How to Value Bonds and Stocks

How to Value Bonds and Stocks

2

What is a Bond?

A bond is a legally binding agreement between a borrower and a lenderIOU

3

Bond Terminology Face value (F) or Principal

For a corporate bond this is generally $1,000 Zero- coupon bond Coupon Rate

This is a Stated Annual RateDetermines the coupon payment

Coupon payment (C ) Rating

Bond Pricing Terminology Par

The price of the bond equals its face value Premium

The price of the bond is greater than its face value Discount

The price of the bond is less than its face value Yield to Maturity

4

5

Yield to Maturity YTM is the return that the bond is offering if

you bought it today and held it till maturity The YTM is determined by the riskiness of

the bond Risk comes from:

1. Risk of default Risk is often measured with bond ratings

Investment Grade / Junk

2. Time to maturity Longer term bonds are riskier

6

Pure Discount Bonds Have no coupon

Sometimes called zeroes, deep discount bonds, or original issue discount bonds (OIDs)

Example: T-Bill

Yield to Maturity comes only from the difference between the purchase price and principal repayment

A pure discount bond cannot sell at a premiumWHY?

7

Pure Discount Bonds

Information needed for valuing pure discount bonds: Time to maturity (T) = Maturity date - today’s date Face value (F) Discount rate (r)

TR

F VP V

)1(

Present value of a pure discount bond at time 0:

0

0$ 0$

2

0$

1T

F$

T

8

Pure Discount Bond: ExampleFind the value of a 30-year zero-coupon bond with a $1,000 par value and a YTM of 6%.

1,000/(1.0630) = 174.11

0

0$ 0$

2

0$

2 9

0 0 0,1$

30

0

0$ 0$

2

0$

29

0 0 0,1$

30

9

Coupon Bonds

Make periodic coupon payments in addition to repaying the principal

Coupon payments are the same each period Typically occur semi-annual

An investor’s return is comprised of:1. Difference between the purchase price & face value 2. Coupon payments

10

Valuing a Coupon Bond

The value of a bond is simply the present value of it’s future cash flows

We value a bond is a package of two investments:

1. Present value of the coupon payments

2. Present value of the principal repayment

Determining Coupon Payments

Coupon ($)= (Principal * Coupon Rate) / FrequencyEx:

8% semi-annual (1,000 * 0.08) / 2 = 40

12% monthly (1,000 * 0.12) / 12 = 10

20% annual (1,000 * 0.20) / 1 = 200

11

12

Coupon Bond Pricing Equation

TT )(1

FV

)(1

11

C Value Bond

RRr

AnnuityCoupon Payments

Lump SumPrincipal

Repayment

13

Coupon Bond Pricing: BA II plus

N = The number of coupon payments I/Y = The rate corresponding to the coupon

frequency PV = The price of the bond today PMT= The amount of the coupon payment FV = The principal that will be repaid

Coupon Example 2

What is the yield to maturity of a 9% 15 year, bond that sells for $1,200%?

N = I/Y = PV = PMT = FV =

14

30 = 15 * 2

3.42%??

-1,200

45 = (1000 * 0.08)/2

1,000

The 3.42% is a 6 month rate, the YTM = 6.84%

Coupon Example 1

What is the present value of a 8% 10 year, bond with the yield to maturity is 12%?

N = I/Y = PV = PMT = FV =

15

20 = 10 * 2

402.44

12

??

40 = (1000 * 0.08)/2

1,000

16

Valuing a Corporate Bond

DuPont issued a 30 year bonds with a coupon rate of 7.95%. Interest is paid semi-annually

These bonds currently have 28 years remaining to maturity and are rated AA.

The bonds have a par value of $1,000 Newly issued AA bonds with maturities greater than

10 years are currently yielding 7.73% What is the value of DuPont bond today?

