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1、Chapter 8:Valuation of Known Cash Flows:BondsObjectiveValuation of fixed income securitiesExplain why bond prices change1Copyright 2009 Pearson Education,Inc.Publishing as Prentice HallChapter 8 Contents8.1 Using Present Value Formulas to Value Known Flows8.2 The Basic Building Blocks:Pure Discount
2、Bonds8.3 Coupon Bonds,Current Yield,and Yield-to-Maturity8.4 Reading Bond Listings8.5 Why Yields for the same Maturity Differ8.6 The Behavior of Bond Prices Over Time2Copyright 2009 Pearson Education,Inc.Publishing as Prentice Hall8.1 Using Present Value Formulas to Value Known FlowsYou have been of
3、fered the opportunity to purchase a mortgage.It was originally part of a creative financing package where the original owner financed the buyerThe remaining life of the mortgage is 60 months,with payment of$400.Your required rate of return is 1.5%/month3Copyright 2009 Pearson Education,Inc.Publishin
4、g as Prentice HallCalculationUsing the present value of an annuity formula discussed in chapter 4,you will pay no more than4Copyright 2009 Pearson Education,Inc.Publishing as Prentice HallFinancial CalculatorAlternatively,using your financial calculator(remember to set the correct defaults)you obtai
5、n5Copyright 2009 Pearson Education,Inc.Publishing as Prentice HallChange in Required RateIf your required rate of return increased to 1.6%/month6Copyright 2009 Pearson Education,Inc.Publishing as Prentice HallUsing Present Value Formulas to Value Known FlowsObserve that the maximum you would pay for
6、 the bond has decreasedAn increase in the required rate of return always leads to a decrease in the value of a fixed income securityThe proof is very easy7Copyright 2009 Pearson Education,Inc.Publishing as Prentice HallBond Prices Rise as the Interest Rates FallWrite the PV of the fixed income secur
7、ity as the sum terms8Copyright 2009 Pearson Education,Inc.Publishing as Prentice HallBond Prices Rise as the Interest Rates FallIf i goes up,1+i goes up,1/(1+i)goes down for i -1,(1/(1+i)j goes down for i 0.So if the payments are positive,then the sum must also go down Similarly,i down-PV up9Copyrig
8、ht 2009 Pearson Education,Inc.Publishing as Prentice HallBond Prices Rise as the Interest Rates FallBasic principle in evaluating known flows A change in market interest rates causes a A change in market interest rates causes a change in the change in the oppositeopposite direction in the direction
9、in the market values of all existing contracts market values of all existing contracts promising fixed payments in the futurepromising fixed payments in the future10Copyright 2009 Pearson Education,Inc.Publishing as Prentice HallNoteVolatile market rates imply volatile market values11Copyright 2009
10、Pearson Education,Inc.Publishing as Prentice HallFinding the Correct Discount RateBond analysis is not as easy as this analysis appears to imply We need an interest rate to use in the We need an interest rate to use in the formulaformula We saw in Chapter 2 that interest rates are a We saw in Chapte
11、r 2 that interest rates are a function of time-to-maturityfunction of time-to-maturity Two default-free bonds with identical Two default-free bonds with identical maturities may have different maturities may have different YTMsYTMs12Copyright 2009 Pearson Education,Inc.Publishing as Prentice HallYie
12、ld CurveA typical yield curve:13Copyright 2009 Pearson Education,Inc.Publishing as Prentice Hall8.2 The Basics Building Blocks:Pure Discount BondsWe can always analyze any fixed income contract into a sum of pure discount bondsA pure discount bond is a security that promises to pay a specified singl
13、e cash payment(face value or par value)at a specified date called its maturity date14Copyright 2009 Pearson Education,Inc.Publishing as Prentice HallPure Discount BondsNote There is no cash flow associated with There is no cash flow associated with interest interest Pure discount bonds are purchased
14、 at a Pure discount bonds are purchased at a discount from their face or par valuediscount from their face or par value15Copyright 2009 Pearson Education,Inc.Publishing as Prentice HallPure Discount BondsThe pure discount bond is an example of the present value of a lump sum equation we analyzed in
15、Chapter 4Solving this,the yield-to-maturity on a pure discount bond is given by the relationship:16Copyright 2009 Pearson Education,Inc.Publishing as Prentice HallPure Discount BondsIn this equation,P is the present value or price of the bondP is the present value or price of the bond F is the face
16、or future value F is the face or future value n is the investment periodn is the investment period i is the yield-to-maturityi is the yield-to-maturity17Copyright 2009 Pearson Education,Inc.Publishing as Prentice HallPure Discount BondsExample You can purchase a pure discount bond for You can purcha
17、se a pure discount bond for$9,000,and it matures in two years with a$9,000,and it matures in two years with a face value of$10,000face value of$10,000 What is the ytm?What is the ytm?18Copyright 2009 Pearson Education,Inc.Publishing as Prentice HallPure Discount Bonds19Copyright 2009 Pearson Educati
