CHAPTER 14

Bond Prices and YieldsBond Prices and Yields

14-2

Bond Characteristics

• Bonds are debt Issuers are

Bond Characteristics

• Bonds are debt. Issuers are borrowers and holders are creditors.

– The indenture is the contract between the issuer and the bondholder.

– The indenture gives the coupon rate, g p ,maturity date, and par value.

14-3

Bond Characteristics• Face or par value is typically $1000; this is

Bond CharacteristicsFace or par value is typically $1000; this is the principal repaid at maturity.

• The coupon rate determines the interest tpayment.

– Interest is usually paid semiannually.– The coupon rate can be zero.– Interest payments are called “couponInterest payments are called coupon

payments”.

14-4

U.S. Treasury BondsU.S. Treasury Bonds

• Bonds and notes may be purchased directly from •Note maturity

is 1 10 years p ythe Treasury.

• Denomination can be as

is 1-10 years

• Denomination can be as small as $100, but $1 000

•Bond maturity is 10 30 years $1,000 is more common.

• Bid price of 100:08

is 10-30 years

Bid price of 100:08 means 100 8/32 or $1002 50$1002.50

14-5

Corporate BondsCorporate Bonds

C ll bl b d b h d b f• Callable bonds can be repurchased before the maturity date.

• Convertible bonds can be exchanged for shares of the firm’s common stockshares of the firm s common stock.

• Puttable bonds give the bondholder the• Puttable bonds give the bondholder the option to retire or extend the bond.

• Floating rate bonds have an adjustable coupon ratep

14-6

Preferred StockPreferred Stock

• Dividends are paid in perpetuity•Equity perpetuity.

• Nonpayment of dividends

q y•Fixed income

does not mean bankruptcy.p y

• Preferred dividends are paid before commonpaid before common.

• No tax break.

14-7

Innovation in the Bond MarketInnovation in the Bond Market

• Inverse FloatersAsset Backed Bonds• Asset-Backed Bonds

• Catastrophe BondsCatastrophe Bonds• Indexed Bonds

–Treasury Inflation Protected Securities (TIPS)Securities (TIPS).

14-8

Table 14.1 Principal and Interest Payments for a Treasury Inflation Protected Security

14-9

Bond PricingT ParValueCBond Pricing

1 (1 )(1 )TB t

t

ParValueCPrr

1 (1 )(1 )t rr

PB = Price of the bondCt = interest or coupon paymentsT = number of periods to maturityr = semi-annual discount rate or the semi-annual

yield to maturity

14-10

Example 14 2: Bond PricingExample 14.2: Bond Pricing

Price of a 30 year, 8% coupon bond.Market rate of interest is 10%.

60

60

0511000$

05140$Price t 1 05.105.1t

71.810$Price

14-11

Bond Prices and YieldsBond Prices and Yields

• Prices and yields (required rates of return) have an inverse relationshipreturn) have an inverse relationship

• The bond price curve (Figure 14.3) is convex.

• The longer the maturity, the moreThe longer the maturity, the more sensitive the bond’s price to changes in market interest ratesmarket interest rates.

14-12

Figure 14.3 The Inverse Relationship Between Bond Prices and Yields

14-13

Table 14.2 Bond Prices at Different Interest Rates

14-14

Yield to MaturityYield to Maturity

• Interest rate that makes the presentInterest rate that makes the present value of the bond’s payments equal to its price is the YTMto its price is the YTM.

Solve the bond formula for rT

TBParValueCP

1 (1 )(1 )Tt

tBP

rr

14-15

Yield to Maturity ExampleYield to Maturity ExampleSuppose an 8% coupon, 30 year bond pp p , yis selling for $1276.76. What is its average rate of return?

1000$40761276$60

g

)1()1(76.1276$ 60

1 rrtt

r = 3% per half year

B d i l t i ld 6%Bond equivalent yield = 6%

EAR = ((1.03)2)-1=6.09%EAR ((1.03) ) 1 6.09%

14-16

YTM vs  Current YieldYTM vs. Current Yield

YTM• The YTM is the bond’s

i t l t f t

Current Yield• The current yield is the

internal rate of return.• YTM is the interest rate

that makes the present

ybond’s annual coupon payment divided by the that makes the present

value of a bond’s payments equal to its price

bond price.• For bonds selling at a

price.• YTM assumes that all

bond coupons can be

premium, coupon rate > current yield>YTM.bond coupons can be

reinvested at the YTM rate.

