Bond Valuation and Interest Rates

Prepared byPrepared byKen Hartviksen and Robert IronsideKen Hartviksen and Robert Ironside

INTRODUCTION TOINTRODUCTION TO CORPORATE FINANCECORPORATE FINANCELaurence Booth Laurence Booth •• W. Sean Cleary W. Sean Cleary

Chapter 6 – Bond Valuation and Interest Chapter 6 – Bond Valuation and Interest RatesRates

CHAPTER 6CHAPTER 6 Bond Valuation and Interest Bond Valuation and Interest

RatesRates

CHAPTER 6 – Bond Valuation and Interest Rates 6 - 3

Lecture AgendaLecture Agenda

• Learning ObjectivesLearning Objectives• Important TermsImportant Terms• Basic Structure of BondsBasic Structure of Bonds• Valuing BondsValuing Bonds• Bond YieldsBond Yields• Interest Rate DeterminantsInterest Rate Determinants• Other Types of Bonds/Debt InstrumentsOther Types of Bonds/Debt Instruments• Summary and ConclusionsSummary and Conclusions

– Concept Review QuestionsConcept Review Questions– Appendix – Bond DurationAppendix – Bond Duration


Learning ObjectivesLearning Objectives

• The basic features of different types of bondsThe basic features of different types of bonds• How to value bonds given an appropriate discount rateHow to value bonds given an appropriate discount rate• How to determine the discount rate or yield given the How to determine the discount rate or yield given the

market value of a bondmarket value of a bond• How market interest rates or yields affect bond investorsHow market interest rates or yields affect bond investors• How bond prices change over timeHow bond prices change over time• The factors (both domestic and global) that affect The factors (both domestic and global) that affect

interest ratesinterest rates


Important Chapter TermsImportant Chapter Terms

• Balloon paymentBalloon payment• BillsBills• Bond indentureBond indenture• Bullet paymentBullet payment• Call pricesCall prices• Callable bondsCallable bonds• Canada Savings BondsCanada Savings Bonds• Collateral trust bondsCollateral trust bonds• CouponsCoupons• Current yieldCurrent yield• DebenturesDebentures• Debt ratingsDebt ratings• Default freeDefault free

• Default riskDefault risk• Discount (premium)Discount (premium)• DurationDuration• Equipment trust certificatesEquipment trust certificates• Expectations theoryExpectations theory• Extendible bondsExtendible bonds• Face valueFace value• Floating rate bondsFloating rate bonds• Interest paymentsInterest payments• Interest rate parity (IRP) Interest rate parity (IRP)

theorytheory• Interest rate riskInterest rate risk• Issue-specific premiumsIssue-specific premiums• Liquidity preference theoryLiquidity preference theory• Maturity valueMaturity value


Important Chapter TermsImportant Chapter Terms

• Mortgage bondsMortgage bonds• Nominal interest ratesNominal interest rates• NotesNotes• PaperPaper• Par valuePar value• Protective covenantsProtective covenants• Purchase fund provisionsPurchase fund provisions• Real return bondsReal return bonds

• Retractable bondsRetractable bonds• Risk-free rateRisk-free rate• Sinking fund provisionsSinking fund provisions• SpreadSpread• Term structure of Term structure of

interest ratesinterest rates• Term to maturityTerm to maturity• Yield curveYield curve• Yield to maturityYield to maturity• Zero coupon bondZero coupon bond

The Basic Structure of BondsThe Basic Structure of Bonds

Bond Valuation and Interest RatesBond Valuation and Interest Rates


What is a Bond?What is a Bond?

• In its broadest sense, a bond is any debt In its broadest sense, a bond is any debt instrument that promises a fixed income instrument that promises a fixed income stream to the holderstream to the holder

• Fixed income securities are often classified Fixed income securities are often classified according to maturity, as follows:according to maturity, as follows:– Less than one year – Bills or “Paper”Less than one year – Bills or “Paper”– 1 year 1 year < Maturity < 7 years – Notes< Maturity < 7 years – Notes– < 7 years – Bonds< 7 years – Bonds


Basic Structure of BondsBasic Structure of Bonds

• A typical bond has the following A typical bond has the following characteristics:characteristics:– A fixed face or par value, paid to the holder of the A fixed face or par value, paid to the holder of the

bond, at maturitybond, at maturity– A fixed coupon, which specifies the interest payable A fixed coupon, which specifies the interest payable

over the life of the bondover the life of the bond• Coupons are usually paid either annually or semi-annuallyCoupons are usually paid either annually or semi-annually

– A fixed maturity dateA fixed maturity date



Note:Note:– The coupon rate, the maturity date, par value are all The coupon rate, the maturity date, par value are all

set (fixed) at the time the bond was originally sold to set (fixed) at the time the bond was originally sold to the marketthe market

– The coupon rate will reflect the required rates of The coupon rate will reflect the required rates of interest at the time of bond issue.interest at the time of bond issue.

– After issue, interest rates, and required rates of return After issue, interest rates, and required rates of return will change. Because everything is fixed except the will change. Because everything is fixed except the required rate of return and the bond price, as rates required rate of return and the bond price, as rates change, so too will bond prices!change, so too will bond prices!


• Bonds may be either:Bonds may be either:– Bearer bondsBearer bonds– Registered bondsRegistered bonds

• Bond indenture - the contract between the Bond indenture - the contract between the issuer of the bond and the investors who hold issuer of the bond and the investors who hold itit

• The market price of a bond is equal to the The market price of a bond is equal to the present value of the payments promised by present value of the payments promised by the bondthe bond

(See the basic pattern of cash flows from a traditional bond on the next slide)



Basic Structure of BondsBasic Structure of BondsCash Flow Pattern for a Traditional Coupon-Paying BondCash Flow Pattern for a Traditional Coupon-Paying Bond

0 1 2 3 … n

I I I I I

F

0 1 2 3 … n

I I I I I

F

6-1 FIGURE

I = interest payments, and F = principal repayment


Cash Flow Pattern of a BondCash Flow Pattern of a Bond

The Purchase Price or Market Price of a bond is simply the present value of the cash inflows, discounted at the bond’s yield-to-maturity

0 2 3 4 n1

Coupon Coupon Coupon Coupon Coupon +Face Value

Purchase Price

Cash Inflows to the Investor

Cash Outflows to the Investor

Bond Features and ProvisionsBond Features and Provisions



Bond IndentureBond Indenture

• The bond indenture is the contract between The bond indenture is the contract between the issuer and the holder. It specifies:the issuer and the holder. It specifies:– Details regarding payment termsDetails regarding payment terms– CollateralCollateral– Positive & negative covenantsPositive & negative covenants– Par or face value (usually increments of $1,000)Par or face value (usually increments of $1,000)– Bond pricing – usually shown as the price per $100 of Bond pricing – usually shown as the price per $100 of

par value, which is equal to the percentage of the par value, which is equal to the percentage of the bond’s face valuebond’s face value


More Bond TerminologyMore Bond Terminology

• Term-to-maturityTerm-to-maturity – the time remaining to the – the time remaining to the bond’s maturity datebond’s maturity date

• Coupon rateCoupon rate – the annual percentage interest – the annual percentage interest paid on the bond’s face value. To calculate paid on the bond’s face value. To calculate the dollar value of the annual coupon, the dollar value of the annual coupon, multiply the coupon rate times the face value.multiply the coupon rate times the face value.– If the coupon is paid twice a year, divide the annual If the coupon is paid twice a year, divide the annual

coupon by twocoupon by two– Example: A $1000 bond with an 8% coupon rate will Example: A $1000 bond with an 8% coupon rate will

have an $80 coupon if paid annually or a $40 coupon have an $80 coupon if paid annually or a $40 coupon if paid semi-annually.if paid semi-annually.