17

DuPont example (continued)

Annual interest ($) = Semiannual coupon payment = Semiannual discount rate = Number of semiannual periods= PV=

18

DuPont example (continued)

Annual interest ($) = 0.0795*1000 =79.50 Semiannual coupon payment = Semiannual discount rate = Number of semiannual periods= PV=

19

DuPont example (continued)

Annual interest ($) = 0.0795*1000 =79.50 Semiannual coupon payment = 79.5/2= 39.75 Semiannual discount rate = Number of semiannual periods= PV=

20

DuPont example (continued)

Annual interest ($) = 0.0795*1000 =79.50 Semiannual coupon payment = 79.5/2= 39.75 Semiannual discount rate = 0.0773/2 =0.03865 Number of semiannual periods= PV=

21

DuPont example (continued)

Annual interest ($) = 0.0795*1000 =79.50 Semiannual coupon payment = 79.5/2= 39.75 Semiannual discount rate = 0.0773/2 =0.03865 Number of semiannual periods= 28*2 = 56 PV=

N = ??, I/Y = ??, PV= ????, PMT =??, FV=??

22

DuPont example (continued)

Annual interest ($) = 0.0795*1000 =79.50 Semiannual coupon payment = 79.5/2= 39.75 Semiannual discount rate = 0.0773/2 =0.03865 Number of semiannual periods= 28*2 = 56 PV=

N= 56, I/Y = ??, PV= ????, PMT =??, FV=??

23

DuPont example (continued)

Annual interest ($) = 0.0795*1000 =79.50 Semiannual coupon payment = 79.5/2= 39.75 Semiannual discount rate = 0.0773/2 =0.03865 Number of semiannual periods= 28*2 = 56 PV=

N = 56, I/Y = 3.865, PV= ????, PMT = ??, FV= ??

24

DuPont example (continued)

Annual interest ($) = 0.0795*1000 =79.50 Semiannual coupon payment = 79.5/2= 39.75 Semiannual discount rate = 0.0773/2 =0.03865 Number of semiannual periods= 28*2 = 56 PV=

N = 56, I/Y = 3.865, PV= ????, PMT = 39.75, FV= ??

25

DuPont example (continued)

Annual interest ($) = 0.0795*1000 =79.50 Semiannual coupon payment = 79.5/2= 39.75 Semiannual discount rate = 0.0773/2 =0.03865 Number of semiannual periods= 28*2 = 56 PV=

N = 56, I/Y = 3.865, PV= ????, PMT = 39.75, FV = 1,000

26

DuPont example (continued)

Annual interest ($) = 0.0795*1000 =79.50 Semiannual coupon payment = 79.5/2= 39.75 Semiannual discount rate = 0.0773/2 =0.03865 Number of semiannual periods= 28*2 = 56 PV= 1,025.06

The bond is currently selling for 1,025.06

N = 56, I/Y = 3.865, PV= ????, PMT = 39.75, FV= 1,000

27

Level Coupon Bond: Example (Given) Consider a U.S. government bond with a 6 3/8%

coupon that expires in December 2010. The Par Value of the bond is $1,000. Coupon payments are made semi-annually (June 30 and

December 31 for this particular bond). Since the coupon rate is 6 3/8%, the payment is $31.875. On January 1, 2006 the size and timing of cash flows are:

The require annual rate is 5%

0 6/1/1

8 7 5.3 1$

0 6/3 0/6

8 7 5.3 1$

0 6/3 1/1 2

8 7 5.3 1$

1 0/3 0/6

8 7 5.0 3 1,1$

1 0/3 1/1 2

28

Level Coupon Bond: Example (Given) Coupon Rate 6 3/8%, pay semi-annually

10 Semi-Annual Payments of $31.875.

Maturity December 2010, Start Jan. 2006 The Par Value of the bond is $1,000. The require annual rate is 5% N = 10, I/Y = 2.5, PV=???, PMT = 31.875,

FV=1,000::: PV = $1,060.17

1010 025).(1

1,000

025).(1

11

025.0

31.875 Value Bond

29

Valuing a Corporate Bond (Given) Value a bond with the following

characteristics (calculator): Face value: $1,000Coupon rate (C ): 8%Time to maturity: 4 yearsDiscount rate: 9%Present Value: $967.02

You should know how to get any one of these numbers given the other 4.