• For discount bonds, relationships are reversed.

14-17

Yield to CallYield to Call

• If interest rates fall, price of straight bond can rise considerably.y

• The price of the callable bond is flat over a range of low interest rates because therange of low interest rates because the risk of repurchase or call is high.

• When interest rates are high, the risk of call is negligible and the values of thecall is negligible and the values of the straight and the callable bond converge.

14-18

Figure 14.4 Bond Prices: Callable and Straight Debt

14-19

Realized Yield versus YTMRealized Yield versus YTM

• Reinvestment AssumptionsReinvestment Assumptions• Holding Period Return

– Changes in rates affect returns– Reinvestment of coupon paymentsReinvestment of coupon payments– Change in price of the bond

14-20

Figure 14 5 Growth of Invested FundsFigure 14.5 Growth of Invested Funds

14-21

Figure 14.6 Prices over Time of 30‐Year Maturity, 6.5% Coupon Bonds

14-22

YTM vs  HPRYTM vs. HPR

YTM• YTM is the average

HPR• HPR is the rate of return g

return if the bond is held to maturity.

over a particular investment period.

• YTM depends on coupon rate, maturity, and par value

• HPR depends on the bond’s price at the end of

value.• All of these are readily

observable

the holding period, an unknown future value.

observable.• HPR can only be

forecasted.

14-23

Figure 14.7 The Price of a 30‐Year Zero‐Coupon Bond over Time

14-24

Default Risk and Bond Pricing

R i i

Default Risk and Bond Pricing

• Rating companies:– Moody’s Investor Service, Standard & y

Poor’s, Fitch• Rating CategoriesRating Categories

– Highest rating is AAA or AaaI t t d b d t d BBB– Investment grade bonds are rated BBB or Baa and above

– Speculative grade/junk bonds have ratings below BBB or Baa.

14-25

Factors Used by Rating CompaniesFactors Used by Rating Companies

• Coverage ratios• Leverage ratios• Leverage ratios• Liquidity ratios• Profitability ratios• Cash flow to debtCash flow to debt

14-26

Table 14.3 Financial Ratios and Default Risk by Rating Class, Long‐Term Debt

14-27

Protection Against Default

• Sinking funds – a way to call bonds

Protection Against Default

Sinking funds a way to call bonds earlyS b di ti f f t d bt t i t• Subordination of future debt– restrict additional borrowing

• Dividend restrictions– force firm to retain assets rather than paying them p y gout to shareholders

• Collateral a particular asset• Collateral – a particular asset bondholders receive if the firm defaults

14-28

Default Risk and YieldDefault Risk and Yield

• The risk structure of interest rates refersThe risk structure of interest rates refers to the pattern of default premiums.Th i diff b t th i ld• There is a difference between the yield based on expected cash flows and yield

fbased on promised cash flows.• The difference between the expected p

YTM and the promised YTM is the default risk premium.default risk premium.

14-29

Figure 14 11 Yield Spreads Figure 14.11 Yield Spreads 

CHAPTER 15

The Term Structure of InterestThe Term Structure of Interest Rates

15-31

Overview of Term Structure

• The yield curve is a graph that displays

Overview of Term Structure

The yield curve is a graph that displays the relationship between yield and maturitymaturity.

• Information on expected future short term rates can be implied from the yield p ycurve.

15-32

Figure 15 1 Treasury Yield CurvesFigure 15.1 Treasury Yield Curves

15-33

Bond PricingBond Pricing

• Yields on different maturity bonds are notYields on different maturity bonds are not all equal.W d t id h b d h• We need to consider each bond cash flow as a stand-alone zero-coupon bond.

• Bond stripping and bond reconstitution• Bond stripping and bond reconstitution offer opportunities for arbitrage.

• The value of the bond should be the sum of the values of its parts.p

15-34

Table 15.1 Prices and Yields to Maturities on Zero‐Coupon Bonds ($1,000 Face Value)

15-35

Example 15 1 Valuing Coupon BondsExample 15.1 Valuing Coupon Bonds

• Value a 3 year, 10% coupon bond using discount rates from Table 15.1:

1100$100$100$Price 32 07.106.105.1Price

• Price = $1082.17 and YTM = 6.88%• 6.88% is less than the 3-year rate of 7%.