Security & Protective ProvisionsSecurity & Protective Provisions

• Mortgage bondsMortgage bonds – secured by real assets – secured by real assets• DebenturesDebentures – either unsecured or secured – either unsecured or secured

with a floating charge over the firm’s assetswith a floating charge over the firm’s assets• Collateral trust bondsCollateral trust bonds – secured by a pledge – secured by a pledge

of financial assets, such as common stock, of financial assets, such as common stock, other bonds or treasury billsother bonds or treasury bills

• Equipment trust certificatesEquipment trust certificates – secured by a – secured by a pledge of equipment, such as railway rolling pledge of equipment, such as railway rolling stockstock



• CovenantsCovenants– Positive covenantsPositive covenants – things the firm agrees – things the firm agrees toto dodo

• Supply periodic financial statementsSupply periodic financial statements• Maintain certain ratiosMaintain certain ratios

– Negative covenantsNegative covenants – things the firm agrees – things the firm agrees not to not to dodo

• Restrictions on the amount of debt the firm can take onRestrictions on the amount of debt the firm can take on• Prevents the firm from acquiring or disposing of assetsPrevents the firm from acquiring or disposing of assets


More Bond Features More Bond FeaturesMore Bond Features More Bond Features

• Call featureCall feature – allows the issuer to redeem or – allows the issuer to redeem or pay off the bond prior to maturity, usually at a pay off the bond prior to maturity, usually at a premiumpremium

• Retractable bondsRetractable bonds – allows the holder to sell – allows the holder to sell the bonds back to the issuer before maturitythe bonds back to the issuer before maturity

• Extendible bondsExtendible bonds – allows the holder to – allows the holder to extend the maturity of the bondextend the maturity of the bond

• Sinking fundsSinking funds – funds set aside by the issuer – funds set aside by the issuer to ensure the firm is able to redeem the bond to ensure the firm is able to redeem the bond at maturity at maturity



• Convertible bondsConvertible bonds – can be converted into – can be converted into common stock at a pre-determined common stock at a pre-determined conversion priceconversion price

Bond ValuationBond ValuationAnnual Coupon PaymentsAnnual Coupon Payments



Bond ValuationBond Valuation

• The value of a bond is a function of:The value of a bond is a function of:– The bond’s par (face) valueThe bond’s par (face) value– Term to maturityTerm to maturity– Coupon rateCoupon rate– Investor’s required rate of return (discount rate is also Investor’s required rate of return (discount rate is also

known as the bond’s yield to maturity)known as the bond’s yield to maturity)


Bond ValueBond ValueGeneral FormulaGeneral Formula

)k(

Fk

)k(IB

nbb

nb

1

111

1

[ 6-1]

Where:I = interest (or coupon ) paymentskb = the bond discount rate (or market rate)n = the term to maturityF = Face (or par) value of the bond


Bond Valuation: ExampleBond Valuation: Example

• What is the market price of a ten year, $1,000 bond What is the market price of a ten year, $1,000 bond with a 5% coupon, if the bond’s yield-to-maturity is with a 5% coupon, if the bond’s yield-to-maturity is 6%?6%?

10

10

1 1

1

1 1.06 1,00050

0.06 1.06

$926.40

n

bn

b b

k FB I

k k

Calculator Approach:1,000 FV50PMT10NI/Y 6CPT PV 926.40

Bond ValuationBond ValuationSemi-Annual Coupon PaymentsSemi-Annual Coupon Payments



Bond Valuation: Semi-Annual CouponsBond Valuation: Semi-Annual Coupons

• So far, we have assumed that all bonds have annual So far, we have assumed that all bonds have annual pay coupons. While this is true for many Eurobonds, it pay coupons. While this is true for many Eurobonds, it is not true for most domestic bond issues, which have is not true for most domestic bond issues, which have coupons that are paid semi-annuallycoupons that are paid semi-annually

• To adjust for semi-annual coupons, we must make To adjust for semi-annual coupons, we must make three changes:three changes:– Size of the coupon payment (divide the annual coupon payment Size of the coupon payment (divide the annual coupon payment

by 2 to get the cash flow paid each 6 months )by 2 to get the cash flow paid each 6 months )– Number of periods (multiply number of years to maturity by 2 to Number of periods (multiply number of years to maturity by 2 to

get number of semi-annual periods)get number of semi-annual periods)– Yield-to-maturity (divide by 2 to get the semi-annual yield)Yield-to-maturity (divide by 2 to get the semi-annual yield)– Once you solve for the semi-annual yield, you will want to Once you solve for the semi-annual yield, you will want to

convert it back to an annualized rate of return (YTM).convert it back to an annualized rate of return (YTM).


Bond Valuation: Semi-Annual CouponsBond Valuation: Semi-Annual Coupons

For example, suppose you want to value a 5 year, For example, suppose you want to value a 5 year, $10,000 Government of Canada bond with a 4% $10,000 Government of Canada bond with a 4% coupon, paid twice a year, given a YTM of 6%.coupon, paid twice a year, given a YTM of 6%.

2

2

2 5

2 5

1 12

212 2

.061 1

400 10,00020.062 .06

12 2

$9,146.98

n

b

nb b

x

x

kI F

Bk k

Calculator Approach:10,000 FV400 ÷ 2 = PMT5 x 2 = N6 ÷ 2 = I/YCPT PV 926.40

Factors Affecting Bond PricesFactors Affecting Bond Prices



Factors Affecting Bond PricesFactors Affecting Bond Prices

• There are three factors that affect the price There are three factors that affect the price volatility of a bondvolatility of a bond– Yield to maturityYield to maturity– Time to maturityTime to maturity– Size of couponSize of coupon

• We will look at each of these in turn.We will look at each of these in turn.


Inverse Relationship Between Bond Inverse Relationship Between Bond Prices and Yields to MaturityPrices and Yields to Maturity

• When interest rates (required rate of return When interest rates (required rate of return on the bond) increase, bond prices fall.on the bond) increase, bond prices fall.

