1

Chapter 8: Valuation of Chapter 8: Valuation of Known Cash Flows: Known Cash Flows: BondsBonds

Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley

ObjectiveValuation of fixed income securities

Explain why bond prices change

2

Chapter 8 ContentsChapter 8 Contents

• 1 Using Present Value Formulas to Value Known 1 Using Present Value Formulas to Value Known FlowsFlows

• 2 The Basic Building Blocks: Pure Discount Bonds2 The Basic Building Blocks: Pure Discount Bonds

• 3 Coupon Bonds, Current Yield, and Yield-to-3 Coupon Bonds, Current Yield, and Yield-to-MaturityMaturity

• 4 Reading Bond Listings4 Reading Bond Listings

• 5 Why Yields for the same Maturity Differ5 Why Yields for the same Maturity Differ

• 6 The Behavior of Bond Prices Over Time6 The Behavior of Bond Prices Over Time

3

8.1 Using Present Value 8.1 Using Present Value Formulas to Value Known Formulas to Value Known FlowsFlows• You have been offered the opportunity You have been offered the opportunity

to purchase a mortgage. It was to purchase a mortgage. It was originally part of a creative financing originally part of a creative financing package where the original owner package where the original owner financed the buyerfinanced the buyer

• The remaining life of the mortgage is 60 The remaining life of the mortgage is 60 months, with payment of $400. Your months, with payment of $400. Your required rate of return is 1.5% / monthrequired rate of return is 1.5% / month

4

CalculationCalculation

• Using the present value of an Using the present value of an annuity formula discussed in chapter annuity formula discussed in chapter 4, you will pay no more than4, you will pay no more than

11.752,15$

015.1

11

015.0

400

1

11

60

n

ii

pmtPV

5

Financial CalculatorFinancial Calculator

• Alternatively, using your financial Alternatively, using your financial calculator (remember to set the calculator (remember to set the correct defaults) you obtaincorrect defaults) you obtain

N I PV PMT FV

60 1.5% ?15,752.11

-400 0

6

Change in Required RateChange in Required Rate

• If your required rate of return If your required rate of return increased to 1.6% / monthincreased to 1.6% / month

N I PV PMT FV

60 1.6% ?15,354.66

-400 0

7

Using Present Value Using Present Value Formulas to Value Known Formulas to Value Known FlowsFlows

• Observe that the maximum you Observe that the maximum you would pay for the bond has would pay for the bond has decreaseddecreased

• An increase in the required rate of An increase in the required rate of return always leads to a decrease in return always leads to a decrease in the value of a fixed income securitythe value of a fixed income security

• The proof is very easyThe proof is very easy

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Bond Prices Rise as the Bond Prices Rise as the Interest Rates FallInterest Rates Fall

• Write the PV of the fixed income Write the PV of the fixed income security as the sum termssecurity as the sum terms

n

n

n

n

n

j

j

j

ipmt

ipmt

ipmt

ipmt

ipmtPV

1

1*

1

1*...

1

1*

1

1*

1

1*

1

1

2

2

1

1

1

9

Bond Prices Rise as the Bond Prices Rise as the Interest Rates FallInterest Rates Fall

• If i goes up, 1+i goes up, 1/(1+i) If i goes up, 1+i goes up, 1/(1+i) goes down for i > -1, (1/(1+i))goes down for i > -1, (1/(1+i))jj goes down for i > 0. So if the goes down for i > 0. So if the payments are positive, then the payments are positive, then the sum must also go down sum must also go down

• Similarly, i down -> PV upSimilarly, i down -> PV up

10

Bond Prices Rise as the Bond Prices Rise as the Interest Rates FallInterest Rates Fall

• Basic principle in evaluating known Basic principle in evaluating known flowsflows– A change in market interest rates A change in market interest rates

causes a change in the causes a change in the oppositeopposite direction in the market values of all direction in the market values of all existing contracts promising fixed existing contracts promising fixed payments in the futurepayments in the future

