cfa® preparation fixed income

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FIXED INCOME

Reading Number Reading Title Study Session

32 The Term Structure and Interest Rate Dynamics

1233 The Arbitrage-Free Valuation Framework

34 Valuation and Analysis of Bonds with EmbeddedOptions

1335 Credit Analysis Models

36 Credit Default Swaps


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FIXED INCOME

The Term Structure and InterestRate Dynamics

Study Session 12

Reading Number 32


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FI – The Term Structure and Interest Rate Dynamics

LOS32.a:Describerelationships among spotrates,forwardrates,yield tomaturity, expected andrealized returns on bonds,and the shape ofthe yield curve

SPOTRATES Ø Annualizedmarket interest rate for asimplepayment tobereceived inthe future

Ø Normally we usespotrates for government securities togenerate the spotrate curve

Ø Spotratescanbeinterpretedasthe yieds on zero-coupon bonds (sometimes arereferredaszero-coupon rates)

Priceofazero-coupon bond(discount factor)

P" =#

#$&' '

P" =discount factor(price today ofa1$parzero-coupon bond)S" =spotrate (yield tomaturity)T=maturity

Thetermstructureofspotrates (graphofthe spotrateS" versusthematurity T)is known asthe spotyield curveor spotcurve

Shape ofspotcurvechangescontinously with the market prices ofthe bonds


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FORWARDRATES Ø Annualized interest rate onaloantobeinitiated atafutureperiod

Ø The termstructureofforwardrates is called the forwardcurve

Ø Forwardcurvesandspotcurvesaremathematically related (we canderiveone from the other)

f(j, k) =the annualized interest rate applicable on ak-year loanstarting injyears

F(/,0) =#

#$1(/,0) 2

F(/,0) =forwardprice ofa$1parzero-coupon bondmaturing attimej+k delivered attime j(discount factorassociatedwiththe forwardrate)

t =0 t =j t =j+k

k


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THESPOTRATEFORAGIVENMATURITYCANBEEXPRESSEDASAGEOMETRICAVERAGEOFTHEONEPERIODSPOTRATEANDASERIESOFONEPERIODFORWARDRATES

S0 3 =S# xf(1,1) x f(2, 1) xf(3,1) x …...................x f(k− 1, 1)

S# f(2, 1)f(1, 1) f(3, 1)

t =0 t =1 t =kt =2 t =3 t =4 t =k- 1

f(𝑘 − 1, 1)….........................................................................................

S0 3


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YIELDTOMATURITY(YTM) Ø Is the yield tomaturity ofabondpurchased atmarket price

Ø If the spotrate curveis not flat,YTMwill not bethe same asthe spotrate

EXPECTEDRETURNONBOND Ø Ex-anteholdingperiod return that abondinvestor expect toearn

Ø Will beequal tothe bond´syield only when:

1. The bondis held tomaturity

2. All payments (coupon andprincipal)aremade intimeandinfull

3. All coupons arereinvested atthe originalYTM

REALIZEDRETURNONBOND Ø Actualreturn that the investor experiences over the investment´sholdingperiod


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Ifspot rate curvewould beflat(𝑆#=𝑆:=𝑆;)

YTM=SpotRate


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LOS32.b:Describethe forwardpricing and forwardrate models andcalculate forwardandspotprices and rates using those models

FORWARDPRICINGMODEL Forwardpricing model values forwardcontractsbased on arbitrage - freepricing

P(/$0) =P/ x F(/,0) F(/,0) =<(=>2)<=

t =0 t =j t =j+k

P/ F(/,0)

P(/$0)

Remember: P/ =#

#$&== F(/,0) =

##$1(/,0) 2and

If there is noarbitrage then

Knowing the spotcurve,we cancalculate any forwardprice(considering noarbitrage)


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t =0 t =2 t =5

P: F(:,;)

P(?)

P/ =#

#$&==

F(/,0) =<(=>2)<=

F(:,;) =<@<A


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FORWARDRATEMODEL Relatesforwardandspotrates

1 + 𝑓(/,0)3=

#$&(=>2)(=>2)

(#$&=)=

t =0 t =j t =j+k

S/ 𝑓(/,0)

S(/$0)

1 + S(/$0)(/$0)

=(1 + S/)/ x 1 + 𝑓(/,0)3

This equations derivefrom the equations ofthe previous slide

Ifthe yield curveis upward sloping S(/$0) >S/ 𝑓(/,0) >S(/$0)



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t =0 t =2 t =5

S: f(:,;)

S?

