Chapter 12 FIXED-INCOME ANALYSIS. Chapter 12 Questions What different bond yields are important to investors?What different bond yields are important

Chapter 12Chapter 12

FIXED-INCOME ANALYSISFIXED-INCOME ANALYSIS

Chapter 12 QuestionsChapter 12 Questions

• What different bond yields are important to What different bond yields are important to investors?investors?

• How are the following major yields on bonds How are the following major yields on bonds computed: current yield, yield to maturity, computed: current yield, yield to maturity, yield to call, and compound realized (horizon) yield to call, and compound realized (horizon) yield?yield?

• What factors affect the level of bond yields at What factors affect the level of bond yields at a point in time?a point in time?

• What economic forces cause changes in the What economic forces cause changes in the yields on bonds over time?yields on bonds over time?


• When yields change, what characteristics of When yields change, what characteristics of a bond cause differential price changes for a bond cause differential price changes for individual bonds?individual bonds?

• What do we mean by the duration of a bond, What do we mean by the duration of a bond, how is it computed, and what factors affect how is it computed, and what factors affect it?it?

• What is modified duration and what is the What is modified duration and what is the relationship between a bond’s modified relationship between a bond’s modified duration and its volatility?duration and its volatility?


• What is the convexity for a bond, what What is the convexity for a bond, what factors affect it, and what is its effect on factors affect it, and what is its effect on a bond’s volatility?a bond’s volatility?

• Under what conditions is it necessary to Under what conditions is it necessary to consider both modified duration and consider both modified duration and convexity when estimating a bond’s convexity when estimating a bond’s price volatility?price volatility?

The Fundamentals of Bond The Fundamentals of Bond ValuationValuation

• Like other financial assets,the value of Like other financial assets,the value of a bond is the present value of its a bond is the present value of its expected future cash flows:expected future cash flows:

VVjj = = CFCFtt/(1+k)/(1+k)tt

The Fundamentals of Bond The Fundamentals of Bond ValuationValuation

• To incorporate the specifics of bonds:To incorporate the specifics of bonds:

P = P = (C(Cii/2)/(1+Y/2)/(1+Ymm/2)/2)tt + P + Pp p /(1+Y/(1+Ymm/2)/2)2n2n • This is the present value model where:This is the present value model where:

– PP is the current market price of the bondis the current market price of the bond– n is the number of years to maturityn is the number of years to maturity– CCii is the annual coupon payment is the annual coupon payment – YYmm is the yield to maturity of the bond is the yield to maturity of the bond– PPpp is the par value of the bond is the par value of the bond

Bond Price/Yield Bond Price/Yield RelationshipsRelationships

• Bond prices change as yields change, and Bond prices change as yields change, and have the following relationships:have the following relationships:– When yield is below the coupon rate, the bond will When yield is below the coupon rate, the bond will

be priced at a premium to par valuebe priced at a premium to par value– When yield is above the coupon rate, the bond will When yield is above the coupon rate, the bond will

be priced at a discount from its par valuebe priced at a discount from its par value– The price-yield relationship is not a straight line, The price-yield relationship is not a straight line,

but rather convex (This is convexity)but rather convex (This is convexity)• As yields decline, prices increase at an increasing rateAs yields decline, prices increase at an increasing rate• As yield increase, prices fall at a declining rateAs yield increase, prices fall at a declining rate

The Yield ModelThe Yield Model

The yield on the bond may be computed The yield on the bond may be computed when we know the market pricewhen we know the market price

t

n

itt Y

CP)1(

1

Where:

P = the current market price of the bond

Ct = the cash flow received in period t

Y = the discount rate that will discount the cash flows to equal the current market price of the bond

Computing Bond YieldsComputing Bond YieldsYield Measure PurposeYield Measure PurposeCoupon rate Measures the coupon rate or the percentage

of par paid out annually as interest

Current yield Measures current income rate

Promised yield to maturity Measures expected rate of return for bond held to maturity

Promised yield to call Measures expected rate of return for bond held to first call date

Realized (horizon) yield Measures expected rate of return for a bond likely to be sold prior to maturity. It considers specified reinvestment assumptions and an estimated sales price. It can also measure the actual rate of return on a bond during some past period of time.

