CHAPTER II: FIXED INCOME SECURITIES AND MARKETS

Transcription

1 CHAPTER II: FIXED INCOME SECURITIES AND MARKETS 30

2 FIXED INCOME PORTFOLIO MANAGEMENT A: TYPES OF FIXED INCOME SECURITIES I terms of dollar volume, the U.S. markets for debt istrumets are larger tha for ay other type of security. Debt markets, icludig markets for mortgages, are more tha twice as large as stock markets. Debt securities are IOUs issued by a variety of types of orgaizatios, icludig federal, state ad local govermets ad their agecies as well as by corporatios ad other istitutios. Debt securities are sold for the purpose of raisig moey. A debt security represets a claim o the issuer for a fixed series of future paymets. For example, a debt security might specify for its ower to receive a stated series of iterest paymets util the istrumet matures. The istrumet may also provide for pricipal repaymet whe the istrumet matures. These securities are issued i primary markets ad the traded i secodary markets, just as are other fiacial istrumets. Debt securities will ormally specify terms of paymet, icludig amouts ad dates, collateral ad priority (who is to paid first i the evet of issuer fiacial distress). Amog the various types of debt securities are bods, otes, mortgages ad treasury istrumets. May of the fixed icome securities with shorter terms to maturity are cosidered to be moey market istrumets. Treasury Securities ad Markets The Uited States Treasury is the largest issuer of debt securities i the world. The federal govermet raises (borrows) moey through the sale of U.S. Treasury issues icludig Treasury Bills (T-Bills), Treasury Notes ad Treasury Bods. By purchasig Treasury issues, a ivestor is loaig the govermet moey. The Uited States govermet has prove to be a extremely reliable debtor (at least it makes good o all of its Treasury obligatios). Treasury issues are fully backed by the full faith ad credit of the U.S. govermet which has substatial resources due to its ability to tax citizes ad create moey. Thus, these securities have lower default risk tha the safest of corporate bods or short term otes. The treasury obligatios with the shortest terms to maturity are Treasury Bills. They typically mature i less tha oe year (3, 26 or 52 weeks). These issues are sold as pure discout debt securities, meaig that their purchasers receive o explicit iterest paymets. Such pure discout istrumets are also kow as zero coupo issues. I primary markets, T-Bills are sold to the public through a auctio process maaged by district Federal Reserve baks. Auctios of three ad six moth bills are typically aouced o Tuesdays, coducted the followig Moday ad settled o the followig Thursday. There are two ways to purchase T-Bills. The first is to eter a competitive bid at the auctio where the biddig istitutio competes for a give dollar amout of the ew issue based o how much it is willig to pay. Secodly, o-competitive bids ca be tedered where the prospective purchaser states how may bills he would like to purchase at the average price of accepted competitive bids. Bidders for T-Bills geerally eter their bids just before the deadlie to participate i the auctio. No-competitive bids are satisfied at the average price of successful competitive bids. The treasury determies the dollar amout of competitive bids that it wishes to satisfy by subtractig the face values of the o-competitive bids from the level of bills that the Treasury wishes to sell. Successful competitive bids are selected by rakig them, startig with the highest bid. Successful bidders competitors obtai their bills at the prices that they bid; the lowest bid is referred to as the stop-out price. Highly liquid secodary markets exist for Treasury Bills. A ivestor ca easily purchase ad sell Treasury Bills through a broker i the Over The Couter Markets. Oe variatio of a T-Bill issue is a so-called Strip Issue. Strips are portfolios of T-Bills sold by the Treasury i blocks with varyig maturities. For example, a block of five strips maturig at the ed of a give year i a five year period may provide for a paymet of $,000 at the ed of a give period. The idividual strips ca be "stripped" from the block ad sold i secodary markets. I additio to the short-term pure discout istrumet issues discussed above, the Treasury also offers a umber of loger term coupo issues. For example, Treasury Notes (T-Notes) have maturities ragig from oe to te years ad make semi-aual iterest paymets. Similarly, Treasury Bods (T-Bods) typically rage i maturity from te to thirty years ad make semi-aual iterest paymets. These T-Bods are frequetly callable, meaig that the Treasury maitais a optio to repurchase them from ivestors at a stated price. The Uited States Treasury also offers o-marketable issues such as Series EE U.S. Savigs Bods ad Series H U.S. Savigs Bods. These savigs bods are ormally issued oly to idividuals ad caot be traded amog ivestors. Such issues are ofte subjected to certai restrictios (such as a $5,000 maximum level of 3