YTM and Bond Prices

How are prices and YTM related?Inversely, as one goes up the other goes downAs you pay more for the bond you earn a lower

return

30

31

Coupon Rate and YTM Coupon rate = YTM

Price = Face, Bond is selling at Par Coupons provide all the required return

Coupon rate > YTM Price > Face, Bond is selling at a Premium Coupons provide more than the required return

Coupon rate < YTM Price < Face, Bond is selling at a Discount Coupons do not provide the required return need

to increased the return by paying less

32

YTM and Bond Value

800

1000

1100

1200

1300

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Discount Rate

Bon

d V

alu

e

6 3/8

When the YTM < coupon, the bond trades at a premium.

When the YTM = coupon, the bond trades at par.

When the YTM > coupon, the bond trades at a discount.

Coupon Rate

33

Computing Yield to Maturity Finding the YTM requires trial and error if you

do not have a financial calculator If you have a financial calculator, enter N, PV,

PMT, and FV,Remembering the sign convention

PMT and FV need to have the same sign, PV the opposite sign

34

YTM with Semiannual Coupons A bond has a 10% coupon rate, 20yrs to maturity, makes coupon

payments semi-annually, a $1,000 face, and is selling at $1,197.93 Is the YTM more or less than 10%?

LESSWhat is the semi-annual coupon payment?

(1,000 * 0.10) / 2 = $50How many periods are there?

20 * 2 = 40

What is the YTM? N= 40,I/Y = ?, PV= -1197.93, PMT = 50, FV= 1,000→ 3.99%YTM = 7.99998011%

35

YTM with Annual Coupons (Given)

Consider a bond with a 10% annual coupon rate, 15 years to maturity, and a par value of $1,000. The current price is $928.09. Will the YTM be more or less than 10%?

MORE What is the YTM?

N = 15 I/Y = ???? = 11% PV = 928.09 PMT = 100 FV = 1000

36

Rate Changes and Bond Prices Known as interest rate risk Consider two identical 8% coupon bonds except

that one matures in 4 years, the other matures in 10 years

Calculate the change in the price of each bond if:Interest rates fall from 8% to 6%Interest rates rise from 8% to 10%