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NoteNote

• Volatile market rates imply volatile Volatile market rates imply volatile market valuesmarket values

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Finding the Correct Finding the Correct Discount RateDiscount Rate

• Bond analysis is not as easy as this Bond analysis is not as easy as this analysis appears to implyanalysis appears to imply– We need an interest rate to use in the We need an interest rate to use in the

formulaformula

– We saw in Chapter 2 that interest rates We saw in Chapter 2 that interest rates are a function of time-to-maturityare a function of time-to-maturity

– Two default-free bonds with identical Two default-free bonds with identical maturities may have different YTMsmaturities may have different YTMs

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Yield CurveYield Curve

• A typical yield curve:A typical yield curve:

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US Treasury Yiled Curve, Jan 97

4.50

5.00

5.50

6.00

6.50

7.00

7.50

0 5 10 15 20 25 30

Years to Maturity

An

nu

aliz

ed Y

ield

(%

)

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8.2 The Basics Building 8.2 The Basics Building Blocks: Pure Discount Blocks: Pure Discount BondsBonds

• We can always analyze any fixed We can always analyze any fixed income contract into a sum of income contract into a sum of pure pure discount bondsdiscount bonds

• A pure discount bond is a security that A pure discount bond is a security that promises to pay a specified single promises to pay a specified single cash payment (cash payment (face valueface value or or par par valuevalue) at a specified date called its ) at a specified date called its maturity datematurity date

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Pure Discount BondsPure Discount Bonds

• NoteNote– There is no cash flow associated with There is no cash flow associated with

interest interest

– Pure discount bonds are purchased at a Pure discount bonds are purchased at a discount from their face or par valuediscount from their face or par value

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Pure Discount BondsPure Discount Bonds

• The pure discount bond is an The pure discount bond is an example of the present value of a example of the present value of a lump sum equation we analyzed in lump sum equation we analyzed in Chapter 4Chapter 4

• Solving this, the yield-to-maturity on Solving this, the yield-to-maturity on a pure discount bond is given by the a pure discount bond is given by the relationship:relationship: 11

1

nn

P

FiiPF

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Pure Discount BondsPure Discount Bonds

11

1

nn

P

FiiPF

• In this equation,In this equation,– P is the present value or price of the P is the present value or price of the

bondbond

– F is the face or future value F is the face or future value

– n is the investment periodn is the investment period

– i is the yield-to-maturityi is the yield-to-maturity

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Pure Discount BondsPure Discount Bonds

• ExampleExample– You can purchase a pure discount bond You can purchase a pure discount bond

for $9,000, and it matures in two years for $9,000, and it matures in two years with a face value of $10,000with a face value of $10,000

– What is the ytm?What is the ytm?

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Pure Discount BondsPure Discount Bonds

%41.519000

100001

2

11

n

P

Fi

N I PV PMT FV

2 ?5.41%

9,000 0 -10,000

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8.3 Coupon Bonds, 8.3 Coupon Bonds, Current Yield, and Yield to Current Yield, and Yield to MaturityMaturity

• A coupon bond obligates the issuer toA coupon bond obligates the issuer to– make periodic payments of interest make periodic payments of interest

(called (called coupon paymentscoupon payments) to the bond ) to the bond holder until the bond matures holder until the bond matures

– at which time the face value of the bond at which time the face value of the bond is also paid to the bond holderis also paid to the bond holder

– and the contract is satisfiedand the contract is satisfied

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Coupon RateCoupon Rate

• The The coupon ratecoupon rate is the interest rate is the interest rate applied to the face value to applied to the face value to compute the coupon paymentcompute the coupon payment– A bond with a face value of $1,000 and A bond with a face value of $1,000 and

a coupon rate of 10% pays an annual a coupon rate of 10% pays an annual coupon of $100coupon of $100

– At maturity, the payment is $1,000+At maturity, the payment is $1,000+$100$100

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Par, premium, and Par, premium, and Discount BondsDiscount Bonds

• A coupon bond with its current price A coupon bond with its current price equal to its par value is a par bondequal to its par value is a par bond

• If it is trading below par it is a If it is trading below par it is a discount bonddiscount bond

• If it is trading above par it is a If it is trading above par it is a premium bond premium bond (not to be confused with the (not to be confused with the U.K. lottery bond of the same name!)U.K. lottery bond of the same name!)