3 purchases per year). These bods ca be purchased through most baks ad savigs istitutios, ad may busiesses maitai plas through which their employees ca purchase savigs bods through payroll deductio programs. Agecy Issues The Uited States federal govermet has created ad sposored a umber of istitutios kow as agecies. These agecies eable the govermet to make fuds available for a umber of fuctios. Amog the oldest of these agecies is The Federal Natioal Mortgage Associatio (FNMA or Faie Mae) which is curretly a federally sposored private corporatio with shareholders ad other security holders. FNMA creates mortgage backed securities by purchasig residetial mortgages from baks ad thrift istitutios. I effect, FNMA purchases the mortgage obligatios held by baks ad thrifts, repackages them as debt security portfolios, isures them ad resells them to the geeral public. These portfolios of mortgage backed securities are also pass- through securities. FNMA ca obtai moey directly from the U.S. Treasury should it eed to do so. The Govermet Natioal Mortgage Associatio (GNMA or Giie Mae) ad Federal Home Loa Mortgage Corporatio (FHLMC or Freddie Mac) also create, isure ad sell pass-through securities related to residetial mortgages. The Studet Loa Marketig Associatio (SLMA or Sallie Mae creates, isures ad sells pass-through securities related to studet loas. Muicipal Securities ad Markets The muicipal securities markets owe much of their success to U.S. federal taxatio code which permits ivestors i muicipal istrumets to omit from their taxable icome ay iterest paymets received o these issues. Thus, iterest received o muicipal bods eed ot be declared as part of icome subject to federal icome taxatio. This feature makes muicipal bods more attractive to ivestors, eablig issuers to offer these bods at a reduced iterest rate. Several types of muicipal issues are offered by state ad local govermets. The first, so-called Geeral Obligatio Bods are full faith ad credit bods. This meas that the issuer backs the bods to the fullest extet possible, give its assets ad other obligatios. Limited Obligatio Bods provide for the issue to be backed oly by specific resources or assets. For example, a reveue bods may be backed oly be the cash flows geerated by a specific asset such as a toll bridge. Some muicipal issues are isured by private isurace istitutios. This isurace is iteded to reduce the default risk associated with the issue, ad make them more attractive to ivestors. This reduced risk would eable issuers to offer bods with reduced iterest rates to the public. Amog the larger isurers of muicipal bods are The America Muicipal Bod Isurace Associatio ad the Muicipal Bod Isurace Associatio. Fiacial Istitutio Issues No-govermet fiacial istitutios are also importat participats i primary markets for debt istrumets. For example, the Federal Fuds markets allow baks ad other depository istitutios to led to oe aother to meet federal reserve requiremets. Essetially, this market provides that excess reserves of oe bak may be loaed to other baks for satisfactio of reserve requiremets. The rate at which these loas occur is referred to as the Federal Fuds Rate. Normally, bak accouts are ot regarded as marketable securities. Oe exceptio to this are Negotiable Certificates of Deposit (also kow as Jumbo C.D.s). These are depository istitutio certificate of deposit accouts with deomiatios exceedig $00,000. The amouts by which these jumbo C.D.s exceed $00,000 are ot subject to FDIC isurace. Moey Market Mutual Fuds are created by baks ad ivestmet istitutios for the purpose of poolig together depositor or ivestor fuds for the purchase of moey market istrumets (short term, highly liquid low risk debt securities). Bakers Acceptaces are origiated whe a bak accepts resposibility for payig a cliets loa. Because the bak is likely to be regarded as a good credit risk, these acceptaces are usually easily marketable as securities. Repurchase Agreemets (Repos) are issued by fiacial istitutios (usually securities firms) ackowledgig the sale of assets ad a subsequet agreemet to re-purchase at a higher price i the ear term. This agreemet is essetially the same as a collateralized short term loa. The couterparty istitutio buyig the securities with the agreemet to resell them is said to be takig a reverse repo. 32

4 Corporate Bods ad Markets Corporatios are also importat issuers of debt securities. Large, well-kow, credit-worthy firms eedig to borrow for a short period of time may issue large deomiatio short otes frequetly referred to as Commercial Paper. Well-developed markets exist for these short-term promissory otes. Firms requirig fuds for loger periods of time may issue corporate bods. These loger term istrumets are ofte issued with a variety of features, icludig callability, covertibility, sikig fud provisios, etc. There are a large umber of differet types of corporate bods. The terms of the bod will be specified i a cotract kow as a bod ideture. I additio, firms may make bak commercial loas, though secodary markets for bak loas ted to be limited i size ad scope. Callable bods may be called by the issuig istitutio at its optio. This meas that the issuig istitutio has the right to pay off the callable bod before its maturity date. The callable bod typically has a call date associated with it as well as a call price. The call date is the first date (ad perhaps oly date) that the bod ca be repurchased by the issuig istitutio. The call price is ormally set higher tha the bod s par value ad represets the price that the issuig istitutio agrees to pay the bod owers. Because the issuig istitutio retais the optio to force early retiremet of callable debt, the call provisio ca be expected to reduce the market value of the callable bod relative to otherwise comparable o-callable bods. Covertible bods ca be covertible by bod holders ito equity or other securities. This ormally meas that the covertible bodholder has the right to exchage the covertible bod for a specified umber of shares of commo stock or some other security. The covertibility provisio of such a bod ehaces its value relative to otherwise comparable o-covertible bods. Debetures are ot backed by collateral. May other bods are either backed by collateral or have some other device such as Sikig fud provisios to provide for additioal safety for bod holders. Oe type of sikig fud provisios provides for the issuig istitutio to place specified sums of moey ito a fud at specified dates that will be accumulated over time to esure full satisfactio of the firm s obligatio to bodholders. I some istaces, sikig fuds will be used to retire associated debt early. Serial bods are issued i series with staggered maturity dates. May more iovative bods have bee offered i the market. Floatig rate bods have coupo rates that rise ad fall with market iterest rates; reverse floaters have coupo rates that move i the opposite directio of market iterest rates. Idexed bods have coupo rates that are tied to the price level of a particular commodity like oil or some other value like the iflatio rate. Catastrophe bods make paymets that deped o whether some disaster occurs, like a earthquake i Califoria or a hurricae i Florida. These catastrophe bods provide a sort of isurace for issuers agaist the occurrece of the disaster. I some respects, purchasers of these bods are providig isurace to the issuers. Most corporate bods are rated by well-kow agecies with respect to aticipated default risk. Corporatios pay istitutios like Stadard & Poor s ad Moody s to rate the riskiess of their issues. Other ratig agecies iclude Fitch, A.M. Best, Duff ad Phelps ad Du ad Bradstreet. Bods without ratigs assiged by these agecies are very difficult to sell; i fact, may istitutios face restrictios o purchasig bods that are either urated or have ratigs below a give level. Stadard & Poor s ad Moody s use the ratig schemes depicted i Table. Descriptio Stadard & Poor s Moody s Least likely to default AAA Aaa High quality AA Aa Medium grade ivestmet quality A A Low grade ivestmet quality BBB Baa High grade speculative quality BB Ba Speculative B B Lower grade speculative CCC Caa Highly speculative CC Ca Likely bakruptcy C C Already i default D D Table : Stadard & Poor s ad Moody s Corporate Bod Ratigs 33