Rate Change and Bond Pricing 4 years @ 6%, 8% Coupon

N=_, I/Y = _, PV=_, PMT = _, FV = _

4 years @ 10%, 8% CouponN=_, I/Y = _, PV=_ PMT = _, FV = _

10 years @ 6%, 8% CouponN=_, I/Y = _, PV=_ PMT = _, FV = _

10 Years @ 10%, 8% CouponN=_, I/Y = _, PV=_ PMT = _, FV = _

37

Rate Change and Bond Pricing 4 years @ 6%, 8% Coupon

N=8, I/Y = _, PV=_,PMT = _, FV = _

4 years @ 10%, 8% CouponN=_, I/Y = _, PV=_ PMT = _, FV = _

10 years @ 6%, 8% CouponN=_, I/Y = _, PV=_ PMT = _, FV = _

10 Years @ 10%, 8% CouponN=_, I/Y = _, PV=_ PMT = _, FV = _

38

Rate Change and Bond Pricing 4 years @ 6%, 8% Coupon

N=8, I/Y = 3, PV=_, PMT = _, FV = _

4 years @ 10%, 8% CouponN=_, I/Y = _, PV=_ PMT = _, FV = _

10 years @ 6%, 8% CouponN=_, I/Y = _, PV=_ PMT = _, FV = _

10 Years @ 10%, 8% CouponN=_, I/Y = _, PV=_ PMT = _, FV = _

39

Rate Change and Bond Pricing 4 years @ 6%, 8% Coupon

N=8, I/Y = 3, PV=?, PMT = _, FV = _

4 years @ 10%, 8% CouponN=_, I/Y = _, PV=_ PMT = _, FV = _

10 years @ 6%, 8% CouponN=_, I/Y = _, PV=_ PMT = _, FV = _

10 Years @ 10%, 8% CouponN=_, I/Y = _, PV=_ PMT = _, FV = _

40

Rate Change and Bond Pricing 4 years @ 6%, 8% Coupon

N=8, I/Y = 3, PV=?, PMT = 40, FV = _

4 years @ 10%, 8% CouponN=_, I/Y = _, PV=_ PMT = _, FV = _

10 years @ 6%, 8% CouponN=_, I/Y = _, PV=_ PMT = _, FV = _

10 Years @ 10%, 8% CouponN=_, I/Y = _, PV=_ PMT = _, FV = _

41

Rate Change and Bond Pricing 4 years @ 6%, 8% Coupon

N=8, I/Y = 3, PV=?, PMT = 40, FV = 1,000

4 years @ 10%, 8% CouponN=_, I/Y = _, PV=_ PMT = _, FV = _

10 years @ 6%, 8% CouponN=_, I/Y = _, PV=_ PMT = _, FV = _

10 Years @ 10%, 8% CouponN=_, I/Y = _, PV=_ PMT = _, FV = _

42

Rate Change and Bond Pricing 4 years @ 6%, 8% Coupon

N=8, I/Y = 3, PV=? PMT = 40, FV = 1,000 PV =$1,070.20

4 years @ 10%, 8% CouponN=8, I/Y = 5, PV=? PMT = 40, FV = 1,000

PV = $935.37

10 years @ 6%, 8% CouponN=_, I/Y = _, PV=_, PMT = _, FV = _

10 Years @ 10%, 8% CouponN=_, I/Y = _, PV=_, PMT = _, FV = _

49

Rate Change and Bond Pricing 4 years @ 6%, 8% Coupon

N=8, I/Y = 3, PV=? PMT = 40, FV = 1,000 PV =$1,070.20

4 years @ 10%, 8% CouponN=8, I/Y = 5, PV=? PMT = 40, FV = 1,000

PV = $935.37

10 years @ 6%, 8% CouponN=20, I/Y = _, PV=_ PMT = _, FV = _

10 Years @ 10%, 8% CouponN=_, I/Y = _, PV=_, PMT = _, FV = _

50

Rate Change and Bond Pricing 4 years @ 6%, 8% Coupon

N=8, I/Y = 3, PV=? PMT = 40, FV = 1,000 PV =$1,070.20

4 years @ 10%, 8% CouponN=8, I/Y = 5, PV=? PMT = 40, FV = 1,000

PV = $935.37

10 years @ 6%, 8% CouponN=20, I/Y = 3, PV=_ PMT = _, FV = _

10 Years @ 10%, 8% CouponN=_, I/Y = _, PV=_, PMT = _, FV = _

51

Rate Change and Bond Pricing 4 years @ 6%, 8% Coupon

N=8, I/Y = 3, PV=? PMT = 40, FV = 1,000 PV =$1,070.20

4 years @ 10%, 8% CouponN=8, I/Y = 5, PV=? PMT = 40, FV = 1,000

PV = $935.37

10 years @ 6%, 8% CouponN=20, I/Y = 3, PV=? PMT = _, FV = _

10 Years @ 10%, 8% CouponN=_, I/Y = _, PV=_, PMT = _, FV = _

52

Rate Change and Bond Pricing 4 years @ 6%, 8% Coupon

N=8, I/Y = 3, PV=? PMT = 40, FV = 1,000 PV =$1,070.20

4 years @ 10%, 8% CouponN=8, I/Y = 5, PV=? PMT = 40, FV = 1,000

PV = $935.37

10 years @ 6%, 8% CouponN=20, I/Y = 3, PV=? PMT = 40, FV = _

10 Years @ 10%, 8% CouponN=_, I/Y = _, PV=_, PMT = _, FV = _

