Reading 27: Valuation and Analysis of Bonds with Emdedded Options

University:
California State University, Northridge
Course:
ACCT 230 | Introduction to Managerial Accounting
Academic year:

2023
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463

Pages:

35
Author:

SophGr

Page No. 7747Nov' 2023 ThuesdayDate / /Heeding 27: Valuation Andanalysis of Bonds WithEmbedded Options* Types of Embedded OptionsI Simple optionsI Complex options,Simple OptionsCallable bonds 1SlakoutPutable bonds $Per is10Issue has right to callInvestor has right to sellback the bondthe bond back to theissuerInvestor has shortposition on call optionInvestor has long position onMost Callable bonds have aput optionlockeat period (period duringOptions (un be European ,which the issuer Cannot Call the bondAmerican , BermudanOption can be European t AmericanBermudian Page No. 49Date//Valuing Embedded OptionsEmbedded Calls :Investors get a disc equal to call value :longVcall = Vstraight - VallableCallable = Straight - CallshortEmbedded Puts :Investors pay a prem equal to put value:Vput = Vistable -straightputable = Staaight t putshort longCB = OFB - CallCall = OFB -CBPB = OFB + PutPut = PB - OFBValuation of Bonds with embedded OptionsUsual backward induction process in a binomial treebe the call price (put price) if the computed value exceedsValue of Gall any node where the bond is callable (putable) must(is below) the call price (put price) b = LN (upper nodelower node)Page No. 50Date//2Assuming the option Can be exercised.--Value at any node is avg PV of two tve valuesfrom next period or the call price, whichever is lewer(for cullable)celing-Value at any node is avg PV of two possible valuesSfrom next period or the put price, whichever is higherfor putable)flourBackward Induction Methodology: Example7% 2-yr bond Callable In 1yr at parForT1$100To$???$7$ 77.1826 7.???$ 1004.574%$???$ 7$7.5.32107.K$100V1,4=100+7= 99.83$71.071826V2,L = ( 1.05321 100 +7 = 101.59Vo=99-83 + 101.59+7=10321.045749 1.095749for putabe bonoPage No. 51Date//100$99.83$1000 t $7$7103.087.1826$ 102.24$100 + $745744%5.3210%$101.59$ 1004.$ 100+ $7$ 7.0ToT,T2Vcall = Vstraight - Vallable -Vput = Vputable - Vstraight$0.76 = $103 - $ 102.240.08 = $ 103.08 $103Example 2: Find the price of a 2-yr 5% bondcallable in=yrat $ 100, if Coupon =6%Step3%Step 247101.483799.0566 +100)15105 =99.056621.061-03105 = 100-9615104We will replace 100.9015 with 100 asbond will be calledtal 11a) for guloV Corporate bond :- - Path indipendentPage No. 5eMBS / Pool of :- - Path dependentDate//BondsVolatility & Embedded OptionsStraightbondvaluesareNOTaffectedbyvalatilityPoint:EmbeddedoptionsAREaffectedbyvolatility- If - : Ratevolatilityincrease- Then:ValueofembeddedCall / putincreaseValueofcallablebondfallsValue of Call flutValueofputablebondincreasesDecreasesVallable = Vstaight - VcallIncreasesV.IncreasePutable= Vstraight tVputPVt6s=502inCtpCB = OFB-CallTPVioTxe20.50%i,3CtPPB = OFB+PutPVTLevel and Shape of Yield CarveUpsideonacallablelimitedwhenratesdeclineDownsideonaputablelimitedwhenlatesincreaseGale have gone upWhen yieldcurvesteepens,ICall&lowoptionsiswhiletheunlikelytobeexercisedhasvalueputoptionisvaluableValue of Call / PutstepnessVolatility1RR BPJ Page No. 53Date//RJBRIWhentheyieldCurveflattensthecall+optianbecomesmore valuablewhiletheput optionlosesvaluesPOption freebenvalue ofX= 100callcallX= 100Valueof 1011fatePatalowhighYlowhighEffectofintratevolatilitythevalueofcallable/putablebondInt Rate VolatilityValue of call optionValue of put optionInt Rate volatilityEValue of callable bondValue of putable bond(B-C)(B+P)Effect of change in the level and shape of yield curveon the valueStraight of Bond acallable / putable bondInt Rateunderlying asset = Call