5 Bods rated BBB (or Baa) ad higher are typically referred to as ivestmet grade bods while bods below this level are cosidered to be of speculative grade. Speculative grade bods are ofte called juk bods. There is sigificat evidece that these bod ratigs are highly correlated with icidece of default, suggestig that these agecies are least reasoably i forecastig default ad measurig default risk. Furthermore, it is fairly uusual for ratigs provided by these agecies to differ by more tha oe grade. Bod markets seem to agree with these statistical fidigs, pricig bods such that their yields are strogly iversely correlated with bod ratigs. Bod ratig agecies make extesive use of fiacial statemet ad ratio aalysis to compute their ratigs. Such aalyses are frequetly supplemeted by statistical techiques such as Multi-discrimiate Aalysis, Probit ad Logit modelig. Eurocurrecy Istrumets ad Markets Eurodollars are freely covertible dollar-deomiated time deposits outside the Uited States. The baks may be o-u.s. baks, overseas braches of U.S. baks or Iteratioal Bakig Facilities (ot subject to reserve requiremets). Eurodollar markets bega after World War II whe practically all currecies other tha the U.S. dollar were perceived as ustable. Thus, most foreig trade betwee coutries was deomiated i U.S. dollars. However, the Soviet Uio ad Easter Europeas were cocered that their dollars held i U.S. baks might be attached by U.S. residets i litigatio with these coutries. Thus, they dealt ot with actual U.S. dollars, but merely deomiated their debits ad credits with dollars. Moies owed to them were simply offset by moies that they owed. I a sese, they dealt with "fake" euro-dollars, but sice their tradig parters did also, ad their accouts teded to "zero out" over time, this did ot create sigificat problems. Their euro-dollars were left i Wester Europea baks. Durig the 960s ad 970s, these markets thrived due to regulatios imposed by the U.S. govermet such as Regulatio Q (iterest ceiligs), Regulatio M (reserve requiremets), the Iterest Equalizatio Tax imposed begiig i 963 to tax iterest paymets o foreig debt sold i the U.S. ad restrictios placed o the use of domestic dollars outside the U.S. More geerally, eurocurrecies are loas or deposits deomiated i currecies other tha that of the coutry where the loa or deposit is created. Approximately 65% of eurocurrecy loas are deomiated i dollars. Eurocredits (e.g: Eurodollar Credits) are bak loas deomiated i currecies other tha that of the coutry where the loa is exteded. They are attractive due to very low iterest rate spreads which are possible due to the large size of the loas ad the lack of reserve, FDIC ad other requiremets directly or idirectly with domestic loas ad deposits. Their rates are geerally tied to LIBOR (the Lodo Iterbak Offered Rate) ad U.S. rates. Loa terms are usually less tha five years, typically for six moths. Euro-Commercial paper are short-term (usually less tha six moths) otes issued by large, particularly "credit-worthy" istitutios. Most commercial paper is ot uderwritte. The otes are geerally very liquid ad most are deomiated i dollars. Euro-Medium Term Notes (EMTNs), ulike Eurobods, are usually issued i istallmets. Agai, most are ot uderwritte. Eurobods are geerally uderwritte, bearer bods deomiated i currecies other tha that of the coutry where the loa is exteded. Eurobods ofte have call ad sikig fud provisios as well as other features foud i bods publicly traded i America markets. Euro-Commercial paper is the term give to short-term (usually less tha six moths) otes issued by large, particularly "credit-worthy" istitutios. Most commercial paper is ot uderwritte. The otes are geerally very liquid (much more so tha Sydicated loas) ad most are deomiated i dollars. They are usually pure discout istrumets. Euro-Medium Term Notes (EMTNs) iterest-bearig istrumets usually issued i istallmets. Most are ot uderwritte. Eurobods are geerally uderwritte, bearer bods deomiated i currecies other tha that of the coutry where the loa is exteded. Eurobods typically make aual coupo paymets ad ofte have call ad sikig fud provisios. 34

6 B. Bod Yields, Rates ad Sources of Risk Assume that we wish to aalyze a bod maturig i periods with a face value (or priciple amout) equal to F payig iterest aually at a rate of c. The aual iterest paymet is rate c multiplied by face value F (or cf) ad the bod makes a sigle paymet i time equal to F. Usig a stadard preset value model discoutig cash flows at a rate of k, the bod is evaluated as follows: PV j t cf t F (%k) t% (%k) () For example, let c equal.0, F equal $000, k equal.2 ad equal 2. The preset value of this bod is computed as follows: PV (%.2) % (%.2) 2% (%.2) (2) Preset Value is used to determie the ecoomic worth of a bod; the retur of a bod measures the profit relative to the ivestmet of a bod. There are several measures of bod retur icludig curret yield ad yield to maturity. The more simple measuremet, curret yield, is cocered with aual iterest paymets relative to the iitial ivestmet required by the bod ad is measured as follows: (3) cf P 0 If the bod used i the example above may be purchased for $986.48, its curret yield is simply 0.4%. Fiacial pages i ewspapers frequetly quote yields for treasury bills ad other pure discout istrumets usig the bak discout method: F & P 0 F 360 where is the umber of days before the bod matures. This measure of the bod s ecoomic efficiecy is rather odd for several reasos. First, it is based o simple rather tha compoud iterest. Secod, it assumes a 360 day yield. Fially, it measures retur as a proportio of face value rather tha the sum ivested. The bod equivalet yield represets a slight improvemet over the bak discout formula: F & P P 0 Oe ecouters two problems usig curret yield as a retur measure. First, the curret yield does ot accout for ay capital gai or loss (F - P 0 ) that may accrue whe the bod matures. Secod, curret yield does ot accout for the time value of moey or compoudig of the cash flows associated with the bod. Hece, oe may wish to compute the bods iteral rate of retur, which is geerally referred to as yield to maturity (y): 35

