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1、Chapter 12FIXED-INCOME ANALYSISChapter 12 QuestionsWhat 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:
2、 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 chan
3、ges in the What economic forces cause changes in the yields on bonds over time?yields on bonds over time?Chapter 12 QuestionsWhen 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
4、 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 it?how is it computed, and what factors affect it?What is modified duration and what is the What is modified duration and what is the r
5、elationship between a bonds modified relationship between a bonds modified duration and its volatility?duration and its volatility?Chapter 12 QuestionsWhat is the convexity for a bond, what factors affect it, and what is its effect on a bonds volatility?Under what conditions is it necessary to consi
6、der both modified duration and convexity when estimating a bonds price volatility?The Fundamentals of Bond ValuationLike other financial assets,the value of a bond is the present value of its expected future cash flows:Vj = SCFt/(1+k)tThe Fundamentals of Bond ValuationTo incorporate the specifics of
7、 bonds:P = S(Ci/2)/(1+Ym/2)t + Pp /(1+Ym/2)2n This is the present value model where:P P is the current market price of the bondis the current market price of the bondn is the number of years to maturityn is the number of years to maturityC Ci i is the annual coupon payment is the annual coupon payme
8、nt Y Ymm is the yield to maturity of the bond is the yield to maturity of the bondP Pp p is the par value of the bond is the par value of the bondBond Price/Yield RelationshipsBond prices change as yields change, and Bond prices change as yields change, and have the following relationships:have the
9、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 When yield is above the coupon rate, the bond will be pr
10、iced at a discount from its par valuewill be 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
11、 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 rateThe Yield ModelThe yield on the bond may be computed when we know the market priceWhere:P = the current market price of t
12、he bondCt = the cash flow received in period tY = the discount rate that will discount the cash flows to equal the current market price of the bondComputing Bond YieldsYield Measure PurposeCoupon rateMeasures the coupon rate or the percentage of par paid out annually as interestCurrent yieldMeasures
13、 current income ratePromised yield to maturityMeasures expected rate of return for bond held to maturityPromised yield to callMeasures expected rate of return for bond held to first call dateRealized (horizon) yieldMeasures expected rate of return for a bond likely to be sold prior to maturity. It c
14、onsiders 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 YieldSimilar to dividend yield for stocks, this Similar to dividend yield for stocks, this measure is important to income oriented
15、 measure is important to income oriented investorsinvestorsCY = 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 marke
16、t price of the bondPromised Yield to MaturityWidely used bond yield figureAssumesInvestor holds bond to maturityInvestor holds bond to maturityAll the bonds cash flow is reinvested at All the bonds cash flow is reinvested at the computed yield to maturitythe computed yield to maturitySolve for Y tha
17、t will equate the current price to all cash flows from the bond to maturity, similar to IRRPromised Yield to MaturityFor zero coupon bonds, the only cash flow is the par value at maturity. This simplifies the calculation of yield.P = 1,000/(1+Ym/2)2nWhere n is the number of years to Where n is the n
18、umber of years to maturity.maturity.Promised Yield to CallWhen a callable bond is likely to be called, yield to call is the more appropriate yield measure than yield to maturityAs 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 val
19、ue 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 Yield to CallWhere:Where:P P = market price of the bond= market price of the bondC Ct t = annual co
20、upon payment = annual coupon paymentncnc = number of years to first call = number of years to first callP Pc c = call price of the bond = call price of the bondRealized YieldThe horizon yield measures yield when the investor expects to sell the bond (for a price of Pf in hp time periods) prior to ma
21、turity or callCalculating Future Bond PricesExpected future bond prices are an important calculation in several instances:When computing horizon yield, we need When computing horizon yield, we need an estimated future selling pricean estimated future selling priceWhen issues are quoted on a promised
22、 When issues are quoted on a promised yield, as with municipalsyield, as with municipalsFor portfolio managers who frequently For portfolio managers who frequently trade bondstrade bondsCalculating Future Bond PricesWhere:Where:P Pf f = estimated future price of the bond= estimated future price of t
23、he bondC Ci i = annual coupon payment = annual coupon paymentn n = number of years to maturity = number of years to maturityhphp = holding period of the bond in years = holding period of the bond in yearsY Ym m = expected semiannual rate at the end of the holding = expected semiannual rate at the en
24、d of the holding periodperiodAdjusting for Differential Reinvestment RatesThe 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 yieldIf expectations are no