53

Rate Change and Bond Pricing 4 years @ 6%, 8% Coupon

N=8, I/Y = 3, PV=? PMT = 40, FV = 1,000 PV =$1,070.20

4 years @ 10%, 8% CouponN=8, I/Y = 5, PV=? PMT = 40, FV = 1,000

PV = $935.37

10 years @ 6%, 8% CouponN=20, I/Y = 3, PV=? PMT = 40, FV = 1,000

10 Years @ 10%, 8% CouponN=_, I/Y = _, PV=_, PMT = _, FV = _

54

Rate Change and Bond Pricing 4 years @ 6%, 8% Coupon

N=8, I/Y = 3, PV=? PMT = 40, FV = 1,000 PV =$1,070.20

4 years @ 10%, 8% CouponN=8, I/Y = 5, PV=? PMT = 40, FV = 1,000

PV = $935.37

10 years @ 6%, 8% CouponN=20, I/Y = 3, PV=? PMT = 40, FV = 1,000

PV = $1,148.77

10 Years @ 10%, 8% CouponN=_, I/Y = _, PV=_, PMT = _, FV = _

55

Rate Change and Bond Pricing 4 years @ 6%, 8% Coupon

N=8, I/Y = 3, PV=? PMT = 40, FV = 1,000 PV =$1,070.20

4 years @ 10%, 8% CouponN=8, I/Y = 5, PV=? PMT = 40, FV = 1,000

PV = $935.37

10 years @ 6%, 8% CouponN=20, I/Y = 3, PV=? PMT = 40, FV = 1,000

PV = $1,148.77

10 Years @ 10%, 8% CouponN=20, I/Y = _, PV=_ PMT = _, FV = _

56

Rate Change and Bond Pricing 4 years @ 6%, 8% Coupon

N=8, I/Y = 3, PV=? PMT = 40, FV = 1,000 PV =$1,070.20

4 years @ 10%, 8% CouponN=8, I/Y = 5, PV=? PMT = 40, FV = 1,000

PV = $935.37

10 years @ 6%, 8% CouponN=20, I/Y = 3, PV=? PMT = 40, FV = 1,000

PV = $1,148.77

10 Years @ 10%, 8% CouponN=20, I/Y = 5, PV=_ PMT = _, FV = _

57

Rate Change and Bond Pricing 4 years @ 6%, 8% Coupon

N=8, I/Y = 3, PV=? PMT = 40, FV = 1,000 PV =$1,070.20

4 years @ 10%, 8% CouponN=8, I/Y = 5, PV=? PMT = 40, FV = 1,000

PV = $935.37

10 years @ 6%, 8% CouponN=20, I/Y = 3, PV=? PMT = 40, FV = 1,000

PV = $1,148.77

10 Years @ 10%, 8% CouponN=20, I/Y = 5, PV=? PMT = _, FV = _

58

Rate Change and Bond Pricing 4 years @ 6%, 8% Coupon

N=8, I/Y = 3, PV=? PMT = 40, FV = 1,000 PV =$1,070.20

4 years @ 10%, 8% CouponN=8, I/Y = 5, PV=? PMT = 40, FV = 1,000

PV = $935.37

10 years @ 6%, 8% CouponN=20, I/Y = 3, PV=? PMT = 40, FV = 1,000

PV = $1,148.77

10 Years @ 10%, 8% CouponN=20, I/Y = 5, PV=? PMT = 40, FV = _

59

Rate Change and Bond Pricing 4 years @ 6%, 8% Coupon

N=8, I/Y = 3, PV=? PMT = 40, FV = 1,000 PV =$1,070.20

4 years @ 10%, 8% CouponN=8, I/Y = 5, PV=? PMT = 40, FV = 1,000

PV = $935.37

10 years @ 6%, 8% CouponN=20, I/Y = 3, PV=? PMT = 40, FV = 1,000

PV = $1,148.77

10 Years @ 10%, 8% CouponN=20, I/Y = 5, PV=? PMT = 40, FV = 1,000

60

Rate Change and Bond Pricing 4 years @ 6%, 8% Coupon

N=8, I/Y = 3, PV=? PMT = 40, FV = 1,000 PV =$1,070.20

4 years @ 10%, 8% CouponN=8, I/Y = 5, PV=? PMT = 40, FV = 1,000

PV = $935.37

10 years @ 6%, 8% CouponN=20, I/Y = 3, PV=? PMT = 40, FV = 1,000

PV = $1,148.77

10 Years @ 10%, 8% CouponN=20, I/Y = 5, PV=? PMT = 40, FV = 1,000

PV = $875.38 61

62

Interest Rates and Time to Maturity

The longer a bond has till maturity, the greater the price impact of a change in interest rates

WHY? Longer maturity bond have more payments

affected by the rate change, and the principal repayment is further away so it will be more heavily discounted

63

Interest Rates and Bond Prices Bond Prices and Interest Rates have an Inverse

Relationship

64

Pricing Stocks

Remember: The value of any asset is the present value of its expected future cash flows.