1 + 𝑓(/,0)3=

#$&(=>2)(=>2)

(#$&=)=

1 + 𝑓(:,;);=

# $&@@

(#$&A)A


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LOS32.c:Describehow zero-coupon rates (spotrates)may beobtained from the parcurveby bootstrapping

PARRATE Ø Yield tomaturityofabondtradingatpar

Ø Parrates for bonds with differentmaturities makeupthe parrate curve

(by definition,parrate =coupon rate ofthe bond)

By bootstrapping,spotsrates (or zero-coupon rates)canbederived from the parcurve


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LOS32.c:Describehow zero-coupon rates (spotrates)may beobtained from the parcurveby bootstrapping

1.- The one year spot rate =one year parrate 𝑺𝟏 =1,00%

2.- We canvalue the 2years bondusing parrates: 100 = #,:?(#,G#:?)

+ #G#,:?(#,G#:? )A

Alternatively,we canalso value the twoyearsbond using spotrates 100 = #,:?

(#$HI)+ #G#,:?

(#$HA)Aintroducing𝑆# =1,00%we get 𝑺𝟐 =1,252%

3.- Similarly for the three years bond: 100 = #,?G(#$HI)

+ #,?G(#$HA)A

+ #G#,?G(#$HK)K

introducing𝑆# and𝑆: weget 𝑺𝟑 =1,51%


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LOS32.d:Describethe assumptions concerning the evolution ofspotrates inrelation toforwardrates implicit inactivebondportfoliomanagement

Upward sloping spotcurve

Downward sloping spotcurve

Forwardcurvewill beabove spotcurve

Forwardcurvewill bebelow spotcurve

Upward sloping spotcurveDownward sloping spotcurve

Forwardrates rises asjincreasesForwardrates declinesasjincreases

Ex.SpotandforwardcurvesasofJuly 2013

1 + S" M =(1+S# ) x(1+f(1,1)) x(1+f(2,1)) x…...................x(1+f(T − 1, 1))

1+S#

0 1 T2 3 T- 1

(1+f(T− 1,1))….................................... ....... ...

1+S" M

(1+f(1,1)) (1+f(2,1))

From the forwardratemodel:

Spotrate for along maturity equal the geometricmeanoftheone period spotrate andaseriesofone-year forwardrates

RELATIONSHIPBETWEENSPOTANDFORWARDRATES


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FORWARDPRICEEVOLUTION

Ø If the future spotratesactuallyevolve asforecastedby the forwardcurve,the forwardprice will remain unchanged

Ø Therefore,achange inthe forwardprice,indicatesthat the future spotrate(s)did not conform the forwardcurve

Whenspotratesturn out tobe OPQRSTUVTRS

thanimplied by the forwardcurveforwardprice will UWXSRYZR[RXSRYZR

Atrader expecting lower future spotrates (than impliedby the current forwardrates)would purchase the forwardcontract toprofitfrom its operation

An activeportfoliomanagerwill try tooutperform the overall bondmarket by predicting how the future spotrates will differ fromthose predicted by the current forwardcurve

“For abondinvestor,the return ofabondover one year horizon (independently from the maturityofthe bond)isalwaysequal tothe one year risk freerate if the spotratesevolve aspredictedby today´sforwardcurve. If not,the

returnover the one year period will differ depending on the bond´smaturity”


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A)We calculate the predicted forwardrates 𝒇(𝟏,𝟏) and𝒇(𝟏,𝟐) based on the actual spot rates (values ofthe forward curve)

1 + 𝑓(/,0)3=

#$&(=>2)(=>2)

(#$&=)=

1 + 𝑓(#,#)#=

#$&AA

(#$&I)I= # ,G]

A

(#,G;)I𝑓(# ,# ) =0,0501

1 + 𝑓(#,:):=

# $&KK

(#$&I)I= # ,G?

K

(# ,G;)I𝑓(# ,: ) =0,0601

Aswe get the same values for the predicted forwardrates than the spot rates after one year,we canconclude that the spot rates have evolved as

predicted by forward rates

B)Now we calculate the one year holding period return ofeach bond

1.- The price ofaone year zero coupon bond given by the one year spot rate of3%is P" = ##$&' ' = #

#$G,G; I =0,9709

After one year the bond is atmaturity andpays $1,then the holding period return = #G,^_G^

- 1=3%


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B)Now we calculate the one year holding period return ofeach bond (continue)

2.- The price ofatwo years zero coupon bond given by the two years spotrate of4%is P" = ##$&' ' = #

#$G,G] A =0,9246

After one year the bond will have one year remaining tomaturity, andbased on aone year expected spotrate of5,01%, the bond´s price will be#