25、t 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 f
26、ind the yield based on an “Ending wealth” measure, which is calculated “Ending wealth” measure, which is calculated from the differential ratesfrom the differential ratesYield Adjustments for Tax-Exempt BondsIn order to compare taxable and tax-exempt bonds on an “equal playing field” for an investor
27、, we calculate the fully taxable equivalent yield (FTEY) for tax-free bonds based on their returnsFTEY = Tax-Free Annual Return/(1-T)Where T is the investors marginal tax rateWhat Determines Interest Rates?Inverse relationship with bond pricesChanges in interest rates have an impact Changes in inter
28、est rates have an impact on bond portfolios, in particular rising on bond portfolios, in particular rising interest ratesinterest ratesIt 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
29、to gain some insight as to forecasting future interest insight as to forecasting future interest ratesratesForecasting interest ratesInterest 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 t
30、hat 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
31、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
32、 ratesleast be protected from adverse changes in ratesDeterminants of Interest RatesNominal 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-fre
33、e rate of interestRFR = real risk-free rate of interest I = expected rate of inflationI = expected rate of inflation RP = risk premiumRP = risk premiumThe key is to anticipate changes in any of The key is to anticipate changes in any of these factorsthese factorsDeterminants of Interest RatesAlterna
34、tively, 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
35、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
36、risk (exchange rate risk and country risk)foreign bond risk (exchange rate risk and country risk)Determinants of Interest RatesTerm structure of interest ratesTerm structure of interest rates One important source of interest rate variability is One important source of interest rate variability is th
37、e 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 timeYield curve shapesYield curve shapes Rising curve (common) when r
38、ates 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
39、decline Note: usually relatively flat beyond 15 yearsNote: usually relatively flat beyond 15 yearsDeterminants of Interest RatesTerm Structure Theories (what explains the Term Structure Theories (what explains the changing shape of the yield curve?)changing shape of the yield curve?)Expectations hyp
40、othesisExpectations 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 expe
41、ctations 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 maturit
42、ies continually adjusting to estimates of future interest ratesto estimates of future interest ratesDeterminants of Interest RatesTerm Structure TheoriesTerm Structure TheoriesLiquidity preference hypothesisLiquidity preference hypothesis Indicates that long term rates have to pay a Indicates that l
43、ong 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 Bor
44、rowers 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 upw
45、ard slope of hypothesis to explain the normal upward slope of the yield curvethe yield curveDeterminants of Interest RatesTerm Structure TheoriesSegmented market hypothesisAsserts that different investors, in Asserts that different investors, in particular institutions, have different particular ins
46、titutions, have different maturity needs, so have “preferred maturity needs, so have “preferred habitats” along the yield curvehabitats” along the yield curveInterest rates in differentiated maturity Interest rates in differentiated maturity markets are determined by unique supply markets are determ
47、ined by unique supply and demand factors in those marketsand demand factors in those marketsDeterminants of Interest RatesTerm Structure and TradingKnowledge of the term structure can aid in Knowledge of the term structure can aid in bond market trading strategiesbond market trading strategies For e
48、xample, 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 advan
49、tage of price appreciation as interest rates fall in the futurerates fall in the futureDeterminants of Interest Rates Yield SpreadsYield SpreadsBond investing strategies can focus on Bond investing strategies can focus on predicting various changing yield spreads, predicting various changing yield s
50、preads, which exist between:which exist between: Segments: government bonds, agency bonds, Segments: government bonds, agency bonds, and corporate bondsand corporate bonds Sectors: prime-grade municipal bonds versus Sectors: prime-grade municipal bonds versus good-grade municipal bonds, AA utilities
51、 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 sectorBond Price VolatilityAs inte
52、rest rates and bond yields change, so As interest rates and bond yields change, so do bond prices (thats we weve been talking do bond prices (thats we weve been talking about interest rates!)about interest rates!)What determines how much a bonds price What determines how much a bonds price will chan
53、ge 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 Bon
54、dDeterminants of Bond Price VolatilityFour factors determine a Four factors determine a bonds price volatility bonds price volatility to changing interest to changing interest rates:rates:1.1.Par valuePar value2.2.CouponCoupon3.3.Years to maturityYears to maturity4.4.Prevailing level of Prevailing l