optionWhen int rateCall option in callable bond limits thebonds upside potentialWhen int ratePut option inputablebond hedge's againstthelossinvaluePB = SBIH + po(t)straightBindUnderlyingPut optionValue of call optionsasthe yward sloping yield curve flattenValue of put optionsastheupwardyiddcurveopingflattchanges in lul ofStraight bond is affected by ^ Int rate A, & not affectedby volatility of int late Page No. 55Date//TOASundervaluedattractiveinvest"volatilityOASforcallablebondOASforputablebondVolatilityValueOASOASallputCallputscallablePutableHighHighHighLowHighLowHighLowLowLowHighLowHighLowExample : A $100 -par , 3-yr, I 6% annual-pay ABC Inc . Callable bondtrades at$99.95 Theunderlyingcalloptionisabermudan-styleoptionthat can be exercisedinoneor two yr at parRates are as followsYear DYear 2Year 22Value using benchmalk3%tree =$101.775.7883%10.78373.8800%7.198174.82507.T2T3(95.72498.8146RT1$95.72$97.65$6$106To$610.738%5.7883%$101.77$ 98.88 + $f3%K3-88%71981$106$161.54.8250 %$100$101.12can't exeed 100$ 100$ 6$1066 12 /10%goesmodel pricePage No. 56ByDate//1% =OASToget Maket priceGoat seaker$ 94.86of $99.95, add 100 bps$95.91$6$ 106$610-7383% +1%5.7883+ 1%$99.5$97.97$ 1063% + M$6$100+719817 + 1%$ 60100.17 $100OAS = 1%$6$6$10633.887. +1%4-8250% + 1%ToT,TzT3*OAS&Volatility IMPTheassumedlvlaffects theCalculatedofvolatilityinbinomialintratetreevalueoftheBothcallandputoptionsunderlyingoptionsincreaseinvaluewhenvolatilityisassumedtobehigherIftheoptionsvalueishigherthe+Callablecomputedvaluebondwillbelowertheofa&putablebendwillbehighervalue ofaCB= OFB - Call0yT PB= OFB tPutNE88F.2Model price18PLF0258001 Black - -scholes modelGood forConvertablePage No. 57-Bad forCallable (putableDate / /Keeping market price of the bond constant , t the highervolatility results in lower computed OAS for a callable bondand higher Computed OAS for a putable bondHence a higher volatility assumption may make a riskycallable bond less attractive & a risky putable bondmore attractiveAssumed lulValue ofValue ofOASCallable Bondof volatilityCall optionstraightbond uneffectedAssumed lulValue ofValue ofOASPutable Bondof voletilityput optionunaffected straightbondStopcurratureEffective Duration & ConvexitypDay +=DAy 7 1/2 C sy2StepsIUsing backward induction CalculateOAS of the bond.2Shock yield by Ay , add OASfrom step 1 & Calculate BV AY & BVsy-3Use formulae to Compute ED & ECYMust come from treeED =BVA- BV.AEC = BV-AY + BV4sy - (2x BVo)2x BV, X Dy 2BV. X Dy2 Page No. 58Mod durInt ratoTJCouponConvexityDate//sensitivityYTMlonger MaturityitEffective duration of a Callable / putable bondEffective duration = V- -Vt2AyV & V+ are calculated using binomial int rate tree flameworkCalculating effective duration & effective convexity for bonds withembedded options is a complicated undertaking because we mustCalculate values of BV1sy & BVStep I:Given assumption about benchmark int rate, int ratevolatility & any calls & lor puts, Calculate the OAS for theissue using the Current market price and the binomia (model12Impose a small parallel shift in the benchmark yield curveby an amt equal to + - AY3Build a new bino minal int rate tree using the new yield (ul VcGAddtheOASfrom1 to each of the one-yo rates in the int ratetree to get a "modified" tree,5Compute BV1sy using this modified int rate tree.6Repeat2to obtain a value of BV Aythrough 5 using a parallel rate Shift of Ayn DulationPage No. 592nd Nov' 2023 FridayDate/DurtaasInt futestraightEDurationBundeffective Div)key Rate Durat15yr bond (callable / Puttable 10thy)Strainght bondLow Coupon BondCallablePutablePar bondPrem DisEarlier KeyFV=100 MV=00BondRates (411,2,5) Low CouponHigh Couponwill haveonly one kid15th yr-ve KRdsUnlikelyLikely to@ yr 15Krd willto bebe calledall othe