fixed income

7/17/2019 fixed income

http://slidepdf.com/reader/full/fixed-income-568e1dfd2fbc1 1/22

1

Introduction to the

Measurement of InterestRate Risk



2

Interest Rate Risk

Relation between interest rate and bond value

Approaches to measuring interest rate risk 1. Full Valuation Approach

2. Duration/Conveit! Approach

Bond values are inversely proportional to interest rate.Bond values are inversely proportional to interest rate.



"

Full Valuation Approach

It is a process in #hich the position of $ond isevaluated for various scenarios of chan%e ininterest rates.

Step in Full Valuation Approach

1. Identi&cation of the current price and !ield of a$ond or $ond portfolio.

2. Determination of possi$le !ield chan%escenarios.

". Calculate the $ond price for each ne# scenario.

'. Calculate the percenta%e chan%e in the price ofthe $ond or $ond portfolio for each scenario.

Note:Each security in the portfolio is valuedseparately under the full valuationapproach, and portfolio value is

determined as the sum of constituent parts.



'

Full Valuation Approach

ExampleCurrent (osition of $ond) *+ coupon 1, !ear $ond

-option free

Current Market (rice) Rs.12,.0

ield to Maturit!) ,.,+(ar value of $ond) Rs.1

Market Value of the position) Rs.12,2,

Solution



,

(rice Volatilit! Characteristics of 3ption4Free 5onds

Facts• (rice of option free $ond moves

in opposite direction to a chan%e

in the $ond6s !ield.

• 7he relationship is not linear i.e.

not a strai%ht line relationship.

• 7he price4!ield relationship foran! option free $ond is referred as

conve.



8

(rice Volatilit! Characteristics of 3ption4Free 5onds

Properties• (ercenta%e chan%e in price is not the same for all

the $onds for a chan%e in !ield.

• (ercenta%e chan%e in price for a %iven $ond isapproimatel! the same -#hether increase ordecrease in !ield for a small chan%e in !ield.

• For a lar%e chan%e in !ield the percenta%e pricechan%e increase in %reater that the percenta%eprice decrease.



(rice Volatilit! of 5onds #ith :m$edded 3ptions

Components of price of bond with embeddedoption

• 7he price of the same $ond considerin% it to $e anoption free $ond.

• Value of the em$edded option.

!pes of "ptions

• Call Option (Prepay): It is the ri%ht to the issuer to call$ack the issue or repa! the de$t prior to the principalpa!ment date.

• Put Option: It is the ri%ht #ith the investor to demand

for the prepa!ment of the de$t.

Value of bond with embedded option # Valueof option free bond $ Value of the embedded

option


http://slidepdf.com/reader/full/fixed-income-568e1dfd2fbc1 8/22*

5onds #ith Call or (repa! 3ptionsFacts

• A calla$le $ond ehi$its positive conveit! at hi%h!ield levels and negative convexity at lo yield levels.

• ;e%ative conveit! means that for a lar%e chan%e ininterest rates the amount of the price appreciation is

less than the amount of the price depreciation.• <hen the re=uired !ield for the calla$le $ond is hi%her

than its coupon rate the $ond is unlikel! to $e called. 7herefore the calla$le $ond #ill have similarprice/!ield relationship -positive conveit! as a

compara$le option4free $ond.• <hen the re=uired !ield $ecomes lo#er than the

coupon rate the value of the call option increases$ecause it is %ettin% more and more likel! that the

$ond ma! $e retired at the call price.



5onds #ith :m$edded (ut 3ption

Facts

• 5onds #ith puta$le options can $e redeemed $! the$ondholder on the dates and at the put pricementioned in the indenture.

• If $ond value > put price in case the !ield rises put

option ma! %et eercised $! the $ondholder.• If put price is par value and !ield ? coupon rate put

option m! %et eercised.

• Value of puta$le $ond @ value of option free $ond

value of put option.• If the !ield is lo# price of puta$le $ond @ price ofoption free $ond.

• If !ield rises decline in the price of puta$le $ond isless as compare the decline in price of option free

$ond.



Duration

Facts• It is the approimate price sensitivit! to the

chan%e in interest rate.

• It can also $e interpreted as percenta%e chan%e

in price for a 1 $ps chan%e in interest rate.

