Reading 25 The Term Structure & Int Rate Dynamics

University:
California State University, Northridge
Course:
ACCT 220 | Introduction to Financial Accounting
Academic year:

2023
Views:

130

Pages:

24
Author:

SophGr

togsPage No. I25th Friday Nov 2023Date / /Reading 25 The termStructer & 1nt Rate DynamicsM21.1 : Spot & Forward RatesIspot ruteq90C100toSpot Rate - YTM on ZCBSpot rate is the rate used to discount a singlecash flow + it is the YTM of a coupon bond(Single CF)The price today of $1 par, ZCB is Known as thedisc factor e which we call PTAsit is a ZCB + the spot rate is the YTM of thebond, , which we represent as STPT = I(1+ST)Spot Curve :Yield carveGraph of the spot rate ST Vs. Maturity L CoupounYTM will be closer to last spotPage No. 2Date / /ofForwaid notForward turve Rate (r,5,)02Theannualizedintrateonaloantobeinitiatedatafuture periodForward CurvestartYYdurationF(2,3)1=FvForwardratesVsMaturity2SForwardCurvesandspotCurvesareNotation : Ratemathematicallyrelated- F(dd) = TheannualizedintrateloanapplicationonaK-periodstartingattime-F(JIK)period=Forwardjpriceattimejofa$1parzero-we to future (disc 1$ by F(2,3)) for given period K) I FC213) 2Price get that 1$ Coupon in bond maturity at time Jtk5YTM :Avgannualizedyieldearnedonabond((RR)40104007%3SestEx:3-yr4%annual-+pay$1,000/parbondS1= = 5% S2 = 6% & S3= 7%TStep1OFrespectfullyPricehighestUpward sloopingofthebondspot Curveusingthespotratecurve.Price = 40 + 40 + 1040 = $922 64(1.05) (1.06)2 (1.07)S,CY367.6xx5 = 30%]2Trick(6%.XS)3 = 1.23726(42-X2)31((2.3) 1.237263f(2,3)= 733% Approx(2f(2,3) = 7.35%F(2,3) = $18 08-0 - (E,S)71.073508-0 x asp.o F(2,1) = (1+f(2)!))Page No. 5Date / /When the spot Curve is flat, forward rate l= Spot RateWhen the spot curve is upward sloping forward rate > Spot RateWhen the spot Carve is downward sloping t forward rate < Spot RateEx: FV=1000(= 6% semi-annual M = 2yrsTimeSpot30 + 30 + 30 + 1030 = 930.40.5711.0351.0421.04531.05"I8%1.59%210%7%9%223rd Yr Spot Rate = ?YTM of 3 yr maturity bond = ?MaturityCoupon RateMVI00.9125ZCBe2O0.8415300.8182400.7936YTM of 3yr maturity Bond is 6.91 %0.8182 = 0 +( +IMagic Method(1+S2)364 A4+3\=F(2,3) = Ps6 +6+6+6+6 =30P24 +4+3 +3 7?=308 + 31 = 30223?=30-8=7.333? = 22 Note Long telm Rate Can be viewed as geometric avg of short term sheet Rate 6SpotPage No.Date//*Spot and ForwardRateRelationshipsFor anupward - slopingyieldCurve ,theforwardratef(d,k)risesasjincrases6s455Spot Curves & forward303Curves as of July 20132.52.1.5F(0,1) = 4%0.5G=5%f(1,1) 67.20142016201820202024202620285%4%6%2Because,theyieldCurveisupwardcarves lie above the spot curveslopping,theforwardPar RatesParrateistheYTMforabondPar rate = Coupon ratetrading at parFor a bond with a single CF remaining li.e / lastspotCoupon t principal), par lates are used to generateratesusingbootstrappingEnext1-yr2-yr & 3-yrparrateare1%t 1-25% & 1.5% respectivelyUsingbootstrappingCalulateS, S24531501.50t+101.503-yr bond did100sft1.01(1.0252)2(1+53)38970517 101.505-08S3(10458)=C(Its,)SS Spot Rate Curve Is flat then forward Curve in coincident withif spet Rate (aive is upwaid slopping then forward carve is also No. 7spoping & lies above itDate/if spot Rate calve is inverted then forword curve is also invered 4 tiestellow it.BootstrapingSpotRates:Example1st par rate = 1st spot rate1-yr 2 - yr & 3-yr par rates are % e 257 P, & 1.5% P, respectivelyUsingbootstropcalculateS,S24S3:+get this value VM theuesS, 1-Yr par rate 1.237 = 1% by definition 100-1-237 PV 98.76, N=2=2 Yr bond : 100 =1.2598.76FV=101-2501.25=101.25+1-01(1+52)21(1+52)22[CPT.144=1-252This has to behigher than P2I101-25 = (1+5.)2(S2 = (1-0252)due to98.76=1.252 %-upwold stopingBoststropping as involves to using the output of one steplate (ZCB) of yr-1 to calculate the spot rate (ZCB) ofan input the next step i.e using spotyr2ParRate : It is that Coupon Rate which forces the bond tobe pricedatparEg "BootstrappingParTimePar RateSpot Raterate118%7%of212%of12%318%117 7Time F(213) Ps P2Page No. 8Date / /If lyr Par Rate is 7% then FV=.1000 (=7.7: MV= 1000 M=1000 = 1000+70Spot,=77.