#$G,G?G# I =0,9523

Then, the holding period return =G,^?:;G,^:]` - 1=3%

3.- The price ofathree years zero coupon bond given by the three years spot rate of5%is P" = ##$&' ' = #

#$G,G? K =0,8638

After one year the bond will have two years remaining tomaturity, andbased on atwo years expected spot rate of6,01%, the bond´s price will be#

#$G,G`G# A =0,8898

Then, the holding period return =G,aa^aG,a`;a - 1=3%

Hence, regardless ofthe maturity ofthe bond, the one year holdingperiod return will bethe one year spotrate asthe spotrates haveevolved consistent with the forwardcurve (asit existed when the trade was initiated)


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LOS32.e:Describethe strategy ofriding the yield curve

Ø The most straightforwardstrategy for abondinvestor isMaturityMatching purchasing bondwith the samematurity tothe investor´sinvestment horizon

Ø However,with an upward sloping interest rate termstructure,investorscould obtain superiorreturn by“riding the curveyield”(also known asrolling down the curveyield)

Under this strategy,investor purchase bonds with longer maturities than the investment horizon

If theyield curveremains unchanged during the investment period riding the yield curvewill producehigher returns via two effects:

1.- getting moreyield by purchasing abondwith longer maturity (longermaturity bonds have higher yield when term structure is upward sloping)

2.- when selling the bond,we also obtain acapitalprofit

However,this strategy increases interest risk


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LOS32.e:Describethe strategy ofriding the yield curve

Example ofriding the yield curve

Given this yield curve,andinvestor withone year horizon could:

1.- Purchase aone year maturity bondandget areturn of6%(maturitymatching)

2.- Follow ariding the yield curvestrategyby purchasing a2years maturity bondandselling it after one year.If we dothat,the totalreturn will be:

(a)7%return for the first year +(b)the benefit ofselling the bondafter one year

Supposing the parvalue ofthe bond is $1000:

(a) 7%return for the first year =$70(7%)

(b) the benefit ofselling the bondafter one year (*)=1.009,43 - 1.000=$9,43Totalreturn =$79,43(7,943%)

(*) Afterone year (considering that the yield curveremains the same),the value of

the bondwill be:#G_G#,G`

=1.009,43 (after one year the interest rate for aone year bondwill be6%while mybondhasayield of7%)

Int rate

Years1 2

6%7%

0


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LOS32.f:Explain the swaprate curveandwhy andhow market participants useit invaluation

Ø Inaplain vanilla interest rate swap• One party makes payments on afixed rate• Counterparty makes payments based on floating rate

Ø The fixed rate inan interest rate swapis called swapfixed rate or swaprate

Ø Swaprates versusmaturities swaprate curve

Ø Market participants prefer swaprate curveasabenchmark interest rate curverather than agovernment bondyield curve for the followingreasons:

• Swapsrates reflect credit risk ofcommercial banks rather than the credit risk ofgovernments

• Swapmarket is not regulated by governments,which makes swaprates indifferent countries morecomparable

• Swapcurvehasyield qoutes atmany maturities

Ø The SFR" (swapfixed rate for tenorT)canbecomputed using relevant (LIBORnormally)spotrate curveas: cSFR"

(1 + Sd)d+

1(1 + S" )"

"

de#

=1

Inthe equation,SFRcanbethought ofasthe coupon rate ofa$1parvalue bondgiven the underlying spot rate curve


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LOS32.f:Explain the swaprate curveandwhy andhow market participants useit invaluation

Ø For aone year swapcontract SFRis 3%Ø For atwo years swapcontract SFRis 3,98%Ø For athree years swapcontract SFRis 4,93%

The SFRis the fixed interest rate that one partymust pay,while the counterparty pays the LIBOR

(underlying)interest rate

cSFR"

(1 +Sd)d+

1(1+S")"

"

de#=1

SFRcanbethought asthe coupon ofa$1parvalue bond given the underlyingspot rate curve(LIBOR normally)


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LOS32.g:Calculate and interpret the swapspreadfor agiven maturity

Ø SWAPSPREAD Amount by which the swaprate exceedsthe yield ofagovernment bondwith the same maturity

swapspreadd =swaprated - Treasuryyieldd

The LIBORswapcurveis the most commonly used interest rate curve,it roughly reflects the defaultrisk ofacommercial bank

Ø I- SPREAD I– spreadfor acredit-risky bondis the amount by which the yield ofthe risky bondexceeds theswaprate for the samematurity

I − spreadd =yieldoftheriskybondd - swaprated

Investors useI-spreadtoseparate the timevalue portion ofabondyield from the risk premiafor credit andliquidity (i.e.While abondyieldreflects timevalue aswell ascompensation for credit and liquidity risk,I-spreadonly reflects compensation for credit and liquidity risk)