18、on,Inc.Publishing as Prentice Hall8.3 Coupon Bonds,Current Yield,and Yield to MaturityA coupon bond obligates the issuer to make periodic payments of interest(called make periodic payments of interest(called coupon paymentscoupon payments)to the bond holder until)to the bond holder until the bond ma
19、tures the bond matures at which time the face value of the bond is at which time the face value of the bond is also paid to the bond holderalso paid to the bond holder and the contract is satisfiedand the contract is satisfied20Copyright 2009 Pearson Education,Inc.Publishing as Prentice HallCoupon R
20、ateThe coupon rate is the interest rate applied to the face value to compute the coupon payment A bond with a face value of$1,000 and a A bond with a face value of$1,000 and a coupon rate of 10%pays an annual coupon coupon rate of 10%pays an annual coupon of$100of$100 At maturity,the payment is$1,00
21、0+$100At maturity,the payment is$1,000+$10021Copyright 2009 Pearson Education,Inc.Publishing as Prentice HallCash Flows for 10%$1,000 Coupon Bond22Copyright 2009 Pearson Education,Inc.Publishing as Prentice HallPar,premium,and Discount BondsA coupon bond with its current price equal to its par value
22、 is a par bondIf it is trading below par it is a discount bondIf it is trading above par it is a premium bond(not to be confused with the U.K.lottery bond of the(not to be confused with the U.K.lottery bond of the same name!)same name!)23Copyright 2009 Pearson Education,Inc.Publishing as Prentice Ha
23、llBonds Trading at Par Bond Pricing Principle#1:(Par Bonds)If a bonds price equals its face value,then If a bonds price equals its face value,then its yield-to-maturity=current yield=its yield-to-maturity=current yield=coupon rate.coupon rate.Proof:Proof:24Copyright 2009 Pearson Education,Inc.Publis
24、hing as Prentice HallCoupon Bonds,Current Yield,and Yield-to-MaturityThe yield-to-maturity is the discount rate that makes the present value of the cash flows from the bond equal to the current price of the bondAn excellent way to compute the ytm is given in Chapter 425Copyright 2009 Pearson Educati
25、on,Inc.Publishing as Prentice HallUsing Pure Discount Bonds to Value other BondsValue a bond that pays its$100 coupon at the end of each year for 3-years,and its par value of$1,000 in 3-years You have discovered three pure discount You have discovered three pure discount bonds(each with a$1,000 par
26、value)that bonds(each with a$1,000 par value)that mature in 1,2,and 3 years,and that are mature in 1,2,and 3 years,and that are trading at$960,$890,and$810 respectivelytrading at$960,$890,and$810 respectively26Copyright 2009 Pearson Education,Inc.Publishing as Prentice HallFirst Solution Method27Cop
27、yright 2009 Pearson Education,Inc.Publishing as Prentice HallSecond Solution Method28Copyright 2009 Pearson Education,Inc.Publishing as Prentice HallConclusionThe first method uses the fact that a coupon bond is the sum of pure discount bonds it is fast and directit is fast and directThe second meth
28、od first determines the yields-to-maturity of each discount bond cash flows are then evaluated using themcash flows are then evaluated using them29Copyright 2009 Pearson Education,Inc.Publishing as Prentice HallThe YTM of the Coupon BondWe have the price of the coupon bond,and the timing and magnitu
29、de of its future cash flows,so we can determine its YTMWe use the financial calculator,but a numerical method was provided in chapter 4 for this class of problems30Copyright 2009 Pearson Education,Inc.Publishing as Prentice HallThe YTM of the Coupon Bond31Copyright 2009 Pearson Education,Inc.Publish
30、ing as Prentice HallObservationThe yield to maturity on the 3-year pure discount bond was 7.28%and the yield-to-maturity on the 3-year coupon bond was 7.10%The yield-curve for default-free bonds is not a unique value32Copyright 2009 Pearson Education,Inc.Publishing as Prentice HallBond Pricing Princ
31、iple#2&3Bond Principle#2:Premium Bondsbond price face value bond price face value ytm current ytm current yield coupon rateyield coupon rateBond Principle#3:Discount Bondsbond price face value bond price current ytm current yield coupon rateyield coupon rate33Copyright 2009 Pearson Education,Inc.Pub
32、lishing as Prentice HallProof of Relationship between YTM and Current YieldFor coupon bonds,we have the following relationships Note the(sensible)restrictions on the Note the(sensible)restrictions on the variable rangesvariable ranges Note that 1/(1+i)n-1)is always positiveNote that 1/(1+i)n-1)is al
33、ways positive34Copyright 2009 Pearson Education,Inc.Publishing as Prentice Hall35Copyright 2009 Pearson Education,Inc.Publishing as Prentice HallProof of Relationship between Current and Coupon Yields For coupon bonds,we have the following relationship derived from the bond formula Note that the dif