((+S))If 2 yr Par Rate is 12% then FV=1000 (=12% MV =1000 M=21000 = 120 1120(1+7%) + (Its2)2887.85 = 1120 #Spotz = 12.31%(1+S2)3 yr par Rate is 187. then FV = 1000 C=18% M=3 MV=10001000 = 180+(1+7%)(1+12.31%)2 + (1+53) 1180180689.1 = 1180(1+53)3Spotz = 19-67"Formulo"SpotParParBootstrappingRateRatesRatesSpotRates1-24Z,+zztZ3tZqZ= Dis factore.g Z3 =I(1+52)" IFPage No. 9Date//EgIPar rate for Yr 4:=2Verify3Parlurvet ??YearsSpot17%I Z,=Zz 1_23124=1212%1.071.1221.1531.24315%420%Par,=1-24Zitzz+Z3tZ4= 18.032 VerificationFV = 1000C= 18.03% M=hy yrMV=?YearsSpotI7%1000 = 180.3 180.3 t 180.3 + 1180.3212%1-07'1.1221-1531.24315%420%1000 = 180.3 + 1000 y Z43ParValueYearsSpotParI7%7%212%11.7%+314.337. =11515%420%18.03%1+t1-071-1221-15gettfrigastaforin 1:49P2 100 = 90.7T2 = 5%1-052ofofPage No. 10F(1,1)==6%P.=100=94.34a+1.00Date//M25.20HPR =94.34-l=47.90-7f=els)Spot & Forward Rate RelationshipsReturn on a bond over one yr is always equal tothetoday's forward Curveone - yr Rf late if Spot rates develop as predicted byAn activity portfolio manager will try to outperform themarket by predicting how the future spot rates willdiffer from those predicted by the Current forward curveIf the future spot rates are below the current forwardrates, thethan the one-yr Rf fate.folio manager will earn a return greaterAssumptions Concerning the evolution of Spot ratesinrelationto forward rate implicit in active bondportfolio managementForwalFuture stops spot curve below Current forward Rate5prBond is undervalue (active portfolio manager will buy) if,Expected future spot rate < Current forward rateBond is overvalued (active portfolio manager will sell) ifExpected future spot Rate > Current forward rate.SpotForwardSteeping of the yield cuive an manger would reduce the durationoffolio ForwardspotVTMPage No. 11Yom/ParpotForwardDate//upward slopingFintreeDownwardSwap Rate CurveSloppingSwap Rate = Par Rate= Int rate for fixed Rate leg of on int rateSwap rate : Fixed rate in an int rate swapswapSwap markets are highly liquidLIBOR interest / swap ratio rate curve is the most widely usedrate curvebecauseitRiskcommercialreflectsthecreditofbanks&swapmarkets areunregulatedmarketing them comparableacross countriesWhole salebanks use swap CuNes to value theirassets&liabilitiesLIBORRetail bankuseaIn dicatesgoverment spot curve as benchmarkthepremfortimevalueofmoney#spreadRiding or the YieldCurvediffmaturitingRollingIf the yield Curve is upward- sloping + investor willpurchasebondswithholding periodhighermaturity than theirAs time passes and maturity shortens the bond's cashWill flows will be discounted at successively lower yield,.producesuperiorreturnsiftheyieldcurveremainsstableovertheinvestment horizonDisadvantage : IncreasesIntlate risk<=4isF e e s[=5yone Yr" 4%P=100=96.15f(1,1) = 6"bterP2 = 100 =90.710411.05HPR9615-6%190.7 IPage No. 1326th Saturday 2022 SaturdayDate//M.25.3 : The SWAP RATE CURVEReflects - illiquidity & Mispricing in the marketSwap rate Spread : Amt fixed the - rate payer sidee of an intswap exceeds yield of agovermentbond with the same maturityincreaseIt will Increase with an in the creditusually of Swap Spread = Swap fixed Rate - Treasury Yieldspread(Reflects diff in demand & supply)embedded+vein the (efI- Spread (interpolated spread) = Risky bond risk - Swap RateYield Curve: Typically based on Treasury SecuritiesSwap rate Curve : Yield curve based on series of fixed - ratequotes on interest rate swaps ( swap ratesreflects credits risk) Wholesale Bank - Swap curveRetail Bank - Govt bond yield