professor del hawley finance 634 fall 2003 stock and bond valuation chapter 4

Professor Del HawleyFinance 634

Professor Del HawleyFinance 634

Fall 2003Fall 2003

Stock And Bond ValuationStock And Bond Valuation

Chapter 4Chapter 4

Valuation FundamentalsValuation Fundamentals

• Value of any financial asset is the PV of future cash flows– Bonds: PV of promised interest & principal payments– Stocks: PV of all future dividends– Patents, trademarks: PV of future royalties

• Valuation is the process linking risk & return– Output of process is asset’s expected market price

• A key input is the required [expected] return on an asset– Defined as the return an arms-length investor would require

for an asset of equivalent risk– Debt securities: risk-free rate plus risk premium(s)

• Required return for stocks found using CAPM or other asset pricing model – Beta determines risk premium: higher beta, higher reqd return

The Basic Valuation ModelThe Basic Valuation Model

Can express price of any asset at time 0, P0, mathematically as Equation 4.1:

Where:

P0 = Price of asset at time 0 (today)

CFt = cash flow expected at time t

r = discount rate, reflecting asset’s risk

n = number of discounting periods (usually years)

)1.4.(Eq)r +(1

CF + . . . + )r +(1

CF + )r +(1

CF = P nn

22

11

0

Illustration Of Simple Asset ValuationIllustration Of Simple Asset Valuation

Assume you are offered a security that promises to make four $2,000 payments at the end of years 1-4.

If the appropriate discount rate for securities of this risk is 2%, what price should you pay for this security?

$7,615.53

+++

.( +

.( +

.( +

).( =

).( +

).( +

).( +

).(

,$ = P 20

75.847,1$66.884,1$34.922,1$78.960,1$

)08241

000,1$

)06121

000,1$

)04041

000,2$

021

000,2$

021

000,2$

021

000,2$

021

000,2$

021

000243

Security would be worth $7,615.53 each.

Illustration Of Simple Asset ValuationIllustration Of Simple Asset Valuation

With most debt securities, the cash flows are smooth (equal amounts at equal time intervals, except possibly the last cash flow) and so it can be treated like an annuity or an annuity with a balloon.

PV = Price or value of the security

FV = Maturity or par value (usually $1000)

n = Number of remaining interest payments

i = Market required return per payment period

PMT = Periodic interest payment (Coupon Rate x 1000)

FV = 0 n = 4 I = 2 PMT = 2000 PV = $7,615.53

Illustration Of Bond ValuationUsing U.S. Treasury SecuritiesIllustration Of Bond ValuationUsing U.S. Treasury Securities

The simplest debt instruments to value are U.S. Treasury securities since there is no default risk.

Instead, the discount rate to use (rf) is the pure cost of borrowing.

rf = Real Rate of Interest + Inflation Premium

The Fisher Effect And Expected InflationThe Fisher Effect And Expected Inflation

• The relationship between nominal (observed) and real (inflation-adjusted) interest rates and expected inflation is called the Fisher Effect (or Fisher Equation).

• Fisher said the nominal rate (r) is approximately equal to the real rate of interest (a) plus a premium for expected inflation (i).

– If the real rate equals 3% (a = 0.03) and expected annual inflation rate equals 2% (i = 0.02), then:

r a + i 0.03 + 0.02 0.05 5%

• The true Fisher Effect is multiplicative, rather than additive:

(1+r) = (1+a)(1+i) = (1.03)(1.02) = 1.0506; so r = 5.06%

Illustration Of Bond ValuationUsing U.S. Treasury SecuritiesIllustration Of Bond ValuationUsing U.S. Treasury Securities

Assume you are asked to value two Treasury securities, when rf is 1.75 percent:

– A (pure discount) Treasury bill with a $1,000 face value that matures in three months, and

– A 1.75% coupon rate Treasury note, also with a $1,000 face value, that matures in three years.

Illustration Of Bond Valuation Using U.S. Treasury Securities (Continued)Illustration Of Bond Valuation Using U.S. Treasury Securities (Continued)

• The 3-month T-Bill pays no interest; return comes from the difference between purchase price and maturity value.