34、ferences between the Note that the differences between the reciprocals have the same sign,so the actual reciprocals have the same sign,so the actual differences also have the same signdifferences also have the same sign Note that size relationship is determined by Note that size relationship is dete
35、rmined by the discount factor which is always 1the discount factor which is always 136Copyright 2009 Pearson Education,Inc.Publishing as Prentice Hall37Copyright 2009 Pearson Education,Inc.Publishing as Prentice HallHow to Remember PrinciplesImagine that the bond was issued at par the yield-to-matur
36、ity moves from the coupon the yield-to-maturity moves from the coupon yield in the opposite direction to priceyield in the opposite direction to price the coupon rate is unchangingthe coupon rate is unchangingThis diagram may help:38Copyright 2009 Pearson Education,Inc.Publishing as Prentice Hall39C
37、opyright 2009 Pearson Education,Inc.Publishing as Prentice HallHigh Yield T-Bond FundsYield curves with large positive slopes make longer-term T-bonds tempting because,like T-bills,they are default-free The above diagram was based on:par=The above diagram was based on:par=$1000,coupon=$100,n=10-year
38、s,flat$1000,coupon=$100,n=10-years,flat Observe the large effect of modest changes Observe the large effect of modest changes in interest on capitalin interest on capital A close up is given belowA close up is given below40Copyright 2009 Pearson Education,Inc.Publishing as Prentice Hall41Copyright 2
39、009 Pearson Education,Inc.Publishing as Prentice HallClarificationThe last example used a flat yield curveLet us look at an example with short-term rates remaining fixedshort-term rates remaining fixed longer-term rates rising on increased longer-term rates rising on increased expectation of a gener
40、al rise in interest ratesexpectation of a general rise in interest rates42Copyright 2009 Pearson Education,Inc.Publishing as Prentice Hall43Copyright 2009 Pearson Education,Inc.Publishing as Prentice HallInvestment ImplicationsAssume a 20-year bond with a coupon rate of 6%Purchase for$1016.54 when t
41、he lower curve Purchase for$1016.54 when the lower curve prevailsprevails When yield curve rises,the bond is worth When yield curve rises,the bond is worth only$814.05only$814.05 This is a massive capital risk This is a massive capital risk Additionally,long-term rates are more volatile Additionally
42、,long-term rates are more volatile than short-term ratesthan short-term rates44Copyright 2009 Pearson Education,Inc.Publishing as Prentice Hall8.4 Reading Bond ListingsThere are traditions for reporting yields and computing earned interest that need to be understood before trading Coupon bonds are o
43、ften quoted in terms of Coupon bonds are often quoted in terms of the annual rate compounded semi-annuallythe annual rate compounded semi-annually T-bills are often quoted on a discount basisT-bills are often quoted on a discount basis e.g.,a 1 year T-bill has 364 days outstanding,e.g.,a 1 year T-bi
44、ll has 364 days outstanding,but a year has only 360 days(it gets nasty)but a year has only 360 days(it gets nasty)45Copyright 2009 Pearson Education,Inc.Publishing as Prentice HallReading Bond ListingsTake care that the fractional part of a number is understood Is it 16ths,32nds,64ths,100ths or some
45、 Is it 16ths,32nds,64ths,100ths or some other convention?other convention?Ask price:dealers selling priceBid price:dealers buying price46Copyright 2009 Pearson Education,Inc.Publishing as Prentice Hall8.5 Why Yields for the same Maturity Differ The fundamental building block of bonds is The fundamen
46、tal building block of bonds is the pure discount bond:Coupon bonds may the pure discount bond:Coupon bonds may be viewed as a portfolio of discount bondsbe viewed as a portfolio of discount bonds The rule of one price applies to bonds The rule of one price applies to bonds through pure discount bond
47、sthrough pure discount bonds It is a mistake to assume that coupon bonds It is a mistake to assume that coupon bonds with the same life have the same yield-their with the same life have the same yield-their coupon rates differ,leading to a different%coupon rates differ,leading to a different%mix of
48、discount bondsmix of discount bonds47Copyright 2009 Pearson Education,Inc.Publishing as Prentice HallMovement of a Pure Discount Bonds Price over Time48Copyright 2009 Pearson Education,Inc.Publishing as Prentice Hall8.6 The Behavior of Bond Prices Over TimeThe expected price of pure discount bonds r
49、ises exponentially to the face value with time,and the actual price never exceeds parCoupon bonds are more complex,and their price may exceed their par value,but at maturity they reach their par value49Copyright 2009 Pearson Education,Inc.Publishing as Prentice HallBond Prices at Alternative Yields5
50、0Copyright 2009 Pearson Education,Inc.Publishing as Prentice HallBond Price Sensitivity to Yield Changes51Copyright 2009 Pearson Education,Inc.Publishing as Prentice HallBond Price TrajectoryThe following diagram shows the dynamic nature of the yield curve as it passes through time Think of a Think