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Bonds Trading at Par Bonds Trading at Par

• Bond Pricing Principle #1: (Par Bonds)Bond Pricing Principle #1: (Par Bonds)– If a bond’s price equals its face value, If a bond’s price equals its face value,

then its yield-to-maturity = current yield = then its yield-to-maturity = current yield = coupon rate. coupon rate. Proof:Proof:

Fi

pmtP

ii

pmt

iP

FPi

Fii

pmtP

nn

nn

1

11

1

11

&1

1

1

11

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Coupon Bonds, Current Coupon Bonds, Current Yield, and Yield-to-Yield, and Yield-to-MaturityMaturity

• The The yield-to-maturityyield-to-maturity is the discount is the discount rate that makes the present value of rate that makes the present value of the cash flows from the bond equal the cash flows from the bond equal to the current price of the bondto the current price of the bond

• An excellent way to compute the An excellent way to compute the ytm is given in Chapter 4ytm is given in Chapter 4

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Using Pure Discount Using Pure Discount Bonds to Value other Bonds to Value other BondsBonds

• Value a bond that pays its $100 Value a bond that pays its $100 coupon at the end of each year for 3-coupon at the end of each year for 3-years, and its par value of $1,000 in 3-years, and its par value of $1,000 in 3-yearsyears– You have discovered three pure discount You have discovered three pure discount

bonds (each with a $1,000 par value) that bonds (each with a $1,000 par value) that mature in 1, 2, and 3 years, and that are mature in 1, 2, and 3 years, and that are trading at $960, $890, and $810 trading at $960, $890, and $810 respectivelyrespectively

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First Solution MethodFirst Solution Method

00.1076$

10010001000

810100

1000

890100

1000

960

P

P

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Second Solution MethodSecond Solution Method

91.075,1$0728.1

1001000

0600.1

100

0417.1

100

%28.71810

000,1

%00.61890

000,1

%17.41960

000,1

32

3

1

3,0

2

1

2,0

1

1

1,0

P

P

i

i

i

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ConclusionConclusion

• The first method uses the fact that a The first method uses the fact that a coupon bond is the sum of pure discount coupon bond is the sum of pure discount bondsbonds

– it is fast and directit is fast and direct

• The second method first determines the The second method first determines the yields-to-maturity of each discount bondyields-to-maturity of each discount bond

– cash flows are then evaluated using themcash flows are then evaluated using them

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The YTM of the Coupon The YTM of the Coupon BondBond

• We have the price of the coupon We have the price of the coupon bond, and the timing and magnitude bond, and the timing and magnitude of its future cash flows, so we can of its future cash flows, so we can determine its YTMdetermine its YTM

• We use the financial calculator, but We use the financial calculator, but a numerical method was provided in a numerical method was provided in chapter 4 for this class of problemschapter 4 for this class of problems

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The YTM of the Coupon The YTM of the Coupon BondBond

N I PV PMT FV

3 ?7.10%

-1076 100 1000

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ObservationObservation

• The yield to maturity on the 3-year The yield to maturity on the 3-year pure discount bond was 7.28% and pure discount bond was 7.28% and the yield-to-maturity on the 3-year the yield-to-maturity on the 3-year coupon bond was 7.10%coupon bond was 7.10%

• The yield-curve for default-free The yield-curve for default-free bonds is not a unique valuebonds is not a unique value

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Bond Pricing Principle #2 Bond Pricing Principle #2 & 3& 3

• Bond Principle # 2: Premium BondsBond Principle # 2: Premium Bonds

bond price > face value bond price > face value ytm < current ytm < current yield < coupon rateyield < coupon rate