Calculation of %uration



Duration

Example

Consider a 9+ coupon 1, !ear option free $ond sellin% atRs.10.9122 to !ield 8+. Calculate the duration for a 2,$ps chan%e in interest rateB

Solution&

(rice if !ield declines $! 2, $ps @ Rs.112."'11

(rice if !ield rises $! 2, $ps @ Rs.19.1889

Initial price @ Rs.10.9122

Chan%e in !ield in decimal @ .2,

7herefore

Duration @ -112."'11 19.1889/-2 10.9122 .2,

@ 0.'"+

'nterpretation& For a 1 $ps chan%e in !ield the price ofthe $ond chan%es $! 0.'"+



Approimatin% the percenta%e price chan%eusin% duration

iven the duration of a $ond the percenta%echan%e in the price of a $ond can $e approimatedas)

!pproximate " price change # $ %uration x change

in yield x &''

Example

Consider the 9+ 1,4!ear $ond tradin% atRs.10.9122 #hose duration #e calculated inprevious slide is 0.'"+. For a 1 $asis pointincrease in !ield the approimate percenta%e pricechan%e is

40.'" .1 1 @ 4.0'"+.

;ote) 7he ne%ative si%n $eforeduration is due to the inverserelationship $et#een price

and !ield chan%e


http://slidepdf.com/reader/full/fixed-income-568e1dfd2fbc1 13/221"

Modi&ed Duration

• It is the approimate percenta%e chan%e in a $ondEsprice for a 1 $asis point chan%e in !ield assumin% thatthe $ondEs epected cash o#s donEt chan%e #hen the!ield chan%es.

• Modi&ed duration sho#s ho# $ond prices move

proportionall! #ith small chan%es in !ields.

PP x &'' # $%mod x i

#here)

G( @ chan%e in price for the $ond

4Dmod @ the modi&ed duration for the$ond

Gi @ !ield chan%e in $asis points divided

$! 1

( @ $e%innin% price for the $ond

;ote)Modi&ed duration cannot

$e used to measure theinterest rate risk for$onds #ith em$eddedoptions $ecause achan%e in !ield ma!si%ni&cantl! aHect the

epected cash o#s of$onds #ith em$edded


http://slidepdf.com/reader/full/fixed-income-568e1dfd2fbc1 14/221'

Macaula! Duration

It is calculated as)*acaulay %uration # *od. %uration (& + yield)

-here,

# num$er of periodsield # !ield to maturit! of the $ond

Example

A $ond #ith a Macaula! duration of , !ears a !ield tomaturit! of 9+ and semiannual pa!ments #ill have amodi&ed duration of)

Dmod @ ,/-1 .9/2

@ ,/1.", @ '.*" !ears

;ote)As Modi&ed duration

cannot $e used tomeasure the interestrate risk for $onds #ithem$edded optionssame holds true forMacaula! Duration too.


http://slidepdf.com/reader/full/fixed-income-568e1dfd2fbc1 15/221,

Interpretations of duration

1



(ortfolio Duration

It is o$tained $! calculatin% the #ei%hted avera%eof the duration of the $onds in the portfolio.

Portfolio %uration # &%& + /%/ + 0%0 + 111

n%n

(here)

#1#2#n @ #ei%ht of individual $onds in portfolio

D1 D2 Dn @ duration of individual $onds in

portfolioNote:2or the duration measure to 3euseful, the change in yield for eachof the 3ond in the portfolio should3e e4ual, i.e. there should 3e a

parallel shift in the yield curve.



(ortfolio Duration

Example

Market Value @ '"*,, 0218* " @Rs.1882"

#1 @ '"*,,/1882" @ .28'2

#2 @ 0218*/1882" @ .,,,2

#" @ "/1882" @ .1*9

(ortfolio Duration @ #1D1 #2D2 #"D"

@ -.28'2 8.""02 -.,,,2 *.",'" -.1*9 ".8',"

@ 8.0911+


http://slidepdf.com/reader/full/fixed-income-568e1dfd2fbc1 18/221*



10

Conveit! measure

It is used to approimate the chan%es in price thatis not eplained $! duration.

Convexit! *easure # C x +,! - ./ x 011

<here

G!J @ the chan%e in !ield for #hich the percenta%e

price chan%e is sou%ht.



2

Modi&ed conveit! and eHective conveit!

*odi2ed convexit!• It is the conveit! measure investors o$tain if the!

assume that !ield chan%es have no eHect on the$ondEs epected cash o#s.

• It does not consider the eHect of em$edded optionson epected cash o#s.

E3ective convexit!

• It includes the eHects of !ield chan%es on the casho#s.

• It re=uires an adKustment in the estimated $ond toreect an! chan%e in estimated cash o#s due tothe presence of em$edded options.



21

(rice value of a $asis point -(V5(

• It is the a$solute chan%e in the price of a $ond for a 1$asis point chan%e in !ield.

P56P # L 7nitial price $ price if yield is changed 3y & 3asis point L

PV4P di3ers from traditional duration as &

• It is identical for $oth increases and decreases in !ield$ecause it eplains ho# price chan%es due to ver! small

interest rate shifts -one $asis point. <hen interestrates are adKusted $! this small amount the price4!ieldschedule #ould $e approimatel! linear.

• It sho#s the dollar chan%e in price rather thanpercenta%e chan%e.



22

Thank You…

Documents

fixed income