(See Figure 6 – 2 on the next slide)


Factors Affecting Bond PricesFactors Affecting Bond PricesBond Price-Yield CurveBond Price-Yield Curve

Market Yield (%)

6 - 2 FIGURE

Price ($)


Bond ConvexityBond Convexity

• The convexity of the price/YTM graph reveals The convexity of the price/YTM graph reveals two important insights:two important insights:– The price rise due to a fall in YTM is greater than the The price rise due to a fall in YTM is greater than the

price decline due to a rise in YTM, given an identical price decline due to a rise in YTM, given an identical change in the YTMchange in the YTM

– For a given change in YTM, bond prices will change For a given change in YTM, bond prices will change more when interest rates are low than when they are more when interest rates are low than when they are high high


Coupon Rate Relationship to Yield-to- Coupon Rate Relationship to Yield-to- MaturityMaturity

• The relationship between the coupon rate and the The relationship between the coupon rate and the bond’s yield-to-maturity (YTM) determines if the bond’s yield-to-maturity (YTM) determines if the bond will sell at a premium, at a discount or at parbond will sell at a premium, at a discount or at par

If Then Bond Sells at a:

Coupon < YTMCoupon < YTM Market < FaceMarket < Face DiscountDiscount

Coupon = YTMCoupon = YTM Market = FaceMarket = Face ParPar

Coupon > YTMCoupon > YTM Market > FaceMarket > Face PremiumPremium


Factors Affecting Bond PricesFactors Affecting Bond PricesInverse Relationship Between Yields and PricesInverse Relationship Between Yields and Prices

• Yield to maturity (investor’s required return)Yield to maturity (investor’s required return)– Bond prices go down when the YTM goes upBond prices go down when the YTM goes up– Bond prices go up when the YTM goes downBond prices go up when the YTM goes down

• Look at the graph on the next slide. It shows how Look at the graph on the next slide. It shows how the price of a 25 year, 10% coupon bond changes the price of a 25 year, 10% coupon bond changes as the bond’s YTM varies from 1% to 30%as the bond’s YTM varies from 1% to 30%

• Note that the graph is not linear – instead it is Note that the graph is not linear – instead it is said to be convex to the originsaid to be convex to the origin


Price & Yield: 25 Year Bond, 10% CouponPrice & Yield: 25 Year Bond, 10% Coupon

Price/Yield Relationship

0

50

100

150

200

250

300

350

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Percent YTM

Pri

ce p

er

$100 o

f F

ace

Valu

eAs interest rates

increase, bond prices fall, but fall at a decreasing rate


Other Factors Affecting Bond PricesOther Factors Affecting Bond PricesTerm to Maturity and Size of CouponTerm to Maturity and Size of Coupon

• Term to maturityTerm to maturity - long bonds have greater - long bonds have greater price volatility than short bondsprice volatility than short bonds

• Size of couponSize of coupon – low coupon bonds have – low coupon bonds have greater price volatility than high coupon greater price volatility than high coupon bondsbonds


Other Factors Affecting Bond PricesOther Factors Affecting Bond PricesTime to MaturityTime to Maturity

• Time to maturityTime to maturity– Long bonds have greater price volatility than short Long bonds have greater price volatility than short

bondsbonds• The longer the bond, the longer the period for which the The longer the bond, the longer the period for which the

cash flows are fixedcash flows are fixed• More distant cash flows are affected more in the More distant cash flows are affected more in the

discounting process (remember the exponential nature of discounting process (remember the exponential nature of compounding…and that discounting is the inverse of compounding…and that discounting is the inverse of compounding)compounding)

• The most distant cash flow from a bond investment is the The most distant cash flow from a bond investment is the most important (it is the face value of the bond) and this most important (it is the face value of the bond) and this cash flow is affected the greatest in the discounting cash flow is affected the greatest in the discounting process.process.


Other Factors Affecting Bond PricesOther Factors Affecting Bond PricesSize of the Coupon RateSize of the Coupon Rate

• Size of couponSize of coupon– Low coupon bonds have greater price volatility Low coupon bonds have greater price volatility

than high coupon bondsthan high coupon bonds• High coupons act like a stabilizing device, since a High coupons act like a stabilizing device, since a

greater proportion of the bond’s total cash flows occur greater proportion of the bond’s total cash flows occur closer to today & are therefore less affected by a change closer to today & are therefore less affected by a change in YTMin YTM

• The greatest price volatility is found with stripped bonds The greatest price volatility is found with stripped bonds (no coupon payments)(no coupon payments)


Interest Rate Risk & DurationInterest Rate Risk & Duration

• The sensitivity of bond prices to changes in The sensitivity of bond prices to changes in interest rates is a measure of the bond’s interest rates is a measure of the bond’s interest rate riskinterest rate risk

• A bond’s interest rate risk is affected by:A bond’s interest rate risk is affected by:– Yield to maturityYield to maturity– Term to maturityTerm to maturity– Size of couponSize of coupon

• These three factors are all captured in one These three factors are all captured in one number called number called DurationDuration


DurationDuration

• Duration is a measure of interest rate riskDuration is a measure of interest rate risk• The higher the duration, the more sensitive The higher the duration, the more sensitive

the bond is to changes in interest ratesthe bond is to changes in interest rates• A bond’s duration will be higher if its:A bond’s duration will be higher if its:

– YTM is lowerYTM is lower– Term to maturity is longerTerm to maturity is longer– Coupon is lowerCoupon is lower

(See the Appendix to this slide set for a complete discussion of duration)(See the Appendix to this slide set for a complete discussion of duration)

Bond QuotationsBond Quotations



Bond PricesBond PricesDiscount and Premium Priced BondsDiscount and Premium Priced Bonds

• Bonds trading at prices > par - premium Bonds trading at prices > par - premium pricedpriced

• Bonds trading at prices < par – discount Bonds trading at prices < par – discount pricedpriced


Bond QuotationsBond Quotations

Issuer Coupon Maturity Price Yield

Canada 5.500 2009-Jun-01 103.79 4.16

The issuer of the bond. The coupon rate in percent.

The bond’s maturity date. It takes on meaning on the

bond reporting page because there will be a

current date. The difference between the two

dates is the number of years to maturity.

The bond’s price is in dollars assuming a par

value of $100. Since this bond price is greater than

$100, it is called a premium priced bond.

The bond equivalent yield to maturity (BEY)

expressed in percent on an annualized basis.


Cash Versus Quoted PricesCash Versus Quoted Prices

• The quoted price is the price reported by the The quoted price is the price reported by the mediamedia

• The cash price is the price paid by an investorThe cash price is the price paid by an investor– The cash price includes both the quoted price plus The cash price includes both the quoted price plus

any interest that has accrued since the last coupon any interest that has accrued since the last coupon payment datepayment date


Cash Versus Quoted Price: ExampleCash Versus Quoted Price: Example

• Assume that you want to purchase a $1,000 Assume that you want to purchase a $1,000 bond with a 5% coupon, paid semi-annually. bond with a 5% coupon, paid semi-annually. Today is July 15Today is July 15thth. The last coupon was paid . The last coupon was paid June 30June 30thth. If the quoted price is $902, how . If the quoted price is $902, how much is the cash price? much is the cash price?