• 3-year T-Note makes two end-of-year $17.5 coupon payments (CF1=CF2=$17.5), plus end-of-year 3 payment of interest plus principal (CF3 = $1,017.5)

• Can value both with variation of Equation 4.1:

$1,000=$965.9+ $16.9 + $17.2 =

1.0175

$1,017.5 +

1.0175

$17.5+

1.0175

$17.5 = P

$995.68=1.00434

$1,000=

1.0175

$1,000=

r +1CF = P

32NoteT

0.250.25

0.25BillT


For the 3-month Treasury Bill:

PV -995.68

FV 1000

N 1/4

I 1.75

PMT 0


For the 3-year Treasury Note:

PV -1000

FV 1000

N 3

I 1.75

PMT 17.5



FV 1000n 3i 1.75PMT 17.5

PV ($1,000.00)

For Excel: =PV(rate,nper,PMT,FV,Type)

Bond Valuation FundamentalsBond Valuation Fundamentals

Most U.S. corporate bonds:

– Pay interest at a fixed coupon interest rate

– Have an initial maturity of 10 to 30 years, and

– Have a par value (also called face or principal value) of $1,000 that must be repaid at maturity.

Bond Valuation FundamentalsBond Valuation Fundamentals

The Sun Company, on January 3, 2004, issues a 5 percent coupon interest rate, 10‑year bond with a $1,000 par value

– Assume annual interest payments for simplicity– Will value later assuming semi-annual coupon payments

Investors in Sun Company’s bond thus receive the contractual right to:

– $50 coupon interest (C) paid at the end of each year and – The $1,000 par value (Par) at the end of the tenth year.

• Assume required return, r, also equal to 5%

• The price of Sun Company’s bond, P0, making ten (n=10) annual coupon interest payments (C = $50), plus returning $1,000 principal (Par) at end of year 10, is determined as:

Bond Valuation Fundamentals (Continued)Bond Valuation Fundamentals (Continued)

00.000,1$05.

050,1$

05.

50$

05.

50$

05.

50$

05.

50$

05.

50$

05.

50$

05.

50$

05.

50$

05.

50$

10987

6543

=)(1

+)(1

+ )(1

+)(1

+

)(1

+ )(1

+)(1

+ )(1

+ )(1

+ )(1

= P 20


PV -1000

FV 1000

N 10

I 5

PMT 50

For Excel: =PV(rate,nper,PMT,FV,Type)


A bond’s value has two separable parts:

(1) PV of stream of annual interest payments, t=1 to t=10

(2) PV of principal repayment at end of year 10.

Therefore, we can also value a bond as the PV of an annuity plus the PV of a single cash flow.

PV -386.09 -613.91

FV 0 1000

N 10 10

I 5 5

PMT 50 0

Bond Values If Required Return Is Not Equal To The Coupon RateBond Values If Required Return Is Not Equal To The Coupon Rate

• Whenever the required return on a bond (r) differs from its coupon interest rate, the bond's value will differ from its par, or face, value.

– Will only sell at par if r = coupon rate

• When r is greater than the coupon interest rate, P0 will be less than par value, and the bond will sell at a discount.

– For Sun, if r >5%, P0 will be less than $1,000

• When r is below the coupon interest rate, P0 will be greater than par, and the bond will sell at a premium.

– For Sun, if r <5%, P0 will be greater than $1,000

• Value Sun Company, 10-year, 5% coupon rate bond if required return, r =6% and again if r = 4%.

Bond Values If Required Return Is Not Equal To The Coupon Rate (Continued)

Bond Values If Required Return Is Not Equal To The Coupon Rate (Continued)

At r = 6%, the bond sells at a discount of $1,000 - $926.405 = $73.595

At r = 4%, the bond sells at a premium of $1,081.45 - $1,000 = $81.45

Premiums & discounts change systematically as r changes

PV -926.405 -1081.45

FV 1000 1000

N 10 10

I 6 4

PMT 50 50

0 1 2 3 4 5 6 7 8

Required Return, r (%)

Mar

ket

Val

ue

of

Bo

nd

P0 ($

)

1,200

1,100

1,081

926900

800

Premium

Par

Discount

Bond Value & Required Return, Sun Company’s 5 % Coupon Rate, 10-year, $1,000 Par, January 1, 2004 Issue Paying Annual Interest

Bond Value & Required Return, Sun Company’s 5 % Coupon Rate, 10-year, $1,000 Par, January 1, 2004 Issue Paying Annual Interest

1,000

The Dynamics Of Bond Valuation Changes For Different Times To Maturity


• Whenever r is different from the coupon interest rate, the time to maturity affects bond value even if the required return remains constant until maturity.