Current YieldCurrent Yield

• Similar to dividend yield for stocks, this Similar to dividend yield for stocks, this measure is important to income oriented measure is important to income oriented investorsinvestors

CY = C/P CY = C/P • where: where:

– CY = the current yield on a bondCY = the current yield on a bond– C = the annual coupon payment of the bondC = the annual coupon payment of the bond– P = the current market price of the bondP = the current market price of the bond

Promised Yield to MaturityPromised Yield to Maturity

• Widely used bond yield figureWidely used bond yield figure

• AssumesAssumes– Investor holds bond to maturityInvestor holds bond to maturity– All the bond’s cash flow is reinvested at All the bond’s cash flow is reinvested at

the computed yield to maturitythe computed yield to maturity

tm

n

itt Y

CP)1(

1

Solve for Y that will equate the current price to all cash flows from the bond to maturity, similar to IRR

Promised Yield to MaturityPromised Yield to Maturity

• For zero coupon bonds, the only cash For zero coupon bonds, the only cash flow is the par value at maturity. This flow is the par value at maturity. This simplifies the calculation of yield.simplifies the calculation of yield.

P = 1,000/(1+YP = 1,000/(1+Ymm/2)/2)2n2n

– Where n is the number of years to Where n is the number of years to maturity.maturity.

Promised Yield to CallPromised Yield to Call

• When a callable bond is likely to be When a callable bond is likely to be called, yield to call is the more called, yield to call is the more appropriate yield measure than yield to appropriate yield measure than yield to maturitymaturity– As a rule of thumb, when a callable bond is As a rule of thumb, when a callable bond is

selling at a price equal to par value plus selling at a price equal to par value plus one year of interest, the value should be one year of interest, the value should be based on yield to callbased on yield to call

Calculating Promised Calculating Promised Yield to CallYield to Call

Where:Where:

P P = market price of the bond= market price of the bond

CCtt = annual coupon payment = annual coupon payment

ncnc = number of years to first call = number of years to first call

PPcc = call price of the bond = call price of the bond

ncc

cnc

tt

c

t

Y

P

Y

CP

2

2

1 )2/1()2/1(

2/

Realized YieldRealized Yield

hpR

fhp

tt

R

t

Y

P

Y

CP

2

2

1 )2/1()2/1(

2/

• The horizon yield measures yield when The horizon yield measures yield when the investor expects to sell the bond (for the investor expects to sell the bond (for a price of Pa price of Pff in hp time periods) prior to in hp time periods) prior to

maturity or callmaturity or call

Calculating Future Bond Calculating Future Bond PricesPrices

• Expected future bond prices are an Expected future bond prices are an important calculation in several important calculation in several instances:instances:– When computing horizon yield, we need When computing horizon yield, we need

an estimated future selling pricean estimated future selling price– When issues are quoted on a promised When issues are quoted on a promised

yield, as with municipalsyield, as with municipals– For portfolio managers who frequently For portfolio managers who frequently

trade bondstrade bonds

Calculating Future Bond Calculating Future Bond PricesPrices

Where:Where:

PPff = estimated future price of the bond= estimated future price of the bond

CCii = annual coupon payment = annual coupon payment

nn = number of years to maturity = number of years to maturityhphp = holding period of the bond in years = holding period of the bond in years

YYmm = expected semiannual rate at the end of the holding = expected semiannual rate at the end of the holding periodperiod

hpnm

hpn

tt

m

i

f YY

CP

22

p22

1 )2/1(

P

)2/1(

2/

Adjusting for Differential Adjusting for Differential Reinvestment RatesReinvestment Rates

• The yield calculations implicitly assume The yield calculations implicitly assume reinvestment of early coupon payments at reinvestment of early coupon payments at the calculated yieldthe calculated yield

