Principles of Bond and Stock Valuation

Estimating value by discounting future cash flows

Bond Price (semiannual coupons)

Tr

FC

r

C

r

C

P22 )

21(

2...)2

1(

2

)2

1(

2

• P = bond price• C = annual coupon ($)• F = face value (par, principal)• r = yield (annual)• T = years to maturity

Bond Price Relative to Par

• C/F > r Bond sells above par (premium bond)

• C/F = r Bond sells at par (par bond)

• C/F < r Bond sells below par (discount bond)

Zero Coupon Bonds

TrF

P2)

21(

n

n rz

)2

1(

1

• Zeros make only one payment at maturity• zn is the price today of $1 to be delivered n

semiannual periods from today• We can represent any bond price in terms of

zero coupon bond prices

Recall Applying Discount Factors to Cash Flow Streams

TT

TT

CFr

CFr

CFr

CF

r

CF

r

CF

r

CFCFP

)1(

1...

)1(

1

1

1

)1(...

)1()1(

2210

221

0

• Discount factors are like prices (exchange rates)

Price of Coupon Bond in Terms of Zeros

F

Cz

Cz

CzP T 2

...22 221

Common Stock Valuation

• I buy a stock now for P0

• I expect to sell one year from now for P1

• I collect the dividend DIV1 paid in Year 1• My opportunity cost rate of return is r

r

PDIVP

1

110

The One-Year Rate of Return

0

01

0

1

P

PP

P

DIVr

• First term represents dividend yield, second term represents capital gains

• Stock will be priced so that investors can expect to earn their opportunity cost rate of return

What Determines Future Stock Prices?

333

221

033

2

2221

022

1

)1()1(11

)1(11

r

PDIV

r

DIV

r

DIVP

r

PDIVP

r

PDIV

r

DIVP

r

PDIVP

The Dividend Discount Model

• Carrying this process on out indefinitely:

10 )1(t

tt

r

DIVP

But how can we estimate all future dividends?

Constant Growth Dividend Discount Model

• Suppose dividends grow at a constant rate g each year forever:

gr

DIVP

1

0

Stock Price Grows at rate g in Constant Growth Model

012

1 )1()1(

Pggr

DIVg

gr

DIVP

Dividends Growing at Sustainable Growth Rate

• If dividends grow because the firm pays out the fraction (1-b) of each year t’s earnings Et as dividends and retains the fraction b, reinvesting to earn the rate ROE, dividends will grow at the sustainable rate = bROE:

bROEr

EbP

10

)1(

Price-Earnings Ratio

bROEr

b

E

P

1

1

0

• PE ratio as discount rate , growth rate , and dividend payout , other things equal

• However, other things are not equal. An increase in payout lowers the growth rate

Investment Opportunities, Growth and Stock Prices

Dividend Discount Model

00 )1(t

tt

r

DIVP

gr

DIVP

1

0

• Left-hand equation is general version of Dividend Discount Model (DDM)

• Right-hand equation is special case of DDM when there is constant perpetual growth

General Case Constant Growth Case

Dividends Growing at Sustainable Growth Rate

• If dividends grow because the firm pays out the fraction (1-b) of each year t’s earnings Et as dividends and retains the fraction b, reinvesting to earn the rate ROE, dividends will grow at the sustainable rate = bROE:

bROEr

EbP

10

)1(

Growth Opportunities Model

VGOr

EP 10

• Growth Opportunities Model is an alternative but equivalent model to the DDM

• First term is the value of the earnings stream from existing assets

• VGO is value of growth opportunities

Growth Opportunities Model

1

10 )1(t

tt

r

NPV

r

EP

gr

NPV

r

EP

11

0

Second term in both expressions above is VGO (PV of NPVs of all future investments)

Value is added from positive-NPV future projects rather than a higher growth rate per se

General Case Constant Growth Case

Equivalent Approaches to Stock Valuation

1

10 )1(t

tt

r

NPV

r

EP

gr

NPV

r

EP

11

0

00 )1(t

tt

r

DIVP

gr

DIVP

1

0

Growth Opportunities Approach

General Case Constant Growth Case

Dividend Discount Approach