KRDbe theCalledwill be zeromost Impkid of 10thKRd15thyrwill be mostmost impImpLow CouponHigh Couponlikely to be CallputUnlikely to be calledKid of 15th yr HighestKid of 10th yr Highest Page No. 72Date//Value of Convertible bond:Value of SB + value of Call option on equityValue of Callable bond Convertable bond :Value of SB + value of Call option(aquity Value of Call option onbondValue of Callabble & Puttable Convertable bondValue of SB t V of Call option (Eq) - Value of call option (bund)+Value of put option on bondHazard rate =ST.YrHazard ratePDSurvivalI5r.5%95725%95%x57.=4.75%95% x(1-5%) =40.2535%(1-5)X5%95% x (1-sx)2 = 85.7345%95% - x (1-54) = 81.45* 4.5195% x95%X4-75%Cortelation[PD & Recovy bak is IVC Page No. 74G credit spread ; YTM on risk ZCB -YTM on Rf ZCBDate//8Credit Valuation AdjustmentSum of the PV of expectedloss for each period.CVA = Value of Rf bonds - Value of Risky bondGBOXless Security EAG X LGD% X PO9PV of expected loss: Priced into loanHighest price an investor would pay toan insurer to entirely removethe credit risk ofthe investment.- Using Risk- neutral probabilities f &- Induding YTMPV of expected loss = Value of Rf bond - Value of risky bondPV of expected loss= Expected loss + Rf prem - Time value discPVCVAExample91.5189.36A 2% 3-yr t $100 Its par, ZCB (Corporate) has a hazard rate ofrateper yr. recovery rate is 60%& the benchmarkcurveisflatat37for all40% loss SeventyCalculate the expected exposure, probability of survival probmaturity3t 2of defalt loss given defalt &CVAnsol" the Exposere bond at the end of 3 yr is the face value of($100) Exposure at the end of yr 2 ispresent value of $100 disc for period at 37( = 100/1.03) Exposure at end of yr is the year 1is the PV of desc for 2 period (= 100/1.032) LGD liven Calculate PDPD Given Calculate LGPCredit Risk = LGD x PDPage No. 75Date//spleadPDTRRTLGDLGD = Exposure x (1-0.60)tverelationship given credit spreadFactorDFisthePVof$Year 2DF==0.942596(1.03)2Probability of survival (PS) for yr & 3 are calculated as(1-0.02) = 0.98(1-0.02)2=0.9604(1-0-02)3=0.9412Probability of default is 2% in the first yr & decreaseswiththePSPOforyr2=2%ofPSforyrI=0.02x0.98=0.0196or1.96%SimilarlythePDTforyr3 = 0-02 X 0.4604= 0.019208 or 1.92087.100/1.03407-x9426 (1- hazard rate2% x37.71.03PV of ExpectedYearExposureLGDPSPDExpectedDFLosslossExpected X DF\94.26037.70498%2%0.7540770.9708740.732113100297.087 10338.83596.04%1.96%0.7611650.9425940.7174713100.0040.0094.114%1.9208%0.768320.9151420.703122Total-5.88CVA2.152706Hazar XPsyr2floard rate X 4r1 PSRisk-NeutralProbabilityofDefaultAkaimpliedprobIstheprobofdefaultimpliedInthecurrentmarketpricegiventhemarketprice&hencethecreditspreadIs calculated by assuming a certain recovery rateAssumedrecoveryrate&risk-neutralprobabilityofdefaultaletuely correctatedPSXFV + PDXL60-1)andthendiscto appropriateRisk NeutralPD 8 Recovery Rates Page No. 76Date//market price = 79.61 t I yr ZCB FV = wo 1 r= 31 RRate 407.91.9930%1004040is70% - PS X100 + PD x 04 x 100 =74.61601.03(1+3x)81.998181881998 =41-99-70=30GoExample Expected Rate of Return 60. conditional pichA 3-yr , $100 par, ZCB (corporate) bond has a hazard rateof 2% per yr Given a recovery rate of 60% & a flatbenchmark Curve at 3% € the CVA is $2.15 Exposureat the end of yr 1,2 &3 is $94.26 $ 97.09 & $100respectivelyCalculate the IRR on the invest if the bond defaults ineach of the 3yr as well as if it does not defaultsol Atthe end of yr I t there are two possible outcomes,of exposere of 0.6 x 9426 - $56.56 = , occursdefault or no defalt If default occurs, a recovery of 60%are $58.25 & $60 respectivelySimilary in yr 2 & 3, in case of default the recovery - amtIf default occurs in yr IPV = -89.36 , N=1 , + FV = 56.56 T (PT 1/4 = -36.71%, CVAprice