• If expectations are not consistent with this If expectations are not consistent with this assumption, we can compound early cash assumption, we can compound early cash flows at differential rates over the life of the flows at differential rates over the life of the bond and then find the yield based on an bond and then find the yield based on an “Ending wealth” measure, which is calculated “Ending wealth” measure, which is calculated from the differential ratesfrom the differential rates

Yield Adjustments for Tax-Yield Adjustments for Tax-Exempt BondsExempt Bonds

• In order to compare taxable and tax-In order to compare taxable and tax-exempt bonds on an “equal playing exempt bonds on an “equal playing field” for an investor, we calculate the field” for an investor, we calculate the fully taxable equivalent yield (FTEY) for fully taxable equivalent yield (FTEY) for tax-free bonds based on their returnstax-free bonds based on their returnsFTEY = Tax-Free Annual Return/(1-T)FTEY = Tax-Free Annual Return/(1-T)

• Where T is the investor’s marginal tax Where T is the investor’s marginal tax raterate

What Determines Interest What Determines Interest Rates?Rates?

• Inverse relationship with bond pricesInverse relationship with bond prices– Changes in interest rates have an impact Changes in interest rates have an impact

on bond portfolios, in particular rising on bond portfolios, in particular rising interest ratesinterest rates

– It is therefore important to learn about what It is therefore important to learn about what determines interest rates and to gain some determines interest rates and to gain some insight as to forecasting future interest insight as to forecasting future interest ratesrates

Forecasting interest ratesForecasting interest rates

• Interest rates are the cost of borrowing Interest rates are the cost of borrowing money, or the cost of “loanable funds”money, or the cost of “loanable funds”

• Factors that affect the supply of loanable Factors that affect the supply of loanable funds (through saving) and the demand for funds (through saving) and the demand for loanable funds (borrowing) affect interest loanable funds (borrowing) affect interest ratesrates– The goal is to monitor these factors, and to The goal is to monitor these factors, and to

anticipate changes in interest rates and to be well-anticipate changes in interest rates and to be well-positioned to either benefit from the forecast or at positioned to either benefit from the forecast or at least be protected from adverse changes in ratesleast be protected from adverse changes in rates

Determinants of Interest Determinants of Interest RatesRates

• Nominal interest rates (i) can be broken down Nominal interest rates (i) can be broken down into the following components:into the following components:

i = RFR + I + RPi = RFR + I + RPwhere:where:– RFR = real risk-free rate of interestRFR = real risk-free rate of interest– I = expected rate of inflationI = expected rate of inflation– RP = risk premiumRP = risk premium

• The key is to anticipate changes in any of The key is to anticipate changes in any of these factorsthese factors


• Alternatively, we can break down interest rate Alternatively, we can break down interest rate factors into two groups of effects:factors into two groups of effects:– Effect of economic factorsEffect of economic factors

• real growth ratereal growth rate• tightness or ease of capital markettightness or ease of capital market• expected inflationexpected inflation• supply and demand of loanable fundssupply and demand of loanable funds

– Impact of bond characteristicsImpact of bond characteristics• credit qualitycredit quality• term to maturityterm to maturity• indenture provisionsindenture provisions• foreign bond risk (exchange rate risk and country risk)foreign bond risk (exchange rate risk and country risk)


Term structure of interest ratesTerm structure of interest rates– One important source of interest rate variability is One important source of interest rate variability is

the time to maturitythe time to maturity– The yield curve shows the relationship between The yield curve shows the relationship between

bond yields and time to maturity bond yields and time to maturity at a point in timeat a point in time

• Yield curve shapesYield curve shapes– Rising curve (common) when rates are modestRising curve (common) when rates are modest– Declining curve when rates are relatively highDeclining curve when rates are relatively high– Flat curves can happen any timeFlat curves can happen any time– Humped when high rates are expected to declineHumped when high rates are expected to decline– Note: usually relatively flat beyond 15 yearsNote: usually relatively flat beyond 15 years


Term Structure Theories (what explains the Term Structure Theories (what explains the changing shape of the yield curve?)changing shape of the yield curve?)