55、evel of market interest ratemarket interest rateDeterminants of Bond Price Volatility Malkiels five bond relationships:Malkiels 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 y
56、ields, 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 maturity3. Price volatility increases at a diminishing
57、 rate as term to 3. Price volatility increases at a diminishing rate as term to maturity increasesmaturity increases4. 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
58、 not symmetrical5. 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 inverse
59、ly related to couponDeterminants of Bond Price VolatilityThe maturity effectThe longer the time to maturity, the greater The longer the time to maturity, the greater a bonds price sensitivitya bonds price sensitivityPrice volatility increases at a decreasing Price volatility increases at a decreasin
60、g rate with maturityrate with maturityThe coupon effectThe greater the coupon rate, the lower a The greater the coupon rate, the lower a bonds price sensitivitybonds price sensitivityDeterminants of Bond Price VolatilityThe yield level effectFor the same change in basis point yield, For the same cha
61、nge in basis point yield, there is greater price sensitivity of lower there is greater price sensitivity of lower yield bondsyield bondsSome trading implicationsIf our interest rate forecast is for lower If our interest rate forecast is for lower rates, invest in bonds with the greatest rates, inves
62、t 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 ratesDeterminants of Bond Price VolatilityThe Duration MeasureSince price volatility of a bond varies Since price volatility of a
63、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 objectiv
64、etwo variables to achieve your objectiveA composite measure considering both A composite measure considering both coupon and maturity would be beneficial, coupon and maturity would be beneficial, and thats what this measure providesand thats what this measure providesDeterminants of Bond Price Volat
65、ility Developed by Frederick R. Macaulay,1938Developed by Frederick R. Macaulay,1938Where:Where: t t = = time period in which the coupon or principal payment occurs time period in which the coupon or principal payment occursC Ct t = = interest or principal payment that occurs in period interest or p
66、rincipal payment that occurs in period t t Y Ym m = = yield to maturity on the bondyield to maturity on the bondDeterminants of Bond Price Volatility Characteristics of Macaulay DurationCharacteristics of Macaulay Duration Duration of a bond with coupons is always less Duration of a bond with coupon
67、s 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 bonds duration equals its maturityA zero-coupon bonds duration equals its maturity There is an inverse rel
68、ation 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 decreas
69、ing rate with maturityrate with maturityDeterminants of Bond Price VolatilityCharacteristics of Macaulay DurationThere is an inverse relation between YTM There is an inverse relation between YTM and durationand durationSinking funds and call provisions can have Sinking funds and call provisions can
70、have a dramatic effect on a bonds durationa dramatic effect on a bonds durationDuration and Bond Price VolatilityAn adjusted measure of duration can be used to approximate the price volatility of a bondWhere:m = number of payments a yearYm = nominal YTMDuration and Bond Price VolatilityBond price mo
71、vements 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:Where:P = change in price for the bondP = beginning price for the bondDmod = the modified duration of the bondYm = yiel
72、d change in basis points divided by 100Trading Strategies Using DurationLongest-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
73、 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 minimi
74、ze your price the average duration to minimize your price declinedeclineDuration 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 por
75、tfolioBond ConvexityThe 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 yieldsyieldsPrice changes ar
76、e not linear, but a curvilinear Price changes are not linear, but a curvilinear (convex) function(convex) functionBond Convexity 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 curviline
77、ar 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
78、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
79、 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 yieldBond ConvexityThe convexity is the measure of the The convexity is the measure of the curvature and is the second der
80、ivative of curvature and is the second derivative of price with resect to yield (price with resect to yield (d d2 2P/diP/di2 2) )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 yieldBond ConvexityDeterminants of
81、 ConvexityInverse relationship between coupon and Inverse relationship between coupon and convexityconvexityDirect relationship between maturity and Direct relationship between maturity and convexityconvexityInverse relationship between yield and Inverse relationship between yield and convexityconve
82、xityModified Duration-Convexity EffectsChanges in a bonds price resulting from a Changes in a bonds price resulting from a change in yield are due to:change in yield are due to: Bonds modified durationBonds modified duration Bonds convexityBonds convexityRelative effect of these two factors depends
83、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 changeConvexity 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