15-36

Two Types of Yield CurvesTwo Types of Yield CurvesOn-the-run YieldPure Yield Curve

• The pure yield curve

On-the-run Yield Curve

Th th i ldp yuses stripped or zero coupon Treasuries.

• The on-the-run yield curve uses recently i d b dp

• The pure yield curve may differ significantly

issued coupon bonds selling at or near par.

may differ significantly from the on-the-run yield curve

• The financial press typically publishes on-yield curve. the-run yield curves.

15-37

Yield Curve Under CertaintyYield Curve Under Certainty

• Suppose you want to invest for 2• Suppose you want to invest for 2 years. – Buy and hold a 2-year zero-or-or– Rollover a series of 1-year bonds

• Equilibrium requires that bothEquilibrium requires that both strategies provide the same return.

15-38

Figure 15.2 Two 2‐Year Investment Programs

15-39

Yield Curve Under CertaintyYield Curve Under Certainty

• Buy and hold vs rollover:• Buy and hold vs. rollover:2

2 1 2(1 ) (1 ) (1 )y r x r

12

2 1 21 (1 ) (1 )y r x r

• Next year’s 1-year rate (r2) is just enough to make rolling over a series of 1-year bonds equal to investing in y q gthe 2-year bond.

15-40

Spot Rates vs  Short RatesSpot Rates vs. Short Rates

• Spot rate – the rate that prevails today for a given maturityg y

• Short rate – the rate for a given maturity (e g one year) at different points in time(e.g. one year) at different points in time.

• A spot rate is the geometric average of its component short rates.

15-41

Short Rates and Yield Curve Slope

• When next year’s short rate, r2 , is

• When next year’s short rate, r2 , is less

greater than this year’s short rate, r1,

than this year’s short rate, r1, the yield

the yield curve slopes up.

curve slopes down.– May indicate rates

– May indicate rates are expected to

are expected to fall.

rise.

15-42

Figure 15 3 Short Rates versus Spot RatesFigure 15.3 Short Rates versus Spot Rates

15-43

Forward Rates from Observed RatesForward Rates from Observed Rates

1)1()1()1(

n

nn

n yyf

1)1( ny

f = one-year forward rate for period nfn one year forward rate for period n

yn = yield for a security with a maturity of n

)1()1()1( 11 n

nn

nn fyy

15-44

Example 15 4 Forward RatesExample 15.4 Forward Rates

• The forward interest rate is a forecast of a future short rate.

• Rate for 4-year maturity = 8%, rate for 3-year maturity = 7%year maturity = 7%.

1106108.11144

4

yf

1106.1

07.111 33

34

yf

%.f 06114

15-45

Interest Rate UncertaintyInterest Rate Uncertainty

S %• Suppose that today’s rate is 5% and the expected short rate for the following p gyear is E(r2) = 6%. The value of a 2-year zero is: 1000$zero is:

47.898$06.105.1

1000$

• The value of a 1-year zero is:y

38.952$051

1000$

05.1

15-46

Interest Rate UncertaintyInterest Rate Uncertainty

• The investor wants to invest for 1 year. – Buy the 2-year bond today and plan toBuy the 2 year bond today and plan to

sell it at the end of the first year for $1000/1 06 =$943 40$1000/1.06 =$943.40.

– 0r-– Buy the 1-year bond today and hold to

maturitymaturity.

15-47

Interest Rate UncertaintyInterest Rate Uncertainty

• What if next year’s interest rate is more (or less) than 6%? )

Th t l t th 2 b d i– The actual return on the 2-year bond is uncertain!

15-48

Interest Rate UncertaintyInterest Rate Uncertainty• Investors require a risk premium to• Investors require a risk premium to

hold a longer-term bond.

• This liquidity premium compensatesThis liquidity premium compensates short-term investors for the uncertainty about future pricesabout future prices.

15-49

Theories of Term StructureTheories of Term Structure

• ExpectationsLi idit P f• Liquidity Preference– Upward bias over expectationsUpward bias over expectations

15-50

Expectations TheoryExpectations Theory

• Observed long-term rate is a function of t d ’ h t t t d t dtoday’s short-term rate and expected future short-term rates.

