: Differential Growth

)(1DivDiv 101 g

Assume that dividends will grow at rate g1 for N years and grow at rate g2 thereafter

210112 )(1Div)(1DivDiv gg

NNN gg )(1Div)(1DivDiv 1011

)(1)(1Div)(1DivDiv 21021 ggg NNN

...

...

Case 3: Differential Growth

)(1Div 10 g

Dividends will grow at rate g1 for N years and grow at rate g2 thereafter

210 )(1Div g

Ng )(1Div 10 )(1)(1Div

)(1Div

210

2

gg

gN

N

…0 1 2

…N N+1

…

Case 3: Differential GrowthWe can value this as the sum of:

an N-year annuity growing at rate g1

T

T

A r

g

gr

CP

)1(

)1(1 1

1

plus the discounted value of a perpetuity growing at rate g2 that starts in year N+1

NB r

grP

)1(

Div

2

1N

Case 3: Differential GrowthTo value a Differential Growth Stock, we can use

NT

T

r

gr

r

g

gr

CP

)1(

Div

)1(

)1(1 2

1N

1

1

Or we can cash flow it out.

A Differential Growth ExampleA common stock just paid a dividend of $2. The

dividend is expected to grow at 8% for 3 years, then it will grow at 4% in perpetuity.

What is the stock worth if the rate of return is 12%?

With the Formula

NT

T

r

gr

r

g

gr

CP

)1(

Div

)1(

)1(1 2

1N

1

1

3

3

3

3

)12.1(

04.12.)04.1()08.1(2$

)12.1(

)08.1(1

08.12.

)08.1(2$

P

3)12.1(

75.32$8966.154$ P

31.23$58.5$ P 89.28$P

A Differential Growth Example (continued)

08).2(1$ 208).2(1$…

0 1 2 3 4

308).2(1$ )04.1(08).2(1$ 3

16.2$ 33.2$

0 1 2 3

08.

62.2$52.2$

89.28$)12.1(

75.32$52.2$

)12.1(

33.2$

12.1

16.2$320

P

75.32$08.

62.2$3 P

The constant growth phase

beginning in year 4 can be valued as a

growing perpetuity at time 3.

Supernormal Growth Stock

• Supernormal growth of 30% for 3 years, and then long-run constant g = 6%.

• Can no longer use constant growth model.

• However, growth becomes constant after 3 years.

Nonconstant growth followed by constant growth (D0 = $2):

0

2.3009

2.6470

3.0453

46.1135

1 2 3 4r=13%

54.1067 = P0

g = 30% g = 30% g = 30% g = 6%

2.60 3.38 4.394 4.6576

^P3 = ^ $4.6576

0.13 – 0.06= $66.5371

Estimates of Parameters in the Dividend-Discount Model

• The value of a firm depends upon its growth rate, g, and its discount rate, r. – Where does g come from?– Where does r come from?

Formula for Firm’s Growth Rate

g = Retention ratio × Return on retained earnings

Where does r come from?

• The discount rate can be broken into two parts. – The dividend yield – The growth rate (in dividends)

• In practice, there is a great deal of estimation error involved in estimating r.

Growth Opportunities

• Growth opportunities are opportunities to invest in positive NPV projects.

• The value of a firm can be conceptualized as the sum of the value of a firm that pays out 100% of its earnings as dividends plus the net present value of the growth opportunities.

NPVGOR

EPSP

NPVGO Model: Example

Consider a firm that has forecasted EPS of $5, a discount rate of 16%, and is currently priced at $75 per share.

• We can calculate the value of the firm as a cash cow.

• So, NPVGO must be: $75 - $31.25 = $43.75

25.31$16.

5$EPS0

RP

Retention Rate and Firm Value

• An increase in the retention rate will:– Reduce the dividend paid to shareholders– Increase the firm’s growth rate

• These have offsetting influences on stock price

• Which one dominates?– If ROE>R, then increased retention increases

firm value since reinvested capital earns more than the cost of capital.

Price Earnings Ratio• Many analysts frequently relate earnings per share to

price.• The price earnings ratio is a.k.a the multiple

– Calculated as current stock price divided by annual EPS

– The Wall Street Journal uses last 4 quarter’s earnings

• Firms whose shares are “in fashion” sell at high multiples. Growth stocks for example.