Cash Versus Quoted Price: SolutionCash Versus Quoted Price: Solution

• The cash price is equal to:The cash price is equal to:– Quoted price of $902Quoted price of $902– Plus 15 days of interestPlus 15 days of interest

15902 1,000 0.05

365

902 2.05

$904.05

Cash price = Quoted Price+ Accrued Interest

Bond YieldsBond Yields



Bond YieldsBond YieldsThe Yield to Maturity = Investor’s Required Rate of ReturnThe Yield to Maturity = Investor’s Required Rate of Return

• Yield-to-maturity (YTM) – the discount rate Yield-to-maturity (YTM) – the discount rate used to evaluate bondsused to evaluate bonds– The yield earned by a bond investor who:The yield earned by a bond investor who:

• Purchases the bond at the current market pricePurchases the bond at the current market price• Holds the bond to maturityHolds the bond to maturity• Reinvests all of the coupons at the YTM for the remaining Reinvests all of the coupons at the YTM for the remaining

term to maturity (the reinvestment rate assumption)term to maturity (the reinvestment rate assumption)

– Is the bond’s Internal Rate of Return (IRR) Is the bond’s Internal Rate of Return (IRR)


Bond Yield to MaturityBond Yield to Maturity

• The yield to maturity is that discount rate that causes The yield to maturity is that discount rate that causes the sum of the present value of promised cash flows the sum of the present value of promised cash flows to equal the current bond price.to equal the current bond price.

YTM)(

FYTM

YTM)(IB

n

n

1

111

1

[ 6-2]


Solving for YTMSolving for YTM

• To solve for YTM, solve for YTM in the following To solve for YTM, solve for YTM in the following formula:formula:

• There is a ProblemThere is a Problem: : – You can’t solve for YTM algebraically; therefore, must either You can’t solve for YTM algebraically; therefore, must either

use a financial calculator, Excel, trial & error or approximation use a financial calculator, Excel, trial & error or approximation formula.formula.

1 1

1

n

n

YTM FB I

YTM YTM


Solving for YTMSolving for YTM

• Example: What is the YTM on a 10 year, 5% coupon Example: What is the YTM on a 10 year, 5% coupon bond (annual pay coupons) that is selling for $980?bond (annual pay coupons) that is selling for $980?

10

10

1 1

1

1 1 1,000980 50

1

5.26%

n

n

YTM FB I

YTM YTM

YTM

YTM YTM

YTM

Financial Calculator1,000 FV980 +/- PV50 PMT51 NI/Y 5.26%


Solving for YTM: Semi-annual CouponsSolving for YTM: Semi-annual Coupons

• When solving for YTM with a semi-annual pay When solving for YTM with a semi-annual pay coupon, the yield obtained must be multiplied coupon, the yield obtained must be multiplied by two to obtain the annual YTMby two to obtain the annual YTM

• Example: What is the YTM for a 20 year, Example: What is the YTM for a 20 year, $1,000 bond with a 6% coupon, paid semi-$1,000 bond with a 6% coupon, paid semi-annually, given a current market price of annually, given a current market price of $1,030?$1,030?


Solving for YTM: Semi-annual CouponsSolving for YTM: Semi-annual Coupons

40

40

1 1

1

1 1 1,0001,030 30

1

2.87 2 5.74%

n

n

YTM FB I

YTM YTM

YTM

YTM YTM

YTM x

Financial Calculator1,000 FV1,030 +/- PV30 PMT40 NI/Y 2.87 x

2 =

5.746%

What is the YTM for a 20 year, $1,000 bond with a 6% What is the YTM for a 20 year, $1,000 bond with a 6% coupon, paid semi-annually, given a current market price of coupon, paid semi-annually, given a current market price of $1,030?$1,030?

Using the Approximation Formula to Using the Approximation Formula to Solve for Yield to MaturitySolve for Yield to Maturity



The Approximation FormulaThe Approximation Formula

• This formula gives you a quick estimate of the This formula gives you a quick estimate of the yield to maturityyield to maturity– It is an estimate because it is based on a linear It is an estimate because it is based on a linear

approximation (again you will remember the approximation (again you will remember the exponential nature of compound interest)exponential nature of compound interest)

• Should you be concerned with the ‘error’ Should you be concerned with the ‘error’ inherent in the approximated YTM?inherent in the approximated YTM?– NONO– Remember a YTM is an Remember a YTM is an ex ante ex ante calculation – as a calculation – as a

forecast, it is based on assumptions which may or forecast, it is based on assumptions which may or may not hold in this case, therefore as a forecast or may not hold in this case, therefore as a forecast or estimate, the approximation approach should be fine.estimate, the approximation approach should be fine.


The Approximation FormulaThe Approximation Formula

F = Face Value = Par Value = $1,000F = Face Value = Par Value = $1,000

B = Bond PriceB = Bond Price

I = the semi annual coupon interestI = the semi annual coupon interest

N = number of semi-annual periods left to maturityN = number of semi-annual periods left to maturity

1YTM) annual-semi (1YTM

YTM annual-semi 2YTM2

nB-F

Maturity toYield annual-Semi

2

BF

I


ExampleExample

• Find the yield-to-maturity of a 5 year 6% Find the yield-to-maturity of a 5 year 6% coupon bond that is currently priced at $850. coupon bond that is currently priced at $850. (Always assume the coupon interest is paid (Always assume the coupon interest is paid semi-annually.)semi-annually.)

• Therefore there is coupon interest of $30 paid semi-annuallyTherefore there is coupon interest of $30 paid semi-annually• There are 10 semi-annual periods left until maturity There are 10 semi-annual periods left until maturity


Example – with SolutionExample – with Solution

Find the yield-to-maturity of a 5 year 6% coupon bond that is currently priced at $850. (Always assume the coupon interest is paid semi-annually.)

%97.91)0486.1(1YTM) annual-semi (1YTM

9.3%0.0927320.0486YTM annual-semi 2YTM

0486.0925$

30$15$

2850,1$

30$10

850$000,1$

2

nB-F

Maturity toYield annual-Semi

22

BF

I

The actual answer is 9.87%...so of course, the approximation approach only gives us an approximate answer…but that is just fine for tests and exams.

The actual answer is 9.87%...so of course, the approximation approach only gives us an approximate answer…but that is just fine for tests and exams.


The Logic of the EquationThe Logic of the EquationApproximation Formula for YTMApproximation Formula for YTM

• The numerator simply represents the average semi-annual The numerator simply represents the average semi-annual returns on the investment…it is made up of two returns on the investment…it is made up of two components:components:– The first component is the average capital gain (if it is a discount bond) The first component is the average capital gain (if it is a discount bond)

or capital loss (if it is a premium priced bond) per semi-annual period.or capital loss (if it is a premium priced bond) per semi-annual period.– The second component is the semi-annual coupon interest received.The second component is the semi-annual coupon interest received.

