Fixed Income Portfolio Management Interest rate ...

1 FixedIncomePortfolioManagement Interestratesensitivity, duration,andconvexity passive bondportfoliomanagement Active bondportfoliomanagement Interestrateswaps1 Interestratesensitivity, duration,andconvexity bondprice: Tt 1C 1 y t F 1 y T, wherey YTM whenyieldschange,bondpriceswillchange;th epercentagepricedifferenceswillbelargerw hen: Cis lower(givensameT) Tis larger(givensameC) example( ):YTMT 1 yearT 10yearsT 20years8%CouponBond8%1, , , durationmeasureshow long,onaverage,a bondholdermustwaittoreceivecashpayments a zeromaturinginTperiodshasa durationofT; a couponbondmaturinginTperiodshasduration TsincesomepaymentsarereceivedbeforeT durationis a weightedaveragewheretheweightsaretheprop ortionoftotalbondPVduetothepayments, T t 1tWt whereWt CFt 1 y tP whereCFtdenotescashflow attandPis thepriceofthebond3 durationis relatedtothepricesensitivityofa bondtochangesinyields:P C1 y C 1 y 2 C F 1 y T P y C 1 y 2 2C 1 y 3 T C F 1 y T 1 11 y C1 y 2C 1 y 2 3C 1 y 3 T C F 1 y T DividingbyPgives: P yP 11 y C1 y 2C 1 y 2 T C F 1 y T P 11 y D4 switchingtodiscretechanges: P yP 11 y D PP y1 y D a relatedmeasureis modifiedduration.

2 D D1 y PP D y notethatthisexpressionforthepercentagepr icechangeis anapproximationbasedona first-orderTaylor s seriesexpansion:f x x f x f x x5 lettingP f y , wehaveP y y P y P y y P y y P y P y P y yP y PP P y yP D y D y1 y notethatwecanexpressdurationandmodifiedd urationas:D 1 y P yPD P yP anotherrelatedmeasureis just P y. Forthespecialcasewherethechangeinyieldis 1 basispoint,thisis calledthepricevalueofa basispoint(PVBP).6 ,PVBPis verysimpletocalculate:P0 1 rBDY n360 FP1 1 rBDY 0001 n360 F PVBP P1 P0 0001 n360 F thebasicideaunderlyingD,D , andPVBPis:yP7 wecangeta moreaccurateapproximationbyusinga second-orderexpansion:f x x f x f x x 12 2f x2 x 2 PP P y yP 12 2P y2 y 2P D y 12 convexity y 2whereconvexity 2P y2P Tt 1t t 1 Wt 1 y 2 notethatconvexityfora zeroisT T 1 1 y 2 examplehandout#18 , durationis higherwhenCis , durationis lowerwhenyis , durationis generallyhigherwhenTis higher(exceptforsomedeepdiscountcouponbo nds).

3 PortfolioPcontainingMbondsisDP Mi 1wiDi,wherewiis :ContractDurationZerocouponbondTFlatperp etuity1 yyFlatannuity1 yy T 1 y T 1 Couponbond1 yy 1 y T c y c 1 y T 1 yCouponbondsellingat par1 yy 1 1 1 y T 9 alwayspositive (forbondswithoutoptionfeatures)( ). , convexityis higherwhenCis , convexityis lowerwhenCis lower. notethat,givenD, convexityis valuable:yP10 passive bondportfoliomanagement indexing a bondindex is intendedtotrackbroadmovementsovertimeinf ixedincomesecurities ScotiaMcLeodprovidesfourCanadianbondinde xes:universe,shortterm,midterm,andlongte rm a bondindex portfoliowillhave thesamerisk/returncharacteristicsasthein dex it is basedon inpracticeit is hardtoformbondindex portfoliossince.

4 Therecanbea largenumberofsecuritiesinvolved many securitiesarethinlytraded index compositionchangesfrequently11 thegeneralapproachwhichis usuallyadoptedis thata portfoliocomprisinga representative setofsecurities( )is formedandrebalancedovertimesoastotrackth eindex reasonablyclosely cashflow matching forsomepurposes,it is desirabletoreduceinterestrateriskbelowth ebroadmarket riskthatanindexingstrategyassumes( obligations) thesimplestalternative is cashflow matching,inwhichtheportfoliomanagersimpl ypurchaseszeros(orcouponbonds)oftheappro priatematurityinsufficientamountstomeett heobligations thisis notalwaysfeasibleduetoliquidity/availabi lityconstraints12 immunization considera portfolioV y :V y y V y V y y if V y 0 V y y V y andtheportfoliois saidtobeimmunizedagainstinterestraterisk immunizationcanbedoneonanetworthbasis( )orintermsofatarget date( ,any institutionwitha fixedfutureobligation) immunizationis accomplishedbyequatingthedurationofasset sandliabilities, A DL L examplehandout#213 second-orderexpansion:V y y V y V y y 12 2V y2 y 2 Whathappensif theportfoliois immunized?

5 Since V y 0, 2V y2 0,V y y V y andV y y V y , ineitherdirection! flattermstructureandany ,sotheportfoliomayhave someotherpointsaboutduration/immunizatio n: if thetermstructureis notflat,thentheformulafordurationmustbem odified inparticular, theweightsbecome:Wt CFt 1 yt tP therearealsoextensionsavailabletohandle: non-parallelshiftsinthetermstructure callability defaultrisketc. hedgingwithnon-parallelshiftsintheyieldc urve considera portfolioV n1P1 n2P2. Usingmodifiedduration, V n1D 1P1 y1 n2D 2P2 y2 15 initially, supposetheyieldcurve willonlyshiftina parallelmanner( y1 y2). Thenwecanset V 0 bychoosing:n2 n1D 1P1D 2P2 if y1 y2, thenweneedtospecifya new objective forthehedgingpolicy.

