Lecture 5Lecture 5

How to Value Bonds and Stocks

Valuing Bonds

How to value Bonds

A bondbond is a certificate (contract) showing that a borrower owes a specified sum that will be repaid on a number of specified dates, along with a schedule of interest payments

• Pure discount bonds (zero coupon bonds)• Level coupon bonds• US government bonds• Consoles

Pure discount bonds (zero coupon bonds)

TrF

PV

1

A pure discount bondpure discount bond paying F in T years, when the annual interest rate r in each 1,…,T year will have a value

A discount bonddiscount bond of value PV paying F in T years has spot return

(T -year spot rate)

1

1

T

PV

Fr

A pure discount bondpure discount bond makes one payment (the face value) at a specified date (the maturity date). The face value is also called principal or denomination

Level coupon bonds

TTr

TT

r

FAC

r

F

r

C

r

C

r

CPV

1

1111 2

The value of a level coupon bondlevel coupon bond with face value F, coupon C and a maturity of T years will be, where r is the annual interest rate

Most bonds issued by governments or corporations pay coupons C in addition to a face value F at maturity T

US government bonds

A US government bondUS government bond called “13 of November 1999” will have

- a face value of $1000

- an annual coupon of 13% of the face value $ 130

- coupons paid in May and in November $ 65

until November 1999 when the bond is redeemed for $1000

Suppose

- it is November 1995,

- the stated annual market rate is 10% , and hence the semi annual rate is 5% .

US government bonds (continued)

Date 96.5 96.11 97.5 97.11 98.5 98.11 99.5 99.11Payment 65 65 65 65 65 65 65 65+1000

95.1096

05.1

100065

05.1

1000

05.1

65

05.1

65

05.1

65

88

05.0

882

A

PV

The cash flows from the bond would be

The value of this bond is

Consoles

r

CPV

ConsolesConsoles are bonds with no maturity date.

The value of a consoleconsole with the coupon C at the interest rate r will be

Relationship between Bond values and Interest rates

Determining Yields from Bond Prices: The yield to maturity is the interest rate that equates the PV of the payments on the bond to the current bond price.

Value of a bond depends inverselyinversely on interest rate r.Coupons reflect interest rates at issue time.Coupon rate is the market interest rate at the issue time.If r falls below the coupon rate, the bond sells at premium.If r rises above the coupon rate, the bond sells at discount.

ExampleExample

The value of a 5% coupon 2 year bond with annual payments is

The yield to maturity y on this bond solves

06.914

1.1

100050

08.1

502

PV

%95.9 ;

1

100050

1

5006.914 2

yyy

Suppose that

the current spot rate on a one year discount bond is 8%

the current annual spot rate on a two year zero coupon bond is 10%

I.e., market interest rate for year 1 is r1 = 8% , for year 2 r2 = 10% .

Example (continued)Example (continued)

The value of a 12% coupon 2 year bond with annual payments is

73.1036

1.1

1000120

08.1

1202

PV

The yield to maturity y on this bond solves

%89.9 ;

1

1120

1

12073.1036 2

y

yy

Therefore, higher coupon bonds have lower yield to maturity.

Term Structure of Interest Rates

Recall our earlier example where the one year spot rate r1 = 8% and the annual spot rate (or annual yield to maturity) on a two year zero coupon bond is r2 = 10% .

An individual investing $1 in a 2 year zero coupon bond will receive

210.11$

Notice that 1204.108.11$10.11$ 2

The term structure of interest rates relates the annual spot rates (yields to maturity) on zero-coupon government bonds to their terms to maturity.

Term Structure of Interest Rates (continued)

where fn is a forward rate over n-th year and rn is a n-year spot rate.

An investor in the 2 year bond effectively invest in a 1 year bond at r1 and “locks in” an investment for 1 year at f2. Forward rates for later years can be calculated as :

11

11

1

nn

nn

nr

rf

212

2 111 frr

We can breakdown the 2 year spot rate r2 into one year spot rate r1 and forward rate f2 for next year. More formally,

Estimating the Price of a Bond at a future date

One year spot rate from year1 to year2 is unknown at date 0.

Initial Price Payment at maturity Interest Rate MaturityBond A $1,000 $1,080 8% 1 yearBond B $1,000 $1,210 10% 2 year

One year spot rate from year1 to year2 Price of Bond B at date1

6% 1210 / 1.06 = 1141.517% 1210 / 1.07 = 1130.84

14% 1210 / 1.14 = 1061.40

Estimating the Price of a Bond (continued)

The price of Bond B at date 1 is unknown at date 0. Thus we consider expected value of Bond B at date 1, which is given by

year2over ratespot expected1

$1210

Now consider the following investment strategies at date 0.

I : Buy a 1 year bond at date 0

II : Buy a 2 year bond at date 0 and sell it at date 1

Proceeds from the investment I at date 1 is

108008.11000

Estimating the Price of a Bond (continued)

Proceeds from the investment II (expected) at date 1

year2over ratespot expected1

1.12041.08$1000

year2over ratespot expected1

1.10$1000 2

If f2 (=12.04%)= expected spot rate over year2, then I and II give the same proceed at date 1.

So, the investors should be indifferent.

If f2 > expected spot rate over year 2, then the proceed from II is greater than I.