7 (4) P 0 j t cf t F (%y) t% (%y) Yield to maturity is that value for y which satisfies Equality (4). Usually, a solutio must be obtaied through a iterative process. The yield to maturity (or iteral rate of retur for the bod described above has a yield to maturity of 2%, computed as follows: (5) P (%.2) % (%.2) 2% (%.2) 2 Thus, yield to maturity may be iterpreted as that discout which sets the purchase price of a bod equal to its preset value. The yield to call for a callable bod differs from yield to maturity i two respects:. Cash flows are assumed to cease at the call date rather tha the maturity date ad 2. The call price is used as the bod s fial cash flow rather tha the face value of the bod Bod risk may be categorized as follows:. Default or credit risk: the bod issuer may ot fulfill all of its obligatios 2. Liquidity risk: there may ot exist a efficiet market for ivestors to resell their bods 3. Iterest rate risk: market iterest rate fluctuatios affect values of existig bods. Uited States Treasury Issues are geerally regarded as beig practically free of default risk. Furthermore, there exists a active market for treasury issues, particularly those maturig withi a short period. Thus, treasury issues are regarded as havig miimal liquidity risk. However, all bods are subject to iterest rate risk. Loger term bods are subject to icreased iterest rate risk due to the icreased periods that the yields o loger term bods are likely to differ from ewly issued bods. 36

8 C. The Term Structure of Iterest Rates The Term Structure of Iterest Rates is cocered with the chage i iterest rates o debt securities resultig from varyig times to maturity o the debt. For example, it may be cocered with explaiig why the iterest rate o debt maturig i oe year is 4% versus 7% for debt maturig i twety years. Geerally at a give poit i time, we observe loger term iterest rates exceedig shorter term rates, though this is ot always the case (for example, the years ). Followig are three theories which, either separately or i combiatio, attempt to explai the relatio betwee log ad short term iterest rates. The Pure Expectatios Theory states that log term spot rates (iterest rates o loas origiatig ow) ca be explaied as a product of short term spot rates ad short term forward rates (iterest rates o loas committed to ow but actually origiatig at later dates). Where y t,m is the rate o a loa origiated at time t to be repaid at time m, the Pure Expectatios Theory defies the relatioship betwee log ad short term iterest rates as follows: () (%y 0, ) k t (%y t&,t ) Thus, the log term spot rate y 0,t is defied as th root of the product of the oe period spot rate y 0, ad a series of oe period forward rates y t -,t mius oe. I other words, the log term spot rate y 0, ca be determied based o the short-term spot rate y 0, ad a series of oe period forward rates y t-,t as follows: (2) y 0, k t (%y t&,t ) & Cosider a example where the oe year spot rate y 0, is 5%. Ivestors are expectig that the oe year spot rate oe year from ow will icrease to 6%; thus, the oe year forward rate y,2 o a loa origiated i oe year is 6%. Furthermore, assume that ivestors are expectig that the oe year spot rate two years from ow will icrease to 7%; thus, the oe year forward rate y 2,3 o a loa origiated i two years is 7%. Based o the pure expectatios hypothesis, what is the three year spot rate? This is determied with Equatio (2) as follows: y 0, k t (%y t&,t ) & 3 (%y 0, )(%y,2 )(%y 2,3 ) & 3 (%.05)(%.06)(%.07) & Oe problem with the Pure Expectatios Theory is that it does ot explai why iterest rates o loger term bods ted to exceed short term rates much more tha fifty percet of the time. If oe were to use the Pure Expectatios Theory to explai this pheomeo, oe would coclude that ivestors cosistetly expect that iterest rates will rise. Sice this is probably ot true, it may be useful to propose a alterative theory to explai the term structure of iterest rates. The Liquidity Premium Theory is based o the Pure Expectatios Theory where it is assumed that ivestors required icreased rate premiums to ivest outside of their preferred ivestmet horizos. Sice shorter term debt markets are more liquid, ivestors frequetly require icreased rate compesatio for bearig liquidity risk as loa terms to maturity icrease. Where LP t icreases as t icreases, the Liquidity Premium Theory defies iterest rates as follows: 37

9 (3) (%y 0, ) k t (%y t&,t )%LP t Equatio (3), which is based o Equatio () simply states that ivestors may require additioal iterest for certai maturities as compesatio for illiquidity that may exist i the market. A alterative explaatio (ad perhaps complemetary explaatio) of term structure, The Market Segmetatio Theory states that various types of borrowig ad ledig istitutios have strog debt maturity prefereces. Such istitutios will bid up or dow iterest rates for the various maturity dates based o their prefereces. This theory has attracted more attetio i the professioal tha i the academic literature. This theory is very much to the Preferred Habitat Theory which describes maturity matchig of assets ad liabilities as the basis for much segmetatio. The yield curve ca be obtaied empirically by examiig the payoffs associated with a bod simultaeously with the bod s purchase price. Let D t be the discout fuctio for time t; that is, D t = /(+y 0,t ) t. This meas that a cash flow paid at time t will be discouted by multiplyig it by the discout fuctio D t : A little algebra produces the followig spot rate: PV = CF D t = CF t /(+y 0,t ) t y 0,t = (/D t ) /t - Thus, oe ca obtai the spot rates y 0,t from the bod s curret purchase price P 0 ad expected future cash flows from coupo paymets ad face value CF t. Thus, cosider a $000 face value bod makig a sigle iterest paymet at a aual rate of 5%. Suppose this bod is curretly sellig for 02 (meaig 02% or 020) ad that it matures i oe year whe its coupo paymet is made. The oe year spot rate implied by this bod is determied as follows: 020 = (50 + D = (050)/(+y 0, ) D = 020/050 = ( ) / ; / = y 0, =.0294 Thus, the oe year spot rate is 2.95%. However, a difficulty arises whe the bod has more tha oe cash flow. As spot rates may vary over time, there may be a spot rate for each period, hece, a spot rate for each cash flow. Cosider a $000 face value two year bod makig iterest paymets at a aual rate of 5%. Suppose this bod is curretly sellig for 0.75 (meaig 0.75% or 07.5) ad that it matures i two years whe its secod coupo paymet is made. The two spot rate implied by this bod is bootstrapped from the oe year spot rate as follows: 07.5 = + (50 + D 2 D 2 = [ ] = (/ ) /2 - = y 0,2 =.040 More geerally, this bootstrappig process is applied as follows: PV = D + D 2 + D D - + (cf + D = ΣcF/(+y 0,t ) t + (cf + F) /(+y 0,t ) PV = ΣCF D t + (cf + F) /(+y 0,t ) = ΣCF t /(+y 0,t ) t + (cf + F) /(+y 0,t ) D = [P 0 - ( D t )]/[cf + F] Bootstrappig requires that there be oe bod maturig i each year t so that its D t ca be used to determie (bootstrap) the D t for the bod maturig i oe year subsequet. Thus, oe starts by determiig D, D 2 ad so o util all D t values have bee determied. These expressios are used to bootstrap spot rates from bod prices, 38