• Expectations hypothesisExpectations hypothesis– The shape of the yield curve depends on The shape of the yield curve depends on

expected future interest rates and inflation ratesexpected future interest rates and inflation rates– An upward-sloping curve indicates expectations of An upward-sloping curve indicates expectations of

higher rates in the futurehigher rates in the future– We can use this hypothesis to compute implied We can use this hypothesis to compute implied

future (forward) interest ratesfuture (forward) interest rates– Yields of different maturities continually adjusting Yields of different maturities continually adjusting

to estimates of future interest ratesto estimates of future interest rates


Term Structure TheoriesTerm Structure Theories• Liquidity preference hypothesisLiquidity preference hypothesis

– Indicates that long term rates have to pay a Indicates that long term rates have to pay a premium over short term rates because:premium over short term rates because:

• Investors need a premium to compensate for the added Investors need a premium to compensate for the added price risk associated with long-term bondsprice risk associated with long-term bonds

• Borrowers are willing to pay higher rates on long-term Borrowers are willing to pay higher rates on long-term debt to avoid refinancing riskdebt to avoid refinancing risk

– Works well in combination with the expectations Works well in combination with the expectations hypothesis to explain the normal upward slope of hypothesis to explain the normal upward slope of the yield curvethe yield curve


Term Structure TheoriesTerm Structure Theories• Segmented market hypothesisSegmented market hypothesis

– Asserts that different investors, in Asserts that different investors, in particular institutions, have different particular institutions, have different maturity needs, so have “preferred maturity needs, so have “preferred habitats” along the yield curvehabitats” along the yield curve

– Interest rates in differentiated maturity Interest rates in differentiated maturity markets are determined by unique supply markets are determined by unique supply and demand factors in those marketsand demand factors in those markets


• Term Structure and TradingTerm Structure and Trading– Knowledge of the term structure can aid in Knowledge of the term structure can aid in

bond market trading strategiesbond market trading strategies• For example, if the yield curve is sharply For example, if the yield curve is sharply

downward sloping, rates are likely to fall – downward sloping, rates are likely to fall – lengthen bond maturities to take the most lengthen bond maturities to take the most advantage of price appreciation as interest advantage of price appreciation as interest rates fall in the futurerates fall in the future

Determinants of Interest Determinants of Interest Rates Rates

Yield SpreadsYield Spreads• Bond investing strategies can focus on Bond investing strategies can focus on

predicting various changing yield spreads, predicting various changing yield spreads, which exist between:which exist between:– Segments: government bonds, agency bonds, and Segments: government bonds, agency bonds, and

corporate bondscorporate bonds– Sectors: prime-grade municipal bonds versus Sectors: prime-grade municipal bonds versus

good-grade municipal bonds, AA utilities versus good-grade municipal bonds, AA utilities versus BBB utilitiesBBB utilities

– Different coupons within a segment or sectorDifferent coupons within a segment or sector– Maturities within a given market segment or sectorMaturities within a given market segment or sector

Bond Price VolatilityBond Price Volatility

• As interest rates and bond yields change, so As interest rates and bond yields change, so do bond prices (that’s we we’ve been talking do bond prices (that’s we we’ve been talking about interest rates!)about interest rates!)

• What determines how much a bond’s price What determines how much a bond’s price will change as a result of changing yields will change as a result of changing yields (interest rates)?(interest rates)?