curvePoint : Increasingly preferred as benchmarkSFRT)+=1- Not affected by govt regulationt(1+St)(1+S+)- More comparable across countries- Quotes at more maturities SFR Swap fixed rate fora tenor TYield has quotes to many maturities, while us bond yield curvenumber of maturites.on the run issues trading at only a smallLIBOR It swap curve is the most commonly used int rate cuapprox reflects the default risk of a Commercial bankI-Spread refects only credit & li quidity risk CFA use MRRasgenerectermforreplacementofLLBorPage No. 17Market RefRateDate-TheT-Billareconsidered to be Rf- LIBOR reflects the risk of lending to banksA rising TED Spread suggests banks are increasingly likelyto default on loansArising TED spread = falling liquidityItThewasLIBOR - 015SpreadMRR - 015high in 208Sonia inUKLIBOR = Overnightindexedswap rateEonia for EW015 = OvernightInderedSwapRoughly equals to Fed-Includesminimalcounterpartyriskfund RateLIBOR - 015 spread : Amount by which Libor rate (whichincludes creadit risk) exceeds the015rate (includesonly minimal credit risk)Useful as a measure of Credt risk and an indicationof well being. of bankingsystemConsideredanindicatorofrisk&liquidityofmoneymarket sa minimal Counter partyrisklow LIBOR - as spread is a sigh of high mkt liquidity Page No. 1827th Nov 2023 SundayDate / /Rolling Yield CurveInvestor purchases bond with maturities longerthan investment horizonShorter maturity bonds have lower yields thanlonger maturity bonds in an upward sloping yield CarveAs the bond approaches maturities + it is valued usinglower yield & at high price (i.e it rolls down the yidd CurvGreater the diff bet forward rate & spot rate, and longerthe maturity of the bond, the higher the total returnMaturity Matching : purchasing bonds that have a maturityequal to the investors inut. horizon.Term Structer TheoriesChange in shape of yieldTraditional Theories :1lvlcurve is explained byIn1Unbiased Expectations Theory2 Steepnessorder2Local Excep Expectations Theory3 Curvature3Liquidity Preference Theory4Segmented Markets Theory5Preferred Habitat Theory Page No. 19Forward iates = Expected future spot ratesDate //Investors are expected to be1Unbiased Expectations TheoryRisk neatral,Also Known as pure expectations theory unaffected by Risk uncer taint J&"Investors" expectations determine the Shape of the Prem dony mt existrate term structerForward rate = Expected future Spot rateseLong-term int rate equal the mean of future expectedshort-term rates(geometric)If the yield Curve is upward sloping short -term rate areirve)expected to rise2Local Excepectations Theory Riding the yield culve doesn't workLocal expectations does not say that every maturityStrategyshouldhavethesamereturnoveragiven investment horizonOver longer periods , risk prem existsRisk- neutrality tis preserved for only short -term underlocal expectationsStow Shown not to workshort-holding period returns oflung maturity bonds is higher than short maturity bondsBond maturity does not influence return for short holdingperiodExpected return for every bond is if sate but does not extendthe risk-neulraility assumption to every maturity strategylike pure expectation Page No.21Date//Various market participants only deal in securities ofa particular maturity- Beacuse they are prevented from operating atdiff maturitiesYield determinedby supply & demand5Preferred Habitat TheorySimilar to above theory but recognises that marketparticipants will deviate from their preferred maturityhabitat if Compensated adequatelyAlso proposes that forward rate are expected futurespot rates plus a plem.