10 maturities ad coupo rates i Table 2 ad i Figure mappig out the yield curve. Ay i-year forward rate, y t-i,t, from year t-i to year t is determied from (D t /D t ) /i -. Ask Maturity %Coupo Price D t Spot Rate % / % / % / % / % / % / % / % / % / % / % / % % % % / % Table 2: Bootstrappig Spot Rates 6.00% 5.00% Spot Rate 4.00% 3.00% 2.00%.00% 0.00% Years Figure : Mappig the Yield Curve 39

11 D. Term Structure Estimatio with Coupo Bods The spot rate is the yield at preset prevailig for zero coupo bods of a give maturity. The t year spot rate is deoted here by y 0,t, which represets the iterest rate o a loa to be made at time zero ad repaid i its etirety at time t. Spot rates may be estimated from bods with kow future cash flows ad their curret prices. We are able to obtai spot rates from yields implied from series of bods whe we assume that the Law of Oe Price holds. The yield curve represets yields or spot rates of bods with varyig terms to maturity. For example at a give poit i time, the yield for oe-year bods may be 5% (y 0, =.05), while the yield for five-year bods may be 0% (y 0,5 =.0). This sectio is cocered with how iterest rates or yields vary with maturities of bods. The simplest bods to work with from a arithmetic perspective are pure discout otes, otes which make o iterest paymets. Such otes make oly oe paymet at oe poit i time o the maturity date of the ote. Determiig the relatioship betwee yield ad term to maturity for these bods is quite trivial. The retur oe obtais from a pure discout ote is strictly a fuctio of capital gais; that is, the differece betwee the face value of the ote ad its purchase price. Short-term U.S. Treasury Bills are a example of pure discout (or zero coupo) otes. Coupo bods are somewhat more difficult to work with from a arithmetic perspective because they make paymets to bodholders at a variety of differet periods. Simultaeous Estimatio of Discout Fuctios A coupo bod may be treated as a portfolio of pure discout otes, with each coupo beig treated as a separate ote maturig o the date the coupo is paid. This slightly complicates the process for determiig yields, but is ecessary to avoid associatig wrog yields with give time periods. Cosider a example ivolvig three bods whose characteristics are give i Table. The three bods are tradig at kow prices with a total of eight aual coupo paymets amog them (three for bods A ad B ad 2 for bod C). Bod yields or spot rates must be determied simultaeously to avoid associatig cotradictory rates for the aual coupos o each of the three bills. Table Coupo Bods A, B ad C BOND CURRENT PRICE FACE VALUE COUPON RATE YEARS TO MATURITY A B C Let D t be the discout fuctio for time t; that is, D t = /(+y 0,t ) t. Sice y 0,t is the spot rate or discout rate that equates the preset value of a bod with its curret price, the followig equatios may be solved for discout fuctios the spot rates: D % 00D 2 % 00D D % 20D 2 % 20D D % 00D 2 This system of equatios may be represeted by the followig system of matrices: 40

12 D D D d P 0 To solve this system we first ivert Matrix CF, the use this iverse to premultiply Vector P 0 to obtai Vector d: & & D &.0055 & D & D 3.72 CF P 0 d Thus, we fid from solvig this system for Vector d that D =.9, D 2 =.8 ad D 3 =.72. Sice D t = /(+y 0,t ) t, /D t = (+y 0,t ) t, ad y 0,t = /D /t!. Thus, spot rates are determied as follows: D &. D 2 2 D 3 3 &.80 &.57 Note that there exists a differet spot rate (or discout rate) for each term to maturity; however, the spot rates for all cash flows geerated by all bods at a give period i time are the same. Thus, y 0,t will vary over terms to maturity, but will be the same for all of the bods i a give time period. The origial system of three equatios is solved i a step-by-step process i Appedix A at the ed of the chapter. Regressio Estimatio I the sectio above, we were cocered with the yield curve describig the relatioships amog spot ad forward rates over differet itervals of time. We defied the followig discout fuctio D t : D t ( % y t ) t where y t is the spot rate which varied over time. Our solutio techique for the differet discout fuctios D t ad yields y t required that we aalyze a series of bods maturig ad makig coupo paymets o specific dates. I particular, our solutio techique required that we have at least oe bod for each yield we wished to estimate ad that bods make paymets o idetical dates. I reality, we may have difficulty fidig bods which make paymets o commo dates; furthermore, the bods which we select may ot be priced cosistetly. Our solutio techique would ot imply a spot rate for ay date that would ot be cosistet with at least oe bod paymet. 4

