FixedIncomeSecurities_2

7/30/2019 FixedIncomeSecurities_2

1/29

1 . 0 1Recitation

2a:FixedIncomeSecurities


2/29

LearningObjectivesoSpot/forwardinterestratesoYTMandbondpricing

xamp esoSpot/forwardoYTMand riceoRateofreturn

2010/YichuanLiu 2


3/29

Review:spot/forwardinterestratesForwardrate(ft,T)istheinterestratefortheperiod

(t,T)determinedattime0.Noarbitrageimplies(1+rt)t x(1+ft,T)(Tt)=(1+rT)T.

rt ft Tt=0 t T

$1 $(1+ft,T)(Tt)$1 $(1+rt)t$1 $(1+rT)T

2010/YichuanLiu 3


4/29

Review:zerocouponbondzero

coupon(purediscount)bonds.

Wecancalculatertgiventhepriceofatperiodzerocouponbond:

1

FV FV t t 1 t

(FV=facevalue)AfterwefindrtandrT,wecancalculateft,T.2010/YichuanLiu

Prt

4


5/29

Review:couponbondP or P T Ct t FV T T Ct t FV T

yistheyieldtomaturity(oryield).ItisequaltotheoitisboughtnowatpricePandheldtomaturity,andoallcouponsarereinvestedatratey.

y snotaspotrate.ThereisaonetoonemappingbetweenyandP.2010/YichuanLiu 5


6/29

Example1:spotandforwardrates

Maturity(yrs) Couponrate(%) YTM(%)1 0 5.25 2 5 5.503 6 6.00

a are epr ceso e1year,2yearan 3yeara.noteswithfacevalue=$100?

b. Whatarethespotinterestratesforyear1,2and3?c. Whatistheimpliedforwardrateforyear2toyear3?

2010/YichuanLiu 6


7/29

Example1:spotandforwardratesa. 100P1 $95.01

15.25%1055

P $99.08215.50% 15.50%2

100Pb. r1 5.2500%; r2 5.5063%; r3 6.0359%

1rc. f2,3 1r23 2 17.1032%

2010/YichuanLiu 7


8/29

Example2:YTMandprice (facevalue=$100,annualcouponpayments)iftheyieldtomaturityiso4%?o5%?

Whenisthepriceabove/at/belowpar?

2010/YichuanLiu 8


9/29

9 4

Example2:YTMandpriceAnswer:

TFV

r

FVPFV, r,y t Tt1 1y 1yr 1 1

FVy

1

1yT

1yT .

o4%:$108.11o5%:$100.00o6%:$ 2.6

Priceisabove/at/belowparwhenYTMislowerthan/equalto/higherthanthecouponrate.

2010/YichuanLiu 9


10/29

3 5

Example3:RateofReturn Su osethat oubou hta2 earSTRIP facevalue=$100 a

yearago,

and

the

interest

rates

at

the

time

were

as

follows:

Years Spotrate(%)

1 2.5 2 3

oYousellyourSTRIPrightnow,andtheyieldcurvehappenstobethesameasayearago.Whatistheannua ize returnonyourinvestment?

oWhatistheannualizedreturnifyousellitnextyear?2010/YichuanLiu 10


11/29

Example3:RateofReturnoPurchaseprice=100/(1.03)2=$94.26oCurrentprice=100/1.025=$97.56

e now:rea ze re urn=3.5024 peryearoSellnextyear:return=3%peryear(forsure)

2010/YichuanLiu 11


12/29

Example3:RateofReturn(revisited)ten

yeartreasurybond(facevalue=$100,annualcouponpayments)atayieldof3.5%peryear.

Sincethen,youhavedepositedthecouponsinabankat2%peryear. o ayyouse e on a ay e o 5 peryear.

Whatistheannualizedreturnonyourinvestment?

2010/YichuanLiu 12


13/29

Example3:RateofReturn(revisited)oCumulativevalueofdepositedcoupons=31.22oSellingpricetoday=104.33

o a payo =135.55oPurchaseprice=120.79oAnnualizedrealizedreturn=2.3328%

Followupquestion:oWhyistherealizedreturnsolow?

2010/YichuanLiu 13


14/29

1 . 0 1Recitation2b:FixedIncomeSecurities


15/29

LearningObjectivesoBondarbitrageoDuration/convexitymmun za on

ExamplesoDurationoBondarbitrageoTrue/false

2010/Yichuan

Liu

2


16/29

Review:bondarbitrage

tothePVofpaymentsdiscountedatthespotratesCaveats:othebondmusthavethesameriskcharacteristicsasthe

securitiesfromwhichthespotratesarederived. .

oeachcouponpaymentcanbematchedexactlybyaspotrate;

w

.

