Valuation of Stocks

The Present Valueof Common Stocks

Dividends versus Capital Gains Valuation of Different Types of Stocks

Zero Growth Constant Growth Differential Growth

Dividends versus Capital Gains

r

P

r

DivP

r

P

r

DivP

11

11

221

110

Dividends versus Capital Gains (contd.)

13

32

210

22

221

2210

1111

111

11

1

tt

t

r

Div

r

Div

r

Div

r

DivP

r

P

r

Div

r

Div

r

PDivDiv

rP

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:

Case 2: Constant Growth

)1(DivDiv 01 g

Since future cash flows grow at a constant rate forever, the value of a constant growth stock is the present value of a growing perpetuity:

grP

1

0

Div

Assume that dividends will grow at a constant rate, g, forever. i.e.

2012 )1(Div)1(DivDiv gg

..

3023 )1(Div)1(DivDiv gg

.

Case 3: Differential Growth

Assume that dividends will grow at different rates in the future

One constant growth rate in the foreseeable future A different constant growth 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 thereafter (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 Dividends will grow at rate g1 for N years

and grow at rate g2 thereafter

)(1Div 10 g 210 )(1Div g

…0 1 2

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

)(1Div

210

2

gg

gN

N

…N N+1

…

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

an N-year annuity growing at rate g1

N

N

Ar

g

gr

Div1P)1(

)1(1 1

1

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

NBr

grP

)1(

Div

2

1N

Case 3: Differential Growth

To value a Differential Growth Stock, we can use

NN

N

r

gr

r

g

gr

Div1P)1(

Div

)1(

)1(1 2

1N

1

1

A Differential Growth Example

A 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? The discount rate is 12%.

With the Formula

NN

N

r

gr

r

g

gr

Div1P)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(

108.12.

)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.

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?

Estimating Growth Rate

Earningsnextyear

Earnings

thisyear

+Retainedearningsthis year

×Return onretainedearnings

Increase in earnings

earnings retainedon Return

year thisEarnings

year thisearnings Retained

year thisEarnings

year thisEarnings

year thisEarnings

yearnext Earnings

Estimating Growth Rate (contd.)

1 + g = 1 + Retention ratio × Return on retained earnings

g = Retention ratio × Return on retained earnings

Estimating Discount Rate

The estimation of discount rate can be broken into two parts. Dividend yield (Div/P0) Growth rate of dividends (g)

In practice, there is a great deal of estimation error involved in estimating the discount rate, r

Estimating Discount Rate (contd.)

gP

Divr

gr

DivP

0

0

Value of Stock when Firm Acts as a Cash Cow

r

Div

r

EPSValue

Value of a firm that pays out 100% of its earnings as dividends

Price Earnings Ratio Many analysts frequently relate earnings per

share to price The price earnings ratio or multiple

Calculated as current stock price divided by annual EPS

EPS

shareper Priceratio P/E