ofloePrice of = 91.51 -2.15Rf1-033Risky = $89.36= 91.51 Page No. 79Date/Credit RatingCaparuc PD 4 CGDUsed for companies + ABS , govt & quasi-govt entities.Invest grade: BBB & above Ordinal ratingSpeculative grade (Junk bonds) : Below BBB(redit rating Summarize the results of credit analysisin one simple metric.Credit rating agencies are under pressure from s toThere is an inherent conflict of int is an issues-pay modelmaintain stability in credit ratingsCredit Sconing / Rating - Oridinal rankingCredit scoring & credit rating models are ordinalrankings that categorize credit quality-But, they provide no info on the degree of diffin quality betn rankingsonCredit scoring is used for small biz & individuals- Higher scores indicate better credit qualityCredit ratings are also issued for corporate debt, asset -backedsecurities, & govt & quasi - govt debt.wider variety of credit useThe HB Flco score in the us factors in the age of account,delin quencies, utilization of lines of credit, types of accountand number of hard inquiries Higher the betterCredit scoreSoft inquires Check CIBILyourself Senior securedissue RatingPage No 80cIusser RatingRating of seniorBBBNotchingDate / /(RIL)unsecured Bondssubordinate unsecured bondsNotchingBBNotching in credit ratings C bet several issues of sameentity) accounts for seniority & hence, LGDCapuring LGD effect , Avaibility of collationApart from a letter grade credit ratings also specificoutlook ( +ve -ve or stable)Notching :-Plactice of lowing the rating by one or more Lulsfor more subordinate debt of the issuer - -it Alc forLGD diff bet" classes of debt by same issuer (longer credit historiesabsence of delinquenciesFICO Scorefewer credit enquirieshigher forwide variety of types of (Tedit usedStrengthsCredit RatingWeaknessSimple to understand &Lower correlation with defaultSummarize complex creditprobabilityanalysisStable over time that reducesThey do not explicity depend onvdatility in the debt marketsbiz does cycle whereas default propabilityIssuer pays model may not provideaccurate rating Mod durTypically itt expected returnYTMBondOProb of Page No.81CreditRatingspeadpricechange Date / /is not Symmetricallymore*Example : Credit Migration2 Tin spread is much larger thatordowndyrs than in spread for HproysCredit Migration - Change in ratingCredit rating transitionBiased toward downgradSuppose a bond with a modified duration of 6.32 getsdown graded from RAA to AA . The typical AAA creditspread spread is 60 bps ( while the typical AA credit spreadis 87 bps;AAAAAmoreriskisCompinsatedbySpreadof27bpsCalculatethe%changeinthepriceofthebond,assumingthebondispricedattypicalspread.Sol' Change in spread = ( 87 -60)= 27 bps = 0.0027% A P = - (modified duration of the bond ) X (D Spread)Expected RetuinF% AP + Coupon=-6.32X0.0027=-0.0171or-1.71%Structural ModelsRb & Volatility are constantAre based on structer of a Company's balance sheet &insights from option priceingShareholders have a call option on the Company's assets(AT) with the face value of debt (K) as the strines price:FV of ZCBValue of stock = max Co, AT K)Value of debt= AT - max ( O. Ar -K)= Value of Asset - value of equity= min (Ar,K) Page No. 82Date/Heathy (1)ATroubled (2)LIRE Equity holderi'AL&EE = 20get thisdebt holderA = 120K =100getE = O Equity holde getdebtholderA = 80 thisK = 100Fnothingget thisEquity call = max (O,A-k)AssetLiabibilityD= = A - E - = A- max (0, A-k)ID= 120 - - max (0,120-100) =100202D=80 - max Co 80 - 100) = 80OTM=0Value of = Value of RfEPRisky debtdebtValue of a put options onthe company's assets ,Value of debt at maturity:if At >K (no default) =KPay off on a putoption (X-ST)if AT FV of debtImp inputs in structualmodels.1Expected return on Co's assets2 Assetreturn