Generalstrategy:oBu low sellhi h

2010/Yichuan

Liu

3


17/29

Review:bondarbitrageoScaleavailablepayoffstreamssothatthenetcashflowatt=1,2,isexactlyzero.

us es gnsso a epayo a =0 spos ve.Multiplier

AmB

AssetB

t=0 A AmBPB

t=1A A1

mACA1

t=2 A A1

mBCB2

t=TA AT

mBCBT

mN+ N mNPN0

mNCN10 mNCN20 mNCNT0

2010/Yichuan

Liu

4


18/29

Review:bondarbitrageoArbitragestrategyisnotunique.oGivenanarbitragestrategy(mA,mB,,mN)withprofit0, mA, mB,, mN sa soanar rages ra egywithprofitk0.

oAstrategywherecashflowsatt=0,1,Tareallzeroexceptatt=s>0(whenitispositive)isalsoanarbitrage.oForthepurposeofthiscourse,weonlyconsiderthetype

2010/Yichuan

Liu

5


19/29

Review:duration/convexity

exposuretointerestraterisk:D T

t T ; DP t1 1y 1y 1y

Since MD , asmallchange(y)inYTMP y

P PMD y.Theformulaisnotaccurateforlar echan esin .2010

/Yichuan

Liu

6


20/29

Review:duration/convexityothesecondderivativeofP(y);oameasureofcurvatureofP(y);

esens v yo e ura on oac ange n ey e .1 2P

CX y Abetterapproximation:

PyMDy2 CX.P 2

2010/Yichuan

Liu

7


21/29

=

Immunizationand(1w)onassetYis[w D(X)+(1w) D(Y)].

Institutionssuchasbanks,pensionfundsandinsurancecompaniesarehighlyexposedtointerestrate

fluctuations.

They

would

like

to

insure

or

mmun zeaga ns suc uc ua ons.

Solution:structurethebalancesheetsothat.

Continuousrebalancingisrequiredforperfectimmunization.

2010/Yichuan

Liu

8


22/29

Example1:duration thatpaysanannualcouponof8%.Assumespotratesareflatat5%.a. Findthebondspriceandduration.b. Supposethat10yryieldsincreaseby10bps.Calculate

formulaandthenusingthedurationapproximation.

c. Supposenowthat10yryieldsincreaseby200bps.y u

u

.

2010/Yichuan

Liu

9


23/29

Example1:duration8 8 108

a. P ... $123.161.05 1.052 1.0510

D 2 10 7.54123.16 1.05 1.05 1.05

b. Actualnewprice=$122.28.D 7.54

P P y 123.16 0.001 $0.881y 1.05

. . newc. Actualnewprice= $107.02

Newpriceusingdurationapproximation=$105.472010/YichuanLiu 10
http:///reader/full/$122.28http:///reader/full/$122.28http:///reader/full/$122.28http:///reader/full/$122.28


24/29

= = = =

Example2:bondarbitragerisklessbonds:

A 97 100B 92 100C 87 100D 102 5 5 105

2010/YichuanLiu 11


25/29

=

5

Example2:bondarbitrage

Multiplier Asset t=0 t=1 t=2 t=3x A 97x 100xy B 92y 100yz C 87z 100z

0 0 0

. y = 0.05wz =1.0 w2010/YichuanLiu 12


26/29

Example2:bondarbitrage=

= 1.2wSetw= 1Arbitragestrategy:oLong0.05A

. oLong1.05CoShort1D

Pro it=1.22010/YichuanLiu 13


27/29

Example3:trueorfalseo

Investors

expect

higher

returns

on

long

term

bonds

than

shorttermbondsbecausetheyareriskier.Thus,theterms ruc ureo n eres ra es sa waysupwar s op ng.

oToreduceinterestraterisk,anoverfundedpensionfund,i.e.,afundwithmoreassetsthanliabilities,shouldinvestinassetswithlongerdurationthanitsliabilities.

2010/YichuanLiu 14


28/29

Example3:trueorfalseo

False.

The

term

structure

depends

on

the

expected

path

ofinterestrates(amongotherfactors).Forexample,ifn eres ra esareexpec e o a , e erms ruc urew

bedownwardsloping.oFalse.Tominimizeinterestraterisks,wewantMD(A)xV(A)MD(L)xV(L)=0.IfV(A)>V(L),wewant

MD(A)


29/29

MIT OpenCourseWare

http://ocw.mit.edu

15.401 Finance Theory IFall 2008

For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
http://ocw.mit.edu/http://ocw.mit.edu/termshttp://ocw.mit.edu/termshttp://ocw.mit.edu/

FixedIncomeSecurities_2

Documents

Transcript of FixedIncomeSecurities_2