• Bond Principle # 3: Discount BondsBond Principle # 3: Discount Bonds

bond price < face value bond price < face value ytm > current ytm > current yield > coupon rateyield > coupon rate

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Proof of Relationship Proof of Relationship between YTM and Current between YTM and Current YieldYield

• For coupon bonds, we have the For coupon bonds, we have the following relationshipsfollowing relationships– Note the (sensible) restrictions on the Note the (sensible) restrictions on the

variable rangesvariable ranges

– Note that 1/((1+i)^n - 1) is always Note that 1/((1+i)^n - 1) is always positivepositive

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011

11&

11

111

11

0&0&0&1

1

1

11

n

n

n

n

nn

ip

pf

ippfpx

fip

ixi

xnii

fii

xp

36

Proof of Relationship Proof of Relationship between Current and between Current and Coupon Yields Coupon Yields

• For coupon bonds, we have the For coupon bonds, we have the following relationship derived from the following relationship derived from the bond formulabond formula– Note that the differences between the Note that the differences between the

reciprocals have the same sign, so the reciprocals have the same sign, so the actual differences also have the same signactual differences also have the same sign

– Note that size relationship is determined Note that size relationship is determined by the discount factor which is always < 1by the discount factor which is always < 1

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ytm

1

yieldcoupon

1ytm1

ytm

1

yieldcurrent

1

111

11

0&0&0&1

1

1

11

n

f

x

n

p

x

nn

ii

i

xnii

fii

xp

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How to Remember How to Remember PrinciplesPrinciples

• Imagine that the bond was issued at Imagine that the bond was issued at parpar– the yield-to-maturity moves from the the yield-to-maturity moves from the

coupon yield in the opposite direction coupon yield in the opposite direction to priceto price

– the coupon rate is unchangingthe coupon rate is unchanging

• This diagram may help:This diagram may help:

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Yield Relationships

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

600.00 800.00 1000.00 1200.00 1400.00 1600.00 1800.00

Price

Yie

ld

coupon_y

current_yy_t_m

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High Yield T-Bond FundsHigh Yield T-Bond Funds

• Yield curves with large positive slopes Yield curves with large positive slopes make longer-term T-bonds tempting make longer-term T-bonds tempting because, like T-bills, they are default-freebecause, like T-bills, they are default-free

– The above diagram was based on: par = The above diagram was based on: par = $1000, coupon = $100, n = 10-years, flat$1000, coupon = $100, n = 10-years, flat

– Observe the large effect of modest changes in Observe the large effect of modest changes in interest on capitalinterest on capital

– A close up is given belowA close up is given below

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Yield Relationships

0.07

0.09

0.11

0.13

800.00 1000.00 1200.00

Price

Yie

ld

coupon_y

current_yy_t_m

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ClarificationClarification

• The last example used a flat yield The last example used a flat yield curvecurve

• Let us look at an example withLet us look at an example with– short-term rates remaining fixedshort-term rates remaining fixed

– longer-term rates rising on increased longer-term rates rising on increased expectation of a general rise in interest expectation of a general rise in interest ratesrates

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Two Yield Curves (Pure Discount)

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

7.00%

8.00%

9.00%

0 5 10 15 20

Years to Maturity

Yie

ld t

o M

atu

riry

44

Investment ImplicationsInvestment Implications

• Assume a 20-year bond with a coupon Assume a 20-year bond with a coupon rate of 6%rate of 6%

• Purchase for $1016.54 when the lower Purchase for $1016.54 when the lower curve prevailscurve prevails

• When yield curve rises, the bond is worth When yield curve rises, the bond is worth only $814.05only $814.05

– This is a massive capital risk This is a massive capital risk • Additionally, long-term rates are more Additionally, long-term rates are more

volatile than short-term ratesvolatile than short-term rates

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8.4 Reading Bond Listings8.4 Reading Bond Listings