volatilitySince Company's assets are not traded above parameters arenun observable Therefore we must use implict assumptionproduces (calibration)A"Invest" is a Risky bond can be viewed as purchasingaRf bond & writing put option on Company assets. Page No. 84Date//for default intensityReduced from Models of corporate credit risk.Can be calculated using regression modelsTreats defaults are external rexogenous rate and vol are sto chasticDoes not explain why supple def(randomoccursexplain whenUses default intensity > Key input (est using regression model)Allows linkage of default intensity, Rf rate & Secoveryrateto the state and of the economy.Prob of defalt & RR dependent (non-constantReduced form model uses historical data (as againstcalibrated data as an input to the model)Basing est on historical data is called hazard rate est.Assumption , strengths and weakness of both structure& reduced from model.1Structural modelsAssumptionsCompany's assets trade in a frictionless markets ArbitracedSimple B/S structurefree modilInterest rates are constant over timeAsset's return volatility is constant over timeStrengthsWeakness AssumptionsProvides an option analogy to understandProbability of default & recoverycompany's pro bability of defaultrate depend on assumed B/3& recovery rate21.226of the companyfromJA Page No. 86Date / /StrengthWeaknessHistorical est Can be usedHazard rate est may not beCredit lisk measures reflect changingappropriate to predict futurebiz cycleThis model does not rey company'sdefaultsB/S to have a Simple capitalDo not explain why default ocursstructureTreats defaults as random suprieDo not assume that Companyassets tradeDefaults intensity is allowed tofluctuate as Company fundamentalchange, as well as with biz cycleCredit Risk Measures2CB bond and a risk -free ZCB.Credit Spread is the diff bet" YTM on a credit riskyCredit Spread = YTM of a credit Risky YTM of a risky -freezero Coupon bondZCBVND = Value assuming no defaultValue given no defout: Value of Risky bond, assuming ifdoes not default - calculate using theRf to value the risky bond Page No. 89Date / /ExpectationofanimpendingrecessionincreasesspreadsInverselyrelatedtorecoveryratetuely related to PBLGDSpreadSpreatIf PDCredit curve would be upward slopingHat creditcurvesin dicatestableexpectationovertimein long in credit quality will improveDeterminants of the termStructure of credit spreadLow Rated bendssteeper upwarsleping but am be inverted0Credit Quality - High lated sectorsflatter term structuresAAA termstructurestendtobeflatNIGor slightlyupwardsploingIG2Financial Condition - Steeper when expecting recessionExponsionNassowSpread narrow during economic explan sions &ContistionWidenwiden during syclical down turns3 Market demand - Spread are affected by more liquid sec& SupplyLess liquid maturities would show higher spreadRecent issue (bias)(eveniftheexpectationsforthattimeLess liquidityspreadperiodareStable) moreresentissuasdominatethespread came4Equity Market -VolutilityIncrease in volatility tend to widen spread &influence the Shape of the credit spreadCurve.Equity marke volatilityCled it Spread()PD(st 'u ctufd model) Page No. 9-9Date//Exposure Yr 1:3+3t103=105.4361.0175'1.01752Exposure Yr2:3+103= 104.2241.0175'LGD Yrl = 30% x 105.436= 31.63|EL Yrl= 1.25% X31.63DFyrl = 1/1.0175= 0.395= 0.983PVEL = 0.395 X 0.983 = 0.389ELDFHazard Rate = 1.5% & RR = 60%1.5%98.50Keep this constantYrExposureLGDPDPSELDFPVEL1105.43642.1741.5%x98.50%0.6330.9830.6222104.22941.6911.4787.97-0237.0.6160.9660.5953103.041.201.455%95.567%0.600.9460.569CVA1.78591.4787.K98.5Example : Changing Expectation1Basedon2assumption (a) I is the bond correctly priced?Calculatethecreditspreadonthebondimpliedinthe3current market