• f = E(r ) and liquidity premiums are• fn = E(rn) and liquidity premiums are zero.

15-51

Liquidity Premium Theory

L t b d i k

Liquidity Premium Theory

• Long-term bonds are more risky; therefore, fn generally exceeds E(rn)

• The excess of f over E(r ) is the liquidityThe excess of fn over E(rn) is the liquidity premium.

• The yield curve has an upward bias built into the long-term rates because of the liquidity premium.q y p

15-52

Figure 15 4 Yield CurvesFigure 15.4 Yield Curves

15-53

Figure 15 4 Yield CurvesFigure 15.4 Yield Curves15-54

Interpreting the Term StructureInterpreting the Term Structure

Th i ld fl t t ti f• The yield curve reflects expectations of future interest rates.

• The forecasts of future rates are clouded by other factors, such as liquidity premiums.

• An upward sloping curve could indicate:– Rates are expected to riseRates are expected to rise– And/or

I t i l li idit i– Investors require large liquidity premiums to hold long term bonds.

15-55

Interpreting the Term StructureInterpreting the Term Structure

• The yield curve is a good predictor of the business cycle.– Long term rates tend to rise in anticipation

of economic expansionof economic expansion.– Inverted yield curve may indicate that

interest rates are e pected to fall andinterest rates are expected to fall and signal a recession.

15-56

Figure 15.6 Term Spread: Yields on 10‐year vs. 90‐day Treasury Securities

CHAPTER 16

Managing Bond PortfoliosManaging Bond Portfolios

16-58

Bond Pricing Relationships

1 Bond prices and yields are inversely

Bond Pricing Relationships

1. Bond prices and yields are inversely related.

2 A i i b d’ i ld t t it2. An increase in a bond’s yield to maturity results in a smaller price change than a

fdecrease of equal magnitude.3. Long-term bonds tend to be more price g p

sensitive than short-term bonds.

16-59

Bond Pricing Relationships Bond Pricing Relationships 

4. As maturity increases, price sensitivity increases at a decreasing rate.increases at a decreasing rate.

5. Interest rate risk is inversely related to the bond’s coupon ratethe bond s coupon rate.

6. Price sensitivity is inversely related to the yield to maturity at which the bond is selling.g

16-60

Figure 16.1 Change in Bond Price as a Function of Change in Yield to Maturity

16-61

Table 16.1 Prices of 8% Coupon Bond (Coupons Paid Semiannually)

16-62

Table 16.2 Prices of Zero‐Coupon Bond  (Semiannually Compounding)

16-63

DurationDuration

• A measure of the effective maturity of a bond

• The weighted average of the times until each payment is received, with p ythe weights proportional to the present value of the payment

• Duration is shorter than maturity for all bonds except zero coupon bonds.p p

• Duration is equal to maturity for zero coupon bonds.coupon bonds.

16-64

Duration: CalculationDuration: Calculation

Price)1( yCF t Price)1( yCFwt t

T

twtDT

t

1t

CFt=cash flow at time t

16-65

Duration/Price RelationshipDuration/Price Relationship

Price change is proportional to duration and not to maturityy

(1 )P yDx

D* = modified duration1

DxP y

D = modified duration

P *P D yP

16-66

Example 16 1 DurationExample 16.1 Duration

• Two bonds have duration of 1.8852 years. One is a 2-year, 8% coupon bond with y , pYTM=10%. The other bond is a zero coupon bond with maturity of 1 8852coupon bond with maturity of 1.8852 years.

f 1 88 2 2• Duration of both bonds is 1.8852 x 2 = 3.7704 semiannual periods.p

• Modified D = 3.7704/1+0.05 = 3.591 periodsperiods

16-67

Example 16 1 DurationExample 16.1 Duration

• Suppose the semiannual interest rate increases by 0.01%. Bond prices fall by:y p y

DP * yDPP

• =-3.591 x 0.01% = -0.03591%• Bonds with equal D have the same

interest rate sensitivity.y

16-68

Example 16 1 DurationExample 16.1 Duration

Coupon Bond• The coupon bond,

Zero• The zero-coupon p ,

which initially sells at $964.540, falls to

pbond initially sells for $1,000/1.05 3.7704 = $ ,

$964.1942 when its yield increases to

$ ,$831.9704.