• The denominator represents the average price of the bond.The denominator represents the average price of the bond.• Therefore the formula is basically, average semi-annual Therefore the formula is basically, average semi-annual

return on average investment.return on average investment.• Of course, we annualize the semi-annual return so that we Of course, we annualize the semi-annual return so that we

can compare this return to other returns on other can compare this return to other returns on other investments for comparison purposes.investments for comparison purposes.

Yield To CallYield To Call



Yield to CallYield to Call

• If a bond has a call feature, the issuer can call If a bond has a call feature, the issuer can call the bond prior to its stated maturitythe bond prior to its stated maturity

• To calculate the yield to call, simply replace To calculate the yield to call, simply replace the maturity date with the first call datethe maturity date with the first call date


Yield to CallYield to Call

• The yield to call is that discount rate that causes the The yield to call is that discount rate that causes the present value of all promised cash flows including the present value of all promised cash flows including the call price (CP) to equal the current bond price.call price (CP) to equal the current bond price.

YTC)(

CPYTC

YTC)(IB

n

n

1

111

1

[ 6-3]


Solving for YTC: Semi-annual CouponsSolving for YTC: Semi-annual Coupons

Financial Calculator1,050 FV1,030 +/- PV30 PMT10 NI/Y 3.081 x

2 =

6.16%

YTC on a 20-year 6 percent bond that is callable in five years at a call price of $1,050. The bond pays semi-annual coupons and is selling for $1,030.

%16.62%081.3

%081.3

1

050,1$11

130$030,1$

1

111

1

10

10

YTC

annuallysemiYTC

YTC)(YTCYTC)(

YTC)(

CPYTC

YTC)(IB

n

n

Current YieldCurrent Yield



Current YieldCurrent Yield

• The current yield is the yield on the bond’s current The current yield is the yield on the bond’s current market price provided by the annual couponmarket price provided by the annual coupon– It is not a true measure of the return to the bondholder because It is not a true measure of the return to the bondholder because

it does not consider potential capital gain or capital losses based it does not consider potential capital gain or capital losses based on the relationship between the purchase price of the bond and on the relationship between the purchase price of the bond and it’s par value.it’s par value.

B

interestAnnualCY [ 6-4]


Current YieldCurrent YieldExampleExample

• The current yield is the yield on the bond’s current The current yield is the yield on the bond’s current market price provided by the annual couponmarket price provided by the annual coupon

• Example:Example: If a bond has a 5.5% annual pay coupon If a bond has a 5.5% annual pay coupon and the current market price of the bond is $1,050, and the current market price of the bond is $1,050, the current yield is:the current yield is:

55

1,050

5.24%

Annual CouponCurrent Yield =

Current Market Price

Short-Term Interest RatesShort-Term Interest Rates



Interest Rate DeterminantsInterest Rate Determinants

• Interest is the “price” of moneyInterest is the “price” of money– Interest rate changes are often measured in Basis Interest rate changes are often measured in Basis

points – 1/100 of 1%points – 1/100 of 1%

• Interest rates go:Interest rates go:– Up Up – when the demand for loanable funds rises– when the demand for loanable funds rises– DownDown – when the demand for loanable funds falls – when the demand for loanable funds falls


Risk-free Interest RateRisk-free Interest Rate

• Usually use the yield on short federal Usually use the yield on short federal government Treasury bills as a proxy for the government Treasury bills as a proxy for the risk-free rate (RF)risk-free rate (RF)

• The risk-free rate is comprised of two The risk-free rate is comprised of two components:components:– Real rateReal rate – compensation for deferring consumption – compensation for deferring consumption– Expected inflationExpected inflation – compensation for the expected – compensation for the expected

loss in purchasing power loss in purchasing power

(See Figure 6 – 3 to see rates of inflation and yields on long Canada bonds since 1961)


Inflation and Yields over TimeInflation and Yields over Time

6 - 3 FIGURE


Fisher EquationFisher Equation

• If we call the risk-free rate the nominal rate, then If we call the risk-free rate the nominal rate, then the relationship between the real rate, the the relationship between the real rate, the nominal rate and expected inflation is usually nominal rate and expected inflation is usually referred to as the Fisher Equation (after Irving referred to as the Fisher Equation (after Irving Fisher)Fisher)

inflation Expectedrate RealRF [ 6-5]


Fisher EquationFisher Equation

• When inflation is low, can safely use the When inflation is low, can safely use the approximation formula:approximation formula:

• When inflation is high, use the exact form of the When inflation is high, use the exact form of the Fisher Equation:Fisher Equation:

Nominal RealR = R + Expected Inflation

1 1 1Nominal RealR = R Expected Inflation


Fisher Equation Fisher Equation ExampleExample

• If the real rate is 3% and the nominal rate is 5.5%, If the real rate is 3% and the nominal rate is 5.5%, what is the approximate expected future inflation what is the approximate expected future inflation rate?rate?

5.5 3

2.5%

Nominal RealR = R + Expected Inflation

Expected Inflation

Expected Inflation


Global Influences on Interest RatesGlobal Influences on Interest Rates

• Canadian domestic interest rates are heavily Canadian domestic interest rates are heavily influenced by global interest ratesinfluenced by global interest rates

• Interest rate parity (IRP) theory states that FX Interest rate parity (IRP) theory states that FX forward rates will be established that equalize forward rates will be established that equalize the yield an investor can earn, whether the yield an investor can earn, whether investing domestically or in a foreign investing domestically or in a foreign jurisdictionjurisdiction– A country with high inflation and high interest rates A country with high inflation and high interest rates

will have a depreciating currencywill have a depreciating currency

Term Structure of Interest RatesTerm Structure of Interest Rates(Long-term Interest Rates)(Long-term Interest Rates)



Term Structure of Interest RatesTerm Structure of Interest Rates

• Is that set of rates (YTM) for a given risk-class Is that set of rates (YTM) for a given risk-class of debt securities (for example, Government of debt securities (for example, Government of Canada Bonds) at a given point in time.of Canada Bonds) at a given point in time.

• When plotted on a graph, the line is called a When plotted on a graph, the line is called a Yield CurveYield Curve


Term Structure of Interest RatesTerm Structure of Interest Rates

• The Yield Curve is the graph created by putting term The Yield Curve is the graph created by putting term to maturity on the X axis, YTM on the Y axis and then to maturity on the X axis, YTM on the Y axis and then plotting the yield at each maturity.plotting the yield at each maturity.