Bond cash flows are: Stock produces cash flows from:

DividendsCapital Gains

Coupon & Principal

65

Stock Valuation Terminology

Dt or Divt –dividend expected at time t

P0 – market price of stock at time 0

Pt – expected mkt price of stock at time t

g- expected growth rate of dividends rs or re- required rate of return on equity

D1 / P0 – expected one-year dividend yield

(P1 - P0)/ P0 – expected one year capital gainThe stocks total return = div yield + cap. gain

66

Valuing Common Stock The price of a share is simply the present value of

the expected future cash flowsAn investor planning on selling his share in a

year is willing to pay:The investor buying the share next year plans

on selling it a year later so he is only willing to pay:

67

Valuing Common Stock The price of a share is simply the present value of

the expected future cash flowsAn investor planning on selling his share in a

year is willing to pay: P0=(D1+P1)/(1+R)The investor buying the share next year plans

on selling it a year later so he is only willing to pay:

68

Valuing Common Stock The price of a share is simply the present value of

the expected future cash flowsAn investor planning on selling his share in a

year is willing to pay: P0=(D1+P1)/(1+R)The investor buying the share next year plans on

selling it a year later so he is only willing to pay: P1=(D2+P2)/(1+R)

Therefore: P0

69

Valuing Common Stock The price of a share is simply the present value of the

expected future cash flowsAn investor planning on selling his share in a year is

willing to pay: P0=(D1+P1)/(1+R)The investor buying the share next year plans on selling

it a year later so he is only willing to pay: P1=(D2+P2)/(1+R)

Therefore: P0=(D1+{(D2+P2)/(1+R)})/(1+R)P0=D1 / (1+R) + (D2 + P2)/(1+R)2

70

Keep Going This process can be repeated into the future

Using summation: P0 = H Dh / (1 + r)h + PH / (1 + r)H What happens to PH as H approaches infinity?

The present value becomes insignificant

PD iv

r

D iv

r

D iv P

rH H

H01

12

21 1 1

( ) ( )

. . .( )

71

Dividend Valuation Model

As H approaches infinity PH goes to zeroBecause of this we only need to be concerned with

the stock’s future dividends

The price of a stock is equal to the present value of its expected future dividends

72

Constant Dividend

How do you value a stock that will pay a constant dividend? Hint: what does the cash flow stream look similar

to?Firms are a going concern so treat dividends as

a perpetual cash flowP = D / r

73

Constant Dividend Example

What is the value of a stock that is expected to pay a constant dividend of $2 per share? The required rate of return is 10%P = 2 / 0.1 = 20

74

Growing Dividends

Now we are assuming that the firm’s dividends will grow at a constant rate, g forever

This is similar to a: So the price of a share is:

)1(D ivD iv 01 g2

012 )1(D iv)1(D ivD iv gg 3

023 )1(D iv)1(D ivD iv gg

A Growing Perpetuity

P = D1 / (r-g)

75

Growing Dividend Example

Geneva steel just paid a dividend of $2.10. Dividend payments are expected to grow at a constant rate of 6%. The appropriate discount rate is 12%. What is the price of Geneva stock?

Div1 =

P0 =

76

Growing Dividend Example

Geneva steel just paid a dividend of $2.10. Dividend payments are expected to grow at a constant rate of 6%. The appropriate discount rate is 12%. What is the price of Geneva stock?

Div0 = $2.10 so Div1 = 2.10*(1.06)=$ 2.226

P0 = 2.226 / (0.12- 0.06) = $37.10

77

Valuing Stock with Changing g

1. Find the PV of dividends during the period of non-constant growth, PA

2. Find the price of the stock at the end of the non-constant growth period, PN

3. Discount the price found in 2 back to the present, PB

4. Add the two present values (1+3) to find the intrinsic value (price) of the stock P0 = PA + PB

78

Differential Growth Rates

Dividends will grow at g1 for N years and g2 thereafter

Step 1: An N-year annuity growing at rate g1

Step 2: A growing perpetuity at rate g2

PN = DivN+1 / (R-g2)

Step 3: PB = PN / (1+R)N

Step 4: P0 = PA + PB

TA R

g

gR

CP }{

)1(

)1(1 1

1

79

Non-Constant Growth Example (Given) Websurfers Inc, a new internet firm is expected to do

very well during its initial growth period. Investors expect its dividends to grow at 25% for the next 3 years. Obviously one cannot expect such extraordinary growth to continue forever, and it is expected that dividends will grow at 5% after year 3 in perpetuity. Its current dividend is $1/share. Required rate of return on the stock = 10%. Calculate what the current price should be.