13 Here, we will cosider a alterative techique for mappig out a yield curve. Suppose that a fixed icome maager believes that the followig equatio describes the relatioship betwee bod discout fuctios ad time (t): D t a % b t % b 2 t 2 % ε t where a, b ad b 2 are multiple OLS regressio coefficiets. We ca use the multiple regressio techique to determie spot rates from the data i Table 0 derived from zero coupo bods. Based o a two-idepedetvariable OLS model, what would this fud maager predict the for the yield for a 2.5 year bod? Note that oe of the bods mature or make a coupo paymet i exactly 2.5 years, so that we caot compute D 2.5 usig the solutio techique from above. Thus, to estimate the 2.5 year yield, we shall perform a OLS regressio of D t o t ad t 2. The first step i our computatios is to calculate each value for D t from y t. We fid that D =.9743, D 2 =.82946, D 3 =.7462, D 4 =.6635 ad D 5 = We regress D t agaist t ad t 2 to obtai the followig regressio equatio ad t-statistics: D t.0488 &.09956t & t 2 (552.74) (&33.) (5.99) Table 0 Bod Yields ad Maturity Data Bod Yield t t 2 A.060 B C D E Isertig t = 2.5 ito this equatio, we fid that D 2.5 = This leads to a yield y 2.5 solutio of Our stadard error estimates for a, b ad b 2 are, respectively,.00836,.003 ad Thus, based o resultig t- statistics, our estimates for a, b ad b 2 are statistically sigificat at the.0 level. 42

14 E. Arbitrage With Riskless Bods The example provided above cosists of three priced riskless bods defiig spot rates for all three relevat years. The cash flow structure of ay three-year bod (for example, Bod D) added to the market ca be replicated with some portfolio of bods A, B, ad C as log as its cash paymets to ivestors are o the same dates as those made by at least oe (i this example, two) of the three bods A, B, ad C. For example, assume that there ow exists Bod D, a three-year,.5% coupo bod sellig i this market for $990. This bod will make paymets of $5 i years ad 2 i additio to a $5 paymet i year 3. A portfolio of bods A, B ad C ca be comprised to geerate the exact cash flow series. Thus, Bod D ca be replicated by a portfolio of our first three bods with the followig weights: w A =.25, w B =.75 ad w C =0, which are determied by the followig system of equatios or matrices: 5 = 00w A + 20w B + 00w C 5 = 00w A + 20w B + 00w C 5 = 00w A + 20w B w A w B w C w cf D To solve this system we first ivert Matrix CF, the use it to premultiply Vector cf D to obtai vector w: & &.0055 &.0050 & w A w B w C CF cf D w Thus, we fid from this system that w A =.25, w B =.75 ad w C = 0. We determie the value of the portfolio replicatig Bod D by weightig their curret market prices: $962) + $00.4) = $ Based o the portfolios price, the value of Bod D is $998.3, although its curret market price is $990. Thus, oe gais a arbitrage profit from the purchase of this bod for $990 fiaced by the sale of the portfolio of Bods A ad B at a price of $ Here, we simply swap a portfolio comprised of Bods A ad B for Bod D. Our cash flows i years,2 ad 3 will be zero, although we receive a positive cash flow ow of $8.3. This is a clear arbitrage profit. This arbitrage opportuity will persist util the value of the portfolio equals the value of Bod D. Thus, spot rates must be cosistet for all bods of the same risk class ad maturity. 43

15 F. Fixed Icome Portfolio Dedicatio A fixed icome fud is cocered with esurig the provisio of a relatively stable icome over a give period of time. Typically, a fixed icome fud must provide paymets to its creditors, cliets or owers for a give period. For example, a pesio fud is ofte expected to make a series of fixed paymets to pesio fud participats. Such fuds must ivest their assets to esure that their liabilities are paid. I may cases, fixed icome fuds will purchase assets such that their cash flows exactly match the liability paymets that they are required to make. This exact matchig strategy is referred to as dedicatio ad is iteded to miimize the risk of the fud. The process of dedicatio is much the same as the arbitrage swaps discussed above; the fud maager merely determies the cash flows associated with his liability structure ad replicates them with a series of default risk free bods. For example, assume that a pesio maager eeds to make paymets to pesio pla participats of $,500,000 i oe year; $2,500,000 i two years; ad $4,000,000 i three years. He wishes to match these cash flows with a portfolio of bods E, F ad G whose characteristics are give i Table 3. These three bods must be used to match the cash flows associated with the fuds liability structure. For example, i year, Bod E will pay $00 (000+00), F will pay $20 ad G will pay $00. These paymets must be combied to total $,500,000. Cash flows must be matched i years 2 ad 3 as well. Table 3 Coupo Bods E, F ad G BOND CURRENT PRICE FACE VALUE COUPON RATE YEARS TO MATURITY E F G Oly oe matchig strategy exists for this sceario. The followig system may be solved for b to determie exactly how may of each of the bods are required to satisfy the fuds cash flow requiremets: b E,500,000 2,500,000 4,000,000 b F b G b L Ivertig Matrix CF ad multiplyig by Vector L, we fid that the purchase of Bods E, Bods F ad Bods G satisfy the maagers exact matchig requiremets. The fuds time zero paymet for these bods totals $6,385,