• The shorter is n, the less responsive is P0 to changes in r.

Assume r falls from 5% to 4%

– For n=8 years, P0 rises from $1,000 to $1,067.33, or 6.73%

– For n=3 years, P0 rises from $1,000 to $1,027.75, or 2.775%



• The same relationship holds if r rises from 5% to 6%, (though the percentage decline in price is less than the percentage increase was in the previous example).

For n=8 years, P0 falls from $1,000 to $937.89, or 6.21%

For n=3 years, P0 falls from $1,000 to $973.25, or 2.675%

• Even if r doesn’t change, premiums and discounts will decline towards the bond’s par value as the bond nears maturity.

r

Time to maturity (years)

Mar

ket

Val

ue

of B

ond

P0 ($

)Relation Between Time to Maturity, Required Return & Bond Value,

Sun Company’s 5%, 10-year, $1,000 Par Issue Paying Annual InterestRelation Between Time to Maturity, Required Return & Bond Value,

Sun Company’s 5%, 10-year, $1,000 Par Issue Paying Annual Interest

Premium Bond, Required Return, r = 4%

Par-Value Bond, Required Return, r = 5%

Discount Bond, Required Return, r = 6%

10 9 8 7 6 5 4 3 2 1 0

M

1,100

1,0811,067.3

1,050

1,027.75

1,000

926

900

950

Relationship Between Bond Prices & Yields, Bonds Of Differing Original Maturities But Same Coupon Rates Relationship Between Bond Prices & Yields, Bonds Of Differing Original Maturities But Same Coupon Rates

Bond Prices and Yields

$0

$200

$400

$600

$800

$1,000

$1,200

$1,400

$1,600

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Yield to maturity, %

Bon

d P

rice

10-year bond

2-year bond

Semi-Annual Bond Interest PaymentsSemi-Annual Bond Interest Payments

• Most bonds pay interest semi-annually rather than annually• Can easily modify the basic valuation formula; divide both

coupon payment (C) and discount rate (r) by 2, as in Eq 4.3:

• N is always the number of PAYMENT PERIODS• I is always the required return PER PERIOD

)3.4.()

21(

000,12....

)2

1(

2

)2

1(

2

)2

1(

2Pr2321

Eqr

C

r

C

r

C

r

C

icen

Value a T-Bond with a par value of $1,000 that matures in exactly 2 years and pays a 4% coupon if r = 4.4% per year.

Valuing A Bond With Semi-Annual Bond Interest Payments


43.992$97.934$74.18$15.19$57.19$

)022.1(

020,1$

)022.1(

20$

)022.1(

20$

)022.1(

20$

2044.0

1

000,1240$

2044.0

1

240$

2044.0

1

240$

2044.0

1

240$

432

43210

P



Value a T-Bond with a par value of $1,000 that matures in exactly 2 years and pays a 4% coupon if r = 4.4% per year.

PV -992.43

FV 1000

N 4

I 2.2

PMT 20

The Importance And CalculationOf Yield To Maturity


• Yield to Maturity (YTM) is the rate of return investors earn if they buy the bond at P0 and hold it until maturity.

• YTM is the discount rate that equates the PV of a bond’s cash flows with its price.

• The YTM on a bond selling at par (P0 = Par) will always equal the coupon interest rate. When P0 Par, the YTM will differ from the coupon rate.



Suppose you purchase a T-Bond for $875.00 that has 2 years to maturity and pays its 5% coupon rate in semi-annual payments. What is the YTM for the bond?

PV -875.00

FV 1000

N 4

I 6.117

PMT 25

This is the semi-annual yield on the bond, whereas the YTM is always stated as an annual rate. To annualize the semi-annual yield, simply multiply it by 2. So, the YTM on the bond is 12.23%.