priceAssumingthatthemarketpricechangesconsistentwithassumptions(b) ,calculatethenewcredit+spread Credit spread 2/ LGD XPDPage No. 92Date / /Sor : Based on assumptions (a) t , CVA , = 1-12 (given)Calculate VND using benhmark rates:N= 3 , PMT is 3 t 114 1.75 , FV = 100 , CPT PV = 103.62 =VNDBond value = VND - CVA= 103-62 - 1.12= (02-50Given the market price of $102 the bond is undervaluedSol'2:-.Rf YTM = Par rate = 1.75 (given)Using the market price of the bond C $102). theYTMon the bond is calculated as :PV = -102 N= 3, + PMT = 3 i FV =100 ( CPT 114 = 2.3%credit spread = 2.3% - 1.75%= 0.55 %Sol" 3Rf YTM = Par rate = 1.75 % ( given)Based on assumptions (b) CVA = $179 (given)market plice of risky bond = VND - CVA = 103.62 - 1.79= $101.83PV = -101.83 , N = 3 t , PMT = 3 ,FV = 100 CPT 1/4 = 236 7Credit spread = 2.36-1.75= 0.61%in AfricaPage No. 99Date / /ICollateral Pool:Homogeneity refers to similar assets withinthe pool- granularity refers to transperancy of assetswithin the pool. Non granular / discreteless assethighly granularmany assetShort term granular &Statistical based use stats ofhomogeneous structuredApproach avy asset to evaluatefinancial vehicalMedium term granularPortfolio based& homogeneous oblig"approachDiscrete & non granularIndividual loanlevelportfolioloan - by - loan2Service Quality :Investors face the erational &counterparty risk of the servies3Structure :- Determinesthetranching or other mgt of creditCredit EnhancementInternalEx teralDistribution waterfavTranching of credit risks,Third party guarantesexcessive servicing spread etc,OverCollateralion Page No. 95Date//Covered Bond :-Senior secured bonds backed by a Collateralpool as well as by the issuer (i.e covered bondinvestors have recovere right)Originating in Germany , Covered bond have spreadto the rest of Eu t Asia & AustraliaThey are issued by a financial insitutionSerio While the Collateral types vary by jurisdictionCommon forms are commercial and residential mortgages,as well as public sector bonds*PV of Expected lossHighest price an investor would pay to an issuerto entirely remove the credit of the investment.PV of expected loss = Value of Value ofRf bondRisky bondPV of expected loss can be & established from the Credit Spread*Credit Analysis of ABS / Structural model can 4 Reduce beusedABS do not default but they will loss value as defaultsoccur. in the collateral poolTherefore propability of default does not apply to an ABSABS credit stofe risk is modeled using probability of lossloss given default expected loss, & PV of expected lossReduced & Stiuts form can be used as leny as they takecomplexity into account Greedyxi,hilpPage No. 96credit option risk priving & srutedDatehistorical variable such us rationStructural Form modelReduced form1970's - BSM1990'sJarrows Smaleton & othersModel depends on Structer of BS i.eModel capture biz cycleAsset Liability Equityobservable inputsCompany Value modelCompang default SH defaults)ifvariable that occurs randomTreats default as evogeneousValue of debt>Value of assetExplain When default occurs-ve equityModelled using poisson Sto chasticLimitation: Assumesassets areprocessactively tradedinside themodelsCompany Specific + Macro eco dataTreats defauts as endogéneous variableMake use of regressionExplain Why default. occursDont explain "Why" default occurDifficult to implement whenDefault comes as a supricecompany hide debt, off Bls itemsStructuralFormLong CallEq shareCall valueIwill buy the5050C ==5-BtPcall option3030I own asset -liabilityIown that550Pat option X=50E = 50A = 1000B = 50Sipppsi Beloid>FyStpI BtcE = SoA =100Iown thes&Iassetsam willing to sellB = 50aCall option on assetLong BondX = 50Town a Rf bondShortCallF=50lam selling a putoption X=50 on asset5-C= B -Pa