-Does not sale state that prem is directly relatedto maturity-Investors pref er a particular maturity-Would bewilling to leave preferred maturityhabitatto obtain a lower price Chence higher yield- Preferred habitat theory Can be used to explainalmost any yield curve shopePure expectationBreak - even -Rate : Forward rate that leaves investors indiff betinvesting for the full term of their investment horizon or investingin part of the horizon & Rolling the invest over at the"Break even for ward late for the remainder of the termqlock -in Rate" Forward rates are interpreted as the rate thatCan be locked. -in" for some furue period. Page No 22Date//Modern Term Structer ModelsEquiliblriumArbitrage - freeCox Ingersoll - RossVasicekAssume that securitiesare correctly pricedEconomy has a naturalInt rate volatilitylong run int rate thatlvl is independentHo - leethe short term run rateof lul of shortConverges toterm int rateCalibrated by usingmkt. price to find timedependent drift thatShort term Rate: Linked to uncertaintygenerates current termover monetary policystructerLong Term Rate: Uncertainty related toreal economy & InflationYield Curve ShiftsYieldcurve Shifts can be either simple parallel shifts,or more complexnon parallel shiftsParallel ShiftNonparallel shiftC M change)C Stepness Change)Nonparallel ShiftYieldNew curve(Carvature change)Newforiginal curvecurveoriginaloriginalcurvecurveMaturityMaturityMaturity Page No. 23Date / /Methods of Measuring Yield curve SentitivityIShaping Risk : Sensitivity of a bond's price to change inthe shape of yield Curve.2 G Effective duration: Measures the sensitivity of a bond's price to puralleloptionsusefulshift in yidd (live.3 Key Rate duration: Measures the sensitivity of bond's price to a changein par rate , It captures shaping risk to atimelarge extentn4Sensitivity to parallel A three-factor model that decomposessteepness & Curvature : changes in the yield curve intoMovementchange in lul , steepress & curvature1N90%2LeastMostImp/9vgJIMPEffective DurationEffective durations : measures price risk for smallparallel shift in yield curveProblem :- Most yield Curve shifts have nonparallel charac tersticsSolution :- Use Key rate duration I which measures impact ofnonparallel shifts.Flatting J key rate at the short end of the yield wive makes an upward slopingyieldCurve steeper, Performing the corresponding change in folio value willPage No. 24determine the risk of a steepening yield CarveDate / /Key Rate Duration Change in speafic maturityA more sophisticated method used to quantity pricesensitivity to non-parallel yield curve shifts.Keg rate duration is price Sensitivity to 1% change in asingle pay rate, holding other par rates constantYTMWeighted avy of underlying sput ratehighest weight to last spotSensitivity to level steepness ,and Curvature MovementsDecomposes risk into sensitivity to these categories ofyield curve movementsMost impLevel (AXL): A parallel increase or decrease of int late biSteepness (AXs) : Long maturity int rates increases shortlate decreasetoCurvature (AXc) : Short & long rates increases intermediaterates doesn't changesWe Can then model the change in the Value of ourportfolio as follow:A P a -DL Ax, - DsAXs - Dc A XcP Page No. 25Date / /Sensitivities to changes:levelDL b,SteepnessDs b2CurvatureDc b3*Maturity Structer of Yield CurveVolatilleThe term structer of int rate volatility is the groph ofyield votalility Vs maturity (Risk at diff maturitiesAs shownhere, short - termintratesaremovevolatilethan are long term rates:Short-term rates more volatileIt depicts yidd curve riskMaturityInt rate votality will drive price volatility inafixed - income portfolioThis is especially important when securities haveembedded options (sentivitive to volotility)expectationLong-maturity volatility is associatedwith uncertaintyoftherealeconomyandinflationVolatilityatthexshort - moturityendrefectsrisksregarding monetary policy92091)fintersHob tise 79813 Dis factorPage No. 26=(spotleDate / /Economic