The Importance and Calculation of Yield to Maturity

The Importance and Calculation of Yield to Maturity


PV -875FV 1000n 4PMT 25

i 6.12%

For Excel: =RATE(nper,PMT,PV,FV,<Type>,<guess>)

The “Current Yield” is anApproximation of the YTMThe “Current Yield” is anApproximation of the YTM

In bond quotes, the Current Yield (Cur Yld) is computed as:

Cur Yld = Annual $ of Interest / Price

It is an approximation of the YTM.

The long the time to maturity and the larger the coupon rate, the better the approximation.

See http://www.bondpickers.com/?source=gglBondPrice for detailed price quotes for corporate bonds.

http://www.bondpickers.com/?source=gglBondPrice

Holding-Period ReturnsHolding-Period Returns

If you purchase a bond with YTM = 10% and hold it until maturity, you will earn an average of 10% return on the bond over its life even though the value of the bond will fluctuate throughout its life.

BUT, if you sell the bond prior to maturity your average return will depend on the selling price of the bond, which will depend on the prevailing level of interest rates and the relative risk of the bond at the time you sell it.

This is called “Interest rate risk” or “price risk”. Even “riskless” treasury bonds have this risk.

Holding-Period ReturnsHolding-Period Returns

The value of the bond will move inversely with the prevailing level of interest rates.

Increase in required yield → decrease in value

Decrease in required yield → increase in value

So, if required return on your bond rises while you hold it your selling price will be lower that you expected and your holding period return (the average you earn over the time you hold the bond) will fall. And vice versa.



Suppose you purchase a bond today for $960 that has a 15-year maturity, $1000 face value, 8% coupon rate, and pays interest semi-annually. The next coupon payment is due in exactly six months. Also suppose that you sell the bond four years from now, immediately after the 8th coupon payment you receive. At the time that you sell the bond, its required return in the market is 12%. What is your average annual holding period return on this bond?

Holding Period ReturnsHolding Period Returns

First, find the selling price of the bond in four years.

PV 759.17This is the price that the buyer would pay in four years.

FV 1000The buyer will still value the bond with its terminal value as a cash flow.

N 22The buyer will get 11 more years of semi-annual payments.

I 6The buyer will value the bond based on a 12% required return.

PMT 40The bond has an 8% coupon rate, and that won’t change over time.

Holding Period ReturnsHolding Period Returns

Next, use the price you just found as the FV, and determine the rate that you would earn over four years.

PV -960.00This is the price you paid when you bought the bond.

FV 759.17This is the price that you will sell the bond for after four years.

N 8This is the number of semi-annual periods you will hold the bond.

I 1.70This is the semi-annual return you earned, so double that for the annual return of 3.40%.

PMT 40The bond has an 8% coupon rate, and that won’t change over time.

The Term Structure Of Interest RatesThe Term Structure Of Interest Rates

• At any point in time, there will be a systematic relationship between YTM and maturity for securities of a given risk.

– Usually, yields on long-term securities are higher than the yields on short-term securities.

• The relationship between yield and maturity is called the Term Structure of Interest Rates.

– The graphical depiction of the term structure is called a Yield Curve.

• Yield curves are normally upwards-sloping (long yields > short), but can be flat or even inverted during times of financial stress

• We won’t cover term structure in depth, but three principal “expectations” theories explain the term structure:

– Pure expectations hypothesis: YC embodies prediction

– Liquidity premium theory: Investors must be paid to invest L-T

– Preferred habitat hypothesis: Investors prefer maturity zones

Yield Curves for US Treasury SecuritiesYield Curves for US Treasury Securities

2

4

6

8

10

12

14

16

5 10 15 20 30

Years to Maturity

Inte

res

t R

ate

%

August 1996

October 1993

May 1981

January 1995

1 3

Yield Curve, February 12, 2003From www.cnnfn.com

Yield Curve, February 12, 2003From www.cnnfn.com

% %

Years to maturity

Yield Curve TodayYield Curve Today

%For detailed current information on the yield curve, go to

http://www.bondsonline.com/asp/news/yieldcurve.html

http://www.bondsonline.com/asp/news/yieldcurve.html

Changes In The Shape And Level Of Treasury Yield Curve During Early October 1998

Changes In The Shape And Level Of Treasury Yield Curve During Early October 1998

3.7

3.9

4.1

4.3

4.5

4.7

4.9

5.1

5 10 30

Maturity in Years

Yie

ld %

October 9

October 8

October 2

1

Bond Risk Premiums, February 97-November 98Bond Risk Premiums, February 97-November 98

0

100

200

300

400

500

600

High-yield BondYields less yieldon 10-yearTreasurys inbasis points

9897

Equity ValuationEquity Valuation

• As you learned in MBA 611, the required return on common stock is based on its beta coefficient, which is derived from the CAPM

– Valuing common stock is the most difficult, both practically and theoretically, since nothing (except the current price, is known with certainty.