• At the higher yield ity5.01%

• percentage decline of

At the higher yield, it sells for $1 000/1 053.7704 =percentage decline of

0.0359%.$1,000/1.05 $831.6717. This price also falls by 0 0359%also falls by 0.0359%.

16-69

Rules for DurationRules for Duration

Rule 1 The duration of a zero-coupon bond equals its time to maturityequals its time to maturity

Rule 2 Holding maturity constant a bond’sRule 2 Holding maturity constant, a bond s duration is higher when the coupon rate is lower

Rule 3 Holding the coupon rate constant, a bond’s duration generally increasesa bond s duration generally increases with its time to maturity

16-70

Rules for DurationRules for Duration

Rule 4 Holding other factors constant, g ,the duration of a coupon bond is higher when the bond’s yield to maturity is lowerlower

Rules 5 The duration of a level perpetuityRules 5 The duration of a level perpetuity is equal to: (1+y) / y

16-71

Figure 16.2 Bond Duration versus Bond Maturity

16-72

Table 16.3 Bond Durations (Yield to Maturity = 8% APR; Semiannual Coupons)

16-73

ConvexityConvexity

• The relationship between bond pricesThe relationship between bond prices and yields is not linear.

• Duration rule is a good approximation for only small changes in bond yields.

• Bonds with greater convexity have t i th i i ldmore curvature in the price-yield

relationship.

16-74

Figure 16.3 Bond Price Convexity: 30‐Year Maturity, 8% Coupon; Initial YTM = 8%

16-75

ConvexityConvexity

CF1

n

tt

t tty

CFyP

Convexity1

22 )(

)1()1(1

t yyP 1 )1()1(

C ti f C itCorrection for Convexity:

P 21 [ ( ) ]2P D y Convexity y

P

16-76

Figure 16 4 Convexity of Two BondsFigure 16.4 Convexity of Two Bonds

16-77

Why do Investors Like Convexity?Why do Investors Like Convexity?

• Bonds with greater curvature gain more in price when yields fall than they lose when p y yyields rise.

• The more volatile interest rates the more• The more volatile interest rates, the more attractive this asymmetry.

• Bonds with greater convexity tend to have higher prices and/or lower yields, all elsehigher prices and/or lower yields, all else equal.

16-78

Callable BondsCallable Bonds

• As rates fall, there is a ceiling on the bond’s market price, which cannot risebond s market price, which cannot rise above the call price.N ti it• Negative convexity

• Use effective duration:/Effective Duration = P Pr

16-79

Figure 16.5 Price –Yield Curve for a Callable Bond

16-80

Passive Management

• Two passive bond portfolio strategies:

Passive Management

• Two passive bond portfolio strategies:

1.Indexing2 Immunization2.Immunization

• Both strategies see market prices as being correct but the strategies havebeing correct, but the strategies have very different risks.

16-81

Bond Index FundsBond Index Funds

• Bond indexes contain thousands of issues, many of which are infrequently traded.y q y

• Bond indexes turn over more than stock indexes as the bonds matureindexes as the bonds mature.

• Therefore, bond index funds hold only a representative sample of the bonds in the actual index.actual index.

16-82

Figure 16.8 Stratification of Bonds into Cells

16-83

ImmunizationImmunization

• Immunization is a way to control interest rate risk.

Wid l d b i f d i• Widely used by pension funds, insurance companies, and banks.

16-84

ImmunizationImmunization

• Immunize a portfolio by matching the interest rate exposure of assets and pliabilities.

This means: Match the duration of the assets– This means: Match the duration of the assets and liabilities.P i i k d i t t t i k tl– Price risk and reinvestment rate risk exactly cancel out.

• Result: Value of assets will track the value of liabilities whether rates rise or fall.

16-85

Table 16.4 Terminal value of a Bond Portfolio After 5 Years

16-86

Table 16 5 Market Value Balance SheetTable 16.5 Market Value Balance Sheet

16-87

Figure 16 9 Growth of Invested FundsFigure 16.9 Growth of Invested Funds16-88

Figure 16 10 ImmunizationFigure 16.10 Immunization

16-89

Cash Flow Matching and DedicationCash Flow Matching and Dedication

• Cash flow matching = automatic• Cash flow matching = automatic immunization.

• Cash flow matching is a dedication strategy.gy

• Not widely used because of constraints associated with bondconstraints associated with bond choices.