• The four typical shapes of yield curves:The four typical shapes of yield curves:• Upward sloping (the most common and persistent shape historically Upward sloping (the most common and persistent shape historically

when short-term interest rates and inflation are low)when short-term interest rates and inflation are low)• Downward sloping (occurs at peaks in the short-term interest rate Downward sloping (occurs at peaks in the short-term interest rate

cycle, when inflation is expected to decrease in the future)cycle, when inflation is expected to decrease in the future)• Flat (occurs when rates are transitioning)Flat (occurs when rates are transitioning)• Humped (occurs when rates are transitioning or perhaps market Humped (occurs when rates are transitioning or perhaps market

participants are attracted in large numbers to particular maturity participants are attracted in large numbers to particular maturity segment of the market)segment of the market)

(See Figure 6-4 for Yield curves that existed at various times in Canada)


Historical Yield CurvesHistorical Yield Curves1990, 1994, 1998, 20041990, 1994, 1998, 2004

Pe

rce

nt

Term Left to Maturity

16

14

12

10

8

6

4

2

0

1 mth 3 mths 6 mths 1 yr 2yrs 5 yrs 7 yrs 10 yrs 30 yrs

6 - 4 FIGURE

1990 1994 1998 2004


Theories of the Term StructureTheories of the Term Structure

• Three theories are used to explain the shape Three theories are used to explain the shape of the term structureof the term structure– Liquidity preference theoryLiquidity preference theory

• Investors must be paid a “liquidity premium” to hold less Investors must be paid a “liquidity premium” to hold less liquid, long-term debtliquid, long-term debt

– Expectations theoryExpectations theory• The long rate is the average of expected future short interest The long rate is the average of expected future short interest

ratesrates

– Market segmentation theoryMarket segmentation theory• Distinct markets exist for securities of different maturitiesDistinct markets exist for securities of different maturities


Term Structure of Interest RatesTerm Structure of Interest RatesRisk PremiumsRisk Premiums

• More risky bonds (ie. BBB rated Corporate Bonds) will More risky bonds (ie. BBB rated Corporate Bonds) will have their own yield curve and it will plot at higher have their own yield curve and it will plot at higher YTM at every term to maturity because of the default YTM at every term to maturity because of the default risk that BBBs carryrisk that BBBs carry

• The difference between the YTM on a 10-year BBB The difference between the YTM on a 10-year BBB corporate bond and a 10-year Government of Canada corporate bond and a 10-year Government of Canada bond is called a bond is called a yield spreadyield spread and represents a and represents a default-default-risk premiumrisk premium investors demand for investing in more investors demand for investing in more risky securities.risky securities.

• Spreads will increase when pessimism increases in the Spreads will increase when pessimism increases in the economyeconomy

• Spreads will narrow during times of economic Spreads will narrow during times of economic expansion (confidence)expansion (confidence)


Yield Curves for Different Risk ClassesYield Curves for Different Risk ClassesRisk Premiums (Yield Spreads)Risk Premiums (Yield Spreads)

Pe

rce

nt

Term Left to Maturity

16

14

12

10

8

6

4

2

0

1 mth 3 mths 6 mths 1 yr 2yrs 5 yrs 7 yrs 10 yrs 30 yrs

BBB Corporates Government of Canada Bonds

Yield Spread


Risk PremiumsRisk Premiums

• The YTM on a corporate bond is comprised of:The YTM on a corporate bond is comprised of:

• The maturity yield differential is explained by the The maturity yield differential is explained by the term structureterm structure

• Spread is the additional yield due to default riskSpread is the additional yield due to default risk

Spread aldifferenti yieldMaturity -/ RFkb YTM[ 6-6]


Debt RatingsDebt Ratings

• All publicly traded bonds are assigned a “risk All publicly traded bonds are assigned a “risk rating” by a rating agency, such as Dominion rating” by a rating agency, such as Dominion Bond Rating Service (DBRS), Standard & Bond Rating Service (DBRS), Standard & Poors (S&P), Moodys, Fitch, etc.Poors (S&P), Moodys, Fitch, etc.

• Bonds are categorized as:Bonds are categorized as:– Investment grade – top four rating categories (AAA, Investment grade – top four rating categories (AAA,

AA, A & BBB)AA, A & BBB)– Junk or high yield – everything below investment Junk or high yield – everything below investment

grade (BB, B, CCC, CC, D, Suspended)grade (BB, B, CCC, CC, D, Suspended)


Why do Bonds Have Different Yields?Why do Bonds Have Different Yields?

• Default riskDefault risk – the higher the default risk, the – the higher the default risk, the higher the required YTMhigher the required YTM

• LiquidityLiquidity – the less liquid the bond, the higher – the less liquid the bond, the higher the required YTMthe required YTM

• Call featuresCall features – increase required YTM – increase required YTM• Extendible featureExtendible feature – reduce required YTM – reduce required YTM• Retractable featureRetractable feature – reduce required YTM – reduce required YTM

Other Types of Bonds/Debt InstrumentsOther Types of Bonds/Debt Instruments



Treasury BillsTreasury Bills

• Short-term obligations of government with an initial Short-term obligations of government with an initial term to maturity of one year or lessterm to maturity of one year or less

• Issued at a discount & mature at face valueIssued at a discount & mature at face value• The difference between the issue price and the face The difference between the issue price and the face

value is treated as interest incomevalue is treated as interest income• To calculate the price of a T bill, use the following To calculate the price of a T bill, use the following

formula:formula:

1T Bill

FP

nBEY

B

Where:P = market price of the T BillF = face value of the T BillBEY = the bond equivalent yieldn = the number of days until maturityB = the annual basis (365 days in Canada)


Treasury Bills: ExampleTreasury Bills: Example

• What is the price of a $1,000,000 Canadian T bill with What is the price of a $1,000,000 Canadian T bill with 80 days to maturity and a BEY of 4.5%?80 days to maturity and a BEY of 4.5%?

1

1,000,00080

1 .045365

$990,233.32

T Bill

FP

nBEY

B


Solving for Yield on a T BillSolving for Yield on a T Bill

• To solve for the yield on a T bill, rearrange the To solve for the yield on a T bill, rearrange the previous formula and solve for BEY. previous formula and solve for BEY.

• ExampleExample: What is the yield on a $100,000 T bill with : What is the yield on a $100,000 T bill with 180 days to maturity and a market price of $98,200?180 days to maturity and a market price of $98,200?

100,000 98,200 365

98,200 180

3.72%

F P BBEY

P n


Zero Coupon BondsZero Coupon Bonds

• A zero coupon bond is a bond issued at a A zero coupon bond is a bond issued at a discount that matures at par or face valuediscount that matures at par or face value

• A zero coupon bond has no reinvestment rate A zero coupon bond has no reinvestment rate risk, since there are no coupons to be risk, since there are no coupons to be reinvestedreinvested

• To calculate the price of a zero coupon bond, To calculate the price of a zero coupon bond, solve for the PV of the face amount solve for the PV of the face amount


Zero Coupon BondsZero Coupon Bonds

• Example: What is the market price of a $50,000 Example: What is the market price of a $50,000 zero coupon bond with 25 years to maturity that is zero coupon bond with 25 years to maturity that is currently yielding 6%?currently yielding 6%?