– Preferred stock valuation is much easier (the easiest of all)

• Whenever investors feel the expected return, rˆ, is not equal to the required return, r, prices will react:

– If exp return declines or req’d return rises, stock price will fall

– If exp return rises or req’d return declines, stock price will rise

• Asset prices can change for reasons besides their own risk

– Changes in the asset’s liquidity or tax status can change price

– Changes in market risk premium can change all asset values

• Most dramatic change in market risk: Russian default Fall 98

– Caused required return on all risky assets to rise, price to fall

Preferred Stock ValuationPreferred Stock Valuation

• PS is an equity security that is expected to pay a fixed annual dividend over its (assumed infinite) life.

• Preferred stock’s market price, P0, equals next period’s dividend payment, Dt+1, divided by the discount rate, r, appropriate for securities of its risk class:

• A share of PS paying a $2.3 per share annual dividend and with a required return of 11% would thus be worth $20.90:

r

D = P 1t

0

%0.1111.090.20$

3.2$1 == =P

D = r

0

t

share= =r

D = P t

0 /90.20$11.0

3.2$1

• Formula can be rearranged to compute required return, if price and dividend known:

Common Stock ValuationCommon Stock Valuation

The basic formula for valuing a share of stock easy to state; P0 is equal to the present value of the expected stock price at end of period 1, plus dividends received during the period, as in Eq 4.4:

)4.4.Eq()r1(

DPP 11

0

The problem is how to determine P1.

Common Stock ValuationCommon Stock Valuation

)4.4.Eq()r1(

DPP 11

0

P1 is the PV of expected stock price P2, plus dividends received during period 1. P2 in turn, the PV of P3 plus dividends, and so on.

Repeating this logic over and over, you will find that today’s price equals the PV of the entire dividend stream the stock will pay in the future, as in Eq 4.5:

)5.4.Eq(....)r1(

D

)r1(

D

)r1(

D

)r1(

D

)r1(

DP

55

44

33

2

210

The Zero Growth Valuation ModelThe Zero Growth Valuation Model

• To value common stock, we must make an assumption about the growth rate of future dividends.

• The simplest approach, the zero growth model, assumes a constant, non-growing dividend stream:

D1 = D2 = ... = D

• Plugging the constant value D into Eq 4.5 reduced the valuation formula to the simple equation for a perpetuity:

r

DP 0

The Zero Growth Valuation ModelThe Zero Growth Valuation Model

Assume the dividend of Disco Company is expected to remain at $1.75/share indefinitely, and the required return on Disco’s stock is 15%. The next dividend will be one year from now. P0 is determined to be $11.67 as:

67.11$15.0

75.1$0

r

DP

The Constant Growth Valuation ModelThe Constant Growth Valuation Model

• The most widely used simple stock valuation formula, the constant growth model, assumes dividends will grow at a constant rate, g, that is less than the required return (g<r)

• If dividends grow at a constant rate forever, we can value stock as a growing perpetuity. Denoting next year’s dividend as D1:

• This is commonly called the Gordon Growth Model, after Myron Gordon, who popularized model in the 1960s.

6.4.10 Eq

gr

DP

The Constant Growth Valuation ModelThe Constant Growth Valuation Model

The Gordon Company’s dividends have grown by 7% per year, reaching $1.40 per share this year. This growth is expected to continue, so D1=$1.40 x 1.07=$1.50. If the required return on this stock is 15%, then its market value should be:

6.4.10 Eq

gr

DP

75.18$08.0

50.1$

07.015.0

50.1$10

gr

DP

Valuing Common Stock Using The Variable Growth Model

Valuing Common Stock Using The Variable Growth Model

• Because future growth rates might change, we need to consider a variable growth rate model that allows for a change in the dividend growth rate.