16 G. Bod Duratio Bods ad other debt istrumets issued by the Uited States Treasury are geerally regarded to be free of default risk ad of relatively low liquidity risk. However, these bods, particularly those with loger terms to maturity are subject to market value fluctuatios after they are issued, primarily due to chages i iterest rates offered o ew issues. Geerally, iterest rate icreases o ew bod issues decrease values of bods which are already outstadig; iterest rate decreases o ew bod issues icrease values of bods which are already outstadig. The duratio model is iteded to describe the proportioal chage i the value of a bod that is iduced by a chage i iterest rates or yields of ew issues. May aalysts use preset value models to value treasury issues, frequetly usig yields to maturity of ew treasury issues to value existig issues with comparable terms. It is importat for aalysts to kow how chages i ew-issue iterest rates will affect values of bods with which they are cocered. Bod duratio measures the proportioal sesitivity of a bod to chages i the market rate of iterest. Cosider a two-year 0% coupo treasury issue which is curretly sellig for $ The yield to maturity y of this bod is 2%. Default risk ad liquidity risk are assumed to be zero; iterest rate risk will be of primary importace. Assume that this bods yield or discout rate is the same as the market yields of comparable treasury issues (which might be expected i a efficiet market) ad that bods of all terms to maturity have the same yield. Further assume that ivestors have valued the bod such that its market price equals its preset value; that is, the discout rate k for the bod equals its yield to maturity y. If market iterest rates ad yields were rise for ew treasury issues, the the yield of this bod would rise accordigly. However, sice the cotractual terms of the bod will ot chage, its market price must drop to accommodate a yield cosistet with the market. Assume that the value of a -year bod payig iterest at a rate of c o face value F is determied by a preset value model with the yield y of comparable issues servig as the discout rate k: () PV j t cf (%y) t % F (%y) Assume that the terms of the bod cotract,, F ad c are costat. Just what is the proportioal chage i the price of a bod iduced by a proportioal chage i market iterest rates (techically, a proportioal chage i [+y])? This may be approximated by the bods Macaulay Simple Duratio Formula as follows: (2) PV PV (%y) (%y). Dur dpv PV d(%y) (%y) dpv d(%y) (%y) PV Equatio (2) provides a good approximatio of the proportioal chage i the price of a bod i a market meetig the assumptios described above iduced by a ifiitesimal proportioal chage i ( + y). To compute the bods sesitivity, we first rewrite Equatio () i polyomial form (to take derivatives later) ad substitute y for k (sice they are assumed to be equal): (3) PV j t cf t (%y) t % First, fid the derivative of PV with respect to (+y): (4) Equatio (4) is rewritte: dpv d(%y) j t F (%y) j t cf(%y) &t % F(%y) & &tcf(%y) &t& & F(%y) && 45

17 (5) dpv d(%y) j t &tcf(%y) &t & F(%y) & (%y) Sice the market rate of iterest is assumed to equal the bod yield to maturity, the bods price will equal its preset value. Next, multiply both sides of Equatio (5) by (+y) P 0 to maitai cosistecy with Equatio (2): (6) Dur dpv d(%y) (%y) P 0 j t &tcf(%y) &t & F(%y) & Thus, duratio is defied as the proportioal price chage of a bod iduced by a ifiitesimal proportioal chage i (+y) or plus the market rate of iterest: P 0 (7) Dur dpv d(%y) (%y) P 0 j t &tcf % &F (%y) t (%y) P 0 Sice the market rate of iterest will likely determie the yield to maturity of ay bod, the duratio of the bod described above is determied as follows from Equatio (7): (8) Dur 000 (%.2) % 000 (%.2) 2 % (%.2) 2 &.87 This duratio level of -.87 suggests that the proportioal decrease i the value of this bod will equal.87 times the proportioal icrease i market iterest rates. This duratio level also implies that this bod has exactly the same iterest rate sesitivity as a pure discout bod (a bod makig o coupo paymets) which matures i.87 years. Applicatio of the Simple Macaulay Duratio model does require several importat assumptios. First, it is assumed that yields are ivariat with respect to maturities of bods; that is, the yield curve is flat. Furthermore, it is assumed that ivestors projected reivestmet rates are idetical to the bod yields to maturity. Ay chage i iterest rates will be ifiitesimal ad will also be ivariat with respect to time. The accuracy of this model will deped o the extet to which these assumptios hold. 46

18 H. Fixed Icome Immuizatio Earlier, we discussed bod portfolio dedicatio, which is cocered with matchig termial cash flows or values of bod portfolios with required payouts associated with liabilities. This process assumes that o trasactios will take place withi the portfolio ad that cash flows associated with liabilities will remai as origially aticipated. Clearly, these assumptios will ot hold for may istitutios. Alteratively, oe may hedge fixed icome portfolio risk by usig immuizatio strategies, which are cocered with matchig the preset values of asset portfolios with the preset values of cash flows associated with future liabilities. More specifically, immuizatio strategies are primarily cocered with matchig asset duratios with liability duratios. If asset ad liability duratios are matched, it is expected that the et fud value (equity or surplus) will ot be affected by a shift i iterest rates; asset ad liability chages offset each other. Agai, this simple immuizatio strategy is depedet o the followig:. Chages i ( + y) are ifiitesimal. 2. The yield curve is flat (yields do ot vary over terms to maturity). 3. Yield curve shifts are parallel. 4. Oly iterest rate risk is sigificat. 47