The higher the I-spread, the higher the compensation for credit and liquidity riskFor adefaultfreebond,I-spreadprovides an indication ofliquidity risk


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LOS32.h:Describethe Z-spread

Ø Z- SPREAD The spreadthatwhen added toeach spotrate on the yield curvemakesthe present value ofabond´scashflows equal tothe bond´s market price

§ Z-spreadis aspreadover the entire spotcurve

§ Z(comesfrom zerovolatility) Z-spreadassumes interest rate volatility is zero

Example

• OneyearspotrateS#=4%• Twoyears spotrate S: =5%• Market price ofatwoyear bondwith

annual coupon of8%=$104,12

$104,12 = $a(#$G,G]$x)

+ $#Ga(#$G,G?$x)A

Z=0,008

§ Z-spreadis not appropiate tousetovaluebondswith embedded options (aswithout anyinterest rate volatility,options aremeaningless)

Z-spreadis aconstant,andmeasures the spreadthat an investor will receive over the entirely oftheTreasure yield curve it gives arealistic valuation ofasecurity


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1 + S(/$0)(/$0)

=(1 + S/)/ x 1+ 𝑓(/ ,0)3

Note that for calculating spot rates we usethe formula from the forward rate model (seen before)

LOS32.h:Describethe Z-spread


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LOS32.i:Describethe TEDandLIBOR-OISspreads

Ø TEDSPREAD Amount by which the interest rate on loans between banks (formally,three month LIBOR)exceeds the interest rate on shortterm USgovernment debt

TEDspread=(3-monthLIBORrate)- (3-monthT-bill rate)

T-bills areconsidered risk freewhile LIBORreflects the risk oflending tocommercial banks

TEDspreadis seen asan indication ofthe risk ofinterbak loans

Ø LIBOR-OISSPREAD Amount by which the LIBORrate (which includes credit risk)exceeds the OISrate (whichincludes only minimal credit risk)

OIS(overnightindexed swaprate)rate roughly reflects the federalfunds rate andincludes minimal counterparty risk

LIBOR-OISspreadis auseful measure ofcredit risk andofthe overallwellbeing ofthebanking system


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LOS32.j:Explain traditional theories ofthe term structure ofinterest rates anddescribethe implications ofeach theory for forwardrates and the shape ofthe yield curve

1. Unbiased Expectations Theory

2. LocalExpectations Theory

3. Liquidity Preference Theory

4. Segmented Markets Theory

5. Preferred Habitat Theory


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1.- Unbiased Expectations Theory

§ Investorexpectations determinethe shape ofthe interest rate termstructure

§ Under this theory invest on a5year bond=invest ina2years bondandthen invest ina3yearsbond

§ Also known asunbiased expectationstheory or pure expectationstheory

§ Forwardratesarean unbiased predictoroffuture spotrates

§ The underlying principle ofthis theory is risk neutrality.“investors donot demand arisk premiumfor maturity strategiesthat differ from their investment horizon”

Unbiased expectations Theory says that if one year spot rate is 5%andtwo yearsspot rate is 7%then the one year forward rate inone yearmust be9%

It would bethe same toinvest inatwo years bond than invest first inaone yearbond an then in another year bond (risk neutrality)

Ex

Today 2years1year

5%

7%7%

9%


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2.- LocalExpectations Theory

§ Equal tounbiased expectationstheory but only inthe shortterm:“risk neutrality is assumed only for shortholdingperiodsbut risk premiums should exist for long termperiods”

§ This implies that over shorttimeperiods, every bondshould earn the risk freerate

§ This Theory does not hold asshort-holding-period returns oflong-maturity bonds canbeshown tobehigher than short-holding-period returnon short-maturitybonds

3.- Liquidity Preference Theory

§ Investors demand aliquidity premium that is positively relatedtoabond´smaturity (tocompensate investrors for exposuretointerest rate risk)

§ Forwardratesreflects investor expectations offuture spotratesplusaliquidity premium

§ Liquidity preference theory statesthat forwardratesarebiased estimatesofthe market’s expectationoffuture ratesbecause they include aliquidity premium


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4.- Segment Market Theory

§ The yield curveis determined by the preferencesofborrowers andlenders

§ Yield ateachmaturity is independent ofthe yields ofother maturities

§ Yield is determinedby supply anddemand

5.- Preferred Habitat Theory

§ Forwardrates =expectedspotrate +premium (not directly relatedwith maturity)