25

F

1

50,000

1.06

$11,649.93

n

b

Bk


Floating Rate & Real Return BondsFloating Rate & Real Return Bonds

• Floating rate bondsFloating rate bonds have a coupon that floats have a coupon that floats with some reference rate, such as the yield with some reference rate, such as the yield on T billson T bills– Because the coupon floats, the market price will Because the coupon floats, the market price will

typically be close to the bond’s face valuetypically be close to the bond’s face value

• Real return bondsReal return bonds are issued by the are issued by the Government of Canada to protect investors Government of Canada to protect investors against unexpected inflationagainst unexpected inflation– Each period, the face value of the bond is grossed up Each period, the face value of the bond is grossed up

by the inflation rate. The coupon is then paid on the by the inflation rate. The coupon is then paid on the grossed up face value.grossed up face value.


Canada Savings BondsCanada Savings Bonds

• A Canada Savings Bond (CSB) is a special type of bond A Canada Savings Bond (CSB) is a special type of bond issued by the Government of Canadaissued by the Government of Canada

• It is issued in two forms:It is issued in two forms:– Regular interest – interest is paid annuallyRegular interest – interest is paid annually– Compound interest – interest compounds over the life of the Compound interest – interest compounds over the life of the

bondbond

• CSBs are redeemable at any chartered bank in Canada CSBs are redeemable at any chartered bank in Canada at their face value plus accrued interest (after the first at their face value plus accrued interest (after the first three months after issue)three months after issue)

• There is no secondary market for CSBs (they are ‘non-There is no secondary market for CSBs (they are ‘non-negotiable’ meaning that they cannot be traded in a negotiable’ meaning that they cannot be traded in a market between investors.market between investors.


Summary and ConclusionsSummary and Conclusions

In this chapter you have learned:In this chapter you have learned:

– About the nature of bonds as an investmentAbout the nature of bonds as an investment– How to value a bond using discounted cash flow How to value a bond using discounted cash flow

conceptsconcepts– About the determinants of interest rates and theories About the determinants of interest rates and theories

used to explain the term structure of interest ratesused to explain the term structure of interest rates

Concept Review QuestionsConcept Review Questions



Concept Review Question 1Concept Review Question 1Bonds and MortgagesBonds and Mortgages

In what ways are bonds different from In what ways are bonds different from mortgages?mortgages?

Appendix A – Bond DurationAppendix A – Bond Duration



DurationDuration

• An alternative measure of bond price sensitivity is the bond’s An alternative measure of bond price sensitivity is the bond’s duration.duration.

• Duration measures the life of the bond on a present value Duration measures the life of the bond on a present value basis.basis.

• Duration can also be thought of as the average time to receipt Duration can also be thought of as the average time to receipt of the bond’s cash flows.of the bond’s cash flows.

• The longer the bond’s duration, the greater is its sensitivity to The longer the bond’s duration, the greater is its sensitivity to interest rate changes.interest rate changes.


Duration Rules-of-ThumbDuration Rules-of-Thumb

• Duration of zero-coupon bond (strip bond) = the term left until Duration of zero-coupon bond (strip bond) = the term left until maturity.maturity.

• Duration of a consol bond (ie. a perpetual bond) = 1 + (1/k)Duration of a consol bond (ie. a perpetual bond) = 1 + (1/k)where: k = required yield to maturitywhere: k = required yield to maturity

• Duration of an FRN (floating rate note) = 1/2 yearDuration of an FRN (floating rate note) = 1/2 year


Other Duration Rules-of-ThumbOther Duration Rules-of-Thumb

Duration and MaturityDuration and Maturity Duration Duration increasesincreases with maturity of a fixed-income asset, but at a decreasing with maturity of a fixed-income asset, but at a decreasing

rate.rate.

Duration and YieldDuration and Yield Duration Duration decreasesdecreases as yield increases. as yield increases.

Duration and Coupon InterestDuration and Coupon Interest The higher the coupon or promised interest payment on the security, the lower The higher the coupon or promised interest payment on the security, the lower

its duration.its duration.


Economic Meaning of DurationEconomic Meaning of Duration

• duration is a duration is a direct direct measure of the interest rate sensitivity measure of the interest rate sensitivity or or elasticityelasticity of an asset or liability. (ie. what impact will a of an asset or liability. (ie. what impact will a change in YTM have on the price of the particular fixed-change in YTM have on the price of the particular fixed-income security?)income security?)

• interest rate sensitivity is equal to:interest rate sensitivity is equal to:

dPdP == - D- D [ [ dk/(1+k)dk/(1+k)]]

PPWhere:Where: P =P = Price of bondPrice of bond

C =C = Coupon (annual)Coupon (annual)

k =k = YTMYTM

N =N = Number of periodsNumber of periods

F = F = Face value of bondFace value of bond


Interest Rate ElasticityInterest Rate Elasticity

• the percent change in the bond’s price the percent change in the bond’s price caused by a given change in interest rates caused by a given change in interest rates (change in YTM)(change in YTM)

(The following slide illustrates how bond price sensitivity can be graphed (The following slide illustrates how bond price sensitivity can be graphed against changing discount rates)against changing discount rates)


Price Elasticity of Stripped BondsPrice Elasticity of Stripped Bonds

$0

$5,000

$10,000

$15,000

$20,000

0.0% 5.0% 10.0% 15.0% 20.0%

30 year stripped bond price given different YTM.


Price Sensitivity of a Stripped BondPrice Sensitivity of a Stripped Bond

Take our previous example where a $20,000 30-year stripped Take our previous example where a $20,000 30-year stripped bond has a required rate of return of 12%:bond has a required rate of return of 12%:

PP00 = $20,000(PVIF= $20,000(PVIFn=30, k = 12%n=30, k = 12%))

= $20,000 (.0334)= $20,000 (.0334)

= $668.00= $668.00

Assume now that interest rates fall by 16.7% from 12% to 10%. Assume now that interest rates fall by 16.7% from 12% to 10%. What is the percentage change in price of the bond?What is the percentage change in price of the bond?

PP00 = $20,000(PVIF= $20,000(PVIFn=30, k = 10%n=30, k = 10%))

= $20,000 (.0573)= $20,000 (.0573)

= $1,146.00= $1,146.00

Percentage change in price = ($1,146 - $668) / $668Percentage change in price = ($1,146 - $668) / $668

=71.6%=71.6%This stripped bond had a 71.6% increase in price with a 2% decrease

(200 bp) decrease in required rate of return.


Duration and Coupon RatesDuration and Coupon Rates

• A bond’s duration is affected by the size of the coupon rate A bond’s duration is affected by the size of the coupon rate offered by the bond.offered by the bond.– The duration of a zero coupon bond is equal to the bond’s term to The duration of a zero coupon bond is equal to the bond’s term to

maturity. Therefore, the longest durations are found in stripped bonds maturity. Therefore, the longest durations are found in stripped bonds or zero coupon bonds. These are bonds with the greatest interest rate or zero coupon bonds. These are bonds with the greatest interest rate elasticity.elasticity.

– The higher the coupon rate, the shorter the bond’s duration. Hence the The higher the coupon rate, the shorter the bond’s duration. Hence the greater the coupon rate, the shorter the duration, and the lower the greater the coupon rate, the shorter the duration, and the lower the interest rate elasticity of the bond price.interest rate elasticity of the bond price.