80

Websurfer Inc, Example (Given)

1.PA=[(1.25)/(0.10-0.25)]*[1-{1.25/1.10}3] = 3.90 D1 = D0 * (1 + g1) = 1 * 1.25 = 1.25

2.PN ={2.05}/(0.10-0.05) = 41.00 D4 = D3 * (1 + g2) = 1.95 * 1.05 = 2.05

D4 = D0*(1 + g1)3*(1 + g2)= 1*1.253* 1.05 = 2.05

3.PB =41.00/(1.103) = 30.80

4.P0 = PA + PB = 3.90+ 30.80 = $34.70

0 1 2 3 4 5

1*1.25 = 1.25

1.95*1.05 = 2.05

1.25*1.25

=1.561.56*1.25

= 1.952.05*1.05

= 2.151

81

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%.

0 1 2 3 4 5

2*1.081

=2.16

2*1.083 *1.04 =2.62

2*1.082

=2.332*1.083

=2.52

2*1.083*1.042 = 2.72

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Solution

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. R=12%

1. PA =

2. PN =

3. PB =

4. P0 = PA + PB =

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Solution

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. R=12%1. PA=[(2*1.08)/(0.12-0.08)]*[1-{1.08/1.12}3]=5.58

2. PN =

3. PB =

4. P0 = PA + PB =

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Solution

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. R=12%1. PA=[(2*1.08)/(0.12-0.08)]*[1-{1.08/1.12}3]=5.58

2. PN ={2*1.083*1.04}/(0.12-0.04) = 32.75

3. PB =

4. P0 = PA + PB =

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Solution

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. R=12%1. PA=[(2*1.08)/(0.12-0.08)]*[1-{1.08/1.12}3]=5.58

2. PN ={2*1.083*1.04}/(0.12-0.04) = 32.75

3. PB =32.75/(1.123) = 23.31

4. P0 = PA + PB =

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Solution

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. R=12%1. PA=[(2*1.08)/(0.12-0.08)]*[1-{1.08/1.12}3]=5.58

2. PN ={2*1.083*1.04}/(0.12-0.04) = 32.75

3. PB =32.75/(1.123) = 23.31

4. P0 = PA + PB = 5.58 + 23.31 = $28.89

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Important Parameters

The value of a firm depends on the discount rate, the growth rate, and the initial dividend.

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The Discount Rate

The market consensus of the firm’s required rateThis is the Market Capitalization RateReturn that an investor expects to makeThis is similar to what for a bond?

Yield to Maturity

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Where does “r” come from? We generally estimate r from one of the dividend

valuation models Using constant dividend growth model:

In practice, estimates of r have a lot of estimation error

gP

D R

g-R

DP

0

1

10

Rearrange and solve for R:

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Where does “R” come from?

What is D1/P0?

What is g?

An investor’s return comes from either the dividends received or price appreciation

gP

D R

0

1

Dividend Yield

The growth rate, or the Capital Gains Yield

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Classifying Stocks Firms are often classified based on where

investors expect to earn their return from“Income/Value stocks”: have a higher

dividend yield“Growth stocks”: have a higher growth

component As long as both are equally risky, the

return should be the same

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Where does “g” come from? From analysts' estimates

I/B/E/S, Google, Yahoo, or WSJ From earnings re-investment

g = plowback ratio * ROE How much does the firm reinvest, and what is the return on

the investment

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Link between stock prices and earnings

A “new valuation model” : Consider a firm with a 100% payout ratio, so

Div = EPS and earnings remain flat. P0 = DIV / r Because Div = EPS P0 = EPS / r

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Present Value of all Future Growth Opportunities (PVGO) The price is composed of the value of the

firm’s current assets (100% payout firm) and the firm’s growth opportunitiesGrowth opportunities are opportunities to invest in

positive NPV projects.

P0 =EPS / r + PVGO

EPS / r : This is the value of the firm’s current assets

PVGO : This is the value of what the firm can invest in

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Who cares about PVGO?