• Let g1 = the initial higher growth rate and g2 = the lower subsequent growth rate, and assume a single shift in growth rates from g1 to g2.

• The constant growth rate model can be generalized for two or more changes in growth rates, but let’s keep it simple for now.

• For a single change in growth rates, we can use four-step valuation procedure:

Valuing Common Stock Using The Variable Growth Model (Continued)

Valuing Common Stock Using The Variable Growth Model (Continued)

• Step 1: Find the value of the dividends at the end of each year, Dt, during the initial high-growth phase.

• Step 2: Find the PV of the dividends during this high-growth phase, and sum the discounted cash flows.

• Step 3: Using the Gordon growth model, find the value of the stock at the end of the high-growth phase using the next period’s dividend (after one year’s growth at g2).

– Then compute PV of this price by discounting back to time 0.

• Step 4: Determine the value of the stock today (P0) by adding the PV of the stock price computed in step 3 to the sum of the discounted dividend payments from step 2.

An Example Of Stock Valuation Using The Variable Growth Model

An Example Of Stock Valuation Using The Variable Growth Model

Estimate the current (end‑of‑2003) value of Morris Industries' common stock, P0 = P2003 , using the four‑step procedure presented above, and assuming the following:– The most recent (2003) annual dividend payment of Morris

Industries was $4 per share. – The firm's financial manager expects that these dividends

will increase at an 8 percent annual rate, g1 , over the next three years (2004, 2005, and 2006).

– At the end of the three years (end of 2006) the firm's mature product line is expected to result in a slowing of the dividend growth rate to 5 percent per year forever (noted as g2).

– The firm's required return, r , is 12 percent.

An Example Of Stock Valuation Using The Variable Growth Model (Continued)


• Step 1: Compute the value of dividends in 2004, 2005, and 2006 as (1+g1)=1.08 times the previous year’s dividend:

Div2004= Div2003 x (1+g1) = $4 x 1.08 = $4.32

Div2005= Div2004 x (1+g1) = $4.32 x 1.08 = $4.67

Div2006= Div2005 x (1+g1) = $4.67 x 1.08 = $5.04

• Step 2: Find the PV of these three dividend payments:

PV of Div2004= Div2004 (1+r) = $ 4.32 (1.12) = $3.86

PV of Div2005= Div2005 (1+r)2 = $ 4.67 (1.12)2 = $3.72

PV of Div2006= Div2006 (1+r)3 = $ 5.04 (1.12)3 = $3.59

Sum of discounted dividends = $3.86 + $3.72 + $3.59 = $11.17



• Step 3: Find the value of the stock at the end of the initial growth period (P2006) using constant growth model.

• To do this, calculate next period dividend by multiplying D2006 by 1+g2, the lower constant growth rate:

D2007 = D2006 x (1+ g2) = $ 5.04 x (1.05) = $5.292

• Then use D2007=$5.292, g =0.05, r =0.12 in Gordon model:

60.7507

292.5

2

292.576 $ =

0.

$ =

0.05 -0.1

$ =

g -rD = P

2

200200

• Next, find the PV of this stock price by discounting P2006 by (1+r)3.

81.53405.1

60.75$

)12.1(

60.75$

)1( 33

6 $ = = = r

P =PV 200



• Step 4: Finally, add the PV of the initial dividend stream (found in Step 2) to the PV of stock price at the end of the initial growth period (P2006):

P2003 = $11.17 + $53.81 = $64.98

• The current (end-of-year 2003) stock price is thus $64.98 per share.

Other Approaches To Common Stock Valuation

Other Approaches To Common Stock Valuation

• Book value: simply the net assets per share available to common stockholders after liabilities (and PS) paid in full– Assumes assets can be sold at book value, so may over-

estimate realizable value– Ignores non-balance-sheet assets (value of reputation,

human capital, unrealized gains, etc.)• Liquidation value: is the actual net amount per share likely

to be realized upon liquidation & payment of liabilities– More realistic than book value, but doesn’t consider firm’s

value as a going concern• Price/Earnings (P/E) multiples: reflects the amount

investors will pay for each dollar of earnings per share– P/E multiples differ between & within industries– Especially helpful for privately-held firms

professor del hawley finance 634 fall 2003 stock and bond valuation chapter 4

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