DurationDuration

The numerator of the duration formula represents the present value of The numerator of the duration formula represents the present value of future payments, weighted by the time interval until the payments future payments, weighted by the time interval until the payments occur. The longer the intervals until payments are made, the larger occur. The longer the intervals until payments are made, the larger will be the numerator, and the larger will be the duration. The will be the numerator, and the larger will be the duration. The denominator represents the discounted future cash flows resulting denominator represents the discounted future cash flows resulting from the bond, which is the bonds present value.from the bond, which is the bonds present value.

maturitytoyieldsbondthek

providedarepaymentsthewhichattimethet

bondthebygeneratedpaymentprincipalorcoupontheCwhere

kC

ktC

DUR

t

n

tt

t

n

tt

t

'

:

)1(

)1()(

1

1


Duration Example Duration Example A Formula-based Duration Calculation for a Three Year, 7% Coupon BondA Formula-based Duration Calculation for a Three Year, 7% Coupon Bond

• As an example, the duration of a bond with $1,000 par value and a 7 As an example, the duration of a bond with $1,000 par value and a 7 percent coupon rate, three years remaining to maturity, and a 9 percent coupon rate, three years remaining to maturity, and a 9 percent yield to maturity is:percent yield to maturity is:

years

DUR

80.2

)09.1(1070$

)09.1(70$

)09.1(70$

)09.1()3(1070$

)09.1()2(70$

)09.1(70$

321

321


Duration ExampleDuration ExampleA Formula-based Duration Calculation for a Zero Coupon BondA Formula-based Duration Calculation for a Zero Coupon Bond

• As an example, the duration of a zero-coupon bond with $1,000 par As an example, the duration of a zero-coupon bond with $1,000 par value and three years remaining to maturity, and a 9 percent yield to value and three years remaining to maturity, and a 9 percent yield to maturity is:maturity is:

0.3

)09.1(1000$

)09.1()3(1000$

3

3

years

DUR


Example of a Duration CalculationExample of a Duration CalculationUsing a Spreadsheet ModelUsing a Spreadsheet Model

ExampleAssume a 10% coupon bond with three years left to maturity and a required return of 8%.Coupon Rate = 10.00%Required Return = 8.00%

Time Cashflow PVIF Present Value Weight

Time Weighted

CFs0

0.5 50 0.96225 $48.11 4.55% 0.0227676791 50 0.925926 $46.30 4.38% 0.043816419

1.5 50 0.890973 $44.55 4.22% 0.0632435542 50 0.857339 $42.87 4.06% 0.081141517

2.5 50 0.824975 $41.25 3.90% 0.0975980773 1050 0.793832 $833.52 78.89% 2.36662759

Bond Price = $1,056.60 100.00% 2.675194837 =Duration


Duration of a PortfolioDuration of a Portfolio

• Bond portfolio mangers commonly attempt to immunize their Bond portfolio mangers commonly attempt to immunize their portfolio, or insulate their portfolio from the effects of interest portfolio, or insulate their portfolio from the effects of interest rate movements.rate movements.

• This is a common challenge when the investment portfolio is This is a common challenge when the investment portfolio is ‘dedicated’ to funding a future liability. ‘dedicated’ to funding a future liability.


Duration of a PortfolioDuration of a Portfolio

Insurance Company Example

A life insurance company knows that they need $100 million 30 years from now cover actuarially-determined claims against a group of life insurance policies just no sold to a group of 30 year olds.

The insurance company has invested the premiums into 30-year government bonds. Therefore there is no default risk to worry about. The company expects that if the realized rate of return on this bond portfolio equals the yield-to-maturity of the bond portfolio, there won’t be a problem growing that portfolio to $100 million. The problem is, that the coupon interest payments must be reinvested and there is a chance that rates will fall over the life of the portfolio.

If this happens the portfolio’s terminal value will be less than the liability the insurance company needs to finance. This shortfall in investment returns will have to be borne at the expense of the Insurance company’s shareholders.


Duration of a Portfolio ...Duration of a Portfolio ...Interest Rate RiskInterest Rate Risk

• The life insurance company example illustrates a key risk in The life insurance company example illustrates a key risk in fixed-income portfolio management - interest rate risk.fixed-income portfolio management - interest rate risk.

• The portfolio manager, before-the-fact calculates the bond The portfolio manager, before-the-fact calculates the bond portfolio’s yield-to-maturity. This is an portfolio’s yield-to-maturity. This is an ex anteex ante calculation. calculation.

• As such, a naïve assumption assumption is made that the As such, a naïve assumption assumption is made that the coupon interest received each year is reinvested at the yield-coupon interest received each year is reinvested at the yield-to-maturity for the remaining years until the bond matures.to-maturity for the remaining years until the bond matures.

• Over time, however, interest rates will vary and reinvestment Over time, however, interest rates will vary and reinvestment opportunities will vary from that which was forecast.opportunities will vary from that which was forecast.


Duration of a PortfolioDuration of a PortfolioImmunizationImmunization

• The insurance company will want to IMMUNIZE their portfolio The insurance company will want to IMMUNIZE their portfolio from this reinvestment risk.from this reinvestment risk.

• The simplest way to do this is to convert the entire bond The simplest way to do this is to convert the entire bond portfolio to zero-coupon/stripped bonds. Then the portfolio to zero-coupon/stripped bonds. Then the ex anteex ante yield-to-maturity will equal yield-to-maturity will equal ex postex post (realized) rate of return. (realized) rate of return. (ie. the (ie. the ex anteex ante YTM is locked in since there are no YTM is locked in since there are no intermediate cash flows the require reinvestment).intermediate cash flows the require reinvestment).

• If the bond portfolio manager matches the duration of the bond If the bond portfolio manager matches the duration of the bond portfolio with the expected time when they will require the portfolio with the expected time when they will require the $100 m, then interest rate risk will be largely eliminated.$100 m, then interest rate risk will be largely eliminated.


CopyrightCopyright

Copyright © 2007 John Wiley & Copyright © 2007 John Wiley & Sons Canada, Ltd. All rights Sons Canada, Ltd. All rights reserved. Reproduction or reserved. Reproduction or translation of this work beyond that translation of this work beyond that permitted by Access Copyright (the permitted by Access Copyright (the Canadian copyright licensing Canadian copyright licensing agency) is unlawful. Requests for agency) is unlawful. Requests for further information should be further information should be addressed to the Permissions addressed to the Permissions Department, John Wiley & Sons Department, John Wiley & Sons Canada, Ltd.Canada, Ltd. The purchaser may The purchaser may make back-up copies for his or her make back-up copies for his or her own use only and not for distribution own use only and not for distribution or resale.or resale. The author and the The author and the publisher assume no responsibility publisher assume no responsibility for errors, omissions, or damages for errors, omissions, or damages caused by the use of these files or caused by the use of these files or programs or from the use of the programs or from the use of the information contained herein.information contained herein.

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Bond Valuation and Interest Rates