19 F. Covexity Earlier, we used duratio to determie the approximate chage i a bods value iduced by a chage i iterest rates ( + y). However, the accuracy of the duratio model is reduced by fiite chages i iterest rates, as we might reasoably expect. Duratio may be regarded as a first order approximatio (it oly uses the first derivative) of the chage i the value of a bod iduced by a chage i iterest rates. Covexity is determied by the secod derivative of the bods value with respect to ( + y). The first derivative of the bods price with respect to ( + y) is give: MP 0 () M(%y) j &tcf(%y) &t& & F(%y) && t We fid the secod derivative by determiig the derivative of the first derivative as follows: (2) M 2 P 0 M(%y) 2 j t &t(&t&)cf(%y) &t&2 & (&&)F(%y) &&2 j t (t 2 % t)cf % ( 2 %)F (%y) t%2 (%y) %2 Covexity is merely the secod derivative of P 0 with respect to ( + y) divided by P 0. The first two derivatives may be used i a Taylor series to approximate ew bod prices iduced by chages i iterest rates as follows: P. P 0 % f( % y 0 [ (%y)] % f )) ( % y 0 [ (%y)] 2 (3) P. P 0 % j t &tcf t & F (%y 0 ) t% (%y 0 ) [ y] % 2 j t (t 2 % cf t ( % y 0 ) t%2 % ( 2 F ( % y 0 ) [ y] 2 Cosider a five-year te percet $000-face-value coupo bod curretly sellig at par (face value). We may compute the preset yield to maturity of this bod as y 0 =.0. The first derivative of the bods value with respect to (+y) at y 0 =.0 is foud from Equatio () to be (duratio is 000 = ); the secod derivative is foud from Equatio (2) to be 9, (covexity is 9, = ). If bod yields were to drop from.0 to.08, the actual value of this bod would icrease to , as determied from a stadard preset value model. If we were to use the duratio model (first-order approximatio from the Taylor expasio, based oly o the first derivative), we estimate that the value of the bod icreases to If we use the covexity model secod-order approximatio from Equatio (2), we estimate that the value of the bod icreases to The secod-order approximatio may also be writte as: (4) P P 0 % [ ( % P 0 /( % y)] % ( 2 P ( ( % y))2 Note that this secod estimate with the secod-order approximatio geerates a revised bod value which is sigificatly closer to the bods actual value as measured by the preset value model. Therefore, the duratio ad 48

20 immuizatio models are substatially improved by the secod order approximatios of bod prices (the covexity model). The fud maager wishig to hedge portfolio risk should ot simply match duratios (first derivatives) of assets ad liabilities, he should also match their covexities (secod derivatives). 49

21 EXERCISES. Cosider a example where we ca borrow moey today for oe year at 5%; y 0, =.05. Suppose that we are able to obtai a commitmet to obtai a oe year loa oe year from ow at a iterest rate of 8%. Thus, the oe year forward rate o a loa origiated i year equals 8%. Accordig to the Pure Expectatios Theory, what is the two year spot rate of iterest y 0,2? 2. Suppose that the oe year spot rate y 0, of iterest is 5%. Ivestors are expectig that the oe year spot rate oe year from ow will icrease to 6%; thus, the oe year forward rate y,2 o a loa origiated i oe year is 6%. Furthermore, assume that ivestors are expectig that the oe year spot rate two years from ow will icrease to 7%; thus, the oe year forward rate y 2,3 o a loa origiated i two years is 7%. Based o the pure expectatios hypothesis, what is the three year spot rate? 3. Suppose that the oe year spot rate y 0, of iterest is 5%. Ivestors are expectig that the oe year spot rate oe year from ow will icrease to 7%; thus, the oe year forward rate y,2 o a loa origiated i oe year is 7%. Furthermore, assume that the three-year spot rate equals 7% as well. What is the aticipated oe year forward rate y 2,3 o a loa origiated i two years based o the pure expectatios hypothesis? 4.** Bod A, a three year 7% issue, curretly sells for Bod B is a two year 8% issue curretly sellig for Bod C is a three year 6% issue curretly sellig for Based o this iformatio, aswer the followig: c. What are the oe, two ad three year spot rates of iterest? d. What are the oe- ad two-year forward rates o loas origiatig oe year from ow? e. What is the oe-year forward rate o a loa origiated i two years? 5. Assume that there are two three-year bods with face values equalig $000. The coupo rate of bod A is.05 ad.08 for bod B. A third bod C also exists, with a maturity of two years. Bod C also has a face value of $000; it has a coupo rate of %. The prices of the three bods are $ , $ ad $055.49, respectively. a.** What are the spot rates implied by these bods? b. Fid a portfolio of bods A, B ad C which would replicate the cash flow structure of bod D, which has a face value of $000, a maturity of three years ad a coupo rate of 3%. 6. A life isurace compay expects to make paymets of $30,000,000 i oe year, $5,000,000 i two years $25,000,000 i three years ad $35,000,000 i four years to satisfy claims of policyholders. These aticipated cash flows are to be matched with a portfolio of the followig $000 face value bods: BOND CURRENT PRICE COUPON RATE YEARS TO MATURITY How may of each of the four bods should the compay purchase to exactly match its aticipated paymets to policyholders? 7. Fid the duratio of the followig pure discout bods: a. A $000 face value bod maturig i oe year curretly sellig for $

22 b. A $000 face value bod maturig i two years curretly sellig for $800. c. A $2000 face value bod maturig i three years curretly sellig for $400. d. A portfolio cosistig of oe of each of the three bods listed i parts a, b ad c of this problem. 8. What is the relatioship betwee the maturity of a pure discout bod ad its duratio? 9. Fid the duratio of each of the followig $000 face value coupo bods assumig coupo paymets are made aually: a. 3 year 0% bod curretly sellig for $900 b. 3 year 2% bod curretly sellig for $900 c. 4 year 0% bod curretly sellig for $900 d. 3 year 0% bod curretly sellig for $ Based o duratio computatios, what would happe to the prices of each of the bods i Questio 9 if market iterest rates (+r) were to decrease by 0%?. What is the duratio of a portfolio cosistig of oe of each of the bods listed i problem 9? 2. Fid duratios ad covexities for each of the followig bods: a. A 0% five year bod sellig for $ yieldig 8% b. A 2% five year bod sellig for $000 yieldig 2% 3.a. Use the duratio (first order) approximatio models to estimate bod value icreases iduced by chages i iterest rates (yields) to 0% for each of the bods i Problem 2 above. b. Use the covexity (secod order) approximatio models to estimate bod value icreases iduced by chages i iterest rates (yields) to 0% for each of the bods i Problem 2 above. c. Fid the preset values of each of the bods i Problem 2 above after yields (discout rates) chage to 0%. 4.**The followig table lists five pure discout bods alog with their yields ad terms to maturity. BOND YIELD t A.060 B C.00 3 D.4 4 E.25 5 Based o a multiple regressio model, with t ad t 2 as idepedet variables, what would you predict for the yield of a 4.5-year bod? 5