Estimating the Price of a Bond (continued)

Under Expectation hypothesisExpectation hypothesis :

(investors are assumed to be risk-neutral)

f2 = expected spot rate over year2

Under Liquidity Preference hypothesisLiquidity Preference hypothesis :

(investors are assumed to be risk-averse : in order to induce risk averse investors to hold the riskier two year bonds, the market sets the forward rate f2 over the second year to be above the spot rate expected over year2.)

f2 > expected spot rate over year2

How to value Stocks

Consider a shareholder who intends to hold a stock for 1 year, earn a dividend D1 and sell the stock for an expected price P1.

Fundamental equation of yieldFundamental equation of yield

dividend + expected capital gain = opportunity cost

rPPPDr

P

r

DP

0011

110

11

If the required return on the stock is r, the price of the stock will be

How to value Stocks (continued)

Note that P1 is unknown now, and consequently we need to use its expected value, which can be computed if we know expected values of the dividend in 2 periods D2 and the price of the stock in period 2, P2.

rP

r

DP

1122

1

22

221

2210

111

111

1

1

r

P

r

D

r

D

r

P

r

D

rr

DP

Substituting P1 into the first Fundamental yield equation gives

How to value Stocks (continued)

The current price of the stock P0 can be obtained by repeating the above process.

ii

i

r

D

r

D

r

D

r

DP

1111 33

221

0

All future dividends Di affect the price P0 even if the investor’s investment horizon is only one year.

Some Special Cases

Zero Growth :Zero Growth : the share price of a stock that pays fixed dividend D in perpetuity should be

r

D

r

D

r

D

r

DP

320

111

For example, preferred stocks

Some Special Cases (continued)

Constant Growth : Constant Growth : if the dividends are expected to grow at the constant rate g, then

gr

D

r

g

r

g

r

D

r

gD

r

gD

r

DP

2

2

3

2

20

1

1

1

11

1

1

1

1

1

1

WW is expected to pay per-share dividend of $3 next year, growing at 8% forever. What is the price of the WW stock if the required return is 12% ?

75$08.012.0

3$

gr

DP

Some Special Cases (continued)

Differential Growth : Differential Growth :

A stock has just paid a dividend of $1, which is expected to grow at 20% for 5 years, 15% for 3 years, and then 8% for all future periods. Suppose the discount rate is 10% .

Current stock price = 11.61 + 95.33 = 106.94

PV of the expected dividends for the first 8 years = 11.61

33.951.1

36.204

08.010.0

08.4

1.1

188

PV of the expected dividends from 9 year on

Estimating the dividend growth rate g

Consider a firm with a fixed retention ratio

t

t

t

tt

E

D

E

DE

1

Such a firm would have

t

t

t

t

t

t

t

t

E

E

D

D

E

D

E

D 11

1

1

and this in turn gives

t

tt

t

tt

E

EE

D

DD

11

Estimating the dividend growth rate g (continued)

Earning next year = earning this year + increase in earning

increase in earning = retained earning *

expected gross return on retained earning at t

Now, notice that

ettttt DEEE 11

Then we have

etet

t

tt

t

tetttt

t

tt

E

DE

E

EDEE

E

EEg

11

11

Estimating the dividend growth rate g (continued)

growth rate of earnings (dividends)

= retention ratio * return on retained earnings

1g

use the historical gross return on equity

to approximate the expected gross return at t

t1

et1

Growth Opportunities

Consider a company with a constant stream of earnings in perpetuity.

Now suppose the dividend at date 1 is retained and invested in an investment project. The share value should increase by the NPV of the “growth opportunity”(NPVGO) induced by the investment project.

NPVGOr

EPSP 0

If the firm pays all these earnings out as dividends to shareholders, then at all dates,

earnings per share = EPS = d = dividends per share

The share value at date 0, P0 should be EPS/r.

r

EPSP 0

Growth Opportunities (continued)

Example :Example : Sam shipping with 100,000 shares outstanding expects to earn $1,000,000 per year in perpetuity, if it distributes all its earnings to shareholders. Suppose the appropriate discount rate r = 10% . Then

The firm finds an investment opportunity that will cost $1 million at date 1, but will increase earnings in every subsequent period by $210,000. If the firm decides to retain the earning at date 1 and invest in the project, what is the share price?

1001.0

100

r

EPSP

NPV of the investment opportunity at date 1-1,000,000 + 210,000/0.1 = 1,100,000

NPV at date 0 1,100,000/1.1 = 1,000,000 or 10 per share

Growth Opportunities (continued)

The share price with the investment project

P0 = EPS/r + NPVGO = 100 + 10 = 110

The above share prices can be obtained from calculating PV’s of the future earnings with or without the investment opportunity.

Growth Opportunities (continued)

Price-earning ratioPrice-earning ratio (PER)(PER)

P0 / EPS = 1 / r + NPVGO / EPS

PER depends positively on the growth opportunities.

Hence, the stocks of firms retaining earnings to invest in growth opportunities do have higher PER.

PER depends negatively on the discount rate r.

Firms with risky earnings will therefore have lower PER.

Reported accounting earnings are used.

Conservative accounting rules leads to higher PER’s.

For instances, Japanese firms PER’s