§ An imbalance betweensupply anddemand for funds inagivenmaturity rangewill inducelenders andborrowers toshift from their preferredhabitats (maturity range)toother if theyarecompensated adequately

§ Differentmarket participantsonly deal with securities atdifferent maturities


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LOS32.k:Describemodern term structure models andhow they areused

“Moderntermstructuremodels attempt tocapturethe statistical propertiesofinterest ratemovements”

1)EquilibriumTerm EstructureModels

1.1)The Cox-Ingersoll-RossModel

1.2)The Vasicek Model

2)Arbitrage-FreeModels

2.1)Ho-LeeModel


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1)Equilibrium Term EstructureModels

Attempt todescribechanges inthe term structure through the useoffundamentaleconomic variablesthat driveinterest rates

1.1)The Cox-Ingersoll-RossModel(CIRmodel)

Interest rates aredriven by individuals choosing between consumption today versusinvesting andconsuming atalater time

Assumes that the economy hasalong-runinterest rate (b)towhich the short-termrate converges

dr =a(b-r)dt +𝜎 𝑟 dz• a(b-r)dt :this term forces the interest rate tomeanrevert toward the long-run value (b)ataspeed

determined by the parameter (a)

• 𝜎 𝑟dz :indicates that volatility increases with interest rate (athigh interest rates, the amount ofperiod-over-period fluctuation inrates is also high)

1.2)The Vasicek Model Like CIR,Vacisek model suggest that interest rates aremeanreverting tosome long-runvalue

dr =a(b-r)dt +𝜎 dz• Volatility inthis model does not increase asthe level ofinterest rates increases (no interest rate (r)

term appears inthe second term σ dz)

• The main disadvantage ofthis model is that the model does not force interest rate tobenon-negative


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2)Arbitrage-FreeModels

Begins withobserved market prices andthe assumption that securities arecorrectly priced,andthe model is calibrated tovalue such securitiesconsistent with their market price

This model does not trytojustify the current yield curve,rather, they take this curveasgiven

2.1)The Ho-LeeModel The model assumes that changes inthe yield curveareconsistentwith ano-arbitragecondition

d𝑟{ =𝜃{dt +𝜎dz• 𝜃{:time dependent drift (deviation) term• This model usesthe market prices tofind the time depent drift term (𝜃{)that generates the current term

structure


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LOS32.l:Explain how abond´sexposure toeach ofthe factors driving the yield curvecanbemeasured andhow these exposures canbeused tomanage yieldcurverisks

Yield curverisk • Risk tothe value ofabondportfoliodue tounexpected changes inthe yield curve

• Yield curvesensitivity canbemeasured by effective duration,key ratedurationor athree factormodel (level,steepness andcurvature)

Ø EffectiveDuration Measures the sensitivity ofabond´s price toparallel shifts inthe yield curve

Ø KeyRateDuration Measures bond price sensitivity toachange inaspecific spotrate keeping everithingelse constant

Numerically is defined asthe change inthe value ofabondportfolioinresponseofa1%change inthe corresponding key rate,holdingall other ratesconstant

∆<<

≈ -D# ∆r# - D: ∆r: -D; ∆r;…D� =key rate duration formaturity i(aredatagivenby the model topredict changes inportfolioprice Pwhile changes inkey rates r#,r:,…)


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LOS32.l:Explain how abond´sexposure toeach ofthe factors driving the yield curvecanbemeasured andhow these exposures canbeused tomanage yieldcurverisks

Ø Sensitive toparallel,steepnessandcurvaturemovements

Measures sensitivity tothreedistinct categoriesofchanges inthe shapeofthebenchmark curve

§ Level(∆𝑋�) Aparallel increase or decrease ofinterest rates

§ Steepness(∆𝑋H) Longterm interest rates increase while shorttermrates decrease

§ Curvature (∆𝑋�) Shortandlong term interest rates increase whileintermediate rates donot change

We canmodel the change inthe value ofthe portfolio: ∆<<

≈ -D� ∆X� - D&∆X& -D� ∆X�D�,D&,D� :arethe portfoliosensitivities tochangesinthe yield curve´s level,steepness andcurvature


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LOS32.m:Explain the maturity structure ofyield volatilities andtheir effect on price volatility

The term structureofinterest rate volatility is the graphofyield volatility versusmaturityIt provides an indication ofyield curverisk

Ø Shortterm interest ratesaregenerallymorevolatile than arelong term rates

Ø Volatility atlong term interest rates is relatedtouncertainty regarding the realeconomy andinflation

Ø Volatility atshortterm interest ratesreflects risks regardingmonetarypolicy

Ø Fixed income instruments with embedded options canbespecially sensitive tointerest rate volatility

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