• Percentage Change = (EPB/BPB) – 1Percentage Change = (EPB/BPB) – 1– EPB = Ending Price of the BondEPB = Ending Price of the Bond– BPB = Beginning Price of the BondBPB = Beginning Price of the Bond

Determinants of Bond Determinants of Bond Price VolatilityPrice Volatility

Four factors determine a Four factors determine a bond’s price volatility bond’s price volatility to changing interest to changing interest rates:rates:

1.1. Par valuePar value

2.2. CouponCoupon

3.3. Years to maturityYears to maturity

4.4. Prevailing level of Prevailing level of market interest ratemarket interest rate

Determinants of Bond Determinants of Bond Price Volatility Price Volatility

Malkiel’s five bond relationships:Malkiel’s five bond relationships:1. Bond prices move inversely to bond yields (interest 1. Bond prices move inversely to bond yields (interest

rates)rates)2. For a given change in yields, longer maturity bonds post 2. For a given change in yields, longer maturity bonds post

larger price changes, thus bond price volatility is directly larger price changes, thus bond price volatility is directly related to maturityrelated to maturity

3. Price volatility increases at a diminishing rate as term to 3. Price volatility increases at a diminishing rate as term to maturity increasesmaturity increases

4. Price movements resulting from equal absolute 4. Price movements resulting from equal absolute increases or decreases in yield are not symmetricalincreases or decreases in yield are not symmetrical

5. Higher coupon issues show smaller percentage price 5. Higher coupon issues show smaller percentage price fluctuation for a given change in yield, thus bond price fluctuation for a given change in yield, thus bond price volatility is inversely related to couponvolatility is inversely related to coupon


• The maturity effectThe maturity effect– The longer the time to maturity, the greater The longer the time to maturity, the greater

a bond’s price sensitivitya bond’s price sensitivity– Price volatility increases at a decreasing Price volatility increases at a decreasing

rate with maturityrate with maturity

• The coupon effectThe coupon effect– The greater the coupon rate, the lower a The greater the coupon rate, the lower a

bond’s price sensitivitybond’s price sensitivity


• The yield level effectThe yield level effect– For the same change in basis point yield, For the same change in basis point yield,

there is greater price sensitivity of lower there is greater price sensitivity of lower yield bondsyield bonds

• Some trading implicationsSome trading implications– If our interest rate forecast is for lower If our interest rate forecast is for lower

rates, invest in bonds with the greatest rates, invest in bonds with the greatest price sensitivity, and do the opposite if we price sensitivity, and do the opposite if we expect higher interest ratesexpect higher interest rates


• The Duration MeasureThe Duration Measure– Since price volatility of a bond varies Since price volatility of a bond varies

inversely with its coupon and directly with inversely with its coupon and directly with its term to maturity, it is necessary to its term to maturity, it is necessary to determine the best combination of these determine the best combination of these two variables to achieve your objectivetwo variables to achieve your objective

– A composite measure considering both A composite measure considering both coupon and maturity would be beneficial, coupon and maturity would be beneficial, and that’s what this measure providesand that’s what this measure provides


Developed by Frederick R. Macaulay,1938Developed by Frederick R. Macaulay,1938

Where:Where:

tt = = time period in which the coupon or principal payment occurs time period in which the coupon or principal payment occurs

CCtt = = interest or principal payment that occurs in period interest or principal payment that occurs in period tt

YYmm = = yield to maturity on the bondyield to maturity on the bond

Price

)(

)1(

)1()(

1

1

1

n

tt

n

tt

m

t

n

tt

m

t CPVt

YCYtC

D


• Characteristics of Macaulay DurationCharacteristics of Macaulay Duration– Duration of a bond with coupons is always less Duration of a bond with coupons is always less

than its term to maturity because duration gives than its term to maturity because duration gives weight to these interim paymentsweight to these interim payments

• A zero-coupon bond’s duration equals its maturityA zero-coupon bond’s duration equals its maturity

– There is an inverse relation between duration and There is an inverse relation between duration and the coupon ratethe coupon rate

– A positive relation between term to maturity and A positive relation between term to maturity and duration, but duration increases at a decreasing duration, but duration increases at a decreasing rate with maturityrate with maturity