• Firms whose shares are out of favor sell at low multiples. Value stocks for example.

EPS

shareper Priceratio P/E

PE and NPVGO

• Recall,

• Dividing every term by EPS provides the following description of the PE ratio:

• So, a firm’s PE ratio is positively related to growth opportunities and negatively related to risk (R)

NPVGOR

EPSP

EPS

NPVGO

RPE

1

Other Price Ratio Analysis• Many analysts frequently relate earnings per share to

variables other than price, e.g.:– Price/Cash Flow Ratio

• cash flow = net income + depreciation = cash flow from operations or operating cash flow

– Price/Sales• current stock price divided by annual sales per

share– Price/Book (a.k.a Market to Book Ratio)

• price divided by book value of equity, which is measured as assets - liabilities

Preferred Stock

• Hybrid security.

• Similar to bonds in that preferred stockholders receive a fixed dividend which must be paid before dividends can be paid on common stock.

• However, unlike bonds, preferred stock dividends can be omitted without fear of pushing the firm into bankruptcy.

Expected return on preferred stock, given Vps = $50 and annual dividend =

$5

Vps = $50 =$5

rps

rps

$5

$50= = 0.10 = 10.0%

Are volatile stock prices consistent with rational pricing?

• Small changes in expected g and rs cause large changes in stock prices.

• As new information arrives, investors continually update their estimates of g and r.

• If stock prices aren’t volatile, then this means there isn’t a good flow of information.

What is market equilibrium?

• In equilibrium, stock prices are stable. There is no general tendency for people to buy versus to sell.

• The expected price, P, must equal the actual price, P. In other words, the fundamental value must be the same as the price.

(More…)

What’s the Efficient MarketHypothesis (EMH)?

• Securities are normally in equilibrium and are “fairly priced.” One cannot “beat the market” except through good luck or inside information.

(More…)

Weak-form EMH

• Can’t profit by looking at past trends. A recent decline is no reason to think stocks will go up (or down) in the future. Evidence supports weak-form EMH, but “technical analysis” is still used.

Semistrong-form EMH

• All publicly available information is reflected in stock prices, so it doesn’t pay to pore over annual reports looking for undervalued stocks. Largely true.

Strong-form EMH

• All information, even inside information, is embedded in stock prices. Not true--insiders can gain by trading on the basis of insider information, but that’s illegal.

Markets are generally efficient because:

• 100,000 or so trained analysts--MBAs, CFAs, and PhDs--work for firms like Fidelity, Merrill, Morgan, and Prudential.

• These analysts have similar access to data and megabucks to invest.

• Thus, news is reflected in P0 almost instantaneously.

Stock Market Reporting

52WEEKS YLD VOL NETHI LO STOCKSYMDIV % PE 100s HI LOCLOSE CHG

52.75 19.06 Gap Inc GPS 0.09 0.5 15 65172 20.50 19 19.25 -1.75

Gap has been as high as $52.75 in the last year.

Gap has been as low as $19.06 in the last year.

Gap pays a dividend of 9 cents/share

Given the current price, the dividend yield is ½ %

Given the current price, the PE ratio is 15 times earnings

6,517,200 shares traded hands in the last day’s trading

Gap ended trading at $19.25, down $1.75 from yesterday’s close

Stock Market Reporting

52WEEKS YLD VOL NETHI LO STOCKSYMDIV % PE 100s HI LOCLOSE CHG

52.75 19.06 Gap Inc GPS 0.09 0.5 15 65172 20.50 19 19.25 -1.75

Gap Incorporated is having a tough year, trading near their 52-week low. Imagine how you would feel if within the past year you had paid $52.75 for a share of Gap and now had a share worth $19.25! That 9-cent dividend wouldn’t go very far in making amends.

Yesterday, Gap had another rough day in a rough year. Gap “opened the day down” beginning trading at $20.50, which was down from the previous close of $21.00 = $19.25 + $1.75

Looks like cargo pants aren’t the only things on sale at Gap.

Summary and Conclusions

A stock can be valued by discounting its dividends. There are three cases:

1. Zero growth in dividends

2. Constant growth in dividends

3. Differential growth in dividends

rP

Div0

grP

1

0

Div

NT

T

r

gr

r

g

gr

CP

)1(

Div

)1(

)1(1 2

1N

1

1