FactorsNorminal rate NamultiedTwo - thirds of the variation in short and intermediate -termyields is explained by inflation with the remainingequally explained by economic growth & monetary policyBond Risk Prem = E(R) - Rbexpected holdingPeriod return- E(R) = expected return on a long-term govt bond-Also called duration prem or team premMonetary Policy Effects on yield CurveBearish flatteningopposite of current state Caused by increase in ST rate duringexpansionary times.lighteringBullish steepeningexponsionel yCaused by decrease in ST rates duringrecessiory timeOther Factors1 DeficitsIssuing debtSupplyFiscal Policy : Expersionary Causes increase in yieldDecreasing yield when buget de(ifit fallsMaturity structer : Demand / Supply in market Segmentmarket segmentation greater supply of bond , of long-termmaturity increases the yield in that market sagment.Longer govt dobt maturity Structer predicts greater excess bond retain Greater Domestic & Non - domestic investor demand increases bond prices& reduces the bond risk prem (Bond yield) & Vice versaPage No. 27Pension funds and Insurance cos use long dated govtDate//bondsto match expected future liabilities.Investor demand : Flight to safety during times ofmcrisis leads to a bullish flattening*Portfolio ActionsDurationin RatesInverseDalationRates RelationshipDecrease ( increase) durationinanticipationofrise( fall) in rateslowRisk#Bullet portfolio focusing on the segment expectedtohave adecreaseinyidd.Rotate out of bullet ( & into Barbell ) in expectationofabullishflattingoftheyieldcurveQ0How Key economic factor are used to establish a viewon benchmark rates, spreads & yield curve changesImplied forward rates act as market - neutral referencepoints for fixed income traders.In practice active fixed-income market participantsestablishtheir own views on future int rate developmentandthen position their folio in order to Capitalizeen diff bet" their own rate view & market ConsensusBond Risk Prem / Term Prem :inThe expected excess return of along-term bond is less than that of a similar short -term bondor rate It govtthe period Rf is usually measured wingbonds to capture uncer tainty of default -free rates CFA : Short-term volatility - Monetaly policylong-term voletility - Real GDP & inflationPage No. 28Date//Credit liquidity & other Risks:May indicate increasethe overallRisk prem for a specific bond.Macroeconomic factor influencing bond price & Req returnEconomic Growth Monetary Policy Inflation other factors such as(GDP)(Int Rate)Fiscal policy t Maturitiesstructure of debt &Investor demandShort-Term & IntermediateLong-term Bond Yield Variationsterm bond yield variationsVariations explained byVariations explained byMoretoryPolicyGrowthInflationMonetaryInflation17%GDP33%GDPpolicy16%67%Growth67%3Ox2/3 - -Explained by Inflation1/3 - Roughly equally explained by2/3 - Explained by monetary policy1/3 - largely attributedle to inflationGDP & monetary policy Page No 30Date / /Flattening CurveStepening Curvelong-termlong -termcitishort termShort-TermMaturityMaturityshort-term int rates changemore than long-term int rateLong term int rates change morethan short -term int rate( Flattening )C SteepeningTrader will purchase longe-termbonds & Sell short term bondsTraders will purchases short -termbonds & sell long -term bond sDuration Neutral Portfolio predicts the bond holder fromchanges in the lul of the term mc structerFixed - income investors will long-only investment mandates mayKnownalternate bet" portfolio concentrated in a single maturity /as a Ballet postfolio and those with similar durationthat Combine shat & long maturities t Known as BarbellpostfolioExample, an investor may seek to Capitalized on an expectedbullish flatting of the yield curve by shifting fromBullet to a barbelll position