6 A commonobjective is totominimizethevarianceofchangesinV:Var V n1D 1P1 2 Var y1 n2D 2P2 2 Var y2 2n1n2D 1D 2P1P2 Cov y1 y2 take thederivative ofthis( ) andsetequaltozerotoget: Var V n2 2n2 D 2P2 2 Var y2 2n1D 1D 2P1P2 Cov y1 y2 0 n2 n1D 1D 2P1P2 Cov y1 y2 Var y2 16 notethatintheabove oneofthetwo ,supposea traderwishestobelonga 2 yearT-noteandshorta 30yearT- cashfromrepodealer, purchase2 yearnote,postit the30yearbondfromrepodealer, sellit, thecostsinvolvedinclude: thebid/askspreadandaccruedinterestwhenbu ying/sellingtheTreasuryissues therepoborrowingrateexceedstherepolendin grate margin( haircut ) specialreporates(occasionallydemandforpa rticularissuesbecomesveryhigh, )18 Active bondportfoliomanagement aspectsofactive bondmanagementincludeviewsaboutthelevel ofinterestrates,theshapeofthetermstructu re,andthepricingofindividualbonds forportfolioswithinternationalholdings,e xchangeratesmustbeconsideredinadditionto theabove foreachcountry a simpleexample:pricesoflongtermbondsaremo resensitive tochangesinthegenerallevel ofinterestratesthanpricesofshorttermbond s.

7 Aninvestorwhoanticipatesthatrateswillfal lshouldlengthenportfoliomaturitytogenera tecapitalgains,whereasaninvestorwhobelie vesrateswillriseshouldswitchtoshortermat urityinstrumentstoreducepotentialcapital losses19 forecastinginterestrates onesourceofinformationis thecurrenttermstructure(recalltheexpecta tionshypothesis) if bondmarketsare(semi-strongform)efficient ,allthatispubliclyknownaboutfutureratess houldbeimpoundedintotoday s yieldcurve othertechniquesusedrangefromfairlysimple timeseriesforecaststofullscalemacroecono micmodels(recalllinksbetweentermpremiaan dcreditspreadsandeconomiccycles) availableevidencesuggeststhatnoneofthese methodsworksverywell insomecasesa forecastofthedirectionofinterestratesiss ufficient,buteventhisis hardtodoaccuratelyona consistentbasis20 ridingtheyieldcurve if thecurrenttermstructureslopesupandit is expectedtoremainunchanged,thenbuyinglong ertermbondsproduceshigherreturnsovertime duetoadditionalcapitalgainsastheiryields dropovertimety21 example:supposea 5 yearzerohasa yieldof6%(soa priceof$ )anda 10yearzerohasa ( priceof$ ).

8 Buyingthe5 yearzerolocksina returnof6%.If theyieldcurve remainsunchanged,in5 yearsthe10yearzerowillsellfor$ for5yearsgeneratesa returnof 747 26 485 19 1 5 1 9 02%. notethatif theyieldcurve shiftsupward(evenif itssloperemainsthesame),investinginlongt ermbondswillturnouttohave beenaninferiorchoice alsonotethatthisstrategyimplicitlyassume sthattheexpectationshypothesisdoesnothol d22 thebarbellstrategy basicideais toduplicatethedurationofanexistingbondus ingaportfoliowithoneshortermaturitybonda ndonelongermaturitybondinsucha wayastoincreaseconvexity underlyingassumptionis thatyieldcurve shiftsina parallelway examplefromNovember20,1987usingontherunT reasuries:IssueYieldD supposeaninvestorholds$ ,andusedtopurchase$ $ :5 4 1 78 4 6 6 5410 3 96anditsconvexityis.

9 5 4 0 041 4 6 0 56810 0 283 assuminga parallelshiftintheyieldcurve thebarbellwilloutperformthe5-yearnoteasl ongasthecurve shiftsupordown however, if theyieldcurve steepensthebarbellcansubstantiallyunderp erformthe5-yearnote24 securityselection thebasicbondPVequationPV T t 1C 1 yt t F 1 yT Tcanbeusedtoestimatea valueforany givenbond;thiscanbecomparedtomarket pricestoseeif a bondis overvaluedorundervalued many factorscancomplicatethisanalysis,however : capitalgainstaxesincreasetheattractivene ssoflow-couponbonds,whichmayincreasethei rrelative prices embeddedoptions25 liquidity:on-the-runTreasuriestypicallyh ave moreliquidity( narrowerbid-askspread)andsosellat a slightlyhigherpricethanoff-the-runissues aninvestorwithouttheneedforliquiditymayb ebetteroff sellingon-the-runissuesandpurchasingoff- the-runissues supposethata fundowns$100 Mofanon-the-run7 yearT-note,sellingat parwitha yieldof8%.

10 Supposealsothattwo 7 yearoff-the-runissuesareavailable:(i) ;and(ii)a zerowitha thefundmanagerweretoselltheon-the-runiss ueandpurchasea portfoliowith$79,974, $18,183,739ofthezero,thisportfoliowouldh avethesamecashflowsastheon-the-runnote(i fheldfor7years)butwouldonlycost$98,157, bondswaps(portfoliorebalancingstrategies ) :exchangeonebondforanotherwhichis similarintermsofcoupon,maturity, andcreditqualitybutoffersa spreadswap:if aninvestorbelievesthatspreadsbetweentwod ifferenttypesofbonds( )arenotcurrentlyat normallevelsbutwillreverttothem, :if aninvestorbelievesyieldswillfall(rise),h e/shecanswitchtobondsofhigher(lower) :a simpleswitchtohigheryieldbonds(withoutre gardtoany expectationsaboutchangesinyields,spreads ,etc.)