• Characteristics of Macaulay DurationCharacteristics of Macaulay Duration– There is an inverse relation between YTM There is an inverse relation between YTM

and durationand duration– Sinking funds and call provisions can have Sinking funds and call provisions can have

a dramatic effect on a bond’s durationa dramatic effect on a bond’s duration

Duration and Bond Price Duration and Bond Price VolatilityVolatility

• An adjusted measure of duration can be An adjusted measure of duration can be used to approximate the price volatility used to approximate the price volatility of a bondof a bond

mY

1

durationMacaulay duration Modified

m

Where:

m = number of payments a year

Ym = nominal YTM

Duration and Bond Price Duration and Bond Price VolatilityVolatility

Bond price movements will vary Bond price movements will vary proportionally with modified duration for proportionally with modified duration for small changes in yields:small changes in yields:

mmod Y100

DP

P

Where:

P = change in price for the bond

P = beginning price for the bond

Dmod = the modified duration of the bond

Ym = yield change in basis points divided by 100

Trading Strategies Using Trading Strategies Using DurationDuration

• Longest-duration security provides the Longest-duration security provides the maximum price variationmaximum price variation– If you expect a decline in interest rates, increase If you expect a decline in interest rates, increase

the average duration of your bond portfolio to the average duration of your bond portfolio to experience maximum price volatilityexperience maximum price volatility

– If you expect an increase in interest rates, reduce If you expect an increase in interest rates, reduce the average duration to minimize your price the average duration to minimize your price declinedecline

• Duration of a portfolio is the market-value-Duration of a portfolio is the market-value-weighted average of the duration of the weighted average of the duration of the individual bonds in the portfolioindividual bonds in the portfolio

Bond ConvexityBond Convexity

• The percentage price change formula using The percentage price change formula using duration is a linear approximation of bond duration is a linear approximation of bond price change for small changes in market price change for small changes in market yieldsyields

• Price changes are not linear, but a curvilinear Price changes are not linear, but a curvilinear (convex) function(convex) function

mmod Y100

DP

P


• The graph of prices relative to yields is not a straight The graph of prices relative to yields is not a straight line, but a curvilinear relationshipline, but a curvilinear relationship– This can be applied to a single bond, a portfolio of bonds, or This can be applied to a single bond, a portfolio of bonds, or

any stream of future cash flowsany stream of future cash flows

• The convex price-yield relationship will differ among The convex price-yield relationship will differ among bonds or other cash flow streams depending on the bonds or other cash flow streams depending on the coupon and maturitycoupon and maturity– The convexity of the price-yield relationship declines slower The convexity of the price-yield relationship declines slower

as the yield increasesas the yield increases

• Modified duration is the percentage change in price Modified duration is the percentage change in price for a nominal change in yieldfor a nominal change in yield


– The convexity is the measure of the The convexity is the measure of the curvature and is the second derivative of curvature and is the second derivative of price with resect to yield (price with resect to yield (dd22P/diP/di22))

– Convexity is the percentage change in Convexity is the percentage change in dP/didP/di for a given change in yield for a given change in yield

PdY

Pd2

2

Convexity


• Determinants of ConvexityDeterminants of Convexity– Inverse relationship between coupon and Inverse relationship between coupon and

convexityconvexity– Direct relationship between maturity and Direct relationship between maturity and

convexityconvexity– Inverse relationship between yield and Inverse relationship between yield and

convexityconvexity

Modified Duration-Modified Duration-Convexity EffectsConvexity Effects

• Changes in a bond’s price resulting from a Changes in a bond’s price resulting from a change in yield are due to:change in yield are due to:– Bond’s modified durationBond’s modified duration– Bond’s convexityBond’s convexity

• Relative effect of these two factors depends Relative effect of these two factors depends on the characteristics of the bond (its on the characteristics of the bond (its convexity) and the size of the yield changeconvexity) and the size of the yield change

• Convexity is desirableConvexity is desirable– Greater price appreciation if interest rates fall, Greater price appreciation if interest rates fall,

smaller price drop if interest rates risesmaller price drop if interest rates rise

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Chapter 12 FIXED-INCOME ANALYSIS. Chapter 12 Questions What different bond yields are important to investors?What different bond yields are important