For what type of stock is the PVGO more important?Growth or Value stocks

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Who cares about PVGO?

For what type of stock is the PVGO more important?Growth or Value stocks

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PVGO Example

Assume that a firm has 2 potential projects. Project A & B with NPV’s of $2m, and $3m, respectively. The firm pays out all its earnings as dividends, and paid a dividend of $1/share last year. It has 200,000 shares outstanding. Assume the discount rate is 10%.

What is the share price, if the cash flow from the firm's existing assets are expected to remain the same in perpetuity, and the firm takes on Project A, and B?  

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PVGO Example

P0 = Div / r + PVGO, since Div = EPS

P0 = 1 / 0.1 + PVGO

What are the firm’s growth opportunities? Worth?

Project A & B, $5 million

PVGO per share?

5,000,000 / 200,000 = $25

P0 = 1 / 0.1 + 25 = $35 / share

The firm is worth: 200,000 * 35 = $7million

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Stock Value Represents:

Present value of expected future dividends, Present value of free cash flow, Present value of average future earnings under

a no-growth policy plus the present value of growth opportunities

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Price-Earnings Ratio

The price-earnings ratio is calculated as the current stock price divided by annual EPS.The Wall Street Journal uses last 4 quarter’s

earnings

Many analysts use this to determine how the market feels about a company

E P S

sha rep e r P ricera tio P /E

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Price/Earnings Ratio

Is selling at a high P/E good?   Why might the P/E be high:

1. r is low (investors think the firm is relatively safe)

2. Good growth opportunities (high PVGO)

3. Current EPS is low

Remember, earnings are an accounting measure, which means P/E is an accounting measure

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Problem 1 (Given)

A firm is expected to grow at 25% for the next 3 years. Its growth is expected to decline to 15% for the following 4 years. It is then expected to grow at 5% in perpetuity. Find the current share price if the current dividend is $1 and the discount rate is 10%. 

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Problem 1 (Given)A firm is expected to grow at 25% for the next 3 years. Its growth is expected to

decline to 15% for the following 4 years. It is then expected to grow at 5% in perpetuity. Find the current share price if the current dividend is $1 and the discount rate is 10%.

PA=[(1*1.25)/(0.10-0.25)]*[1-{1.25/1.10}3]=3.89PN1 ={1*1.253*1.15}/(0.10-0.15)]*

[1-{1.15/1.10}4]=8.76PB =8.76/(1.103) = 6.58PN2 ={1*1.253*1.154*1.05}/(0.10-0.05)] = 71.80PC =71.80/(1.107) = 36.83P0 = PA + PB + Pc = 3.89+6.58+36.83 = $47.30

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Problem 2 (Given)

Consider a firm whose dividend growth is expected to decline gradually. For the next two years, the growth is expected to be 20%. In the following years, it is expected to grow at 18%, 13% and 10%. From year 6 onwards, dividends are expected to grow at 5% for perpetuity. Assume the current dividend is $1 and the required rate of return is 10%. What is the current price?

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Problem 2: Phase 1 – Years 1-5 (Given) DIV1 = 1.00 * 1.20 = 1.20 DIV2 = 1.20 * 1.20 = 1.44 DIV3 = 1.44 * 1.18 = 1.70 DIV4 = 1.70 * 1.13 = 1.92 DIV5 = 1.92 * 1.10 = 2.11 PA = 1.2/1.1 + 1.44/1.12 + 1.70/1.13 + 1.92/1.14

+ 2.11/1.15 = 6.18

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Problem 2 Phase 2 – Years 6- (Given)

DIV6 = 2.11 * 1.05 = 2.22

PN = DIV6 / (r-g) = 2.22/(0.10-0.05) = 44.4

PB = 44.4 * 1/1.15 = 27.57

Current Price = 6.18 + 27.57 = $33.75

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Quick Quiz

How do you find the value of a bond, and why do bond prices change?

What is a bond indenture, and what are some of the important features?

What determines the price of a share of stock? What determines g and R in the DGM? Decompose a stock’s price into constant growth and

NPVGO values. Discuss the importance of the PE ratio.

Why We Care

Basic real world application of the time value of money

Foundation of Investment/ Financial Analysis

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