• There are traditions for reporting yields There are traditions for reporting yields and computing earned interest that and computing earned interest that need to be understood before tradingneed to be understood before trading– Coupon bonds are often quoted in terms of Coupon bonds are often quoted in terms of

the annual rate compounded semi-annuallythe annual rate compounded semi-annually

– T-bills are often quoted on a discount basisT-bills are often quoted on a discount basis• e.g., a 1 year T-bill has 364 days e.g., a 1 year T-bill has 364 days

outstanding, but a year has only 360 days…outstanding, but a year has only 360 days…(it gets nasty)(it gets nasty)

46

Reading Bond ListingsReading Bond Listings

• Take care that the fractional part of Take care that the fractional part of a number is understooda number is understood– Is it 16ths, 32nds, 64ths, 100ths or Is it 16ths, 32nds, 64ths, 100ths or

some other convention?some other convention?

• Ask price: dealer’s selling priceAsk price: dealer’s selling price

• Bid price: dealer’s buying priceBid price: dealer’s buying price

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8.5 Why Yields for the 8.5 Why Yields for the same Maturity Differsame Maturity Differ

– The fundamental building block of bonds is The fundamental building block of bonds is the pure discount bond: Coupon bonds may the pure discount bond: Coupon bonds may be viewed as a portfolio of discount bondsbe viewed as a portfolio of discount bonds

– The rule of one price applies to bonds The rule of one price applies to bonds through pure discount bondsthrough pure discount bonds

– It is a mistake to assume that coupon bonds It is a mistake to assume that coupon bonds with the same life have the same yield--with the same life have the same yield--their coupon rates differ, leading to a their coupon rates differ, leading to a different % mix of discount bondsdifferent % mix of discount bonds

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8.6 The Behavior of Bond 8.6 The Behavior of Bond Prices Over TimePrices Over Time

• The The expectedexpected price of pure discount price of pure discount bonds rises exponentially to the face bonds rises exponentially to the face value with time, and the actual price value with time, and the actual price never exceeds parnever exceeds par

• Coupon bonds are more complex, and Coupon bonds are more complex, and their price may exceed their par value, their price may exceed their par value, but at maturity they reach their par but at maturity they reach their par valuevalue

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Bond Price TrajectoryBond Price Trajectory

• The following diagram shows the The following diagram shows the dynamic nature of the yield curve as dynamic nature of the yield curve as it passes through timeit passes through time– Think of a Think of a yield curveyield curve as constant time as constant time

cross-section of a cross-section of a yield surfaceyield surface in time in time • maturity • rate space• maturity • rate space

50

Dymanic Yield Curve

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

7.00%

8.00%

0 5 10 15 20

years to maturity

Yie

ld t

o m

atu

tiry

Current5-year10-year15-Year20-Year

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A Different View of the A Different View of the Yield CurveYield Curve

• The dynamics of the short-term and The dynamics of the short-term and long-term interest rates are shown nextlong-term interest rates are shown next

• Note that the two rates track each Note that the two rates track each other somewhat, but the long-term other somewhat, but the long-term rates have a little more volatilityrates have a little more volatility– the processes the processes areare synthetic, so don’t read synthetic, so don’t read

tootoo much into them (granular, smoothed) much into them (granular, smoothed)

52

Interest Rates

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

7.00%

8.00%

9.00%

0 5 10 15 20

Years

Rat

e

spot

long_forward

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The Non-Stationary Price The Non-Stationary Price Dynamics of a Maturing Dynamics of a Maturing BondBond• The following diagram demonstrates that The following diagram demonstrates that

the price history of a long-term coupon the price history of a long-term coupon bond is quite dynamic in its early days, bond is quite dynamic in its early days, but as the bond matures price but as the bond matures price movements becomes much more sedatemovements becomes much more sedate

• The path is similar to the x-component of The path is similar to the x-component of the random flight of a moth to a flamethe random flight of a moth to a flame

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20-Year Bond Value Over Time

920

940

960

980

1000

1020

1040

1060

05101520

Time to Maturity

Val

ue